Ioannis Karatzas Steven E. Shreve
Brownian Motion and Stochastic Calculus Second Edition With 10 Illustrations
Springer-Verlag
New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona
Ioannis Karatzas Department of Statistics Columbia University
Steven E. Shreve Department of Mathematics Carnegie Mellon University Pittsburgh, PA 15213 USA
New York, NY 10027 USA Editorial Board
J.H. Ewing
Department of Mathematics Indiana University Bloomington, Indiana 47405 USA
F.W. Gehring Department of Mathematics University of Michigan Ann Arbor, Michigan 48109 USA
P.R. Halmos Department of Mathematics University of Santa Clara Santa Clara, California 95053 USA
AMS Classifications: 60G07, 60H05
Library of Congress Cataloging-in-Publication Data Karatzas, Ioannis. Brownian motion and stochastic calculus/ Ioannis Karatzas, Steven
E. Shreve.-2nd ed. p. cm.-(Graduate texts in mathematics; 113) Includes bibliographical references and index. ISBN 0-387-97655-8 (New York).-ISBN 3-540-97655-8 (Berlin) 1. Brownian motion processes. 2. Stochastic analysis. I. Shreve, Steven E. II. Title. III. Series. QA274.75.K37 1991 530.4'75-dc20 91-22775
The present volume is the corrected softcover second edition of the previously published hardcover first edition (ISBN 0-387-96535-1). Printed on acid-free paper. © 1988, 1991 by Springer-Verlag New York, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag, 175 Fifth Avenue, New York, New York 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc. in this publication, even
if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly by used freely by anyone. Typeset by Asco Trade Typesetting Ltd., Hong Kong. Printed and bound by R.R. Donnelley and Sons, Harrisonburg, VA. Printed in the United States of America.
98765432
1
ISBN 0-387-97655-8 Springer-Verlag New York Berlin Heidelberg ISBN 3-540-97655-8 Springer-Verlag Berlin Heidelberg New York
To Eleni and Dot
Preface
Two of the most fundamental concepts in the theory of stochastic processes are the Markov property and the martingale property.* This book is written for readers who are acquainted with both of these ideas in the discrete-time setting, and who now wish to explore stochastic processes in their continuoustime context. It has been our goal to write a systematic and thorough exposition of this subject, leading in many instances to the frontiers of knowledge. At the same time, we have endeavored to keep the mathematical prerequisites as low as possible, namely, knowledge of measure-theoretic probability and some familiarity with discrete-time processes. The vehicle we have chosen for this task is Brownian motion, which we present as the canonical example of both a Markov process and a martingale. We support this point of view by showing how, by means of stochastic integration and random time change, all continuous-path martingales and a multitude of continuous-path Markov processes can be represented in terms of Brownian motion. This approach forces us to leave aside those processes which do not have continuous paths.
Thus, the Poisson process is not a primary object of study, although it is developed in Chapter 1 to be used as a tool when we later study passage times and local time of Brownian motion. The text is organized as follows: Chapter 1 presents the basic properties of martingales, as they are used throughout the book. In particular, we generalize from the discrete to the continuous-time context the martingale convergence theorem, the optional sampling theorem, and the Doob-Meyer decomposition. The latter gives conditions under which a submartingale can be written * According to M. Loeve, "martingales, Markov dependence and stationarity are the only three dependence concepts so far isolated which are sufficiently general and sufficiently amenable to investigation, yet with a great number of deep properties" (Ann. Probab. 1 (1973), p. 6).
viii
Preface
as the sum of a martingale and an increasing process, and associates to every martingale with continuous paths a "quadratic variation process." This process is instrumental in the construction of stochastic integrals with respect to continuous martingales.
Chapter 2 contains three different constructions of Brownian motion, as well as discussions of the Markov and strong Markov properties for continuous-time processes. These properties are motivated by d-dimensional Brownian motion, but are developed in complete generality. This chapter also contains a careful discussion of the various filtrations commonly associated with Brownian motion. In Section 2.8 the strong Markov property is applied
to a study of one-dimensional Brownian motion on a half-line, and on a bounded interval with absorption and reflection at the endpoints. Many densities involving first passage times, last exit times, absorbed Brownian motion, and reflected Brownian motion are explicitly computed. Section 2.9 is devoted to a study of sample path properties of Brownian motion. Results found in most texts on this subject are included, and in addition to these, a complete proof of the Levy modulus of continuity is provided. The theory of stochastic integration with respect to continuous martingales is developed in Chapter 3. We follow a middle path between the original constructions of stochastic integrals with respect to Brownian motion and the more recent theory of stochastic integration with respect to right-continuous martingales. By avoiding discontinuous martingales, we obviate the need to introduce the concept of predictability and the associated, highly technical, measure-theoretic machinery. On the other hand, it requires little extra effort
to consider integrals with respect to continuous martingales rather than merely Brownian motion. The remainder of Chapter 3 is a testimony to the power of this more general approach; in particular, it leads to strong theorems concerning representations of continuous martingales in terms of Brownian motion (Section 3.4). In Section 3.3 we develop the chain rule for stochastic calculus, commonly known as It6"s formula. The Girsanov Theorem of Section 3.5 provides a method of changing probability measures so as to alter the drift of a stochastic process. It has become an indispensable method for constructing solutions of stochastic differential equations (Section 5.3) and is also very important in stochastic control (e.g., Section 5.8) and filtering. Local time is introduced in Sections 3.6 and 3.7, and it is shown how this concept leads to a generalization of the Ito formula to convex but not necessarily differentiable functions. Chapter 4 is a digression on the connections between Brownian motion, Laplace's equation, and the heat equation. Sharp existence and uniqueness theorems for both these equations are provided by probabilistic methods; applications to the computation of boundary crossing probabilities are discussed, and the formulas of Feynman and Kac are established. Chapter 5 returns to our main theme of stochastic integration and differential equations. In this chapter, stochastic differential equations are driven
Preface
ix
by Brownian motion and the notions of strong and weak solutions are presented. The basic Ito theory for strong solutions and some of its ramifications, including comparison and approximation results, are offered in Section 5.2,
whereas Section 5.3 studies weak solutions in the spirit of Yamada & Watanabe. Essentially equivalent to the search for a weak solution is the search for a solution to the "Martingale Problem" of Stroock & Varadhan. In the context of this martingale problem, a full discussion of existence, uniqueness, and the strong Markov property for solutions of stochastic differential equations is given in Section 5.4. For one-dimensional equations it is possible to provide a complete characterization of solutions which exist only up to an "explosion time," and this is set forth in Section 5.5. This section also
presents the recent and quite striking results of Engelbert & Schmidt concerning existence and uniqueness of solutions to one-dimensional equations. This theory makes substantial use of the local time material of Sections 3.6,
3.7 and the martingale representation results of Subsections 3.4.A,B. By analogy with Chapter 4, we discuss in Section 5.7 the connections between solutions to stochastic differential equations and elliptic and parabolic partial differential equations. Applications of many of the ideas in Chapters 3 and 5 are contained in Section 5.8, where we discuss questions of option pricing and optimal portfolio/consumption management. In particular, the Girsanov
theorem is used to remove the difference between average rates of return of different stocks, a martingale representation result provides the optimal portfolio process, and stochastic representations of solutions to partial differential equations allow us to recast the optimal portfolio and consumption
management problem in terms of two linear parabolic partial differential equations, for which explicit solutions are provided. Chapter 6 is for the most part derived from Paul Levy's profound study of Brownian excursions. Levy's intuitive work has now been formalized by such notions as filtrations, stopping times, and Poisson random measures, but the remarkable fact remains that he was able, 40 years ago and working without these tools, to penetrate into the fine structure of the Brownian path and to inspire all the subsequent research on these matters until today. In the spirit
of Levy's work, we show in Section 6.2 that when one travels along the Brownian path with a clock run by the local time, the number of excursions away from the origin that one encounters, whose duration exceeds a specified number, has a Poisson distribution. Levy's heuristic construction of Brownian motion from its excursions has been made rigorous by other authors. We do not attempt such a construction here, nor do we give a complete specification of the distribution of Brownian excursions; in the interest of intelligibility, we content ourselves with the specification of the distribution for the durations of the excursions. Sections 6.3 and 6.4 derive distributions for functionals
of Brownian motion involving its local time; we present, in particular, a Feynman-Kac result for the so-called "elastic" Brownian motion, the formulas of D. Williams and H. Taylor, and the Ray-Knight description of
Preface
x
Brownian local time. An application of this theory is given in Section 6.5, where a one-dimensional stochastic control problem of the "bang-bang" type is solved.
The writing of this book has become for us a monumental undertaking involving several-people, whose assistance we gratefully acknowledge. Foremost among these are the members of our families, Eleni, Dot, Andrea, and Matthew, whose support, encouragement, and patience made the whole en-
deavor possible. Parts of the book grew out of notes on lectures given at Columbia University over several years, and we owe much to the audiences in those courses. The inclusion of several exercises, the approaches taken to a number of theorems, and several citations of relevant literature resulted from discussions and correspondence with F. Baldursson, A. Dvoretzky, W. Fleming, 0. Kallenberg, T. Kurtz, S. Lalley, J. Lehoczky, D. Stroock, and M. Yor. We have also taken exercises from Mandl, Lanska & Vrkoe (1978), and Ethier & Kurtz (1986). As the project proceeded, G.-L. Xu, Z.-L. Ying, and Th. Zariphopoulou read large portions of the manuscript and suggested numerous corrections and improvements. Careful reading by Daniel Ocone and Manfred Schal revealed minor errors in the first printing, and these have been corrected. However, our greatest single debt of gratitude goes to Marc Yor, who read much of the near-final draft and offered substantial mathematical and editorial comments on it. The typing was done tirelessly, cheerfully, and efficiently by Stella De Vito and Doodmatie Kalicharan; they have our most sincere appreciation.
We are grateful to Sanjoy Mitter and Dimitri Bertsekas for extending to us the invitation to spend the critical initial year of this project at the Massachusetts Institute of Technology. During that time the first four chapters were essentially completed, and we were partially supported by the Army Research Office under grant DAAG-299-84-K-0005. Additional financial support was provided by the National Science Foundation under grants DMS84-16736 and DMS-84-03166 and by the Air Force Office of Scientific Research
under grants AFOSR 82-0259, AFOSR 85-0360, and AFOSR 86-0203.
Ioannis Karatzas Steven E. Shreve
Contents
Preface
Suggestions for the Reader
vii
xvii
Interdependence of the Chapters
xi x
Frequently Used Notation
xxi
CHAPTER 1
Martingales, Stopping Times, and Filtrations 1.1. Stochastic Processes and a-Fields 1.2. Stopping Times 1.3. Continuous-Time Martingales A. Fundamental inequalities B. Convergence results C. The optional sampling theorem 1.4. The Doob-Meyer Decomposition 1.5. Continuous, Square-Integrable Martingales 1.6. Solutions to Selected Problems 1.7. Notes
1 1
6 11
12 17 19 21
30 38 45
CHAPTER 2
Brownian Motion
47
2.1. Introduction 2.2. First Construction of Brownian Motion A. The consistency theorem B. The Kolmogorov-entsov theorem 2.3. Second Construction of Brownian Motion
47 49 49
2.4. The Space C[0, co), Weak Convergence, and Wiener Measure A. Weak convergence
53 56 59
60
xii
B. Tightness C. Convergence of finite-dimensional distributions D. The invariance principle and the Wiener measure 2.5. The Markov Property A. Brownian motion in several dimensions B. Markov processes and Markov families C. Equivalent formulations of the Markov property 2.6. The Strong Markov Property and the Reflection Principle A. The reflection principle B. Strong Markov processes and families C. The strong Markov property for Brownian motion 2.7. Brownian Filtrations A. Right-continuity of the augmented filtration for a strong Markov process B. A "universal" filtration C. The Blumenthal zero-one law 2.8. Computations Based on Passage Times A. Brownian motion and its running maximum B. Brownian motion on a half-line C. Brownian motion on a finite interval D. Distributions involving last exit times 2.9. The Brownian Sample Paths A. Elementary properties B. The zero set and the quadratic variation C. Local maxima and points of increase D. Nowhere differentiability E. Law of the iterated logarithm F. Modulus of continuity 2.10. Solutions to Selected Problems 2.11. Notes
Contents 61
64 66 71
72 74 75 79 79 81
84 89 90 93
94 94 95 97 97 100 103 103 104 106 109 111 114 116 126
CHAPTER 3
Stochastic Integration
128
3.1. Introduction 3.2. Construction of the Stochastic Integral A. Simple processes and approximations B. Construction and elementary properties of the integral C. A characterization of the integral D. Integration with respect to continuous, local martingales 3.3. The Change-of-Variable Formula A. The Ito rule B. Martingale characterization of Brownian motion
128 129 132 137
C. Bessel processes, questions of recurrence D. Martingale moment inequalities E. Supplementary exercises 3.4. Representations of Continuous Martingales in Terms of Brownian Motion A. Continuous local martingales as stochastic integrals with respect to Brownian motion
141 145
148 149 156 158 163 167 169
170
Contents
xiii
B. Continuous local martingales as time-changed Brownian motions C. A theorem of F. B. Knight D. Brownian martingales as stochastic integrals E. Brownian functionals as stochastic integrals 3.5. The Girsanov Theorem A. The basic result B. Proof and ramifications C. Brownian motion with drift D. The Novikov condition 3.6. Local Time and a Generalized It8 Rule for Brownian Motion A. Definition of local time and the Tanaka formula B. The Trotter existence theorem C. Reflected Brownian motion and the Skorohod equation D. A generalized Ito rule for convex functions E. The Engelbert-Schmidt zero-one law 3.7. Local Time for Continuous Semimartingales 3.8. Solutions to Selected Problems 3.9. Notes
I73 179 180 185 190 191
193 196 198
20I 203 206 210 212 215 217 226 236
CHAPTER 4
Brownian Motion and Partial Differential Equations
239
4.1. Introduction 4.2. Harmonic Functions and the Dirichlet Problem A. The mean-value property B. The Dirichlet problem C. Conditions for regularity D. Integral formulas of Poisson E. Supplementary exercises 4.3. The One-Dimensional Heat Equation A. The Tychonoff uniqueness theorem B. Nonnegative solutions of the heat equation C. Boundary crossing probabilities for Brownian motion D. Mixed initial/boundary value problems 4.4. The Formulas of Feynman and Kac A. The multidimensional formula B. The one-dimensional formula 4.5. Solutions to selected problems 4.6. Notes
239 240 241
243 247 251
253 254 255
256 262 265 267 268 271
275 278
CHAPTER 5
Stochastic Differential Equations
281
5.1. Introduction 5.2. Strong Solutions A. Definitions B. The Ito theory C. Comparison results and other refinements D. Approximations of stochastic differential equations E. Supplementary exercises
281
284 285 286 291
295 299
xiv
5.3. Weak Solutions A. Two notions of uniqueness B. Weak solutions by means of the Girsanov theorem C. A digression on regular conditional probabilities D. Results of Yamada and Watanabe on weak and strong solutions 5.4. The Martingale Problem of Stroock and Varadhan A. Some fundamental martingales B. Weak solutions and martingale problems C. Well-posedness and the strong Markov property D. Questions of existence E. Questions of uniqueness F. Supplementary exercises 5.5. A Study of the One-Dimensional Case A. The method of time change B. The method of removal of drift C. Feller's test for explosions D. Supplementary exercises 5.6. Linear Equations A. Gauss-Markov processes B. Brownian bridge C. The general, one-dimensional, linear equation D. Supplementary exercises 5.7. Connections with Partial Differential Equations A. The Dirichlet problem B. The Cauchy problem and a Feynman-Kac representation C. Supplementary exercises 5.8. Applications to Economics A. Portfolio and consumption processes B. Option pricing C. Optimal consumption and investment (general theory) D. Optimal consumption and investment (constant coefficients) 5.9. Solutions to Selected Problems 5.10. Notes
Contents
300 301
302 306 308 311
312 314 319 323 325 328 329 330 339 342 35 I
354 355 358 360 361
363 364 366 369 371 371
376 379 381
387 394
CHAPTER 6
P. Levy's Theory of Brownian Local Time
399
6.1. Introduction 6.2. Alternate Representations of Brownian Local Time A. The process of passage times B. Poisson random measures C. Subordinators D. The process of passage times revisited E. The excursion and downcrossing representations of local time 6.3. Two Independent Reflected Brownian Motions A. The positive and negative parts of a Brownian motion B. The first formula of D. Williams C. The joint density of (W(t), L(t), f (t))
399 400
400 403 405 411
414 418 418 421
423
Contents
xv
6.4. Elastic Brownian Motion A. The Feynman-Kac formulas for elastic Brownian motion B. The Ray-Knight description of local time C. The second formula of D. Williams 6.5. An Application: Transition Probabilities of Brownian Motion with Two-Valued Drift 6.6. Solutions to Selected Problems 6.7. Notes
425 426
Bibliography
447
Index
459
430 434 437 442 445
Suggestions for the Reader
We use a hierarchical numbering system for equations and statements. The k-th equation in Section j of Chapter i is labeled (j.k) at the place where it occurs and is cited as (j.k) within Chapter i, but as (i j.k) outside Chapter i. A definition, theorem, lemma, corollary, remark, problem, exercise, or solution is a "statement," and the k-th statement in Section j of Chapter i is labeled j.k Statement at the place where it occurs, and is cited as Statement j.k within Chapter i but as Statement i.j.k outside Chapter i. This book is intended as a text and can be used in either a one-semester or a two-semester course, or as a text for a special topic seminar. The accompany-
ing figure shows dependences among sections, and in some cases among subsections. In a one-semester course, we recommend inclusion of Chapter 1 and Sections 2.1, 2.2, 2.4, 2.5, 2.6, 2.7, §2.9.A, B, E, Sections 3.2, 3.3, 5.1, 5.2, and §5.6.A, C. This material provides the basic theory of stochastic integration, including the Ito calculus and the basic existence and uniqueness results for strong solutions of stochastic differential equations. It also contains matters of interest in engineering applications, namely, Fisk-Stratonovich integrals and approximation of stochastic differential equations in §3.3.A and 5.2.D, and Gauss-Markov processes in §5.6.A. Progress through this material can
be accelerated by omitting the proof of the Doob-Meyer Decomposition Theorem 1.4.10 and the proofs in §2.4.D. The statements of Theorem 1.4.10, Theorem 2.4.20, Definition 2.4.21, and Remark 2.4.22 should, however, be retained. If possible in a one-semester course, and certainly in a two-semester course, one should include the topic of weak solutions of stochastic differential equations. This is accomplished by covering §3.4.A, B, and Sections 3.5, 5.3, and 5.4. Section 5.8 serves as an introduction to stochastic control, and so we recommend adding §3.4.C, D,.E, and Sections 5.7, and 5.8 if time permits. In either a one- or two-semester course, Section 2.8 and part or all of Chapter 4
xviii
Suggestions for the Reader
may be included according to time and interest. The material on local time and its applications in Sections 3.6, 3.7, 5.5, and in Chapter 6 would normally be the subject of a special topic course with advanced students. The text contains about 175 "problems" and over 100 "exercises." The former are assignments to the reader to fill in details or generalize a result, and these are often quoted later in the text. We judge approximately twothirds of these problems to be nontrivial or of fundamental importance, and solutions for such problems are provided at the end of each chapter. The exercises are also often significant extensions of results developed in the text, but these will not be needed later, except perhaps in the solution of other exercises. Solutions for the exercises are not provided. There are some exercises for which the solution we know violates the dependencies among
sections shown in the figure, but such violations are pointed out in the offending exercises, usually in the form of a hint citing an earlier result.
Interdependence of the Chapters
Chapter 1 2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
§2.9.A, B, E
§2.9.C, D, F
4.2
3.2
62
4.3
3.3
3.6
4.4
§3.4.A, B
3.7
§3.4.C, D, E
5.1
35
5.2
5.3
§5.6.A, C
5.4
§5.6.B, D
5.7
55
6.3
[ 5.8,
[ 6.4 L 6.5
1
Frequently Used Notation
I. General Notation Let a and b be real numbers. (1) A means "is defined to be."
(2) anbA min la, bl. (3) a v bA max {a, b}. (4) a+ A max{a, 0}.
(5) a- A max{ -a,0}.
II. Sets and Spaces (1) No A {0,1, 2, ... }.
(2) Q is the set of rational numbers. (3) Q+ is the set of nonnegative rational numbers. (4) Rd is the d-dimensional Euclidean space; R1 = R. (5) B, A {x e Rd; Ilx II <
(p. 240).
(6) (Rd)1°') is the set of functions from [0, co) to Rd (pp. 49, 76). (7) C[0, cc)" is the subspace of (Rd)'0'') consisting of continuous functions; C[0, oo)1 = C[0, co) (pp. 60, 64). (8) D[0, co) is the subspace of RE°'"') consisting of functions which are right continuous and have left-limits (p. 409). (9) 0E), CNE), (E): See Remark 4.1, p. 312. (10) CI 2 ( [0, T) x E), 2 ((3, T) x E): See Remark 4.1, p. 312. (11) .29, Sf(M), .29*, 2) *(M): See pp. 130-131. (12) P1(M), P1 *(M): See pp. 146-147. (13) .1/2(11`2): The space of (continuous) square-integrable martingales (p. 30). (14) ../rf-1.(.11`.1"): The space of (continuous) local martingales (p. 36).
Frequently Used Notation
xxii
III. Functions 1; x > 0, - 1; x < 0.
(1) sgn(x) =
x e A,
(2) 1A(x)
5.1;
10;
(3) p(t; x, y)
x A.
2e-(x-Y)212'; t > 0, x, ye R (p. 52). nt (4) p+ (t; x, y) p(t; x, y) ± p(t; x, - y); t > 0, x, y e R (p. 97). (5) It]] is the largest integer less than or equal to the real number t.
/
IV. a-Fields (1) .4(U): The smallest 6 -field containing all open sets of the topological space U (p. 1). (2) at(c [0, co)), A(C[0, oo)d). See pp. 60, 307. (3) a(W): The smallest o--field containing the collection of sets W. (4) o-(Xs): The smallest a-field with respect to which the random variable X, is measurable. (5) o-(Xs; 0 < s < t): The smallest a-field with respect to which the random variable X, is measurable, V s e [0, t]. A o-(Xs; 0
(6)
s
nc>0.Fr+c,
(7)
t), ..Foo A o-(U,,o,97,): See p. 3. 0-(us
(8) .97T: The a-field of events determined prior to the stopping time T; see p. 8.
+ The a-field of events determined immediately after the optional
(9)
time T; see p. 10.
W A o-(A x B;Ae.F,Bel: The product a-field formed from the
(10) .97
a-fields ,F and W.
V. Operations on Functions d
(1) A A E
02
,: The Laplacian (p. 240).
ax;
(2) .91, sit: Second order differential operators; see pp. 281, 311.
VI. Operations on Processes (1) Os, Os: Shift operator at the deterministic time s and the random time S; see pp. 77, 83.
Frequently Used Notation
xxiii
(2) /1"(X) A j) XS dMs: The stochastic integral of X with respect to M. See p. 141 for M e .41(.2, X e ,99* (M); see p. 147 for M e dr.'" X e 1)*(M). (3) Mt* -A maxs<, I Msl: See p. 163 for M e dif`.1".
(4) <X>: The quadratic variation process of X e d12 (p. 31) or X e dr' (p. 36).
(5) <X, Y>: The cross-variation process of X, Y in /2 (p. 31) or in i`.'" (p. 36). (6) II XL, II XII: See p. 37 for X e .412-
VII. Miscellaneous (1) mT(X, 6) A sup{ I Xs - X,I; 0 -s
-s
61: See p. 62.
(3) D: The closure of the set D Rd. (4) DC: The complement of the set D. (5) OD: The boundary of the set D Rd. (6) ID 4' inf{t > 0; W e Dc}: The first time the Brownian motion W exits from the set D Rd (p. 240). (7) Tb A inf{t > 0; W = 6}: The first time the one-dimensional Brownian motion W reaches the level be (p. 79). (8) 1",(t) 1(0)(Ws)ds: The occupation time by Brownian motion of the positive half-line (p. 273). (9) P. P: Weak convergence of the sequence of probability measures {P};',°=1 to the probability measure P (p. 60). (10) X,, '4 X: Convergence in distribution of the sequence of random variables {X}.°=, to the random variable X (p. 61). (11) P": Probability measure corresponding to Brownian motion (p. 72) or a Markov process (p. 74) with initial position x e Rd. (12) Pµ: Probability measure corresponding to Brownian motion (p. 72) or a Markov process (p. 74) with initial distribution tt. (13) ,P, ,ACP: Collections of PP-negligible sets (p. 89). (14) 1(a), Z(o-): See pp. 331, 332. (15) Id: The (d x d) identity matrix. (16) meas: Lebesgue measure on the real line (p. 105).
CHAPTER 1
Martingales, Stopping Times, and Filtrations
1.1. Stochastic Processes and a-Fields A stochastic process is a mathematical model for the occurrence, at each moment after the initial time, of a random phenomenon. The randomness is captured by the introduction of a measurable space called the sample space, on which probability measures can be placed. Thus, a stochastic process
is a collection of random variables X = {X,; 0 < t < co} on (a .97), which take values in a second measurable space (S, 99), called the state space. For our purposes, the state space (S, 9') will be the d-dimensional Euclidean space equipped with the a-field of Borel sets, i.e., S = Rd, = R(1 W), where 9B(U) will always be used to denote the smallest a-field containing all open sets of a topological space U. The index t e [0, co) of the random variables X, admits a convenient interpretation as time. For a fixed sample point w a 12, the function X,(w); t > 0 is the sample path (realization, trajectory) of the process X associated with w. It provides
the mathematical model for a random experiment whose outcome can be observed continuously in time (e.g., the number of customers in a queue observed and recorded over a period of time, the trajectory of a molecule subjected to the random disturbances of its neighbors, the output of a communications channel operating in noise).
Let us consider two stochastic processes X and Y defined on the same probability space (S2, .97 , P). When they are regarded as functions of t and w, we would say X and Y were the same if and only if Xj(w) = Y,(w) for all t > 0 and all w E a However, in the presence of the probability measure P, we could weaken this requirement in at least three different ways to obtain three related concepts of "sameness" between two processes. We list them here.
1. Martingales, Stopping Times, and Filtrations
2
1.1 Definition. Y is a modification of X if, for every t
0, we have
P[X, = y] = 1. 1.2 Definition. X and Y have the same finite-dimensional distributions if, for < t < cc, and A e .4(Ord), we have:
any integer n > 1, real numbers 0 < t, < t2 <
P[(X,,,... , Xin) e A] =
..., Y,n)EA].
1.3 Definition. X and Y are called indistinguishable if almost all their sample paths agree:
P[X, = Yt; V 0 < t < cc] = 1. The third property is the strongest; it implies trivially the first one, which in turn yields the second. On the other hand, two processes can be modifications of one another and yet have completely different sample paths. Here is a standard example: 1.4 Example. Consider a positive random variable T with a continuous dis0; t T Y is a modification of X, since tribution, put X, 0, and let Y, = 1;
for every t > 0 we have P[Y, = P[Y, Xi; V t 0] = 0.
t=T
= PET z t] = 1, but on the other hand:
A positive result in this direction is the following.
1.5 Problem. Let Y be a modification of X, and suppose that both processes have a.s. right-continuous sample paths. Then X and Y are indistinguishable. It does not make sense to ask whether Y is a modification of X, or whether
Y and X are indistinguishable, unless X and Y are defined on the same probability space and have the same state space. However, if X and Y have the same state space but are defined on different probability spaces, we can ask whether they have the same finite-dimensional distributions.
1.2' Definition. Let X and Y be stochastic processes defined on probability spaces (11, , P) and (S, ., P), respectively, and having the same state space (Rd, g(Old. )) X and Y have the same finite-dimensional distributions if, for any integer n > 1, real numbers 0 < t, < t2 < < t < cc, and A e gi(Ora), we have
, Xt.) e A] =
Y,n)EA].
Many processes, including d-dimensional Brownian motion, are defined in terms of their finite-dimensional distributions irrespective of their probability
1.1. Stochastic Processes and o -Fields
3
space. Indeed, in Chapter 2 we will construct a standard d-dimensional Brownian motion B on a canonical probability space and then state that any process, on any probability space, which has state space (Rd, gi(Rd)) and the same finite-dimensional distributions as B, is a standard d-dimensional Brownian motion.
For technical reasons in the theory of Lebesgue integration, probability measures are defined on a-fields and random variables are assumed to be measurable with respect to these a-fields. Thus, implicit in the statement that a random process X = {Xi; 0 t < co } is a collection of (Rd, .4(Rd))-valued random variables on (fl, F), is the assumption that each X, is /.4(E4d)measurable. However, X is really a function of the pair of variables (t, w), and so, for technical reasons, it is often convenient to have some joint measurability properties.
1.6 Definition. The stochastic process X is called measurable if, for every A e MR"), the set {(t, w); Xt(co)e AI belongs to the product a-field .4([0, oo))
,F; in other words, if the mapping Xt(co): ( [0, co) x
(t,
M( [0, oo))
.F) -÷ (Rd, mold))
is measurable.
It is an immediate consequence of Fubini's theorem that the trajectories of such a process are Borel-measurable functions oft e [0, co), and provided that the components of X have defined expectations, then the same is true for the function m(t) = E Xi; here, E denotes expectation with respect to a probability measure P on (f2, Moreover, if X takes values in R and I is a subinterval of [0, co) such that jj EIX,I dt < co, then
f,
X,' dt < co a.s. P,
and
ji EX, dt = E
X, dt.
There is a very important, nontechnica reason to include a-fields in the study of stochastic processes, and that is to keep track of information. The temporal feature of a stochastic process suggests a flow of time, in which, at every moment t > 0, we can talk about a past, present, and future and can ask how much an observer of the process knows about it at present, as compared to how much he knew at some point in the past or will know at some point in the future. We equip our sample space (f2, ,F) with a filtration, i.e., a nondecreasing family {A; t > 0} of sub-a-fields of .97 for 0 s < t < co. We set d. 6-co =-- a(Ut>o Given a stochastic process, the simplest choice of a filtration is that generated by the process itself, i.e.,
A' A o-(Xs; 0 < s < t), the smallest a-field with respect to which X, is measurable for every s e [0, t].
4
1. Martingales, Stopping Times, and Filtrations
We interpret A E "ix to mean that by time t, an observer of X knows whether or not A has occurred. The next two exercises illustrate this point.
1.7 Exercise. Let X be a process, every sample path of which is RCLL (i.e., right-continuous on [0, co) with finite left -hand limits on (0, co)). Let A be the event that X is continuous on [0, to). Show that A e 1.8 Exercise. Let X be a process whose sample paths are RCLL almost surely, and let A be the event that X is continuous on [0, to). Show that A can fail to be in .??-tox, but if {.";; t > 0} is a filtration satisfying .??7,' t 0, and Fto is
complete under P, then A Eo. Let {,56;; t > 0} be a filtration. We define
cr(Us
of events strictly prior to t > 0 and Ft, nE>0,,,E to be the a-field of events immediately after t > 0. We decree .50_ A .yo and say that the filtration {..F,} is right- (left-)continuous if = (resp., 3,7,_) holds for every t > 0. The concept of measurability for a stochastic process, introduced in Definition 1.6, is a rather weak one. The introduction of a filtration {g} opens up the possibility of more interesting and useful concepts. 1.9 Definition. The stochastic process X is adapted to the filtration {.F,} if, for each t > 0, X, is an ..F- measurable random variable.
Obviously, every process X is adapted to 1,a-el. Moreover, if X is adapted to {A} and Y is a modification of X, then Y is also adapted to {,} provided that .yo contains all the P-negligible sets in Note that this requirement is not the same as saying that .F0 is complete, since some of the P-negligible sets in g" may not be in the completion of go.
1.10 Exercise. Let X be a process with every sample path LCRL (i.e., left continuous on (0, co) with finite right-hand limits on [0, co)), and let A be the event that X is continuous on [0, to]. Let X be adapted to a right-continuous filtration {.Ft}. Show that A e "o. 1.11 Definition. The stochastic process X is called progressively measurable with respect to the filtration {,;} if, for each t > 0 and A e gii(Rd), the set {(s, a)); 0 < s < t, w e SZ, Xs(w) e A} belongs to the product a-field .4([0, t]) ..Ft; in other words, if the mapping (s, co) 1-+ X s(w): ([0, t] x S2, .4([0, t]) 0 g") -+
(Rd M(Rd)) is measurable, for each t
0.
The terminology here comes from Chung & Doob (1965), which is a basic reference for this section and the next. Evidently, any progressively measurable process is measurable and adapted; the following theorem of Chung & Doob (1965) provides the extent to which the converse is true.
1.1. Stochastic Processes and a-Fields
5
1.12 Proposition. If the stochastic process X is measurable and adapted to the filtration {.9;}, then it has a progressively measurable modification.
The reader is referred to the book of Meyer (1966), p. 68, for the (lengthy, and rather demanding) proof of this result. It will be used in this text only in a tangential fashion. Nearly all processes of interest are either right- or leftcontinuous, and for them the proof of a stronger result is easier and will now be given.
1.13 Proposition. If the stochastic process X is adapted to the filtration 1.97,1 and every sample path is right-continuous or else every sample path is leftcontinuous, then X is also progressively measurable with respect to IF J.
PROOF. We treat the case of right-continuity. With t > 0, n > 1, k = 0, ..., 2" - 1, and 0 < s < t, we define: X.1")((0) = X(k+1)02^(W)
for
-kt
2"
1,
<s<- k+lt 2"
as well as Xnw) = Xo(w). The so-constructed map (s, w) X 1") (co) from [0, t] x SI into Rd is demonstrably .4([0, t]) ..Ff-measurable. Besides, by right-continuity we have: lim, )(no) = )(Jo)), V (s, w) E [0, t] x a Therefore, the (limit) map (s,
XS(w) is also 9([0, t])
.F,- measurable.
1.14 Remark. If the stochastic process X is right- or left-continuous, but not necessarily adapted to then the same argument shows that X is measurable.
A random time T is an "--measurable random variable, with values in [0, 09].
1.15 Definition. If X is a stochastic process and T is a random time, we define the function XT on the event { T < co} by XT(w) A XT00(co).
If Xco(co) is defined for all w e SI, then X,- can also be defined on SI, by setting XT(w) A X00(o) on { T = co }.
1.16 Problem. If the process X is measurable and the random time T is finite, then the function XT is a random variable.
1.17 Problem. Let X be a measurable process and T a random time. Show that the collection of all sets of the form {XT e A }; A e R(R), together with the set { T = oo}, forms a sub-a-field of We call this the o--field generated by XT.
6
1. Martingales, Stopping Times, and Filtrations
We shall devote our next section to a very special and extremely useful class of random times, called stopping times. These are of fundamental importance in the study of stochastic processes, since they constitute our most effective tool in the effort to "tame the continuum of time," as Chung (1982) puts it.
1.2. Stopping Times Let us keep in mind the interpretation of the parameter t as time, and of the as the accumulated information up to t. Let us also imagine that we are interested in the occurrence of a certain phenomenon: an earthquake with intensity above a certain level, a number of customers exceeding the safety requirements of our facility, and so on. We are thus forced to pay particular attention to the instant T(w) at which the phenomenon manifests itself for the first time. It is quite intuitive then that the event {o); T(co) < t}, which occurs if and only if the phenomenon has appeared prior to (or at) time t, should be part of the information accumulated by that time. We can now formulate these heuristic considerations as follows: (7-field
2.1 Definition. Let us consider a measurable space (11, S) equipped with a filtration {Fr}. A random time T is a stopping time of the filtration, if the event
{T < t} belongs to the (7-field ,97 for every t > 0. A random time T is an optional time of the filtration, if { T< t}
for every t
0.
2.2 Problem. Let X be a stochastic process and T a stopping time of {,Fix}. Suppose that for any w, ol e LI, we have X,(co) = Xt(o)') for all t e [0, T(w)] n [0, co). Show that T(w) = T(w'). 2.3 Proposition. Every random time equal to a nonnegative constant is a stopping
time. Every stopping time is optional, and the two concepts coincide if the filtration is right-continuous. PROOF. The first statement is trivial; the second is based on the observation { T < t} = Unc°---1 t - (1/n)} e ..Ft, because if T is a stopping time, then {T < t - (l/n) } g --(1/n) for n 1. For the third claim, suppose that T is an optional time of the right-continuous filtration {,F,}. Since {T < =
nE,01T< t + el, we have { T < t} E Ft+, for every t whence T e ..Ft+ =
0 and every e > 0;
2.4 Corollary. T is an optional time of the filtration {..F,} if and only if it is a stopping time of the (right-continuous!) filtration {g",+}.
25 Example. Consider a stochastic process X with right-continuous paths, which is adapted to a filtration {A}. Consider a subset F .4(Fid) of the state
7
1.2. Stopping Times
space of the process, and define the hitting time 11,-(w) = inf { t >_ 0; X, (co) e F}.
We employ the standard convention that the infimum of the empty set is infinity. 2.6 Problem. If the set T in Example 2.5 is open, shoWthat Hr is an optional time. 2.7 Problem. If the set F in Example 2.5 is closed and the sample paths of the
process X are continuous, then 11,- is a stopping time.
Let us establish some simple properties of stopping times. 2.8 Lemma. If T is optional and B is a positive constant, then T + B is a stopping time.
PROOF. If 0 < t < 0, then {T + 0 _. t} = 0 e ...Ft. If t > 0, then
{T + 0 < t} = IT < t - 0} e ,F0_0), g Sr.,. 2.9 Lemma. If T, S are stopping times, then so are T
A
0
S, T v S, T + S.
PROOF. The first two assertions are trivial. For the third, start with the decomposition, valid for t > 0:
{T + S > t} = {T = 0,S > t} u {0 < T < t, T + S > t} u {T > t, S = 0} u {T ._ t, S > 0}.
The first, third, and fourth events in this decomposition are in ., either trivially or by virtue of Proposition 2.3. As for the second event, we rewrite it as:
U ft > T > r, S > t - r}, re Q+
0
where Q+ is the set of rational numbers in [0, co). Membership in ,°?7, is now obvious. CI
2.10 Problem. Let T, S be optional times; then T + S is optional. It is a stopping time, if one of the following conditions holds:
(i) T > 0, S > 0; (ii) T > 0, T is a stopping time. 2.11 Lemma. Let {T} n°11 be a sequence of optional times; then the random times sup Tn, n> 1
inf T, lim T, lim 'T n> 1
n-,:o
n-co
are all optional. Furthermore, if the Tn's are stopping times, then so is supn,
1 Tn.
1. Martingales, Stopping Times, and Filtrations
8
PROOF. Obvious, from Corollary 2.4 and from the identities co
{ sup T,, 5 t} n = n >1
{7,, 5 t}
and
n=1
{ inf n>1
.
T < t} = U I T,, < tl.
El
n=1
How can we measure the information accumulated up to a stopping time T? In order to broach this question, let us suppose that an event A is part of this information, i.e., that the occurrence or nonoccurrence of A has been decided by time T Now if by time t one observes the value of T, which can happen only if T 5 t, then one must also be able to tell whether A has occurred. In other words, A n {T < t} and A' n {T 5 t} must both be Frmeasurable, and this must be the case for any t 0. Since
A' n IT
t} = IT
t} n (An IT
t})`,
it is enough to check only that A n IT < t} e Ft, t > 0. 2.12 Definition. Let T be a stopping time of the filtration {A}. The o--field FT of events determined prior to the stopping time T consists of those events A e F for which An {T < 4 E, for every t > 0.
2.13 Problem. Verify that FT is actually a 6 -field and T is FT-measurable. Show that if T(w) = t for some constant t > 0 and every w e n, then FT = A. 2.14 Exercise. Let T be a stopping time and S a random time such that S > T on n. If S is FT-measurable, then it is also a stopping time. 2.15 Lemma. For any two stopping times T and S, and for any A e Fs, we have A n {S < TI e FT. In particular, if S _-_ Ton Si, we have Fs FT.
PROOF. It is not hard to verify that, for every stopping time T and positive constant t, T A t is an F,- measurable random variable. With this in mind, the claim follows from the decomposition:
A n{ S 5 T} n {T5 t} = [A nIS 5 tnnIT5tInIS A t
TA tl,
which shows readily that the left-hand side is an event in A. 2.16 Lemma. Let T and S be stopping times. Then .."°-"T A s = FT n Fs, and each
of the events
IT < SI, IS < TI, IT 5 SI, IS 5 TI, IT = SI belongs to FT n Fs. PROOF. For the first claim we notice from Lemma 2.15 that FT,,s _c_ FT n Fs.
In order to establish the opposite inclusion, let us take A E .9-'s n FT and
9
1.2. Stopping Times
observe that
An{SA Tst}=An[ISstlulTstl] = [A n IS
tl] u [A n { T
t}] e F
and therefore A e Fs T From Lemma 2.15 we have {S < T} EFT, and thus {S > T} e FT. On the
other hand, consider the stopping time R = S A T, which, again by virtue of Lemma 2.15, is measurable with respect to Fr. Therefore, {S < T} = {R < T} e Fr. Interchanging the roles of S, T we see that {T > S }, {T < S} belong to Fs, and thus we have shown that both these events belong to FT n Fs. But then the same is true for their complements, and consequently also for {S = T}. 2.17 Problem. Let T, S be stopping times and Z an integrable random variable. We have
(i) E[ZIFT] = ECZ I's AT], P-a.s. on {TS S} (ii) E[E(ZIFT)I.F.s] = E[ZLFS A sr], P-a.s. Now we can start to appreciate the usefulness of the concept of stopping time in the study of stochastic processes. 2.18 Proposition. Let X = {X Ft; 0 < t < oo} be a progressively measurable
process, and let T be a stopping time of the filtration IF,I. Then the random variable XT of Definition 1.15, defined on the set {T < co} EFT, is FT-measurable, and the "stopped process" {X7. At .5'-'1; 0 S t < co } is progressively measurable. PROOF. For the first claim, one has to show that for any B e .4(W) and any 0, the event {Xr E B} n {T < t} is in F; but this event can also be written t in the form {XT Ate B} n {T < t}, and so it is sufficient to prove the progressive
measurability of the stopped process. To this end, one observes that the mapping (s, w) i- (T(w) A s, w) of [0, t] x SI
into itself is .4([0, t]) 0 Ft-measurable. Besides, by the assumption of progressive measurability, the mapping (s, w) i- Xs(w): ([0, t] x 12, .4( [0, t]) ® Ft) -. (Rd, .4(W))
is measurable, and therefore the same is true for the composite mapping (s, w)i- X Too (w): ([0, t] x f2, M( [0, t]) 0 Ft) -, (Rd, WV)).
2.19 Problem. Under the same assumptions as in Proposition 2.18, and with f(t, x): [0, Go) x Rd -R a bounded, M([l), co)) ®M(W)- measurable function, show that the process Y, = Po f(s, X3) ds; t > 0 is progressively measurable with
respect to {A}, and YT is an Fr-measurable random variable.
1. Martingales, Stopping Times, and Filtrations
10
2.20 Definition. Let T be an optional time of the filtration {,F,}. The cr-field FT., of events determined immediately after the optional time T consists of those events A E gor' for which A n {T < t} e ..°?7,÷ for every t > 0.
2.21 Problem. Verify that the class ,F is indeed a o--field with respect to which T is measurable, that it coincides with {Ae,F;AnIT < tle.Vt 0}, and that if T is a stopping time (so that both .FT,
2.22 Problem. Verify that analogues of Lemmas 2.15 and 2.16 hold if T and S are assumed to be optional and
stopping time with S < T, and S < Ton {S < co}, then ,s+
2.23 Problem. Show that if { Tn}'_, is a sequence of optional times and
T = inf, T, then gr- T+ = nnc°=1 FT+ Besides, if each Tn is a positive stopping time and T < T on { T < co}, then we have -g"T+ = nr1=1 "T2.24 Problem. Given an optional time T of the filtration Pr., 1, consider the sequence { Tn},`,°=, of random times given by
Tn(w) =
+oo1
on {w; T(w)
T(w); k
on tw;
2'
k-1 2"
< T(w) < -k}
-
2"
for n > 1, k > 1. Obviously Tn > 'Tn+, > T, for every n > 1. Show that each T
is a stopping time, that lim, T = T, and that for every A e FT+ we have A n { T = (k/2")} e
n, k
1.
We close this section with a statement about the set of jumps for a stochastic process whose sample paths do not admit discontinuities of the second kind.
2.25 Definition. A filtration {,F,} is said to satisfy the usual conditions if it is right-continuous and contains all the P-negligible events in
2.26 Proposition. If the process X has RCLL paths and is adapted to the filtration {.97,} which satisfies the usual conditions, then there exists a sequence {Tn}n°°.1 of stopping times of 1.Fil which exhausts the jumps of X, i.e., (2.1)
{ (t, (0) E (0, cc) X f2; X,(w)
X,_ (w)}
U {(t, co) e [0, CO) X 12; Tn(w) = t }. n=1
The proof of this result is based on the powerful "section theorems" of the general theory of processes. It can be found in Dellacherie (1972), p. 84, or Elliott (1982), p. 61. Note that our definition of the terminology "{ T},`;"_, exhausts the jumps of X" as set forth in (2.1) is a bit different from that found
1.3. Continuous-Time Martingales
11
on p. 60 of Elliott (1982). However, the proofs in the cited references justify our version of Proposition 2.26.
1.3. Continuous-Time Martingales We assume in this section that the reader is familiar with the concept and basic properties of martingales in discrete time. An excellent presentation of this material can be found in Chung (1974, §§9.3 and 9.4, pp. 319-341) and we shall cite from this source frequently. Alternative references are Ash (1972) and Billingsley (1979). The purpose of this section is to extend the discrete-time results to continuous-time martingales. The standard example of a continuous-time martingale is one-dimensional Brownian motion. This process can be regarded as the continuous-time ver-
sion of the one-dimensional symmetric random walk, as we shall see in Chapter 2. Since we have not yet introduced Brownian motion, we shall take instead the compensated Poisson process as a continuing example developed in the problems throughout this section. The compensated Poisson process is a martingale which will serve us later in the construction of Poisson random measures, a tool necessary for the treatment of passage and local times of Brownian motion. In this section we shall consider exclusively real-valued processes X = {X1; 0 < t < col on a probability space (0, P), adapted to a given filtration 1,,1 and such that El X,I < co holds for every t > 0.
3.1 Definition. The process {X Ft; 0 < t < col is said to be a submartingale (respectively, a supermartingale) if, for every 0 < s < t < cc, we have, a.s. P: E(X,I.F) X, (respectively, E(Xtig;) < X3). 0 < t < col is a martingale if it is both a subWe shall say that {X martingale and a supermartingale. 3.2 Problem. Let T1, T2, ... be a sequence of independent, exponentially distributed random variables with parameter A > 0:
P[7; E dt] = Ae't dt,
t > 0.
Let Si, = 0 and S = E7=1 Ti; n > 1. (We may think of S as the time at which the n-th customer arrives in a queue, and of the random variables T, i = 1, 2, ... as the interarrival times.) Define a continuous-time, integer-valued RCLL process (3.1)
N, = max {n > 0; S
1};
0
t < co.
(We may regard N, as the number of customers who arrive up to time t.)
(i) Show that for 0 < s < t we have
1. Martingales, Stopping Times, and Filtrations
12
> tig,N] = e-1("), a.s. P. (Hint: Choose Ae AN and a nonnegative integer n. Show that there exists an event A ea(T,,...,Tn) such that A n {N, = n} = An {N, = n}, and use the independence between Tn." and the pair (S, 1A) to establish fin{r,=.}
P[S,, > t1.97,^] dP = e-A(`-s)P[A n {N, = n }].)
(ii) Show that for 0 < s < t, N, -N, is a Poisson random variable with parameter A.(t - s), independent of Fs1.1 . (Hint: With A e AN and n > 0 as before, use the result in (i) to establish
PEN, -N, < k!3 N] L{Ns=n}
= P[A n {N, = n }].
e-'(`-s)
(A(t - s))i
J=--0
j!
for every integer k > 0.)
3.3 Definition A Poisson process with intensity A > 0 is an adapted, integervalued RCLL process N = {N A; 0 < t < oo } such that N0 = 0 a.s., and for 0 < s < t, N, - N, is independent of and is Poisson distributed with mean
2(t - s).
We have demonstrated in Problem 3.2 that the process N = {N AN; 0 < t < co} of (3.1) is Poisson. Given a Poisson process N with intensity A, we define the compensated Poisson process M,
- At, A; 0 < t < co.
Note that the filtrations {A"} and {AN} agree.
3.4 Problem. Prove that a compensated Poisson process {M A; t
0} is a
martingale.
33 Remark. The reader should notice the decomposition N, = M, + A, of the (submartingale) Poisson process as the sum of the martingale M and the increasing function A, = At, t > 0. A general result along these lines, due to P. A. Meyer, will be the object of the next section (Theorem 4.10).
A. Fundamental Inequalities Consider a submartingale {X,; 0 5 t < co}, and an integrable, F,-measurable random variable X,; we recall here that ,F, = a(Uto If we also have, for every 0 5 t < co, X,
a.s. P,
1.3. Continuous-Time Martingales
13
then we say that "{X <9,; 0 < t < co} is a submartingale with last element X,0". We have a similar convention in the (super)martingale case. A straightforward application of the conditional Jensen inequality (Chung (1974), Thm. 9.1.4) yields the following result.
3.6 Proposition. Let {X,, Ft; 0 < t < co} be a martingale (respectively, submartingale), and cp: 01-+ IR a convex (respectively, convex nondecreasing) function, such that Elcp(X,)I < co holds for every t _.. 0. Then {cp(X,), ,Fi; 0 < t < co} is a submartingale.
The method used to prove Jensen's inequality and Proposition 3.6 extends to the vector situation of the next problem.
3.7 Problem. Let {X, = (XP),... ,,q`')), ,97,; 0 < t < col be a vector of martingales, and cp: Rd -> P a convex function with Elcp(X,), < co valid for every t > 0. Then {co(X,), ,i; 0 t < co} is a submartingale; in particular { II X, I1, .9,; 0 < t < co} is a submartingale.
Let X = {X,; 0 < t < co} be a real-valued stochastic process. Consider two numbers a < /3 and a finite subset F of [0, co). We define the number of upcrossings U,(a, 13; X(co)) of the interval [a, /3] by the restricted sample path {X1; t E F} as follows. Set T I (CO) = min{t e F; X,(co)
a},
and define recursively for j = 1, 2, ... 01(w) = min{t E F; t _>_ ti(co), X,(co) > $),
ti,i(co) = mil* e F; t > ori(co), X,(co) < a). The convention here is that the minimum of empty set is +co, and we denote by UF(a, /3; X(w)) the largest integer j for which a(co) < co. If / [0, co) is not necessarily finite, we define Ui(a, 13; X(w)) = sup{ UF(a, /3; X(w)); F
I, F is finite).
The number of downcrossings 1),(a, /3; X(co)) is defined similarly.
The following theorem extends to the continuous-time case certain wellknown results of discrete martingales. 3.8 Theorem. Let {X *-7,; 0 < t < co} be a submartingale whose every path is right-continuous, let [o-, r] be a subinterval of [0, co), and let a < 13, ). > 0 be real numbers. We have the following results:
(i) First submartingale inequality: A
-P[ sup X, > Al 5 E(X,±).
c-[
1. Martingales, Stopping Times, and Filtrations
14
(ii) Second submartingale inequality: 2- P[
inf cr.r
-.1.1 5 E(Xt+) - E(X,).
X, _<
(iii) Uperossings and downcrossings inequalities: laj EU10 1(z /1; X(a))) < E(X`) + , 13 - 2
E(X, - a)'
EIA,.1(2, 13; X (w))
17 -1
(iv) Doob's maximal inequality:
E(
sup X,)p < cr
(
P
r
-p
)P E(Xf),
p > 1,
1
provided X, > 0 a.s. P for every t > 0, and E(Xf) < co. (v) Regularity of the paths: Almost every sample path {X,(w); 0 5 t < cc)} is bounded on compact intervals; is free of discontinuities of the second kind, i.e., admits left -hand limits everywhere on (0, co); and its jumps are exhausted by a sequence of stopping times (Proposition 2.26).
PROOF. Let the finite set F consist of o, T, and a finite subset of [o, 1 n Q. We obtain from Theorem 9.4.1 of Chung (1974): FP[max,EF X, > it] < as well as: liP[min, EF X, < -it] < E(Xt+) - E(X,). By considering an increasing sequence {F} T=1 of finite sets whose union is the whole of ([o-, r] n Q) u {6, T }, we may replace F by this union in the preceding inequalities. The right-continuity of sample paths implies then itP[sup,,,, X, > pi] ._.. E(Xt+)
and itP[inf,,,,, X, < -it] 5 E(XT) - E(X,). Finally, we let it i A to obtain (i) and (ii).
Being the limit of random variables of the form UF(2, /3; X(co)) with finite F, U[c.,i(a, 13; X (a))) is measurable. We obtain (iii), (iv) from Theorems 9.4.2, 9.5.4 in Chung (1974) (see also Meyer (1966), pp. 93-94). For (v), we note first
that the boundedness of (almost all) sample paths on the compact interval [0, n], n > 1, follows directly from (i), (ii); second, we consider the events Art3 A {we I; U10.](a, 13; X (w)) = co},
n > 1, a < 13.
By virtue of (iii), these have zero probability, and the same is true for the union
A(") = U An)13, rt<13
ri,fleQ
which includes the set
.
{co e SI; lim Xs(co) < lim Xs(w), for some t e [0, n]} Sit
51t
Consequently, for every co e il\A(n), the left limit X,_ (w) = limst, Xs(co) exists for all 0 < t 5 n. This is true for every n 1, so the preceding left limit exists for every 0 < t < co, a) e (U ,°,3=, A(n))`.
1.3. Continuous-Time Martingales
15
3.9 Problem. Let N be a Poisson process with intensity 2.
(a) For any c > 0, lim
P[
sup (N, - As)
1-00
(b) For any c > 0, lim
(c) For 0 < o- <
T,
P[ inf (Ns- As) < -c
ost
A-It1
5
1
c/27r
we have
E[ sup
-1
4r2
-T.
(N, t
cr
o-
(Hint: Use Stirling's approximation to show that lim,, (1//Tit)E(N,-10+ = 3.10 Remark. From Problem 3.9 (a) and (b), we see that for each c > 0, there exists 7', > 0 such that Nr
P[
_A >c
3 -t < c.N./27r
,
V t > Tc.
From this we can conclude the weak law of large number for Poisson processes: (N,/t)-* A, in probability as t oo. In fact, by choosing a =r and T = 2' in Problem 3.9 (c) and us'ng ebySev's inequality, one can show
P[
sup
2^12^""
Nt t
4
> s] <
82 E2 2n
for every n > 1, E > 0. Then by a Borel-Cantelli argument (see Chung (1974), Theorems 4.2.1, 4.2.2), we obtain the strong law of large numbers for Poisson processes: (N,/t) = A, a.s. P.
The following result from the discrete-parameter theory will be used repeatedly in the sequel; it is contained in the proof of Theorem 9.4.7 in Chung (1974), but it deserves to be singled out and reviewed. 3.11 Problem. Let {F,,} ,;°_, be a decreasing sequence of sub-a-fields of F (i.e., n > 1} be a backward submartinVn 1), and let {X., 'n+1 g g
gale; i.e., El X ni < co, X" is F-measurable, and E(XIF,, ) X,,,, a.s. P, for E(X)> -co implies that the sequence {X n} n'°-1 every n > 1. Then 1 A is uniformly integrable.
3.12 Remark. If IX Ft; 0
t < ool is a submartingale and {t} T=1 is a nonincreasing sequence of nonnegative numbers, then {X,n, Ft.; n >_ 1} is a backward submartingale.
1. Martingales, Stopping Times, and Filtrations
16
It was supposed in Theorem 3.8 that the submartingale X has rightcontinuous sample paths. It is of interest to investigate conditions under which we may assume this to be the case.
3.13 Theorem. Let X = {X A; 0 < t < co} be a submartingale, and assume the filtration {,97,} satisfies the usual conditions. Then the process X has a rightcontinuous modification if and only if the function t EX, from [0, co) to ER is right-continuous. If this right-continuous modification exists, it can be chosen so as to be RCLL and adapted to {A}, hence a submartingale with respect to {A}.
The proof of Theorem 3.13 requires the following proposition, which we establish first.
3.14 Proposition. Let X = {X ,F,; 0 < t < co} be a submartingale. We have the following: (i)
There is an event Cr e..97 with P(SI*) = 1, such that for every we Sr: the limits X,+(w) g urn Xs(w),
X,_ A lim Xs(co)
st.t seQ
stt
SEQ
exist for all t > 0 (respectively, t > 0). (ii) The limits in (i) satisfy E(Xt+1,Ft)
E(X,I.Ft4'
Xt
a.s. P, V t > 0. a.s. P, V t > 0.
(iii) {X,,, ..97,+; 0 < t < co} is a submartingale with P-almost every path RCLL. PROOF.
(i)
We wish to imitate the proof of (v), Theorem 3.8, but because we have not assumed right-continuity of sample paths, we may not use (iii) of Theorem 3.8 to argue that the events il!,'"),3 appearing in that proof have probability zero. Thus, we alter the definition slightly by considering the submartingale X evaluated only at rational times, and setting A(c3 =
E
Uto,1,Q(ot, /3; X (w)) = co},
n > 1, a <
A(n) =a
Then each AVp has probability zero, as does each A("). The conclusions follow readily. (ii) Let {t}c°_, be a sequence of rational numbers in (t, co), monotonically decreasing to t > 0 as n co. Then IX1., A; n > 11 is a backward sub-
martingale, and the sequence IE(Xj.)1T,_, is decreasing and bounded below by E(X,). Problem 3.11 tells us that IX,nr_1 is a uniformly integrable
1.3. Continuous-Time Martingales
17
sequence. From the submartingale property we have IA X, dP 5 fA X,. dP, for every n > 1 and A e .; uniform integrability renders almost sure into L' -convergence (Chung (1974), Theorem 4.5.4), and by letting n oo we obtain SA X, dP < 1,4 X, dP, for every A e The first inequality in (ii) follows. Now take a sequence ftnIn`c=i in (0, t) n Q, monotonically increasing to t > 0. According to the submartingale property E[X,I.F,.] > X,. a.s. We may let n a) and use Levy's theorem (Chung (1974), Theorem 9.4.8) to obtain the second inequality in (ii). (iii) Take a monotone decreasing sequence ts,,In°3_, of rational numbers, with
0 < s < s < t holding for every n > 1, and limn-. s = s. According to the first part of (ii), E(X,,A.)> Xs. a.s. Letting n co and using Levy's theorem again, we obtain the submartingale property E(X,+I,S+) > Xs+ a.s. It is not difficult to show, using (i), that P-almost every path Xi+ is RCLL. PROOF OF THEOREM 3.13. Assume that the function
EX, is right-continuous;
we show that IX,+, A; 0 < t < co as defined in Proposition 3.14 is a modification of X. The former process is adapted because of the right-continuity of {A}. Given t > 0, let Ign, I be a sequence of rational numbers with q t. Then lim, Xq. = X,+, a.s., and uniform integrability implies that EX,+ = nc,D=
limn, EXq.. By assumption, limn, EXg. = EX and Proposition 3.14 (ii)
gives X,± X a.s. It follows that X = X a.s. Conversely, suppose that {Yet; 0 < t < co} is a right-continuous modification of X. Fix t > 0 and let Itn In`°_, be a sequence of numbers with to l t. We
have P[X, = fe X,. = fe,.; n > 1] = 1 and limn gt. = fe a.s. Therefore, limn, X,. = X, a.s., and the uniform integrability of 1X1.1°°=, implies that EX, = lim,EX,.. The right-continuity of the function t EX, follows.
B. Convergence Results For the remainder of this section, we deal only with right-continuous processes, usually imposing no condition on the filtrations {A}. Thus, the description right-continuous in phrases such as "right-continuous martingale" refers to the sample paths and not the filtration. It will be obvious that the assumption of right-continuity can be replaced in these results by the assumption of right-continuity for P-almost every sample path. 3.15 Theorem (Submartingale Convergence). Let {X ,51;,; 0 5 t < oo} be a right-continuous submartingale and assume C A sup,,,,E(X,+) < co. Then Xao(w) A lim,o0 X,(w) exists for a.e. w E 0, and EIXeol < 00.
PROOF. From Theorem 3.8 (iii) we have for any n > 1 and real numbers a < 13: ELio,n0, /3; X((0)) 5 (E(Xn+) + lal)/(/3 - a), and by letting n co we obtain,
1. Martingales, Stopping Times, and Filtrations
18
thanks to the monotone convergence theorem: EUE0.00(a, /3; X(w)) 5
C + la! # - cc
The events A,,3 gl- {w; U[0,03)(a, /3; X(w)) = col, -co < a < /3 < co, are thus P-negligible, and the same is true for the event A = .J.<13 H A ,p, which contains the set {co; lim, X,(w) > limy. X,(w) }. °,13Q Therefore, for every wen\ A, Xco(co) = lim,._ X,(w) exists. Moreover,
EIX,I = 2E(X7) - E(X,) 5 2C - EX() shows that the assumption sup,>0 E(X,+) < co is equivalent to the apparently stronger one sup,>0 EIX,I < co, which in turn forces the integrability of Xco, by Fatou's lemma.
3.16 Problem. Let {X,, A; 0 < t < col be a right-continuous, nonnegative supermartingale; then X,(w) = lim,, X,(w) exists for P-a.e. (DEO, and {X,, ,F,; 0 < t 5 co} is a supermartingale. 3.17
Definition. A right-continuous, nonnegative supermartingale {Z,, 3,--,;
0 < t < col with lim, E(Z,) = 0 is called a potential. Problem 3.16 guarantees that a potential {Z,, .97,; 0 < t < co } has a last element Zco, and Z,,, = 0 a.s. P.
3.18 Exercise. Suppose that the filtration I.Fil satisfies the usual conditions. Then every right-continuous, uniformly integrable supermartingale {X,F,; 0 5 t < co} admits the Riesz decomposition X, = M, + Z,, a.s. P, as the sum of a right-continuous, uniformly integrable martingale {M,, A; 0 < t < col
and a potential {Z,, .,; 0 < t < col. 3.19 Problem. The following three conditions are equivalent for a nonnegative,
right-continuous submartingale {X,, A; 0 < t < col: (a) it is a uniformly integrable family of random variables;
(b) it converges in L', as t -, co; (c) it converges P a.s. (as t -, co) to an integrable random variable Xoc, such that {X,, ,9;; 0 < t < col is a submartingale. Observe that the implications (a) =. (b) nonnegativity.
(c) hold without the assumption of
3.20 Problem. The following four conditions are equivalent for a rightcontinuous martingale {X,, A; 0 5 t < co}: (a), (b) as in Problem 3.19; (c) it converges P a.s. (as t -, co) to an integrable random variable Xco, such that {X,, ,F,; 0 5 t < co} is a martingale;
1.3. Continuous-Time Martingales
19
(d) there exists an integrable random variable Y, such that X, = E( P, for every t > 0.
a.s.
Besides, if (d) holds and ,c is the random variable in (c), then E(Y1.973) = X.
a.s. P.
3.21 Problem. Let {N .9-;,; 0 5 t < co} be a Poisson process with parameter 1, define the process A > 0. For u e C and i =
X, = exp[iuN, - At(eiu - 1)]; 0 5 t < o0. {Im(X,), .9-;,; 0 < t < co} are martin(i) Show that {Re(X,), .9-;,; 0 < t < gales. (ii) Consider X with u = - i. Does this martingale satisfy the equivalent conditions of Problem 3.20?
C. The Optional Sampling Theorem What can happen if one samples a martingale at random, instead of fixed, times? For instance, if X, represents the fortune, at time t, of an indefatigable gambler (who plays continuously!) engaged in a "fair" game, can he hope to improve his expected fortune by judicious choice of the time to quit? If no clairvoyance into the future is allowed (in other words, if our gambler is restricted to quit at stopping times), and if there is any justice in the world, the answer should be "no." Doob's optional sampling theorem tells us under what conditions we can expect this to be true.
3.22 Theorem (Optional Sampling). Let IX .07-t; 0 < t < col be a rightcontinuous submartingale with a last element X., and let S < T be two optional times of the filtration We have
E(XTIA.) > Xs a.s. P. If S is a stopping time, then .F's can replace 9.'s+ above. In particular, EXT EX0, and for a martingale with a last element we have EXT = EX0. PROOF. Consider the sequence of random times S(w)
if S(w) = +co
S (w) = 2"
if
k
2"
1
S(w) <
2"
and the similarly defined sequences I T,,I. These were shown in Problem 2.24
to be stopping times. For every fixed integer n 1, both S,, and 7,, take on a countable number of values and we also have S,, 5 Tn. Therefore, by the "discrete" optional sampling Theorem 9.3.5 in Chung (1974) we have
1. Martingales, Stopping Times, and Filtrations
20
I A XsndP 5 i A X T. dP for every A e An, and a fortiori for every A a .F5+ = n.c0=. g-s, by virtue of Problem 2.23. If S is a stopping time, then S .. S implies ..Fs g mss as in Lemma 2.15, and the preceding inequality also holds for every Au ..Fs .
It is checked similarly that {X5., An; n > 1} is a backward submartingale, with {E(Xs.)},T=1 decreasing and bounded below by E(X0). Therefore, the sequence of random variables {X,.}`°_, is uniformly integrable (Problem 3.11), and the same is of course true for {X Tn}'_, . The process is right-continuous, so XT(ow) = lim, X Tn(co) and Xs(w) = lim, Xsn(w) hold for a.e. co e D. It follows from uniform integrability that XT, Xs are integrable, and that 1, Xs dP 5 IA XT dP holds for every A a A + . 3.23 Problem. Establish the optional sampling theorem for a right-continuous
submartingale {X 5,; 0 < t < oo} and optional times S < T under either of the following two conditions: (i) T is a bounded optional time (there exists a number a > 0, such that T < a); (ii) there exists an integrable random variable Y, such that X, 5 E(YIA) a.s. P, for every t > 0.
3.24 Problem. Suppose that {X A; 0 < t < co} is a right-continuous submartingale and S < T are stopping times of {,F,}. Then (i) {XT. A t, .; 0 < < Go} is a submartingale; (ii) E[XT,,IA] > Xs," a.s. P, for every t > 0.
3.25 Problem. A submartingale of constant expectation, i.e., with E(X,) = E(X0) for every t > 0, is a martingale.
3.26 Problem. A right-continuous process X = {X ..,; 0 < t < co} with El X,I < co; 0 < t < oo is a submartingale if and only if for every pair S < T of bounded stopping times of the filtration {,F,} we have (3.2)
E(XT) > E(Xs).
3.27 Problem. Let T be a bounded stopping time of the filtration 1,,I, which satisfies the usual conditions, and define . = 'T-Ft; t > 0. Then {A} also satisfies the usual conditions. (i)
(ii)
If X = IX g",; 0 < t < col is a right-continuous submartingale, then so is )7 = {2t '' Kr+t - X T , g',; 0 < t < Op} . If g = {g .; 0 < t < oc } is a right-continuous submartingale with go = 0, a.s. P, then X = {X, A 1(t_7.) 0, ,F,; 0 < t < co} is also a submartingale.
3.28 Problem. Let Z = {Z ,5F,; 0 < t < oo} be a continuous, nonnegative martingale with Zco -4 lim, Z, = 0, a.s. P. Then for every s _. 0, b > 0:
1.4. The Doob-Meyer Decomposition 1
= -bZ,,
(i) P [sup Z,
21
a.s. on {Zs < b}.
t>s
(ii) P [sup Z, t>s
to]
= P[Z,
1
b] + -E[Zslf b
z.'
]
3.29 Problem. Let {X ,F,; 0 < t < co} be a continuous, nonnegative supermartingale and T = inf {t 0; X, = 0}. Show that XT+, = 0; 0
t < co
holds a.s. on IT < co}.
3.30 Exercise. Suppose that the filtration {A} satisfies the usual conditions and let X(") = {X:1", ,F,; 0 < t < co}, n > 1 be an increasing sequence of right-
continuous supermartingales, such that the random variable, lim_co X:") is nonnegative and integrable for every 0 < t < co. Then there exists an RCLL
supermartingale X = {X ,F,; 0 5 t < co} which is a modification of the process
={
Ft; 0
t < co}.
1.4. The Doob-Meyer Decomposition This section is devoted to the decomposition of certain submartingales as the summation of a martingale and an increasing process (Theorem 4.10, already presaged by Remark 3.5). We develop first the necessary discrete-time results. 4.1 Definition. Consider a probability space (It P) and a random sequence {A},7=0 adapted to the discrete filtration {,},T=0. The sequence is called increasing, if for P-a.e. (c) E S/ we have 0 = Ao(w) < A,(w) < - - - , and E(A) < co holds for every n > 1.
An increasing sequence is called integrable if E(A0D) < co, where As, = A. An arbitrary random sequence { },T=0 is called predictable for the filtration {,},7_0, if for every n 1 the random variable is ,_1-measurable.
Note that if A = {A, F; n = 0, 1, ... } is predictable with E I AI < co for every n, and if {M, n = 0, 1, ...} is a bounded martingale, then the martingale transform of A by M defined by (4.1)
Yo =0 and Y "= E A(11/1 - Mk_,);
n >- 1,
k =1
is itself a martingale. This martingale transform is the discrete-time version of the stochastic integral with respect to a martingale, defined in Chapter 3. A fundamental property of such integrals is that they are martingales when parametrized by their upper limit of integration. Let us recall from Chung (1974), Theorem 9.3.2 and Exercise 9.3.9, that any
1. Martingales, Stopping Times, and Filtrations
22
submartingale {X,,, ".; n = 0, 1, ... } admits the Doob decomposition X =
M + An as the summation of a martingale {M, F.} and an increasing sequence {A, ,}. It suffices for this to take Ao = 0 and A+1 = A -X + -X ki, for n 0. This increasing sequence E(X.+1 ign) = =o[E1X k+ is actually predictable, and with this proviso the Doob decomposition of a submartingale is unique. We shall try in this section to extend the Doob decomposition to suitable continuous-time submartingales. In order to motivate the developments, let us discuss the concept of predictability for stochastic sequences in some further detail. 4.2 Definition. An increasing sequence IA, F; n = 0, 1, ...1 is called natural if for every bounded martingale IM,,, .97;; n = 0, 1, ...1 we have (4.2)
E(MA) = E > Mk_i(Ak - A"), do > 1. k=1
A simple rewriting of (4.1) shows that an increasing sequence A is natural if and only if the martingale transform Y = Y1,,'=0 of A by every bounded martingale M satisfies EY = 0, n > 0. It is clear then from our discussion of martingale transforms that every predictable increasing sequence is natural. We now prove the equivalence of these two concepts. 4.3 Proposition. An increasing random sequence A is predictable if and only if
it is natural.
PROOF. Suppose that A is natural and M is a bounded martingale. With Y};,°=0 defined by (4.1), we have
E[A(M - M_1)] = EY - EY_i = 0, n > 1. It follows that
(4.3) E[M{A - E(A1,F_1) 1] = E[(M - M-,)A] + E[M_11A - E(A1,T,_1) 1]
- E[(M -
=0
for every n > 1. Let us take an arbitrary but fixed integer n > 1, and show that the random variable A is ,_1-measurable. Consider (4.3) for this fixed integer, with the martingale M given by
sgn[A - E(A1"_1)], k = n, Mk =
E(Mni,k), We obtain El An -
k > n, k = 0, 1, ..., n.
= 0, whence the desired conclusion.
1.4. The Doob-Meyer Decomposition
23
From now on we shall revert to our filtration {,51;,} parametrized by t e [0, cc) on the probability space (f2, P). Let us consider a process A = IA1; By analogy with Definitions 4.1 and 4.2, we have 0 < t < col adapted to the following: 4.4 Definition. An adapted process A is called increasing if for P-a.e. w a SI we have
(a) Ao(w) = 0 (b) t A,(w) is a nondecreasing, right-continuous function,
and E(A,) < co holds for every te [0, co). An increasing process is called integrable if E(4,0) < co, where Aco =
A,.
4.5 Definition. An increasing process A is called natural if for every bounded,
right-continuous martingale {M ,;(; 0 < t < col we have E .1
(4.4)
Ms dAs = E
1
Ms_ dA,, for every 0 < t <
(041
(0,fl
4.6 Remarks. (i)
If A is an increasing and X a measurable process, then with w e SI fixed, the sample path IX,(w); 0 < t < col is a measurable function from [0, cc) into R. It follows that the Lebesgue-Stieltjes integrals Xs-± (w) dAs(w)
I ,± (w) -4 (o ,ti
are well defined. If X is progressively measurable (e.g., right-continuous and adapted), and if I, = - I,- is well defined and finite for all t > 0, then I is right-continuous and progressively measurable. (ii) Every continuous, increasing process is natural. Indeed then, for P-a.e. E
SI we have
= 0 for every 0 < t < co,
(Ms(co) - Ms_(o.))) dA fo.o
because every path {MA)); 0 < s < col has only countably many discontinuities (Theorem 3.8(v)). (iii)
It can be shown that every natural increasing process is adapted to the filtration {,F,_ } (see Liptser & Shiryaev (1977), Theorem 3.10), provided that {,F,} satisfies the usual conditions.
4.7 Lemma. In Definition 4.5, condition (4.4) is equivalent to (4.4)'
E(M,A,) = E
Ms_ dAs. J
1. Martingales, Stopping Times, and Filtrations
24
PROOF. Consider a partition H = {to, t, , ... , tk} of [0, t], with 0 = to <
--
- < t = t, and define n
Mn = E Mtkl(tk-1,td(S). k=1
The martingale property of M yields E
I
n
NINA, = E E 111,k(Atk - Atk_,) =
(0,1]
k=1
n
n-1
k=1
k=0
E[ E m,,,A,k -E mtk
'
,Atk]
..-1
= E(Mtili) -E E Afk(M,,,,, - MO = E(MiAt). k=1
Now let 111111 A max, , < (tk - tk_, )-* 0, so Mn -* Ms, and use the bounded
convergence theorem for Lebesgue-Stieltjes integration to obtain (4.5)
E(1141A1) = E
1
MsdAs.
D
(o,t]
The following concept is a strengthening of the notion of uniform integrability for submartingales.
4.8 Definition. Let us consider the class Y(.9'.) of all stopping times T of the filtration {,,} which satisfy P(T < co) = 1 (respectively, P(T 5 a) = 1 for a given finite number a > 0). The right-continuous process {X1, A; 0 < t < co} is said to be of class D, if the family {X T} T e y is uniformly integrable; of class DL, if the family {XT} T c 9,. is uniformly integrable, for every 0 < a < oo.
4.9 Problem. Suppose X = {X1, A; 0 < t < co} is a right-continuous submartingale. Show that under any one of the following conditions, X is of class DL.
(a) X, > 0 a.s. for every t _- 0. (b) X has the special form (4.6)
X, = M, + A 0 < t < co
suggested by the Doob decomposition, where {M1, ,97,; 0 < t < co} is a martingale and {A1, A; 0 < t < co} is an increasing process. Show also that if X is a uniformly integrable martingale, then it is of class D.
The celebrated theorem which follows asserts that membership in DL is also a sufficient condition for the decomposition of the semimartingale X in the form (4.6).
4.10 Theorem (Doob-Meyer Decomposition). Let {A} satisfy the usual conditions (Definition 2.25). If the right-continuous submartingale X = {X1, A; 0 5 t < col is of class DL, then it admits the decomposition (4.6) as the summa-
1.4. The Doob-Meyer Decomposition
25
tion of a right-continuous martingale M = {M .51;i; 0 < t < co} and an increas-
t < co}. The latter can be taken to be natural;
ing process A = {Al,
under this additional condition, the decomposition (4.6) is unique (up to indistin-
guishability). Further, if X is of class D, then M is a uniformly integrable martingale and A is integrable.
PROOF. For uniqueness, let us assume that X admits both decompositions X, = M; + A; = Ml + A;', where M' and M" are martingales and A', A" are natural increasing processes. Then {B, A A; - A," = M;' ,; 0 < t < co is a martingale (of bounded variation), and for every bounded and rightcontinuous martingale {L, .f.} we have
E[ ,(A; - A;')] = E
f
d13, = lim E E j=1
"_.co
( 0,t1
j-
[Be.) J
J-
where ri= {t(S),...,t(,")}, n > 1 is a sequence of partitions of [0, t] with = maxi j<,. (ein) -t.(i1) converging to zero as n co, and such that each 1-In+1 is a refinement of H. But now EH,p),(B,(;) - Ber)i)] = 0,
E[ ,(A; - A7)] = 0.
and thus
For an arbitrary bounded random variable we can select R Al to be a right-continuous modification of {EUIFJ, Ft} (Theorem 3.13); we obtain EMA; - AD] = 0 and therefore P(A; = At") = 1, for every t 0. The rightcontinuity of A' and A" now gives us their indistinguishability. For the existence of the decomposition (4.6) on [0, co), with X of class DL, it suffices to establish it on every finite interval [0, a]; by uniqueness, we can then extend the construction to the entire of [0, co). Thus, for fixed 0 < a < co, let us select a right-continuous modification of the nonpositive submartingale
X, - E[XIF], 0
Y,
Consider the partitions 11 = {tr, ei"),
form tr) = (j/2")a, j = 0,
1,
,
t
a.
t(1),} of the interval [0, a] of the
..., 2^. For every n > 1, we have the Doob
decomposition
= M,7, + A
,
j = 0, 1,
, 2"
where the predictable increasing sequence A(n) is given by A(,'(:!)
= AP)i + E J-1
= E E[Yr:li - Ytzo Ftz)], j = 1,...,2n. k=0
Notice also that because MI") = - A!,"), we have (4.7)
Ye?) = /1(,V) - E[A!,n]Ig"00],
j = 0, 1, ..., 2".
We now show that the sequence {4)}`°_1 is uniformly integrable. With A > 0, we define the random times
1. Martingales, Stopping Times, and Filtrations
26
n.) = a A min{t } "),; A t(i'a))> A for some j, 1 < j
, r, and { TP) < a) =
} = {4 > A} E Ay. for j = 1,
We have { T71 <
}, we have E [A(:) I Ard =
{ Ala") > 2 }. Therefore, TA!")E 9. On each set { 71")=
E[A!,") I gTr], so (4.7) implies (4.8)
YTS", = AV^) - EEA(:)1.°17Tri < A - EEA(an)1.°17Tri
on { 71") < a). Thus (4.9)
Al7) dP 5 2P[T1") < a] - I
I
Y7TodP.
Replacing A by A/2 in (4.8) and integrating the equality over the measurable set {
- L171
< a}, we obtain (A(") - A(PLI)dP
Ypn) dP = /2
f{T71
IIn')
(A(Z1) - TA(t) dP > -P [71") < a], 2
A/2
and thus (4.9) leads to A(an) dP < -2
(4.10) AT)> A}
Y (.)d P f{
YT1"d P - f(Tr
-
7'
The family {XT}TE.q is uniform y integrable by assumption, and thus so is {YT }TE.q. But
P[11") < a] =
>A]<
E(A(:))
E( Yo)
A
SO
sup P[71") < a] -> 0 as A
co.
a>1
Since the sequence { YT00}:1.1 is uniformly integrable for every c > 0, it follows from (4.10) that the sequence 1,4("))°11 is also uniformly integrable.
By the Dunford-Pettis compactness criterion (Meyer (1966), p. 20, or Dunford & Schwartz (1963), p. 294), uniform integrability of the sequence IM,")},f_1 guarantees the existence of an integrable random variable Aa, as well as of a subsequence 1/1!,nk)),`,°_i which converges to A. weakly in I):
lim E(A?k)) = E(Aa) k-.co
for every bounded random variable To simplify typography we shall assume henceforth that the preceding subsequence has been relabeled and shall denote it simply as {Anax_1. By analogy with (4.7), we define the process {A..57;,} as a right-continuous modification of (4.11)
A, = 17,
E(A.IA);
0
t
a.
1.4. The Doob-Meyer Decomposition
27
4.11 Problem. Show that if {Aw}c°_, is a sequence of integrable random variables on a probability space (12, .F, P) which converges weakly in 1) to an
integrable random variable A, then for each 6 -field E[A(")1W] converges to E[Al] weakly in L'.
c .F, the sequence
Let n = U-=, nn. Fort E II, we have from Problem 4.11 and a comparison
of (4.7) and (4.11) that lim, E(,V)) = E(At) for every bounded random variable For s, t E n with 0 s < t < a, and any bounded and nonnegative ER(A!") - Ar)] 0, random variable we obtain E[(A, - A,)] = we get AS < A a.s. P. Because II is countable, and by selecting = for a.e. w WE SI the function ti-* At(co) is nondecreasing on II, and right-continuity
shows that it is nondecreasing on [0, a] as well. It is trivially seen that Ao = 0, a.s. P. Further, for any bounded and right-continuous martingale gt, Ftl, we have from (4.2) and Proposition 4.3: 2"
- A:740
EE
E(0A(:))
j =1 2"
= E E toni [Yon) - Ytnni] J2"
=EE
[Air - Aepi],
where we are making use of the fact that both sequences {Ai -Y
{Ar -
for t e n, are martingales. Letting n
and co one obtains by virtue
of (4.5):
E
f
sdA, = E
(0,a]
f
S
dAs,
(0,a1
as well as E
dAs, V t e [0, a], if one remembers that =E a} is also a (bounded) martingale (cf. Problem 3.24). There{SAtI fore, the process A defined in (4.11) is natural increasing, and (4.6) follows with .; 0
Mi
E [X. -
0
t
a.
Finally, if the submartingale X is of class D it is uniformly integrable, hence it possesses a last element Xco to which it converges both in 11 and almost surely as t co (Problem 3.19). The reader will have no difficulty repeating the preceding argument, with a = oo, and observing that E(A,o) < co.
Much of this book is devoted to the presentation of Brownian motion as the typical continuous martingale. To develop this theme, we must specialize the Doob-Meyer result just proved to continuous submartingales, where we discover that continuity and a bit more implies that both processes in the decomposition also turn out to be continuous. This fact will allow us to conclude that the quadratic variation process for a continuous martingale (Section 5) is itself continuous.
1. Martingales, Stopping Times, and Filtrations
28
4.12 Definition. A submartingale {X .; 0 < t < oo} is called regular if for every a > 0 and every nondecreasing sequence of stopping times {
g
with T = limn T,, we have limn E(XT,.) = E(XT). 4.13 Problem. Verify that a continuous, nonnegative submartingale is regular.
4.14 Theorem. Suppose that X = {X,; 0 < t < co} is a right-continuous submartingale of class DL with respect to the filtration {A}, which satisfies the usual conditions, and let A = {A1; 0 < t < oo} be the natural increasing process in the Doob-Meyer decomposition (4.6). The process A is continuous if and only if X is regular.
PROOF. Continuity of A yields the regularity of X quite easily, by appealing
to the optional sampling theorem for bounded stopping times (Problem 3.23(i)).
Conversely, let us suppose that X is regular; then for any sequence {T}n°_, as in Definition 4.12, we have by optional sampling: limn, E(AT.) = limn E(XTn) - E(MT) = E(AT), and therefore AT(w)(w) j AT(.)(w) except for w in a P-null set which may depend on T To remove this dependence on T, let us consider the same sequence {11}n'.1 of partitions of the interval [0, a] as in the proof of Theorem 4.10, and select
a number A > 0. For each interval (tr, tp,), j = 0, 1, ..., 2" - 1 we consider a right-continuous modification of the martingale
t") < t <
= E[A A
This is possible by virtue of Theorem 3.13. The resulting process {V); 0 < t < a} is right-continuous on (0,a) except possibly at the points of the partition, and dominates the increasing process {A A 211; 0 < t < a}; in particular, the two processes agree a.s. at the points tV, el.!. Because A is a natural increasing process, we have from (4.4) Vs") dAs = E
dAs;
j = 0, 1, ..., 2" - 1,
yv,
and by summing over j, we obtain (4.12)
E
Vs") dAs = E
J
0,0
f
Vs"2 dAs,
(0,1]
for any 0 < t < a. Now the process
= 0,
- (A A 241),
t < a, t = a, 0
is right-continuous and adapted to {.97, }; therefore, for any E > 0 the random time
T(s) = a A inf {0
t 5 a; fir >
= a A inf {0 < t < a; 111) - (A A At) >
29
1.4. The Doob-Meyer Decomposition
is an optional time of the right-continuous filtration {.F, }, hence a stopping time in X,' (cf. Problem 2.6 and Proposition 2.3). Further, defining for each n > I the function con( ): [0, a] -+ 11 by (p(t)= tin+)1; ti ") < t < t.11,)1, we have
(p.(7,,(E)) e tea. Because en) is decreasing in n, the increasing limit T = T(E) exists a.s., is a stopping time in 5o, and we also have
T = lim (p.(7,.(0) a.s. P. By optional sampling we obtain now 2 " -1
EN(4.1.)(,)]= E E[E(A. A ile,PV, (' T(e))I {tp)< T(e)5 ty.Vj j=1
= E[2 A Aq,.(7,n(e))],
and therefore E[(A A ilo.(T.(0)) - (A A AT,,(0)] = EN(4.1,!(,) - (2 n AT,,(0)] = E[1{7.(0
(2 A A T,,(e)))]
EP [ Tn(E) < a].
We employ now the regularity of X to conclude that for every E > 0,
P[Q > e].= P[Tn(E)< a] <_!E[(2 n Aon(T.(,))) - (A n AT,(e))]
0
oc, where Q,, supo <1 n) - (2 A A,)I. Therefore, this last sequence of random variables converges to zero in probability, and hence also almost surely along a (relabeled) subsequence. We apply this observation to (4.12), along with the monotone convergence theorem for Lebesgue-Stieltjes integration, to obtain
as n
,7
1
f(A A As)dAs= E E (o,ti
f
(A A As4dAs, 0
< oo,
which yields the continuity of the path ti-* 2 A At(co) for every 2 > 0, and At(co) for P-a.e. w hence the continuity of
4.15 Problem. Let X = {Xt, Ft; 0 < t < co} be a continuous, nonnegative process with X0 = 0 a.s., and A = {A Ft; 0 < t < co} any continuous, increasing process for which (4.13)
E(XT) < E(AT)
holds for every bounded stopping time T of {",}. Introduce the process V, A maxo.(s.,, X5, consider a continuous, strictly increasing function F on [0, oo) with F(0) = 0, and define G(x) g 2F(x) + x Lc' 14' dF(u); 0 < x < oo. Establish the inequalities (4.14)
P[VT
e] <
E(AT) ;
V>0
1. Martingales, Stopping Times, and Filtrations
30
(4.15) (4.16)
P[VT
e, AT < 6]
E(6 A AT) ;
c
V c > 0, 6 > 0
EF(VT) _< EG(AT)
for any stopping time T of {F }. 4.16 Remark. If the process X of Problem 4.15 is a submartingale, then A can be taken as the continuous, increasing process in the Doob-Meyer decomposition (4.6) of Theorems 4.10 and 4.14.
4.17 Remark. The corollary (4.15)'
P[ VT > e] <
E(6 A AT) c
+ PEAT > 6]
of (4.15) is very useful in the limit theory of continuous-time martingales; it is known as the Lenglart inequality. We shall use it to establish convergence results for martingales (Problem 5.25) and stochastic integrals (Proposition 3.2.26). On the other hand, it follows easily from (4.16) that (4.17)
E( VI)
2 1
p
- pE(Ac);
0
holds for any stopping time T of {g; }.
1.5. Continuous, Square-Integrable Martingales In order to appreciate Brownian motion properly, one must understand the role it plays as the canonical example of various classes of processes. One such
class is that of continuous, square-integrable martingales. Throughout this section, we have a fixed probability space (Q, ..F., P) and a filtration {,F} which satisfies the usual conditions (Definition 2.25).
5.1 Definition. Let X = {X gt; 0 < t < oo} be a right-continuous martingale. We say that X is square-integrable if EV < co for every t > 0. If, in addition, X0 = 0 a.s., we write X e.1'2 (or X e .4q, if X is also continuous).
5.2 Remark. Although we have defined JP; so that its members have every sample path continuous, the results which follow are also true if we assume only that P-almost every sample path is continuous. For any X e .Ai 2, we observe that X2 = {V, F; 0 < t < oo } is a nonnegative submartingale (Proposition 3.6), hence of class DL, and so X2 has a unique Doob-Meyer decomposition (Theorem 4.10, Problem 4.9):
V = M, + At; 0
t < oo
1.5. Continuous, Square-Integrable Martingales
31
where M = {Mt, Ft; 0 < t < co} is a right-continuous martingale and A = {A -A; 0 s t < cc} is a natural increasing process. We normalize these processes so that Mo = Ao = 0, a.s. P. If X e Jf2, then A and M are continuous (Theorem 4.14 and Problem 4.13); recall Definitions 4.4 and 4.5 for the terms increasing and natural.
5.3 Definition. For X e./112, we define the quadratic variation of X to be the process <X>, -4- A where A is the natural increasing process in the Doob-
Meyer decomposition of X2. In other words, <X> is that unique (up to indistinguishability) adapted, natural increasing process, for which <X>0 = 0 a.s. and X2 - <X> is a martingale.
5.4 Example. Consider a Poisson process {N A; 0 < t < oo} as in Definition 3.3 and assume that the filtration {,,} satisfies the usual conditions (this can be accomplished, for instance, by "augmentation" of IF,N1; cf. Remark 2.7.10). It is easy to verify that the martingale M, = N, - At, of Problem 3.4 is in ///i'2, and <M>, = At.
If we take two elements X, Y of X2, then both processes (X + Y)2 <X + Y> and (X - Y)2 - <X - Y> are martingales, and therefore so is their + Y> - <X - Y>]. difference 4X Y 5.5 Definition. For any two martingales X, Y in 112, we define their crossvariation process <X, Y> by
<X, Y>, = 1[<X + Y>, - <X - Y>,]; 0 < t < and observe that X Y - <X, Y> is a martingale. Two elements X, Y of .112 are called orthogonal if <X, Y>, = 0, a.s. P, holds for every 0 < t < co. The uniqueness argument in Theorem 4.10 also shows that <X, Y> is, up to indistinguishability, the only process of the form A = A(1) - A(2) with A(j) adapted and natural increasing (j = 1, 2), such that X Y -A is a martingale.
In particular, <X, X> = <X>. For continuous X and Y, we give a different uniqueness argument in Theorem 5.13. 5.6 Remark. In view of the identities
E[(X, - Xs/(Y - Ys/1"-'s] = E[XtYt - X. Ys Fs] = EUX , Y>, - <X, Y>,1 37.j,
valid P a.s. for every 0 < s < t < co, the orthogonality of X, Y in .112 is equivalent to the statements "X Y is a martingale" or "the increments of X and Y over [s, t] are conditionally uncorrelated, given ,Fs". 5.7 Problem. Show that < > is a bilinear form on 12, i.e., for any members X, Y, Z of 12 and real numbers a, )3, we have
1. Martingales, Stopping Times, and Filtrations
32
(i) = a <X, Z> + 13 ,
(ii) <X, Y> = , (iii) loc, Y>12 < <X> (iv) For P-a.e. w e SI, 4'r(w) - 4s(w) < 1-[<X>,(w) - <X>.*0) + < Y>,(co) - < Y>sIcon;
0<s
A <X, Y> on [0, t].
The use of the term quadratic variation in Definition 5.3 may appear to be unfounded. Indeed, a more conventional use of this term is the following. Let X = {X,; 0 < t < co} be a process, fix t > 0, and let 11 = {to, ti,..., t,,,}, with < t, = t, be a partition of [0, t]. Define the p-th variation 0 = to < t1 < t2 .< (p > 0) of X over the partition n to be V(P)(11) = i I Xt, - Xtk, IP. k=1
Now define the mesh of the partition 11 as 111111 = max,
the quadratic variation of X on [0, t]. Our justification of Definition 5.3 for continuous martingales (on which we shall concentrate from now on) is the following result:
5.8 Theorem. Let X be in A. For partitions
11
of [0,t], we have
limiln11-0 V(2)(1) = <X>, (in probability); i.e., for every e > 0, ri > 0 there exists
o > 0 such that linll < (5 implies
P[I v,(2 >(n) - <x),1 > e] < ri.
The proof of Theorem 5.8 proceeds through two lemmas. The key fact employed here is that, when squaring sums of martingale increments and taking
the expectation, one can neglect the cross-product terms. More precisely, if
X e it2 and 0 <s. In the latter case, we note that because
EUX - XJ2 1.°7A = E[X ,2 - 2X E[X 1"] + X,;1.9%]
= E[Xt2 -X,2,I,] = E[ <X> -<X>1], the terms Xv2 - <X> - (X,42 - <X>.) and (X - XJ2 - (<X>,, - <X>,4) have the same conditional expectation given g", namely zero, and thus the expectation of products of such terms over nonoverlapping intervals is still zero.
1.5. Continuous, Square-Integrable Martingales
33
5.9 Lemma. Let X e di2 satisfy IX,1 < K < oo for all s e [0, t], a.s. P. Let 11= {to,ti,..., tm}, with 0 = to 5 t1 5 5 t. = t, be a partition of [0, t]. 6K4.
Then E[Vt(2)(n)12
PROOF. Using the martingale property, we have for 0 < k < m - 1,
E[
(X,,-
1)2
E[{
.7,
j=k+1
j=k+1
(X1, -
X1,_1)} 2
E[(X,2, -
=
Ak]
Ak]
j=k+1
K2
E[X,2,.1,(k]
SO
m-i
m
EE
x,_,)2pck -
E[ k=1 j=k+1
xtk_i>21
m-1
=
(5.1)
E[ E (X- x1,,_1)2 E Euxt,k=1
x1,_1)24-7,3]
j=k+1
m-1 Ic2ELE (Xk
xik_,)21< K4.
k=1
We also have (5.2)
E[
(Xtk - Xtk_i)4]< 4K2 E
[E (X,k
k=1
Xtk -1 )2]
k=1
< 4K4.
Inequalities (5.1) and (5.2 imply
E[1' 2'(n)]2 =
E[
(Xtk -X,
)41
k=1 m
+ 2E E E
x,)2pck- x(k,)2]
k=1 j=k+1
< 610.
5.10 Lemma. Let X e .11'2 satisfy XSI < K < Do for all se[0,t], a.s. P. For partitions n of [0, t], we have lim Ev,(4)(n) = 0. II
bil-o
PROOF. For any partition n, we may write v(4)(H). v,(2)-ff).14(x; 111111), where (5.3)
tn,(X; 6) A sup{ I(X,. - Xs)l; 0
r<s
t, s -r
6}
is measurable because the supremum can be restricted to rational s and r. The
1. Martingales, Stopping Times, and Filtrations
34
Holder inequality implies
(E0,;(2)(n)]2,1/2(E4(x;
Evt(4)(n)
H10)1/2. I
I)
As 111111 approaches zero, the first factor on the right-hand side remains bounded and the second term tends to zero, by the uniform continuity on [0, t] of the sample paths of X and by the bounded convergence theorem.
PROOF OF THEOREM 5.8. We consider first the bounded case: I X,I < K < oo and
<X s> < K hold for all s E [0, a.s. P. For any partition II = {to, tl, t,,,} as earlier we may write (see the discussion preceding Lemma 5.9 and relation (5.3)):
E(v(2)(n)- <x>1)2 =
- <X >,k_,)}]2
{(Xtk - X,k_i)2 -
= E EUX, k k=1
2 k=1
k-I
(XX,, - <X >tk_M2
EUX,k - X,k_, )4 + RX>tk - <X>tk-1)2]
2EV,(4)(11) + 2E[<X>, rni( <X>; 111111)].
As the mesh of II approaches zero, the first term on the right-hand side of this inequality converges to zero because of Lemma 5.10; the second term does as well, by the bounded convergence theorem and the sample path uniform continuity of <X>. Convergence in L2 implies convergence in probability, so this proves the theorem for martingales which are uniformly bounded. Now suppose X e Llz is not necessarily bounded. We use the technique of
localization to reduce this case to the one already studied. Let us define a sequence of stopping times (Problem 2.7) for n = 1, 2, ... by inf It > 0; 1X11 > n or <X>,
7:,
n }.
Now Xr
X,A T. is a bounded martingale relative to the filtration {.97,} - <X>t, r,,, .mot; 0 < t < co} is a bounded martingale. From the uniqueness of the Doob-Meyer decomposition, we see that (Problem 3.24), and likewise IX,2A
(5.4)
(V")>t = <X>tAT-
Therefore, for partitions II of [0, t], we have lim 11n11-0
E[
k=i
(X,k A
Xtk IA T,,)2
<X>t A 7;12 = O.
Since T. T co a.s., we have for any fixed t that P[7,, < t] = 0. These facts can be used to prove the desired convergence of v,(2)(n) to <x>, in probability.
111
1.5. Continuous, Square-Integrable Martingales
35
5.11 Problem. Let {X g'-t; 0 5 t < co} be a continuous process with the property that for each fixed t > 0 and for some p > 0, lira vt(P)(n) = L,
(in probability),
0111-0
where Li is a random variable taking values in [0, co) a.s. Show that for q > p, vi(q)(n) = 0 (in probability), and for 0 < q < limllnllo V( )(n) = GO (in probability) on the event 14 > 01.
5.12 Problem. Let X be in ./C, and T be a stopping time of IFtl. If <X> T = 0, a.s. P, then we have P[XT = 0; d0 < t < co] = 1. The conclusion to be drawn from Theorem 5.8 and Problems 5.11 and 5.12 is that for continuous, square-integrable martingales, quadratic variation is the "right" variation to study. All variations of higher order are zero, and, except in trivial cases where the martingale is a.s. constant on an initial interval, all variations of lower order are infinite with positive probability. Thus, the sample paths of continuous, square-integrable martingales are quite different from "ordinary" continuous functions. Being of unbounded first variation, they cannot be differentiable, nor is it possible to define integrals of the form Ito Ys(w) dXs(w) with respect to X e dr, in a pathwise (i.e., for every
or P-almost every w en), Lebesgue-Stieltjes sense. We shall return to this problem of the definition of stochastic integrals in Chapter 3, where we shall give Ito's construction and change-of-variable formula; the latter is the counterpart of the chain rule from classical calculus, adapted to account for the unbounded first, but bounded second, variation of such processes. It is also worth noting that for X e..11`2, the process <X>, being monotone, is its own first variation process and has quadratic variation zero. Thus, an integral of the forml Yid < X>, is defined in a pathwise, Lebesgue-Stieltjes sense (Remark 4.6 (i)).
We discuss now the cross-variation between two continuous, squareintegrable martingales.
5.13 Theorem. Let X = {X Fi; 0 < t < co} and Y = {X f7i; 0 < t < co} be members of dic2 . There is a unique (up to indistinguishability) {A}-adapted, continuous process of bounded variation {A ..Ft; 0 < t < co} satisfying Ao = 0
a.s. P, such that {X,Yi - A A; 0 < t < co} is a martingale. This process is given by the cross-variation <X, Y> of Definition 5.5. PROOF. Clearly, A = <X, Y> enjoys the stated properties (continuity is a con-
sequence of Theorem 4.14 and Problem 4.13). This shows existence of A. To prove uniqueness, suppose there exists another process B satisfying the conditions on A. Then
M A (X Y - A) - (X Y - B) = B - A is a continuous martingale with finite first variation. If we define
36
1. Martingales, Stopping Times, and Filtrations
T = inf It
0:
=n },
then IM:n)
MIA T, .Ft; 0 5 t < col is a continuous, bounded (hence squareintegrable) martingale, with finite first variation on every interval [0, t]. It follows from (5.4) and Problem 5.11 that <M> t
T,, =
<M() )t = 0 a.s.,
t > 0.
Problem 5.12 shows that M(n) FE 0 a.s., and since 7,, 1. cc as n
co, we conclude
that M= 0, a.s. P. 5.14 Problem. Show that for X, Y e .//t2 and H = {to, t1,..., t,,,} a partition of
[0, a
lim E (xtk -
-
= <X, Y>,
(in probability).
11r111-0 k=1
Twice in this section we have used the technique of localization, once in the proof of Theorem 5.8 to extend a result about bounded martingales to squareintegrable ones, and again in the proof of Theorem 5.13 to apply a result about
square-integrable martingales to a continuous martingale which was not necessarily square-integrable. The next definitions and problems develop this idea formally.
5.15 Definition. Let X = {X
0 < t < col be a (continuous) process with
X0 = 0 a.s. If there exists a nondecreasing sequence { 7,,},T=1 of stopping times of such that A Xt AT , .Ft; 0 < t < co} is a martingale for each n 1
and P[lim,'T = oo] = 1, then we say that X is a (continuous) local martingale and write X e di' (respectively, X e icI if X is continuous). 5.16 Remark. Every martingale is a local martingale (cf. Problem 3.24(i)), but
the converse is not true. We shall encounter continuous, local martingales which are integrable, or even uniformly integrable, but fail to be martingales (cf. Exercises 3.3.36, 3.3.37, 3.5.18 (iii)).
5.17 Problem. Let X, Y be in .41`1". Then there is a unique (up to indistinguishability) adapted, continuous process of bounded variation <X, Y> jrioc. If X = Y, we satisfying <X, Y>0 = 0 a.s. P, such that X Y - <X, write <X> = <X, X>, and this process is nondecreasing. 5.18 Definition. We call the process <X, Y> of Problem 5.17 the cross-variation
of X and Y, in accordance with Definition 5.5. We call <X> the quadratic variation of X.
5.19 Problem. (i) A local martingale of class DL is a martingale. (ii) A nonnegative local martingale is a supermartingale. (iii) If M E ./IrI" and S is a stopping time of then E(MD < E <M)s, where Ai! -4
37
1.5. Continuous, Square-Integrable Martingales
We shall show in Theorem 3.3.16 that one-dimensional Brownian motion IB ..11,; 0 < t < col is the unique member of .11`.1" whose quadratic variation
at time t is t; i.e., B,2 - t is a martingale. We shall also show that d-dimensional Brownian motion {(Bp), ..., iv)), A; 0 < t < col is characterized by the condition , = 5,t, t > 0, where 64 is the Kronecker delta.
5.20 Exercise. Suppose X e di2 has stationary, independent increments. Then
<X>, = t(EXt),t
0.
5.21 Exercise. Employ the localization technique used in the solution of Problem 5.17 to establish the following extension of Problem 5.12: If X e dioc'1" and for some stopping time T of {Ft} we have <X>, = 0 a.s. P, then P[XT, t = 0; V 0 < t < co] = 1. In particular, every X e dr' of bounded first variation is identically equal to zero.
We close this section by imposing a metric structure on di2 and discussing the nature of both dif2 and its subspace .AZ under this metric. 5.22 Definition. For any X e 112 and 0 < t < co, we define 11X
N/E(X().
We also set IIXII
E =,
11X11,, A 1
2"
11, on [0, co) is nondecreasing, Let us observe that the function t because X2 is a submartingale. Further, IIX -Y II is a pseudo-metric on .#2, which becomes a metric if we identify indistinguishable processes. Indeed, suppose that for X, Y e.//l2 we have 11X - Y11 = 0; this implies X = Y,, a.s. P, for every n > 1, and thus X, = E(X1";,) = E(Y.1.-Fr) = Y, a.s. P, for every 0 < t < n. Since X and Y are right-continuous, they are indistinguishable (Problem 1.5).
5.23 Proposition. Under the preceding metric, //2 is a complete metric space, and 12 a closed subspace of A'2. PROOF. Let us consider a Cauchy sequence {X(")},,cc_1
`/#2: hmn,m.x MX(n) -
X(m)l1 = 0. For each fixed t, IAT1)1'_1 is Cauchy in L2(0, y, P), and so has an L2-limit X,. For 0 < s < t < cc and A e A, we have from L2-convergence and the Cauchy-Schwarz inequality that lim,E[1A(X!") - Xs)] = 0, limn E[lA(X") - X,)] = 0. Therefore, E[lAX:")] = E[1AX!")] implies E[lAXJ = E[lAX,], and X is seen to be a martingale; we can choose a right-continuous modification so that X e .4/2. We have II X( ") -X II = 0.
1. Martingales, Stopping Times, and Filtrations
38
To show that 2 is closed, let {X(")},T-1 be a sequence in Jr2 with limit X in di,. We have by the first submartingale inequality of Theorem 3.8:
P[ sup Pq") - Xti >
<
Clt
1
1
- XTI2 =
X`"'- X II -0 0
as n -> co. Along an appropriate subsequence {NM ., we must have
I XP.) - Xti > P[ 0sup
1
1
< y; k> 1
and the Borel-Cantelli lemma implies that .elqnk) converges to X uniformly on [0, T], almost surely. The continuity of X follows.
5.24 Problem. Let M = {M Fr; 0 < t < oo } be a process in 412 u .41`.'" and assume that its quadratic variation process <M> is integrable: E <M>, < co. Then:
(i) M is a martingale, and M and the submartingale M2 are both uniMt exists a.s. P, and formly integrable; in particular, Moc, = EMS = E<M>; (ii) we may take a right-continuous modification of Z, = E(M.021.fl - Mi2; t 0, which is a potential. 5.25 Problem. Let M e .1r I" and show that for any stopping time T of 1.97,1, (5.5)
P[ max 1M,1 0
a]
E(S A <M)T) E
2
+ P[ <M>r
6],
V E > 0, .5 > 0. In particular, for a sequence proln.°_1 c .A'''"c we have (5.6)
max IM(")1 t
<M(")> T
n-boo
5.26 Problem. Let IM A; 0 < t < co and IN %; 0
0.
t < ocl on (S2,3'7, P)
be continuous local martingales relative to their respective filtrations, and assume that .97,0 and are independent. With ff; A u %), show that
IM Yf; 0 < t < col, { N Ye; 0
t < co} and {M,N Yt' 0 < t < co} are
local martingales. If we define iti9, = ns>,a(ffs u Al), where
is the collec-
tion of P-negligible events in F, then the filtration {} satisfies the usual conditions, and relative to it the processes M, N and MN are still local martingales. In particular, <M, N> -a 0.
1.6. Solutions to Selected Problems 1.8. We first construct an example with A
The collection of sets of the form
{(X,,, Xr2,...)E B}, where B E ge(Rd) 0 ./(Ella) and 0
1.6. Solutions to Selected Problems
39
representation; cf. Doob (1953), p. 604. Choose Q = [0, 2), y = i([0, 2)), and P(F) = meas (F n [0, 1]); F E 347, where meas stands for "Lebesgue measure." For CO E [0, 1], define X,(w) = 0, V t > 0; for CO E (1, 2), define X,(w) = 0 if t 0
co,
X,,(w) = 1. Choose t, = 2. If A E y,o, then for some B E AR) 0 AR) 0 and Choose t e (1, 2), some sequence g [0, 2], we have A = {(X,,, X,,, ...) t {t, , t2, ...}. Since to = is not in A and X,k(i) = 0, k = 1, 2, ..., we see that (0, 0, ...) B. But X,,,(co) = 0, k = 1, 2, ..., for all w E [0, 1]; we conclude that [0, 1] n A = 0, which contradicts the definition of A and the construction of X. We next show that if gl,,;`-
and ..97,0 is complete, then A Egi,o. Let N c S2
be the set on which X is not RCLL. Then
A=
(U An
NC,
n=1
where An =
n
_x>
U
m=1 qi,q,
1
1.
Qn10,101
-921<(0.)
2.6. Try to argue the validity of the identity {Hr < t} = UsnQ
{XsE F}, for any
t > 0. The inclusion is obvious, even for sets which are not open. Use rightcontinuity, and the fact that F is open, to go the other way. 2.7. (Wentzell (1981)): For x E Fe, let p(x, r) = inf ilx - yll; ye r}, and consider the nested sequence of open neighborhoods of F given by F,, = {x E ffid; p(x, F) < (1 /n) }. By virtue of Problem 2.6, the times T, n > 1, are optional. They form a nondecreasing sequence, dominated by H = Hr, with limit TA limn. Tn H, and we have the following dichotomy: On {1/ = 0}: T = 0, V n > 1. On {H > 0 }: there exists an integer k = k(w) > 1 such that = 0; V 1 n < k, and 0 < T < T,, < H;
Vn
k.
We shall show that T = H, and for this it suffices to establish T > H on > 0, T < col. On the indicated event we have, by continuity of the sample paths of X: XT = lim, XT. and Xi._ E arm g_ rn; V m > n > k. Now we can let in -> oo, to obtain X TE rn; V n > k, and thus XT E rn = r. We conclude with the desired result Tn < t }, valid for H < T The conclusion follows now from {H < t} = t > 0, and {H = 0} = (X0 E F}.
2.17. For every AET we know that A n < S) belongs to both ..97r (Lemma 2.16) and ys (Lemma 2.15), and therefore also to T ,.S = T n ,o6r.s. Consequently, SA
I{r.,s}E(Zi.Fr,,$)dP =
JAn
<S}dP Z
.1,4,{r,s}E(Z1.97r)dP =
1{T <S} E(Z1217T)dP, so (i) follows.
For claim (ii) we conclude from (i) that 1{T<S}E[E(ZVFT)13"-s] = E[1{r<s}E(ZI.FY1Fs] = E[1{T<s}E(ZISTs,,T)Vor's] = 1{r<s}E[ZI-97s,7-]+
which proves the desired result on the set IT < S}. Interchanging the roles of S and T and replacing Z by E(Z1.97T), we can also conclude from (i) that
40
1. Martingales, Stopping Times, and Filtrations 1{s< T}E[E(Z19'01Fs] = !Is< E[E(Z = !IS<
snTI
n ETZ IFS T1
2.22. We discuss only the second claim, following Chung (1984 For any A e F, we have =CU [An A
EQ
Now A n {S < r < T} = A n {S < r} n {T > r} is an event in .FT, as is easily verified, because A n {S < e.g,. On the other hand, A n {S = ao} = [A n {S = oo}] n {T = col is easily seen to be in T. It follows that A E FT. 2.23. T is an optional time, by Lemma 2.11, and so FT+ is defined and contained in 1. Therefore, .FT+ - 107=1 -5477,+ To go the other way, con.54-7;+ for every n sider an event A such that A n T < t} e .F for every n > 1 and t 0. Obviously
{"T < t})e ,F n'e=i(A (UIT=1 {Tn < t}) = ,T+. The second claim is justified similarly, using Problem 2.22.
then, A n {T < t} = A Ae 3.2.
and thus
(i) Fix s > 0 and a nonnegative integer n. Consider the "trace" a-field g of all sets obtained by intersecting the members of Fs" with the set {Ns = n}. Consider also the similar trace a-field if of a(T,, ,7;,) on {N5 = n}. A generating family for g is the collection of sets of the form {N,, < n1, , /s1,k
P[S,, > tl.FsN] dP firl{N,=n)
= P[{S±, >
= P[{S
n A n {S
T,,,, >
s < S,,}]
n A n {S
=ft-s P[{S > t - u} n A n {S = e- '11` -5)
s}] s}],le-1" du
P[{S + u > s} n A n
s}],le- A" du
= e-mt-s)P[{S + T+, > s} n A n {S = e-m")P[A n {N5 = Summation over n > 0 yields IA P[S,,,.+, >
dP = e-M`-s)P(A) for every
de Fs". (ii) For an arbitrary but fixed k > 1, the random variable Y.. g S -n+ k+I Sn +1 = E.721:1 'I; is independent of a(T,, , T+1); it has the gamma density P[Yke du] = [(Au)" /(k - 1)!]Ae-A" du; u > 0, for which one checks easily the
1.6. Solutions to Selected Problems
41
identity: k-1
(;1.. 0)-i
X
P[Yk > e]
3!
3=o
e';
k
1, 0 > O.
We have, as in (i),
P[N, -
5 klFsn dP
fin{N,=n}
= P[{N,
n
n+
(-) {Ns = n}]
= P[ {Sn +k +i >t }nAn {Ns =n }] = P[ {Sn +1 +Yk >t }nAn {Ns =n }]
=
f
P[{S+, + u > 1-} n A n {Ns = n}] P(Ykedu)
= P[A n {Ns = n}] P(Yk > t s
+ .1o
P[{S
,
-s)
> t - IA} n A n {Ns = n}]P(Yk E du)
= P[A n {Ns = n }] E
e-.1.0-0(.1(t - s))' 1
J =0
.1
.1t-s
+
CA(`-s-")P(YkEdU))
= P[A n (Ns = n }] E e-A0-0(Act J=0 i! Adding up over n > 0 we obtain
J.
P[N, -
k
klei.sndP = P(A) E e-m t-s)
(A(t -
J=0
for every :4 E ,Fsr I and k > 1, and both assertions follow.
for which
3.7. Let 1k; a E AI be a collection of linear functions from EI;Id supc A h5. Then for 0 < s < t we have
E[9(X,)Ifs] > E[h(X,)13r.s] = k,(Xs),
=
V a E A.
Taking the supremum over a, we obtain the submartingale inequality for 9(X). + < cc), so IIXII is a subNow II ' is convex and EllX,11 5_ E(IX11)1 + martingale. II
3.11. Thanks to the Jensen inequality (as in Proposition 3.6) we have that {X+ , n > 1} is also a backward submartingale, and so with ). > 0: ). PO XI > A] < EIXSI = -E(X)+ 2E(X+) 5 -1 + 2E(X < co. It follows that sup,, P[IXSI > )] converges to zero as ). -> co, and by the submartingale property: sfix,..,> A) J
X' dP
f
X; dP {x,;>-).}
dP.
42
1. Martingales, Stopping Times, and Filtrations Therefore, {X,;},`,°=, is a uniformly integrable sequence. On the other hand,
-fXdP > E(X) - I
0 > fn<_A) XdP = E(X)
= E(X) - E(X,)
X , dP,
X,dP
for n > m.
f{x<-,1.)
Given e > 0, we can certainly choose m so large that 0 5_ E(X,,) - E(X) < c/2 holds for every n > m, and for that m we select ). > 0 in such a way that PC,IdP < 2.
sup n>m
f{x<-A}
Consequently, for these choices of m and A we have:
sup
X;. dP < E, and thus pr 1,;°_, is also uniformly integrable.
^>^. f(x,-,> A)
(b): Uniform integrability allows us to invoke the submartingale convergence Theorem 3.15, to establish the existence of an almost sure limit Xo, for {X1; 0 < t < co} as t co, and to convert almost sure convergence into
3.19. (a)
L I- convergence. (b) (c): Let
be the L'-limit of {Xr; 0 < t < co }. For 0 < s < t and A E we have IA Xs dP < f X, dP, and letting t co we obtain the submartingale property IA Xs dP < L X, dP; 0 < s < co, A Fs. (C) (a): For 0 t < c o and A > 0, we have fix,,A} X, dP fixo. A} X0 dP, which converges to zero, uniformly in t, as A I co, because P[X, > < (1 /A)EX, < (1 / A)EX,. ...97,; 0 t < co } to obtain the 3.20. Apply Problem 3.19 to the submartingales equivalence of (a), (b), and (c). The latter obviously implies (d), which in turn
gives (a). If (d) holds, then jA Y dP = f A X, dP, V A E A. Letting t co, we obtain SA Y dP = fA Xco dP. The collection of sets A E 347. for which this equality holds
is a monotone class containing the field U,ZO A. Consequently, the equality holds for every A E Fc, which gives E( Yl..9700) = Xeo, a.s.
3.26. The necessity of (3.2) follows from the version of the optional sampling theorem for bounded stopping times (Problem 3.23 (i)). For sufficiency, consider 0 < s < t < co, A E ..Fs and define the stopping time S(co) A s lA (a)) + t lAc(a)). The condition
E(X,) > E(X s) is tantamount to the submartingale property E[X,1A] > E[Xs1A].
3.27. By assumption, each A contains the P-negligible events of .F. For the rightcontinuity of {g, }, select a sequence {t }'_i strictly decreasing to t; according to Problem 2.23, co
co
n=1
n n=1
3,--(T+0+,
and the latter agrees with ,97T+t = A under the assumption of right-continuity of 1.97,1 (Definition 2.20).
(i) With 0 < s < t < co, Problem 3.23 implies E[fCrigs]
E[XT, -XTIT+.]
X T+s
XT
Xs,
a.s. P.
1.6. Solutions to Selected Problems
43
(ii) Let S, < S2 be bounded stopping times of {A}; then for every t
{(Si - T) v 0 < t} = {Si
T + t}
0,
=
by Lemma 2.16. It develops that (S, - T) v 0 < (S2 - T) v 0 are bounded stopping times of {A} and so, according to Problem 3.26,
EXs = Eg(Ss_ T)v 0
Eg(Si- Dv 0 = EXst.
Another application of Problem 3.26 shows that X is a submartingale. 3.28. (Robbins and Siegmund (1970)): With the stopping time T= inf {t E [S, 00); Z, = b },
the process {ZT A A; 0 < t < co} is a martingale (Problem 3.24 (i)). It follows that for every A e 3,7 t > s:
ZsdP =
ZT AtdP
f
Arn{Z.<6}
Arn{Z.<10)
= b- P[A
{zs < b, T
t}] +
Z,1{T>odP. SAn{4
The integrand Z,1{T>t} is dominated by b, and converges to zero as t co by assumption; it develops then from the dominated convergence theorem that
ZsdP = b P[A n (Zs < b,T < cc] An{2.
=b
fAn{Z,
establishing the first conclusion. The second follows readily.
4.9. (a) According to Problem 3.23 (i) we have X,,dP and P[XT > A ] <
X T dP <
J
E( X T)
<
A
for every a > 0, A > 0, TE
A
and therefore
lim sup ).-
)
E(X
Te
f
XT dP = 0. {xr>.1.}
(b) It suffices to show the uniform integrability of {MT} T5,. Once again, Problem 3.23 (i) yields MT = E(M,71,T) a.s. P, for every TE 99, and the claim then follows easily, just as in the implication (d) =. (a) of Problem 3.20.
This latter problem, coupled with Theorem 3.22, yields the representation XT = E(Xcol'aFT) a.s. P, V T E .9' for every uniformly integrable martingale X, which is thus shown to be of class D. 4.11. For an arbitrary bounded random variable EC
on
P),
E(ie")1W)] = E[E(IW) E(.4(")1W)] = E[A() E(IW)],
which converges to E[A E(11)] = E[E(A1W)].
1. Martingales, Stopping Times, and Filtrations
44
4.15. Define the stopping times H,= inf {
0; X, > e}, S = inf {t
0; A, L. .5} (Prob-
lem 2.7) and T=TAnAH,. We have tP[VTn > c] < E[XT 1{V,. >0] < E(XT,,) < E(AT) < E(AT) co. On the other hand, we have
and (4.14) follows because '1;, T T n H,, a.s. as n < (5]
P[VT
E((5A AT)
E(As,T)
P[VT As L
thanks to (4.14), and (4.15) follows (adapted from Lenglart (1977)). From the identity F(x) = pos 1{,}dF(u), the Fubini theorem, and (4.15)' we have EF(VT) = J
A AT)
P(VT L u)dF(u)
+ P(AT _u)}dF(u)
0 =foopp(AT,u),E(AT,,,,,,)]dF(u)
= E[2F(AT)+
AT
f
1
-dF(U) I = EG(AT) 4r U
(taken from Revuz & Yor (1987); see also Burkholder (1973), p. 26).
5.11. Let H = {to, , t,n}, with 0 = to < t1 < q > p, we have v,(q)(n) < v,(P)(n)
< t. = t, be a partition of [0, t]. For
max IX,- X 1
The first term on the right-hand side has a finite limit in probability, and the second term converges to zero in probability. Therefore, the product converges to zero in probability. For the second assertion, suppose that 0 < q < p, P(L, > 0) > 0 and assume that v,o)(n) does not tend to oo (in probability) as 111111 -0. Then we can find ö > 0, 0 < K < co, and a sequence of partitions {TIn}n',1 such that P(A)L SP(L, > 0), where A= {L, > 0, V(q)(11)
Consequently, with n = {too,.
,
K);
n
I.
t;,t}, we have
v(P)(11.) < Km?-9(X; 1111"o
n > 1.
on A,,;
This contradicts the fact that v,(Pqn) converges (in probability) to the positive random variable L, on {L, > 0}.
5.12. Because <X> is continuous and nondecreasing, we have P[<X>T,, = 0; 0 < t < oo] = 1. An application of the optional sampling theorem to the continuous martingale M A X2 - <X> yields (Problem 3.24 (i)): 0 = EMT,,, = E[XL,EXh,, which implies P[XT A, = 0] = 1, for every 0 < t < oo. The conclusion follows now by continuity. 5.17. There are sequences {Sn}, {T "} of stopping times such that S. 00, T,,T 00, and XI") A Xo.,S, Yi(n) -4 Yr, T are {,,}-martingales. Define Rn A Sn A Tn A inf It
0: I X,I = n
or
I
= nl,
1.7. Notes
45
and set Xr = X in R., g(n) = li R, Note that R T cc a.s. Since XP) =
R.,
and likewise for Y("), these processes are also {,,}-martingales (Problem 3.24), and are in because they are bounded. For m > n, XP) = X:'",,),, and so
- t
A
R. = an.)2 -
(')>t n R.
is a martingale. This implies a*>, = <X(m)>,,, R.,. We can thus decree <X>, A <X(")>, whenever t < R and be assured that <X> is well defined. The process
(X> is adapted, continuous, and nondecreasing and satisfies <X>0 = 0 a.s. Furthermore,
-
R = (X;n))2 - r
is a martingale for each n, so X2 - <X> edlc.'. As in Theorem 5.13, we may now take <X, Y> = -14:[<X + Y> - <X - Y>]. As for the question of uniqueness, suppose both A and B satisfy the conditions
required of (X, Y>. Then M A XY- A and N A XY-B are in .1/', so just as before we can construct a sequence {R} of stopping times with R,, j cc such
that MP) A M, . and N,P) A Al, . are in Consequently Mr - NP) = - Art, ,,E.1r2, and being of bounded variation this process must be identically zero (see the proof of Theorem 5.13). It follows that A = B.
E<M>; VO < t < co. If M E
5.24. If ME 412, then E(W) =
Problem 5.19 (iii) gives E(MS) < E <M> for every stopping time S; it follows that the family {Ms}se 9, is uniformly integrable (Chung (1974), Exercise 4.5.8), i.e., that M is of class D and therefore a martingale (Problem 5.19 (i)). In either case, therefore, M is a uniformly integrable martingale; Problem 3.20 now shows that Moo = limn. M, exists, and that E(Mj,F,) = M, holds a.s. P, for every t > 0. Fatou's lemma now yields (6.1)
E(M!) = E Clim AV) < lim E(A112)= lim E<M>, = E<M>, too
1-co
and Jensen's inequality: M< < E(M.021F,), a.s. P, for every t > 0. It follows that
the nonnegative submartingale M2 has a last element, i.e., that {AV, A; 0 < t < cc} is a submartingale. Problem 3.19 shows that this submartingale is uniformly integrable, and (6.1) holds with equality. Finally, Z, = E(M2.1327,) AV is now seen to be a (right-continuous, by appropriate choice of modification) nonnegative supermartingale, with E(Z,) = E(M,02) - E(M,2) converging to zero as t co. 5.25. Problem 5.19 (iii) allows us to apply Remarks 4.16, 4.17 with X = M2, A = <M>.
1.7. Notes Sections 1.1, 1.2: These two sections could have been lumped together under the rubric "Fields, Optionality, and Measurability" after the manner of
Chung & Doob (1965). Although slightly dated, this article still makes excellent reading. Good accounts of this material in book form have been written by Meyer [(1966); Chapter IV], Dellacherie [(1972); Chapter III and
46
1. Martingales, Stopping Times, and Filtrations
to a lesser extent Chapter IV], Dellacherie & Meyer [(1975/1980); Chapter IV], and Chung [(1982); Chapter 1]. These sources provide material on the classification of stopping times as "predictable," "accessible," and "totally inaccessible," as well as corresponding notions of measurability for stochastic processes, which we need not broach here. A new notion of "sameness" between two stochastic processes, called synonimity has been introduced by Aldous. It was expounded by Hoover (1984) and was found to be useful in the study of martingales. A deep result of Dellacherie [(1972), p. 51] is the following: for every progressively measurable process X and F e ,4(R), the hitting time Hr of Example 2.5 is a stopping time of {Ft}, provided that this filtration is rightcontinuous and that each o--field is complete. Section 1.3: The term martingale was introduced in probability theory by J. Ville (1939). The concept had been created by P. Levy back in 1934, in an attempt to extend the Kolmogorov inequality and the law of large numbers beyond the case of independence. Levy's zero-one law (Theorem 9.4.8 and Corollary in Chung (1974)) is the first martingale convergence theorem. The classic text, Doob (1953), introduced, for the first time, an impressively complete theory of the subject as we know it today. For the foundations of the discrete-parameter case there is perhaps no better source than the relevant sections in Chapter 9 of Chung (1974) that we have already mentioned; fuller accounts are Neveu (1975), Chow & Teicher (1978), Chapter 11, and Hall & Heyde (1980). Other books which contain material on the continuous-, parameter case include Meyer [(1966); Chapters V, VI], Dellacherie & Meyer [(1975/1980); Chapters V-VIII], Liptser & Shiryaev [(1977), Chapters 2, 3] and Elliott [(1982), Chapters 3, 4]. Section 1.4: Theorem 4.10 is due to P. A. Meyer (1962, 1963); its proof was later simplified by K. M. Rao (1969). Our account of this theorem, as well as that of Theorem 4.14, follows closely Ikeda & Watanabe (1981). Section 1.5: The study of square-integrable martingales began with Fisk (1966) and continued with the seminal article by Kunita & Watanabe (1967). Theorem 5.8 is due to Fisk (1966). In (5.6), the opposite implication is also true; see Lemma A.1 in Pitman & Yor (1986).
CHAPTER 2
Brownian Motion
2.1. Introduction Brownian movement is the name given to the irregular movement of pollen, suspended in water, observed by the botanist Robert Brown in 1828. This random movement, now attributed to the buffeting of the pollen by water molecules, results in a dispersal or diffusion of the pollen in the water. The range of application of Brownian motion as defined here goes far beyond a study of microscopic particles in suspension and includes modeling of stock prices, of thermal noise in electrical circuits, of certain limiting behavior in queueing and inventory systems, and of random perturbations in a variety of other physical, biological, economic, and management systems. Furthermore, integration with respect to Brownian motion, developed in Chapter 3, gives us a unifying representation for a large class of martingales and diffusion processes. Diffusion processes represented this way exhibit a rich connection with the theory of partial differential equations (Chapter 4 and Section 5.7). In particular, to each such process there corresponds a second-order parabolic equation which governs the transition probabilities of the process. The history of Brownian motion is discussed more extensively in Section 11; see also Chapters 2-4 in Nelson (1967). 1.1 Definition. A (standard, one-dimensional) Brownian motion is a continuous,
adapted process B = IB A; 0 < t < co }, defined on some probability space (n, .S, P), with the properties that Bo = 0 a.s. and for 0 < s < t, the increment A - B., is independent of 377s and is normally distributed with mean zero and variance t - s. We shall speak sometimes of a Brownian motion B =
{13 ,Ft; 0 < t < T} on [0, T], for some T > 0, and the meaning of this terminology is apparent.
2. Brownian Motion
48
If B is a Brownian motion and 0 = to < t, < < t < co, then the increments {A, - /311_,}7=1 are independent and the distribution of Btu -Btu
depends on t; and
only through the difference ti - tj_i; to wit, it is
normal with mean zero and variance ti . We say that the process B has stationary, independent increments. It is easily verified that B is a squareintegrable martingale and , = t, t 0. The filtration {,F,} is a part of the definition of Brownian motion. However,
if we are given {B,; 0 < t < co} but no filtration, and if we know that B has stationary, independent increments and that B, = B, - Bo is normal with mean zero and variance t, then {B ,F13; 0 < t < co} is a Brownian motion (Problem 1.4). Moreover, if {.97,} is a "larger" filtration in the sense that for t > 0, and if B, -B., is independent of ys whenever 0 < s < t, then {B Ft; 0 < t < co} is also a Brownian motion. It is often interesting, and necessary, to work with a filtration {ffl} which is larger than 1,,B1. For instance, we shall see in Example 5.3.5 that the stochastic differential equation (5.3.1) does not have a solution, unless we take
the driving process W to be a Brownian motion with respect to a filtration which is strictly larger than {Fr }. The desire to have existence of solutions to stochastic differential equations is a major motivation for allowing { in Definition 1.1 to be strictly larger than {.F,11. The first problem one encounters with Brownian motion is its existence. One approach to this question is to write down what the finite-dimensional distributions of this process (based on the stationarity, independence, and normality of its increments) must be, and then construct a probability measure and a process on an appropriate measurable space in such a way that we obtain the prescribed finite-dimensional distributions. This direct approach is the one most often used to construct a Markov process, but is rather lengthy
and technical; we spell it out in Section 2. A more elegant approach for Brownian motion, which exploits the Gaussian property of this process, is based on Hilbert space theory and appears in Section 3; it is close in spirit to Wiener's (1923) original construction, which was modified by Levy (1948) and later further simplified by Ciesielski (1961). Nothing in the remainder of the book depends on Section 3; however, Theorems 2.2 and 2.8 as well as Problem 2.9 will be useful in later developments. Section 4 provides yet another proof for the existence of Brownian motion, this time based on the idea of the weak limit of a sequence of random walks. The properties of the space C[0, co) developed in this section will be used extensively throughout the book.
Section 5 defines the Markov property,, which is enjoyed by Brownian motion. Section 6 presents the strong Markov property, and, using a proof based on the optional sampling theorem for martingales, shows that Brownian motion is a strong Markov process. In Section 7 we discuss various choices of the filtration for Brownian motion. The central idea here is augmentation of the filtration generated by the process, in order to obtain a right-continuous filtration. Developing this material in the context of strong Markov processes requires no additional effort, and we adopt this level of generality.
49
2.2. First Construction of Brownian Motion
Sections 8 and 9 are devoted to properties of Brownian motion. In Section 8 we compute distributions of a number of elementary Brownian functionals;
among these are first passage times, last exit times, and time and level of the maximum over a fixed time-interval. Section 9 deals with almost sure properties of the Brownian sample path. Here we discuss its growth as t co, its oscillations near t = 0 (law of the iterated logarithm), its nowhere differentiability and nowhere monotonicity, and the topological perfectness of the set of times when the sample path is at the origin.
We conclude this introductory section with the Dynkin system theorem (Ash (1972), p. 169). This result will be used frequently in the sequel whenever
we need to establish that a certain property, known to hold for a collection of sets closed under intersection, also holds for the a-field generated by this collection. Our first application of this result occurs in Problem 1.4. 1.2 Definition. A collection g of subsets of a set n is called a Dynkin system if
(i) Q e g, (ii) A, B e g and B s A imply A\B e 9, 9 and Al A2 c imply U;;3=1 Ae 9. (iii) {A};,°=, 13 Dynkin System Theorem. Let (6' be a collection of subsets of f2 which is closed under pairwise intersection. If 9 is a Dynkin system containing (6, then 9 also contains the a-field a(W) generated by (6.
1.4 Problem. Let X = {X1; 0 < t < co} be a stochastic process for which X0, X,1- Xt., ..., X,. - X,._, are independent random variables, for every integer n > 1 and indices 0 = t, < t, < < t. < co. Then for any fixed 0 < s < t < cc, the increment X, -Xs is independent of .97,x.
2.2. First Construction of Brownian Motion A. The Consistency Theorem Let R)c''') denote the set of all real-valued functions on [0, co). An n-dimensional cylinder set in Or°31 is a set of the form (2.1)
C
to) e FRE'''); (a)(ti ),
, w(t,,)) e Al,
n, and A e ,I(R"). Let (6 denote the field of all cylinder sets (of all finite dimensions) in RE''), and let 1/(R)0')) denote the where t, e [0, cc), i = 1,
smallest a-field containing (6.
2.1 Definition. Let T be the set of finite sequences t = (t1,... , t) of distinct, nonnegative numbers, where the length n of these sequences ranges over the set
2. Brownian Motion
50
of positive integers. Suppose that for each t of length n, we have a probability measure Q, on (R", ROW)). Then the collection {12t},ET is called a family of finite-dimensional distributions. This family is said to be consistent provided that the following two conditions are satisfied:
t), then for any
(a) if s = (t11, tif tin) is a permutation of t = (t, , t2, Al e MR), i = 1, , n, we have 121(A1 x A2 x
(b) if t =
t2,..., t) with n
x A,,) = 1, s =
Q,(A x
x Ai2 x
x Ain);
t2,..., tn_, ), and A e M(R"), then = Q2(A).
If we have a probability measure P on (RE'''), M(RE'''))), then we can define a family of finite-dimensional distributions by (2.2)
Qt(A) = P[0) e Rio''); (w(t,), . . . , co(t))e A],
where A e ,I(Rn) and t = (t1,..., t)e T This family is easily seen to be consistent. We are interested in the converse of this fact, because it will enable us to construct a probability measure P from the finite-dimensional distributions of Brownian motion.
2.2 Theorem (Daniell (1918), Kolmogorov (1933)). Let {Qt} be a consistent family of finite-dimensional distributions. Then there is a probability measure P on (IRE", al(RE'''))), such that (2.2) holds for every te T PROOF. We begin by defining a set function Q on the field of cylinders W. If C is given by (2.1) and t = (t1, t2, t)e T, we set (2.3)
Q(C) = Q,(A),
C e 6.
2.3 Problem. The set function Q is well defined and finitely additive on W, with
Q(Ro'')) = 1. We now prove the countable additivity of Q on W, and we can then draw on the Caratheodory extension theorem to assert the existence of the desired extension P of Q to .4(18E0 °°)). Thus, suppose {/3,},71L, is a sequence of disjoint sets in W with B A Iv also in W. Let Cm = B\UT=i 13,,, SO
Q(B) = Q(C.) + X 121130 k=1
Countable additivity will follow from (2.4)
lim Q(Cm) = 0. m-
Now Q(Cm) = Q(C,+,) + Q(B,,i) Q(C,,,), so the limit in (2.4) exists. Assume that this limit is equal to e > 0, and note that n:=1 C, = 0.
2.2. First Construction of Brownian Motion
51
From {C,},'..., we may construct another sequence {D,}MD=, which has D2 2 - , n,-=, Dm= 0, and limm, Q(Dm) = E > 0. Furthermore, each Dm has the form
the properties Di
Dm = {we 01('''); ((oft 1 ), ... , co(tm))e Am}
for some Am eM(Rm), and the finite sequence t A (t1, ..., t,)e T is an extension of the finite sequence t,,,_, A (ti,..., tm_, ) e T, m > 2. This may be accomplished as follows. Each Ck has a form Ck = lw e RE'''); (w(t, ), ... , w(t,k))e ilmk 1;
Am, e .4(Ork),
where tmk = (t1,..., t,de T. Since C, Ck, we can choose these representais an extension of t and Amk +, Am,, x 08mk+1 Mk. Define tions so that t D, = lw; w(t,)eRI,...,Dmi_, = Ico;()(t 1), ... ,co(t,,,j_1))e gr.-11 and Dm, = C1, as well as
Dm. +, = {a);(co(ti),..., w(tm,), a(t,,,i+i))e Am, x RI,..., D,2_ 1 = lw;(w(t 1), . . . , w(t ,,), w(t , +1), . . . , w(t,2_1))e A,, x Rm2-"11-11
and D,2 = C2. Continue this process, and note that by construction n m1 m =, Dm =
n:=1 Cm= 0. 2.4 Problem. Let Q be a probability measure on (R", .4(R")). We say that A e R(111") is regular if for every probability measure Q on (R", ,4(R")) and for
every C > 0, there is a closed set F and an open set G such that FgA G and Q(G\F) < E. Show that every set in M(01") is regular. (Hint: Show that the collection of regular sets is a a-field containing all closed sets.) According to Problem 2.4, there exists for each in a closed set Fm Am such that Q, (Am \Fm) < e/2m. By intersecting Fm with a sufficiently large closed sphere centered at the origin, we obtain a compact set Km such that, with E,,, A {we 0:8[°');(to(t 1), we have Em
'Mt.)) G Km),
D,,, and
Q(Dm \Em) = Q, (Am \Km) <
.
The sequence {Em} may fail to be nonincreasing, so we define Em
= k=1 nm Ek,
and we have Em = la) e RE°.'); (o)(ti ), ... , w(tm))e Km},
where Km = (K1 x Rm .' ) r-) (K2 x glm -2 ) r) -
-
(-) (Km_i x gl) rm Km,
2. Brownian Motion
52
which is compact. We can bound Q, (1Z.) away from zero, since
Q, (gm) = Q(Em) = Q(Dm) - Q(D,\Em)
= Q(Dm)- QU
(Dm\Ek))
k=i
Q(Dm)- Q(U (Dk\Ek)) k=1
s-
m
c O.
k=1 2
Therefore, K. is nonempty for each m, and we can choose (x(r),..., x(7))e Km. Being contained in the compact set K1, the sequence {x(r)1,,,°°=2 must have a convergent subsequence fx(imol,`,°_, with limit x1. But lx(ro, x(2m01;,°_2 is
contained in K2, so it has a convergent subsequence with limit (x2, x2). Continuing this process, we can construct (x x2, ...)eR x R x -, such that (x1,
,
x.) E Km for each m. Consequently, the set S = {co e Rt°'°°): w(ti) = xi, i = 1, 2, ... }
is contained in each Em, and hence in each Dm. This contradicts the fact that n,-.,1 Dm = 0. We conclude that (2.4) holds.
Our aim is to construct a probability measure P on (S/F) A (Rt'''), R(Rt°''))) so that the process B = IB .F13; 0 < t < col defined by B,(w) co(t), the coordinate mapping process, is almost a standard, one-dimensional
Brownian motion under P. We say "almost" because we leave aside the requirement of sample path continuity for the moment and concentrate on the finite-dimensional distributions. Recalling the discussion following Definition 1.1, we see that whenever 0 = so < s1 < s2 < < s, the cumumust be
lative distribution function for
x)
(2.5) Xi
x2
p(s 1; 0, Y i)P(s2 - s1;Y19Y 2)
for (x1, (2.6)
-
.p(s. - s.-i; Y.--i, Y.) dY. dY2 dYi , x)e R", where p is the Gaussian kernel p(t; x, y) A
1
.2irt
e-cx-ypi2r,
t > 0, x, y e R.
The reader can verify (and should, if he has never done so!) that (2.5) is equivalent to the statement that the increments {B., -BSS }7=1 are independent, and Bs, -1135
is normally distributed with mean zero and variance
- Sj_i. Now let t = (t1, t2,
t), where the ti are not necessarily ordered but
2.2. First Construction of Brownian Motion
53
are distinct. Let the random vector (Bt,, B,2, ,Bt,,) have the distribution determined by (2.5) (where the ti must be ordered from smallest to largest to obtain (s1, .. . , s) appearing in (2.5)). For A E .4(R"), let Qt(A) be the probability
under this distribution that (B,1312, - , An) is in A. This defines a family of finite-dimensional distributions {Qt}t T. 2.5 Problem. Show that the just defined family {Qt},e T is consistent.
2.6 Corollary to Theorem 2.2. There is a probability measure P on (01(0''), ,4([111°''))), under which the coordinate mapping process Moo) = co(t);
co eRt°''), t > 0,
has stationary, independent increments. An increment B, - B3, where 0 < s < t, is normally distributed with mean zero and variance t - s.
B. The Kolmogorov-entsov Theorem Our construction of Brownian motion would now be complete, were it not for the fact that we have built the process on the sample space FEE°'"') of all real-valued functions on [0, co) rather than on the space C[0, co) of continuous
functions on this half-line. One might hope to overcome this difficulty by showing that the probability measure P in Corollary 2.6 assigns measure one to C[0, co). However, as the next problem shows, C[0, co) is not in the a-field
,4(0110'')), so P(C[0, co)) is not defined. This failure is a manifestation of the fact that the a-field gi(011°.')) is, quite uncomfortably, "too small" for a space as big as 01[0'`°); no set in .4(0110'"')) can have restrictions on uncountably
many coordinates. In contrast to the space C[0, co), it is not possible to determine a function in 0:11°'') by specifying its values at only countably many coordinates. Consequently, the next theorem takes a different approach, which is to construct a continuous modification of the coordinate mapping process in Corollary 2.6. 2.7 Exercise. Show that the only M(RE°''))-measurable set contained in C[0, co)
is the empty set. (Hint: A typical set in a(fle'')) has the form E = {co e RE°''); (co(t ,), co(t 2), . . .) e AI,
where A e a(fps x 01 x
)).
2.8 Theorem (Kolmogorov,
entsov (1956a)). Suppose that a process X =
{X1; 0 < t < T} on a probability space (LI, 31%, P) satisfies the condition (2.7)
EIX, - XX' < CI t - sl",
0
s, r 5 T,
for some positive constants a, fi, and C. Then there exists a continuous modification
1 = {gt; 0 5 t < T} of X, which is locally Holder-continuous with exponent y
2. Brownian Motion
54 for every ye (0, )3/ cc), i.e., (2.8)
P[co;
It(w)
sup
s(w)i
= 1,
It - slY
o
where h(w) is an a.s. positive random variable and 6 > 0 is an appropriate constant.
PROOF. For notational simplicity, we take T = 1. Much of what follows is a consequence of the ebygev inequality. First, for any e > 0, we have
P[IX, - XsI
E IX,
c]
-XI < Cc'lt - " s
Ex
and so Xs -> X, in probability ass t. Second, setting t = kr,s= (k - 1)/2", and e = 2-Yn (where 0 < y < 13/a) in the preceding inequality, we obtain
P[142. - 4-1)/2.1 > 2-Y1 < C 2-n( 1 +fl-", and consequently, P [ max 142. - X(k_i On I
(2.9)
2-Yn
1 Sk<2^ 2
= PL
I Xk/2"
X(k-1)/2^1 > 2- Yn]
k=1
< C2-"("Y). The last expression is the general term of a convergent series; by the BorelCantelli lemma, there is a set n* e with P(S1*) = 1 such that for each w e Q *, (2.10)
max I X,02^(w) - 4_1)/2.(w)1 < 2-",
Vn
n*(w),
1 sic <2.
where n*(w) is a positive, integer-valued random variable.
For each integer n > 1, let us consider the partition D = {(k/2");k = 0, , 2"} of [0, 1], and let D = U `°=, D, be the set of dyadic rationals in [0, 1]. We shall fix co e n* n > n*(w), and show that for every m > n, we have
1,
(2.11)
I Xt(w) - Xs(w) I
2
E j=n+1
v t,
se D 0 < t - s < 2
For m = n + 1, we can only have t = (k/r), s = ((k - 1)/2'), and (2.11) follows from (2.10). Suppose (2.11) is valid for m = n + 1, M - 1. Take s < t, s, te Dm, consider the numbers t1 = max{u eDm_i; u 5 t} and s1 = min fu e Dm_i; u > s }, and notice the relationships s 5 s1 < t1 < t, s1 - s 2-14, t - t1 5 2'. From (2.10) we have I Xs, (w) - Xs(c0)1 < 2 Ym, Xt(w) (w) I < 2-7m, and from (2.11) with m = M - 1, M -1
I Xit(w) - Xsi(w)I < 2 E
j=n+1
We obtain (2.11) for m = M.
2.2. First Construction of Brownian Motion
55
We can show now that { X,(w); t E DI is uniformly continuous in t for every
to en*. For any numbers s, t e D with 0 < t - s < h(w) g 2-"a", we select n > n*(w) such that 2-("+1) < t -s < 2-". We have from (2.11) (2.12)
I Xi(w) - Xs(w)I
2
bit - slY,
2-Yi
0 < t - s < h(C0),
j=n+1
where .5 = 2/(1 - 2-Y). This proves the desired uniform continuity.
We define 2 as follows. For w n*, set 21(w) = 0, 0 < t < 1. For w e n* and t e D, set 2,(co) = X,(w). For w E fr and t E [0, 1] n DC, choose a sequence
D with s t; uniform continuity and the Cauchy criterion imply that {X,(w)}T=1 has a limit which depends on t but not on the particular sequence {s},T=1 g D chosen to converge to t, and we set 2,(w) = Xs.(co). The resulting process X is thereby continuous; indeed, 2 satisfies (2.12), so {s},,`°_,
(2.8) is established.
To see that 2 is a modification of X, observe that 2, = X, a.s. for teD; for t e [0, 1] n DC and {s}'_,
and X, -51, a.s., so
=
t, we have
D with sn a.s.
X, in probability
2.9 Problem. A random field is a collection of random variables {X1; t Ed}, where d is a partially ordered set. Suppose {X,; t e [0, T]d }, d > 2, is a random field satisfying El X, - X512 < C Ilt
(2.13)
-
for some positive constants ; f3, and C. Show that the conclusion of Theorem 2.8 holds, with (2.8) replaced by (2.14)
+;
sup s,re[0,T[a
I ii(w) - XS(w)1
Ilt - sll.
< .5] = 1.
2.10 Problem. Show that if B, - Bs, 0 < s < t, is normally distributed with mean zero and variance t - s, then for each positive integer n, there is a positive constant C for which E IB, -
clt - sr.
2.11 Corollary to Theorem 2.8. There is a probability measure P on (41[0,-), ; t > 0} on the same space, a(11V0.))), and a stochastic process W = {W such that under P, W is a Brownian motion. PROOF. According to Theorem 2.8 and Problem 2.10, there is for each T > 0 a modification WT of the process B in Corollary 2.6 such that WT is continuous on [0, T]. Let = {w; WT (w) = B,(w) for every rational t e [0, T] },
so P(S1T) = 1. On n A WT (w)
n T =1 nT, we have for positive integers T1 and "2, wt T2((0), for every rational t e [0, T1 A T2].
2. Brownian Motion
56
Since both processes are continuous on [0, T, A T2], we must have W,Ti (co) --
W,T2(co) for every t e[0, T, A T2], to e n. Define W,(w) to be this common value. For co 0 n, set W,(w) = 0 for all t > 0.
2.12 Remark. Actually, for P-a.e. to e FRE'"), the Brownian sample path { W,(w); 0 < t < col is locally Holder-continuous with exponent y, for every y e (0, 1/2). This is a consequence of Theorem 2.8 and Problem 2.10.
2.3. Second Construction of Brownian Motion This section provides a succinct, self-contained construction of Brownian motion. It may be omitted without loss of continuity. Let us suppose that {B ..F,; t > 0} is a Brownian motion, fix 0 < s < t < co, and set 0 A (t + s)/2; then, conditioned on Bs = x and B, = z, the random variable Be is normal with mean it A (x + z)/2 and variance a2 A (t - s)/4. To verify this, observe that the known distribution and independence of the increments B Be - Bs, and B, -Bo lead to the joint density ;x
P[Bse dx, Bo e dy, B, e dz] = p(s;0, x)p
= p(s; 0, x)p(t - s; x, z)
1
a .,/ 2it
exp
{ (Y
y) p
12)2 }
2a2
(t
S
; y,
z) dx dy dz
dx dy dz
in the notation of (2.6), after a bit of algebra. Dividing by P[Bse dx, B, e dz] = p(s; 0, x)p(t - s; x, z) dx dz,
we obtain (3.1)
P[B0+.0/2 E dy IA = x, B, = z] =
1
e-(Y-P)212a2 dy.
a .,/ 2it
The simple form of this conditional distribution for B(, +S) /2 suggests that we can construct Brownian motion on some finite time-interval, say [0, 1], by interpolation. Once we have completed the construction on [0, 1], a simple "patching together" of a sequence of such Brownian motions will result in a Brownian motion defined for all t > 0. To carry out this program, we begin with a countable collection IV,"); k e 1(n), n = 0, 1, ... } of independent, standard (zero mean and unit variance) normal random variables on a probability space (Q,,5--., P). Here 1(n) is the set of odd integers between 0 and 2"; i.e., /(0) = {1}, /(1) = {1}, /(2) = 11, 31, etc. For each n > 0, we define a process B(n) = IBI"); 0 < t 5 1} by recursion and
linear interpolation, as follows. For n > 1, /3,172,, , will agree with B172-.91, k = 0, 1, ..., 2"-`. Thus, for each value of n, we need only specify B,13., for k e 1(n). We set
2.3. Second Construction of Brownian Motion
57
/V) =
Bo) = 0,
If the values of 13172-.1)i , k = 0, 1, ..., 2"-1 have been specified (so $"-1) is defined for 0 < t < 1 by piecewise-linear interpolation) and k E 1(n), we deand o-2 = (t - s)/4 = = (k + 1)/2", p = Rjr-1) note s = (k 1/2'1 and set, in accordance with (3.1), /1 + 6i;k").
B(7+ s)/ 2
We shall show that, almost surely, $10 converges uniformly in t to a continuous
function B and {B ,,11; 0 < t < 1) is a Brownian motion. Our first step is to give a more convenient representation for the processes Boo, n = 0, 1, .... We define the Haar functions by I-11°)(t) = 1, 0 < t < 1, and 1, k e I(n), for n
k-1
2(n-1)/2,
Hr)(t) =
k+ -< r t< r
201-10
r' 1
'
otherwise.
0,
We define the Schauder functions by Sk")(t) =
I
t
He)(u) du,
0 < t < 1, n
0, k e 1(n).
o
Note that S(i())(t) = t, and for n > 1 the graphs of S,1") are little tents of height 2-( " +1)/2 centered at kon and nonoverlapping for different values of k e 1(n). It is clear that BP) = Vi()),S(,°)(t), and by induction on n, it is easily verified that (3.2)
13:.)(c0)= E E
;:n)(w)s,(r)(t),
0 < t < 1, n > 0.
m=0 k /(m)
3.1 Lemma. As n
co, the sequence of functions {Br(w); 0 < t < 1), n > 0,
given by (3.2) converges uniformly in t to a continuous function 113,(w); 0 < t < 1),
for a.e. w En. PROOF. Define bn = maxk e 1(n) I k ") I. For x > 0 (3.3)
PNri > x]
e-"2/2 du
-2
2 7i
-u C"2/2 du
xX
=
- e--x2/2 TC X
which gives
P[bn >
=
Pr keI
>
2" P[VI
n]
, n > 1.
2. Brownian Motion
58
Now E'_, 2"C"212/n < co, so the Borel-Cantelli lemma implies that there is a set n with P(C2) = 1 such that for each w e n there is an integer n(co) satisfying b(co) < n for all n > n(w). But then
E idn'snoi < E n2-("+"2 < co;
E
n=n(w) k E 1(n)
n=n(w)
so for wen, Bp)(0 converges uniformly in t to a limit Bt(co). Continuity of {B,(co); 0 < t < 1} follows from the uniformity of the convergence.
Under the inner product = 14 f(t)g(t)dt, L2[0,1] is a Hilbert space, and the Haar functions IM"); ke 1(n), n > 0} form a complete, orthonormal system (see, e.g., Kaczmarz & Steinhaus (1951), but also Exercise 3.3 later). The Parseval equality CO
= applied to f =
E
n=0 kel(n)
and g = 1[0,4 yields 00
E > sk.)(01.)(s)= S A t;
(3.4)
0
s, t s 1.
n=0 k e 1(n)
3.2 Theorem. With {.13} "1`°_, defined by (3.2) and 13, = lim,13P), the process {B
PROOF. It suffices to prove that, for 0 = to < t1 < < t < 1, the increments {B,, - /31,_1};=1 are independent, normally distributed, with mean zero and variance ti - ti_, . For this, we show that for A.ie 11, j = 1, ... , n and i = j- 1, . . 1
(3.5)
E [exp
= ri exp --2A4t- t-_,) 'J.'
i E Ai(/31, - 131.,_,) .i=1
.
.i=1
Set An+, = 0. Using the independence and standard normality of the random variables {d")}, we have from (3.2)
-
E[expl-i E
E E ckno
= E[expt
m=0 kE/(m)
- aosint,)}]
j=1
E exp - 4") E (Ai+, - aosr(t.,) = m=0 fl k 11 1(m) 1=1
=m=0 n ,E,r(m) n exp [ 2 pi E (Ai+, 1
= exp [ Letting M
- 2Epi E (=i n
n
n
-
}2]
-
co and using (3.4), we obtain
EE
m=0 kcl(m)
Sint,)S,(,m)(t;)].
2.4. The Space C[0, co], Weak Convergence, and Wiener Measure
E[exp li
n
n
E A.,(B,,
-13,,,)}]= E[exp { - i E (Aii., j=1
.1=1
= exp { -
59
n -1
n
1
- '.;)B,, }1
n
E E (2.,+, -1.,)(2,,, - aot, - - E (Ai+, 2 J=1
J=1 i.J+1
n-1
= exp {- E (A.,+, - a..,)(-).,,,)t,
pi
2,)2t.,}
1
-2
(Aj+i - ili)2 til
.-1
= exp
=
E (A. +1 - apt. exp
1,12t 2
-
O
3.3 Exercise. Prove Theorem 3.2 without resort to the Parseval identity (3.4), by completing the following steps.
(a) The increments {Al. - V_10.}1.:1 are independent, normal random variables with mean zero and variance l/r. (b) If 0 = to < t1 < < t 5 1 and each t; is a dyadic rational, then the increments {B,, -Bpi }T=1 are independent, normal random variables with mean zero and variance (tj (c) The assertion in (b) holds even if {t3 }.7,_1 is not contained in the set of dyadic rationals.
3.4 Corollary. There is a probability space (II, , P) and a stochastic process B = {B ..5/7!; 0 < t < co} on it, such that B is a standard, one-dimensional Brownian motion.
PROOF. According to Theorem 3.2, there is a sequence (a,,,
2,
of probability spaces together with a Brownian motion XA P n0 I on each space. Let S2 = S21 x S22 x , = .F2 ' , and P = Pi x P2 x
.
Define B on S2 recursively by
= V1), 0 < t < 1, Bn
Xj(
n < t < n + 1.
This process is clearly continuous, and the increments are easily seen to be independent and normal with zero mean and the proper variances.
2.4. The Space C[0, co), Weak Convergence, and the Wiener Measure The sample spaces for the Brownian motions we built in Sections 2 and 3 were, respectively, the space W.') of all real-valued functions on [0, co) and a space S2 rich enough to carry a countable collection of independent,
2. Brownian Motion
60
standard normal random variables. The "canonical" space for Brownian motion, the one most convenient for many future developments, is C[0, co), the space of all continuous, real-valued functions on [0, co) with metric (4.1)
p(a) , co2) =
co
E
1
max 0(01(0 - w2(t)I A 1). n=1 2 0
In this section, we show how to construct a measure, called Wiener measure, on this space so that the coordinate mapping process is Brownian motion. This construction is given as the proof of Theorem 4.20 (Donsker's invariance principle) and involves the notion of weak convergence of random walks to Brownian motion.
4.1 Problem. Show that p defined by (4.1) is a metric on C[0, co) and, under p, C[0, co) is a complete, separable metric space. 4.2 Problem. Let W(A) be the collection of finite-dimensional cylinder sets of the form (2.1); i.e., (2.1)'
C= {Co G C[0, GO; (0)(ti ),...,(2)(4))e AI; n> 1, A EAR"),
where, for all i = 1, n, tie [0, co) (respectively, tie [0, t]). Denote by 5(%) the smallest a-field containing (6(A). Show that = .4(C[0, co)), the Borel a-field generated by the open sets in C[0, co), and that = (.11(C[0, co))) A .4,(C[0, co)), where cp,: C[0, co) C[0, co) is the mapping (cp,a))(s) = co(t A s); 0 < s < o o.
Whenever X is a random variable on a probability space (SI, F, P) with values in a measurable space (S, M(S)), i.e., the function X: SI S is . F / .4(S)measurable, then X induces a probability measure PX-1 on (S, .4(S)) by (4.2)
PX-1(B)= P {wen; X(co)e B},
B e M(S).
An important special case of (4.2) occurs when X = {Xi; 0 < t < co} is a continuous stochastic process on (Q, .F, P). Such an X can be regarded as a random variable on (n, .F, P) with values in (C[0, oo),M(C[0, co))), and P X-1 is called the law of X. The reader should verify that the law of a continuous process is determined by its finite-dimensional distributions.
A. Weak Convergence The following concept is of fundamental importance in probability theory. 4.3 Definition. Let (S, p) be a metric space with Borel a-field .4(S). Let {Pn},;°_1
be a sequence of probability measures on (S, M(S)), and let P be another measure on this space. We say that {Pn}n'_1 converges weakly to P and write Pn P, if and only if
2.4. The Space C[0, co], Weak Convergence, and Wiener Measure
lim
n,
f(s)dP(s) =
61
f f(s)dP(s)
for every bounded, continuous real-valued function f on S. It follows, in particular, that the weak limit P is a probability measure, and that it is unique. 4.4 Definition. Let {(S.1, .93, P)}`°_, be a sequence of probability spaces, and on each of them consider a random variable X with values in the metric space P) be another probability space, on which a random variable (S, p) . Let X with values in (S, p) is given. We say that {Xn}'_, converges to X in distribution, and write Xn 4 X, if the sequence of measures {PnXn-1 }nc°=, converges weakly to the measure PX -1.
Equivalently, X 4 X if and only if
lim Ef(Xn) = Ef(X) n-.co
for every bounded, continuous real-valued function f on S, where En and E denote expectations with respect to Pn and P, respectively.
Recall that if S in Definition 4.4 is Old, then x X if and only if the sequence of characteristic functions pn(u) A E. exp{i(u, Xn)} converges to qi(u) A E exp{i(u, X)}, for every u e 11". This is the so-called Cramer-Wold device (Theorem 7.7 in Billingsley (1968)). The most important example of convergence in distribution is that provided
by the central limit theorem. In the Lindeberg-Levy form used here, the theorem asserts that if { fl}nc"Li is a sequence of independent, identically distributed random variables with mean zero and variance 62, then {Sn} defined by 1
`Sn=
L4
0-v n k =1
converges in distribution to a standard normal random variable. It is this fact which dictates that a properly normalized sequence of random walks will converge in distribution to Brownian motion (the invariance principle of Subsection D). 4.5 Problem. Suppose {Xn}nc°_1 is a sequence of random variables taking values
in a metric space (S, , pi) and converging in distribution to X. Suppose (S,,p,) is another metric space, and cp: Si - S, is continuous. Show that Yn A cp(Xn)
converges in distribution to Y
qi(X).
13. Tightness The following theorem is stated without proof; its special case S = IR is used to prove the central limit theorem. In the form provided here, a proof can
62
2. Brownian Motion
be found in several sources, for instance Billingsley (1968), pp. 35-40, or Parthasarathy (1967), pp. 47-49. 4.6 Definition. Let (S, p) be a metric space and let 1-1 be a family of probability measures on (S,M(S)). We say that 1-1 is relatively compact if every sequence
of elements of II contains a weakly convergent subsequence. We say that
II is tight if for every E> 0, there exists a compact set K g S such that P(K) 1 - c, for every P ell. If {X,,}QE A is a family of random variables, each one defined on a prob-
ability space (14, g; Pi) and taking values in S, we say that this family is relatively compact or tight if the family of induced measures {PX,,-1}E A has the appropriate property. 4.7 Theorem (Prohorov (1956)). Let 11 be a family of probability measures on a complete, separable metric space S. This family is relatively compact if and only if it is tight.
We are interested in the case S = C[0, co). For this case, we shall provide a characterization of tightness (Theorem 4.10). To do so, we define for each w e C[0, co), T > 0, and S > 0 the modulus of continuity on [0, T]: (4.3)
max lw(s) - w(t)I.
mT(to,S) L4
Is-115_6
c)..s,t
4.8 Problem. Show that mT (co, (5) is continuous in co e C[0, co) under the metric p of (4.1), is nondecreasing in S, and limo,tomT(a),(5) = 0 for each co e C[0, co).
We shall need the following version of the Arzeld-Ascoli theorem. 4.9 Theorem. A set A g C[0, co) has compact closure if and only if the following two conditions hold: sup Ico(0)1 < co,
(4.4)
weit
(4.5)
lim sup mT(co, S) = 0 blo coeA
for every T > 0.
PROOF. Assume that the closure of A, denoted by A, is compact. Since A is contained in the union of the open sets G,, = {w e C[0, co); Ico(0)1 < n},
n = 1, 2, ...
it must be contained in some particular G, and (4.4) follows. For E > 0, let Kb = Ico e A ; mT (co, .5) > E }. Each Kb is closed (Problem 4.8) and is contained
in A, so each K,, is compact. Problem 4.8 implies n b>0 Kb = 0, so for some (5(E) > 0, we must have K," = 0. This proves (4.5).
2.4. The Space C[0, x], Weak Convergence, and Wiener Measure
63
We now assume (4.4), (4.5) and prove the compactness of A. Since C[0, op)
is a metric space, it suffices to prove that every sequence iconbf-i g A has a convergent subsequence. We fix T > 0 and note that for some (5, > 0, we have mT(co,(5,) < 1 for each w e A; so for fixed integer m > 1 and t e (0, T] with (rn - 1)(51 < t < //161 A T, we have from (4.5):
,-i 1001 < 1(0(0)1 + E ko(k(1) - co((k - 0(51)1 + w(t) - w((m - 061)1 k=1
kw(C)1 + m.
It follows that for each r e Q +, the set of nonnegative rationals, {w,,(r)}Th is bounded. Let Iro, r,, r2,... I be an enumeration of Q +. Then choose {4)1,7_1, a subsequence of {w,,} ;°_, with o$,°)(ro) converging to a limit denoted w(r0). From 14)}'_,, choose a further subsequence {(1)(n1)},7-1 such that co;,1)(r1) converges to a limit co(ri). Continue this process, and then let {(7),,}°°_, = 14)1,7_, be the "diagonal sequence." We have ain(r) co(r) for each re Q+
Let us note from (4.5) that for each E > 0, there exists SW > 0 such that a whenever 0 < s, t < T and Is - tI < (5(a). The same in16.(s) equality, therefore, holds for w when we impose the additional condition s, t e Q+ . It follows that w is uniformly continuous on [0, T] n 12÷ and so has an extension to a continuous function, also called w, on [0, T]; furthermore,
lw(s) - w(t)I < a whenever 0 < s, t < T and Is - tI < Ma). For n sufficiently large, we have that whenever t e [0, T], there is some rk e Q+ with k < n and It -rk S(a). For sufficiently large M > n, we have Ith,,,(rJ) - co(ri)1 5 a for all j = 0, 1, , n and m > M. Consequently,
Ith.(t) -
- 65.0.01 + Idim(rk) - co(rk)1 + ko(rk) - w(t)I
< 3E, Vm> M,0 0, so {(7)1,7_, converges uniformly on bounded intervals to the function we C[0, co). I=1
4.10 Theorem. A sequence IP1,1'._, of probability measures on (C[0, x), a(c[o, co))) is tight if and only if (4.6)
lim sup P,,[(0; At
(4.7)
> A] = 0,
n> 1
lim sup P[w; mT(w,(5) > a] =0; V T > 0, a > 0. blo n>
PROOF. Suppose first that {P.},-1 is tight. Given ri > 0, there is a compact set K with PP(K) > I - ry, for every n > 1. According to Theorem 4.9, for sufficiently large 1. > 0, we have 103(0)1 < 2 for all we K; this proves (4.6). According to the same theorem, if T and a are also given, then there exists So such that mT(w, (5) < a for 0 < S < (50 and we K. This gives us (4.7). Let us now assume (4.6) and (4.7). Given a positive integer T and n > 0, We choose
> 0 so that
64
2. Brownian Motion
1/2'.
sup P[(o; 10)(0)1 > n >1
We choose (5k > 0, k = 1, 2, ... such that sup P,,[(2.); mT (co, bk) >
1
qi2T+k+1.
r1.1
Define the closed sets 1
AT = {(0,10)(0)1
A, mT (co, Sk)
k = 1, 2, ...},
A= n
AT,
T=1
so Pn(AT) > 1 - E,T=o 17/27--f-k+i = 1 - 17/2T and PP(A) > 1 - n, for every n By Theorem 4.9, A is compact, so {P},°,p_1 is tight.
1. 1=1
4.11 Problem. Let {X(m)}:=1 be a sequence of continuous stochastic processes X(m) = IX:m); 0 < t < co} on (SI, F, P), satisfying the following conditions:
(i) sup,i EIXr1" A M < oo, (ii) supm,i El X; m) - X.1m)12 < CT1t
V T > 0 and 0 < s, t < T
s11+13;
for some positive constants a, /3, v (universal) and CT (depending on T > 0). Show that the probability measures Pm P(X(m))-1; m > 1 induced by these processes on (C[0, oo), .4(C [0, co))) form a tight sequence. (Hint: Follow the technique of proof in the Kolmogorov-Centsov Theorem 2.8, to verify the conditions (4.6), (4.7) of Theorem 4.10).
4.12 Problem. Suppose {P}n'=, is a sequence of probability measures on (C[0, oo),M(C[0, co))) which converges weakly to a probability measure P. Suppose, in addition, that If1,;°_1 is a uniformly bounded sequence of realvalued, continuous functions on C[0, co) converging to a continuous function f, the convergence being uniform on compact subsets of C[0, co). Then (4.8)
lim n
f CIO ,
f(co)dP,,(w) = )
f
f(w)dP(w).
Cio . 00)
4.13 Remark. Theorems 4.9, 4.10 and Problems 4.11, 4.12 have natural extensions to C[0, oor, the space of continuous, Rd-valued functions on [0, co). The proofs of these extensions are the same as for the one-dimensional case.
C. Convergence of Finite-Dimensional Distributions Suppose that X is a continuous process on some (n,
P). For each co, X,(w) is a member of C[0, co), which we denote by X(w). Since Al(C[0, co)) is generated by the one-dimensional cylinder sets and X1() is .F-measurable for each fixed t, the random function X: S2 -> C[0, oo) is .97,4(C[0, co))-measurable. Thus, if {X(")}'_1 is a sequence of continuous processes (with each X(") defined on a perhaps distinct probability space the function t
2.4. The Space C[0, co], Weak Convergence, and Wiener Measure
65
(0.,:y7, Pa)), we can ask whether V") X in the sense of Definition 4.4. We can also ask whether the finite-dimensional distributions of {X(")}x_i converge to those of X, i.e., whether (X7), , )0")) 4 (x,,, xt2,..., xtd).
The latter question is considerably easier to answer than the former, since the convergence in distribution of finite-dimensional random vectors can be resolved by studying characteristic functions. For any finite subset { t1, Rd as td: C[0, CO)
,
td} of [0, co), let us define the projection mapping = (co(t
),
co(td)).
If the function f: Rd IR is bounded and continuous, then the composite mapC[0, co) -.CI enjoys the same properties; thus, X(n) 4: X ping f 0 implies lim En fM7),
,
n-,co
lim En(f 0 n
= E(f 0
= Ef(X,,,
, Xid).
In other words, if the sequence of processes {X(")}`°_1 converges in distribution
to the process X, then all finite-dimensional distributions converge as well. The converse holds in the presence of tightness (Theorem 4.15), but not in general; this failure is illustrated by the following exercise. 4.14 Exercise. Consider the sequence of (nonrandom) processes 1q") = nt
(t1+ (1 - nt)
1[0, 1/2n]. ,
1,112n, lin](t);
0 < t < oo, n > 1 and let X, = 0, t > 0. Show that all finite-dimensional distributions of X(n) converge weakly to the corresponding finite-dimensional distributions of X, but the sequence of processes {X(")}'_, does not converge in distribution to the process X. 4.15 Theorem. Let {X(")}°°_, be a tight sequence of continuous processes with < td < co, then the sequence of random the property that, whenever 0 < t, < vectors X1,1,))1°11 converges in distribution. Let Pn be the measure induced on (C[0, co), Ac[o, co))) by V"). Then {Pn},T=1 converges weakly to a measure P, under which the coordinate mapping process W,(w) w(t) on C[0, co) satisfies (X:7),
,
X:d")) 4 (w,,,..., w,), 0 < ti <
< td < oo, d
1.
PROOF. Every subsequence {X( "°} of {X(")} is tight, and so has a further subsequence {g(")} such that the measures induced on C[0, co) by {g(")} converge weakly to a probability measure P, by the Prohorov theorem 4.7. If a different subsequence {X( ")} induces measures on C[0, oo) converging to a probability measure Q, then P and Q must have the same finite-dimensional distributions, i.e.,
66
2. Brownian Motion
P [co e C[0, co); (w(t,), ..., w(td))e A] = Q[co e C[0, cc); (co(ti ), .
0
< td < cc,
. .
, co(td)) E
A],
A e M(l), d > 1.
This means P = Q. Suppose the sequence of measures {Pn};°_, induced by {x(^)}-_, did not converge weakly to P. Then there must be a bounded, continuous function f: C[0, co) -> R such that lim .1 f(co)P,,(dco) does not exist, or else this limit exists but is different from f f(co)P(dco). In either case, we can choose a subsequence {Pn},T,...1 for which lim, i f(w)P,,(dco) exists but is different from f f(w)P(dw). This subsequence can have no further subsequence {P},,'_1 with Li Pn 4.' P, and this violates the conclusion of the previous paragraph.
We shall need the following result.
4.16 Problem. Let {X(")},T-1, {Y(n)},f-1, and X be random variables with values in a metric space (S, p); we assume that for each n > 1, X(n) and Y(") are defined on the same probability space. If X(") - X and p(X("), Y(")) --> 0 in
probability, as n - co, then Y(n) 4 x as n --> co.
D. The Invariance Principle and the Wiener Measure Let us consider now a sequence gilT,i of independent, identically distributed random variables with mean zero and variance a2, 0 < o-2 < co, as well as the sequence of partial sums So = 0, Sk = El=i , k > 1. A continuous-time process Y = { Y; t 0} can be obtained from the sequence {Sk},T=0 by linear interpolation; i.e., (4.9)
Y, = ST,1 + (t - TtKp.,,
t
0,
where N] denotes the greatest integer less than or equal to t. Scaling appropriately both time and space, we obtain from Y a sequence of processes pOnl: (4.10)
X: n) =
aJ Y. 1
t>
0.
2.4. The Space C[0, co], Weal. Convergence, and Wiener Measure
67
Note that with s = k/n and t = (k + 1)/n, the increment Xr -Xr = (1/o-ji)k+i is independent of
=
1,
Furthermore, Xr -
,
has zero mean and variance t - s. This suggests that {X:"); t > 01 is approximately a Brownian motion. We now show that, even though the random variables are not necessarily normal, the central limit theorem dictates that the limiting distributions of the increments of X(n) are normal.
4.17 Theorem. With {VI defined by (4.10) and 0 < t1 <
< td < co, we
have
where {B
t
0} is a standard, one-dimensional Brownian motion.
PROOF. We take the case d = 2; the other cases differ from this one only by being notationally more cumbersome. Set s = t1, t = t2. We wish to show (X!"), XI")) 4 (Bs, Be).
Since 1
1
XI")
Slim]
we have by the Cebygev inequality,
P[ as n
1
X:")
>
o-,/n
<
0
82n
co. It is clear then that 1
(Xs"', Xr)
-* 0
S( isn], SitI)
in probability,
tT
so, by Problem 4.16, it suffices to show n(Sp1, Sim) --0 (B3, Be). n
From Problem 4.5 we see that this is equivalent to proving itn]
1
E o-F j=1
E
-4 (Bs, Be - Bs).
j=[sn]+1
The independence of the random variables {i}.71_1 'mplies
lim E [exp
iu
l[sn]
E
+
iv 1[4 isn
(4.11)
= lim E exp n-.00
iu
to-N/n.fri
j
lim E exp n
CO
iv +1
68
2. Brownian Motion
provided both limits on the right-hand side exist. We deal with limn_o, E[exp{(iu/6 .1r-t)E;s:}]; the other limit can be treated similarly. Since Isrq
L, C.1
0"f j =1
o- ,A[sn] J=1
-0 in probability,
and, by the central limit theorem, (fs/o- ,A[sn]j)E.Ig converges in distribution to a normal random variable with mean zero and variance s, we have iu
lim E [exp
e -"2s/2
1[511
o-,/n J=1
n-vco
Similarly, lim
E[exp
n-oo
= e-v2"-s)/2.
iv
6 1;1 j= Fil+1
Substitution of these last two equations into (4.11) completes the proof. Actually, the sequence {X(")} of linearly interpolated and normalized random walks in (4.10) converges to Brownian motion in distribution. For the tightness required to carry out such an extension (recall Theorem 4.15), we shall need two auxiliary results.
4.18 Lemma. Set Sk = EI)=1 where { .,}72,1 is a sequence of independent, identically distributed random variables, with mean zero and finite variance 62 > 0. Then, for any s > 0, lim lim
1 P[
610 n-co 0
max 1
= O.
ISJI >
.1.11161 4-1
PRooF. By the central limit theorem, we have for each S > 0 that (1/6En61 + 1 )SEnol+, converges in distribution to a standard normal random variable Z, whence (1/6 ,/n6)S1n61 +1 4 Z. Fix A > 0 and let {(pk}i.,°:=1 be a sequence of bounded, continuous functions on 1 1 1 with p i , j 1(_co, We
have for each i, lim P[ISin,q+i n-.aD
Let k (4.12)
AciiTtS] < lirn Ecpkl n -co
Slna1 +1J = E(pk(Z). NPU5
co to conclude lim P[I,Sin6+11
Ao-,1/41(5] < P[IZI > A] < 7EIZ13,
A > 0.
We now define i = min{ j 1; I.SJI > co-,Ft}. With 0 < 6 < E2/2, we have (imitating the proof of the Kolmogorov inequality; e.g., Chung (1974), p. 116):
2.4. The Space C[0, op], Weak Convergence, and Wiener Measure
P[
(4.13)
max
69
ISJ1> Ea Nit;
0 5:1<no]+1
a Nfl(E - ,./2(5)]
P [IS[[no]+11 [[no]
+ 1 P[ISPIO1+11 < a ji(e - ,./26)1T = j] P[T = .nj=i
But if T = j, then IS[nall-il< ajz(t - .126) implies 1Si -Spop-il> 6 ,./2n6. By the ebygev inequality, the probability of this event is bounded above by 1
2nba
[ma+
1
E[(S;
1 5 j < Pl.
= 2nSo-2 E
Swil+i)21T =
Returning to (4.13), we may now write max
P[ 0 <j.[[nE5.]+1
ISJI >
E6j11
fn(v -
<
26)] + iP[t < Ifi51] 1
PCIS[[..3+11
6.,F4E -126)] + P[
max
ISJI > ECI li-1 1 ,
0 5j5 [[nr3. +1
from which follows
P[
max
ISJI >
win] 5 2P [1 ST(014-11
0\Fi(e
-
Setting A = (E - ,./Y,(5)/,./3 in (4.12), we see that lim
1
- P[
n-m 6
max
ISJI >
0 5/5[n51+1
to-jd <
2 ,F5
E1Z13,
(E - N/245)3
and letting 6.1 0 we obtain the desired result. 4.19 Lemma. Under the assumptions of Lemma 4.18, we have for any T > 0,
lim lim P[ 6.1.o
max
ISf+k - Sk >
cajd = 0.
J5[[no]]-1-1
Tnn+1
PROOF. For 0 < b < T, let m = m(S) > 2 be the unique integer satisfying T/m < b < T /(m - 1). Since lim
[[nT1] + 1
T
[[0]] + 1
6
< m,
We have [[nT1] + 1 < an6]] + 1)m for sufficiently large n. For such a large
2. Brownian Motion
70
.11-1 for some k, 0 5 k < TnT1] + 1, and some j, 1 < j < [[n.511 + 1. There exists then a unique integer p, 0 5 p m - 1, such that n, suppose 1Si +1, - Ski >
(Tn.51 + 1)p < k < (Tn.51 + 1)(p + 1).
There are two possibilities for k + j. One possibility is that (Tn.51 + 1)p < k + j
(11n61 + 1)(p + 1),
in which case either iSk - Sq,l+opi > ice INAz, or else iSk+; - Sanol+opi > lerrin. The second possibility is that (Tn.511 + 1)(p + 1) < k + j < (Tn611 + 1)(p + 2), in which case either iSk - S(Qn,31+1)pl > iEcr \At, IS(Qnol+i)p - S([[no1+1)(p+1)1 >
Sk+ii > icer\Fi. In conclusion, we see that
jeQ NIP-1, or else IS rr(0,31+1)(p+1)
ISJ,k - Ski > errin
max 1
1.
j<[[nol+1
0 5_k 51[171+1
(
m
g- p=0 U
1
max
I Spi-pqnaii+i) - S panO11+1)1 > -3EGr AI
1 ..j .[[no11+1
But 1
P[
max
S 1)&61+1)1 > -3E6r NA?
ISi+ pano]] +1)
1 5..1[[11.511+1
=
p[
max 1 ..1
1
IS.I > -ErrNAti, 3 _I
and thus:
P[ 0
max [[Ifoll+1
1
Si+k -Sk I > Err\Ad
(m + 1)P[
max 1
S[[1u31+1
S.I > _I
3 Err\Ail
It.[[n711+1
Since m < (TM) + 1, we obtain the desired conclusion from Lemma 4.18.
We are now in a position to establish the main result of this section, namely the convergence in distribution of the sequence of normalized random walks in (4.10) to Brownian motion. This result is also known as the invariance principle.
4.20 Theorem (The Invariance Principle of Donsker (1951)). Let (n, P) be a probability space on which is given a sequence gitti of independent, identically distributed random variables with mean zero and finite variance a2 > 0. Define X(n) = PO"); t > 0} by (4.10), and let Pn be the measure induced by X(n) on (C[0, co), .4(C[0, co))). Then {Pn}n'=1 converges weakly to a measure P*,
2.5. The Markov Property
71
under which the coordinate mapping process W(co) standard, one-dimensional Brownian motion.
w(t) on C[0, co) is a
PROOF. In light of Theorems 4.15 and 4.17, it remains to show that {X(")}x_1 is tight. For this we use Theorem 4.10, and since X' = 0 a.s. for every n, we need only establish, for arbitrary s > 0 and T > 0, the convergence
lim sup
(4.14)
64.0 n> 1
P[
max PC.!") - XP) I > El = 0.
Is-tl<3
0 <s,t < T
We may replace supn >1 in this expression by lim, since for a finite number of integers n we can make the probability appearing in (4.14) as small as we choose, by reducing 6. But
P[
max
1X1n)
- x:")I > 61= fi[ max
I Ys -
> so- ,A11,
rib
0 <s,i< T
0 <s,t
and max
max
I Ys - Y,I <
0 <s,t
Is-ti
max
YS - Ygi <
ISi+k - Ski,
1 <j S1n511+1 0
where the last inequality follows from the fact that Y is piecewise linear and changes slope only at integer values oft. Now (4.14) follows from Lemma 4.19.
4.21 Definition. The probability measure on (C[0, co), ,I(C[0, co))), under which the coordinate mapping process No)) w(t), 0 < t < co, is a standard, one-dimensional Brownian motion, is called Wiener measure.
4.22 Remark. A standard, one-dimensional Brownian motion defined on any
probability space can be thought of as a random variable with values in C[0, co); regarded this way, Brownian motion induces the Wiener measure on (C[0, co), ,I(C[0, co))). For this reason, we call (C[0, co), R(C[0, co)), Ps), where pi, is Wiener measure, the canonical probability space for Brownian motion.
2.5. The Markov Property In this section we define the notion of a d-dimensional Markov process and cite d-dimensional Brownian motion as an example. There are several equivalent statements of the Markov property, and we spend some time developing them.
72
2. Brownian Motion
A. Brownian Motion in Several Dimensions 5.1 Definition. Let d be a positive integer and it a probability measure on (Rd, .4(01d)) Let B = {B F; t _. 0} be a continuous, adapted process with values in Rd, defined on some probability space (SI, ..571, P). This process is called a d-dimensional Brownian motion with initial distribution t, if
(i) P[B0 e 11 = it(F), V F e ,4(Rd); (ii) for 0 < s < t, the increment A - B3 is independent of g' and is normally distributed with mean zero and covariance matrix equal to (t - s)Id, where Id is the (d x d) identity matrix.
If it assigns measure one to some singleton {x}, we say that B is a ddimensional Brownian motion starting at x.
Here is one way to construct a d-dimensional Brownian motion with initial distribution t. Let X(coo) = coo be the identity random variable on (01d9
(old), it),
and for each i = 1, ... , d, let /II° = lir, AB"); t > 01 be a
standard, one-dimensional Brownian motion on some (0"), .9""), P")). On the product space (Rd
nu)
not) ,4(01d)
®y(1) ® ... ® F(d),
x
p(1 x ... x p(d),
define Bt(w) A X(coo) + (R1)(coi )9 ... 9 Rd)(cod))9
and set .9; = g-,B. Then B = {B1, Ft; t > 0} is the desired object.
There is a second construction of d-dimensional Brownian motion with initial distribution t, a construction which motivates the concept of Markov family, to be introduced in this section. Let P"), i = 1, ... , d be d copies of Wiener measure on (C[0, co), .(C[0, co))). Then P° A P(1) x x P(d) is a measure, called d-dimensional Wiener measure, on (C[0, oo)d, .4(C [0, 03)d)). Under P°, the coordinate mapping process Bt(co) A co(t) together with the filtration WI is a d-dimensional Brownian motion starting at the origin. For x e Rd, we define the probability measure Px on (C[0, oo)d,M(C[0, co)d)) by (5.1)
Px(F) = P°(F - x), F e .4(C[0, oo)a),
where F -x = {we C[0, co)d; u)() + x e F}. Under Px, B A {B ,,B; t > 0} is a d-dimensional Brownian motion starting at x. Finally, for a probability measure it on (Rd, 2(Rd)), we define P" on a(c[o, co)d) by (5.2)
P4(F) = J
Px(F)u(dx). ;la
Problem 5.2 shows that such a definition is possible.
5.2 Problem. Show that for each F e g(C[0, co)d), the mapping x i- Px(F) is 1(Rd)/.4([0, 1])-measurable. (Hint: Use the Dynkin System Theorem 1.3.)
2.5. The Markov Property
73
5.3 Proposition. The coordinate mapping process B = IB gor,,B; t > 01 on (C[0, 09)d, .4(C[0, cor), P") is a d-dimensional Brownian motion with initial distribution u.
5.4 Problem. Give a careful proof of Proposition 5.3 and the assertions preceding Problem 5.2.
gr.,; 0 < t < oc,} be a d-dimensional Brownian motion. Show that the processes
5.5 Problem. Let {B, =
MP) A BP) - Bg), .9;;
t < oo, 1 < i < d
0
are continuous, square-integrable martingales, with <M('), Mti)>, = 654.; 1 < , M(d)) is independent of .yo.
j < d. Furthermore, the vector of martingales M = (M(1),
5.6 Definition. Given a metric space (S, p), we denote by AS)" the completion
of the Borel o-field .4(S) (generated by the open sets) with respect to the finite measure ji on (S, .4(S)). The universal o--field is ail(S) A 11,,,R(S)P, where
the intersection is over all finite measures (or, equivalently, all probability measures) u. A ail(S)/.4(R)-measurable, real-valued function is said to be universally measurable.
5.7 Problem. Let (S, p) be a metric space and let f be a real-valued function defined on S. Show that f is universally measurable if and only if for every finite measure g on (S, a(S)), there is a Borel-measurable function g: S such that pfx ES; f(x) g (x)} = 0. 5.8 Definition. A d-dimensional Brownian family is an adapted, d-dimensional
process B = {B
t > 0} on a measurable space (S/,.F), and a family of
probability measures {Px Ix c Re, such that
(i) for each F e Px(F) is universally measurable; the mapping (ii) for each x e Rd, /3' [Bo = x] = 1;
(iii) under each Px, the process B is a d-dimensional Brownian motion starting at x.
We have already seen how to construct a family of probability measures {Px} on the canonical space (C[0, oor,.4(C[0, °o)")) so that the coordinate mapping process, relative to the filtration it generates, is a Brownian motion
starting at x under any Px. With
= .4(C[0, cor), Problem 5.2 shows
that the universal measurability requirement (i) of Definition 5.8 is satisfied. Indeed, for this canonical example of a d-dimensional Brownian family, the mapping xi- Px(F) is actually Borel-measurable for each F e ,F. The reason we formulate Definition 5.8 with the weaker measurability condition is to allow expansion of .F to a larger o--field (see Remark 7.16).
2. Brownian Motion
74
B. Markov Processes and Markov Families Let us suppose now that we observe a Brownian motion with initial distribution y up to time s, 0 < s < t. In particular, we see the value of Bs, which we call y.
Conditioned on these observations, what is the probability that B, is in some set f e MO:id)? Now B, = (B, - Bs) + Bs, and the increment B, -Bs is independent of the observations up to time s and is distributed just as B,_, is under P°. On the other hand, Bs does depend on the observations; indeed,
we are conditioning on B, = y. It follows that the sum (B, - Bs) + Bs is distributed as B,_, is under P. Two points then become clear. First, knowledge of the whole past up to time s provides no more useful information about B, than knowing the value of Bs; in other words, (5.3)
P"[B, e F I.Fs] = P"[B, e Fl Bs],
0 < s < t, F e a(Rd).
Second, conditioned on Bs = y, B, is distributed as B,_, is under PY; i.e., (5.4)
P"[B,e rIB, = y] = PY [B,_, e fa o
s < t, re.4(01d).
5.9 Problem. Make the preceding discussion rigorous by proving the following. If X and Y are d-dimensional random vectors on (52,.f, P), W' is a sub-a-field of F, X is independent of W and Y is p-measurable, then for every rel/(01°): (5.5)
P[X + Ye rm = P[X + Ye TI Y], a.s. P;
(5.6)
P[X + Y e ri Y = y] = P[X + y e r], for PY-1-a.e. y e Old
in the notation of (4.2).
5.10 Definition. Let d be a positive integer and it a probability measure on (Rd, Al(Rd)). An adapted, d-dimensional process X = {X .f;; t > 0} on some probability space (52,.x, P ") is said to be a Markov process with initial distribution it if
(i) P"[X0 e r] = p(r), vre M(Rd); (ii) for s, t _. 0 and re A(V),
P"[X,e ri,Fs] = PP[X,e rixj,
P"-a.s.
Our experience with Brownian motion indicates that it is notationally and conceptually helpful to have a whole family of probability measures, rather than just one. Toward this end, we define the concept of a Markov family. 5.11 Definition. Let d be a positive integer. A d-dimensional Markov family
is an adapted process X = {X Ft; t > 0} on some (S2,,), together with a family of probability measures {Px} x, Re on (SI, F), such that (a) for each F e .9 ", the mapping x i-3 Px(F) is universally measurable; (b) Px[X0 = x] = I, V x e Rd;
2.5. The Markov Property
75
(c) for x E Rd, s, t> 0 and T e .4(Rd), Px
(d) for x e Rd, s, t
e F .Fs] = Px [XH, e r I xs],
Px-a.s;
0 and F e *Rd),
Px[X,+se F I X, = y] = PY [Xi e
y
in the notation of (4.2). The following statement is a consequence of Problem 5.9 and the discussion preceding it.
5.12 Theorem. A d-dimensional Brownian motion is a Markov process. A d-dimensional Brownian family is a Markov family.
C. Equivalent Formulations of the Markov Property The Markov property, encapsulated by conditions (c) and (d) of Definition 5.11,
can be reformulated in several equivalent ways. Some of these formulations amount to incorporating (c) and (d) into a single condition; others replace the evaluation of X at the single time s + t by its evaluation at multiple times after s. The bulk of this subsection presents those formulations of the Markov property which we shall find most convenient in the sequel. Given an adapted process X = {X t 0} and a family of probability measures {Px gld on (S/, g;), such that condition (a) of Definition 5.11 is satisfied, we can define a collection of operators { Ut}t, 0 which map bounded, Borel-measurable, real-valued functions on R' into bounded, universally measurable, real-valued functions on the same space. These are given by (5.7)
(Utf )(x) A Ex f (X
In the case where f is the indicator of F E M(Rd), we have Exf(X,) = Px [X, e F], and the universal measurability of U, f follows directly from Definition 5.11 (a);
for an arbitrary, Borel-measurable function f, the universal measurability of EU is then a consequence of the bounded convergence theorem. 5.13 Proposition. Conditions (c) and (d) of Definition 5.11 can be replaced by:
(e) For x e
s, t > 0 and T e M(Rd),
Px[X,,,e1-1,s] =
Px-a.s.
PROOF. First, let us assume that (c), (d) hold. We have from the latter:
Px[X,,serix, = y] = (U, 1 )(y) for PxXs-l-a.e. y
Rd.
If the function a(y) g (U,1,-)(y): Rd [0, 1] were AR ")- measurable, as is the case for Brownian motion, we would then be able to conclude that, for all
76
2. Brownian Motion
x e Rd, s 0: Px[Xt+serixs] = a(Xs), a.s. Px, and from condition (c): Px[X,e I" L. s] = a(Xs), a.s. Px, which would then establish (e). However, we only know that Utl r( ) is universally measurable. This means (from Problem 5.7) that, for given s, t > 0, x e Rd, there exists a Borel-measurable function g: Rd --, [0, 1] such that (5.8)
(u, 1r) (y) = g(y),
for PxXs 1-a.e. y e Rd,
whence
(Utir)(Xs) = g(Xs),
(5.9)
a.s. Px.
One can then repeat the preceding argument with g replacing the function a. Second, let us assume that (e) holds; then for any given s, t > 0 and x e Rd, (5.9) gives (5.10)
Px[X,e F I's] = g(X,),
a.s. Px.
It follows that Px [X,,, e I- 1,:j has a o-(Xs)-measurable version, and this establishes (c). From the latter and (5.10) we conclude
Px[X,,se rix, = y] = g(y) for PxX;-1-a.e. ye Rd, and this in turn yields (d), thanks to (5.8). 5.14 Remark on Notation. For given w e SI, we denote by X.(w) the function
t 1- X,(w). Thus, X. is a measurable mapping from 0,39 into ((Rd)1°'), ge(oRdyo,c.),)( the space of all Rd-valued functions on [0, co) equipped with
the smallest a-field containing all finite-dimensional cylinder sets. 5.15 Proposition. For a Markov family X, (II, .x), {Px}xe [Iv, we have:
(c') For x e Rd, s> 0 and F e A(Rdy°,03)), Px [Xs+. e Figs] = Px[Xs+. e FI Xs], Px-a.s; (d') For x e Rd, s > 0 and F e.4((Rd)1° "))),
Px[Xs+.e FIX, = y] = PY[X.e F], PxXs-l-a.e. y. (Note: If f E .4(Rd) and F = {w e(Rd)1°''); w(t)ET }, for fixed t and (d') reduce to (c) and (d), respectively, of Definition 5.11.)
0, then (c')
The collection of all sets F e .4((Rd)""3)) for which (c') and (d') hold forms a Dynkin system; so by Theorem 1.3, it suffices to prove (c') and (d') for finite-dimensional cylinder sets of the form PROOF.
F = {0.) e(Old)(°'x'); w(to) e ro,
.
., co(tn_1)er,,_1,
w(tn)er},
where 0 = to < ti < - < t, Tie mold), i = o, 1, ..., n, and n _. O. For such an F, condition (c') becomes
2.5. The Markov Property (5.11)
77
Px[Xs efo, ... , Xs+,_, e rn_ Xs+t.E rn IA]
= P[Xsero, ..., Xs+,._,ern_i, Xs+,. 6 rnIxs], P-a.s. We prove this statement by induction on n. For n = 0, it is obvious. Assume it true for n - 1. A consequence of this assumption is that for any bounded, Borel-measurable cp: IRd" -, 01,
(5.12) .Ex[cp(Xs, ..., Xs+,._,)Igs] = Ex[p(Xs, ..., Xs+,._,)1X,],
P-a.s.
Now (c) implies that (5.13)
P[Xsero, ..., Xs+tn_ie rn_i, Xs+,. e rn I gs] = Ex[1{x.e 1-0,...,xs*,._1 er_,}Px[Xs+t e rnigs+tn_jigs] = Ex [1(x. e ro,...,x.._, er_,}Px[Xs+t,, 6 TnIxs+t,,_,] Igs].
Any gx.+, -measurable random variable can be written as a Borel-measurable function of Xs+,._i (Chung (1974), p. 299), and so there exists a Borel-measurable
function g: Rd -- [0, 1], such that Px[Xs+,. e 1-.1Xs+/_, ] = g(Xs+in_i), a.s. P.
Setting cp(x0, ... , x_1) g'-- 100... 1,-;_i(x_,)g(xn_i), we can use (5.12) to replace gs by Q(Xs) in (5.13) and then, reversing the previous steps, to obtain (5.11). The proof of (d') is similar, although notationally more complex.
It happens sometimes, for a given process X = {X gt; t _. 0} on a measurable space (SI, g), that one can construct a family of so-called shift operators Os: SI - SI, s > 0, such that each Os is g/g--measurable and (5.14)
Xs,(co) = X, (Cisco);
V co e SI,
s, t > 0.
The most obvious examples occur when SI is either (W)1°.') of Remark 5.14 or C[0, co)" of Remark 4.13, .97 is the smallest o--field containing all finitedimensional cylinder sets, and X is the coordinate mapping process Xt(co) = (0(4 We can then define 00a) = co(s + ), i.e., (5.15)
(Osco)(t) = co(s + t),
t > 0.
78
2. Brownian Motion
When the shift operators exist, then the function Xs+.(w) of Remark 5.14
"
is none other than X.(0sw), so {X.eF} = BS {X. e F}. As F ranges over {X. a F} ranges over 'cx. Thus, (c') and (d') can be reformulated as follows: for every F and s > 0, Px[tV1FI.Fs] = Px[tV1FIXs],
(c")
(d")
Px[0;1 FIX, =
= PY [F],
PXXs I a.e. y.
In a manner analogous to what was achieved in Proposition 5.13, we can capture both (c") and (d") in the requirement that for every F e F cox and s > 0,
Px[VFIF] = Px.(F),
(e")
Px-a.s.
Since (e") is often given as the primary defining property for a Markov family, we state a result about its equivalence to our definition.
5.16 Theorem. Let X = {X
t > 0} be an adapted process on a measurable space (52,,), let it-Px xe Rd be a family of probability measures on (SI, .F), and
let {0,},0 be a family of .97/,-measurable shift-operators satisfying (5.14). Then X, (1/,.F), {Px }Re is a Markov family if and only if (a), (b), and (e") hold.
5.17 Exercise. Suppose that X, (11,,), {Px}xE Rd is a Markov family with shift-operators {Os}s,o. Use (c") to show that for every x e Rd, s > 0, GE",
and FED, px[G
osiFixs]
px[Gixipx[os-iFixj,
px_a.s.
We may interpret this equation as saying the "past" G and the "future" 0;1F are conditionally independent, given the "present" X. Conversely, show that (c'") implies (c").
We close this section with additional examples of Markov families.
5.18 Problem. Suppose X = {X
t > 0} is a Markov process on (CI, P) Rd and T: [0, co) L(l V,OV), the space of linear transformations from Rd to Rd, are given (nonrandom) functions with (p(0) = 0 and T(t) and gyp: [0, co)
nonsingular for every t > 0. Set Y, = (p(t) + T(t)X Then Y = is also a Markov process.
Yi,
t
0}
5.19 Definition. Let B = {13,, t > 0 }, (S2,F), {Px}xE Rd be a d-dimensional Brownian family. If /I E Rd and a n L(Rd, Rd) are constant and a is nonsingular, then with Y, + o-B we say Y = Y t 0), (C/,37), {P° lx}xRd is a d-dimensional Brownian family with drift pi and dispersion coefficient a.
This family is Markov. We may weaken the assumptions on the drift and diffusion coefficients considerably, allowing them both to depend on the location of the transformed process, and still obtain a Markov family. This is the subject of Chapter 5 on stochastic differential equations; see, in particular, Theorem 5.4.20 and Remark 5.4.21.
2.6. The Strong Markov Property and the Reflection Principle
79
5.20 Definition. A Poisson family with intensity A > 0 is a process N = IN ,Ft;
t > 01 on a measurable space (Si, .F) and a family of probability measures { px}xER, such that
(i) for each E e F, the mapping x 1-3 Px(E) is universally measurable; (ii) for each x e gri, Px [No = x] = 1; (iii) under each Px, the process IN, = N, - No, ,Ft: t > 01 is a Poisson process with intensity A.
5.21 Exercise. Show that a Poisson family with intensity A > 0 is a Markov family. Show furthermore that, in the notation of Definition 5.20 and under any I', the 6- fields Fco and Fo are independent.
Standard, one-dimensional Brownian motion is both a martingale and a Markov process. There are many Markov processes, such as Brownian motion with nonzero drift and the Poisson process, which are not martingales. There are also martingales which do not enjoy the Markov property.
5.22 Exercise. Construct a martingale which is not a Markov process.
2.6. The Strong Markov Property and the Reflection Principle Part of the appeal of Brownian motion lies in the fact that the distribution of certain of its functionals can be obtained in closed form. Perhaps the most fundamental of these functionals is the passage time 'I', to a level b e H, defined by (6.1)
Tb(co) = inf { t > 0; 13,(w) = b }.
We recall that a passage time for a continuous process is a stopping time (Problem 1.2.7).
We shall first obtain the probability density function of Tb by a heuristic argument, based on the so-called reflection principle of Desire Andre (Levy
(1948), p. 293). A rigorous presentation of this argument requires use of the strong Markov property for Brownian motion. Accordingly, after some motivational discussion, we define the concept of a strong Markov family and
prove that any Brownian family is strongly Markovian. This will allow us to place the heuristic argument on firm mathematical ground.
A. The Reflection Principle Here is the argument of Desire Andre. Let IB Ft; 0 5 t < col be a standard, one-dimensional Brownian motion on (0, ,F, P°). For b > 0, we have
2. Brownian Motion
80
p0 [Tb < t]
pOrL 1 b < t, B, > b] + P°[Tb < t, B, < b].
Now P° [Tb < t, A > b] = P° [B, > b]. On the other hand, if Tb < t and B, < b, then sometime before time t the Brownian path reached level b, and then in the remaining time it traveled from b to a point c less than b. Because of the symmetry with respect to b of a Brownian motion starting at b, the "probability" of doing this is the same as the "probability" of traveling from
b to the point 2b - c. The heuristic rationale here is that, for every path which crosses level b and is found at time t at a point below b, there is a "shadow path" (see figure) obtained from reflection about the level b which exceeds this level at time t, and these two paths have the same "probability." Of course, the actual probability for the occurrence of any particular path is zero, so this argument is only heuristic; even if the probability in question were positive, it would not be entirely obvious how to derive the type of "symmetry" claimed here from the definition of Brownian motion. Nevertheless, this argument leads us to the correct equation P ° [ Tb < t, B, < b] = P°[Tb < t, B, > b] = P°[B, > IA,
2b -
which then yields 0,
(6.2)
P° [Tb < t] = 2P° [B, > b] =
7t1bt -1/2
e-x212 dx.
Differentiating with respect to t, we obtain the density of the passage time (6.3)
P°[Tbe d[] =
,J2irt3 /27rt3
e-b2/2`dt;
t > 0.
.,
The preceding reasoning is based on the assumption that Brownian motion "starts afresh" (in the terminology of Ito & McKean (1974)) at the stopping
2.6. The Strong Markov Property and the Reflection Principle
81
time Tb, i.e., that the process {Bt +Tb - BTh; 0 < t < co} is Brownian motion, independent of the cr-field ,Tb. If T6 were replaced by a nonnegative constant, it would not be hard to show this; if Tb were replaced by an arbitrary random time, the statement would be false (cf. Exercise 6.1). The fact that this "starting afresh" actually takes place at stopping times such as '1,, is a consequence of the strong Markov property for Brownian motion.
6.1 Exercise. Let {B ",; t > 0} be a standard, one-dimensional Brownian motion. Give an example of a random time S with P[0 < S < co] = 1, such that with W, A Bs+, - Bs, the process W = {Wt, Fr; t > 0} is not a Brownian motion.
B. Strong Markov Processes and Families 6.2 Definition. Let d be a positive integer and ti a probability measure on (Rd, .4(Rd)). A progressively measurable, d-dimensional process X = {X .F,; t > 0} on some (S2, F, P") is said to be a strong Markov process with initial distribution t if
(i) Pu[Xoe 1] =µ(r), V I" e .4(01°); (ii) for any optional time S of {Ft}, t
0 and I" eR(Rd),
r[Xs+, erls+] = PP[xs÷terixs],
P" -a.s. on {S < co}.
6.3 Definition. Let d be a positive integer. A d-dimensional strong Markov family is a progressively measurable process X = {X ,F,; t > 0} on some (C/, F), together with a family of probability measure {Px}xeRa on (SZ, g"), such that: (a) for each F e.F, the mapping x 1-* Px(F) is universally measurable; (b) Px[Xo = x] = 1, V x elle; (c) for x e Rd, t > 0, r eM(Rd), and any optional time S of {Ft},
Px[Xs+teri,s+] = Px[xs+ieriXs], Px-a.s. on {S < co}; (d) for x e Rd, t > 0, F e .4(Rd), and any optional time S of {A},
Px[xs+,e I-I Xs = y] = PY [X, e 11 PxXV -a.e. y.
6.4 Remark. In Definitions 6.2, 6.3, {Xs+, e I"} A {S < co, Xs,,e1"} and PxXV (Rd) = Px(S < co). The probability appearing on the right-hand side of Definition 6.2(ii) and Definition 6.3(c) is conditioned on the cr-field generated by Xs as defined in Problem 1.1.17. The reader may wish to verify in this connection that for any progressively measurable process X,
PPLXs+terls+i = PilXs+teriXs] = 0, P" -a.s. on {S = co}, and so the restriction S < co in these conditions is unnecessary.
2. Brownian Motion
82
6.5 Remark. An optional time of I.F,1 is a stopping time of {,t+} (Corollary 1.2.4). Because of the assumption of progressive measurability, the random
variable Xs appearing in Definitions 6.2 and 6.3 is A,-measurable (Proposition 1.2.18). Moreover, if S is a stopping time of {F,}, then Xs is Fsmeasurable. In this case, we can take conditional expectations with respect to A on both sides of (c) in Definition 6.3, to obtain
Px[Xs+i E FIFO = Px[Xs+IEFIXA,
Px-a.s. on IS <
Setting S equal to a constant s > 0, we obtain condition (c) of Definition 5.11. Thus, every strong Markov family is a Markov family. Likewise, every strong
Markov process is a Markov process. However, not every Markov family enjoys the strong Markov property; a counterexample to this effect, involving a progressively measurable process X, appears in Wentzell (1981), p. 161.
Whenever S is an optional time of IA and u > 0, then S + u is a stopping time of (Problem 1.2.10). This fact can be used to replace the constant s
in the proof of Proposition 5.15 by the optional time S, thereby obtaining the following result.
6.6 Proposition. For a strong Markov family X = IX Ft; t > 0 }, (S2, F), IPx}x. Re, we have
(c') for x c Rd, F c a((R' r,-)), and any optional time S of
Px[Xs+.E FIAJ = Px[Xs+.E PIXs], Px-a.s. on {S < oo 1; (d') for x E Rd, F E a((Rdr,-)), and any optional time S of {A}, Px[xs+. E FIRS = y] = PY [X. E F],
PxXS-1-a.e. y.
Using the operators IU,1, in (5.7), conditions (c) and (d) of Definition 6.3 can be combined. 6.7 Proposition. Let X = {X ,E; t > 0} be a progressively measurable process on (52,,9), and let {Px Re be a family of probability measures satisfying (a) and (b) of Definition 6.3. Then X, (52, 3T), kPx, 1 xERd is strong Markov if and only if for any {3 }- optional time t > 0, and x e Rd, one of the following equivalent conditions holds: (e)
for any r E a(OBd),
Px[X,EFIAJ =
r)(Xs),
Px-a.s. on {S < col;
(e') for any bounded, continuous f: Rd -> Ex[f(Xs+f)IFs+] = (U1.1)(Xs),
Px-a.s. on {S <
PROOF. The proof that (e) is equivalent to (c) and (d) is the same as the proof of the analogous equivalence for Markov families given in Proposition 5.13. Since any bounded, continuous real-valued function on Rd is the pointwise
2.6. The Strong Markov Property and the Reflection Principle
83
limit of a bounded sequence of linear combinations of indicators of Borel sets, (e') follows from (e) and the bounded convergence theorem. On the other hand,
if (e') holds and F g_ Rd is closed, then 1r is the pointwise limit of {f} °11, y E Fl. Each f is bounded and continuous, so (e) holds for closed sets F. The collection of sets F e .4(Ra) for which (e) holds forms a Dynkin system, so, by Theorem 1.3, (e)
-
where f(x) = [1 - np(x,F)] v 0 and p(x, n = holds for all F e M(Rd).
6.8 Remark. If X = { X
; t > 0}, (R.F),
is a strong Markov
family and tt is a probability measure on (Rd, .4(Rd)), we can define a probability measure P" by (5.2) for every F e.517, and then X on (SI, , P") is a strong Markov process with initial distribution it. Condition (ii) of Definition 6.2 can be verified upon writing condition (e) in integrated form:
(Utlr)(Xs)dPx = Px[X,s+ter, F];
FE F +,
SF
and then integrating both sides with respect to it. Similarly, if X, (52, 5), {Px}xeRd is a Markov family, then X on (S2,..97,P") is a Markov process with initial distribution it.
It is often convenient to work with bounded optional times only. The following problem shows that stating the strong Markov property in terms of such optional times entails no loss of generality. We shall use this fact in our proof that Brownian families are strongly Markovian. 6.9 Problem. Let S be an optional time of the filtration {Ft} on some (C/,..F, P).
(i) Show that if Z1 and Z2 are integrable random variables and Z1 = Z2 on some ,975.4.-measurable set A, then
E[Z,1,53-5+] = E[Z21F.j,
a.s. on A.
(ii) Show under the conditions of (i) that if s is a positive constant, then
=
-
tS 4+1 a.s. on {S < s} n A.
(Hint: Use Problem 1.2.17(i)).
(iii) Show that if (e) (or (e')) in Proposition 6.7 holds for every bounded optional time S of
then it holds for every optional time.
Conditions (e) and (e') are statements about the conditional distribution of X at a single time S + t after the optional time S. If there are shift operators Als satisfying (5.14), then for any random time S we can define the random shift Os: IS < oo} -*CI by
= 0, on {S = s}. In other words, 0 is defined so that whenever S(w) < co, then
2. Brownian Motion
84
Xs(.)-Fi(w) = Xi(Os(w))
{X. e E}, and (c') and (d') are, respecIn particular, we have {Xs+. e E} = tively, equivalent to the statements: for every x e Rd, Fegicx, and any optional time S of {A}, px[os-iFIAA
(c")
Px-a.s. on {S < col;
Px [os'Fl Xs = y] = PY (F),
(d")
Px
y.
Both (c") and (d") can be captured by the single condition: (e") for x e Rd,
F
e 'co', and any optional time S of {A}, px[vFigFs+]
=
pxs,-kr,)
Px-a.s. on {S < co}.
Since (e") is often given as the primary defining property for a strong Markov family, we summarize this discussion with a theorem.
6.10 Theorem. Let X = IX ,Ft; t > 01 be a progressively measurable process on (SI, F), let {Px}xE Rd be a family of probability measures on (e, F), and let {0,}s>o be a family of F/g-measurable shift operators satisfying (5.14). Then X , (Si,
{Px} x E Rd is a strong Markov family if and only if (a), (b), and (e") hold.
6.11 Problem. Show that (e") is equivalent to the following condition:
(e"') For all x e gl, any bounded, gc,X-measurable random variable Y, and any optional time S of {gt}, we have Ex[Y 0 Osigs+] = Exs(Y),
Px-a.s. on {S < oo}.
(Note: If we write this equation with the arguments filled in, it becomes Ex [ Y 0 OsIgs+] (w) =
Y(w')Pxs(-)(')(dw'),
Px-a.e. w e {S <
where (Y Os)(w") g Y10s(.-)(w")).)
C. The Strong Markov Property for Brownian Motion The discussion on the strong Markov property for Brownian motion will require some background material on regular conditional probabilities. 6.12 Definition. Let X be a random variable on a probability space (e,F, P) taking values in a complete, separable metric space (5, M(S)). Let be a sub-a-field of F. A regular conditional probability of X given is a function Q: SZ x a(S) -> [0, 1] such that (i) for each wee, Q(co; ) is a probability measure on (S,M(S)), (ii) for each E e M(S), the mapping Q(w; E) is W-measurable, and (iii) for each E e .4(S), P[X e Elfl(co) = Q(w; E), P-a.e. w.
2.6. The Strong Markov Property and the Reflection Principle
85
Under the conditions of Definition 6.12 on X, (S2, .y, P), (S, AS)), and , a regular conditional probability for X given W exists (Ash (1972), pp. 264-265, or Parthasarathy (1967), pp. 146-150). One consequence of this fact is that
the conditional characteristic function of a random vector can be used to determine its conditional distribution, in the manner outlined by the next lemma.
6.13 Lemma. Let X be a d-dimensional random vector on (CI, 37, P). Suppose is a sub-a-field of F and suppose that for each w e SI, there is a function p(w; -): Rd C such that for each u e Rd, cp(w; u) = E[ei(".x)ifl(w),
P-a.e. w.
If, for each w, cp(co; ) is the characteristic function of some probability measure P`d on (Rd ,.4(Rd)), i.e.,
(p(p; u) =
f
el("P'(dx),
Rd
where i = .J -1, then for each T a (Rd), we have
P[X el- l5](w) = P'(F), P-a.e. w. PROOF. Let Q be a regular conditional probability for X given fixed u a Rd we can build up from indicators to show that (6.4)
cp(w; u) = E[ei("m151(w) = I
ei(u.x)Q(a); dx),
, so for each
P-a.e. w.
Rd
The set of w for which (6.4) fails may depend on u, but we can choose a countable, dense subset D of Rd and an event n EY; with p(n) = 1, so that (6.4) holds for every w e n and u e D. Continuity in u of both sides of (6.4) allows us to conclude its validity for every wen and ue Rd. Since a measure is uniquely determined by its characteristic function, we must have P' = Q(w; ) for P-a.e. w, and the result follows.
Recall that a d-dimensional random vector N has a d-variate normal distribution with mean p e Rd and (d x d) covariance matrix / if and only if it has characteristic function (6.5)
Eei(u.N) = e1(")-(1"uo;
u e Rd.
Suppose B = IB A; t > 0 }, (52, 37), {Px}Egid is a d-dimensional Brownian family. Choose u a Rd and define the complex-valued process M, A exp [i(u, B,) + -1102 ], 2
t > 0.
We denote the real and imaginary parts of this process by R, and l respectively.
2. Brownian Motion
86
6.14 Lemma. For each x e Rd, the processes {R A; t > 0} and {I ;; t
0}
are martingales on 4,, Pl. PROOF. For 0 < s < t, we have PEA '1,1.9';] = Ex[Ms exp (i(u,
A -A ) + t
= Ms Ex [exp (i(u, A
- A) + t
S
-
II
u112) F1
S 11U112)]
= Ms,
where we have used the independence of B, -Bs and Fs, as well as (6.5). Taking real and imaginary parts, we obtain the martingale property for IR .; t ..._. 01
0
and II A; t > 01.
6.15 Theorem. A d-dimensional Brownian family is a strong Markov family. A d-dimensional Brownian motion is a strong Markov process. PROOF. We verify that a Brownian family B = {B ..Ft; t .. 0 }, (SI, ..F), NP{x1,.ER.
satisfies condition (e) of Proposition 6.7. Thus, let S be an optional time of {F,}. In light of Problem 6.9, we may assume that S is bounded. Fix x e Rd. The optional sampling theorem (Theorem 1.3.22 and Problem 1.3.23 (i)) applied to the martingales of Lemma 6.14 yields, for Px-a.e. w e a Ex [exp(i(u, Bs+,))1..Fs,](w) = exp [i(u, Bs,,s (w)) )
2
II
ul12] .
Comparing this to (6.5), we see that the conditional distribution of Bs, given Fs+, is normal with mean Bsos)(w) and covariance matrix t/d. This proves (e).
0 We can carry this line of argument a bit further to obtain a related result. 6.16 Theorem. Let S be an a.s. finite optional time of the filtration {. } for the d-dimensional Brownian motion B = {B A; t > 0}. Then with Wt -'4- Bs+t - Bs, the process W = {147i, Fri; t > 0} is a d-dimensional Brownian motion, independent of Fs+.
PROOF. We show that for every n u e Rd, we have a.s. P: (6.6) E [exp (i kEi (uk, "tk - w,,,_,))
1, 0 < to <
< t < oo, and ui , ... ,
Fs,]= fl exp -Alit k=1 1
4-1)0402
;
thus, according to Lemma 6.13 and (6.5), not only are the increments {W,,, - W,k_, },",,_1 independent normal random vectors with mean zero and covariance matrices (tk - 4_04, but they are also independent of the a-field Fs+ This substantiates the claim of the theorem.
2.6. The Strong Markov Property and the Reflection Principle
87
We prove (6.6) for bounded, optional times S of {,F, }; the argument given in Solution 6.9 can be used to extend this result to a.s. finite S. Assume (6.6) holds for some n, and choose 0 < to < < t,, < t,.,. Applying the equality in the proof of Theorem 6.15 to the optional time S + t, we have
ElexPri(uni-i, Wt.,
(6.7)
Wt)71:F(s+i)+1
= Elexp[i(t4d+1, Bs+t,,)71,-*;-(s+to+1 exp [ -i(u+,, Bs+,)]
exp[-i(t.+1
- t.)11/1+1112],
P-a.s.
Therefore, n+1
E [exp
(i E (uk, w,k k=1
- wtk_i)) -Fs+
= E [exp Ci E (uk, wt, - Wtk_i )) k=1
E lexp(i(ud.+1, W,*, - Wt))1,(s+,)+ 1 -Fs+
.
= exp [ -2 (tn+ i - tn) II 140-1 112
E exp i E (4,, wtk -
2
Wtk_ 1)
k=1
n+1
1
k=1
2
= ri exp - - (tk - tk_ 1 ) 11 ukil 2
,
P-a.s.,
which completes the induction step. The proof that (6.6) holds for n = 1 is obtained by setting t = 0 in (6.7). In order to present a rigorous derivation of the density (6.3) for the passage time Tb in (6.1), a slight extension of the strong Markov property for rightcontinuous processes will be needed.
6.17 Proposition. Let X = {X
t > 0 },
{Px}x. Rd be a strong Markov
family, and the process X be right-continuous. Let S be an optional time of {,} and T an °T's., -measurable random time satisfying T(w) S(w) for all wen. Then, for any xe Rd and any bounded, continuous f : Rd R, (6.8)
Ex f(XT) I-Fs+ ha)) = (Ur(.)-s(a)(Xs(A(0)),
PROOF.
Px-a.e. co e IT <
For n > 1, let
S + -1 (P"(T - S)1 + 1), if T < oo, 2"
= oo,
if T = co,
2. Brownian Motion
88
so that Tn = S + k2-" when (k - 1)2-" < T -S < k2 -". We have Tn1 T on {T < co}. From (e') we have for k > 0, Ex [f (s+k2-.)1.975+] = (Uk/2.f )(Xs),
Px-a.s. on {S < co},
and Problem 6.9 (i) then implies Ex [ f (X T)1.-Cr's+]((o) = (UT,,(.)-smf)(Xs(o((0)),
Px-a.e. co e {T < co}.
The bounded convergence theorem for conditional expectations and the rightcontinuity of X imply that the left-hand side converges to Ex[f(Xr)l,s+] (w) as n -, co. Since (U, f )(y) = EV (X,) is right-continuous in t for every y e Rd, the right-hand side converges to (UT,,o_smf)(Xs(,)(0)). 6.18 Corollary. Under the conditions of Proposition 6.17, (6.8) holds for every bounded, .4([18°)/M(P0-measurable function f. In particular, for any F e .4(0V) we have for Px-a.e. w e{T < co): r [XT E F I Fs+ ] (0) = (UT(.)-s(w11r)(Xs(w)(0)).
PROOF. Approximate the indicator of a closed set F by bounded, continuous
functions as in the proof of Proposition 6.7. Then prove the result for any F e R(W), and extend to bounded, Borel-measurable functions. 6.19 Proposition. Let {B .97,; t > 0} be a standard, one-dimensional Brownian motion, and for b 0, let Tb be the first passage time to b as in (6.1). Then Tb has the density given by (6.3). PROOF. Because { -B ,57;,; t > 0} is also a standard, one-dimensional Brownian
motion, it suffices to consider the case b > 0. In Corollary 6.18 set S = Tb, ft
T=
oo
if S < t, if S > t,
and F = ( - oo, 6). On the set {T < co} -- {S < t}, we have Bs(w)(a)) = b and (Urm_s(a,)1,-)(Bs(w)(w)) = 1. Therefore, P °[Tb
< t, B, < b] =
f
P°[137- e I- 1 Fs+] dP"
1
= -2 P° [Tt, < t].
(rb
It follows that
P°['Ti, < t] = P° [Tb < t, Et, > b] + P°[7j, < t, B, < b] = P° [B, > b] + I P° [T,, < t], and (6.2) is proved.
6.20 Remark. It follows from (6.2), by letting t -, co, that the passage times are almost surely finite: P° [Tb < co] = 1.
2.7. Brownian Filtrations
89
6.21 Problem. Recall the notions of Poisson family and Poisson process from
Definitions 5.20 and 1.3.3, respectively. Show that the former is a strong Markov family, and the latter is a strong Markov process.
2.7. Brownian Filtrations In Section 1 we made a point of defining Brownian motion B = {B A; t > 0}
with a filtration {Ft} which is possibly larger than {,-,8} and anticipated some of the reasons that mandate this generality. One reason is related to the fact that, although the filtration {..Fel} is left-continuous, it fails to be right-continuous (Problem 7.1). Some of the developments in later chapters require either right or two-sided continuity of the filtration {A}, and so in this section we construct filtrations with these properties. Let us recall the basic definitions from Section 1.1. For a filtration {9;; t > 0}
on the measurable space (0.., ,), we set A+ = nE>0 A,, for t > 0, ._ = a(U,, A) for t > 0, moo_ = "0, and .94-x, = 6(Uto-5-4-t). We say that {A} is right- (respectively, left-) continuous if Ft+ = . (respectively, ._ = .Ft) holds for every 0 < t < co. When X = {X..F,x;t > 0} is a process on (SI, ..F), then left-continuity of {,x} at some fixed t > 0 can be interpreted to mean that
X, can be discovered by observing X,, 0 < s < t. Right-continuity means intuitively that if X, has been observed for 0 < s < t, then nothing more can be learned by peeking infinitesimally far into the future. We recall here that .Ftx = o-(Xs; 0 < s < t).
7.1 Problem. Let {X ..Fx; 0 < t < co} be a d-dimensional process. (i) Show that the filtration {Fix., } is right-continuous.
(ii) Show that if X is left-continuous, then the filtration {.97x}
is left-
continuous.
(iii) Show by example that, even if X is continuous, {Fix} can fail to be right-continuous and 19,, I can fail to be left-continuous. We shall need to develop the important notions of completion and augmen-
tation of .7-fields, in the context of a process X = {X Fix; 0 < t < co} with
initial distribution p on the space (0, ',I PI, where Pu[X0 en = p(F); re M(Rd). We start by setting, for 0
t
co,
Xf A {F g 0; 3 G eg-x with F g G, P"(G) = o}. 'V
will be called "the collection of fm-null sets" and denoted simply by .Aco.
7.2 Definition. For any 0 < t < co, we define
(i) the completion:." ''-- o-(F,x u XII,
(ii) the augmentation:. " A o-(,x u XI
and
2. Brownian Motion
90
of the a-field ,97,x under P. For t = oo the two concepts agree, and we set simply FP -4 o-GFccx u
The augmented filtration 1.3-711 possesses certain desirable properties, which will be used frequently in the sequel and are developed in the ensuing problems and propositions.
7.3 Problem. For any sub-a-field / of
= {F
define /1' =
u .A(g) and
3GE1 such that FAGE.Aig }.
Show that /" = Yt. We now extend Pg by defining Pg(F) A Pg(G) whenever FE1", and GE1 is chosen to satisfy FAGE.Aig. Show that the probability space (Q,
Pg) is complete:
F El", Pg(F)= 0, D g_ F
DElg.
7.4 Problem. From Definition 7.2 we have ..F,P Fp, for every 0 < t < co. Show by example that the inclusion can be strict.
7.5 Problem. Show that the a-field FP of Definition 7.2 agrees with g;f0 A a (U g;g)t>o
7.6 Problem. If the process X has left-continuous paths, then the filtration 1,11 is left-continuous.
A.
Right-Continuity of the Augmented Filtration for a Strong Markov Process
We are ready now for the key result of this section.
7.7 Proposition. For a d-dimensional strong Markov process X = {Xt, Fir; t > 0} with initial distribution it, the augmented filtration 1,971 is rightcontinuous.
Pg) be the probability space on which X is defined. Fix s > 0 and consider the degenerate, {,F,,x}-optional time S = s. With 0 < to < t, < < tn < s < t,,,, < < t. and ro, F, in *Rd), the PROOF. Let (SI,
strong Markov property gives [X,0 E ro, = 1{Xt0Ero,
Xt,,,E F, fix], Pg [X tn Ern,
G rdxs],
P" -a.s. It is now evident that Pg[X toc To, ..., X tn,E1-.1,97,x+] has an ,Fsx-
2.7. Brownian Filtrations
91
measurable version. The collection of all sets Fegl for which PP[FI9,x+] has an ,sx-measurable version is a Dynkin system. We conclude from the Dynkin System Theorem 1.3 that, for every FE the conditional probability PP[FI,Fsx,] has an .Fsx-measurable version.
Let us take now FEX _c .91; we have P[F137,x+] = IF, a.s. I", so IF has an 37sx-measurable version which we denote by Y. Because G A { Y = 1} e
.f and F AG g_ {1, Y} we have Feg: and consequently ..sf, g:; s > 0. Now let us suppose that FE. for every integer n > 1 we have Fes1,11,,,), as well as a set Gne.Fx, - ,-(1/n) such that F A G EXP. We define G A n:=1U c°=m G, and since G= n := m U ,T= m G,, for any positive integer M, ,°"Fr. To prove that Fe,f, it suffices to show F A GeXP. we have GeFs'f,_ Now
G\F g.
03
00
n =1
n=1
(U G)\F = U (G\F) e X"-
On the other hand,
F\G = F n (() 0 G,,)c = F n (0 () G,`,) m=1 n=m
co r
co
= m=i U [ F n ( n=m n
m=1 n=m CO
CO
Gf)] s U (F n G,`) = U (F\Gm)e .A7. m=1
m=1
It follows that F e
7.8 Corollary. For a d-dimensional, left-continuous strong Markov process X = {X ; t > 0} with initial distribution p, the augmented filtration 1.Ffl is continuous.
7.9 Theorem. Let B = {B .F,B; t > 0} be a d-dimensional Brownian motion with initial distribution p on (f2, .9";c0B, Pl. Relative to the filtration {,F,"}, {B t > 0} is still a d-dimensional Brownian motion. PROOF. Augmentation of a-fields does not disturb the assumptions of Definition 5.1.
7.10 Remark. Consider a Poisson process IN .F11; 0 < t < ool as in Definition 1.3.3 and denote by {,F,} the augmentation of {..Fin. In conjunction with Problems 6.21 and 7.3, Proposition 7.7 shows that { F,} satisfies the usual conditions; furthermore, {N, ,Ft; 0 < t < oo} is a Poisson process.
Since any d-dimensional Brownian motion is strongly Markov (Theorem 6.15), the augmentation of the filtration in Theorem 7.9 does not affect the strong Markov property. This raises the following general question. Suppose {X ,,x; t > 0} is a d-dimensional, strong Markov process with initial distri-
2. Brownian Motion
92
bution u on (SI, Flo`, P4). Is the process {X Ft"; t > 0} also strongly Markov?
In other words, is it true, for every optional time S of {Fil}, t > 0 and F e R(W), that (7.1)
1"[XS+t E r 1 -91+ ] = 1"[Xs+t E r 1 Xs],
PP-a.s. on {S < co }?
Although the answer to this question is affirmative, phrased in this generality the question is not as important as it might appear. In each particular case, some technique must be used to prove that {X ,97,x; t > 0} is strongly Markov in the first place, and this technique can usually be employed to establish the strong Markov property for {X F,P; t > 0) as well. Theorems 7.9 and 6.15 exemplify this kind of argument for d-dimensional Brownian motion. Nonetheless, the interested reader can work through the following series of exercises to verify that (7.1) is valid in the generality claimed.
In Exercises 7.11-7.13, X = {X ..f.fx ; 0 < t < co} is a strong Markov process with initial distribution it on (n, 'cox, P4).
7.11 Exercise. Show that any optional time S of {F,P} is also a stopping time of this filtration, and for each such S there exists an optional time T of {ix } with {S 0 TI e .AfP. Conclude that ..fli., = ,fl = .FT, where Ff is defined to be the collection of sets A e FP satisfying A nIT < tl e F,P, V 0 < t < co. 7.12 Exercise. Suppose that T is an optional time of {Fix }. For fixed positive integer n, define
on IT = col T. = 2 "'
on
{ic
- 1 < T < k} 2"
2"
-
Show that 1,, is a stopping time of {Fix}, and Ff- g o-(gq. u Al"). Conclude that 9-74! g o-(FIL u .A14). (Hint: Use Problems 1.2.23 and 1.2.24.)
7.13 Exercise. Establish the following proposition: if for each t > 0, f ea(ild), and optional time T of {Fix}, we have the strong Markov property (7.2)
PP[X,.. e FIFIQ = 134[XT.eFIXT], PP-a.s. on {T < co I,
then (7.1) holds for every optional time S of {Fr }. This completes our discussion of the augmentation of the filtration generated
by a strong Markov process. At first glance, augmentation appears to be a rather artificial device, but in retrospect it can be seen to be more useful and natural than merely completing each a -field Fix with respect to PTM. It is more
natural because it involves only one collection of PP-null sets, the collection we called ,A14, rather than a separate collection for each t > O. It is more useful because completing each a-field Fix need not result in a right-continuous filtration, as the next problem demonstrates.
93
2.7. Brownian Filtrations
7.14 Problem. Let {Bt; t > 01 be the coordinate mapping process on (C [0, 04
4(C[0, oo))), P° be Wiener measure, and Fi denote the completion of under P°. Consider the set F = lw e C[0, co); w is constant on [0, s] for some e > 01.
Show that: (i) P °(F) = 0, (ii) F e F:+, and (iii) F A.
B. A "Universal" Filtration The difficulty with the filtration Ign, obtained for a strong Markov process with initial distribution ft, is its dependence on it. In particular, such a filtration
is inappropriate for a strong Markov family, where there is a continuum of initial conditions. We now construct a filtration which is well suited for this case.
Let {X
0}5 (S2, 2,1), f DX
JxeOld be a d-dimensional, strong Markov family. For each probability measure on (Rd, ARd)), we define PP as in (5.2):
PP(F)=
f
Px(F)p(dx),
VFe
,
and we construct the augmented filtration {Fr } as before. We define (7.3)
A A n FtP, 0 < t < 00,
where the intersection is over all probability measures it on (Rd, M(Rd)). Note that Fix 0 < t < CO for any probability measure jt on (Rd, gi(Rd)); therefore, if {X t > 0} and {X t z 0} are both strongly Markovian
under Po, then so is {X gt; t > 0}. Because the order of intersection is interchangeable and {Ft"} is right-continuous, we have
= s>tn n p
Thus
=n ns>tgp= n
=
} is also right-continuous.
7.15 Theorem. Let B = IA, FtB; t > ol {Px}xco, be a d-dimensional Brownian family. Then {B gi; t > 0}, (S/, g), IPx}.E gad is also a Brownian family.
PROOF. It is easily verifed that, under each Px,{B
,; t
0} is a d-dimensional
Brownian motion starting at x. It remains only to establish the universal measurability condition (i) of Definition 5.8. Fix F e Ao. For each probability measure it on (Rd,M(Rd)), we have F e..F", so there is some G e.: F! with F AG eAr". Let N e satisfy FAG N and Pv(N)= 0. The functions g(x) g PX(G) and n(x) g PX(N) are universally measurable by assumption. Furthermore,
2. Brownian Motion
94
n(x)ii(dx) = r(N)= 0, The nonnegative functions hi (x) Px(F\G) and h2(x) so n = 0, Px(G\F) are dominated by n, so h, and h2 are zero ft-a.e., and hence h, and h2 are measurable with respect to .4(Rd)", the completion of AR") under /.1. Set f(x) Px(F). We have f(x) = g(x) + hi(x) - h2(x), so f is also .4(Rd)"measurable. This is true for every it; thus, f is universally measurable.
7.16 Remark. In Theorem 7.15, even if the mapping x 1--.Px(F) is Borel(c.f. Problem 5.2), we can conclude only its measurable for each F e universal measurability for each F e yam. This explains why Definition 5.8 was designed with a condition of universal rather than Borel-measurability.
C. The Blumenthal Zero-One Law We close this section with a useful consequence of the results concerning augmentation.
,),
7.17 Theorem (Blumenthal (1957) Zero-One Law). Let IB A; t is given by (7.3). If {Px })C Rd be a d-dimensional Brownian family, where F e go, then for each x e Rd we have either Px(F) = 0 or Px(F) = 1. PROOF. For F ego and each x e Rd, there exists G E .a)7011 such that Px(F A G) = 0.
But G must have the form G = {Boer} for some T eM(Rd), so Px(F) = r(G) = Px {Bo e F} = 1,-(x).
7.18 Problem. Show that, with probability one, a standard, one-dimensional Brownian motion changes sign infinitely many times in any time-interval [0, s], e > 0.
7.19 Problem. Let
Wt, .Ft; 0 < t < co} be a standard, one-dimensional
Brownian motion on (S1,
P), and define
Sb = inf It
0; Wt > bl;
b
O.
(i) Show that for each b > 0, P['T,, 0 Sb] = 0.
(ii) Show that if L is a finite, nonnegative random variable on (SI, which is independent of Fm, then {TL 0 SL} E.9 and P[7i,
P)
SL] = 0.
2.8. Computations Based on Passage Times In order to motivate the strong Markov property in Section 2.6, we derived the density for the first passage time of a one-dimensional Brownian motion from the origin to b 0 0. In this section we obtain a number of distributions
2.8. Computations Based on Passage Times
95
related to this one, including the distribution of reflected Brownian motion, Brownian motion on [0, a] absorbed at the endpoints, the time and value of the maximum of Brownian motion on a fixed time interval, and the time of the last exit of Brownian motion from the origin before a fixed time. Although derivations of all of these distributions can be based on the strong Markov property and the reflection principle, we shall occasionally provide arguments based on the optional sampling theorem for martingales. The former method yields densities, whereas the latter yields Laplace transforms of densities (moment generating functions). The reader should be acquainted with both methods.
A. Brownian Motion and Its Running Maximum Throughout this section, { W .Ft; 0 < t < (S/, 37;), {Px}xcgg will be a onedimensional Brownian family. We recall from (6.1) the passage times 0; W, = b};
Tb = inf {t
be
and define the running maximum (or maximum-to-date)
M, = max Ws.
(8.1)
8.1 Proposition. We have for t > 0 and a < b, b > 0: (8.2)
P°[W, e da, M, e db] =
2(2b - a)
2irt3
exp
{
(2b -
2)
da db.
2t
PROOF. For a < b, b > 0, the symmetry of Brownian motion implies that
(U,,1(_,1)(b) A Pb[W,,
2b - a]
a] =
A (Ut-s1[2b-a,0)(b);
0 < s < t.
Corollary 6.18 then yields P°[Wt < al
(Ut- Tb1[26-a,c0)(b)
a.s. P° on {T6 < t }. 2b Integrating both sides of this equation over {Tb < t} and noting that
= P° [Wt
{Tb 5 t} = {M, > b}, we obtain P° [ W,
a, M,
b] = P°[W, = P° [W,
Differentiation leads to (8.2).
2b - a,
26 - a] =
b] 1
2rct
e-x2/2t dx. 26-a
2. Brownian Motion
96
8.2 Problem. Show that for t > 0, (8.3)
P° [Mt e db] = P° [IWt 1 e db] = P° [Mt -W, e db] 2
/2nt
b > 0.
e-b212' db;
8.3 Remark. From (8.3) we see that (8.4)
t] = P° [Mt > 1)] =
P°
2
n'
/2,7c J
Cx2/2 dx;
b > 0.
By differentiation, we recover the passage time density (6.3):
P°[Tbe dt] =
(8.5)
,
b
e- 1'212' dt;
b > 0, t > 0.
2nt3
For future reference, we note that this density has Laplace transform Eoe-arb
(8.6)
e-b,f2;
b > 0, a > 0.
By letting t r co in (8.4) or a 10 in (8.6), we see that P° [T1' < co] = 1. It is clear from (8.5), however, that E° 'Tb = co.
8.4 Exercise. Derive (8.6) (and consequently (8.5)) by applying the optional sampling theorem to the {. }-martingale X, = exp{.1W, - IA2t}; 0
(8.7)
t < co,
> 0.
with A =
The following simple proposition will be extremely helpful in our study of local time in Section 6.2. 8.5 Proposition. The process of passage times T = {T, .97,-.4.; 0 < a < co} has
the property that, under P° and for 0 < a < b, the increment Tb -T, is independent of FT.., and has the density P° [Tb - 'Tae dt] =
b -a e-(6-0212t dt; ,
/ 2nt3
0 < t < co.
In particular, (8.8)
Eo[e-ct(Tb-T.)1,T.+]
e-(b-a) .12a;
0.
PROOF. This is a direct consequence of Theorem 6.16 and the fact that Tb -7; = inf ft 0; WT.+t WT. = b al.
2.8. Computations Based on Passage Times
97
B. Brownian Motion on a Half-Line When Brownian motion is constrained to have state space [0, oo), one must specify what happens when the origin is reached. The following problems explore the simplest cases of absorption and (instantaneous) reflection. 8.6 Problem. Derive the transition density for Brownian motion absorbed at the origin {W, To, ,Ft; 0 < t < 00 }, by verifying that (8.9)
I' [W, e dy, T, > t] = p _(t; x, y)dy -41- [p(t; x, y) - p(t; x, -y)] dy;
t > 0, x, y > O.
8.7 Problem. Show that under P°, reflected Brownian motion 'WI A {114I, A; 0 < t < co} is a Markov process with transition density (8.10)
P° El wt+sl e 411 Wt I = x] = p +(s; x, y) dy
A [p(s; x, y) + p(s; x, - y)] dy;
s > 0, t _.. 0 and x, y _. O.
8.8 Problem. Define Y, A M, - KO < t < co. Show that under P°, the process Y = {Y,, .9,; 0 < t < co} is Markov and has transition density (8.11)
P° [Y,+, e dy I Y, = x] = p +(s; x, y) dy;
s > 0, t > 0 and x, y .. 0.
Conclude that under P° the processes IWI and Y have the same finitedimensional distributions.
The surprising equivalence in law of the processes Y and I WI was observed by P. Levy (1948), who employed it in his deep study of Brownian local time (cf. Chapter 6). The third process M appearing in (8.3) cannot be equivalent in law to Y and I WI, since the paths of M are nondecreasing, whereas those
of Y and I WI are not. Nonetheless, M will turn out to be of considerable interest in Section 6.2, where we develop a number of deep properties of Brownian local time, using M as the object of study.
C. Brownian Motion on a Finite Interval In this subsection we consider Brownian motion with state space [0, a], where
a is positive and finite. In order to study the case of reflection at both endpoints, consider the function co: IR -+ [0, a] which satisfies co(2na) = 0, cp((2n + 1)a) = a; n = 0, ± 1, ±2, ..., and is linear between these points.
8.9 Exercise. Show that the doubly reflected Brownian motion {9(W), A; 0 < t < col satisfies X
Pq9(W,) e dy] = E p +(t; x, y + 2na) dy; 0 < y < a, 0 < x < a, t > O. n = -x
2. Brownian Motion
98
The derivation of the transition density for Brownian motion absorbed at 0 and a i.e., { WtA To A Ta, d't; 0 < t < co}, is the subject of the next proposition.
8.10 Proposition. Choose 0 < x < a. Then for t > 0, 0 < y < a: (8.12)
> t] = j p_(t; x, y + 2na) dy.
Px[W, a dy, To A
n=-3c
PROOF. We follow Dynkin & Yushkevich (1969). Set a0 A 0, to -4 To, and define recursively an -A inf{t > Tn_1; 147, = in = inf It > an; W, = 0 }; n = 1, 2, .... We know that Px[to < co] = 1, and using Theorem 6.16 we can show by induction on n that an - T_1 is the passage time of the standard Brownian to a, T -an is the passage time of the standard motion -a, and the sequence of differences a, - To, Brownian moton W.". -W ri - at, a2 - t1, t2 - a21 consists of independent and identically distributed random variables with moment generating function e-°/' (c.f. (8.8)). It follows that to - To, being the sum of 2n such differences, has moment generating function e-2na,hat, and so
-
Px[T - To
t] = P °[T2n° < t].
We have then
lim Px[T < t] = 0; 0 < t < GO.
(8.13)
n-oo
For any y e (0, co), we have from Corollary 6.18 and the symmetry of Brownian motion that = Px[W,
Px[W,
-yl",.+]
on {T < t },
and so for any integer n > 0, (8.14)
13' [W,
Similarly, for y
y, T
t] = Px[W,
- y, T.
t] = Px[W,
-y, an
t].
- 09, a), we have
= Px[W, > 2a -y1,97,+]
Px[W,
on {an 5_ t },
whence (8.15)
Px[W,
y, an
t] = Px[W,
2a - y, an 5_ t]
= Px[W, > 2a - y, tn_i < t];
n > 1.
We may apply (8.14) and (8.15) alternately and repeatedly to conclude, for 0 < y < a, n > 0: Px CW1
to
Px[Wi
)7, an
= Px[W, < -y - 2na], = Px[Wr y- 2na],
and by differentiating with respect to y, we see that
2.8. Computations Based on Passage Times
Px[Hie dy,
(8.16)
to
99
< t] = p(t; x, -y - 2na)dy,
[W, e dy, on 5 t] = p(t; x, y - 2na)dy.
(8.17)
Now set no = 0, Po = T., and define recursively nn
= inf {t > p,,_1; 14/, = 0}, p = inf { t > re.; W = a}; n = 1, 2,
....
We may proceed as previously to obtain the formulas
t] = 0; 0 < t <
lim Px[p
(8.18)
11-.09
Px[W, e dy, p
(8.19) (8.20)
Px [ W, e dy, nn
t] = p(t; x, -y + (2n + 1)a) dy; 0 < y < a, n
0,
< t] = p(t; x, y + 2na) dy; 0 < y < a, n > 0.
It is easily verified by considering the cases To < Ta and To > T. that r_1 V p_, = cr. A n and on v 7r,, = to A p,,; n 1. Consequently, Px[Wte dy, r_1 ^ Pn -i < t] = 13' [W, e dy, -c_, < t] + Px[W, e dy, Pn -i < t] - Px[W, e dy,
(8.21)
on A nn
< t],
and
t] = Px[W, e dy, a
Px[Wtedy, a A 7c.
(8.22)
- Px [W te dy,
t] + Px[Wte dy, 7t
Ap
t]
t].
Successive application of (8.21) and (8.22) yields for every integer k
1:
(8.23)
Px[141,e dy, To A Po < t] = E IPx[Wte dy, t_1
t] + Px[Wte dY, Pa-i < t]
n=1
-
[147, e dy, on < t] - Px[Wie dy,
nn
< t]l
+ Px[W, e dy, t k n Pk < t].
Now we let k tend to infinity in (8.23); because of (8.13), (8.18) the last term converges to zero, whereas using (8.16), (8.17) and (8.19), (8.20), we obtain from
the remaining terms: Px[Wie dy, To A
Ta
> t] = Px[Wie dy] - Px[Wie dy, To A Po < t] CO
= E p_(t; x, y + 2na)dy; 0 < y < a, t > O. 8.11 Exercise. Show that for t > 0, 0 < x < a: (8.24)
Px[ To A
e dt] =
,1
E
+ x) exp
(2na
x)2 1
\/2/rt3 n=
+ (2na + a - x) ex p jl
(2na + a - x)2 1 ] 2t
dt.
2. Brownian Motion
100
It is now tempting to guess the decomposition of (8.24): (8.25)
Px [To E dt, To < To] =
1
(2na + x)2 dt, 2t
(2na + x) exp 1
\/27rt3 n=-co
(8.26)
e dt, 7; < To] =
1
2nt3
e°
E (2na + a
(2na + a - x)2
x) exp
2t
dt.
Indeed, one can use the identity (8.6) to compute the Laplace transforms of the right-hand sides; then (8.25), (8.26) are seen to be equivalent to (8.27)
Ex [e'T°1(ro
(8.28)
Ex [e'Tal{Ta
sinh((a - x)
2a)
sinh(a) sinh(x,/2a) sinh(a.,./244).
We leave the verification of these identities as a problem. Note that by adding (8.27) and (8.28) we obtain the transform of (8.24): cosh ((x (8.29)
- 2)
) 2a)
Ex[e-'('°^2.°)] = cosh (a2
f2a)
This provides an independent verification of (8.24).
8.12 Problem. Derive the formulas (8.27), (8.28) by applying the optional sampling theorem to the martingale of (8.7). 8.13 Exercise. Show that for a > 0, 0 < x < a:
<7;;]
a -x
=x a
8.14 Problem. Show that Ex(To A T.) = x(a - x); 0 < x < a.
D. Distributions Involving Last Exit Times Proposition 8.1 coupled with the Markov property enables one to compute distributions for a wide variety of Brownian functionals. We illustrate the method by computing some joint distributions involving the last time before t that the reflected Brownian motion Y of Problem 8.8 is at the origin. Note that such last exit times are not stopping times.
2.8. Computations Based on Passage Times 8.15
101
Proposition. Define
0, A sup{0 < s 5 t; Ws = Mt}.
(8.30)
Then for a e IR, b
a +, 0 < s < t, we have:
(8.31)
b(b - a)
e[W,eda, Mte db, 0,e ds] -
,
n / s3 (t
PROOF. For (8.32)
exp
< Ms
2s
2(t - s)
and 0 < s < we have
b > 0, s > 6 > 0, x > 0, a < b
P°
(b -a )2 db ds.
s)3
-
b2
t,
< b + 6, Ws e b - dx,
Lb
max W.
.,.c,
< b, 147,e da]
< P°[b < M, < b + 6, 0, < s, Ws e b - dx, W, e da]
< P°[b < Ms
< b + 6, Ws e b - dx,
max W s5 s,
< b + E, W, e dd.
Divide by 6 and let 6 i 0, s i 0 (in that order). The upper and lower bounds in the preceding inequalities converge to the same limit, which is (8.33)
P°[Mte db, 0, < s, Ws e b - dx, 141,e da]
Ws e b - dx,
= P° [Ms
Ws e b - dx] Pb-x[M,_,
e db,
_
lc / s3 (t
- exp
(8.2)
.L
x)exp
a2}
dx da db,
and
b(t - s) + (a - b)s
n,
=
a
,
S(t - s) t
t
In terms of 1(z) A (1/,/27r)Sz_co e-x212 (b +
f
2f
2o:2
where we have used
2t
(x + /24] 2
b, 141,_se da]
(2b - a)21
2a2
- s)
{
< b, W, e da]
p+)2 [expt (x +
b + x
=
..1:
max W. Ss.,,,
= P°[Mse db,
(x ±
11-1-)} 2
we may now evaluate the integrals
dx
dx = a2e-P2t/2°2
2a2
+
-s (b + (b
-
a))a.12fr - 40 (-111-),
a
t
and so integrating out x in (8.33) and using the equality p2+
(8.34)
2a2
+
b2
(b + (b - a))2 =.
2t
2s
+
(b - a)2 2(t - s)'
2. Brownian Motion
102
we arrive at the formula P° [M, e db,
2
[0( 1)(26 2iit 3
E da] -
s,
2t
o-
-
- a)21.
(2b
a) exp
(- l'±=) a exp {- cf -}1 da db 2
o-
Note that a / as( - 1.4/a) = 112o-((b/s) ± (b - a)/(t - s)), and so
-a p(
s, W,E da]
e db, 0,
as
b(b - a) { b2 exp 2s Tr,/ s3(t - s)3
(b - a)2} da db ds. 2(t - s)
8.16 Remark. If we define A inf {0 < s < t; WS = AI to be the first time W attains its maximum over [0, t], then (8.32) is still valid when A is replaced by
A. Thus, et and A have the same distribution, and since Of < e we see that P°[A = A] = 1. In other words, the time at which the maximum over [0, t] is attained is almost surely unique. 8.17 Problem. Show that for b > 0, 0 < s < t: [M, E db, 0,e ds] =
b
its3(t - s)
e'212s db ds,
whence ds
P° [et e ds] =
s(t - s)
,
P°[M,E dbl
=
=
b
2
12s db.
In particular, the conditional density of M, given A does not depend on t. We say that 0, obeys the arc-sine law, since
P°[0, < s]
2
- arcsin -; 0 < s < t, t > 0.
8.18 Problem. Define the time of last exit from the origin before t by (8.35)
y,
sup{0
s
t; W = 0}.
Show that y, obeys the arc-sine law; i.e.,
P°[y, e ds] -
ds
/ s(t - s)
;
0 < s < t.
(Hint: Use Problem 8.8.)
8.19 Exercise. With y, defined as in (8.35), derive the quadrivariate density
2.9. The Brownian Sample Paths
103
PqW,e da, M,Edb, y,e ds, 0,e du]
-2ab 2 (2nu(s - u)(t - s))312
a
exp - 2u(sub2- u)
2(t-2
da db ds du;
0 < u < s < t, a < 0 < b.
2.9. The Brownian Sample Paths We present in this section a detailed discussion of the basic absolute properties
of Brownian motion, i.e., those properties which hold with probability one (also called sample path properties). These include characterizations of "bad" behavior (nondifferentiability and lack of points of increase) as well as "good" behavior (law of the iterated logarithm and Levy modulus of continuity) of the
Brownian paths. We also study the local maxima and the zero sets of these paths. We shall see in Section 3.4 that the sample paths of any continuous martingale can be obtained by running those of a Brownian motion according to a different, path-dependent clock. Thus, this study of Brownian motion has much to say about the sample path properties of much more general classes of processes, including continuous martingales and diffusions.
A. Elementary Properties We start by collecting together, in Lemma 9.4, the fundamental equivalence transformations of Brownian motion. These will prove handy, both in this section and throughout the book; indeed, we made frequent use of symmetry in the previous section. 9.1 Definition. An Rd-valued stochastic process X = {X,; 0 < t < co} is called < t, < co, Gaussian if, for any integer k > 1 and real numbers 0 < t, < t2 <
the random vector (X,,, X,,, , X,,) has a joint normal distribution. If the distribution of (X,,,,, , X,,,k) does not depend on t, we say that the process is stationary. The finite-dimensional distributions of a Gaussian process X are determined by its expectation vector m(t) EX,; t > 0, and its covariance matrix
p(s, t) A E[(X, - m(s))(X, - m(t))T]; s, t
0,
where the superscript T indicates transposition. If m(t) = 0; t > 0, we say that X is a zero-mean Gaussian process.
9.2 Remark. One-dimensional Brownian motion is a zero-mean Gaussian process with covariance function (9.1)
p(s,t) = s
t;
s, t
0.
2. Brownian Motion
104
Conversely, any zero-mean Gaussian process X = {X ,x; 0 S t < col with a.s. continuous paths and covariance function given by (9.1) is a onedimensional Brownian motion. See Definition 1.1.
Throughout this section, W = {W, gt; 0 5 t < co } is a standard, onedimensional Brownian motion on (SI, Sri, P). In particular Wo = 0, a.s. P. For fixed w e SI, we denote by W(w) the sample path t W(co).
93 Problem (Strong Law of Large Numbers). Show that (9.2)
lim
-Wt
t-,a)
t
= 0,
a.s.
(Hint: Recall the analogous property for the Poisson process, Remark 1.3.10.)
9.4 Lemma. When W = {W,
0 < t < co} is a standard Brownian motion,
so are the processes obtained from the following "equivalence transformations":
(i) Scaling: X = {X Fet; 0 < t < co} defined for c > 0 by
=
(9.3)
(ii) Time-inversion: Y = { (9.4)
1
Wc,
0
..F,1; 0 < t < co } defined by ftWil,
= 10
0 < t < oo, t = O.
(iii) Time-reversal: Z = {Z ..57",,z; 0 < t < co} defined for T > 0 by (9.5)
Z, = WT - WT_t
0 < I < 'T.
(iv) Symmetry: -W = -W gi; 0 < t < col. PROOF. We shall discuss only part (ii), the others being either similar or completely evident. The process Y of (9.4) is easily seen to have a.s. continuous paths; continuity at the origin is a corollary of Problem 9.3. On the other hand, Y is a zero-mean Gaussian process with covariance function 1
E(Ys Y,) = st (-s
1
A
t) = S A t; S, t > 0
and the conclusion follows from Remark 9.2.
9.5 Problem. Show that the probability that Brownian motion returns to the origin infinitely often is one.
B. The Zero Set and the Quadratic Variation We take up now the study of the zero set of the Brownian path. Define (9.6)
2's (t, OE [0, CO x SI; W,(w) = 0 1,
2.9. The Brownian Sample Paths
105
and for fixed we f2, define the zero set of W.(w):
Y, s {0 < t < co; W(w) = 0 }.
(9.7)
9.6 Theorem. For P-a.e. wen, the zero set (i) has Lebesgue measure zero, (ii) is closed and unbounded,
(iii) has an accumulation point at t = 0, (iv) has no isolated point in (0, co), and therefore (v) is dense in itself.
PROOF. We start by observing that the set .2° of (9.6) is in ,4([0, co)) 0 .F, because W is a (progressively) measurable process. By Fubini's theorem,
E[meas(.2°,)] = (meas x P)(t) =
f
P[W, = 0]dt = 0,
and therefore meas(Y,)) = 0 for P-a.e. w e f2, proving (i); here and in the sequel, meas means "Lebesgue measure." On the other hand, for P-a.e. w e f2 the mapping t W(w) is continuous, and s, is the inverse image under
this mapping of the closed set {0 }. Thus, for every such w, the sets, is closed, is unbounded (Problem 9.5), and has an accumulation point at t = 0 (Problem 7.18). For (iv), let us observe that {w en.; 2°,,, has an isolated point in (0, co)} can be written as (9.8)
fa.) e SI; there is exactly one s e (a, b) with Ws(w) = 01
U
a,beQ
0
where Q is the set of rationals. Let us consider the family of almost surely finite optional times /3,
inf{s > t; Ws = 0};
t > O.
According to (iii) we have fio = 0, a.s. P; moreover, = inf{s > A((4); Ws(w) = 0} = /33(w) + inf {s > 0; Ws+13,(0)(w) -
=
= /3,(w)
-
for P-a.e. w ea because { Ws+p, 0 < s < co} is a standard Brownian motion (Theorem 6.16). Therefore, for 0 < a < b < co, {w e SI; there is exactly one s e (a, b) with W(w) = 01 g {w e f2; fia(w) < b and flt,a( )(w) > b}
has probability zero, and the same is then true for the union (9.8).
9.7 Remark. From Theorem 9.6 and the strong Markov property in the form of Theorem 6.16, we see that for every fixed b e R and P-a.e. w e C2, the level set
2. Brownian Motion
106 .2",,,(b)
{0
t < oo; Wi(w) = b}
is closed, unbounded, of Lebesgue measure zero, and dense in itself. The following problem strengthens the result of Theorem 1.5.8 in the special case of Brownian motion. 9.8 Problem. Let {11},T=1 be a sequence of partitions of the interval [0, t] with limn, 111-1n11 = 0. Then the quadratic variations v,(2)(1-1.) A
k=1
I Wks - WC,12
of the Brownian motion W over these partitions converge to t in L2, as n cc. If, furthermore, the partitions become so fine that E`°_.1 1111,11 < co holds, the preceding convergence takes place also with probability one.
C. Local Maxima and Points of Increase As discussed in Section 1.5, one can easily show by using Problem 9.8 that for almost every we Si, the sample path W.(w) is of unbounded variation on every
finite interval [0, t]. In the remainder of this section we describe just how oscillatory the Brownian path is. 9.9 Theorem. For almost every we O., the sample path W.(w) is monotone in no interval.
PROOF. If we denote by F the set of w ell with the property that W(w) is monotone in some interval, we have
F=
U
{(0e0; W.(w) is monotone on [s,t] },
eQ
0 < s
since every nonempty interval includes one with rational endpoints. Therefore, it suffices to show that on any such interval, say on [0, 1], the path W(w)
is monotone for almost no w. By virtue of the symmetry property (iv) of Lemma 9.4, it suffices then to show that the event A -4 {w e
is in
W.(w) is nondecreasing on [0, l] }
and has probability zero. But A = nr-i A, where n-1 An
n {a) ea
i=o
has probability P (A ) = P(A) = 0.
- wi,n(w) P[W0_01,, -
0} e
0] = 2 -". Thus, P(A) <
2.9. The Brownian Sample Paths
107
In order to proceed with our study of the Brownian sample paths, we need a few elementary notions and results concerning real-valued functions of one variable.
9.10 Definition. Let f: [0, co) -Fl be a given function. A number t > 0 is called
(i) a point of increase of size 6, if for given .5 > 0 we have f(s) < f(t) < f(u) for every s e [(t - S) +, t) and u E (t, t + 6]; a point of strict increase of size & if the preceding inequalities are strict; (ii) a point of increase, if it is a point of increase of size 6 for some S > 0; a point of strict increase, if it is a point of strict increase of size 6 for some 6 > 0; (iii) a point of local maximum, if there exists a number 6 > 0 with f(s) < f(t)
valid for every s e[(t t + (5]; and a point of strict local maximum, if there exists a number 6 > 0 with f(s) < f(t) valid for every SE - (5)+ t + oh{t} 9.11 Problem. Let f: [0, co)
ER be continuous.
(i) Show that the set of points of strict local maximum for f is countable. (ii) If f is monotone on no interval, then the set of points of local maximum for f is dense in [0, co). 9.12 Theorem. For almost every wen, the set of points of local maximum for the Brownian path W(w) is countable and dense in [0, co), and all local maxima are strict. PROOF. Thanks to Theorem 9.9 and Problem 9.11, it suffices to show that the set
A = {w e SI; every local maximum of W.(w) is strict}
includes an event of probability one. Indeed, A includes the (countable) intersection of events of the type A,,,...,,4 A
(9.9)
e
max 14/,(w) - max Wt(w) 0 0 },
13(5t4
ittst2
taken over all quadruples (t, , t2, t3, t4) of rational numbers satisfying 0 < t1 < t2 < t3 < t,< co. Therefore, it remains to prove that for every such quadruple, the event in (9.9) has probability one. But the difference of the two random variables in (9.9) can be written as
(Wt, -
+ min [147t2(w) - W(w)] + max [W(w) - Wt,(u))], t35..tst4 titst2
and the three terms appearing in this sum are independent. Consequently,
2. Brownian Motion
108
P[A,,
ta]
Jo
f°-00 P[N -
P[ because P[W,, - Wi3
1,1i
0 x + y]
P[
min (147,2 -
e dx]
max (W, - W,3)edy] = 1
0
x + y] = 1.
Let us now discuss the question of occurrence of points of increase on the Brownian path. We start by observing that the set A = {(t,00)e [0, co) x f2; t is a point of increase of W(co)}
is product-measurable: A e.4([0, oo)) ®F. Indeed, A can be written as the countable union A = Um =1 A(m), with A(m)
{(t, co)e [0, co) x LI;
max
0-(1/m))-st
Ws(w) = WW(w) =
min Ws(co)}, tsst+0./m)
and each A(m) is in B([0, co)) 0 F. We denote the sections of A by A, -A {co e i2; (t, co)e A},
A. g {t e [0, co); (t, co)e Al,
and A,(m), A.(m) have a similar meaning. For 0 < t < co,
P[At(m)] < P[Ws+t -H7t. 0; V s e [0, l/m]] = 0 because {W +, -
s > 0} is a standard Brownian motion (Problem 7.18);
now A, = Um =1 A,(m) gives also (9.10)
P(A,) = 0; 0 < t < co
as well as
meas(A.) dP = (meas x P)(A) =
f
P(At) dt = 0
from Fubini's theorem. It follows that P [co e meas(A.) = 0] = 1. The ques= 0] = 1, tion is whether this assertion can be strengthened to P [co e or equivalently (9.11)
P[co e i2; the path W(co) has no point of increase] = 1.
That the answer to this question turns out to be affirmative is perhaps one of the most surprising aspects of Brownian sample path behavior. We state this result here but defer the proof to Chapter 6.
9.13 Theorem (Dvoretzky, Erdos, & Kakutani (1961)). Almost every Brownian sample path has no point of increase (or decrease); that is, (9.11) holds.
9.14 Remark. We have already seen that almost every Brownian path has a dense set of local maxima. If T(w) is a local maximum for W.(co), then one
2.9. The Brownian Sample Paths
109
might imagine that by reflection (replacing W,(w) - WT(,,,)(w) by -(W,(w) w,(,,,)(w)) for t > T(w)), one could turn the point T(w) into a point of increase for a new Brownian motion. Such an approach was used successfully at the beginning of Section 6 to derive the passage time distribution. Here, however, it fails completely. Of course, the results of Section 6 are inappropriate in this context because T(co) is not a stopping time. Even if the filtration {.:97,} is rightcontinuous, so that {w e SI; W(w) has a local maximum at t} is in for each
t > 0, it is not possible to define a stopping time T for {F,} such that W.(0)) has a local maximum at T(w) for all w in some event of positive probability. In other words, one cannot specify in a "proper way" which of the numerous times of local maximum is to be selected. Indeed, if it were possible to do this, Theorem 9.13 would be violated. 9.15 Remark. It is quite possible that, for each fixed t > 0, a certain property
holds almost surely, but then it fails to hold for all t > 0 simultaneously on an event whose probability is one (or even positive!). As an extreme and rather trivial example, consider that P[co e W,(w) # 1] = 1 holds for every W,(co) # 1, for every t c [0, co)] = 0. The point here 0 < t < co , but P [a) is that in order to pass from the consideration of fixed but arbitrary t to the consideration of all t simultaneously, it is usually necessary to reduce the latter consideration to that of a countable number of coordinates. This is precisely the problem which must be overcome in the passage from (9.10) to (9.11), and the proof of Theorem 9.13 in Dvoretzky, Erdos & Kakutani (1961) is demandingt because of the difficulty of reducing the property of "being a point of increase" for all t > 0 to a description involving only countably many coordinates. We choose to give a completely different proof of Theorem 9.13 in Subsection 6.4.B, based on the concept of local time. We do, however, illustrate
the technique mentioned previously by taking up a less demanding question, the nondifferentiability of the Brownian path.
D. Nowhere Differentiability 9.16 Definition. For a continuous function f: [0, co) (9.12)
f(t) = lim
R, we denote by
f(t + h) - f(t) h
+
the upper (right and left) Dini derivates at t, and by (9.13)
D +f(t) = lim
f(t + h) - f(t)
h-40+
h
the lower (right and left) Dini derivates at t. The function f is said to be See, however, Adelman (1985) for a simpler argument.
2. Brownian Motion
110
differentiable at t from the right (respectively, the left), if D+ f(t) and D f(t) (respectively, D- f(t) and D_ f(t)) are finite numbers and equal. The function f is said to be differentiable at t > 0 if it is differentiable from both the right and the left and the four Dini derivates agree. At t = 0, differentiability is defined as differentiability from the right. 9.17 Exercise. Show that for fixed t e [0, co),
P[co eft D+ No)) = co and D÷W,(w)= -cc] = 1.
(9.14)
9.18 Theorem (Paley, Wiener & Zygmund (1933)). For almost every w E Si, the Brownian sample path W,(w) is nowhere differentiable. More precisely, the set
{wet; for each t e [0, co), either D+14(,(w) = co or D,147,(w) =
(9.15)
contains an event F e F with P(F) = 1.
9.19 Remark. At every point
t
of local maximum for W.(w) we have
Wg(o)) < 0, and at every point s of local minimum, D,W5(w) > 0. Thus, the "or" in (9.15) cannot be replaced by "and." We do not know whether the set of (9.15) belongs to "cow.
PROOF. It is enough to consider the interval [0, 1]. For fixed integers j > 1, k > 1, we define the set (9.16)
Aik
= U
to.0,11
n
{wen; iw,,,,(w) - wt(coi
he[0.11k1
Certainly we have
{we); -co < D+147,(w)
147,(w) < co, for some t e [0, 1]} = U U 1=1 k=1
and the proof of the theorem will be complete if we find, for each fixed j, k, an event C e ,F with P(C) = 0 and Aik C.
Let us fix a sample path co e Aik, i.e., suppose there exists a number t e [0,1] with 1147,(w) - 14/,(w)1 < jh for every 0 < h < 1/k. Take an integer n > 4k. Then there exists an integer i, 1 < i < n, such that (i - 1)/n < t < i/n, and it is easily verified that we also have ((i + v)/n)) - t < (v + 1) /n 1/k, for v = 1, 2, 3. It follows that I
W1 +11/n(w) - Wi/n(a))1
W, +11 /l(w) - Wt(0)1 + Wirn(w) - W(w)1
< 2jn- -nj
3j n
The crucial observation here is that the assumption w e Aik provides information about the size of the Brownian increment, not only over the interval [i /n, (i + 1)/n], but also over the neighboring intervals [(i + 1) /n, (i + 2)/n] and
2.9. The Brownian Sample Paths
111
[(i + 2)/n, (i + 3)/n]. Indeed, Wo+2)/n(0) - W(i+l)in(W)1
WII+1K+1)In
1W(i+3)In(0) - WO-F 2)1n(a*
WtI +IWO+ 2)In
3j
2j
n
n
Hi d <41:1-
n
n
=
=
15 , n
n
.
Therefore, with 3
Cr A
(.)
2V ± 1 W(i+v-1)/n(0)1
GO; I W(i+v)In(W)
=1
n
j}
we have observed that AJk g_ U7-,1 On) holds for every n > 4k. But now j1(Wy+v)In
Zv;
W(t+v-1)1n)
V = 1, 2, 3
are independent, standard normal random variables, and one can easily verify
the bound P[IZI < a] < a. It develops that
P(Cr)) <
(9.17)
We have AJk
05/3
= 1, 2, ..., n.
;
n3/2
C upon taking n
CA
(9.18)
n u on) G.F, n=4k i=1
and (9.17) shows us that P(C) < inf> 4k P(U7=1 Cin)) = 0.
0
9.20 Remark. An alternative approach to Theorem 9.18, based on local time, is indicated in Exercise 3.6.6.
9.21 Exercise. By modifying the preceding proof, establish the following stronger result: for almost every cu e S2, the Brownian path 14/.(u)) is nowhere Holder-continuous with exponent y > I. (Hint: By analogy with (9.16), consider the sets (9.19) AJk A {w e Lr2;
I Wt,(w) - W(w)1 < jhY for some t e [0,1] and all h e [0,1 /1c]l
and show that each
is included in a P-null event.)
E. Law of the Iterated Logarithm Our next result is the celebrated law of the iterated logarithm, which describes the oscillations of Brownian motion near t = 0 and as t o o. In preparation for the theorem, we recall the following upper and lower bounds on the tail of the normal distribution.
2. Brownian Motion
112
9.22 Problem. For every x > 0, we have x
(9.20)
1
-x2/2 < j.co e -"2
1+x
< -e-x212. x
x
9.23 Theorem. (Law of the Iterated Logarithm (A. Hin6in (1933))). For almost every w e O., we have Wt((0)
(i) lim .%/ 2t
log log(1 /t)
Wi(w)
(iii) lim
t-.
(ii) lim
= 1,
Wt(w)
Wi(w)
(iv) lira
= 1,
1,
.,./2t log log(l/t)
=
t-. .\/2t log log t
2t log log t
1.
9.24 Remark. By symmetry, property (ii) follows from (i), and by time-inversion,
properties (iii) and (iv) follow from (i) and (ii), respectively (cf. Lemma 9.4). Thus it suffices to establish (i).
PROOF. The submartingale inequality (Theorem 1.3.8 (i)) applied to the exponential martingale {X 5c; 0 t < oo} of (8.7) gives for A > 0,11 > 0: (9.21)
P max Ws - -sA2 > / 3 = P max Xs > elfl 0.,5..t
< e-lfl.
ri._.5.i
With the notation h(t)-
72t log log(l/t) and fixed numbers 0, .5 in (0, 1), we choose A --= (1 + .5)0-"h(0"), 13 = lh(0"), and t = 0" in (9.21), which becomes
P[
max (Ws o<s<en
-
>
<
1
(n log(1/0))' ;
n > 1.
The last expression is the general term of a convergent series; by the BorelCantelli lemma, there exists an event Ste,, e of probability one and an integer-valued random variable No, so that for every w e 0 we have max [Ws(w)
1 + s0-"h(0")1< 1
-2 h(0");
2
o<s<0.
n
110,(o))
V]:
Thus, for every t e
W(w) < max Ws(co) < (1 + -2) h(19")
(1 +
-2) O-112h(t).
ct <sson
Therefore,
sup
On +1
On
h(t)
< (1 ±
-6) 2
n> Noo(w)
holds for every w e Sleo, and letting n T oo we obtain -lim "ih(d° t)
< (1 + -6)0-112, 2
a.s. P.
2.9. The Brownian Sample Paths
113
By letting 610, 0 i 1 through the rationals, we deduce W
- < 1. tt.o h(t) - '
lim
(9.22)
a.s. P.
In order to obtain an inequality in the opposite direction, we have to employ the second half of the Borel-Cantelli lemma, which relies on independence. We introduce the independent events
.11 -0 h(0")}; n= 1, 2, ... ,
A = { Won - We.,
again for fixed 0 < 0 < 1. Inequality (9.20) with x = .,./2 log n + 2 log log(1/0) provides lower bounds on the probabilities of these events:
P(iln)= P[Wen -We -1 > ,./0" - On+1
xi>
e- x212
>
const.
n>
-
.,./27c(x + 1/x)
n
1
log 0
Now the last expression is the general term of a divergent series, and the second half of the Borel-Cantelli lemma (Chung (1974), p. 76, or Ash (1972), p. 272)
guarantees the existence of an event n,e.F with Pod = 1 such that, for every co e 00 and k 1, there exists an integer m = m(k, co) ._ k with (9.23)
14/0-((o) - Wo-,,(0)
-11 - 011(0m).
On the other hand, (9.22) applied to the Brownian motion -W shows that there exist an event SI* e g" of probability one and an integer-valued random variable N*, so that for every w e SP (9.24)
- W,..i(w) <
._ 401/2h(0");
n .. N*(w).
From (9.23) and (9.24) we conclude that, for every w e n, n C/* and every integer k > 1, there exists an integer m = m(k, w) > k v N *(w) such that
,./1 -0 - 4j9.
WO-(w)
h(r)
By letting m -4 oo, we conclude that lima° (Wt/h(t)) _. . ,/1 a.s. P, and letting 010 through the rationals we obtain W
lim -- > 1; a.s. P. ,lo h(t)
-0 -140 holds D
We observed in Remark 2.12 that almost every Brownian sample path is locally Holder-continuous with exponent y for every y e(0, I), and we also saw in Exercise 9.21 that Brownian paths are nowhere locally Holder-continuous
for any exponent y > f. The law of the iterated logarithm applied to {Wt+h - Wh; 0 < h < co} for fixed t (9.25)
-limI Wt+h -,_ h 1. 0
\ /it
0 gives W ti
= co, P-a.s.
Thus a typical Brownian path cannot be "locally Hi5Ider-continuous with
2. Brownian Motion
114
exponent y = 4" everywhere on [0, co); however, one may not conclude from (9.25) that such a path has this property nowhere on [0, co); see Remark 9.15 and the Notes, Section 11.
F. Modulus of Continuityt Another way to measure the oscillations of the Brownian path is to seek a modulus of continuity. A function g) is called a modulus of continuity for the function f: [0, T] --) 11-,R if 0 s < t T and It - sl 6 imply If(t) - f(s)I g(5), for all sufficiently small positive S. Because of the law of the iterated logarithm, any modulus of continuity for Brownian motion on a bounded interval, say [0, 1], should be at least as large as ,./2.5 log log(1/6), but because
of the established local Holder-continuity it need not be any larger than a constant multiple of SY, for any y00, 1/2). A remarkable result by P. Levy (1937) asserts that with g(6)
(9.26)
.\./2.5 log(1/5);
6 > 0,
cg(6) is a modulus of continuity for almost every Brownian path on [0, 1] if c > 1, but is a modulus for almost no Brownian path on [0, 1] if 0 < c < 1. We say that g in (9.26) is the exact modulus of continuity of almost every Brownian path. The assertion just made is a straightforward consequence of the following theorem. 9.25 Theorem (Levy modulus (1937)). With g: (0, 1] -) (0, oo) given by (9.26), we have
P
(9.27)
1
max I Wt - WS1 = 11 = 1. g(u) (:).s
t-sb
PROOF. With n > 1, 0 < B < 1, we have by the independence of increments and (9.20):
Pr
max I To. - N-iv2.1
(1
1si5..2n
where
A 213[2no
(1
- 0)11202-nd
(1 - c),-,2" ' exp( - 0,"),
- 0)1/22nagtz,,..-")] > 2e-x2121,./27t(x + 1/x) and
x = .\/(1 - 0)2n log 2. It develops easily that for n > 1, we have where a > 0, and thus
P [ max I Wia. - Wu- 1)/2 1 1 < j<2^
> a2-"(1-0)
(1 - 0)1/2g(2') < exp( - a2").
By the Borel-Cantelli lemma, there exists an event no e
with P(09) = 1 and
This subsection may be omitted on first reading; its results will not be used in the sequel.
2.9. The Brownian Sample Paths
115
an integer-valued random variable No such that, for every co e Cle, we have 1
0/2.(01
ima x I Wi12,,(w) -
n > Ne(w).
0;
sjs
Consequently, we obtain 1,
- 0,
lim max 1141, - Wsl alo 9(0) oss
and by letting 010 along the rationals, we have 1
lim
max
I147, -W I > 1,
alo 9(0) oss
a.s. P.
t-s
For the proof of the opposite inequality, which is much more demanding, we select 0 e (0, 1) and 6 > ((1 + 0)I(1 - 0)) - 1, and observe the inequalities (9.28)
1
P [ max . n, I Wp2. - Wonl _>_ 1 + E 2. g(ICZ.- ) 0 k=j-i2.9 2,01
EP
max
-Wod r
I Wa,
0 < i
k=1 E[2.0]
< 2" E P k=i
ki2
Ik2-"
(1 +
E)g(2"k )1
4n
> (1 +
The probability in the last summand of (9.28) is bounded above, thanks to (9.20), by a constant multiple of n-112(k2')" "2, and 2,0+1
x(1"2 dx = k=1
(2" + 1)v V
0
where v = 1 + (1 + 6)2. Therefore,
P[
max 0 < j< 2. k=j-i52.°
1W.i/2"
Wi/2.I
9(k /2")
> 1 + 61 <
cons!.
2-nn,
with p = (1 - 0) (1 + e)2 - (1 + 0), a positive constant by choice of E. Again by the Borel-Cantelli lemma, we have the existence of an event no e g" with P(00) = 1, and of an integer-valued random variable N, such that 1 2-(1-e)No wi < _;
(9.29)
v co EDO
e
and (9.30)
max 0 < i< j< 2.
k=i--ts2^'
Wi/2.(0 - Wi/24(0)1 < 1 + E; g(k/2n)
n > No(w),
2. Brownian Motion
116
9.26 Problem. Consider the set D = U;;D=14 of dyadic rationals in [0, 1], with D = {kr"; k = 0, 1, ...,2^}. For every w e Slo and every n N8(w), the inequality
No)) - Ws(w)I < (1 + s)[2
(9.31)
g(2-j) + g(t j=n+1
is valid for every pair (s, t) of dyadic rationals satisfying 0 < t - s < 2-"" -8).
(Hint: Proceed as in the proof of Theorem 2.8 and use the fact that g() is strictly increasing on (0, 1/e].)
Returning to the proof of Theorem 9.25, let us suppose that the dyadic rationals s, t in (9.31) are chosen to satisfy the stronger condition
2-("+"(' -6) < 6 L_ t- s< 2-n(1-8).
(9.32)
But then (9.29) implies 2-n(1-8) < 1/e; n > No(w), and because g is increasing on (0, 1/e] we have
E g(2-j)
cg(2-n-1)
j=n+1
-
29
2g( )
holds for an appropriate constant c > 0. We may conclude from (9.31), (9.32), and the continuity of W.(w) that for every w elle and n Ne(w), 1
max Wi(w) - W.,(01 gly 0<s
(1 + s)[1 +
t-s=8
holds for all 6 e [2-("+1)(1-8),2-"" -0)). Letting n
2c 2-8(n+11/2
1 -0
JJ
co, we obtain
1
lim max 1147,(w) - Ws(w)I .510 9(0) o<s
1 + c,
v.-n=6
and because g is strictly increasing on (0, 1/e] we may replace the condition t - s = 6 by t - s < 6 in the preceding expression. It remains only to let 010 (and hence simultaneously a 4.0) along the rationals, to conclude that 1
max 1W(w) - Ws(w)l < 1; 64,0 061 o<s
t
a.s. P.
-ssa
The proof is complete.
2.10. Solutions to Selected Problems 1.4. For fixed 0 5 s < t < (x), and arbitrary integer n 1 and indices 0 = so < s, < < sn= s, the a-field er(Xo, Xso. , Xs) = cr(Xo, X,, Xs - x._,) is
independent of X, - X,. The union of all a-fields of this form (over n > 1,
2.10. Solutions to Selected Problems
117
(s1,... , s"_ 1) as described) constitutes a collection '' of sets independent of X, - X5, which is closed under finite intersections. Now 9, the collection of all sets in .3 which are independent of X, -X is a Dynkin system containing W. From Theorem 1.3 we conclude that sx = a(W) is contained in g. In fact, "-sx
g.
2.3. Suppose that there exist integers n > 1, m > 1, index sequences t = (t1,... , tn), s = (s1,...,s.), and Borel sets A ER(R"), BE /(1r) so that CE (If admits both representations C = {w E 0810 ((Mt! 0..)(4,))E B). It has to be shown that (2.3)'
, 0)(t,,))E Al = {CO E 0810
(0)(S1),
,
Qt(A) = Q!(B)
Case 1 : m = n, and there is a permutation (i 1_ , ik) of (1,...,n) so that si = 1<j
n.
In this case, A = {(x1,... , X.) E 0:1"; (x,,,... , Xi.) E BI. Both sides of (2.3)' are measures on At CR"), and to prove their identity it suffices to verify that they agree on sets of the form A = Al x x A, .4(08) (hence B = A,1 x x But then (2.3)' is just the first consistency condition (a) in Definition 2.1.
Case 2: m > n and {ti,...,t} g_ Without loss of generality (thanks to Case 1) we may assume that I.; = 1 S j < n. Then we have B = A x le with k = m -n > 1, and (2.3)' follows from the second consistency condition (b) of Definition 2.1. Case 3: None of the above holds.
We enlarge the index set, to wit: {q1,...,q,} = {t1,... ,t"} u {s1,.. .,s,}, with m v n
(w(qi ),
, w(q,))E E},
By the same reasoning as before, we may assume that
Lril-" and, by Case 2, Q,(A) = Qq(E) with q = (q1,
EE.I(081).
=
1 s j < n. Then E = A x qd. A dual argument shows
Q1(B) = Qn(E)
The preceding method (adapted from Wentzell (1981)) shows that Q is well defined by (2.3). To prove finite additivity, let us notice that a finite collection {C.,}7=1 g_ if may be represented, by enlargement of the index sequence if necessary, as =_- {CO E REC"D); (0)(t1), ... , CO(t,)) E Al }, Ai .4(01")
for every 1 < j < in. The Al's are pairwise disjoint if the Cis are, and finite additivity follows easily from (2.3), since
U c= fweffe°,-); (co(ti), .i=1
, co(tnfle U A; }. .i=1
Finally, RE°'") = {COE till°'°°°; (0(t)e ER) for every t > 0, and so Q(081°.'")) = Q,(R) = 1.
2.4. Let 327 be the collection of all regular sets. We have OESc. To show .547 is closed under complementation, suppose A E 37, so for each e > 0, we have a closed set F
and an open set G such that F g A g_ G and Q(G\F) < E. But then Fc is open, G` is closed, Gr c A' S F`, and Q(P\G`) = Q(G\F) < E; therefore, ACE F. To show ,F is closed under countable unions, let A = (.);,°=1 Ak, where Ak E3-4.- for each k. For each E > 0, there is a sequence of closed sets {Fk} and a sequence of k = 1, 2, .... Let open sets {Gk} such that Fk g Ak g Gk and Q(Gk \Fk)<
2. Brownian Motion
118
<
G = Ur=1Gk and F = Uk Fk, where m is chosen so that Q(Ulf._, Fk\Ur=, E/2. Then G is open, F is closed, F g A g G, and 1
Q(G\F)
Q
(G\ U
Fk) +
E
Q(6° Fk\F) < E
k=1
E
2
k=1
k=1
= E.
Therefore, 3,7 is a a-field. Now choose a closed set F. Let Gk = {X E RI"; II
Y < 11k, some y F}. Then each Gk is open, G1 2 G2 2 , and nio_i Gk = F. Thus, for each e > 0, there exists an m such that Q(G,\F) < E. It follows that F e ST; . Since the smallest a-field containing all closed sets is .4(R"), we have = .4(R").
2.5. Fix t =
t2, , t), and let s = (t,i, t,2,.. , t,n) be a permutation of t. We have constructed a distribution for the random vector (B,i,131, ..... BO under which
x A) = P[(13,, .....
Q1(A1 x A2 x
.....
=
Ai, x
= Qs(Ai, x Furthermore, for A E g(R"-1) and s' = (t1,t2,.
Qt(A x l) = P[(B,,,
,
x An]
x
x
x
,t_,),
B1_,)e 24] =
2.9. Again we take T = 1. It is a bit easier to visualize the proof if we use the maximum norm ill(t, , t2, , td)ill A max , ,, I t,1 in Rd rather than the Euclidean norm.
Since all norms are equivalent in Rd, it suffices to prove (2.14) with It - sil replaced by III t - s ill. Introduce the partial ordering in Rd by (s,, s2, (t,, t2,... , td) if and only if s, < t i = 1, ,d. Define the lattices
sd)
co
L = y,; k = 0, 1,
, 2" - 1} ; n
1,
L= U L, n=1
and for s E L, define N(s) = {t E L; s < t, IIt - sil = 2'}. Note that L has 2nd elements, and for each s e L, N(s) has d elements. For s L and t o Nn(s), Cebygev's inequality applied to (2.13) gives (with 0 < y < GO), P[I X, - XsI > 2-Yn] < C2-^(d+9 -2Y),
and consequently,
P [max IX, -X sl >
2-Y" < dC2-"(13 -27) .
se L
t e N(s)
The Borel-Cantelli lemma gives an event Sr E 3'7. with P(S1*) = 1, and a positive, integer-valued random variable n*, such that max I X,(0)) - Xs(co)I < 2-", SE
L
n > n(a)), a) E ir.
IE &(s)
Now let Rn(s) = {t e L"; s
a sequence s Consequently,
s1,
t, lilt - s ill = 2-11. For s E Ln and t E Rn(S), there is
, s, = t with m < d and s, N(s,_,), i = 1, ..., m.
2.10. Solutions to Selected Problems
119
max IX,(co) - Xs(co)I 5 d2"",
(10.1)
n* (co), weir.
n
re R,,(s)
We now fix w
n > n*(w), and show that for every m > n, we have 171
(10.2)
IX,(co) - Xs(co)I 5 2d E 2-7i; V t, s E L s < t, lilt - sill < j=n+1
For in = n + 1, we can only have tE R,(s), and (10.2) follows from (10.1). Suppose (10.2) is valid for m = n + 1, , M - 1. Take t, s E Lm, s t. There is a vector t1 t. s' E L,_, n R m(s) and a vector ti E Lf_, with t E R f(t') such that s From (10.1) we have,
IX,,(co) - Xs(co)1 _5 dr'', PC, (co) - X,i(o.))1 5_ d2"m, and from (10.2) with in = M - 1, we have m-i
PC,i(a)) - Xsi(co)1 5_ 2d E 2i=n+1
We obtain (10.2) for in = M. For any vectors s, t E L with s t and 0 < lilt - sill < h(co) A 2 ""), we select n > n*(w) such that 2-("+" 5_ lilt - sill < 2-". We have from (10.2) 2 YJ 5_ Slllt -
I Xi(co) - Xs(co)I 5_ 2d j=n+1
where (5 = 2d/(1 - 2-Y). We may now conclude as in the proof of Theorem 2.8.
4.2. The n-dimensional cylinder sets are generated by those among them which are n-fold intersections of one-dimensional cylinder sets; the latter are generated by sets of the form H = {we C[0, co); w(t,) G}, where G is open in R. But H is open in C[0, co), because for each coc,eli, this set contains a ball B(coo, 8) A {we C[0, oo); p(co,coo) < El, for suitably small E > 0. It follows that W .4(C[0, co)). Because C[0, co) is separable, the open sets are countable unions of open balls of the form B(coo,E) as previously. Let Q be the set of rationals in [0, co). We have {(.0:
E - sup (lco(t) - wo(01 A 1) < El. EB(0)0,0=
n=1 2" 0
The set on the right is W-measurable, so ,I(C[0, co)) c For the second claim, notice that with any cylinder set C of the form (2.1)' we have ,(cp,w)(t))E A}
cptI(C) = Ico E C[0, co); (0),(0)(t ,),
= Ico C[0, co); (o)(t h t1),
co(t
e
, to in [0, t] and
so cp,-1(f) g_ (et. On the other hand, for any Cegt we have t A EM(6") so that
C = {we
= and thus Ce,
oo);(w(t,), C[0, co); (co(t
Eic,
co(t.))E A} co(t
t,,)) E A =
cp,
1 (C),
(if). It follows that Ce, = (pt-1(W), which establishes the claim.
2. Brownian Motion
120
4.11. We have to show
lim sup P[IXr1 > A.] = 0,
(4.6)'
and
A-.x mZ I
(4.7)'
lim sup P I max
- n")I > E I = 0
I
It-sl.p
P10 m21
0 .cr,s
for all positive numbers t, T. Relation (4.6)' is an immediate consequence of (i) and of the Ceby§ev inequality. It suffices to prove (4.7)' for T = 1. Let ri > 0 and E > 0 be given. We denote X(")) simply by X, and employ the notation of the proof of Theorem 2.8. With
ni g ñ
{ max I Xkir,
n=1
X(k -1)/2.1
1 ..ck .c 2^
< 2-Y"
we have from (2.9): PA) 5 C,E`:_,2-"(13-") < n, provided 1 is a large enough integer. Now for every CO e (21 and n > 1, we have from (2.12):
max I X,(a.)) - Xs(w)I < 62-7", t,se D
where .5
2/(1 - 2-7). It follows that, given e > 0, n > 0, there exists an integer
n = n(e,n) such that for every to E Stn:
I Xr(o.)) - n")(0))1 < e,
max
Vm>1
O
(we have used the continuity of the sample path ti-o.")(o.))), and consequently
sup P
,,,1
[ max IXr - .nn)I > e
< P(SY) < ti.
t-m<2-^
()./,s51
4.16. Let (Q,,, .0)--; Fi.) denote the space on which X and Y are defined, and let E denote expectation with respect to P. Let X be defined on (II, 37;,P). We are given that lim,,
= Ef(X) for every bounded, continuous f: S -0 FI and that
p(X("),r")) .0 in probability. To prove Y(") 4 X, it suffices to show
lim E[f(X(")) - f(r"))] = 0 n-ce
whenever f is bounded and continuous. Let such an f be given, and set M = suP..es If(x)1 < co. Since {X("1°'_, is relatively compact, it is tight; so for each E > 0 there exists a compact set K c S, such that P [X(M)E K] > 1 - e/6M, V n > 1. Choose 0 < S < 1 so I f(x) - f(y)I < c/3 whenever XE K and p(x, y) < S.
Finally, choose a positive integer N such that P[p(X("),Y(")) > (5] < e/6M, Vn N We have
< - P [X(") e K, p(X("), V")) < .5]
-3
n
+ 2M P[X(")
IC]
+ 2M P[p(X("), r")) > (5] < E. 5.2. The collection of sets F E a(C. [0, 00 )d ) for which x 1- P'(F) is .4(08°)/,4([0, 1])measurable forms a Dynkin system, so it suffices to prove this measurability for
2.10. Solutions to Selected Problems
121
all finite-dimensional cylinder sets F of the form F = {aye CTO, oo)d; co(t 0) E FP,
where 0 = t, < t, < P'(F) = lro(x)
< t, re.4)(Rd), i = 0, 1,
Jr,r,
I
Jr
Pe(t 1; x, Yi)
, co(t.)e F} , , n. But
Pa(t. -4 -1; Y.-1, Y,OdY.
dYi,
where pd(t; x, y) A (2nt)-da exp{ -(II x - y112 /2t) }. This is a Borel-measurable function of x. 5.7. If for each finite measure p on (S, AS)), there exists g,, as described, then for each cc e R, {x ES;f (x) < a} 0 {x E S; g ,,(x) < a} has p-measure zero. But {go e .4(S), so {f e At(S)". Since this is true for every p, we have If eaW(S). For the converse, suppose f is universally measurable and let a finite measure p be given. For r e Q, the set of rationals, let U(r) = {x e S; f(x) Then f(x) = inf {r e Q; x e U (r)} . Since U(r) E AST', there exists B(r)E R(S) with p[B(r),a,U(r)] = 0, r e Q. Define g m(x) A inf {r e Q; x e B(r)} = inf (1),(4 reQ
where (p,.(x) = r if X E B(r) and (p,(x) = oo otherwise. Then gu: S
measurable, and {x ES; f(x) # g,(x)}
IR is Borel-
U,e Q[B(r) 0 U(r)], which has p-
measure zero.
5.9. We prove (5.5). Let us first show that for D E *Rd x Rd), we have (10.3)
P[(X, 11E1)0] = PUX, Y)EDIn
If D = B x C, where B, C E R(M), then
E[1{xee}1{YeC" = 1{yec}E{1{XeB" = i{yc}P[XE B].
= 1{yc}P[X en so (10.3) holds for this special case. The sets D for which (10.3) holds form a Dynkin system For the same reasons, E[1{x E,3}1{y Ec} I
containing all measurable rectangles, so (10.3) holds for every D E .41(Rd x Rd). To prove (5.5), set D = {(x,y); x + ye r }. A similar proof for (5.6) is possible. 6.9. (ii) By Corollary 1.2.4, S is a stopping time of {#;.,.}. Problem 1.2.17 (i) implies
E[Z2IFs+] = EV21,(5,,o+], a.s. on {S This equation combined with (i) gives us the desired result.
(iii) Suppose that S is an optional time of {A}, and that (e) holds for every bounded optional time of {A}. Then for each s > 0, Px[X(ss)+,e FI.Fas)+] = (U,1,)(Xs s),
a.s.
But on {S < s}, we have X(s,,o+, = Xs+ so (ii) implies
Px[Xs+te rIFs+] = Px[x(s,,$),-,ErIF(s,,o+] =(U,i,-)(xs,$)= (U,1,)(Xs), 13' a.s. on {S Now let s T co to obtain (e) for the (possibly unbounded) optional time S. The argument for (e') is the same.
2. Brownian Motion
122
7.1.
=
= ns>tn.>03,sL=
(1)
(ii) The a-field Ax is generated by sets of the form F = {(X,... , 'COE r}, where
0 = t, <
< t = t and I e AR"). Since X, =
Xs_ for any sequence
g [0, t) satisfying smT t, we have F
(iii) Let X be the coordinate mapping process on C[0, co). Fix t > 0. The nonempty set F A {w E C[0, co); w has a local maximum at t} is in for each n > 0,
since
co
F =U m=n
n
{w; w(t)
w(r)} E
reQ
lirst
On the other hand, a typical set in A has the form G = {we C[0, CO 00(t1), (0(t2), )eI' }, for {ti} g [0, t] and F e ( R x R x ). We claim that
F = G cannot hold for any G e Ax. Indeed, suppose we have F n G for some G e Ax. Then given any weFnG, the function
0 < s < t, co(t) + s - t; s
6(s)
t,
is in G but not in F, so F cannot agree with any G E AX. This shows that the filtration 1,,x1 is not right-continuous. With ter, = A:+c, we have
=
n
e>0
0-(u F(f+,),2)= s
so the failure of {V} to be right-continuous implies the failure of {..f;.'} to be left-continuous. 7.3. Clearly, Ye is a a-field: QS E F e dr implies that there exists an event G with P A GC = FA GE Aim, and finally for any sequence {F}°°_, g_ le we have a companion sequence {G},;°_, g_ 1 such that
( U F) A ( U G ) s U (F A G ) E ,A°, n=1
which yields
FE 112.
Further, the observation FAG = N =. F= GAN yields the characterization Ye = g C2; 3 G el, N E.41" such that F = G A N}. It follows that ?° g 5". On the other hand, Jr contains both / and X°, and since it is a a-field:
5" = cr( L.) ") g Jr. For completeness, let us observe that the requirements F
Pm(F) = 0 imply
the existence of GE4 such that N = FA Ge X° and P"(G) = 0. Now F is contained in N v G and hence F is in .AI", as is any subset of F.
7.4. Let {st; t > 0} be the coordinate mapping process on C[0, co). Let P" on (C[0, cc), ,4(C[0, co))) be the probability measure under which B = {B,, AB; > 0} is a one-dimensional Brownian motion with initial distribution p. The set F = {w; w(1) = 0} has Po-measure zero, so F E .F4.. If F is also in the completion ,01' of under Pi', then there must be some G e ,013 with F s G and P"(G) = 0. Such a G must be of the form G = {w: w(0) e F} for some F E R(R), and the only way G can contain F is to have F = R. But then Pu(G) # 0. It follows that F is not in Foi`.
2.10. Solutions to Selectee Problems
123
7.5. Clearly, AP g_ Fu holds for every 0 < t < o o , so ..f-sg, g_ F. For the opposite inclusion, let us take any F ..F0; Problem 7.3 guarantees the existence of an event G54...,ox such that N = PAGE Ac". But now .Foox .97,,;
for every 0 < t < co), and thus G E
.FX, (we have .97,x g
N e Ar" c ,9 ; imply
F=G.LNE,Fog. 7.6. Repeat the argument employed in Solution 7.5, replacing ..F" by and ..FX by .5,'! and using the left-continuity of the filtration {V} (Problem 7.1).
7.18. Let {B t > 0 }, (0,1"), {P'}11 be a one-dimensional Brownian family. For r E a(Q8), define the hitting time Hr(co) = inf It 0; B,(a)) According to Problem 1.2.6, H(O,) is optional, so {H(,,,s) = 0} is in .510, = A. Likewise, li,,mE37:0. Because of the symmetry of Brownian motion starting at the origin, P°[H,0,, = 0] = = 0]. According to the Blumenthal zeroone law (Theorem 7.17), this common value is either zero or one. If it were zero, then P° [B, = 0, V 0 < t < s for some E> 0] = 1, but this contradicts Problem 7.14 (i). Therefore, P° [H(0a) = 0] = P°[H,_0,,o) = 0] = 1, and for each we {H(0.0,) = 0} n {11(_..0)= 0}, there are sequences s 10, tn 10 with Bs.(co) > 0, Bin(co) < 0 for every n
1.
7.19. For fixed wee, Tb(w) is a left-continuous function of b and Sb(w) is a rightcontinuous function of b. For fixed b E FR, "T is a stopping time and Sb is an optional time (Problem 1.2.6), so both are 37"..w-measurable. According to Remark 1.1.14, the set A = {(b, co) E [0, CC) x 0; Tb(w) Sb(w)} is in .4([0, cc)) (3) ,F07. Furthermore, A g {(0 e f2; (b, 0)) E Al is included in the set {we SI; BA(0)
wrb(o+t(w) - Wrb(w)(0) < 0 for some E> 0 and all t E
en,
which has probability zero because {B F,B; 0 < t < co} is a standard Brownian motion (Remark 6.20 and Theorem 6.16), and Problem 7.18 implies that B takes positive values in every interval of the form [0,E] with probability one. This establishes (i). For (ii), it suffices to show OD
P[wES); (L(co),co)e A] = J
P(Ab)P[L E db].
If A were a product set A = C x D; C E .4([0, co)), D e..Fw, this would follow from the independence of L and The collection of sets A e .4([0, co)) (3).F,w for which this identity holds forms a Dynkin system, so by the Dynkin system Theorem 1.3, this identity holds for every set in 98([0, op)) ®
8.8. We have for s > 0, t > 0, b > a, b > 0; P°[W,±s
a, M,s < b1.5rJ = P°[W,±s < a, M, < b, max W,, < b
ou5s
= 1 { ,,,,,}
P° [ W.s < a, max W, < b oso5s
The last expression is measurable with respect to the a-field generated by the pair of random variables (W M,), and so the process {(W,, M,); .F.,; 0 is Markov. Because V, is a function of (W Me), we have
P°[Y,A,EFIA] = P°[Yt+serlWr,
rEge(R)-
t<
}
2. Brownian Motion
124
In order to prove the Markov property for Y and (8.11), it suffices then to show
that r[Yr-FsE
Wr = W9
Mr = m] = P+(s; m - w,Y)dY
Indeed, for b > m > w, b > a, m > 0 we have
[Wr. da, Mr.
W, = w, M, = m]
= P°[W,,,E da, max W,. E dbl W, = w, M, =
13.5
[Ws da - w, Mse db - w]
Pw[WE da, Ms db] =
=
2(26 -a - w)
(26 -a - w)2}
exp
2s
.12rz.s3
da db,
thanks to (8.2), which also gives r[Wr+s E da, Mt+s = ml W, = w,M, = m] ll
= r[W,+se da, max W +SmIW = w,Mr =mJ = =
e da, Ms 1
t
[W da - w, Ms
m] =
-2s
exp
2its
m -w]
(2m -a -
exp
2s
Therefore, P°[Y,+se dy1W, = w, M, = m] is equal to
P°[WE b - dy, 111,+ sE dbl
1N,
Air
= m] db
f(m, co)
+ P°[W,+sE m - dy, M, +s = mIW, = w, Mr = m] = p+(s; m - w, y)dy. Since the finite-dimensional distributions of a Markov process are determined by the initial distribution and the transition density, those of the processes I WI and Y coincide. 8.12. The optional sampling theorem gives
= Ex
= Ex X, ,., T ATa =
[exp
Since W(AMA T. is bounded, we may let t = Ex [exp JI WT. A T -
W.,ATOAT-
-1,12(t A To A TO)].
co to obtain
1:1.2(To
'TX]
= Ex[1{7-0<,-.}e- '111'0/2] + e"Ex[1{T.<,-.}e-'2T-12].
By choosing = ±
2a, we obtain two equations which can be solved simultaneously and yield (8.27) and (8.28).
9.3. For every 0 < a < i, Doob's maximal inequality (Theorem 1.3.8 (iv)) gives
E[
sup er
(-)
and with r = 2"+1 = 2a,
t
2
W2)
< -2 E( sup
cf
4 0-2
4T
2.10. Solutions to Selected Problems
sup
P[ 2^<(<2 ^.
125
I
t
--8 2-"
> E] <
is seen to hold for every E > 0, n > 1. The result (9.2) follows now from the BorelCantelli lemma.
9.5. Use Problem 7.18 and Lemma 9.4 (ii).
9.8. If II = {to,
t,} is a partition of [0, t], we write Vi(2)(11) - t =
- (tk - 4_0} as a sum of independent, zero-mean random variables. Therefore, E(/,(2)(11) -
=E k=1
- tk_02E[
2
11 s tE(Z2 - 1)2111111,
tk - tk-1
where Z is a standard normal random variable. The first assertion follows readily. For the second, we observe that const.
E P[iv,(2)01)- tl >
2
n=1
E
E 1111Il < co n=1
holds for every E > 0, and by the Borel-Cantelli lemma, P[IV,(2)(11) - t I > E, infinitely often] = 0. The conclusion follows. 9.11. (D. Freedman (1971)) (i) Each point of the set Mn =
It E [0, cc); f(s) < f(t), V SE Rt --1),
is isolated, so Mn is countable. But U:3-1 Mn is the set of points of strict local maximum for f.
(ii) It suffices to show that f has a local maximum in an arbitrary, nonempty interval (a, b) S [0, cc). Let us begin by assuming f(a) < f(b), and let t be the largest number in [a, b) with f (t) = f (a). Because f is not monotone such a t exists, and in (t, b) there are two numbers r and s with r < s and f(r) > f(t) v f(s). Being continuous, f must have a maximum over [t, s], which is a local maximum in the sense of Definition 9.10 (iii). If f(a) > f(b), we apply the preceding argument in reverse, defining t to be the smallest number in (a, b] with f(t) = f(b). If f(a) = f(b), then f must have a maximum over [a, b] at some r e (a, b) and this r is also a point of local maximum. 9.22. Use integration by parts (Chung (1974), p. 231; McKean (1969), p. 4).
9.26. It suffices to show that for every m > n (10.4)
I W(w) - Ws(co)I < (1 +
No(co), we have
[2
g(2-j) + g(t =n+1
valid for every s, t E D, satisfying 0 < t - s < 2-11(1 -9). For m = n, (10.4) follows from (9.30). Let us assume that (10.4) holds for m = n, . . . , M - 1. With s, t E DM
and 0 < t -s < 2-"("), we consider, as in the proof of Theorem 2.8, the numbers t' = max {u E Dm_i; U < t} and sl = min{u E Dm_ ; u
s} and observe the
2. Brownian Motion
126
relations t - tt < 2-1", s1 - s < 2-m, and 0 <11 - s1 < t - s < 2-"" -8) < 1/e, thanks to (9.29). We have M -1
Wo(a)) - Ws.(01 5 (1 + E)[2
E g(2-j) + g(t' -
j=n+I
by the induction assumption, and 1W, co) - W,,(01 5 (1 + e)g(2') as well as 1Ws(w) - W.,((.0)1 < (1 + E)g(2') because of (9.30). Since g(t1 - s') 5 g(t - s), we conclude that (10.4) holds with m = M.
2.11. Notes Section 2.1: The first quantitative work on Brownian motion is due to Bachelier (1900), who was interested in stock price fluctuations. Einstein (1905) derived the transition density for Brownian motion from the molecular-kinetic theory of heat. A rigorous mathematical treatment of Brownian motion began with N. Wiener (1923, 1924a), who provided the first existence proof. The most profound work in this early period is that of P. Levy (1939, 1948);
he introduced the construction by interpolation expounded in Section 2.3, studied in detail the passage times and other related functionals (Section 2.8), described in detail the so-called fine structure of the typical sample path (Section 2.9), and discovered the notion and properties of the mesure du voisinage or "local time" (Section 3.6 and Chapter 6). Most amazingly, he carried out this program without the formal concepts and tools of filtrations, stopping times, or the strong Markov property. Section 2.2: The construction of a probability measure from a consistent family of finite-dimensional distributions is clearly explained in Kolmogorov (1933); Daniell (1918/1919) had constructed earlier an integral on a space of sequences. The existence of a continuous modification under the conditions of Theorem 2.8 was established by Kolmogorov (published in Slutsky (1937));
Loeve ((1978), p. 247) noticed that the same argument also provides local Holder-continuity with exponent y for any 0 < y < Pc. For related results, see also Centsov (1956a). The extension to random fields as in Problem 2.9 was carried out by Centsov (1956b).
Section 2.3: The Haar function construction of Brownian motion was originally carried out by P. Levy (1948) and later simplified by Ciesielski (1961). For a similar construction of Brownian motion indexed by directed sets, see Pyke (1983). Yor (1982) shows that the choice of the complete, ortho-
normal basis in L2[0, 1] is not important for the construction of Brownian motion. Section 2.4 is adapted from Billingsley (1968). The original proof of Theorem
4.20 is in Donsker (1951), but the one offered here is essentially due to Prohorov (1956). It is also possible to construct a probability space on which all the random walks are defined and converge to Brownian motion almost surely, rather than merely in distribution (Knight (1961)).
2.10. Solutions to Selected Problems
127
Sections 2.5, 2.6: The Markov property derives its name from A. A. Markov, whose own work (1906) was in discrete time and state space; in that context, of course, the usual and the strong Markov properties coincide. It was not immediately realized that the latter is actually stronger than the former; Ray ((1956), pp. 463-464) provides an example of a continuous Markov process which is not strongly Markov. It seems rather amazing today that a complete and rigorous statement about the strongly Markovian character of Brownian motion (Theorem 6.16) was proved only in 1956; see Hunt (1956). A Markov family for which the function xl-- Ex f(X,) is continuous for any bounded, continuous f : -4 ill and t e [0, co) is said to have the Feller property, and a right-continuous Markov family with the Feller property is strongly Markovian. Very readable introductions to Markov process theory can be found in Dynkin & Yushkevich (1969), Wentzell ((1981), Chapters 8-13), and Chung (1982); more comprehensive treatments are those by Dynkin (1965), Blumenthal & Getoor (1968), and Ethier & Kurtz (1986). Markov processes with continuous sample paths receive very detailed treatments in the monographs by ItO & McKean (1974), Stroock & Varadhan (1979), and Knight II
(1981).
Section 2.7: The concept of enlargement of a filtration has become very important in Markov process theory and in stochastic integration. There is a substantial body of theory on this topic, which we do not take up here; we instead send the interested reader to the articles in the volume edited by Jeulin & Yor (1985). Sections 2.8, 2.9: The material here comes mostly from P. Levy (1939, 1948).
Section 1.4 in D. Freedman (1971) was our source for Theorems 9.6, 9.9 and 9.12, and can be consulted for further information on this subject matter. Our discussion of the law of the iterated logarithm and of the Levy modulus follows McKean (1969) and Williams (1979). Theorem 9.18 was strengthened by Dvoretzky (1963), who showed that there exists a universal constant c > 0 such that -,-1W+h((o) - 14/11041 > c, V t e [0, op)] = 1. P[co al; lim h40
Nfil
For every w e 0, S,,, -4 It E [0, CO); liMh10 (1 Wt+17(04 - Wt(0))01) < 091 has been called by Kahane (1976) the set of slow points from the right for the path W.(w). Fubini's theorem applied to (9.25) shows that meas(S,o) = 0 for P a.e.
wee, but, for a typical path, S,,, is far from being empty; in fact, we have P1
En; inf lim cl.,
1W
H-h
(0) - W(01 =
11 = 1.
112
This is proved in B. Davis (1983), where we send the interested reader for more information and references on this subject.
Chung (1976) and Imhof (1984) offer excellent follow-up reading on the subject matter of Section 2.8.
CHAPTER 3
Stochastic Integration
3.1. Introduction A tremendous range of problems in the natural, social, and biological sciences came under the dominion of the theory of functions of a real variable: when Newton and Leibniz invented the calculus. The primary components of this invention were the use of differentiation to describe rates of change, the use of integration to pass to the limit in approximating sums, and the fundamental theorem of calculus, which relates the two concepts and thereby makes the latter amenable to computation. All of this gave rise to the concept of ordinary
differential equations, and it is the application of these equations to the modeling of real-world phenomena which reveals much of the power of calculus.
Stochastic calculus grew out of the need to assign meaning to ordinary differential equations involving continuous stochastic processes. Since the most important such process, Brownian motion, cannot be differentiated, stochastic calculus takes the tack opposite to that of classical calculus: the stochastic integral is defined first, in Section 2, and then the stochastic differential is given meaning through the fundamental "theorem" of calculus. This "theorem" is really a definition in stochastic calculus, because the differential has no meaning apart from that assigned to it when it enters an integral. For this theory to achieve its full potential, it must have some simple rules for computation. These are contained in the change of variable formula (Ito's rule), which is the counterpart of the chain rule from classical calculus. We present it, together with some important applications, in Section 3. Section 4 advances our recurrent theme that "Brownian motion is the fundamental martingale with continuous paths" by showing how to represent continuous, local martingales in terms of it, either via stochastic integration
3.2. Construction of the Stochastic Integral
129
or via time-change. We also establish the representation of functionals of the Brownian path as stochastic integrals. The important Theorem 4.13 of F. B. Knight is established as an application of these ideas. Stochastic calculus has a fundamental additional feature not found in its classical counterpart, a feature based on the Girsanov change of measure (Theorem 5.1). This result provides a device for solving stochastic differential equations driven by Brownian motion by changing the underlying probability measure, so that the process which was the driving Brownian motion becomes, under the new probability measure, the solution to the differential equation.
This profound idea is first presented in Section 5, but does not reach its culmination until the discussion of weak solutions of stochastic differential equations in Chapter 5. In some cases, this device is merely a convenient way of finding the distribution of a functional of an already existent stochastic process; such an example is provided by the computations related to Brownian motion with drift in subsection 5.C. In other cases, the change of measure provides us with a proof of the existence of a solution to a stochastic differential equation, when the more standard methods fail. This is discussed in subsection 5.3.B. A particularly nice application of the Girsanov theorem appears in Section 5.8, where it is used in a model of security markets to remove the differences among the mean rates of return of the securities. This reduction of the model permits a complete analysis by martingale methods. Although "optional" in the sense that stochastic calculus can (and did for 20 years) exist and be useful without it, the Girsanov theorem today plays such a central role in further developments of the subject that the reader would be remiss not to come to grips with this admittedly difficult concept. In Section 6 we employ the stochastic calculus in the study of P. Levy's mesure du voisinage or local time, a device for measuring the "amount of time spent by the Brownian path in the vicinity of a certain point." This concept has become exceedingly important in both theory and applications; we examine its connections with reflected Brownian motion, extend with its help the Ito rule to functions which are not necessarily twice continuously differentiable, and use it as a tool in the study of certain additive functionals of the Brownian path. Finally, Section 7 extends the notion of local time to general, continuous semimartingales.
3.2. Construction of the Stochastic Integral Let us consider a continuous, square-integrable martingale M = {M P) equipped with the filtration 0 < t < oo } on a probability space (K2, 1.4r 1, which will be assumed throughout this chapter to satisfy the usual conditions of Definition 1.2.25. We have shown in Section 2.7 how to obtain such a filtration for standard Brownian motion. We assume Mo = 0 a.s. P. Such a process M e .i/4 is of unbounded variation on any finite interval [0, T]
3. Stochastic Integration
130
(cf. Problems 1.5.11, 1.5.12, and the discussion following them), and consequently integrals of the form
/T(X) = I
(2.1)
Xt(w)dM,(w)
cannot be defined "pathwise" (i.e., for each w e SI separately) as ordinary Lebesgue-Stieltjes integrals. Nevertheless, the martingale M has a finite second
(or quadratic) variation, given by the continuous, increasing process <M>; cf. Theorem 1.5.8. It is precisely this fact that allows one to proceed, in a highly nontrivial yet straightforward manner, with the construction of the stochastic integral (2.1) with respect to the continuous, square-integrable martingale M, for an appropriate class of integrands X. The construction is due to ItO (1942a),
(1944) for the special case that M is a Brownian motion and to Kunita & Watanabe (1967) for general Me di2. We shall first confine ourselves to M e."2, and denote by <M> the unique (up to indistinguishability) adapted, continuous, and increasing process, such that IM,2 - <M> A; 0 < t < ool is a martingale (cf. Definition 1.5.3 and Theorem 1.5.13). The construction will then be extended to general continuous, local martingales M. We now consider what kinds of integrands are appropriate for (2.1). We first define a measure gm on ([0, oo) x S2, 8([0, co)) <91 by setting IM(A) = E J
(2.2)
1A(t,a) d<M>,(w).
We shall say that two measurable, adapted processes X = IX ,Fg; 0 < t < col and Y = { Y A; 0 < t < col are equivalent if X,(w) = Y,(co);
µM -a.e. (t, w).
This introduces an equivalence relation. For a measurable, 1,97,1-adapted process X, we define (2.3)
[X]T °= E I
X2 d<M>
provided that the right-hand side is finite. Then [X]j, is the L2-norm for X, regarded as a function of (t, w) restricted to the space [0, T] x CI, under the measure We have [X = 0 for all T > 0 if and only if X and Y are equivalent. The stochastic integral will be defined in such a manner that whenever X and Y are equivalent, then 1(X) and 1(Y) will be indistinguishable:
PUT(X) = /T(Y),
V0
T < oo] = 1.
2.1 Definition. Let 29 denote the set of equivalence classes of all measurable,
IFJ-adapted processes X, for which [X],- < co for all T > 0. We define a metric on 2' by [X - Y], where 2-"(1 A [X].).
[X] -4-n=i
3.2. Construction of the S:ochastic Integral
131
Let ..r* denote the set of equivalence classes of progressively measurable processes satisfying [X]T < oo for all T > 0, and define a metric on .2" in the same way. We shall follow the usual custom of not being very careful about the distinction between equivalence classes and the processes which are members of those equivalence classes. For example, we will have no qualms about saying, ".29* consists of those processes in .29 which are progressively measurable." Note that .29 (respectively, .2") contains all bounded, measurable, {,F, }adapted (respectively, bounded, progressively measurable) processes. Both 2' and 2* depend on the martingale M = {M t > 0}. When we wish to indicate this dependence explicitly, we write 2'(M) and 2' *(M). If the function ti- <M>,(co) is absolutely continuous for P-a.e. to, we shall be able to construct P., X, dM, for all X e and all T > 0. In the absence of this condition on <M>, we shall construct the stochastic integral for X in the slightly smaller class .29*. In order to define the stochastic integral with respect to general martingales in A/2 (possibly discontinuous, such as the compensated Poisson process), one has to select an even narrower class of integrands among the so-called predictable processes. This notion is a slight extension of leftcontinuity of the sample paths of the process; since we do not develop stochastic integration with respect to discontinuous martingales, we shall forego further discussion and send the interested reader to the literature: Kunita & Watanabe (1967), Meyer (1976), Liptser & Shiryaev (1977), Ikeda & Watanabe (1981), Elliott (1982), Chung & Williams (1983). Later in this section, we weaken the conditions that M e.ifz and [X]i < co,
V T > 0, replacing them by M e Jr' and T
P
d<M>, <
co]=
1,
VT
0.
This is accomplished by localization. We pause in our development of the stochastic integral to prove a lemma we will need in Section 4. For 0 < T < co, let .27 denote the class of processes X in ..r* for which Xt(to) = 0; V t > T, co en. For T = co, ..r; is defined as the class of processes X e .2" for which E ft; X,2 d<M>, < co (a condition we already have for T < co, by virtue of membership in _2"). A process X e 2,;!' can be identified with one defined only for (t, w) e [0, T] x n, and so we can regard 4` as a subspace of the Hilbert space (2.4)
YfT -4 L2([0, T] x
a([0, T])
T Pm)
More precisely, we regard an equivalence class in A9T as a member of .27 if
it contains a progressively measurable representative. Here and later we replace [0, T] by [0, co) when T = co. 2.2 Lemma. For 0 < T < co, .27 is a closed subspace of .09T. In particular, .,27 is complete under the norm [X]i, of (2.3).
3. Stochastic Integration
132
PROOF. Let {X(")},111 be a convergent sequence in .27 with limit X EST. We may extract a subsequence, also denoted by {X(")},,m11, for which { (t, co) e [0, T] x
lim X: "°(co) 5 Xt(co)} = O. n-,co
By virtue of its membership in Yer, X is ,4([0, T]) not be progressively measurable. However, with
.F-measurable, but may
A -A {(t,co)e [0, T] x i2; lim X;"°(w) exists in 51),
the process )7, (w) _4_
lim XP)(co);
(t, w) e A
0;
(t, CO) A
{n-co
inherits progressive measurability from {X(")}c°_, and is equivalent to X.
0
A. Simple Processes and Approximations 2.3 Definition. A process X is called simple if there exists a strictly increasing sequence of real numbers {tn}n°3_0 with to = 0 and lim, t = oo, as well as a sequence of random variables g1,110 with supn,oln(01 < C < co, for every we 1, such that is Ft.-measurable for every n > 0 and CO
i(co)10,,,,,i(t);
X,(co) = 0(co) 1{0} (t) +
0 < t < co, co e
i=o
The class of all simple processes will be denoted by Yo. Note that, because members of ..F0 are progressively measurable and bounded, we have .290 g ...?*(M) c .29(M).
Our program for the construction of the stochastic integral (2.1) can now be outlined as follows: the integral is defined in the obvious way for X e as a martingale transform: n-1
(2.5)
it(X) A E
- m) + um, - 4)
co
= E um, A ti+, - MIA t,), i=0
t<
where n > 0 is the unique integer for which t < t < tn+1. The definition is then extended to integrands X e 21* and X e .29, thanks to the crucial results which show that elements of .21* and _V' can be approximated, in a suitable sense, by simple processes (Propositions 2.6 and 2.8). 2.4 Lemma. Let X be a bounded, measurable, {g;}-adapted process. Then there exists a sequence {X('")}:=1 of simple processes such that
3.2. Construction of the Stochastic Integral
sup lira E
(2.6)
T>0 m-.co
f
133
T
RO'n) - X,12 dt = 0.
0
PROOF. We shall show how to construct, for each fixed T > 0, a sequence {x(n,T)} flop_ of simple processes so that lim E
f
I XIn'T) - X,I2dt = 0.
o
Thus, for each positive integer m, there is another integer n,,, such that m
1,0"-m) - X,I2 dt < -1
E
m
o
and the sequence {X("-m)},',., has the desired properties. Henceforth, T is a fixed, positive number. We proceed in three steps. (a) Suppose that X is continuous; then the sequence of simple processes 2.-1
)0 n)(C0) A X0(01{0}(t) + E X0-724(0) laT/2",(k+1)T/2.1(t); n _. 1, k=0
satisfies lim, E f o IX; ") - x,12 dt = 0 by the bounded convergence theorem. (b) Now suppose that X is progressively measurable; we consider the continuous, progressively measurable processes t
(2.7) F,(w)
J.
T
XJw) ds; fes'n)(co)
m[Ft(w) - Fo_(1 /m)".(co)]; m
1,
for t > 0, w e f2 (cf. Problem 1.2.19). By virtue of step (a), there exists, for
each m > 1, a sequence of simple processes {g(m")},T=1 such that limn E - -1:"1)12 dt = 0. Let us consider the A[0, n) .FTmeasurable product set A A {(t, w) e [0, T] x f2; lim it'n(co) = Xi(w)lc.
For each w e f2, the cross section A.
{t e [0, T]; (t,
e A} is in
a([0, T]) and, according to the fundamental theorem of calculus, has Lebesgue measure zero. The bounded convergence theorem now gives lim,n_,. E fpfon)- X,I2 dt = 0, and so a sequence {g(m "-)}:=1 of bounded, simple processes can be chosen, for which T
I i{"-) - X,I2 dt = 0.
lim E m-
(c)
o
Finally, let X be measurable and adapted. We cannot guarantee immediately that the continuous process F = {Ft; 0 < t < co} in (2.7) is progressively measurable, because we do not know whether it is adapted. We do
134
3. Stochastic Integration
know, however, that the process X has a progressively measurable modification Y (Proposition 1.1.12), and we now show that the progressively measurable process {G, g po^ T Ysds, A; 0 5 t 5 T} is a modification of F. For the measurable process ;Mu)) = 0 < t < T, w en, we have from Fubini: E (T)ri,(w)dt = 11; P[x,(0)) Y,(w)]dt = 0. Therefore, 14' ri,(w)dt = 0 for P-a.e. wen. Now {F, 0 G,1 is contained in the event Iw; f o ri,(w) dt > 0 }, G, is A- measurable, and, by assumption, A contains all subsets of P-null events. Therefore, F, is also A- measurable. Adaptivity and continuity imply progressive measurability, and we may now repeat verbatim the argument in (b). 25 Problem. This problem outlines a method by which the use of Proposition 1.1.12, a result not proved in this text, can be avoided in part (c) of the proof of Lemma 2.4. Let X be a bounded, measurable, {. }- adapted process. Let 0 < T < co be fixed. We wish to construct a sequence {X(k)},T_I of simple processes so that T
lim E
(2.8)
k
1)0k) - X,I2 dt = 0.
co
To simplify notation, we set X, = 0 for t < 0. Let cp:
Or"; j =
0, + 1, + 2, ...1 be given by (0 .(t)
for
2"
1
2"
(a) Fix s > 0. Show that t - (1/2") < cp(t - s) + s < t, and that t>0 is a simple, adapted process. (b) Show that lim,40 E 14. X, - X,_I2 dt = 0. (c) Use (a) and (b) to show that T
1
lim E n-
1,O"'s) - X,I2 ds dt = 0. 0
0
(d) Show that for some choice of s > 0 and some increasing sequence Ink1,°_, of integers, (2.8) holds with X(k) = X("k's). This argument is adapted from Liptser and Shiryaev (1977).
2.6 Proposition. If the function t
<M>,(w) is absolutely continuous with
respect to Lebesgue measure for P-a.e. wen, then .290 is dense in .29 with respect to the metric of Definition 2.1. PROOF.
(a) If X e .29 is bounded, then Lemma 2.4 guarantees the existence of a bounded sequence {V")} of simple processes satisfying (2.6). From these
3.2. Construction of the Stochastic Integral
135
we extract a subsequence {X(mk)}, such that the set {(t, w) e [0, co) x SI; lim Xrnkga)) = Xt(co)}`
has product measure zero. The absolute continuity of t
<M>,(co) and
the bounded convergence theorem now imply [X(mk) - X] -+ 0 as k -+ co. (b) If X e .2 is not necessarily bounded, we define )(riga)) -4 Xi(a))1{1x,(,01.(n);
0 < t < co, a)E12,
and thereby obtain a sequence of bounded processes in 2. The dominated convergence theorem implies [X(n)
T
-
E
X,21{
nao 0
for every T > 0, whence lim, [X(n) - X] = 0. Each X(") can be approximated by bounded, simple processes, so X can be as well.
When t H <M>, is not an absolutely continuous function of the time variable t, we simply choose a more convenient clock. We show how to do this in slightly greater generality than needed for the present application. 2.7 Lemma. Let {A1; 0 < t < co} be a continuous, increasing (Definition 1.4.4) process adapted to the filtration of the martingale M = 1111.F,,; 0 < t < col. If X = {X..F,; 0 < t < oo} is a progressively measurable process satisfying T
E
X,2 dA, < oo
for each T > 0, then there exists a sequence {X(")}°11 of simple processes such that
sup lim E
1XP) - X,I2 dA, = 0.
T>0 nom
0
PROOF. We may assume without loss of generality that X is bounded (cf. part (b) in the proof of Proposition 2.6), i.e., (2.9)
IXt(o)))
C < co; V t > 0, a) ED.
As in the proof of Lemma 2.4, it suffices to show how to construct, for each fixed T > 0, a sequence {X(")},;°_, of simple processes for which T
lim E n-
!XI") - X,I2 dA, = O. 0
Henceforth T > 0 is fixed, and we assume without loss of generality that (2.10)
Xi(w) = 0; V t > 7', a) e
3. Stochastic Integration
136
We now describe the time-change. Since A,(w) + t is strictly increasing in t > 0 for P-a.e. w, there is a continuous, strictly increasing inverse function T(w), defined for s > 0, such that V s > 0.
A ,;(,)(w) + Ts(w) = s;
In particular, Ts < s and {T, < t} = {A, + t > s} E. Thus, for each s > 0, Ts is a bounded stopping time for Taking s as our new time-variable, we define a new filtration IC by
S>0,
= and introduce the time-changed process
s > 0, wen,
YS(w) = XT,(0,)(0));
which is adapted to {Ws} because of the progressive measurability of X (Proposition 1.2.18). Lemma 2.4 implies that, given any e > 0 and R > 0, there is a simple process { V, Ws; 0 s < col for which E
(2.11)
- Y.,12 ds < 6/2.
But from (2.9), (2.10) it develops that CO
EJ
CO
1;2 ds = E
= E j'AT-1-7. X?..ds < C(EAT + T)< op, o
so by choosing R in (2.11) sufficiently large and setting V = 0 for s > R, we can obtain CO
E
J
IYS`- Y1Zds <e.
Now V is simple, and because it vanishes for s > R, there is a finite partition
< s < R with
0 = so < s, <
n
0 < s < 00,
Y:(0)) = o((.0)1{0}(5) + E
where each
s,
is measurable with respect to Ws, = FT, and bounded in
absolute value by a constant, say K. Reverting to the original clock, we observe
that Xt A Yi+A,
= 01{0}(t) + E j =1
is measurable and adapted, because (Lemma 1.2.15). We have
,
1(T
s -
T.J 1(4
0
t < co,
restricted to { Ts, < t} is g,-measurable
3.2. Construction of the Stochastic Integral
E
f
IX: - X,I2 dA, < E
137
ixf - X,12(dA, + dt)
J CO
E
- 1;12 ds < E.
f
0
The proof is not yet complete because X' is not a simple process. To finish it off, we must show how to approximate 0 < t < co, co ED,
,(0))1(7;,(.),7;(01(t);
)1t(w)
by simple processes. Recall that Ts,_, < taking sfri = 1, = 2. Set 1 + 2" k Ti(m)(w) =
kZ.1
1
2m
< s; and simplify notation by
TS
((k -1)/2 P", k/2^)( T(W)),
i = 1, 2
and define q!'")((o) A
(01(71-00o, 4m )(.)1(t) 2^' +I
= k=1 E
IT <(k-1)/2'n T2)(W) I ((k -1)/2", k/2",1(t).
Because {T1 < (k - 1)/2m < T2} e 37;(k_i vv. and restricted to {T1 < (k - 1)12m} is 5"(k -1)/2,measurable, ri(m) is simple. Furthermore, 1
E
f
In!'") - 71,12 dA, < K2[E(Ann) - AT2) + E(Avr - AT1)] .7-,03 O.
0
2.8 Proposition. The set Yo of simple processes is dense in Y* with respect to the metric of Definition 2.1. PROOF. Take A = <M> in Lemma 2.7.
B. Construction and Elementary Properties of the Integral We have already defined the stochastic integral of a simple process X e ..To by the recipe (2.5). Let us list certain properties of this integral: for X, Y e SP, and
0 < s < t < co, we have (2.12) (2.13) (2.14) (2.15)
10(X) = 0, a.s. P
EU,(X)IFJ = 4(X), a.s. P E((X))2 I =E 111(X)II
ft
[X]
d<M>.
3. Stochastic Integration
138
E[(I,(X) - Is(X))21.F.J = E [ft
(2.16)
d<M> .Fs], a.s. P
1(aX + 13Y) = al (X + 131(Y);
(2.17)
a, /3e R.
Properties (2.12) and (2.17) are obvious. Property (2.13) follows from the fact that for any 0 < s < t < co and any integer i > 1, we have, in the notation of (2.5),
- Mt At,)Igs] = i(A4sAt,, - MsAt,), a.s. P; this can be verified separately for each of the three cases s < ti, t1 < s < ti+1, Thus, we see that I(X) = and ti+, < s by using the gt,-measurability of 11,(X),,,; 0 < t < co }is a continuous martingale. With 0 < s < t < oo and m and n chosen so that tm_i < s < tm and t,, < t < tn+1, we have (cf. the discussion preceding Lemma 1.5.9) (2.18)
E[(I,(X) - 1,(X))21.F,s]
= E[{,_1(Mtm
-
n-1
-A41,)+ utii-min)}2
+ n-1
- mo2 + a(vit - Nit)2
= E[a-i(Mtm - Ms)2 + E
-<M>) +
= + =E
M<M>t,
i=m
<M)t,)
(<M>, -
[I X,; d<M> gsl
This proves (2.16) and establishes the fact that the continuous martingale 1(X) is square-integrable: I(X)e,11, with quadratic variation (2.19)
, =
J
t
Xu d<M>.
Setting s = 0 and taking expectations in (2.16), we obtain (2.14), and (2.15) follows immediately, upon recalling Definition 1.5.22. For X E 2' *, Proposition 2.8 implies the existence of a sequence {V)},;°_, g ..ro such that [X(") - X] 0 as n co. It follows from (2.15) and (2.17) that
III(x("))-1(x(m))11 = III(x(n) - x(m))11 = [X(n) -x(m)].o as n, m co. In other words, 11(X("))1;°_1 is a Cauchy sequence in .t.. By Proposition 1.5.23, there exists a process I(X) = {I,(X); 0 < t < co} in
3.2. Construction of the Stochastic Integral
139
defined modulo indistinguishability, such that II/(X(")) - /(X)I1
0 as n
co.
Because it belongs to Atl2, I(X) enjoys properties (2.12) and (2.13). For 0 < s < t < co, {/s(V)));°-1 and {h(X("))),`,ci converge in mean-square to IS(X) and I,(X), respectively; so for A e gs, (2.16) applied to {V^)}:=1 gives (2.20)
E[lA(h(X) - 1s(X))2] = lim E[1A(I,(X(n)) - is(X(n)))2] n-.co
= lim E[1A ft (X,(,"))2 d<M>d n-.co
s
d<M>d,
= E[lA
where the last equality follows from [V) - X], = 0. This proves that I(X) also satisfies (2.16) and, consequently, (2.14) and (2.15). Because X and
M are progressively measurable, Its
d<M>. is A- measurable for fixed
0 < s < t < co, and so (2.16) gives us (2.19). The validity of (2.17) for X, Y e also follows from its validity for processes in Yo, upon passage to the limit.
The process 1(X) for X e 2* is well defined; if we have two sequences
{X(")},711 and { Pnl,°,11 in 2'0 with the property lim, [X" - X] = 0, - X] = 0, we can construct a third sequence {Z(1:11 with this property, by setting Z(2"-1) = X(n) and Z(2") = Y(n), for n > 1. The limit 1(X)
of the sequence {/(Z("))},7°_, in dic2 has to agree with the limits of both sequences, namely {/(X("))}:=, and 11(P"))1°11.
2.9 Definition. For X e...9", the stochastic integral of X with respect to the martingale M e .112 is the unique, square-integrable martingale I(X) =
{Ii(X),g;; 0 < t < col which satisfies limn II/(X(")) - 1(X)II = 0, for every sequence {Vnl,;°_, g 210 with lim, [X(n) - X] = 0. We write .1,(X) = .1' XsdMs;
0
2.10 Proposition. For M e,46 and X e.r*, the stochastic integral I(X) = {Ii(X),A; 0 5 t < 00} of X with respect to M satisfies (2.12)-(2.16), as well as (2.17) for every Ye 21*, and has quadratic variation process given by (2.19). Furthermore, for any two stopping times S < T of the filtration {A} and any number t > 0, we have
ER T(X)j,,s] =
(2.21)
s(X),
a.s. P.
With X, Y e 21* we have, a.s. P: (2.22)
EUlt T(X) -
A s(X))(I t A AY)
- t A s(Y))1-Fs]
E[ft A T
=
t As
XYd<M> g.s],
and in particular, for any number s in [0, t],
3. Stochastic Integration
140
(2.23)
E[(I,(X) - 15(X))(1,(Y) - Is(Y))1.F5] =
E[ ft X.Y.d<M>. gd. s
Finally,
1,,,T(x) = It(g) a.s.,
(2.24)
where 51,(w) g Xt(a))1{tr( .)} PROOF. We have already established (2.12)-(2.17) and (2.19). From (2.13) and the optional sampling theorem (Problem 1.3.24 (ii)), we obtain (2.21). The same
result applied to the martingale {/t2.(X) -Po XZ' d<M>., gr; t the identities
0} provides
E[(I(AT(X) -I , A s(X))2I.Fs] = EMT(X) - It2As(X)1,0
=
E[f
t AT
X! d<M>. .Fs], P-a.s.
I As
Replacing X in this equation, first by X + Y and then by X -Y, and subtracting the resulting equations, we obtain (2.22). It remains to prove (2.24). We write
/, A T(X) - Mg) = /tA T(X -JO -uf(S')_ it, Tan. Both {/tA T(X -fa A; t 0} and fli(.1) - 1,,,,,( ),gt; t 0} are in dic2; we show that they both have quadratic variation zero, and then appeal to Problem 1.5.12. Now relation (2.22) gives, for the first process,
E[(I(AT(X - i?) -Is T(X - 2))2I-Fs] tA r
= ELI
(X. - g.)2 d<M>. sAT
g;1= 0
a.s. P, which gives the desired conclusion. As for the second process, we have ,
E [(mg) -it A T(g))2] = E
[ftn.,' g! d<M>d = 0,
and since this is the expectation of the quadratic variation of this process, we again have the desired result.
2.11 Remark. If the sample paths t 1-- <M>,(co) of the quadratic variation process <M> are absolutely continuous functions of t for P-a.e. w, then Proposition 2.6 can be used instead of Proposition 2.8 to define 1(X) for every X E Y. We have I(X)e.112 and all the properties of Proposition 2.10 in this case. The only sticking point in the preceding arguments under these condi-
tions is the proof that the measurable process f; --4-- Ito Xs2 d<M>5 is {A}adapted. To see that it is, we can choose Y, a progressively measurable modification of X (Proposition 1.1.12), and define the progressively measur-
3.2. Construction of the Stochastic Integral
141
able process G, A Po Ys2 d<M>s. Following the proof of Lemma 2.4, step (c),
we can then show that P[F, =
= 1 holds for every t > 0. Because G, is A-measurable, and A contains all P-negligible events in F (the usual conditions), F is easily seen to be adapted to {A}. In the important case that M is standard Brownian motion with <M>, = t, the use of the unproven Proposition 1.1.12 can again be avoided. For bounded X, Problem 2.5 shows how to construct a sequence {X(k) }f_, of bounded, simple processes so that (2.8) holds; in particular, there is a subsequence, also called {Xm}f_, , such that for almost every to [0, T] we have Xs' ds = lim
F, 0
k-.0z,
f (X''))2 ds,
a.s. P.
0
Since the right-hand side is A-measurable and A contains all null events in F, the left-hand side is also A- measurable for a.e. t e [0, T]. The continuity of the samples paths of {Ft; t > 0} leads to the conclusion that this process is .y,- measurable for every t. For unbounded X, we use the localization technique employed in the proof of Proposition 2.6.
We shall not continue to deal explicitly with the case of absolutely continuous <M> and X e .29, but all results obtained for X e .29* can be modified
in the obvious way to account for this case. In later applications involving stochastic integrals with respect to martingales whose quadratic variations are absolutely continuous, we shall require only measurability and adaptivity rather than progressive measurability of integrands. 2.12 Problem. Let W = {W, A; 0 < t < co} be a standard, one-dimensional Brownian motion, and let T be a stopping time of {A } with ET < co. Prove the Wald identities E(WT) = 0, E(WT) = ET
(Warning: The optional sampling theorem cannot be applied directly because
W does not have a last element and T may not be bounded. The stopping time t n T is bounded for fixed 0 < t < co, so E(W,,,T) = 0, E(W t2 T) = E(t n T), but it is not a priori evident that (2.25)
lim E(W( A T)
=
EWT,
lim E(14
t-co
T) = E(WT2)-)
2.13 Exercise. Let W be as in Problem 2.12, let b be a real number, and let Tb be the passage time to b of (2.6.1). Use Problem 2.12 to show that for b 0 0,
we have ET = co.
C. A Characterization of the Integral Suppose M = {M
0
} and N = {N.F.,; 0 < t < co} are in M,
and take X e 2 *(M), Y e Y*(N). Then IM(X) .A SO Xs dMs,
on A SO YS dNs
3. Stochastic Integration
142
are also in .I12 and, according to (2.19),
, = f t XJ d<M>, , = f r Y2 d. We propose now to establish the cross-variation formula , = J r X Yd <M,N) ;
(2.26)
0, P-a.s.
t
0
If X and Y are simple, then it is straightforward to show by a computation similar to (2.18) that for 0 < s < t < co,
E[e (X) - InX))(INY) -Is Y))1,9;]
(2.27)
=
E[1, X d<M, N> gd;
P-a.s.
This is equivalent to (2.26). It remains to extend this result from simple processes to the case of X e -99*(M), Y e.29*(N). We carry out this extension in several stages, culminating in Propositions 2.17 and 2.19 with a very useful characterization of the stochastic integral. 2.14 Proposition (An Inequality of Kunita & Watanabe (1967)). If M, N e dr2, X e Y*(M), and Y e Y*(N), then a.s.
E
Ysl d4,
(f
Xs
d<M>s)1/2
(f
1/2
ds)
where 4, denotes the total variation of the process
;
0 < t < co,
A <M, N> on [0,s].
PROOF. According to Problem 1.5.7 (iv), (w) is absolutely continuous with respect to cp(co) A A<M> ] (co) for every co e n with P(C1) = 1, and for every such w the Radon-Niko4m theorem implies the existence of functions fie , w): [0, co) -> 111; i = 1, 2, 3, such that <M> (w) =
(s,
cicos(0),
,(co) = <M, N>,(co) =
Consequently, for a, /3 elA and w e
(co) =
f f3 (s, co) dcps(w); t
f2(s,(0)403(c0),
0 < t < co.
C satisfying P(Cap) = 1, we have
0 < ,(co) - (co)
=
(cef,(s, w) + 2aflf3(s, w) + fl2f2(s, w)) chps(co); 0 < u < t < co.
This can happen only if, for every co en, there exists a set 7;3(co)e a([o, 00)) with IT.ocodcpi(co) = 0 and such that
3.2. Construction of the Stochastic Integral (2.28)
143
c(2./1(t, co) + 20f3(t, co) + ig2f21t, (0)
0
holds for every t T(w). Now let a _4 n,,EQQ,, poi, A U243 7 (04 so that P(C') = 1, $T(,,,) dpi(w) = 0; V wen. Fix wen; then (2.28) holds for every t T(w) and every pair (a, /1) of rational numbers, and thus also for every t T(w), (1,13)e R2; in particular, a 2 I XtMl 2f1 (t,
0;
2ocl X ,(a) Y(0)11 f 3(t, co)i + I Y*012 f 2(t, (0) Vt
T(w).
Integrating with respect to dp, we obtain s12 d<m>s
oz2
20c s
0
YI s C14 s +
0
1Y 12 d5
0;
0 < t < CO,
0
almost surely, and the desired result follows by a minimization over a. 2.15 Lemma. If M, N e diec2, X e .29*(M), and {X(")};'11 S .29*(M) is such that
for some T > 0, lim n
f
co
I
Xi(in) -
2 d<M> = 0;
a.s. P,
o
then
lira , = (1(X), N>,;
0 < t < T, a.s. P.
n -0 co
PROOF. Problem 1.5.7 (iii) implies for 0 < t < T,
KI(X(")) - I(X),N>,12
, , 1XL°
Xu12 d<M>.' r.
2.16 Lemma. If M, N 6.41'2 and X e S9 *(M), then (2.29)
, = J
t
Xud<M,N>u; VOt< op, a.s.
PROOF. According to Lemma 2.7, there exists a sequence {X"} ,;°_, of simple processes such that
sup lim E T>0
JT
IXL ")- XI2d <M>u =O.
co
Consequently, for each T > 0, a subsequence {;Y-(n)},;°_, can be extracted for which T
lim .1
n-:o
- X12 d<M>u = 0,
a.s.
3. Stochastic Integration
144
But (2.26) holds for simple processes, and so we have , = f i
g;,") d<M, N>; 0 S t< T
0
almost surely; letting n -* co we obtain (2.29) from Lemma 2.15 and the Kunita-Watanabe inequality (Proposition 2.14). 2.17 Proposition. If M, N e dll, X e_r*(M), and Y e 2*(N), then the equivalent formulas (2.26) and (2.27) hold.
PROOF. Lemma 2.16 states that d<M, IN(Y)>,, = y, d<M, NX. Replacing N in (2.29) by /N(Y), we have , = f t X,,d<M, IN(Y)> o ft
=
X,,Y,,d<M, N>u;
t _. 0, P-a.s.
o
2.18 Problem. Let M = {M .9;; 0 S t < co} and N, = {Ng",; 0 < t < co} be in dr2 and suppose X e .9':,(M), Y e ..rt(N). Then the martingales /"'(X), /NY)
are uniformly integrable and have last elements km(X), /NY), the crossvariation , converges almost surely as t -) co, and E[P0M,(X)IZ (Y)] = E
c°
X, Y, d <M , N X.
o
In particular, E(fX,
o
dM,)2 = E
X,2 d<MX. o
2.19 Proposition. Consider a martingale M e ..e2 and a process X E ..r*(M). The stochastic integral Pm (X) is the unique martingale 40 e.11z which satisfies (2.30)
<01), N>, =
f t X d<M, N>; 0 < t < co, a.s. P, o
for every N e dic2.
PROOF. We already know from (2.29) that (1) = /m(X) satisfies (2.30). For uniqueness, suppose (1) satisfies (2.30) for every N e ./11. Subtracting (2.29) from (2.30), we have
<0 - /M(X), NX = 0; 0 < t < oo, a.s. P. Setting N = 41 - IM (X), we see that the continuous martingale (1) - /m(X) has quadratic variation zero, so (1:0 = IM(X).
3.2. Construction of the Stochastic Integral
145
Proposition 2.19 characterizes the stochastic integral IM(X) in terms of the more familiar Lebesgue-Stieltjes integral appearing on the right-hand side of (2.30). Such an idea is extremely useful, as the following corollaries illustrate. In the shorthand "stochastic differential" notation, the first of these states that if dN = X dM, then Y dN = X Y dM. 2.20 Corollary. Suppose M e .ill, X E 2*(M), and N -1=-' IM(X). Suppose further that Y e ..r*(N). Then X Y e ..?*(M) and IN(Y) = IM(X Y). PROOF. Because , = Po Xs2d<M>s, we have fT
X,2Y,2 d<M>, = E
E
o
j
T
l',2 d, < co o
for all T > 0, so X Y e 1 "(M). For any F/ e .1,1, (2.26) gives d, =
Xsd<M,R> and thus , =
f
i
XsYsd<M, R>,
o
=
ft Ysd, = ,. o
According to Proposition 2.19, IM(X Y) = IN(Y).
2.21 Corollary. Suppose M, R e ./e2, X e Y*(M), and 2 e 2 *(1C1), and there exists a stopping time T of the common filtration for these processes, such that for P-almost every co, X, A T(04(W) = it A T(c))(W),
Mt A T(u))(W) = lilt A T(u)/(W);
0 < t < CO.
Then
I7 T(,)(X)(co) = Ii 7-00(50(w);
0 5 t < 00, for P-a.e. w.
PROOF. For any N e ..11, we have <M - fa, N>t A T = 0; 0 < t < co, and so (2,29) implies , A T = 0; 0 < t < co. Setting N = IM(X) Im (1) and using Problem 1.5.12, we obtain the desired result. 111
D. Integration with Respect to Continuous, Local Martingales Corollary 2.21 shows that stochastic integrals are determined locally by the local values of the integrator and integrand. This fact allows us to broaden
the classes of both integrators and integrands, a project which we now undertake.
Let M = {M Ft; 0 5_ t < co} be a continuous, local martingale on a probability space (a, ,, P) with Mo = 0 a.s., i.e., M e dr I" (Definition 1.5.15). Recall the standing assumption that {F,} satisfies the usual conditions. We
3. Stochastic Integration
146
define an equivalence relation on the set of measurable, {g,}-adapted processes just as we did in the paragraph preceding Definition 2.1. 2.22 Definition. We denote by 91 the collection of equivalence classes of all measurable, adapted processes X = {X.Fi; 0 < t < col satisfying (2.31)
P[f
d<M>, < co] = 1
for every T e [0, co).
We denote by 91* the collection of equivalence classes of all progressively measurable processes satisfying this condition. Again, we shall abuse terminology by speaking of 91 and g* as if they were classes of processes. As an example of such an abuse, we write g* g g, and if M belongs to lfZ (in which case both 2' and 2)* are defined) we write g g and ..c," g". We shall continue our development only for integrands in g*. If a.e. path t H <M>,(w) of the quadratic variation process <M> is an absolutely continuous function, we can choose integrands from the wider class g. The reader will see how to accomplish this with the aid of Remark 2.11, once we complete
the development for g". Because M is in .41"", there is a nondecreasing sequence {S},°=, of stopping
times of {A}, such that lim, S = oo a.s. P, and kmtAs,,,,Ft; 0 < t < col is in .41. For X e g", one constructs another sequence of bounded stopping times by setting R,,(w)
n A inf {0
t < co;
XR(o)d <M>s(co)
n}.
R = co,
This is also a nondecreasing sequence and, because of (2.31), a.s. P. For n > 1, co e Q, set (2.32) (2.33)
T(w) = R(w) A Sn(w), M")(w) A Mt A "jai)/ )011)(W) = Xt(01
TWO> t} ;
t<
Then WI) e and X(")e ..9"(M(")), n > 1, so InX(")) is defined. Corollary 2.21 implies that for 1 < n < m,
/r)(X(")) = /r""(X(''')), 0 < t < T, so we may define the stochastic integral as (2.34)
/,(X) A /r)(X("))
on {0 < t < T}.
This definition is consistent, is independent of the choice of {S},fi mines a continuous process, which is also a local martingale.
and deter-
2.23 Definition. For M e ../1"°e and X e 9", the stochastic integral of X with respect to M is the process 1(X) {I,(X),,,; 0 < t < col in de.'" defined by (2.34). As before, we often write co Xs di I 1 instead of /,(X).
3.2. Construction of the Stochastic Integral
147
When M e di'mc and X e4)*, the integral I (X) will not in general satisfy conditions (2.13)-(2.16), (2.21)-(2.23), or (2.27), which involve expectations at fixed times or unrestricted stopping times. However, the sample path properties (2.12), (2.17), (2.19), (2.24), and (2.26) are still valid and can be easily proved by localization. We also have the following version of Proposition 2.19.
2.24 Proposition. Consider a local martingale M e dr' and a process X e go*(M). The stochastic integral IM(X) is the unique local martingale (De dr" which satisfies (2.30) for every N e de, (or equivalently, for every N e
2.25 Problem. Suppose M, N e d' and X e .9* (M) n .9* (N). Show that for every pair (a, /3) of real numbers we have
rm+PN(X) = a/m(X) + 08(X). 2.26 Proposition. Let M e
{X(")};,°_,, g .9* (M), X e g* (M) and suppose that for some stopping time T of {F,} we have II; I XI") -X, I2 d <M>, = 0, in probability. Then
X(s") dM, - J X,dMs
sup 0.1EST
11-.00
*
PROOF. The proof follows immediately from Problem 1.5.25 and Proposition 2.24.
2.27 Problem. Let M e dic'" and choose X e g*. Show that there exists a sequence of simple processes {X(")},;°_, such that, for every T > 0, T
X: ")- X,I2 d<M>, =0
lim n- o0
0
and lim n-
sup I /,(X(")) - 4(X)I = 0
051ST
hold a.s. P. If M is a standard, one-dimensional Brownian motion, then the preceding also hold with X e .9.
2.28 Problem. Let M = W be standard Brownian motion and X e g. We
define for 0 <s
Cf(X) A
f
t
XdW - -1
2
du;
(,(X)
CP(X)
s
The process lexp((,(X)).Ft; 0 < t < co} is a supermartingale; it is a martingale if X e.ro. Can one characterize the class of processes X e Y*, for which the exponential supermartingale {exp (MX)), .9;; 0 t < co) of Problem 2.28 is in fact a
3. Stochastic Integration
148
martingale? This question is at the heart of the important result known as the Girsanov theorem (Theorem 5.1); we shall try to provide an answer in Section 5.
2.29 Problem. Let W be a standard Brownian motion, e a number in [0, 1], and ri = Ito,t..., tn, a partition of [0, t] with 0 = to < t, < < tm = t. Consider the approximating sum rn -1
(2.36)
-
um -4 1=0 E
for the stochastic integral Sto Ws dWs. Show that
lim Sc(I1) =
(2.37)
- 14/,' + (c - -2) t, 1
1
2
where the limit is in L2. The right-hand side of (2.37) is a martingale if and only if e = 0, so that W is evaluated at the left-hand endpoint of each interval [t1, t 1] in the approximating sum (2.36); this corresponds to the Ito integral. With e = i we obtain the Fisk-Stratonovich integral, which obeys the usual rules of calculus such as ro Ws 0 dW, = 1147,2; we shall have more to say about
this in Problems 3.14, 3.15. Finally, e = 1 leads to the backward Ito integral (McKean (1969), p. 35). The sensitivity of the limit in (2.37) to the value of e is a consequence of the unbounded variation of the Brownian path. We know all too well that it is one thing to develop a theory of integration in some reasonable generality, and a completely different task to compute the integral in any specific case of interest. Indeed, one cannot be expected to repeat the (sometimes arduous) process which fortunately led to an answer in the preceding problem. Just as we develop a calculus for the Riemann integral, which provides us with tools necessary for more or less mechanical computations, we need a stochastic calculus for the Ito integral and its extensions. We take up this task in the next section. 2.30 Exercise. For M a die' Inc, X ego *, and Z an .Fs-measurable random variable, show that
ZX.dM = Z ft X.dM.; s < t < co, a.s. S
3.3. The Change-of-Variable Formula One of the most important tools in the study of stochastic processes of the martingale type is the change-of-variable formula, or Ito's rule, as it is better known. It provides an integral-differential calculus for the sample paths of such processes.
3.3. The Change-of-Variable Formula
149
Let us consider again a basic probability space (0, P) with an associated filtration {,F,} which we always assume to satisfy the usual conditions.
3.1 Definition. A continuous semimartingale X = {X A; 0 < t < oo} is an adapted process which has the decomposition, P a.s.,
X, = X0 + M, + Bt;
(3.1)
0 < t < co,
where M = {M
0 < t < co} e .11'''" (Definition 1.5.15) and B = {B.9",; 0 < t < co} is the difference of continuous, nondecreasing, adapted processes
{A,±,,,; 0 < t < co}:
B, = A;F. -,c; 0 < t < GO,
(3.2)
with A,4-- = 0, P a.s. We shall always assume that (3.2) is the minimal decomposition of B; in other words, A,'" is the positive variation of B on [0, t] and 2=1- is the negative variation. The total variation of B on [0, t] is then A,+ .
The following problem discusses the question of uniqueness for the decomposition (3.1) of a continuous semimartingale.
3.2 Problem. Let X = {X,,; 0 < t < co } be a continuous semimartingale with decomposition (3.1). Suppose that X has another decomposition
+
X, =
+111,;
0 < t < co,
where M e../1'1" and is a continuous, adapted process which has finite total variation on each bounded interval [0, t]. Prove that P-a.s.,
M, = -M B, =
0
A. The ItO Rule Ito's formula states that a "smooth function" of a continuous semimartingale is a continuous semimartingale, and provides its decomposition.
3.3 Theorem. (Ito (1944), Kunita & Watanabe (1967)). Let f:[18-[18 be a function of class C2 and let X = {X A; 0 < t < co} be a continuous semimartingale with decomposition (3.1). Then, P-a.s., (3.3)
f(X,) = f(X0)+
f+ 2
0
f'(Xs)dM, +
f "(X )d<M> s, s
f f'(Xs)dBs
0< t<
co .
3.4 Remark. For fixed co and t> 0, the function Xs(w) is bounded for 0 < s 5 t, so f ' (X s(co)) is bounded on this interval. It follows that ft° f ' (X s) dM s is defined
150
3. Stochastic Integration
as in the last section, and this stochastic integral is a continuous, local martingale. The other two integrals in (3.3) are to be understood in the Lebesgue-Stieltjes sense (Remark 1.4.6 (i)), and so, as functions of the upper limit of integration, are of bounded variation. Thus, {f (X,),,; 0 < t < col is a continuous semimartingale.
3.5 Remark. Equation (3.3) is often written in differential notation: (3.3)'
df(X,) = f '(X,)dM, + f ' (X,) dB, +
-1
2
f "(X,) d Of> ,
= f(Xt) dXt + -2 f "PO d 00,, 0 < t < oo. This is the "chain-rule" for stochastic calculus. PROOF OF THEOREM 3.3. The proof will be accomplished in several steps.
Step 1: Localization. In the notation of Definition 3.1 we introduce, for each n _>_ 1, the stopping time 0;
if I X0 I
T = { inf{ t co;
.. n,
0; I Mt 1
n or 4 _. n or <M>,
if I Xo I < n and { ... }
n};
if I Xo I < n,
= 0.
The resulting sequence is nondecreasing with lim, T. = co, P-a.s. Thus, if we can establish (3.3) for the stopped process X:0 .4--, --i X T ,: t > 0, then we obtain the desired result upon letting n co. We may assume, therefore, that X0(w) and the random functions M ,(co), 13,(w), and <M>,(w) on [0, co) x ("/ are
all bounded by a common constant K; in particular, M is then a bounded martingale. Under this assumption, we have I X*0)1 < 3K; 0 < t < co, wee, so the values of f outside [ - 3K, 3K] are irrelevant. We assume without loss of generality that f has compact support, and so f, f', and f" are bounded. Step 2: Taylor expansion. Let us fix t > 0 and a partition II = {to,ti, ...,t,,,} of [0, t], with 0 = to < t, < < t, = t. A Taylor expansion yields
.
./(Xt) -f(x0)=k=1 E {f(xt,)- fix,_,)}
. l = kE f'(x,_,)(x,- x,_,)+ 7,kE=1 f "(nk)(xt, =1
x(,_,)2,
where 1h(w) = X,k_t(w) + 0(co)(X,k(w) - X,k_i(w)) for some appropriate Ok (CO) satisfying 0 < Ok(w) < 1, co E n. We conclude that (3.4)
where
ffX() - fiXo) = -11 (II) + J2(1-1) + 4J3(H),
3.3. The Change-of-Variable Formula
J1(1-1) sE mE f
151
'
- Btk ,), -
.12(11) A E
k=1
E flnk)(xt, - x( 1)2.
J3(11)
k=1
It is easily seen that .11(11) converges to the Lebesgue-Stieltjes integral fof '(Xs)dBs, a.s. P, as the mesh 111111 = maxi <,n 1 tk - tk_11 of the partition
decreases to zero. On the other hand, the process
Ys(w) 9 f'(Xs(co)); 0< s< t, (yen, is in ..r* (adapted, continuous, and bounded); we intend to approximate it by the simple process Kn(w)
f'(x0(0)1{0}(S)
Indeed, we have E1,2(Y" - Y) = EP° 1 Ys' - K12 d<M>, the bounded convergence theorem, and so J2(11) =
Jo
YsndMs
0 as 111111 - 0, by
KdMs
in quadratic mean. Step 3: The quadratic variation term. J3(II) can be written as J3(11) = 4(11) + J5(11) + J6(11), where m
E flIk)(13k=1
4(11)
m
J5(r1) 'A 2
flnk)(13,- Aki)(111,- m(k_,),
J6(11) A E f"(N)(Altkk=1
Because B has total variation bounded by K, we have 14(11)1 + 1J5(11)1 21(11f "11 co
( max
15k5nt
I Bfk - Bik_ii + max
1
I Mtk
-
II)
and, thanks to the continuity of the processes B and M, this last term converges
0 (as well as in L (II, "", P), because of the bounded convergence theorem). As for J6(II), we define
to zero almost surely as uno
3. Stochastic Integration
152
.
'A E flx,_,)(mtk- mik 1)2 k=1
and observe
14(11) - J6(n)I < vi(2)(n)- max If"(h)- f"(x,_1)I, 1 Sk
where V,(2)(11) is the quadratic variation of M over the partition n (c.f. Theorem 1.5.8 and the discussion preceding it). According to Lemma 1.5.9 and the Cauchy-Schwarz inequality, 2
EIJ:(11) - J6(II)I
.,/48K4.1E ( max If"(nk) - f."(Xti,_ i )1 ) 1 'icSm
and this is seen to converge to zero as II nil -+ 0 because of the continuity of
the process X and the bounded convergence theorem. Thus, in order to establish the convergence of the quadratic variation term J3(11) to the integral SI, f "(Xs) d<M>, in 1.1 (SI, .97 , P) as 111111 - 0, it suffices to compare J:(11) to the approximating sum
.17(11) A i f"(x,_,)(<m),k- <m>,_,) k=i
Recalling the discussion just before Lemma 1.5.9, we obtain
WWI) -.17(1)12 =E
=
2
Lm f"(xi, )1(14,,- 14,_,)2 -(04),- <M>tk_,)}
k=1
E[X___1
[f "(Xtk)]2
- Alti,_,)2 - (<m)t - <m>tk_1)}2]
.
211f"ii2
.
'E[I, (M -M )4 +k=1E Km\ - <m>, k=1
211.i" II 2.D'E
[ Vt(4(1-1) + OW >t
i
)21
max (Oki/Xi, - <M>tk id'
1
and Lemma 1.5.10 together with the bounded convergence theorem shows that the last term in the preceding equations goes to zero as 1111 11 -, 0. Since convergence in L2 implies convergence in L1, we conclude that J3(11) 11-=-+.0 Jo f"(Xs)d<M>,
in LI (11,,, P).
Step 4: Final Touches. If Inool-_, is a sequence of partitions of [0, t] with 1111 °'l
_-77- o, then for some subsequence In(^01;10_1 we have, P-a.s.,
lim J10^0) = f i f'(Xs)d135,
k-kco
0
3.3. The Change-of-Variable Formula
153
lim .12(-100) = k-oo
f f'(Xs)dMs, 0
lim J3(fl(k)) = ft f"(Xs)d<M>s
k-.0o
0
Thus, passing to the limit in (3.4), we see that (3.3) holds P-a.s. for each 0 < t < co. In other words, the processes on the two sides of equality (3.3) are
modifications of one another. Since both of them are continuous, they are indistinguishable (Problem 1.1.5).
I=1
We have the following, multidimensional version of Ito's rule.
3.6 Theorem. Let {M, A (Mr),
Mr)), ,F,; 0 < t < col be a vector of local ,Br)), g;; 0 5 t < col a vector of adapted processes of bounded variation with Bo = 0, and set X, = X0 + M, + B,; 0 < t < co, where X0 is an go- measurable random vector in Rd. Let f(t,x): [0, co) x Rd R be of class C"2. Then, a.s. P, ,
martingales in .11`''", {B, A (BP),
t
(3.5)
d
f -atf(s, Xs)ds + E
f(t, X,) = f(0, X0) +
o
a
d
+E 1=1
Oxi
1d
d
ox,
f(s, Xs)dBli)
xodmv 32
+5EE - i=1 j=1
f(s, X s) d <M"), Mul>s,
0
co.
0
3.7 Problem. Prove Theorem 3.6.
3.8 Example. With M = W = Brownian motion, X0 = 0, B, = 0 and f(x) = x2, we deduce from (3.3): Wi2 = 2
I
WsdW,
t.
Compare this with Problem 2.29.
3.9 Example. Again with M = W = Brownian motion, let us consider X EY and recall the exponential supermartingale of Problem 2.28: Z,
exp(C,);
0
t < co
where = C(X) of (2.35). We now check by application of ItO's rule that this process satisfies the stochastic integral equation (3.6)
Z, = 1 +
ZsX,dWs;
0 < t < co.
154
3. Stochastic Integration
Indeed, {(t; .9;; 0 < t < co} is a semimartingale, with local martingale part Yo X, dW, and bounded variation part B, we have
Z` = f(Cr) = f(C0) + f t f '(C)dN, +
ft f'gs)dB, +1 ft f"((s)d,
o
=1+
2
o
f Z,X,d4 + ft
Zs
o
o
=1+
-z f o X,2 ds. With f(x) = ex,
0
+
2
!2 f Z,X2 ds 0
f' Zs Xs d W. 0
The replacement of dN, by X,dW, in this equation is justified by Corollary 2.20 (actually, the extension of Corollary 2.20 to allow for the present case of M = W and X e1). It is usually more convenient to perform computations like this using differential notation. We write = X, dW, - IX,2 dt,
and, to reflect the fact that the martingale part of ( has quadratic variation with differential X,2 dt, we let (c/(,)2 = X,2 dt. One may obtain this from the formal computation (c/(,)2 = (X, dW, - 1-X,2 dt)2
= X,2(dW,)2 - X,3 dW, dt + 4V(dt)2 = X,2 dt,
using the conventional "multiplication table"
dt dW,
dW
dt
dW,
0 0 0
dt
0 0
0
dt
0
where W, W are independent Brownian motions (recall Problem 2.5.5). With this formalism, Ito's rule can be written as df((t) = f ' (Cdcg, + -if " (C,) (dCt)2,
and with f(x) = ex, we obtain dZ, = Z,X, dW, - 4Z,X,2 dt +
dt
= Z,X,dW,. Taking into account the initial condition Z0 = 1, we can then recover (3.6).
3.3. The Change-of-Variable Formula
155
3.10 Problem. With {Z1; 0 5 t < co} as in Example 3.9, set Y = 1/4 0 5 t < oo, which is well defined because P[info ,, < T 4 > 0] = P[info -OA = 1. Show that Y satisfies the stochastic differential equation
T Ct >
dy = Y, X,2 dt -Y, X, dW Yo = 1. 3.11 Example. One of the motivating forces behind the ItO calculus was a desire to understand the effects of additive noise on ordinary differential equations. Suppose, for example, that we add a noise term to the linear, ordinary differential equation
4(t)= a(t)(t) to obtain the stochastic differential equation
4 = a(t)e, dt + b(t)dW where a(t) and b(t) are measurable, nonrandom functions satisfying
JT la(t)Idt + 1o
f
T
b2(t) dt < oo; 0 < T < co,
o
and W is a Brownian motion. Applying the HO rule to XP)XP) with )01) 4 exp[Sto a(s) ds] and X:2) = 4 + Po b(s) exp[ -sso a(u) du] dWs, we see that , = XP'XP) solves the stochastic equation. Note that is well defined because, for 0 < T < co:
b (s)exp [ -2 o
f
s
a(u)du]ds
exp [2
o
f
T
o
T
ja(u)I did .1 b2(s)ds < co. o
A full treatment of linear stochastic differential equations appears in Sec. 5.6.
3.12 Problem. Suppose we have two continuous semimartingales (3.7)
X, = Xo + M, + B Y, = Yo + N, + c; 0 < t < oo ,
where M and N are in dr.'" and B and C are adapted, continuous processes of bounded variation with Bo = Co = 0 a.s. Prove the integration by parts formula (3.8)
f
t
o
Xsdy= X, Y, -X 0 Yo
-f t ydX - <M, N>,. o
The Ito calculus differs from ordinary calculus in that familiar formulas, such as the one for integration by parts, now have correction terms such as <M, N>, in (3.8). One way to avoid these corrections terms is to absorb them into the definition of the integral, thereby obtaining the Fisk-Stratonovich integral of Definition 3.13. Because it obeys the ordinary rules of calculus (Problem 3.14), the Fisk-Stratonovich integral is notationally more convenient than the HO integral in situations where ordinary and stochastic calculus interact; the primary example of such a situation is the theory of diffusions on
3. Stochastic Integration
156
differentiable manifolds. The Fisk-Stratonovich integral is also more robust under perturbations of the integrating semimartingale (see subsection 5.2.D), and thus a useful tool in modeling. We note, however, that this integral is defined for a narrower class of integrands than the HO integral (see Definition 3.13) and requires more smoothness in its chain rule (Problem 3.14). Whenever the Fisk-Stratonovich integral is defined, the Ito integral is also, and the two are related by (3.9).
3.13 Definition. Let X and Y be continuous semimartingales with decompositions given by (3.7). The Fisk-Stratonovich integral of Y with respect to
Xis (3.9)
1
01
Ysdill, +
lc 0 dX, -A
YsdB, + 2 <M, N>,;
o
0
t < co,
o
where the first integral on the right-hand side of (3.9) is an Ito integral.
3.14 Problem. Let X =
,
V)) be a vector of continuous semi-
martingales with decompositions
= Xg) + Ati) + where each Mo) e class C3, then (3.10)
i=
BP;
d,
I" and each B(1) is of the form (3.2). If f: l -R is of
f(x,)= fixo) +
=i
ft ,a
0 ox;
fajodxr.
3.15 Problem. Let X and Y be continuous semimartingales and n = {t0,t1,...,tm} a partition of [0, t] with 0 = t0 < t1 <
<
= t. Show that
the sum m-i
Y'+` + 2
IC'
(X`'`
converges in probability to Po Yso dX, as 111111
X`') 0.
B. Martingale Characterization of Brownian Motion In the hands of Kunita and Watanabe (1967), the change-of-variable formula (3.5) was shown to be the right tool for providing an elegant proof of P. Levy's celebrated martingale characterization of Brownian motion in Rd. Let us recall jr); F; 0 < t < col is a d-dimensional Brownian here that if {B, Ai); motion on (0, P) with P[B0 = 0] = 1, then <01°, Bul>, = bkit; 1 < k, j < d,
0 < t < cc (Problem 2.5.5). It turns out that this property characterizes Brownian motion among continuous local martingales. The compensated Poisson process with intensity A = 1 provides an example of a discontinuous,
3.3. The Change-of-Variable Formula
157
square-integrable martingale with <M>, = t (c.f. Example 1.5.4 and Exercise 1.5.20), so the assumption of continuity in the following theorem is essential.
0 < t < co} V)), 3.16 Theorem (P. Levy (1948)). Let X = {X, = (V), be a continuous, adapted process in Rd such that, for every component 1 < k < d, the process ,
.
Ar)
4k);
iylk)
0 < t < 00,
is a continuous local martingale relative to {,F,}, and the cross-variations are given by (3.11)
<AP), 0-5>t = Skit;
1 < k, j < d.
Then {X1,.97,; 0 < t < oo} is a d-dimensional Brownian motion.
PROOF. We must show that for 0 < s < t, the random vector X, -Xs is independent of and has the d-variate normal distribution with mean zero and covariance matrix equal to (t - s) times the (d x d) identity. In light of Lemma 2.6.13, it suffices to prove that for each u e Rd, with i = I-1, E[ei( = e-(1/2)iiuli2o-s), a.s. P. (3.12) For fixed u = (u a
ax.
, u) e Rd, the function f(x) = ei("'') satisfies a2
f(x) =
ax.J ax k
f(x) =
Applying Theorem 3.6 to the real and imaginary parts of f, we obtain (3.13)
d
E
ei( xt)
j=1
_1
E
uj
el("'x ) dv. s
2.1=1
s
Now If(x)1 < 1 for all xe Rd and, because <Mo>, = t, we have Mti) Thus, the real and imaginary parts of {r, eio"-)divt-h, A; 0 < t < co} are not only in .1r1, but also in .1'. Consequently,
E[ f t eR.,xv) dmy)
= 0,
P-a.s.
For A E., we may multiply (3.13) by e-1".)1A and take expectations to obtain E[ein,,xt-xoi
'A] =
P(A) - -2
E[ei(". x '4) 1A] dv.
This integral equation for the deterministic function t
E[er( ",x,-X.)1A] is
readily solved:
E[eu,xi-xd.A, and (3.12) follows.
= P(A )e-(1124.112(i-s),
V
e
158
3. Stochastic Integration
3.17 Exercise. Let W = (W('), W,(2), Wi(3)) be a three-dimensional Brownian motion starting at the origin, and define 3
X = n sgn(WP), Mu)
wi(i),
m:2)
pvt(2)
and
= M:3)
X W,(3).
Show that each of the pairs (M(1), M(2)), (M(1), M(3)) and (M(2), M(3)) is a two-dimensional Brownian motion, but (Mu), m(2), m(3)) is not a threedimensional Brownian motion. Explain why this does not provide a counterexample to Theorem 3.16, i.e., a three-dimensional process which is not a Brownian motion but which has components in d)" and satisfies (3.11).
3.18 Problem. Let W = 1W, = (Wu), ... , W(d)), g,; 0 < t < cdo I be a d-
dimensional Brownian motion starting at the origin, and let Q be a d x d orthogonal matrix (QT = 12-1). Show that 11/, A QW, is also a d-dimensional Brownian motion. We express this property by saying that "d-dimensional Brownian motion starting at the origin is rotationally invariant."
C. Bessel Processes, Questions of Recurrence Another use of the P. Levy Theorem 3.16 is to obtain an integral representation for the so-called Bessel process. For an integer d _. 2, let W = { W, = (IV,(1), ... , Wt(d)), ,i; 0 < t < col, {Px1,.. Rd be a d-dimensional Brownian family on some measurable space (SI, F). Consider the distance from the origin (3.14)
R, A II Hitli = Wt(1)/2 +
+ (Wi(d))2;
0 < t < oo,
so Px [Ro= Ilxli] = 1. If x, y e Rd and Ilxil = OA, then there is an orthogonal
matrix Q such that y = Qx. Under Px, (V = 111/, A QW ..F,; 0 < t < ool is a d-dimensional Brownian motion starting at y, but II afili = II Will, so for any F EM(C[0, co)), we have (3.15)
Px[R.e F] = Px[110<11 e n = PY[R.e F].
In other words, the distribution of the process R under P depends on x only through Ilx11-
3.19 Definition. Fix an integer d > 2, and let W = { W, .57;,; 0 < t < co}, { Px}x.R. be a d-dimensional Brownian family on (SI, ,). The process R = {R, = 11W,II, ..F,; 0 < t < co} together with the family of measures {P' }, >0 {Po .o.....o»>0 i (S/,.F) is called a Bessel family with dimension d. For fixed r > 0, we say that R on (SI, g", Pr) is a Bessel process with dimension d starting at r. .
3.3. The Change-of-Variable Formula
159
3.20 Problem. Show that for each d > 2, the Bessel family with dimension d
is a strong Markov family (where we modify Definition 2.6.3 to account for the state space [0, co)). 3.21 Proposition. Let d > 2 be an integer and choose r > 0. The Bessel process
R with dimension d starting at r satisfies the integral equation 'd
R, ---- r + fo
(3.16)
-
1
2R,
ds + Bt;
0 < t < co,
where B = {B..; 0 5 t < co} is the standard, one-dimensional Brownian motion B Q > 13(1)
(3.17)
with
BY) A
I0' We R,
1 < i < d.
dWP);
PROOF. We use the notation of Definition 3.19, except we write P instead of P. Note first of all that R, can be at the origin only when Ws") is, and so the Lebesgue measure of the set {0 < s < t; Rs= 0} is zero, a.s. P (Theorem 2.9.6).
Consequently, the integrand (d - 1)/2R3 in (3.16) is defined for Lebesgue almost every s, a.s. P. Each of the processes B(i) in (3.17) belongs to .At e2, because E
f ' (-1Rs W( i))
2
ds < t;
0 < t < co.
0
Moreover, , =
1
J o R52
r
1
. . wscowsco d<wo, Ivo>5 = 45_____ wo Ivo) ds , u D2s s
0 ls
which implies
, =-- E
and we conclude from Theorem 3.16 that B is a standard, one-dimensional Brownian motion. It remains to prove (3.16). A heuristic derivation is to apply Ito's rule (Theorem 3.6) to the function f(x) -A 11x11 = xi + [0, co, for + ald which oxi
f(x)
,
i axax;
f(x) =
c5 Ilx,11
xx 1139 4-1
1 5 i, j 5_ d,
hold on le \ {0}. Then R, = f(W,) and (3.16) follows from (3.5). The difficulty here is that f is not differentiable at the origin, and so Theorem 3.6 cannot be applied directly to f. This problem is related to our uneasiness about whether the integral in (3.16) is finite. Here is a resolution. Define
3. Stochastic Integration
160
11147,112 =
Yr
and use Ito's rule to show that d
= r2 + 2E 1=1
Wswd147,(i)+ td.
J. `
Let g(y) = .67, and for e > 0, define 3
/-
+
gjy) =
3
,y 4,./e
1
y2;
8e,fi y
fi;
6,
SO g, is of class C2 and lime4.0 g,(y) = g(y) for all y > 0. Now apply Ito's rule
to obtain e(e) + .1,(e) + K,(e),
ge(Y,) = ge(r2) +
(3.18)
1=1
where 1
[1{r >0 -+ 1{r.<0
44(0
Rs
d
(3
- rs)] Wsoldwp,
1
2Rs ds'
jt(g) g Kt(e) A
1
2 Ve
f 0
1 ,[3d - (d + 2) ]ds.
4/s
8
We now show that, as el 0, (3.18) yields (3.16). From the monotone convergence theorem, we see that {y.,0}d2 -
lim .1,(e) =
Rs
1
1
t
ds =
0
-
d 2R51 ds,
a.s.
We also have 0 < EK,(e) < (3d/4,./i)PoP[Y, < el ds. The probability in the integrand is bounded above by
.1;
2ir
(yvs(2))2
< e] =
1
-2itse-P212spdpd0,
Jo
0
and so the integral becomes, upon using Fubini's theorem and the change of variable = p/fs: t
P[Y < &]ds 5
p o
=2 f
o
-e-P212s ds dp os
p(f
if
-e-4212 d)dp.
3.3. The Change-of-Variable Formula
161
But now it is easy to see that this expression is o(ft) as t 10, using the rule of l'HOpital. Therefore, lime,1,0 EK,(E) = 0. Finally
r
E[BP) - IP)(En2 = E
=E
1
(3
1
ifr.<0ks
it
__
Jo
_171(3
-
Ys
)i2(,(0,2 ds
_ 114(ws"))2 ds
P[Ys < E] ds = 0(10
E /J
/
asel O.
This establishes (3.16).
Let {R .9;; 0 < t < co} be a Bessel process with dimension d 2 starting at r > 0. Then, for each fixed t > 0, it is clear from (3.14) that P[R, > 0] = 1. A more interesting question is whether the origin is nonattainable:
P[R, > 0; V 0 < t < oo] = 1.
(3.19)
The next proposition shows that this is indeed the case. Of course the situation is drastically different in one dimension, since P[I W:(1)1 > 0; V 0 < t < cc] = 0 (Remark 2.9.7).
3.22 Proposition (Nonattainability of the Origin by the Brownian Path in Dimension d > 2). Let d > 2 be an integer and r > 0. The Bessel process R with dimension d starting at r satisfies (3.19). PROOF.
(w(1))2
It is sufficient to treat the case d = 2, since, for larger d, (wt(d),) 2 can reach zero only if (W;"))2 + (147,(2))2 does.
We consider first the case r > 0. For positive integers k satisfying (1/k)k < r < k and n > 1, define stopping times
7', = inf T
0; Ri = (-1y} k
S = inf {t
0; R, = k},
k
tk= TkASknn.
Because P-almost every Brownian path is unbounded (Theorem 2.9.23), we have CO
(3.20)
P
[n
k=1
{sk < cc} n thin
(X)}] = 1.
Sk
k-co
Using (3.16), apply Ito's rule to log(Rt) to obtain tk
log R, = log r+
1
-dB,.
o Rs
This step is permissible because log is of class C2 in an open interval containing [(1/k)k, k]. For 0 < s < Tk, 11/Rsi is bounded, and since Tk is also bounded, we have E frel/Rs)dB, = 0. Therefore,
3. Stochastic Integration
162 (3.21)
log r = E [log Rrk] = - k(log k) P[Tk < Sk A n]
+ (log k) P[Sk < Tk A n] + E[(log R)1- tn<Si, A TO]. 1, log R on {n < Sk A TO is bounded between - k(log k) and co we have P[n < Sk A Ti] 0. Thus, letting n -co in (3.21), we obtain For every n
log k. According to (3.20), as n
log r = - k(log k) P[Tk < Sk] + (log k) P[Sk < Tk]. If we divide by k(log k) and let k -* cc, we see that
lim P[Tk < Sk] = 0.
(3.22)
k -. co
Now set T = inf{t > 0; R, = 0 }, so that Tk < T for every k > 1. From (3.20) and (3.22), we have (3.23)
P[T < co] = lim PET < Sk] < lim P[Tk < Sk] = 0. k -.co
k - co
It follows that P[R, > 0,V0 < t < co] = 1. Finally, we consider the case r = 0. Recalling the indexing of probability measures in Definition 3.19, we have from Problem 3.20:
PqR, > 0; V s < t < co] =
oE { pR,,R,
i
> 0; V 0 < t < co]} = 1
for any fixed s > 0, by what was just proved and the fact that P° [R, > 0] = 1. Letting a 4 0, we obtain the desired result.
3.23 Problem. Let R = {R,F,; 0 t < col be a Bessel process with dimension d > 2 starting at r > 0, and define m=
inf R,. :)1<0,
(i) Show that if d = 2, then m = 0 a.s. P. (ii) Show that if d > 3, then m has the beta distribution
cy- 2 P[m < c] = G
0 < c < r.
(Hint: Adapt the proof of Proposition 3.22. For (ii), an appropriate substitute for the function f(r) = log r must be used.)
Proposition 3.22 says that, with probability one, a two-dimensional Brownian motion never reaches the origin. Problem 3.23 (i) shows, however, that it comes arbitrarily close. By translation, we can conclude that for any given point ze R2, a two-dimensional Brownian path, with any starting position different from z, never reaches the point z, but does reach every disc of positive radius centered at z. In the parlance of Markov chains, one says that "every singleton is nonrecurrent," but that "every disc of positive radius is recurrent." For a Brownian motion of dimension 3 or greater, Problem 3.23
3.3. The Change-of-Variable Formula
163
(ii) shows that, once it gets away from the origin, almost every path of the process remains bounded away from the origin; this lower bound depends, of course, on the particular path. Thus, d-dimensional spheres are nonrecurrent for d-dimensional Brownian motion when d > 3.
3.24 Problem. Let R be a Bessel process with dimension d > 3 starting at r > 0. Show that P[Iim,-, R, = oc] = 1.
D. Martingale Moment Inequalities As a final application of Ito's rule in this section, we derive some useful bounds on the moments of martingales. The following exercise illustrates the technique.
3.25 Exercise. With W = W, .97,; 0 < t < oo a standard, one-dimensional Brownian motion and X a measurable, adapted process satisfying E
(3.24)
J T
IX,12mdt < co
for some real numbers T > 0 and m > 1, show that (3.25)
f: X tdW,
2m
(m(2m - 1))mrn-lE
(Hint: Consider the martingale {Mt = f o Xs dW,
1X,12'n dt.
0 < t < T }, and apply
Ito's rule to the submartingale IMt12m .)
Actually, with a bit of extra effort, we can obtain much stronger results. We 12m) be the increasing functions E(1 and E(<M>r), with the convention shall show, in effect, that for any M E (3.26)
itit* -A max IMsl,
o<st
have the same growth rate on the entire of [0, co), for every m > 0. This is the subject of the Burkholder-Davis-Gundy inequalities (Theorem 3.28). We present first some preliminary results. 3.26 Proposition (Martingale Moment Inequalities [Millar (1968), Novikov (1971)]). Consider a continuous martingale M which, along with its quadratic variation process <M>, is bounded. For every stopping time T, we have then (3.27)
E(1/147-12"1) < C;E(<M)7);
m>0
(3.28)
BmE(<M)7) < E(IMr12m);
m > 1/2
(3.29)
B,E(<M>7) _5 EUM1)21 5 CmE(<M>7); m > 1/2
3. Stochastic Integration
164
for suitable positive constants B C., C,, which are universal (i.e., depend only on the number m, not on the martingale M nor the stopping time T). PROOF. We consider the process
Y, -A 6 + t<M>, + M? = 6 + (1 + t)<M>, + 2
f Ms c/M 0
t < oo,
o
where 6 > 0 and E z 0 are constants to be chosen later. Applying the changeof-variable formula to f(x) = xm, we obtain (3.30)
Y,'" = 6m + m(1 + E)
il Ysm-1 d<M>s + 2m(m - 1).1o Yr2M: d<M> o
+ 2m f i
YriMsdMs; 0
t < co.
o
Because M, Y, and <M> are bounded and Y is bounded away from zero, the last integral is a uniformly integrable martingale (Problem 1.5.24). The Optional Sampling Theorem 1.3.22 implies that E 11; Yri Ms dM, = 0, so taking expectations in (3.30), we obtain our basic identity T
(3.31)
EYE = 6m + m(1 + OE 1
Ysm-1 d<M>s
o T
+ 2m(m - 1)E f yr2ms2 d<m>s. o
Case 1: 0 < m < 1, upper bound: The last term on the right-hand side of (3.31) is nonpositive; so, letting 610, we obtain T
(3.32)
E[E<M>T + Mflm < m(1 + E)E J
(E<M>s + M:)"1-1 d<M>, 0
<M>;' d <M>s
MO + c)cm -1 E I o
= (1 + E)E'E(<M>7). The second inequality uses the fact 0 < m < 1. But for such m, the function f(x) = xm; x > 0 is concave, so (3.33)
x > 0, y > 0,
2"1-1(xm + ym) < (x + y)m;
E
and (3.32) yields: Em EVM >7) + E(IMTI2m) (3.34)
E(I MTI2m)
(1 + E)G
[(1 + E) (-21 m
E(< M >7), whence
- em1E(<M>7).
Case 2: m > 1, lower bound: Now the last term in (3.31) is nonnegative, and the direction of all inequalities (3.32)-(3.34) is reversed:
3.3. The Change-of-Variable Formula
E(IMT121
165
[(1 + E) (c/m -1
- EmiE(01)7).
Here, c has to be chosen in (0,(2'1 Case 3:1 < m < 1, lower bound: Let us evaluate (3.31) with E = 0 and then let 810. We obtain (3.35)
f
E(IMTI2m) = 2m(m -
IM,12(m-1)d<M>s.
On the other hand, we have from (3.33), (3.31):
2m-1[emE(<M>7) + E(5 + Mi)m] E[E<M>T +
+ MD]m T
6" +
±
(6 + Ms2)m-1 d<M>so
Letting S 4 0, we see that
(3.36) 2' CemE(01>7) +
f
I MT 12m)] < m(1 +
I Ms12("1-1) d<M>s.
Relations (3.35) and (3.36) provide us with the lower bound -1
E(1/1/7.12m)
cm (12+:211 m
1)
E1<m>71
valid for all c > 0. Case 4: m > 1, upper bound: In this case, the inequality (3.36) is reversed, and we obtain E(IMTI2m) < Ern((1
+ c)21-m
2m - 1
1)-1 E(<M>7!),
where now c has to satisfy E > (2m - 1)2' - 1. This analysis establishes (3.27) and (3.28). From them, and from the Doob maximal inequality (Theorem 1.3.8) applied to the martingale IMTAt, -Ft; 0 < t < oo}, we obtain for m > 1/2: B,E(<M>TA
E(IMT A il2m) < EUM1 A t/21
( 2m ) 2171
- 1)2
1"
IMTAtI2m)
< C; (2m 2m_ 1)21"E(<M>7,); which is (3.29) with T replaced by T n t. Now let t and use the monotone convergence theorem.
co in this version of (3.29) 11:1
3. Stochastic Integration
166
3.27 Remark. A straightforward localization argument shows that (3.27), (3.29) are valid for any M edrI". The same is true for (3.28), provided that the additional condition E( <M >) < co holds. We can state now the principal result of this subsection. 3.28 Theorem (The Burkholder-Davis-Gundy Inequalities). Let M e dric'c and recall the convention (3.26). For every m > 0 there exist universal positive constants km, Km (depending only on m), such that (3.37)
kmE(<M >) < E[(M1)21 < KmE(<M>7)
holds for every stopping time T PROOF. From Proposition 3.26 and Remark 3.27, we have the validity of (3.37) for m > 1/2. It remains to deal with the case 0 < m < 1/2; we assume without loss of generality that M, <M> are bounded. Let us recall now Problem 1.4.15 and its consequence (1.4.17). The righthand side of (3.29) permits the choice X = (M*)2, A = C, <M> in the former, and we obtain from the latter
2 -m
EL(MP)21 <
1 -m
CrE(<M>7)
for every 0 < m < 1. Similarly, the left-hand side of (3.29) allows us to take X = B, <M >, A = (M*)2 in Problem 1.4.15, and then (1.4.17) gives for 0 < m < 1:
1 -m
2 -m
0
BTE(<M>7) <
3.29 Problem. Let M = (M"),..., M(d)) be a vector of continuous, local martingales, i.e., M") e dr I°c, and denote d
II M ii ;'`
'''' max IlMs11,
At
0,s,,
-' --
E or>,; 0 ,=,
t < cc.
Show that for any m > 0, there exist (universal) positive constants ,I.m, Am such
that (3.38)
AmE(AT) < E(11M111)2m < AmE(AT)
holds for every stopping time T 3.30 Remark. In particular, if the AO in Problem 3.29 are given by MP )
=E tr=1 jot
n,Pwsch d,
where { W = (W,(1), ... , W,(')), .Ft; 0 < t < co} is standard, r-dimensional
Brownian motion, {X, = (VP); 1
__
i < d, 1 < j 5 r,0 < t < co} is a matrix
3.3. The Change-of-Variable Formula
167
of measurable processes adapted to (3.38) holds with (3.39)
and 11Xt112 A Ef.-.1
(XP.J.))2, then
AT = o
E. Supplementary Exercises 331 Exercise. Define polynomials H(x, y); n = 0, 1, 2, ... by 0"
H(x, y) =
ace
exp ax - -1 (x2 y) 2
x, y e 11;t
a=0
Ho(x, y) = 1, H, (x, y) = x, 112(x, y) = x2 - y, H3(x, y) = x3 - 3xy, H4(x, y) = x4 - 6x2y + 3y2, etc.). These polynomials satisfy the recursive relations
(e.g.,
-
(3.40)
=
ax
n = 1, 2, ...
y);
as well as the backward heat equation (3.41)
a
az H,,(x, v)
+
axe
n = 0, 1, ...
= 0;
H.(x,
For any M e.A,,,k,c, verify
(i) the multiple Ito integral computation t1
o
dMin
41
o
.
. dMt2 dMt, =
1
n!
H,,(M <M>,),
(ii) and the expansion exp (xM,
-
2
<M>, )
ot.
= E -H(M<M>,). ,, =0
n!
(The polynomials H(x, y) are related to the Hermite polynomials h ^(x)
(-1r ex2/2 p-ir
dn
dx"
e-x2a
by the formula H(x, y) = .111! y"/2 h(x/fy).) 3.32 Exercise. Consider a function a:
(0, co) which is of class C1 and such
that 1/a is not integrable at either +co. Let c, p be two real constants, and introduce the (strictly increasing, in x) function f(t, x) = e" f o dy /a(y); 0 < t < co, x e II and the continuous, adapted process = 4 + p roe" ds + c'oe" dW, A; 0 < t < co. Let g(t, -) denote the inverse of f(t, -). Show that the
3. Stochastic Integration
168
process X, = g(t,i) satisfies the stochastic integral equation X, = X0 + 1 t b(Xs) ds +
(3.42)
ft 01)0 dWs;
0 < t < oo
o
o
for an appropriate continuous function b: gl -> DI, which you should determine.
333 Exercise. Consider two real numbers .5, it; a standard, one-dimensional
Brownian motion W; and let W,(") = W, + pt; 0 < t < co. Show that the process
X, = f t exp[ol W,(P) - Ws(P)} - +62(t - s)]ds;
0 < t < co
o
satisfies the Shiryaev-Roberts stochastic integral equation
X, =
it (1 + .3,1Xs)ds + .3 o
f
XsdWs.
o
334 Exercise. Let W be a standard, one-dimensional Brownian motion and 0 < T < co. Show that lim sup p-00 05t5T e-st
fefts
diV,
= 0, a.s.
o
335 Exercise. In the context of Problem 2.12 but now under the condition E.,/T < co, establish the Wald identities E(WT) = 0,
E(W7) = ET.
336 Exercise (M. Yor). Let R be a Bessel process with dimension d z 3, starting at r = 0. Show that {M, A (1/Rd,-2); 1 < t < co} (i) (ii)
(iii)
is a local martingale,
satisfies sups ,,<, E(Mr) < co for every 0 < p < d/(d - 2) (and is thus uniformly integrable), is not a martingale.
3.37 Exercise (M. Yor). Let R be a Bessel process with dimension d = 2 starting at r = 0. Show that {X, = - log Rt; 1 < t < co} is a local martingale with Ee'x, < co for -co < a < 2, t > 1, but X is not a martingale. 338 Exercise (Yor, Stricker). Let X be a continuous process and A a continuous, increasing process with X0 = A0 = 0, a.s. (i) Suppose that for every OE 01, the process
V) A exp(OX, - 102A,); 0 < t < co
is a local martingale. Prove that X E.i1°c and <X> = A.
3.4. Representations of Ccntinuous Martingales in Terms of Brownian Motion 169
(ii) Suppose that both X and Z") = exp(X - IA) are local martingales. Then again <X> = A.
3.39 Exercise (Wong & Zakai (1965b)). Let X be a continuous semimartingale of the form (3.1), and {/34")}Th a sequence of processes of bounded variation, such that P Dim
.1r) = X,] = 1 holds for every finite t > 0. If the is of class 0(R), show that
function f: R
-..7)
f(Xs)dX, + - ft f'(Xs)d<M>,
f(g.))dgio.
lim
0
0
2
0
holds a.s. P, for every fixed t > 0.
3.4. Representations of Continuous Martingales in Terms of Brownian Motion In this section we expound on the theme that Brownian motion is the fundamental continuous martingale, by showing how to represent other continuous martingales in terms of it. We give conditions under which a vector of d continuous local martingales can be represented as stochastic integrals with respect to an r-dimensional Brownian motion on a possibly extended probability space. Here we have r < d. We also discuss how a continuous local
martingale can be transformed into a Brownian motion by a random timechange. In contrast to these representation results, in which one begins with a continuous local martingale, we will also prove a result in which one begins
with a Brownian motion W = { W Ft; 0 < t < co} and shows that every continuous local martingale with respect to the Brownian filtration {..F,} is a stochastic integral with respect to W. A related result is that for fixed 0 < T < co, every g-r-measurable random variable can be represented as a stochastic integral with respect to W. We recall our standing assumption that every filtration satisfies the usual conditions, i.e., is right-continuous, and ..F0 contains all P-negligible events.
4.1 Remark. Our first representation theorem involves the notion of the extension of a probability space. Let X = {X.Ft; 0 < t < co} be an adapted process on some (0,
P). We may need a d-dimensional Brownian motion
independent of X, but because (0,,,P) may not be rich enough to support this Brownian motion, we must extend the probability space to construct this.
Let (0, 327, P) be another probability space, on which we consider a ddimensional Brownian motion B----{B A; 0 < t < co }, set nAo.n,
g
"ost-,15.4p x P, and define a new filtration by gt
g; ®A. The
latter may not satisfy the usual conditions, so we augment it and make it right-continuous by defining
3. Stochastic Integration
170
A A n aosu s>,
where Ai is the collection of P-null sets in g. We also complete g by defining
= cr(g u .4'). We may extend X and B to {A}-adapted processes on P) by defining for (w,
E
= ). Then 11 = {R A; 0 < t < col is a d-dimensional Brownian motion, independent of 5C-, A; 0 < t < col. Indeed, B is independent of the exten= x ,(0)),
gi(0),
A(co
sion to a of any ,-measurable random variable on U. To simplify notation, we henceforth write X and B instead of fe and 11 in the context of extensions.
A. Continuous Local Martingales as Stochastic Integrals with Respect to Brownian Motion Let us recall (Definition 2.23 and the discussion following it) that if W = ig .Ft; 0 < t < col is a standard Brownian motion and X is a measurable, adapted process with P[yo Xs ds < oo] = 1 for every 0 < t < co, then the stochastic integral /,(X) = Po X sc11415 is a continuous local martingale with quadratic variation process <1(X)>1 ro X s2 ds, which is an absolutely con-
tinuous function of t, P a.s. Our first representation result provides the converse to this statement; its one-dimensional version is due to Doob (1953).
4.2 Theorem. Suppose M = {M, = (MP), , AV)), A; 0 < t < col is defined on (a, , P) with Mt') E e' I", 1 < i < d. Suppose also that for 1 < i,j < d, the cross-variation <M(i), M(j)>,(w) is an absolutely continuous function of t for P-almost every w. Then there is an extension P) of (S2, P) on which is defined a d-dimensional Brownian motion W = {W, = (W,(1), , W(°)), A; 0 < t < col, and a matrix X = {(X;'" )°,k =1, A; 0 < t < co} of measurable, adapted processes with (4.1)
p
[f
col=
(xli,k))2 ds
1<
k < d; 0 < t < oo,
such that we have, P-a.s., the representations (4.2)
(4.3)
Ed
t
k =1
0
<M(i), Mti)>,=
E k =1 d
x , k) d ws(k);
Ji'0
1 < I < d, 0 < t < co,
x!")XP'k) ds ;
1 < j < d,
0
t < oo.
PROOF. We prove this theorem by a random, time-dependent rotation of coordinates which reduces it to d separate, one-dimensional cases. We begin by defining
3.4. Representations of Continuous Martingales in Terms of Brownian Motion 171
(4.4)
. .
=
=d
dt
= lim nUM(`), Mtn >, - <M(i), Mcn>0_0 f)).],
so that the matrix-valued process Z = {4 = (4.1;=,,A; 0 < t < col is symmetric and progressively measurable. For a = (a,,..., ota)e Rd , we have d
d
Ed
E ccizIja = - E aior) i=i dt
0,
so 4 is positive-semidefinite for Lebesgue-almost every t, P-a.s. Any symmetric, positive-semidefinite matrix Z can be diagonalized by an = QT, so that ZQ = A and A is diagonal orthogonal matrix Q, i.e., with the (nonnegative) eigenvalues of Z as its diagonal elements. There are several algorithms which compute Q and A from Z, and one can easily verify that these algorithms typically obtain Q and A as Borel-measurable functions
of Z. In our case, we start with a progressively measurable, symmetric, positive-semidefinite matrix process Z, and so there exist progressively measurable, matrix-valued processes {Qt(co) = (q1-1(co))1.;_,; gf,; 0 < t < co} and {A,(w) = (.5u4(a)))1,;=, , 0 < t < co} such that for Lebesgue-almost every t, we have
E
gc.i1k.J
k=1 d
(4.6)
E d
d
(4.5)
1:5_
Si;
k=1
d
EE
(5v1.1 > 0 ;
1
j < d,
k=1 1=1
a.s. P. From (4.5) with = j we see that (e1)2 'i ft
<1,s0
(e.`)2 d<M(k)>s < <Mm>, < co,
and we can define continuous local martingales by the prescription (4.7)
Nio) A
1 < i < d, 0 < t < co.
dIVIlk); k=1
0
From (4.4) and (4.6) we have, a.s. P, d
(4.8)
d
<0),Na)>, = E E k=1 1=1 Jo d
k .i qs qs j d<M(k), Al")).s
.t
d
=EE
q! i 4 I 11.! i ds
k=1 1=1
= Si;
We see, in particular, that
Aids.
3. Stochastic Integration
172
(4.9)
Asi ds = , < o o .
We now represent the vector of local martingales N = MP), , Nt(d)), <9;; 0 < t < col as a vector of stochastic integrals on an extended probability space 0, g, F), which supports a d-dimensional Brownian motion B = -
{13, = (BP), . . . , MI)), .A; 0 < t < co 1 independent of N (cf. Remark 4.1). Since 1
t
d s =
1{,1,0}
t,
1 {,1).0) ds o
we can define continuous, local martingales 1
1{4,0}
Wto g'--
d li) +
1{4.0}
N(4.10)
Joo
di3.11);
1 < i < d.
Jo
.1.1
From (4.8) and Problem 1.5.26 we have
1 < i, j < d;
<147(i), Wu) >, = but,
0 < t < cc,
so, according to Theorem 3.16, W = {W = (W,(1), ..., Wi(d)), gi; 0 < t < col is a d-dimensional Brownian motion. Moreover, t
(4.11)
J
.1,17, dWso)
=
f
t
1 < i < d; 0 < t < co,
11A1,.01dNr = NP,
0
ID
because the martingale So 1{,i,o} dNr, having quadratic variation so
1{4=0} d , =
5
1 {,11.0} As ds = 0,
o
is itself identically zero. Having thus obtained the stochastic integral representation (4.11) for N in
terms of the d-dimensional Brownian motion W, we invert the rotation of coordinates (4.7) to obtain a representation for M. Let us first observe that for 1 < i, k < d,
j.0(qi;k12 4 ds <
f
Ask ds < co;
0
t < co
0
by (4.9), so with XP'k) A ql'k (4.7), and (4.5) imply
, condition (4.1) holds. Furthermore, (4.11),
d
(4.12) k=1
dWs(k) = t j'X!") p
qs kdNsk) k=1 d
0 d
q!, k tip k d
= j=1 E k=1 E =
s
0
t
J=1
ditt) = MP),
0
which establishes (4.2). Equation (4.3) is an immediate consequence of (4.2). LI
3.4. Representations of Continuous Martingales in Terms of Brownian Motion 173
4.3 Remark. If for P-a.e. w E SI, the matrix-valued process Zt(co) = (4'i(co))1,;,,
has constant rank r,1 5 r 5 d, for Lebesgue-almost every t, then the Brownian motion W used in the representation (4.2) can be chosen to be r-dimensional,
and there is no need to introduce the extended probability space (n,,, P). Indeed, we may take At 5... , to be the r strictly positive eigenvalues of Z and replace (4.10) by
=
(4.10)'
1
"
dN(i)-
Jo
r.
i
1
Since /VP = 0; r + 1 5 i 5 d, 0 5 t < co (witness (4.9)), (4.12) becomes r
(4.12)'
E k
t
d
xv,k)dws(k) = 0
fg" i
k =1
=
1
d.
o
W without reference to the Brownian motion B, there is no need to extend the original probability space. Because (4.10)' defines
The following exercise shows that any vector of continuous local martingales can be transformed by a random time-change into a vector of continuous local martingales satisfying the hypotheses of Theorem 4.2.
4.4 Exercise. Let {M = (MP . , gd)), .5r.t; 0 < t < co} be a vector of conP), and define tinuous local martingales on some (SI,
A"
d
<M(i),M(i)>,
d
E E AV(0),
Ai(w)
1=1 ;=,
where AP) denotes total variation of A(Li) on [0, t]. Let 7;(0)) be the inverse of the function At(w) + t, i.e., il,s(0(co) + Ts(co) = s; 0 5 s < co. (i) Show that for each s, T., is a stopping time of (ii) Define Ws A ..f.T.; 0 5 s < co. Show that if {.Fi} satisfies the usual conditions, then {Ws} does also. (iii) Define
Nr
MP,
15i
d;
0 < s < oo.
Show that for each 1 5 i 5 d: N(i) drloc, and the cross-variation <MI), /NM>, is an absolutely continuous function of s, a.s. P.
B. Continuous Local Martingales as Time-Changed Brownian Motions The time-change in Exercise 4.4 is straightforward because the function A, + t
is strictly increasing and continuous in t, and so has a strictly increasing, continuous inverse T. Our next representation result requires us to consider the inverse of the quadratic variation of a continuous local martingale; be-
3. Stochastic Integration
174
cause such a quadratic variation may not be strictly increasing, we begin with a problem describing this situation in some detail.
4.5 Problem. Let A = IA(t); 0 < t < co } be a continuous, nondecreasing function with A(0) = 0, S -4a', A(co) 5 co, and define for 0 < s < co: T(s) =
05s<S
{in* > 0; A(t) > s };
s > S.
oo;
The function T = { T(s); 0 5 s < co } has the following properties:
(i) T is nondecreasing and right-continuous on [0, S), with values in [0, co). If A(t) < S; V t .- 0, then limsts T(s) = co. (ii) A(T(s)) = s A S; 0 _< s < op. (iii) T(A(t)) = supIr t: A(r) = A(t)}; 0 5 t < co. (iv) Suppose cp: [0, co) -> R is continuous and has the property A(t1) = A(t)
cp(ti)= p(t).
for some 0 < t1 < t
Then p(T(s)) is continuous for 0 < s < S, and (4.13)
p(T(A(t))) = cp(t);
0 < t < c0.
(v) For 0 5 t, s < co: s < A(t).. T(s) < t and T(s) S t s < A(t). (vi) If G is a bounded, measurable, real-valued function or a nonnegative, measurable, extended real-valued function defined on [a, b] c [0, co), then A(b)
(4.14)
G(t) dA(t) =
1
G(T(s))ds.
A(a)
4.6 Theorem (Time-Change for Martingales [Dambis (1965), Dubins & Sch-
warz (1965)]). Let M = {M.9;;0 5 t < co } e../r1" satisfy lim,o, <M>, = co, a.s. P. Define, for each 0 < s < co, the stopping time (4.15)
T(s) = inf {t > 0; <M>, > s }.
Then the time-changed process (4.16)
B, '' MT(,),
-41.. LFT(s);
0 ''.
S < 00
is a standard one-dimensional Brownian motion. In particular, the filtration {W,} satisfies the usual conditions and we have, a.s. P: (4.17)
M, = B<m>,;
0 5 t < co.
PROOF. Each T(s) is optional because, by Problem 4.5 (v), { T(s) < t} = I<M>, > sl e .5c and {..F,} satisfies the usual conditions; these are also satisfied by Al just as in Problem 4.4 (ii). Furthermore, for each t, <M>, is a stopping time for the filtration Is} because, again by Problem 4.5 (v), { <M>, 5 s} = { T(s)
t} egT(s) .--- Is;
0 ..' S < 0 0 .
3.4. Representations of Continuous Martingales in Terms of Brownian Motion 175
Let us choose 0 5 s, < s2 and consider the martingale {R, = M T(s2), .57t; 0 < t < co}, for which we have , = <M>, A T(S2)
<M>T(s2) = s2;
0 5- t < GO
by Problem 4.5 (ii). It follows from Problem 1.5.24 that both Cf and SI2 - are uniformly integrable. The Optional Sampling Theorem 1.3.22 implies, a.s. P:
E[Bs2 -
=
- KIT(si).LFT(s,)] = 0,
E[(13,2 - Bs,)20si] = EURT(s2)- MT00 1
I .-Fro,)]
= ECps2)- T401,-FT(so] = S2 -S I s < co} is a square-integrable martingale with quadratic variation , = s. We shall know that B is a standard Brownian Consequently, B = {Bs, Ws; 0
motion as soon as we establish its continuity (Theorem 3.16). For this we shall use Problem 4.5 (iv). We must show that for all co in some Sr f2 with P(S2*) = 1, we have: (4.18)
<M>ti(w) = <M>,(w) for some 0 < t, < t
mii(w) = mt(c0).
If the implication (4.18) is valid under the additional assumption that t, is rational, then, because of the continuity of <M> and M, it is valid even without
this assumption. For rational t > 0, define o- = inf t > t,: <M>, > <M>,, I,
N, = M(,,), - M,,, 0 < s < co, so {N,, ..F,1; 0 < s < co} is in dr'1" and , =
- <M>,, = 0, a.s. P.
It follows from Problem 1.5.12 that there is an event f2(t, ) s 12 with P(C1(t,))= 1 such that for all co e S1(t,), <M>,,(co) = <M>,(co), for some t > t,
M,,(w) = Mi(co).
The union of all such events fl(t, ) as t, ranges over the nonnegative rationals will serve as fr, so that implication (4.18) is valid for each co E sr. Continuity of B and equality (4.17) now follow from Problem 4.5 (iv). 11:1
4.7 Problem. Show that if P[S Q <M>
co] > 0, it is still possible to define a Brownian motion B for which (4.17) holds. (Hint: The time-change T(s) is now given as in Problem 4.5; assume, as you may, that the probability space has been suitably extended to support an independent Brownian motion (Remark 4.1).)
The proof of the following ramification of Theorem 4.6 is surprisingly technical; the result itself is easily believed. The reader may wish to omit this proof on first reading.
3. Stochastic Integration
176
4.8 Proposition. With the assumptions and the notation of Theorem 4.6, we have
the following time-change formula for stochastic integrals. If X = {X.; 0 < t < co} is progressively measurable and satisfies
.
(4.19)
X; d<M>t < co
a.s.,
then the process (4.20)
X2(,), W's;
Ys
0
<s
is adapted and satisfies, almost surely: J0Y52ds
(4.21)
<m>,
X dM
(4.22)
v
v
=-
17d13.;
0 < t < co,
d13.;
< s < cc.
o
f's
j.T(s)
(4.23)
X dm, =
PROOF. The process Y is adapted to {WO because of Proposition 1.2.18. Relation (4.21) follows from (4.19) and (4.14).
Consider the continuous local martingale 1.4
pc, XvdMv,..F,; 0 < t < col.
If (4.24)
<M>t,(0)) = <M>t(w)
for some 0 < t, < t,
then XR(D)d<M>,#0) = io XR(0)d<M>v(w) = <J>t(co)
<J>,,(co) = o
Applying to J the argument used to obtain (4.18), we conclude that for all w in some n* g SI with F(K-e)- 1, (4.24) implies the identity Jti(w)= Ji(co). According to Problem 4.5 (iv), we have that (4.25)
JT(s)
X dMv;
0 < s < co
is continuous, and (4.26)
.7<m>t = 17.(of)t) = ft;
0 < t < cc,
almost surely. Let ; = inf {0 < t < co; <J>, > n}, so {J, Ft; 0 < t < co} is a martingale. For 0 < s, < s2 and n > 1, we have from the optional sampling theorem: E[Js2 A <m),,,,0 s i] = EUT(s2)A n = JT(si)A n
Each
rI'T(s,)]
= si
a.s.
is a stopping time of the filtration {A} because t<M>. A
s}
3.4. Representations of Continuous Martingales in Terms of Brownian Motion 177
= {n A t Lc. T(s)} = {n 5 T(s), <J>m) <M>n
E 'T(s).
By
assumption,
= oo a.s.; it follows that .7 is in Jr' (relative to {C).
Using this fact, we may repeat the preceding argument to show that Joy>, = J, is a continuous local martingale relative to the filtration Mm>,}, which contains and may actually be strictly larger. In fact, we can choose an arbitrary continuous local martingale N relative to {Ws} and construct Art = Rof>,, a continuous local martingale relative to
{W00,}. If we take N = B, then N = M from (4.17) and so M is in dr'" relative to {<,%}. We now establish (4.23) by choosing an arbitrary N = {fs?5s; 0 < s < co} e.4/"c and showing that rs
<J, N>s
(4.27)
Yud;
0<s<
holds a.s. (see Proposition 2.24). Let N, = Si<m> fix t1, and set Mti = M1+1,
- 41,
Nil
=
+, - Nit; 0 < t < co.
Both M', N' (respectively; M, N) are local martingales relative to {Wow>,,,} (resp. {500,}). We may compute cross-variations thus: KM, N>t+ - <M, N>t, I = I <M1,Ni>,1
N/<M1>t,
= NA<M>1.4-1, - <M>ii)(:+1, - 1),
where Problem 1.5.7 (iii) has been invoked. We see that if <M> is constant on an interval, so is <M, N>. From Problem 4.5 (iv) we conclude that <M, N>T(s) is almost surely continuous. Because <M, N>, has finite total variation for t in compact intervals, the composition <M, N>T(,) has finite total variation for s in compact intervals. Finally, Bs Ns
- <M,N>Tco= MT(S)NT(S) - <M N> T(s)
is in dr' relative to {Ws}, so (4.28)
, = <M,N>T(s);
0 < s < co, a.s. P.
Setting s = <M>, in (4.28), we obtain (4.29)
olf>,
<M, N>T(<m>) = <M,N>,;
0 < t < co
almost surely; we have used Problem 4.5 (iv) with cp = <M, N>. Choose w e CI for which (4.29) holds. Then c. 1 fo,b) (v)d<M
N>v ((o) - <M N>b(w) 5
<M,N>.(w)
= 00,(.#0)- <m>.(.)(0))
=
f
1(<m).(.),<m>b(w )(u)d
=
J
1(a,b)(T(u,w))d(co),
Mco)
3. Stochastic Integration
178
by virtue of Problem 4.5 (v) in the last step. Thus we have for step functions, and consequently for all Borel-measurable functions G: [0, oo) -> [0, co], that G(v)d <M, N> =
(4.30)
G(T(u)) d.
holds a.s. Returning to (4.27), we can use the optional sampling theorem and (4.30) to write >s = <JT()1NT()>s = <J1 N>T(s) T(s)
=
Xvd<M,N> =
Yud;
0
s < co, a.s.
This concludes the proof of (4.23). Replacing s by <M>, in (4.23) and using (4.26), we obtain (4.22).
4.9 Remark. If, in the context of Theorem 4.6, we have P[S A <M>co < co] > 0, we may take a Brownian motion B on an extended probability space for which (4.17) holds (Problem 4.7), and the conclusions of Proposition 4.8 are still valid, except now we must define Y by (4.20)'
Y
f XT(s);
0
10;
S
s < S, s < co.
The proof is straightforward but tedious, and it is omitted.
4.10 Remark. Levy's characterization of Brownian motion (Theorem 3.16) permits a bit more generality than expressed in Theorem 4.6. If X = {X ..F;: 0 < t < co} is an adapted process with M <M>, = co a.s., then the time-changed process X0 + MT(s),
Ws A. LFT(s);
0
A X - Xoe Jr' and
s < co
is a one-dimensional Brownian motion with initial distribution PXV. In particular, X0 is independent of Bs
= MT(s) ,
oksB ;
0 < s < co
(Definition 2.5.1). Similar assertions hold in the context of Problem 4.7, Proposition 4.8, and Remark 4.9.
4.11 Problem. We cannot expect to be able to define the stochastic integral Jo XsdW, with respect to Brownian motion W for measurable adapted processes X which do not satisfy 14 Xs ds < co a.s. Indeed, show that if
P[f Xs ds then
oo]=
1,
for 0 < t < 1
and E A
lc
ds = co},
3.4. Representations of Continuous Martingales in Terms of Brownian Motion 179
lim
tti
f X, dW,
-lim I Xs dW, = +a), a.s. on E.
t
tt
o
o
4.12 Problem. Consider the semimartingale X, = x + M, + C, with xe Me.11`.1", C a continuous process of bounded variation, and assume that there exists a constant p > 0 such that IC,C + <M>, < pt, V t 0 is valid almost surely. Show that for fixed T > 0 and sufficiently large n > 1, we have F'[max
IX,I > n] < exp {
0
-n2 1.
18p T
C. A Theorem of F. B. Knight Let us state and discuss the multivariate extension of Theorem 4.6. The proof will be given in subsection E.
4.13 Theorem (F. B. Knight (1971)). Let M = {M, = (MP), t < co} be a continuous, adapted process with M") diecmc, a.s. P, and <M"),110>, = 0;
(4.31)
1
,
Md)),F,; 0 < = 00;
j < d, 0 < t < CO.
Define
Ti(s) = ingt
0; <M")>, > s};
0 < s < co, I < i < d,
so that for each i and s, the random time Ti(s) is a stopping time for the (right-continuous) filtration {.9;;}. Then the processes
Br
M)(s);
0 < s < co,
1
d,
are independent, standard, one-dimensional Brownian motions.
Discussion of Theorem 4.13. The only assertion in Theorem 4.13 which is not
already contained in Theorem 4.6 is the independence of the Brownian motions B(i); 1 < i < d. Theorem 4.6 states, in fact, that B(1) is a Brownian motion relative to the filtration {(e) -4- ,T4 ),,o, but, of course, these filtrations are not independent for different values of i because g!;) = Fes; 1 < i < d. , For are independent, The additional claim is that the a-fields Fr, where {.F.,B1 is the filtration generated by B(i). This would follow easily if the assumption (4.31) were sufficient to guarantee the independence of M(I), Mu)
for i j; in general, however, this is not the case. Indeed, if W = { 0 < t < co} is a standard Brownian motion, then with AV)
1{w,> o} dWs,
we have M(I), M(2) e ..fiq and
0
t < co,
3. Stochastic Integration
180
<M(1), M(2)>, =
f
1 {w,} 1 {w.<0} ds = 0;
0 5 t< co.
o
But M") and M(2) are not independent, for if they were, <M")> and <M(2)> would also be independent. On the contrary, we have <M")>, + <M(2)>, =
1{,.>
ds +
0 < t < aD.
1{w.<0} ds = t
F. B. Knight's remarkable theorem states that when we apply the proper time-changes to these two intricately connected martingales, and then forget the time-changes, independent Brownian motions are obtained. Forgetting the time-changes is accomplished by passing from the filtrations 1101 to the less informative filtrations {.FsIr')}.
We shall use this example in Section 6.3 to prove the independence of the positive and negative excursion processes associated with a one-dimensional Brownian motion.
D. Brownian Martingales as Stochastic Integrals In preparation for the proof of Theorem 4.13, we consider a different class of representation results, those for which we begin with a Brownian motion rather than constructing it. We take as the integrator martingale a standard,
one-dimensional Brownian motion W = W ";; 0 < t < ao on a probability space (SI, .F, P), and we assume satisfies the usual conditions. For 0 < T < oo, we recall from Lemma 2.2 that 4 is a closed subspace of the Hilbert P) preserves inner
space YfT. The mapping X 1- IT(X) from .r.P to L2 (n, products (see (2.23)): E
f
X, Y, dt = E[IT(X)IT(Y)]
Since any convergent sequence in (4.32)
MT -4 {IT(X); X E
is also Cauchy, its preimage sequence in ..r? must have a limit in .r.P. It follows that MT is closed in L2(0., ,T,P), a fact we shall need shortly. Let us denote by 111 the subset of .11`2. which consists of stochastic integrals
/,(X) =
Xs dWs;
0 < t < a),
0
of processes X e (4.33)
llZ
{/(X); X e ..T*}
di`j g
2.
Recall from Definition 1.5.5 the concept of orthogonality in .112. We have the following fundamental decomposition result.
3.4. Representations of Continuous Martingales in Terms of Brownian Motion 181
4.14 Proposition. For every M e .112, we have the decomposition M = N + Z, where N Z e .112, and Z is orthogonal to every element of
PROOF. We have to show the existence of a process Y e 2 " such that M I( Y) + Z, where Z e A2 has the property
= 0; V X e
(4.34)
Such a decomposition is unique (up to indistinguishability); indeed, if we have
M = /( Y') + Z' = /(Y") + Z" with Y', Y" e .29* and both Z' and Z" satisfy (4.34), then
Z
Z" - Z' = I( Y'
Y")
and = = 0. It follows from Problem 1.5.12 that P[Z, = 0, V 0 < t < oo] 1. It suffices, therefore, to establish the decomposition for every finite timeinterval [0, T]; by uniqueness, we can then extend it to the entire half-line [0, co). Let us fix T > 0, let gt, be the closed subspace of P) defined is in
by (4.32), and let giq- denote its orthogonal complement. The random variable MT is in /2(52, 9T, P), so it admits the decomposition (4.35)
MT = IT(Y)
ZT,
where Y e 2' and ZT e L2 (SI, .97T, P) satisfies
E[ZTIT(X)] = 0; V X e
(4.36)
Let us denote by Z = {Z Ft; 0 < t < col a right-continuous version of the martingale E(ZTigc)(Theorem 1.3.13). Note that Z, = ZT for t > T Obviously Z e.42 and, conditioning (4.35) on .F we obtain Ait = 4(Y) + Zt;
(4.37)
0 < t < T, a.s. P.
It remains to show that Z is orthogonal to every square-integrable martingale of the form I(X); X e-C1, or equivalently, that {Z,/,(X), A; 0 < t < T} is a martingale. But we know from Problem 1.3.26 that this amounts to having E[ZsIs(X)] = 0 for every stopping time S of the filtration {A}, with S Lc. T. From (2.24) we have Is(X) = IT(X), where Xi(w) = Xi(o.01{, <s(,)} is a process in Therefore,
E[ZsIs(X)] = E[E(ZTI,$)ls(X)] = E[ZTIT(X)] = 0 by virtue of (4.36).
It is useful to have sufficient conditions under which the classes llZ and ./#1 actually coincide; in other words, the component Z in the decomposition of Proposition 4.14 is actually the trivial martingale Z = 0. One such condition is that the filtration {A} be the augmentation under P of the filtration {.F,W} generated by the Brownian motion W. (Recall from Problem 2.7.6 and Proposition 2.7.7 that this augmented filtration is continuous.) We state and
3. Stochastic Integration
182
prove this result in several dimensions. A martingale relative to this augmented filtration will be called Brownian.
4.15 Theorem (Representation of Brownian, Square-Integrable Martingales W,(d)),..F,; 0 < t < ool be a as Stochastic Integrals). Let W = tW, = (W,"), d-dimensional Brownian motion on (S/,..F, P), and let {..F,} be the augmentation under P of the filtration {..F,w } generated by W. Then, for any square-integrable
martingale M = {M, 0 < t < co) with Mo = 0 and RCLL paths, a.s., there exist progressively measurable processesYoe Y = {Y,(1), g. 0 < t < col such that (4.38)
E
j
(Y,i)2 dt < oo;
*
j< d
1
for every 0 < T < co, and t
(4.39)
M, = E
ysodwp; 0 < t < CO .
J=1
0
In particular, M is a.s. continuous. Furthermore, if fth; 1 < j < d, are any other progressively measurable processes satisfying (4.38), (4.39), then
f
- fru) 12 dt = 0,
I
a.s.
.i=1
PROOF. We first prove by induction on m, where m = processes Y(1),..., Y(m) in .29* such that
d, that there are
m
(4.40)
Z, A M, - E
o < t < co
ysti)dwp;
.i=1
is orthogonal to every martingale of the form ET_1 f o XS dW,,u), where X(J e
2*; 1 _< j m. If m = 1, this is a direct consequence of Proposition 4.14. Suppose such processes exist for m - 1, i.e.,
Z A M,
-
m-i
ypdwp; o < t < cc,
E
J=1
0
is orthogonal to ET_-11 Ito xpdwso for all Xul e 2*; 1 sjsm - 1. Apply Proposition 4.14 to write
Z=
fYs(m) d Ws(m) + 4;
0 < t < oo,
0
for some 11(" e 2*, where Z is orthogonal to 1.0 gn) d Ws(m) for all X(m) e
2*. For 1 s j< m- 1 and X01 e 2*, we have = <2, I') (X(i))> - <1'"(Y(m)), lw())(X0)> = 0. Thus, we have the decomposition (4.40) for M. In particular, with m = d,
3.4. Representations of Continuous Martingales in Terms of Brownian Motion 183
0 5 t < co,
<M, Hi(h>, = I Yu) ds;
(4.41)
1 < j < d.
0
Following Liptser & Shiryaev (1977), pp. 162-163, we now show that, in the notation of (4.40) with m = d, we have P-a.s. that Z,
0 < t < co.
0;
First, we show by induction on n that if 0 = so < s, < < s < t, and if the functions fk: C, 0 < k < n are bounded and measurable, then
E[Z, fl fk(Wsk)] = 0.
(4.42)
k=0
When n = 0, (4.42) can be verified by conditioning on .90 and using the fact Zo = 0 a.s. Suppose now that (4.42) holds for some n, and choose s < t. For 1: 0 = (0,, . . ,0)E Rd fixed and s. < s < t, define with i = co(s) -4
E[Z,
n fk(Wsdeimwd= E[Z k=0 n fk(wsdeimwd.
k=0
Using Ito's rule to justify the identity el("'w.)
=
d
s
+ E ie.
110112
ei(e'w-i dly(i)
2
.i=--1
s
e"'w-) du,
s"
we may write (4.43)
E[Zse"-woi,s.] = zs.ei(t 3 'W..) d
+ E le i E[Zs i=i
f
s
ei(e'w-) dWu)
3,,
110112 Er Zs 2 L
f eimw.)d
j
,
u
But (4.40) and (4.41) imply
E[Z, f
s
et(9'w-) dW,P)
grd = E [(Z., - Zs)
Js
eimw-) dWli) .FS.1 = 0.
Multiplying (4.43) by FP,:_o f(Wsk) and taking expectations, we obtain (4.44)
co(s) = Os.)
110112 f s
2
s
E[Zs- fl f5(14/5,)e"w-)]du k =1
110112
= co(s.) -
2
4910 du; s
s
t.
By our induction hypothesis, cp(s) = 0, and the only solution to the integral equation (4.44) satisfying this initial condition is 9(s) = 0; s < s < t. Thus
3. Stochastic Integration
184
(4.45)
= 0;
E
v e Rd.
k=o
With D A Zt irk'=o fk(Wsk), we define two measures on (Old, M(Rd)) by
11±(F) = ELD± ir(Ws)i;
Fe M(fre).
Equation (4.45) implies c."'x)
e'(°xl
(dx)
(dx);
V0
fo;sa
and by the uniqueness theorem for Fourier transforms, we see that it+ --= Thus
E[D+ f (Ws)] = E[D f(Ws)]; s < 4+1 g s < t, for any bounded, measurable f:01" -> C. This proves (4.42) for n + 1 and completes the induction step. A standard argument using the Dynkin System Theorem 2.1.3 now shows that we have
E[Z,fl = 0
(4.46)
for every A w-measurable indicator and thus, for every 547,w-measurable, bounded Since A differs from Aw only by P-null sets, (4.46) also holds for every A-measurable, bounded Setting = sgn(Z,), we conclude that Z, = 0 a.s. P for every fixed t, and by right-continuity of Z, for all t e [0, oo) simultaneously. The uniqueness assertion in the last sentence of the theorem is proved by observing that the martingale El=, f o (Yso )dHis(i) is identically zero, and so is its quadratic variation. 4.16 Problem. Let W = {W, =
,
Wi(d)), A; 0 < t < a)} be a d-
dimensional Brownian motion as in Theorem 4.15. Let M = {M A; 0 < t < oo } be a local martingale with M, = 0 and RCLL paths, a.s. Then there exist progressively measurable processes Y(i) {Y,(j), A; 0 < t < co} such that ('T (Y,
dt < co;
<j
d,
0
T < op ,
and
lye = E
yso) dwsu);
J'io In particular, M is a.s. continuous.
0 < t < co.
J=1 a
4.17 Problem. Under the hypotheses of Theorem 4.15 and with 0 < T < co, let c be an .97T-measurable random variable with EV < co. Prove that there
3.4. Representations of Continuous Martingales in Terms of Brownian Motion 185
are progressively measurable processes Y"),... , I'm satisfying (4.38), and such
that d
(0 + E = E(0
(4.47)
Ystb d wsch ;
J=1
a.s. P.
o
E. Brownian Functionals as Stochastic Integrals We extend Problem 4.17 to include the case T = co. Recall that for M e we denote by ..C1(M) the class of processes X which are progressively measurable with respect to the filtration of M and which satisfy E f o X,2 d<M>, < oo. According to Problem 1.5.24, when X e Yt(M), we have .1`6° X dM, defined a.s.
P. If W is a d-dimensional Brownian motion, we denote by ..r:(W) the set of processes X which are progressively measurable with respect to the (augmented) filtration of W and which satisfy E roc' X,2 dt < co.
4.18 Proposition. Under the hypotheses of Theorem 4.15, let be an measurable random variable with EV < oo. Then there are processes Y(1), ..., Y(') in ..TVW) such that d
CO
= E() + E I .i=i
yso d Ws° ;
a.s. P.
o
PROOF. Assume without loss of generality that E() = 0, and let M, be a right-continuous modification of E(ig;). According to Theorem 4.15, there exist progressively measurable Y"),..., Y(d) satisfying (4.38) and (4.39). Jensen's
inequality implies MI S E(215";), so E i=1
i't (Ysti))2 ds = E<M>, = E(M,2)
E(V) < co; 0 < t < co.
o
Hence, Yti)E .29,(W) and M.
1.3.20 shows that M. =
f;°, Ysth MP.) is defined for 1 < j < d. Problem
= c.
We leave the proof of uniqueness of the representation in Proposition 4.18 as Exercise 4.22 for the reader. In one dimension, there is a representation result similar to that of Proposi-
tion 4.18 in which the Brownian motion is replaced by a continuous local martingale M. This result is instrumental in our eventual proof of Theorem 4.13. Uniqueness is again addressed in Exercise 4.22.
4.19 Proposition. Let M = {M; 0 < t < co} be in Jr' and assume that = co, a.s. P. Define T(s) by (4.15) and let B be the one-dimensional Brownian motion Mr(s), es;
0
s < co
3. Stochastic Integration
186
as in Theorem 4.6, except now we take the filtration AI to be the augmentation
with respect to P of the filtration {AB} generated by B. Then, for every go,-measurable random variable satisfying E2 < oo, there is a process X e 43.(4) for which CO
(4.48)
= E()
J. X, dM,;
a.s. P.
PROOF. Let Y = { Ys, gs; 0 < s < co} be the progressively measurable process of Proposition 4.18, for which we have
f
(4.49)
E
(4.50)
= E() +
ys2 ds < co,
J
Ys dBs.
Define X,= Yol>,; 0 t < We show how to obtain an {.97,}-progressively measurable process X which
is equivalent to X. Note that because {A} {,T(01 contains {,23} and satisfies the usual conditions (Theorem 4.6), we have es A; 0 5 s < oo. Consequently, Y is progressively measurable relative to If Y is a simple process, it is left-continuous (c.f. Definition 2.3), and it is straightforward to show, using Problem 4.5, that { Yoki>,; 0 < t < cc} is a left-continuous process adapted to {.F(}, and hence progressively measurable (Proposition 1.1.13). In the general case, let { Y('')}c°_, be a sequence of progressively measurable (relative to {6'0), simple processes for which
lim E n- co
I Ys(") -Ys 12 ds = O. o
(Use Proposition 2.8 and (4.49)). A change of variables (Problem 4.5 (vi)) yields (4.51)
lim E
IX:") -
J
,1 2 d<M>, = 0,
o
where X:n) A Vn>,. In particular, the sequence {X(")},13_, is Cauchy in 2 t(M), and so, by Lemma 2.2, converges to a limit X e _KAM). From (4.51) we must have E
12, - X,12 d<M>, = 0,
which establishes the desired equivalence of X and X. It remains to prove (4.48), which, in light of (4.50), will follow from foc°
Ys dB, =
f
X, dM,;
a.s. P.
This equality is a consequence of Proposition 4.8.
3.4. Representations of Continuous Martingales in Terms of Brownian Motion 187 PROOF OF F. B. KNIGHT'S THEOREM 4.13. Our proof is based on that of Meyer
(1971). Under the hypotheses of Theorem 4.13, let {dl')} be the augmentation
of the filtration {..fn generated by B"); 1 5 i 5 d. All we need to show is that el'2), ... , dr are independent. For each i, let V') be a bounded, (IV-measurable random variable. According to Proposition 4.19, there is, for each i, a progressively measurable process
X") = {X1'),.; 0 < t < co} which satisfies 00
f(r)2 d<M(')>, < co; 1 < i < d, E
o
and for which
(i) = E(V)) + foo XP) dMP;
1 5i < d.
o
Let us assume for the moment that
1 5 i < d,
E(V)) = 0;
(4.52)
and define the (5,}-martingale ) -4-
ft Xli) dM11);
0 < t < co ,
1 < i < d.
0
ItO's rule and (4.31) imply that d
n v)= E
(4.53)
1=1
i=1 d
n epxrdmv; 0 _, t< 5'.) i#i
co.
In order to let t -+ co in (4.53), we must show that E
(4.54)
f (nJot es.h.x-r)
2
dop>s< co; 1 < i, d.
o
Repeated application of Holder's inequality yields: f1
E
0
2
( n ix)) d<m(0>s Joi
(VI)2] ,E{n[ Joi sup :)..s..t
-
<e>,}
1/2
[E sup
0 <s <,
sup (V))8
(V))4] IE
[E sup (V))21
0
2-,
ossst
.5.r
[E<e>712; 0 < t t < oo.
For m > 1, Doob's maximal inequality (Theorem 1.3.8 (iv)), and Jensen's inequality (Proposition 1.3.6) give
3. Stochastic Integration
188
( 2m 2m- 1
E[ 11:),s..t sup Ill)12"11
El ai)12m
2m
2m
(2m
2m
- 1)
El Vi)I2m < a).
We have from Proposition 3.26 (see also Remark 3.27): I
B E
I
for some positive constant B which does not depend on t. Thus (4.54) holds, and letting t oo in (4.53) we obtain the representation
= Ed
d
1=1
1=1
m
,v)x,i)dm).
0
The right-hand side, being a sum of martingale last elements (Problem 2.18), has expectation zero. Thus, under the assumption (4.52), we have Err, = 0. Equivalently, we have shown that for any set of bounded random variables V1),..., V), where each V) is 61-measurable, the equality (4.55)
E fl [Vi) - E(V))] = 0 i=1
holds. Using (4.55), one can show by a simple argument of induction on d that d
E ji i=1
d
=
Evi). i=1
Taking Vi) = 1,; Ai e 8,;;), 1 < i < d, we conclude that the a-fields 6V, ), are independent.
, crt')
What happens if the random variable in Problem 4.17 is not squareintegrable, but merely a.s. finite? It is reasonable to guess that there is still a representation of the form (4.47), where now the integrands Y(1), , Y(d) can only be expected to satisfy T
In fact, an even stronger result is true. We send the interested reader to Dudley
(1977) for the ingenious proof, which uses the representation of stochastic integrals as time-changed Brownian motions. There is, however, no uniqueness in Dudley's theorem; see Exercise 4:22 (iii).
4.20 Theorem (Dudley (1977)). Let W = 0 < t < co} be a standard, one-dimensional Brownian motion, where, in addition to satisfying the usual conditions, {9,} is left-continuous. If 0 < T < co and is an 9,-measurable, a.s. finite random variable, then there exists a progressively measurable process
3.4. Representations of Continuous Martingales in Terms of Brownian Motion 189 Y={
-.Ft; 0 < t 5 T} satisfying 17,2 dt < co;
a.s. P,
o
such that T
=
Yt dllit;
a.s. P.
4.21 Remark. Note in Theorem 4.20 that the filtration 1.97,1 might be generated by a d-dimensional Brownian motion of which W is only one component. For an amplification of this point, see Emery, Stricker & Yan (1983).
4.22 Exercise.
(i) In the setting of Proposition 4.18, show that the processes y"), Y(d) are unique in the sense that any other processes P"), f(d) in .29t(W) which permit the representation d
CO
= E(0 + > f J---1
cc(i)dw,u),
a.s.
0
must also satisfy CO
Jo
d
E yi(j) - i7;012 dt = 0,
a.s.
;=-1
(ii) In the setting of Proposition 4.19, show that X is unique in the sense that any other process X E .99*(M) which permits the representation
= E() + J
17, dM a.s., 0
must also satisfy ro°
IX: - g:12 d<M>: = 0,
a.s.
(iii) Find a progressively measurable process Y such that 1
0 < j. Y,2 dt < oo, a.s., o
1
but
Y, dWi = 0,
a.s.
o
In particular, there can be no assertion of uniqueness of Y in Theorem 4.20.
4.23 Exercise. Is the following assertion true or false? "If M = {M .Ft; 0 5 t < co} and N = {N Ft; 0 < t < co} are in ../1"" and <M> , then M and N have the same law."
3. Stochastic Integration
190
4.24 Exercise (Hajek (1985)). Consider the semimartingales
X, = X0 + j't ji ds + j't as dWs,
m(s)ds +
Y = Yo +
p(s)dVS
05t
mW,
thisl
atl
[R,
+ m(s) + p2(s)} ds < co, V0
almost surely. If X0 < Y0 also holds a.s. and f is a nondecreasing, convex show that function on P(X, > c) < 2P(Y, > c); V c e Ef(X,) < Ef(Y)
hold for every t > 0. (Hint: By extending the probability space if necessary, take W to be a Brownian motion independent of W, and consider the continuous semimartingales Y(i) A Y0 +
J
m(s)ds +
J
2 (s)
as dl/K + (-1)i
0.s2 dws;
i = 1, 2.)
3.5. The Girsanov Theorem In order to motivate the results of this section, let us consider independent Z,, on (52,,F, P) with EZi = 0, EZ? = 1. normal random variables Z1, Given a vector (pi,. .., un)egr, we consider the new probability measure P on (s2,9%) given by 1
n
piZi(w) - - E
P(dw) = exp
2 t =1
id
P(dw).
Then 15[Z le dz1,.. .,Zne dzn] is given by exp [tE
- - E id 1
= (2n)-"12 exp [ -
1
n
P [Z1 e dz 1,
- E (zi - pi)2]dz 2 i=t
, Zn e dzn]
dzn.
Therefore, under P the random variables Z1, Zn are independent and normal with EZi = pi and £[(Z1 - yi)2] = 1. In other words, {Z = Zi 1 < i < n} are independent, standard normal random variables on (s2, y, P). The Girsanov Theorem 5.1 extends this idea of invariance of Gaussian finitedimensional distributions under appropriate translations and changes of the
3.5. The Girsanov Theorem
191
underlying probability measure, from the discrete to the continuous setting. Rather than beginning with an n-dimensional vector (Z1,...,4) of independent, standard normal random variables, we begin with a d-dimensional Brownian motion under P, and then construct a new measure P under which a "translated" process is a d-dimensional Brownian motion.
A. The Basic Result Throughout this section, we shall have a probability space (n, P) and a d-dimensional Brownian motion W = {W = Wm), A; 0 < t < co} defined on it, with P[Wo = 0] = 1. We assume that the filtration {F,} satisfies the usual conditions. Let X = {X, = (V), , V)), A; 0 < t < ool be a vector of measurable, adapted processes satisfying (5.1)
dt < co ]=
P
1 < i < d, 0 5 T < co.
1;
Then, for each *, the stochastic integral /"(X(')) is defined and is a member of eirioc. We set (5.2)
Zi(X)
1
it t=1 a
exp [
dWs(i)
2
f
11Xsil 2
o
dd.
Just as in Example 3.9, we have =1
(5.3)
d
(1),
WZ,(X)
which shows that Z(X) is a continuous, local martingale with Zo(X) = 1. Under certain conditions on X, to be discussed later, Z(X) will in fact be a martingale, and so EZ,(X) = 1; 0 < t < co. In this case we can define, for each 0 < T < co, a probability measure PT on FT by (5.4)
E[lAZT(X)]; AE FT T.
PT(A)
The martingale property shows that the family of probability measures {PT; 0 < T < col satisfies the consistency condition PT(A)
(5.5)
A(A); AE
0 < t < T.
5.1 Theorem (Girsanov (1960), Cameron and Martin (1944)). Assume that Z(X) defined by (5.2) is a martingale. Define a process W = {W, =
Ito)), A; 0 < t < col by (5.6)
It(l)
Wt(i) -
ds;
1 < i < d, 0 < t < co.
For each fixed T e [0, co), the process {It, A; 0 < t < T} is a d-dimensional Brownian motion on (CI, FT, PT).
3. Stochastic Integration
192
The preparation for the proof of this result starts with Lemma 5.3; the reader may proceed there directly, skipping the remainder of this subsection on first reading. Discussion. Occasionally, one wants 1V, as a process defined for all t e [0, op), to be a Brownian motion, and for this purpose the measures {PT; 0 < T < co} are inadequate. We would like to have a single measure P defined on so that P restricted to any ..FT agrees with PT; however, such a measure does not exist in general. We thus content ourselves with a measure P defined only on Fcw, the o--field generated by W, such that P restricted to any ..*-TW agrees with PT, i.e., (5.7)
P(A) = E[1AZT(X)]; AE
0 5_ T < co.
,
If such a P exists, it is clearly unique. The existence of P follows from the Daniell-Kolmogorov consistency Theorem 2.2.2. We show this when d = 1; only notational changes are required for the multidimensional case. Let t = (t . . . , t) be a finite sequence of distinct, nonnegative numbers, as in Definition 2.2.1, and let t = max {t,,... , to. Define
Qt(r) =
[WE CI; Mit i((0),
Wtn(a)))
r];
re.6$(R ").
Then {$21} is a consistent family of finite-dimensional distributions, so there is a probability measure Q on (RE°''), gi(Rtc"'))) such that Q,(1") = Q[w e
(w(t ,),
, w (t )) e 11;
But the typical set in ..ny has the form {co e
e .4(01").
W(w)e B}, where B e Mi(Rt"')).
Consequently, Q induces a probability measure P on F' defined by Flu) e n; WOO e B] A Q(B);
B e .4(011°'')),
and this measure satisfies (5.7).
The process 1;17 in Theorem 5.1 is adapted to the filtration {",}, and so is the process {ft, Xr ds; 0 < t < co}; this can be seen as in part (c) of the proof of Lemma 2.4, which uses the completeness of .97,. However, when working with the measure P which is defined only on Fes, we wish 1,r to be adapted to Ign. This filtration does not satisfy the usual conditions, and so we must impose the stronger condition of progressive measurability on X. We have the following corollary to Theorem 5.1.
5.2 Corollary. Let W = {W, gft; 0 < t < co} be a d-dimensional Brownian motion on (12, P) with P[Wo = 0] = 1, and assume that the filtration {.F,} satisfies the usual conditions. Let X = {X
; 0 < t < co} be a d-dimensional, progressively measurable process satisfying (5.1). If Z(X) of (5.2) is a martingale, then 1,V = {14-7 ; 0 < t < co) defined by (5.6) is a d-dimensional Brownian motion on (11," cow , P).
3.5. The Girsanov Theorem
PROOF. For 0 5 ti <
193
< t S t, we have
PC(1171,...,
G
ai)e
= Pt[(147/t.
r EAR").
The result now follows from Theorem 5.1.
Under the assumptions of Corollary 5.2, the probability measures P and P are mutually absolutely continuous when restricted to Fr ; 0 < T < cc. However, viewed as probability measures on P and 15 may not be mutually absolutely continuous. For example, when d = 1 and X, = it, a nonzero constant, then Z,(X) = exp[i2147, -
0 < t < cc
t];
is easily seen to be a martingale. Corollary 5.2 and the law of large numbers imply
1P lim -W, = pl = P lim -W, = 01=
,t 1
,-00
P[lim -W, = pi= 1
1,
O.
t.."D t
In particular, the P-null event flim, (1/t) W, = /II is in FT for every 0 _5 T < co so 15 and PT cannot agree on T. This is the reason we require (5.7) to hold only for A E ..FTw.
B. Proof and Ramifications We now proceed with the proof of Theorem 5.1. We denote by ET (E.) the expectation operator with respect to PT (/-5).
5.3 Lemma. Fix 0 < T < co and assume that Z(X) is a martingale. If 0 < s _5 t < T and Y is an .5(7,-measurable random variable satisfying ETjYI < co, then we have the Bayes' rule:
G[YI T's] = (ma Yzi(x)IFJ,
a.s. P and PT.
PROOF. Using the definition of E T, the definition of conditional expectation, and the martingale property, we have for any A e
ETI1A Zs(X)EEYZt(X)IFJ} = EflAE[YZt(X)1Fs]} = E[IA YZ,(X)] = ET[lA
We denote by J1;4' the class of continuous local martingales M = IM P) satisfying P[MO = 0], and define similarly, 0 5 t < T1 on (12, with P replaced by PT.
3. Stochastic Integration
194
5.4 Proposition. Fix 0 S T< co and assume that Z(X) is a martingale. If M e..16:1" then the process
A A Mt
(5.8)
-
i't XV d<M, W(II> s, .97,; 0 5 t< T 1=1
o
is in ,thl". If N e .11,:l" and
IC I, A N, -i
ft XV d s;
i=1
0 5 t 5 T,
o
then <1171,R>, = <M, N>,;
0 < t < T, a.s. P and PT,
where the cross-variations are computed under the appropriate measures.
PROOF. We consider only the case where M and N are bounded martingales with bounded quadratic variations, and assume also that Z,(X) and 11=i ro (XS )2 ds are bounded in t and w; the general case can be reduced to this one by localization. From Proposition 2.14 2 ft
< <M>, 1 t
XV d<M,14/IiI>,
(X.1`))2 ds,
o
and thus SI is also bounded. The integration-by-parts formula of Problem 3.12 gives
Z,(X)S1, =
1
Zu(X)d111,, + > i=1
o
ft 1171.XLIZAX)d14;V, o
which is a martingale under P. Therefore, for 0 < s < t < T, we have from Lemma 5.3:
EAR tl.F.,1 =
Z,(X)
[E Z,(X)1171,1,j = las,
a.s. P and 15T.
It follows that AI e 2'1". The change-of-variable formula also implies:
AR, - <M, N>, =
ft M dN, + ft IV. dM, -i [ ft 111XLII d. o
+
i=1
o
o
ft I1 XV d<M, Ww>.1 o
as well as Z,(X)[1171,1C,
- <M, N> J =
ft Zu(X)111dN. + ft Zu(X)R. dM, o
o
f t [M.N. - <M, N >] XV 4,(X) dW,V .
+ i=1
0
3.5. The Girsanov Theorem
195
This last process is consequently a martingale under P, and so Lemma 5.3
implies that for 0 .s5t5T
-
EAR iNt - t1.97s] =
N> s;
a.s. P and PT.
This proves that <M, N>, = <M, N),; 0 5 t < T, a.s. PT and P. PROOF OF THEOREM 5.1. We show that the continuous process Won (II, ,FT, PT)
satisfies the hypotheses of P. Levy's Theorem 3.16. Setting M = Wth in Proposition 5.4 we obtain SI = IP) from (5.8), so Wu) e i/V". Setting N = HM), we obtain <1170 VV(k)>
Let {M
<W-1),W(``)>, = Sj.kt;
0 < t < T, a.s. PT and P.
0 < t < T} be a continuous local martingale under P. With
the hypotheses of Theorem 5.1, Proposition 5.4 shows that M is a continuous semimartingale under PT. The converse is also true; if {114 .Ft; 0 < t <
is a continuous martingale under PT, then Lemma 5.3 implies that for 0 < s < t < T: E[Z,(X)1171,IF] = Zs(X)ET[1171,13U = Zs(X)M- s;
a.s. P and PT,
so Z(X)/171 is a martingale under P. If SI EMI", a localization argument shows that Z(X)M edie;21". But Z(X) e dr, and so Ito's rule implies that Al =
[z(X)iii]/Z(X) is a continuous semimartingale under P (cf. Remark 3.4). Thus, given gi eA:'°c, we have a decomposition git = Mt + Bt;
0 < t < 'T,
where M e .11P° and B is the difference of two continuous, nondecreasing adapted processes with Bc, = 0, P-a.s. According to Proposition 5.4, the process
- (M, - E d
i=1
=A+if
d<M, Wffi>
)
o
d
i=1
t
d<M,W")>s;
0
t 5 T,
o
is in A:1", and being of bounded variation this process must be indistinguish-
able from the identically zero process (Problem 3.2). We have proved the following result.
5.5 Proposition. Under the hypotheses of Theorem 5.1, every M e 22;1'°` has the representation (5.8) for some M e
We note now that integrals with respect to dlit(1) have two possible interpretations. On the one hand, we may interpret them by replacing dl,Pi(l) by d147,") - XY)dt so as to obtain the sum of an Ito integral (under P) and a Lebesgue-Stieltjes integral. On the other hand, 1,V(`) is a Brownian motion
196
3. Stochastic Integration
under PT, so we may regard integrals with respect to dii as Ito integrals under PT. Fortunately, these two interpretations coincide, as the next problem shows. 5.6 Problem. Assume the hypotheses of Theorem 5.1 and suppose Y = { Yt, "t; 0 < t < co} is a measurable adapted process satisfying P[11; Y2 dt < co] = 1;
0 < T < co. Under P we may define the Ito integral Sto I's dWs"), whereas under PT we may define the Ito integral Po Ys dlik,(1), 0 < t < T Show that for 1 < i < d, we have t
jot Ys dW(i) --
=
I
Ys
dW(i) - UV ds;
0
0 < t < T, a.s. P and PT.
o
(Hint: Use Proposition 2.24.)
C. Brownian Motion with Drift Let us discuss a rather simple, but interesting, application of the Girsanov theorem: the distribution of passage times for Brownian motion with drift. We consider a Brownian motion W = { 14/ ,F,; 0 < t < co} and recall from Remark 2.8.3 that the passage time Tb to the level b 0 has density and moment generating function, respectively: (5.9)
P[Tb e dt] = / 27tt3
(5.10)
exp [--b2] dt; 2t
t > 0,
Ee-2Tb = e-lbl; a > 0.
For any real numbers 0, the process W = { IT - pt, gir; 0 < t < col is a Brownian motion under the unique measure P(") which satisfies
P(n(A) = EDAM; A e where Z, A exp(gW, - iii2t) (Corollary 5.2). We say that, under 1341), 14/, = pt + i1 is a Brownian motion with drift p. On the set { Tb < t} e n = 3'7,w Tb we have Z, A 71, = ZTb, so the optional sampling Theorem 1.3.22 and Problem 1.3.23 (i) imply (5.11)
P(u)[Tb < t] =
{Tb,,}Z,] = E[1{4.,,}E(Z,Ig7,,,,b)]
= E[1{Tb<,}4,rb] = E[l{rbst}ZTb]
=
(7.b <06,0-012w2Tb]
= J t exp (µb
- 2 µ2s) P[T, e ds]. -1
The relation (5.11) has several consequences. First, together with (5.9) it yields the density of Tb under P"):
3.5. The Girsanov Theorem
197
P(*)[Tbedt] =
(5.12)
I
bI
2 t3
Second, letting t
exp [
(b
Pt)] dt, t> 0.
2t
co in (5.11), we see that
P(")[T < co] = e"E[exp( -114270], and so we obtain from (5.10):
/34")[Tb < co] = exp[ab -
(5.13)
In particular, a Brownian motion with drift p # 0 reaches the level b 0 0 with probability one if and only if p and b have the same sign. If It and b have opposite
signs, the density in (5.12) is "defective," in the sense that P(")[Tb < co] < 1.
5.7 Problem. Let T be a stopping time of the filtration {..fr} with P[T < co] = 1. A necessary and sufficient condition for the validity of the Wald identity
E[exp(pWr - 1/42 T)] = 1,
(5.14)
whereµ is a given real number, is that
P(PIT < co] = I.
(5.15)
In particular, if be fR and pb < 0, then this condition holds for the stopping time S,
(5.16)
inff t
0; W, - pt =- 131.
5.8 Problem. Denote by h(t;b, p)
Ibl ,
exp
7-ct3
(b - 2 t > 0, b 2ttit)];
0, p e 1,
the (possibly defective) density on the right-hand side of (5.12). Use Theorem 2.6.16 to show that h(
; b, + b2, p) = h(
;hi,
* h(
h2, P);
bi b2 > 0,
eR,
where * denotes convolution. 5.9 Exercise. With kt > 0 and W* inft>0 Wt, under P(") the random variable -W* is exponentially distributed with parameter 2a, i.e., P(*)[- W* e db] = 2pe-2*b db,
b > 0.
5.10 Exercise. Show that
E(u)e-ar, = exp(ab - Ibl,//22 + 2a), a > 0. 5.11 Exercise (Robbins & Siegmund (1973)). Consider, for v > 0 and c > 1, the stopping time of {77,"1:
3. Stochastic Integration
198
=
R, = inf {t > 0; exp(v W,
Show that
P[R, < co]
1
E(Y)R, -
2log c
c
.
v2
D. The Novikov Condition In order to use the Girsanov theorem effectively, we need some fairly general conditions under which the process Z(X) defined by (5.2) is a martingale. This process is a local martingale because of (5.3). Indeed, with T. A inf It > 0; max i
f (Z,(X)Xli))2 ds = n} 0
the "stopped" processes Z(") A {Z:") A 4,., T.(x), .Ft; 0 < t < co} are martingales. Consequently, we have
E[ZtATI's] =
A
;
0 < S < t, n > 1,
and using Fatou's lemma as n co, we obtain E[Z,(X)I.Fs] < 4; 0 s s s_ t. In other words, Z(X) is always a supermartingale and is a martingale if and only if EZ1(X) = 1;
(5.17)
0 < t < co
(Problem 1.3.25). We provide now sufficient conditions for (5.17).
5.12 Proposition. Let M = {M gr,; 0 < t < oo} be in Jr' and define
4 = exp[M, - i<M>1]; 0 < t < co.
If (5.18)
E[exp{ <M>,}] < oo; 0 < t <
then EZ, = 1; 0 < t < PROOF. Let T(s) = inf {t > 0; <M>, > s }, so the time-changed process B of (4.16) is a Brownian motion (Theorem 4.6 and Problem 4.7). For b < 0, we define the stopping time for IC as in (5.16):
S, = infls > 0; Bs - s = bl Problem 5.7 yields the Wald identity E[exp(Bsb - ISO] = 1, whence E[exp(lS,)] = e-b. Consider the exponential martingale lc A exp(B, (s /2)), 0 < s < ool and define {N, A Y -3Asb, 4s; 0 < s < co}. According to Problem 1.3.24 (i), N is a martingale, and because P[Sb < co] = 1 we have
N. = lim Ns = exp(Bsb - iSb). CO
3.5. The Girsanov Theorem
199
It follows easily from Fatou's lemma that N = {Ns, '.,; 0 < s < co) is a super-
martingale with a last element. However, ENS = 1 = EN0, so N = IN 's; 0 < s < co) has constant expectation; thus N is actually a martingale with a last element (Problem 1.3.25). This allows us to use the optional sampling Theorem 1.3.22 to conclude that for any stopping time R of the filtration {/s}:
E[exp{Bsb - l(R A Sz, ) } ] = 1. Now let us fix t e [0, 00) and recall, from the proof of Theorem 4.6, that <M>, is a stopping time of {/s}. It follows that for b < 0: (5.19)
E[1(sb <,,,,),) exp(b + ISM + E[1(0,0,<sbi exp(M, - i<M>,)] = 1.
The first expectation in (5.19) is bounded above by ebE[exp(Z <M>,)], which
converges to zero as b1 -co, thanks to assumption (5.18). As b. -co, the second expectation in (5.19) converges to EZ, because of the monotone convergence theorem. Therefore, EZ, = 1; 0 < t < co.
5.13 Corollary (Novikov (1972)). Let W = {W, = (W,(1),..., W,(d)), A; 0 r)), t < co) be a d-dimensional Brownian motion, and let X = pc, = (Xi A; 0 < t < co) be a vector of measurable, adapted processes satisfying (5.1). If (5.20)
E[exp (-21 f To
11Xs 112
ds)]< co;
0 < T < co,
then Z(X) defined by (5.2) is a martingale.
5.14 Corollary. Corollary 5.13 still holds if (5.20) is replaced by the following assumption: there exists a sequence {t},T=0 of real numbers with 0 = to < < tni co, such that t1 < (5.21)
E [exp (-1 f in 2
IlXs112ds)1 < oo;
do
1.
tn -1
PROOF. Let X,(n) = (X)1)1i,n_1.,)(t),..., Xrlitn_i.t.)(0), so that Z(X(n)) is a martingale by Corollary 5.13. In particular, E[Z,.(X (n))1,,._i] = Z,,,_i(X(n)) = 1;
n > 1.
But then,
E[Z,(X)] = E[Z,_1(X)EIZi(X(n))1A_,}i = E[4_1(X)], and by induction on n we can show that E[Z,.(X)] = 1 holds for all n > 1. Since E[Z,(X)] is nonincreasing in t and limn_o, t = co, we obtain (5.17). CI
5.15 Definition. Let C[0, cod be the space of continuous functions x: [0, oo) ->
Rd. For 0 < t < co, define % A r(x(s); 0 < s < t), and set / = /co (cf. Problems 2.4.1 and 2.4.2). A progressively measurable functional on C[0, ce)d is a
3. Stochastic Integration
200
mapping /2: [0, oo) x C[0, oar -> IR which has the property that for each fixed 0 5 t < co, ti restricted to [0, t] x C[0, oor is .4([0, t]) %/.4([11)-measurable.
Ifµ =
tion) is a vector of progressively measurable functionals on
C[0, °or and W =
Wt(d)), 37".,; 0 < t < co} is a d-dimensional
= (W,(1),
Brownian motion on some
P), then the processes
,qi)(u)) A p(l)(t, W(oo));
(5.22)
0 < t < co,
1 < i < d,
are progressively measurable relative to 5.16 Corollary (Beneg (1971)). Let the vector pi =
,
!PI)) of progressively
measurable functionals on C[0, cor satisfy, for each 0 < T < oo and some KT > 0 depending on T, the condition ti(t, x)II < K T(1 + x*(t));
(5.23)
0
Ilx(s)11. Then with X, = where x*(t) g maxo Z(X) of (5.2) is a martingale.
,
Xr) defined by (5.22),
PROOF. If, for arbitrary T > 0, we can find {to, , tnal} such that 0 = to < = T and (5.21) holds for 1 n < n(T), then we can construct t1 < < a sequence ItnI,T.0 satisfying the hypotheses of Corollary 5.14. Thus, fix T > 0. We have from (5.22), (5.23) that whenever 0 < ti_, < t,, < T, then
t
11)(3112 ds < (t,, - 4_1)1(1(1 + WT*)2, Itt,_,
where W; A maxor II NI. According to Problem 1.3.7, the process Y, A exp[(t. )K + II Wt )2/4] is a submartingale, and Doob's maximal
-
inequality (Theorem 1.3.8(iv)) yields
E exp[f(t -
)1q(1 + WT*)2] = EI max Yi2 < 4EY,4, 0
which is finite provided that t,, - t_, _< 1/Tiq. This allows us to construct {to,
,
tn(T)} as described previously.
El
5.17 Remark. Liptser & Shiryaev (1977), p. 222, show that with d = 1 and 0 < e < 1, one can construct a process X satisfying the hypotheses of Corollary 5.13 but with (5.20) replaced by the weaker condition E [exp
IG
- e) fo
dt}1 < co; 0 < T < co,
such that Z(X) is not a martingale. The next exercise, taken from Liptser & Shiryaev (1977), p. 224, provides a simple example in which Z(X) is not a martingale. In particular, it shows that a local martingale (cf. (5.3)) need not be a martingale.
3.6. Local Time and a Generalized ItO Rule for Brownian Motion
201
5.18 Exercise. With W = { 14/,, .; 0
0 < t < 1,
(1- t)2
t = 1.
0;
(i) Prove that P[T < 1] = 1, and therefore .1,`, X,2 dt < cc a.s. (ii) Apply Ito's rule to the process {(41/1(1 - t))2; 0 < t < 1} to conclude that X, d147,
1 ft
dt
0 T o
1
(1 -
031Wt2
(iii) The exponential supermartingale {Z,(X), ,F,; 0 5 t II is not a martingale; however, for each n > 1 and an = 1 - ( 1 /.\Ai), {Z, (X), .??7,; 0 < t < 1} is a martingale.
5.19 Exercise. Let W = {W, g",; 0 < t < co} be a Brownian motion on (0,F, P) with P[Wo = 0] = 1, and assume that {Ft} is the augmentation under P of the Brownian filtration {,,W}. Suppose that, for each 0 < T < oo,
there is a probability measure PT on .FT which is mutually absolutely continuous with respect to P, and that the family of probability measures {PT; 0 < T < oo} satisfies the consistency condition (5.5). Show that there exists a measurable, adapted process X = {X 317,; 0 < t < co satisfying (5.1), such that Z(X) defined by (5.2) is a martingale and (5.4) holds for 0 < T < co. (Hint: Apply Problem 4.16 to the Radon-Nikodj/m derivative process dP, /dP.)
5.20 Exercise. Suppose that {L .Ft; 0 < t < op} e Jr' is such that Z, A exp[L, -1,] is a martingale under P, and define the new probability E(1,4); A e FT. Establish the following generalization of Proposition 5.4 and of the Girsanov theorem: if M e ../r1", then measure PT(A)
/17f, A Mt
- , = M -
d o Zs "
,
;007", ;
0
t
T
is in di'. (Hint: Imitate the proof of Proposition 5.4.)
3.6. Local Time and a Generalized ItO Rule for Brownian Motion In this section we devise a method for measuring the amount of time spent by the Brownian path in the vicinity of a point x e R. We saw in Section 2.9 that the Lebesgue measure of the level set .1°.(x)= {0 5 t < oo; 14/,(co) =
3. Stochastic Integration
202
turns out to be zero, i.e., meas 2',,,(x) = 0,
(6.1)
for P-a.e. co ER
yielding no information whatsoever about the amount of time spent in the vicinity of the point x (Theorem 2.9.6 and Remark 2.9.7). In search of a nontrivial measure for this amount of time, P. Levy introduced the twoparameter random field 1
(6.2)
Lt(x) = lim - meas {0 < s < t; I Ws - xi < e}; t e [0, oo), x e gl a 0 4a
and showed that this limit exists and is finite, but not identically zero. We shall show how Lt(x) can be chosen to be jointly continuous in (t, x) and, for fixed x, nondecreasing in t and constant on each interval in the complement of the closed set 2',,,(x). Therefore, (d/dt)L,(x) exists and is zero for Lebesgue almost every t; i.e., the function t i- Lt(x) is singularly continuous. P. Levy
called Lt(x) the mesure du voisinage, or "measure of the time spent by the Brownian path in the vicinity of the point x." We shall refer to Lt(x) as local time.
This new concept provides a very powerful tool for the study of Brownian sample paths. In this section, we show how it allows us to generalize It6's change-of-variable rule to convex but not necessarily differentiable functions, and we use it to study certain additive functionals of the Brownian path. These functionals will be employed in Chapter 5 to provide solutions of stochastic differential equations by the method of random time-change. Local time will
be further developed in Chapter 6, where we shall use it to prove that the Brownian path has no point of increase (Theorem 2.9.13). In this section, the reader can appreciate the application of local time to the study of Brownian sample paths by providing a simple proof of their nondifferentiability (Exercise 6.6). This exercise shows that jointly continuous local time cannot exist for processes whose sample paths are of bounded variation on bounded intervals. Throughout this section, 1147 A; 0 < t < col, (f2,F), {Pz }Z Ft denotes the one-dimensional Brownian family on the canonical space f2 = C[0, co). This
assumption entails no loss of generality, because every standard Brownian motion induces Wiener measure on C[0, co) (Remark 2.4.22), and results proved for the latter can be carried back to the original probability space. We take the filtration {..F, } to be {A} defined by (2.7.3), and we set .F = .7'03. This
filtration satisfies the usual conditions. In this situation, PZ is just a translate of P°, i.e., (6.3)
P(F) = P°(F - z); F e g ,
(cf. (2.5.1)). We also have at our disposal the shift operators {0,},,,, defined by (2.5.15).
6.1 Definition. A measurable, adapted, real-valued process A = { A .; 0 t < co} is called an additive functional if, for every z e 18 and Pz-a.e. co e f2, .-
3.6. Local Time and a Generalized Ito Rule for Brownian Motion
203
we have (6.4)
At+s(w) = As(co) + At(Osto);
0 < s, t < co.
6.2 Example. For every fixed Borel set Be .41([11), we define the occupation time
of B by the Brownian path up to time t as (6.5)
co) -4 J t
1B( Ws)ds = meas {0 Lc. s Lc t; Ws e B};
0 ...5_ t < co,
o
where meas denotes Lebesgue measure. The resulting process F(B) --- {F1(B), ,F;; 0 < t < co} is adapted and continuous, and is easily seen to be an additive functional.
A. Definition of Local Time and the Tanaka Formula Equation (6.2) indicates that doubled local time 24(x) should serve as a density
with respect to Lebesgue measure for occupation time. In other words, we should have (6.6)
TAB, co) =
i 24(x, w) dx;
0 < t < co, B e .4(01).
B
We take this property as part of the definition of local time. 63 Definition. Let L = {L,(x, w); (t, x) e [0, oo) x R, to ell} be a random field with values in [0, co), such that for each fixed value of the parameter pair (t, x)
the random variable Lt(x) is .- measurable. Suppose that there is an event ll* e g" with Pz(11*) = 1 for every z e 0:8 and such that, for each w e fe, the func-
tion (t, x)i-- Lt(x, co) is continuous and (6.6) holds. Then we call L Brownian local time.
6.4 Remark. There is no universal agreement in the literature as to whether L in Definition 6.3 or 2L is to be called local time. We follow the normalization (6.6), used by Ikeda & Watanabe (1981).
6.5 Remark. With L as in Definition 6.3 and w e 11*, one can immediately derive (6.2) from (6.6) and the continuity of x 1-- Lt(x, co). Further, L(a) = {L1(a), g;; 0 < t < co} is easily seen to inherit the additive functional property (6.4) from its progenitor, the occupation time I" (Example 6.2). 6.6 Exercise. Assume that Brownian local time exists and show that for each w ell* of Definition 6.3, the sample path t F-4 Wt(o)) cannot be differentiable anywhere on (0, co). (Hint: If t'-' Wt(w) is differentiable at t, then for some sufficiently large C and sufficiently small S > 0 we must have iWt+h(w) W,(w)1 _._ Ch; 0 5 h 5 b.)
3. Stochastic Integration
204
6.7 Problem.
(i) Show that the validity of (6.6) is equivalent to (6.7)
fo f(147,(co)) ds = 2
f
t < co,
0
(x) Lt (x, co) dx;
for every Borel-measurable function f : l -4 [0, co). (ii) Let ft' be the class of continuous functions h: [11 - [0, 1] of th,:. form 0;
x -q i
(6.8)
x
q1 ,
q1
x
q2,
q2
x
q3,
q3
x
q4,
x
q4,
q1
1; h(x) = .112
q4 -x q4 - q3 0;
where q1 < q2 < q, < q4 are rational numbers. 1
q2
(13
q4
Show that if (6.7) holds for all h E .°, then it holds for every Borelmeasurable function f : [18 -, [0, co).
Our plan is the following: we shall assume in the present subsection that Brownian local time exists, and we shall derive a convenient representation for it, the Tanaka formula (6.11). In the next subsection it will be shown that the right-hand side of (6.11) leads to a random field which satisfies the requirements of Definition 6.3, thus establishing existence.
Let us fix a number a e R, and take f(x) in (6.7) to be the Dirac delta evaluated at x - a, and derive formally the representation (6.9)
Lt(a, co) =
1f'
6 (117 s(co) - a) ds.
But the integral on the right-hand side is only formal. In order to give it meaning, we consider the nondecreasing, convex function u(x) = (x - a) +, which is continuously differentiable on R \Ial and whose second derivative in the distributional sense is u"(x) = 5(x - a). Bravely assuming that Ito's rule can be applied in this highly irregular situation, we write
3.6. Local Time and a Generalized Ito Rule for Brownian Motion
(6.10)
(Wt - a)+ - (z - a)+ = f
1(,,00)(14/3)dW, +
0
2
f
205
(5(Ws - a) ds,
0
and in conjunction with (6.9), we have (6.11)
L,(a) = (W, - a)+ - (z - a)+
-f I 1(..00(4 )dWs;
0 ._ t < co
0
Pz-a.s. for every z E R. Despite the heuristic nature of both (6.9) and (6.10), the representation (6.11) for local time is valid and will be established rigorously. 6.8 Proposition. Let us assume that Brownian local time exists, and fix a number a e R. Then the process L(a) = {L,(a), ,F,; 0 _<_ t < co} is a nonnegative, continuous additive functional which satisfies (6.11) and the companion representations (6.12)
L,(a) = (W, - a)- - (z - a)- +
ft 1(,,,l(Ws)dWs;
0 < t < co,
0
(6.13)
24(a) = I W, - al - lz - al - ft sgn(Ws - a) digs;
0
t < oo
o
a.s. Pz, for every z E R.
6.9 Remark. Any of the formulas (6.11), (6.12), or (6.13) is referred to as the Tanaka formula for Brownian local time. We need establish only (6.11); then (6.12) follows by symmetry and (6.13) by addition, since Pz[i.
1{.}(Ws) dW, = 0;
V0 < t < oo]=--
1;
V z e R.
o
In particular, it does not matter how we define sgn(0) in (6.13); we shall define sgn so as to make it left-continuous, i.e., (6.14)
sgn(x) =
1;
x>0 x _. 0.
6.10 Remark. The process { (W, - a) +, ,; 0 < t < col is a continuous, nonnegative submartingale (Proposition 1.3.6); it admits, therefore, a unique Doob-Meyer decomposition (under 1', for any z e R): (6.15)
(W, - a)+ = (z - a)+ + M,(a) + A,(a); 0 < t < co,
where A(a) is a continuous, increasing process and M(a) is a martingale (see Section 1.4). The Tanaka formula (6.11) identifies both parts of this decomposition, as A,(a) = L,(a) and t
(6.16)
M,(a) = 1. 1(,,o)(Ws) dWs; 0 5 t < co. o
Similar remarks apply to the representations (6.12) and (6.13).
3. Stochastic Integration
206
PROOF OF PROPOSITION 6.8. In order to make rigorous the heuristic discussion
which led to (6.11), we must approximate the Dirac delta 5(x) by a sequence
of probability densities with increasing concentration at the origin. More specifically, let us start with the C' function p(x)
(6.17)
cexp[(x
1
0 < x < 2,
- 1)2 - 1];
otherwise
0;
which satisfies f°20, p(x) dx = 1 by appropriate choice of the constant c, and use it to define the probability density functions (called mollifiers) (6.18)
/Mx) -4- np(nx)
as well as un(x)
xeIll, n > 1.
Pn(z - a) dz dy;
We observe that un(x) = f X p(z - a) dz , and so we have the limiting relations
lim u'(x) = 1(,,,n)(x),
lim u(x) = (x - a) +,
x e141.
We now choose an arbitrary z e R. According to Ito's rule, (6.19)
un(W,) - u,,(z) =
un(14/s)dW, +
2
ft, pn(W, - a) ds;
t < co,
0
a.s. Pz. But now from (6.7) and the continuity of local time, t
p(14!, - a) ds = 2
pn(x - a)Li(x)dx
2L,(a);
a.s. P2.
On the other hand, Ez
j; un(Ws)dW = Ez
j . lun (Ws)
2
1o,)(Ws) 0, dW, - 10,,)(Ws)12 ds
Pz[IW,
- al
-2
ids,
n
which converges to zero as n co. Therefore, for each fixed t, the stochastic integral in (6.19) converges in quadratic mean to the one in (6.16), and (6.11) for each fixed t follows by letting n co in (6.19). Because of the continuity of the processes in (6.11), we obtain that, except on a Pz-null event, (6.11) holds
for 0 < t < co.
B. The Trotter Existence Theorem We can employ now the Tanaka representation (6.11) to settle the question of existence of Brownian local time. This is in fact the only result proved in this subsection.
3.6. Local Time and a Generalized Ito Rule for Brownian Motion
207
6.11 Theorem (Trotter (1958)). Brownian local time exists.
PROOF. We start by showing that the two-parameter random field, obtained by setting z = 0 on the right-hand side of (6.11), admits a continuous modification under P°. The term (W, - ( -a)+ is obviously jointly continuous in the pair (t, a). For the random field {M,(a); 0 5 t < oo, a E RI in (6.16) we
have, with a < b, 0 <s 1: E° I M1(a) - Ms(b)I2" 2n
< 4" {E°
+ E°
f
s
< 4"Cn[E° (ft 1.(,,,,)(Wn)du)n + E°
1(..b)(W.)dW.
(c
thanks to (3.37). The first expectation is bounded above by (t - s)", whereas the second is dominated by
e[f
1,0.6,(W,)dt
0
T
T
=
E° [1 (a , b]( Wt 1,
1 (a, bl(W tn)]
dt.
dt,
r = n!
[1(a, 11( WI, ) 10
Wt2). . . 10,,bi(Win)]dtn...dt2dt1.
ti
With 0 < t < 0 < T, we have for every y E R,
P° [a < Wo < blW, = y] < P° [ a < Wo
=-f
(6-- 012.18 -t
e-Z212
dZ <
o
b
W=
a + bl
2J
b -a / -t
and so [T 1(,,,,,i(W)dt
E°
o
(b -a en
n
T
T
T
Cti(t2 - ti)...(tn -
)]-1/2 dt .
dt2 dti
(b - a)",
where en, T is a constant depending on n and T but not on a or b. Therefore,
witha
E°14(a) - 1115(b),' 5 Cn,
5
C., T
- sr + (b - 47] a) - (s, b)11"
3. Stochastic Integration
208
for some constant C,,.. By the version of the Kolmogorov-Centsov theorem for random fields (Problem 2.2.9), there exists a two-parameter random field Ilt(a); (t, a) e [0, GO) x RI such that for P°-a.e. wee, the mapping (t, a)1-. I t(a, w) is locally Holder-continuous with exponent y, for any y e (0,1), and for each fixed pair (t, a) we have
P° [It(a) = Mt(a)] = 1.
(6.21)
Now we define
Lt(a) A (W - a)+ - (- a)+ - I t(a);
0 <_ t < GO, a e R.
For fixed (t, a), Lt(a) is an ,t-measurable random variable, and the random field L is P°-a.s. continuous in the pair (t, a). Indeed, because 14/, and I t(a) are both locally HOlder-continuous for any exponent y e (0,1), the local time L also has this property: for every y e (0,1), T > 0, K > 0, there exists a P°-a.s. positive random variable h(w) and a constant 6 > 0 such that P° co [ e SI;
(6.22)
sup
I Lt(a, co) -L s(b, (0)1
o<11060-(s,b)11
Il(t, a) - (s, b)IlY
<
6]=
1.
0 <s,t< T
-K
Our next task is to show that the random field Lt(a) satisf es the identity (6.6), or equivalently (6.7). For every function h in the class Yt of Problem 6.7, define
- u)+ du =
H(x) A I 'D h(u)
x
Ih(u) du dy; Y
xeR
and observe the identities x
H' (x) = 1
h(u)1 (,,,c0(x) du = 1
H"(x) = h(x).
h(u) du,
By virtue of It6's rule and Problem 6.12, we have P°-a.s. for fixed t > 0: 1
2
1 t h(W )ds 0
s
= H(W) - H(0) -
ft H' (Ws) dWs 0
1.0
= -.0 fx'
= -.0 = fm But
t
h(a) {(Wt - a)+ - (- a)+ 1 da
-f (J
.0
h(a)1(a,c0)(Ws)da)dWs
o
h(a){(Wt
- a)+ - (- a)+ -
ft 1(a,c0)(Ws)dWs} da o
+ fm
- Mt(a)} da.
E° ITco (I t(a) - Mt(a))2 da = Icf,, E° (L(a) - Mt(a))2 da = 0
by
(6.21).
3.6. Local Time and a Generalized It6 Rule for Brownian Motion
209
Thus, for each fixed t > 0, we have for P°-a.e. w e S2 (6.23)
co h(Ws(co))ds = 2
i
h(x)L,(x, to) dx .
- -co
Since both sides of (6.23) are continuous in t and Jr is countable, it is possible to find an event M e.F with P°(123) = 1 such that for each w e 03, (6.23) holds for every h e Ye' and every t > 0. Problem 6.7 now implies that for each w e 03, (6.7) holds for every Borel function f: fll -) [0, co). Recall finally that LI = C[0, co) and that P assigns probability one to the event S2z -A {w e SI; w(0) = z }. We may assume that C23 g 00, and redefine Lt(x, w) for w 0 Slo by setting
Li(x, w) A Lt(x - w(0), w - w(0)).
We set SP = {w ell; w - w(0) e M}, so that MCP) = 1 for every z e DI (c.f. (6.3)). It is easily verified that L and Q* have all the properties set forth in Definition 6.3. 1:1 6.12 Problem. For a continuous function h: R -> [0, co) with compact support, the following interchange of Lebesgue and Ito integrals is permissible: (6.24)
f 1 h(a)( f :
=
1 (,,,,,,)(Ws) d Ws)
da
ft (..ix' h(a)1()(Ws)da)dWs, a.s. P°. -co
o
6.13 Problem. We may cast (6.13) in the form (6.25)
I Hit - al = I z - al - 13,(a) + 24(a); 0 < t < co,
where B,(a) A -Po sgn( Ws - a) digs, for fixed a e R.
(i) Show that for any z e R, the process B(a) = {B,(a), .97i; 0 .< t < col is a Brownian motion under P, with Pz [B0(a) = 0] = 1. (ii) Using (6.25) and the representation (6.2), show that L(a) = {L,(a), ,-,; 0 < t < co} is a continuous, increasing process (Definition 1.4.4) which satisfies (6.26)
J
101\{.}(NdLi(a) = 0;
a.s. P=.
1o
In other words, the path t'--> L,(a, w) is "flat" off the level set Y,,,(a) = t < co; W(w) =a }. {0 (iii) Show that for P°-a.e. w, we have L,(0, w) > 0 for all t > 0. (iv) Show that for every z e R and Pz-a.e. w, every point of Yo,(a) is a point of strict increase of ti--+ L ,(a , w).
3. Stochastic Integration
210
C. Reflected Brownian Motion and the Skorohod Equation Our goal in this subsection is to provide a new proof of the celebrated result of P. Levy (1948) already discussed in Problem 2.8.8, according to which the processes (6.27)
{4w - w, A max W- W; 0 -t
have the same laws under P°. In particular, we shall present the ingenious method of A. V. Skorohod (1961), which provides as a by-product the fact that the processes (6.28)
{Mr A max Ws; 0 < t < a)} and ossst
{2L,(0); 0 < t < op}
also have the same laws under P°.
6.14 Lemma (The Skorohod equation (1961)). Let z > 0 be a given number and y() = {y(t); 0 t < co} a continuous function with y(0) = 0. There exists a unique continuous function k() = {k(t); 0 5 t < co}, such that
(i) x(t) A z + y(t) + k(t), 0; 0
t < aD, (ii) k(0) = 0, k() is nondecreasing, and (iii) k() is flat off {t 0; x(t) = 0 }; i.e., l'o° 1{xop.0}dk(s) = 0. This function is given by (6.29)
k(t) = max [0, max { -(z + y(s)) }], 1:..,..1
0 5 t < co.
PROOF. To prove uniqueness, let k() and ic() be continuous functions with
properties (i)-(iii), where x() and Si) correspond to k() and k(-), respectively. Suppose there exists a number T > 0 with x(T) > i(T), and let T A- max {0 < t < T; x(t) - Z(t) = 0} so that x(t) > 5e(t) > 0, V t e(T, T]. But k() is flat on {u 0; x(u) > 0 }, so k(t) = k(T). Therefore,
0 < x(T) - z(T) = k(T) - k(T) < k(t) - kW = x(t) - 54-r), a contradiction. It follows that x(T) < Z(T) for all T > 0, so k 5 k. Similarly, k > k. We now take k( ) to be defined by (6.29). Conditions (i) and (ii) are obviously
satisfied. In order to verify (iii), it suffices to show Jo 1{x(s),,} dk(s) = 0 for every E > 0. Let (t t2) be a component of the open set {s > 0; x(s) > el and note that -(z + y(s)) = k(s) - x(s) _-_ k(t2) - E; t, .5 s 5 t2. But then k(t2) = max [k(t,), max {-(z + y(s)) }] 5 max[k(ti), k(t2) - a],
t,...st,
which shows that k(t2) = k(t,).
3.6. Local Time and a Generalized Ito Rule for Brownian Motion
211
6.15 Remark. For every z > 0 and y() e C[0, co) with y(0) = 0, we denote by if the class of functions k e C[0, co) which satisfy conditions (i) and (ii) of Lemma 6.14 and introduce the mappings (6.30)
T,(z; y) A max [0, max 1 -(z + y(s)) }1; o<s
(6.31)
R,(z; y) A z + y(t) + T(z; y);
0 < t < co
0 5 t < co.
In terms of these, the solution to the Skorohod equation is given by k(t) = 7;(z; y),
(6.32)
and T(z; y) is the minimal element of the proof of Lemma 6.14.
x(t) = R,(z;y), , as can be seen from the first part of
6.16 Proposition. Let z > 0 be a given number, and B = {B 1,; 0 < t < 00} a Brownian motion on some probability space (0,1,Q) with Q[B0 = 0] = 1. We suppose there exists a continuous process k = {k1,; 0 < t < co} such that, for Q-a.e. 0 e 0, we have (i) X JO)
z-
(ii) ko(0) = 0, t
+ k,(0) 0; 0 _5 t < co , k,(0) is nondecreasing, and
(iii) 1,`;) 10.00)(X 5(0)) dk,(0) = 0.
Then X = {Xt; 0 < t < 00} under Q has the same law as 1141 = {I H1,1; 0 s t < col under Pz. PROOF. The law of the pair (k, X) is uniquely determined, since by Lemma 6.14
k,(0) = T(z; - B.(0)), X,(8) = R,(z; - B.(0)); 0 < t < co, for Q-a.e. B e O. It suffices, therefore, on our given measurable space (S1,") equipped with the Brownian family I/V .9,; 0 < t < co}, {Px}),14, to exhibit a standard Brownian motion B = {B 0 < t < co} and a continuous nondecreasing process
k = {k
0 _5 t < 00} such that, for Pz-a.e. w E
0 < t < co,
W(w)1 = z - Bt(w) + kt(w); (6.33)
ko(o)) = 0,
t
k,(co)
is nondecreasing, and
I EiR,{0}(ws.)dks.)=0. But this has already been accomplished in Problem 6.13 (relations (6.25), (6.26) with a = 0), if we make the identifications B, Er- -
t
sgn(Ws)dVV
kt
= 2L,(0).
6.17 Theorem (P. Levy (1948)). The pairs of processes {(MR' -
Mr);
t < co} and {(1W,1,24(0)); 0 5 t < co} as in (6.27), (6.28) have the same laws under P°. 0
212
3. Stochastic Integration
PROOF. Because of uniqueness in the Skorohod equation, we have from (6.33) (6.34)
24(0, co) = MP(to),
I No41 = MN(0) - B,(0)); 0 < t < 00
for P ° -a.e. co e S/, upon observing that
MP(co) = max lits(w)= 'T,(0; -B(w))
(6.35)
o<s
(Remark 6.15). The assertion follows, since both W and B are Brownian motions starting at the origin under P°. We also notice the useful identity, valid for every fixed t e [0, co): 1
AV = lim - meas {0 < s < t; Mr - B5 < c}, a.s. P°.
(6.36)
.i.1) 2s
Li
6.18 Problem. Show that for every pair (a, z)eg12 we have Pz [co e SI, lim Lf(a,w)=
co] = 1.
1-.00
D. A Generalized ItO Rule for Convex Functions The functions fl (x) = (x - a)+, f2(x) = (x - a)- , and f3(x) = Ix - al in the Tanaka formulas (6.11)-(6.13) share an important property, namely convexity: (6.37)
f().x + (1 - A)z)
x < z, 0 < A < 1,
Af(x) + (1 - A)f(z);
which can be put in the equivalent form (6.38)
z -y f(x) + y -x f(z); z -x z -x
f (Y)
x
upon substituting y = Ax + (1 - 01.)z. Our success in representing f(W) explicitly as a semimartingale, for the particular choices f(x) = (x - a)± and
f(x) = Ix - al, makes us wonder whether it might be possible to obtain a generalized Ito formula for convex functions which are not necessarily twice differentiable. This possibility was explored by Meyer (1976) and Wang (1977). We derive the pertinent Ito formula in Theorem 6.22, after a brief digression on the fundamental properties of convex functions on R. 6.19 Problem. Every convex function f: R -> ER is continuous. For fixed x e ER,
the difference quotient (6.39)
A
Af (x; h) =
f(x + h)- f(x) h
;
h00
is a nondecreasing function of he R \ 101, and therefore the right- and leftderivatives
3.6. Local Time and a Generalized Ito Rule for Brownian Motion
(6.40)
213
1
lim - [f(x + h) - f(x)] h-o+ h
DIf(x)
exist and are finite for every x e R. Furthermore, (6.41)
D' f(x) < D f(y) < D'f(y); x < y,
and WA) (respectively, D-f()) is right- (respectively, left-) continuous and nondecreasing on R. Finally, there exist sequences fa1,°;'=, and {/3};,°=, of real numbers, such that (6.42)
f(x) = sup (anx + A); x e R. n>1
(Hint: Use (6.38) extensively.)
6.20 Problem. Let the function (p: R
O
be nondecreasing, and define
(p+(x) = lim ('(y), 1'(x) =
x
(u) du.
d0
y--uc±
(i) The functions cp.,. and (p_ are right- and left-continuous, respectively, with (6.43)
x e R.
(p_(x) < cp(x) < (p_,_(x);
(ii) The functions cp., have the same set of continuity points, and equality holds in (6.43) on this set; in particular, except for x in a countable set N, we have (14 (x) = p(x). (iii) The function cl) is convex, with D-(1)(x) = (p_(x) < cp(x) < (p_,.(x) = D+ 10(x);
(iv) If f: R (6.44)
x e R.
R is any other convex function for which D- f(x) < cp(x) < D+ f(x);
x e R,
then we have f(x) = f(0) + (1)(x); x e R.
R, there is a countable set 6.21 Problem. For any convex function f : O N l such that f is differentiable on R \ N, and (6.45)
f '(x) = D+ f(x) = D-f(x);
x e R\ N
Moreover (6.46)
f(x) - f(0) =
f f '(u) du = J x D-1 f(u) du;
x e R.
The preceding problems show that convex functions are "essentially" differ-
entiable, but Ito's rule requires the existence of a second derivative. For a convex function f, we use instead of its second derivative the second derivative measure p on (R, AR)) defined by (6.47)
p([a, b)) g D-f(b) - D- f(a);
-co < a < b < co.
214
3. Stochastic Integration
Of course, if f " exists, then it(dx) = f "(x) dx. Even without the existence of f ", we may compute Riemann-Stieltjes integrals by parts, to obtain the formula g(x)/1(dx) =
(6.48)
for every function g R
-f
g'(x)D-f(x)dx and has compact support.
R which is piecewise
R 6.22 Theorem (A Generalized Ito Rule for Convex Functions). Let f: be a convex function and it its second derivative measure introduced in (6.47).
Then, for every z e R, we have a.s. Pz: (6.49)
f(W) = f(z) +
D f (Ws) digs +
J
0 < t < co.
Li(x)it(dx)
J
o
PROOF. It suffices to establish (6.49) with t replaced by t n T_ n 1,, and by such a localization we may assume without loss of generality that D+f is of (6.18) to obtain uniformly bounded on R. We employ the mollifiers convex, infinitely differentiable approximations to f by convolution: (6.50)
f
fn(x)
CO
p(x - y)f(y)dy;
n
1.
CO
It is not hard to verify that fn(x) = f°200 p(z)f(x - (z /n)) dz and (6.51)
lim fn(x) = f(x),
lim f,;(x) = Erf(X)
n-,co
n-co
hold for every x e R. In particular, the nondecreasing functions D f and {f'} 11_, are uniformly bounded on compact subsets of R. If g: R C1 and has compact support, then because of (6.48), lim n- co
J-.
g(x)f"(x)dx = -lim n-.co
=-
R is of class
g'(x)f,,'(x)dx -co co
g'(x)D-f(x)dx =
J
g(x)it(dx).
A continuous function g with compact support can be uniformly approximated
by functions of class 0, so that (6.52)
lim n- co
1
g(x)f"(x)dx = 1
-co
g(x)p(dx).
-co
We can now apply the change-of-variable formula (Theorem 3.3) to f(Ws), and obtain, for fixed t e (0, co): `f
1f
in(Wt) - fn(z) = .1on1WOdWs + -2
0
fn" (Ws) ds,
a.s. Pz.
When n -, co, the left-hand side converges almost surely to f(W,) - f(z), and the stochastic integral converges in L2 to fic, a 1(Ws)d Ws because of (6.51) and
3.6. Local Time and a Generalized Ito Rule for Brownian Motion
215
the uniform boundedness of the functions involved. We also have from (6.7) and (6.52):
f,"(x)Li(x)dx = 2
fn"(Ws)ds = 2 lim
lira "OD
L,(x)p(dx),
a.s. P2
n-.co
0
because, for P2-a.e. co GS-2, the continuous function x w) has support on the compact set [min, Ws(co), Ws(co)]. This proves (6.49) for each fixed t, and because of continuity it is also seen to hold simultaneously for all t e [0, co), a.s. PZ. El
6.23 Corollary. If f: R
R is a linear combination of convex functions, then (6.49) holds again for every z e R; now, p defined by (6.47) is in general a signed measure with finite total variation on each bounded subinterval of R.
6.24 Problem. Let a, < a2 < {a, , .
< an be real numbers, and denote D =
P is continuous and f' and f" exist and are continuous on R \D, and the limits , an}
.
Suppose that f:111
f ' (a, ±) s lim f '(x), f "(a, ±) = lim f "(x) x-wk +
x--*ak ±
exist and are finite. Show that f is the difference of two convex functions and, for every z e R, (6.53)
f(W,) = f(z) +
f'(W)dW, + -1 ft f"(W)ds 2
0
+ E Li(a,)[f ' (a, +) -f ' (a, - )]]; 0 < t < co, a.s. P . k =1
6.25 Exercise. Obtain the Tanaka formulas (6.11)-(6.13) as corollaries of the generalized Ito rule (6.49).
E. The Engelbert-Schmidt Zero-One Law Our next application of local time concerns the study of the continuous, nondecreasing additive functional A1(w) = fg f(Wn(co))ds;
0
t < co,
0
where f: P
[0, co) is a given Borel-measurable function. We shall be interested in questions of finiteness and asymptotics, but first we need an auxiliary result.
6.26 Lemma. Let f: P
[0, cc) be Borel-measurable; fix x e P, and suppose there exists a random time T with
3. Stochastic Integration
216 T
Pq0 < T < co] = 1, P° [ I. f(x + Ws) ds < col>
O.
o
Then, for some e > 0, we have
Lf(x + y)dy < co.
(6.54)
PROOF. From (6.7) and Problem 6.13 (iii), we know there exists an event SP
with infe) = 1, such that for every co en*:
I T" f(x + Ws(w)) ds = 2 o
a°
--.
f(x + y)LT(a,)(y, co) dy
and LT())(0, co) > 0. By assumption, we may choose co e SI* such that fl-"f(x + Ws(w)) ds < cc as well. With this choice of w, we may appeal to the continuity of LT(0)( , w) to choose positive numbers c and c such that LT(.)(y, co)
c whenever IA 5 E. Therefore, T(w)
I.e
f(x + y) dy 5 j. 2c
-E
f(x + Ws(co))ds < co,
0
D
which yields (6.54).
6.27 Proposition (Engelbert-Schmidt (1981) Zero-One Law). Let f: l [0, co) be Borel-measurable. The following three assertions are equivalent:
(i) P°[IL f(His)ds < co; V 0 -_ t < co] > 0, (ii) P° [I:, f(Ws)ds < oo; V0 5 t < oo] = 1, (iii) f is locally integrable; i.e., for every compact set K c R, we have JKf(y)dy < cc.
PROOF. For the implication (i) (iii) we fix be R and consider the first passage time Tb. Because P° [Tb < co] = 1, (i) gives P°[ f o Tb f(Ws)ds < co; V 0 < t < co] > 0. But then ,4- ori,coo
,-4- Tb(eol
f(47(0))ds > JO
f(Ws(w))ds = j. f(b + Bs(w))ds, fTb(w)
0
where Bs(w) A 14/,+71(.)(co) - b; 0 < s < oo is a new Brownian motion under P°. It follows that for each t > 0, P°[ f 10 f(b + Bs) ds < co] > 0, and Lemma 6.26 guarantees the existence of an open neighborhood U(b) of b such that Su(of (Y1dY < oo. If K R is compact, the family {U (0}4. lc, being an open covering of K, has a finite subcovering. It follows that Ix f 61 dy < co. For the implication (iii) (ii) we have again from (6.7), for P°-a.e. w e ft Jo
f(Ws(w))ds = 2 I
-.
5[
f(y)Lt(y, co) dy = 2
I
mt(co)
f(y)Lt(y, w) dy
m,(.) Mt(w)
max
?NO)).
A 1 t(0))
24(y, co)]
-
.1 m,(0))
f(y)dy; 0 5 t < co,
3.7. Local Time for Continuous Semimartingales
217
where ?Ow) = min° ss Ws(co), Mt(co) = max05,,, Ws(w). The last integral is finite by assumption, because the set K = [mi(co), A li(co)] is compact. 6.28 Corollary. For 0 < a < co, we have the following dichotomy: p0 [f E
wdSla
V 0 < t < 00 =
< 00;
1;
if 0 < a < 11
0;
if a _._
f
1
6.29 Problem. The conditions of Proposition 6.27 are also equivalent to the following assertions: (iv) P° [po f (Ws) ds < co] = 1, for some 0 < t < co; (v) Px[fo f(Ws) ds < co; V 0 < t < co] = 1, for every x e IR;
(vi) for every x e R, there exists a Brownian motion IB g,; 0 < t < col and a random time S on a suitable probability space (0, , Q), such that Q [Bo = 0, 0 < S < co] = 1 and
f Q[
s
f(x + 13s)ds <
ool> O.
o
(Hint: It suffices to justify the implications (ii) = (iv) the first and last of which are obvious.)
(vi)
(iii)
(v)
(vi),
630 Problem. Suppose that the Borel-measurable function f: R --9 [0, oo) satisfies: measty e 01; f(y) > 01 > 0. Show that (6.55)
Px[u) e SI;
f
.3
f (Ws(w))ds =
xi=
1
o
holds for every x e Fl. Assume further that f has compact support, and consider the sequence of continuous processes 1
X:n)
A.\/n
f(Ws)ds;
0
t < co, n > 1.
o
Establish then, under P°, the convergence (6.56)
X (^) --. X
in the sense of Definition 2.4.4, where X, -4 2 V f ll 1 L,(0) and Vfil, A ff. f(y) dy > 0.
3.7. Local Time for Continuous Semimartingalest The concept of local time and its application to obtain a generalized Ito rule can be extended from the case of Brownian motion in the previous section to that of continuous semimartingales. The significant differences are that f This section may be omitted on first reading; its results will be used only in Section 5.5.
3. Stochastic Integration
218
time-integrals such as in formula (6.7) now become integrals with respect
to quadratic variation, and that the local time is not necessarily jointly continuous in the time and space variables. We shall use the generalized Ito rule developed in this section as a very important tool in the treatment of existence and uniqueness questions for one-dimensional stochastic differential equations, presented in Section 5.5. Let
X, = X0 + M, + Vt; 0 < t < oc
(7.1)
be a continuous semimartingale, where M = IM Ft; 0 < t < col is in dr', V = { V g;; 0 < t < oo} is the difference of continuous, nondecreasing, adapted processes with Vo = 0 a.s., and {,,} satisfies the usual conditions. The results of this section are contained in the following theorem and are inspired by a more general treatment in Meyer (1976); they say in particular that convex functions of continuous semimartingales are themselves continuous semimartingales, and they provide the requisite decomposition. 7.1 Theorem. Let X be a continuous semimartingale of the form (7.1) on some probability space (Q, .0-i-, , P). There exists then a semimartingale local time for
X, i.e., a nonnegative random field A = lAt(a,w);(t,a)e [0, co) x R, wen} such that the following hold: (i) The mapping (t, a, w) F- At(a,w) is measurable and, for each fixed (t, a), the random variable Ada) is .97,-measurable.
(ii) For every fixed a e R, the mapping ti--11,(a,w) is continuous and nondecreasing with Ao(a, w) = 0, and (7.2)
fom 1 R\ {}(X,(co)) d A,(a, co) = 0,
for P-a.e. wen.
[0, op), the identity
(iii) For every Borel-measurable k: IR t
(7.3)
1 k(Xs(co))d<M> (co) = 2 1
k(a)A,(a, co) da;
0 < t < co
o
holds for P-a.e. we a (iv) For P-a.e. we Q, the limits lim At(b, co) = At(a, co)
and
At(a -, w) '' lim At(b, co)
r-,1 b.i.a
S -./
bt a
exist for all (t, a) e [0, cc) x R. We express this property by saying that A is a.s. jointly continuous in t and RCLL in a. (v) For every convex function f:R R, we have the generalized change of variable formula t
(7.4)
f(X,) =
f(X0) +
1 Df(Xs)dM, + 1 Df(Xs)dV, o
o
CO
+I
At(a)tt(da);
0 < t < oo, a.s. P,
3.7. Local Time for Contiauous Semimartingales
219
where D- f is the left -hand derivative in (6.40) and II is the second derivative measure (6.47).
7.2 Corollary. If f: R -- R is a linear combination of convex functions, (7.4) still holds. Now p defined by (6.47) is a signed measure, finite on each bounded subinterval of R.
7.3 Problem. Let X be a continuous semimartingale with decomposition (7.1) and let f: R --- FR be a function whose derivative is absolutely continuous. Then f" exists Lebesgue-almost everywhere, and we have the ItO formula: t
fiXt) = MO
:
+ f f(Xs)dM, + f r(X0dV, o
1
+2
o
0 s t<
f"(Xs)d<M>s;
co, a.s. P.
0
7.4 Remark. In the setting of Theorem 6.17, we observe that the reflected Brownian motion X, A- 1W,1 = -B, + 2L,(0);
0 s t < co
is a semimartingale with X0 E-..- 0, M E-..-. -B, V :--.4 2L(0) under P°. The general-
ized IVO rule (7.4) applied to this semimartingale with f(x) = Ix I gives in conjunction with (6.26): X, = -B, - 2L,(0) + 2A,(0), and therefore A,(0) = 2L,(0); 0 < t < oo, a.s. P°. In other words, the semimartingale local time of reflected Brownian motion at the origin is twice the Brownian local time at the origin, as one would expect intuitively.
By way of preparation for the proof of Theorem 7.1, we provide a construction similar to that used in the proof of Theorem 6.22. Let f : 01 -. 011 be convex. Thanks to the usual localization argument, we may assume that D-f
is uniformly bounded on R and <M>, and V, are uniformly bounded in 0 s t < oo and welt. Applying Ito's rule to the smooth function f, of (6.50), we obtain (7.5)
f(X,) = f,,(X0) + f 'f:(X.,)dMs + o
ft f,;(Xs)dV, + On);
0 S t < co,
o
where
On) = -1 ft f,,(Xs)d<M>s; 2
0 5 t < co
0
is a continuous, nondecreasing (by the convexity of f,,), and {,,}-adapted process. As in Theorem 6.22, we have as n -+ oo: t
t
f(X,)- f(X,) and f f;(Xs)d1/, --4 f o
o
D-f(Xs)dVs,
a.s.
3. Stochastic Integration
220
ft D-f(Xs)dM
.1of;(Xs)d111,
in probability
o
for every fixed t. It follows that the remaining term CP) in (7.5) must also converge in probability to a limit Ct(f), and t
t
(7.6)
f(Xt)= f(X0)+ 1 D-f(Xs)dills+ 1 Df(Xs)dV, + Ct(f); o
0
t < oo.
o
Now f(Xt) is continuous in t and both integrals have continuous modifications, so we may and do choose a continuous modification of Ct(f). Each C") is nondecreasing and adapted to {.Ft}; the limit C(f) inherits both these properties. With a e R fixed, we apply (7.6) to the functions fl(x) &` (x - a)+, f2(x) = (x - a)-, and f3(x) = Ix - al to obtain the Tanaka-Meyer formulas t
(7.7)
(Xt - a)± = (X0 - a)± + 1
1.0,,c0)(Xs)dills +
1
(7.8)
(Xt - a)- = (X0 - a)
t
-1
lt_com(Xs)dMs-
1
o
lt_com(Xs)dVs+ Ct-(a)
o
t
(7.9)
1.0,,c0)(Xs)dV, + CC (a)
o
o
t
sgn(X, - a)dM, +
I Xt - al = 1 X0 - al + o
sgn(X, - a)dV, I
o
+ 2At(a),
with the conventions (6.14), Ct±(a)
Ct(ii), CC (a) .'` Ct(12), and 2At(a) A Ct(f3). The processes Ct± (a), At(a) are adapted, continuous, and nondecreasing
in t, and the random field At(a, w) will be our candidate for the local time of X. Now (7.7) and (7.8) yield CC (a) = CC (a) (upon subtraction), as well as (7.10)
At(a) = Ct± (a);
0
t < cx), a e R
(upon addition and comparison with (7.9)). Although the process {At(a); 0 < t < ool is continuous for every fixed a, we do not yet have any information about the regularity of At(a) in the pair (t, a). We approach this issue by studying the regularity of the other terms appearing in (7.7).
7.5. Lemma. Let X be a continuous semimartingale with decomposition (7.1). Define It(a) A 1 t 10,,,o(Xs) dMs; 0< t< co, a e R. o
The random field I = {L(a), ,t; 0 < t < co, a e RI has a continuous modification.
In other words, there exists a random field r= {rt(a),,t; 0 < t < 00, a e RI such that:
3.7. Local Time for Continuous Semimartingales
221
(i) For P-a.e. well, the mapping (t, a)I-* 1,(a, co) is continuous on [0, oo) x R. (ii) For every t a [0, co) and a a 11-8, we have 1,(a) = 11(a), a.s. P.
PROOF. By the usual localization argument, we may assume without loss of generality that there is a constant K for which sup
(7.11)
I Xr1
K, <MX, 5 K, Pc, < K ,
0 < ( < co
where R., is the total variation of V on [0, co). According to Remark 4.9, we may choose a Brownian motion B for which we have the equations Of>,
,
(7.12)
I,(a) -4
j.
1.0,,,o(X)(//11 =
1(.,o(Y)dB; 0 < t < co,
o
(7.13)
Hs(a) A
f
s
T(s)
1(s,))(11)dB. =
0 < s < co,
14,, 00(X) dM;
o
o
where T() is given by (4.15), lc -A XT(S) for 0 s < <MX, and I', is chosen so that 10,0,1( Ys) = 0 for s <M> b in some neighborhood of a (cf. (4.20)', Remark 4.9). We shall prove the existence of a continuous modification
of H by using the extension of the Kolmogorov-entsov theorem (Problem 2.2.9). According to the latter, it suffices to show (7.14)
Ellk(a) - Hs1(b)1' 5 C[(s2 - s1)2 + (b - a)271+s;
0<s, <s2< co, a,beR for suitable positive constants a, /3, and C. Note that (7.15)
E I 11,2(a) -Hs 1(b)18 < 2' E I 11,2(a) - 11,2(b)11 + 2' E 111, 2(b) -It i(b)18,
and, according to the martingale moment inequality (3.37), we may bound the latter expectation by E111,2(b) - 11,,(b)12
C.E[M(b)>s, - <1-1(b)>sj212 5 Cs(s2 - s1)a'2.
When a > 4, this bound is of the type required by (7.14). Thus, it remains only to deal with the first expectation on the right-hand side of (7.15), i.e., to show (7.16)
Ellis(a) - 115(b)12 5 C(b - a)2("); 0 < s < co, -co < a < b < co,
where a > 4, /3 > 0, and C is a positive constant. We fix a < b and introduce the convex function f (x) =
f
X
o
4,,,i(z)dz dy, o
for which I f 'I is bounded by b -a and 1)--f' = 10,,bi. In particular, passage to the limit in (7.5) yields
i f'(X0)d1148 + f f'(X0)dV0 +,
f(X,) = f(X0) +
0
,
o
2
' J 0
10, bi(X8)d<M>o. '
3. Stochastic Integration
222
Assumption (7.11) and Problem 1.5.24 show that X, and all the preceding integrals have limits as t oo, so we may replace t by T(s) (which may be infinite) to obtain, for every k > 1, the bound (7.17) rT(s)
T(s)
k
G 6k I f(XT(s))- f (x 0)1k + 6k j.
k
f'(X9)dMo
Jo rT(s)
k
f'(X,i)dii
+ 6k
Jo
We bound the terms on the right-hand side of (7.17). The mean-value theorem implies I f (x T(o)) -f (x 0)1k < (b - a)k I XT(S) -X Olk
2k Kk (b
- a)k,
and it is also clear that T(s)
k
f'(X9)d1/9 < Kk(b - a)k.
io
Applying the martingale moment inequality (3.37) to the stochastic integral in (7.17), we obtain the bound To)
E
J.0
k
k
f ' (X v)ditiv < lim E fo(s)At f'(XJdMt, G Ck E
[1
oo
I f ' (x v)I2 d<M>v o
< CkKki2(b - a)k. We conclude from these considerations that there exists a constant C, depending
only on k and on the bound K in (7.11), such that T(s)
k
1(,,,)(Xt,)d<M>t,
E
fo
C(b - a)k;
0 < s < co,
-co < a < b < co.
Now (3.37) can be invoked again to establish (7.16) with a = 2k and 2(1 + /3) = k, provided k > 2. From (7.12), (7.13) we see that It(a)= Hoo,(a), and since <M> is continuous, the existence of a continuous modification for H implies the same for I. The Lebesgue-Stieltjes integral (7.18)
J,(a) --4 1[ 1(,00)(Xs)dVs o
appearing in (7.7) can fail to be jointly continuous in (t, a); see Exercise 7.9. However, it is jointly continuous in t and RCLL in a; the proof is left as a problem for the reader.
3.7. Local Time for Continuous Semimartingales
223
7.6 Problem. Let X be a continuous semimartingale with decomposition (7.1). For P-almost every co e 1, we have for all (t, a) e [0, co) x (7.19)
lim .1,(b, co) = b
=
lim
(7.20)
tt
f
lia,c0(X,(0)dVs(co).
0
bfa
PROOF OF THEOREM 7.1. Using Lemma 7.5 and Problem 7.6, we choose a modification of Cf+ (a) in (7.7) which is jointly continuous in t and RCLL in a. We take Ai(a) and C, ±(a) to satisfy (7.10) for every t, a, and co. In particular, Ai(a, co) is jointly a( [0, co)) 0 MR) F-measurable and satisfies (iv), and the other measurability claims of (i) and (ii) hold. In particular, it follows from (7.19), (7.20) that
A,(a) - A,(a -) = fo 1{}(Xs)dVs.
For the proof of (7.2), consider any two rational numbers 0 < u < v < co and the event Ha A {co ED; Xs(co) < a,
V se [u, v] }.
From relation (7.7) we have on H: Av(a, co) = A (a, co), except for co in a null set N,,,,. Let N M Na and fix co e 12\N. The set S(a, co) A ,_, 0 u,v e Q {0 < t < co; X,(co) < a} is open and, as such, is the countable union of disjoint open intervals. Let (a, /3) be such an interval. If a < u < v < /1, where u and v are rational, then A(a, co) = A (a, co). It follows that 1,.13) d A,(a, co) = 0, and thus IW 1,_,,o(X,(co))d A,(a, co) = 0.
(7.21)
0
A similar argument based on (7.8) shows that (7.22)
f.
1,,)(X,(co))dA,(a, co) = 0,
a.s.
which establishes (7.2).
For the proof of (7.4), we may assume, by the usual localization argument, that there exists a constant K > 0 such that (7.11) holds. Consequently, we
may also assume without loss of generality that D-f is constant outside (- K, K), so the second derivative measure p has support on [ - K, K). Let x e [ - K, K] be fixed and introduce the function
a < -K - 1, (x + K)(a + K + 1); -K - 1 < a < -K, 0;
g(a) =
x - a;
-K < a < x,
0;
x < a.
3. Stochastic Integration
224
According to (6.48), (6.46), we have for -K < x < K, (x - a) +µ(da) =
(7.23)
J
-f
g'(a)D- f (a) da -K
= -(x + K)
= -(x
D- f (a) da + -K-1
J
X
D- (a) da
-K
K) D-f(- K) + f(x) - f(- K),
and from the definition of it: (7.24)
K, x)) = D- f(x) - D-f(- K).
1,,,,,,o(x)y(da) =
cc'
We may now integrate with respect to it in the Tanaka-Meyer formula (7.7) and use (7.23) to obtain (7.25)
f(X,) = (X, - X0) D- f( - K) + f(X0) +
J
J t
-co co
+
i(,00(Xs)dVsit(da) + J _co
f
10,,,,)(X0d114,p(da)
0
co
Ai(a)it(da).
-co
0
Fubini's theorem and (7.24) allow us to write (7.26)
fc° f t -a)
10,,00)(Xs)dV, Oda) = f i 1)- f (X ,) dils -V, 1)- f ( - 10.
0
0
A similar interchange of the order of integration in the integral ('t
1(..)(X,) dM, y(da) =
(7.27) 00
D- f(Xs) d114, -M, D- f(- K)
0
is justified by Problem 7.7 following this proof. Substitution of (7.26), (7.27) into (7.25) results in (7.4). Finally, let us consider (7.4) in the special case f(x) = x2. Then µ(da) = 2 da, and comparison of (7.4) with the result from the usual Ito rule reveals that 00
<M>, = 2
J
Ai(a)da; 0 < t < co, a.s.
Thus, for any measurable function h(s, co): [0, co) x CO
OD
OD
h(s, co) d <M>,(co) = 2 o
holds for P-a.e. wee. Now if k:
[0, co,
h(s, co) d As(a, co) da -co
o
-* [0, co) is measurable, we may take
h(s,(o) = 110,11(s)- k(X Jo))) and obtain for P-a.e. to:
3.7. Local Time for Cc ntinuous Semimartingales
k(X (w))d<M>s(co)= 2
J o
s
_.
=2J
225
k(Xs(w))dAs(a,w)da J.` k(a)Ai(a, co) da;
0 5 t< co,
thanks to (7.2). This completes the proof. 7.7 Problem. Let X be a continuous semimartingale with decomposition (7.1), it be a 6- finite measure on (DI, .4(01)), and h: R [0, oo) be a continuous function with compact support. Then
.1°3 h(a)(1 10,,,)(Xs)dMs)p(da)=
(.1
h(a)10,.)(Xs)p(da))01,.
7.8 Remark. The proof of Theorem 7.1 shows that the semimartingale local time Azi" (a) for a continuous local martingale M is jointly continuous in (t, a), because the possibly discontinuous term .1,(a) of (7.18) is not present. In particular, (7.9) becomes then a.s. P: (7.28)
IM, - al =1M0 -
+
sgn(Ms - a)dMs + 2Ana); 0 < t < co.
Comparison of (7.28) with the Tanaka formula (6.13) shows that the semimartingale local time A"(a) for Brownian motion W coincides with the local time L(a) of the previous section. If M e,C, then for any stopping time t, (7.29)
BIM, Arl = 2E[e, r(0)]; 0 < t < co.
7.9 Exercise. Show by example that Jt(a) defined by (7.18) can fail to be continuous in a.
7.10 Exercise. Let X be a continuous semimartingale with decomposition (7.1). Show that for every a e R,
1{.}(Xs)d<M>s= 0, a.s. P. 0
7.11 Exercise. Show that the semimartingale local time of a continuous process of bounded variation is identically zero.
7.12 Exercise (LeGall (1983)). Let X be a continuous semimartingale with decomposition (7.1), and suppose that there exists a Borel-measurable function k: (0, co) (0, co) such that .1(0,0 (du /k(u)) = po,V s > 0, but for every t e (0, co) we have
226
3. Stochastic Integration
it d<M> jo k(X5) i{"} < c°'
a.s. P.
Then the local time A(0) of X at the origin is identically zero, almost surely.
7.13 Exercise. Consider a continuous local martingale M and denote St 4 maxo <s <, Ms, L, A 2A,(0). Suppose now that f(t,x,y): R3 -0 J is a function of class C2 (R3) which satisfies
of 102f + at 2 ax2
=0
in R3 and
of ay
Of
-(t, 0, y) + - (t, 0, y) = 0 ax
for every (t, y) e 1112. Show then that the processes f(< MX, 141,4) and .R<M>t,
S, - Mt, St) are local martingales. Deduce also the following:
(i) The process (S, - M,)2 - <M>, is a local martingale. (ii) For every real-valued function g of class C1(R), the processes
g(L) - 141 04)
and
g(S,) - (S, - M,) g'(S,)
are local martingales.
7.14 Exercise. For a nonnegative, continuous semimartingale X of the form (7.1) with X0 -..= 0, the following conditions are equivalent: (i) 1/ is flat off {t > 0; X, = 0}. (ii) The process P01{,s00} dXs; 0 < t < co belongs to .ilic I". (iii) There exists N edr.1" such that X, = maxo <,.,, Ns - Ali.
3.8. Solutions to Selected Problems 2.5. (a) It is easily verified that u - (1/2") < 9(u) < u. Consequently, XI") is ,-tmeasurable, and since gyp" takes only discrete values, V"51 is simple. (b) The procedure (2.7) results in measurable (but perhaps not adapted) processes {gopoopm= i 1, such that
f
T
lim E .0151im) - XtI2 dt = 0. m-00
Thus, for given e > 0, we can find m > 1 so that with X' A g(m) we have E Jo IX: - x(12 dt < O. The Minkowski inequality leads then to
3.8. Solutions to Selected Problems
(E fr
227
- Xt_hl2 dE)"2
0 1/2
(E
IX, - XN2 dE)
+ E
IX _h -
dt)
(E
f
- X:_hI2 dt
13
< 2E +
(E
- X:_h12 CIEY I)
We can now let h j 0 and conclude, from the continuity of X' and the bounded convergence theorem, that T
liM E /40
j. IX, -
dt < 4E2.
0
(c) Let i be any nonnegative integer. As s ranges over [1/2", (1 + 1)/2"), tp(t - s) + s ranges over [t - (1/2"), t). Therefore, T
T
IX, -
- X,I2 ds dt = 2"E
E 0
JO
0
= 2"
dhdt
0
s2- [E f T IX, - Xr_hI2 dtidh 0
13
T
max E
0112-.
IX, - X,_12 dt, 0
which converges to zero as n -0 co because of (b). (d) From (c) there exists a sequence {n, }ck°=, of integers, increasing to infinity
as k
co, such that for meas x meas x P-a.e. triple (s,t,co) in [0,1] x
[0, T] x SI, where meas means "Lebesgue measure," we have
lim IXPk')(co) - X,0)I2 = 0.
(8.1)
k-.co
Therefore, we can select se [0,1] such that for meas x P-a.e. pair (t, co) in [0, T] x fl, we have (8.1). Setting X('`) 9 X("ks), we obtain (2.8) from the bounded convergence theorem. 2.18. By assumption, we have
E = E
d<M>, < co.
Uniform integrability and the existence of a last element for IM(X) follow from Problem 1.5.24, as does uniform integrability of (1m(X))2; similarly for /N( Y). Applying Proposition 2.14 with X., Y. replaced by X.1{.). 71, Y.1{, T}, respectively, we obtain X. Ysd <M, N>,
(f
T+I
d<M>.
f
T+t
1/2
Y.2
ds)
,
3. Stochastic Integration
228 whence
XsYsd<M,N>s <
I
(f
d<M>s XS
T
f
Ys2
ds)112
a.s. P. As T -0 co, the right-hand side of this inequality converges to zero; therefore, Y)>, =
f
Xs Ysd<M, N>s
converges as t -0 co and is bounded by the integrable random variable
(f: X d<M>.
0
a.s. P. The dominated convergence theorem gives then
lim ER' (X)INY)] = lim E, = Es, fro.
=E
f
X sYsd <M, N>s.
We also have
E[e,f(X)C,(Y)] = E[(1:,4, (X) - r (X))(IVY) -Ii (Y))]
+ E[IP4(X)(IVY) -Ii (Y))] + E[I7(Y)(e,o4(X) - ir(x))]
+ E[Ir(X)17(Y)]. We have just shown that the fourth term on the right-hand side converges to (X), IN (Y)>. as t -0 co. The other three terms converge to zero because of Holder's inequality and the uniform integrability of Om( X))2 and ON Y))2. E
2.27. With X E 9"(M), we construct the sequence of bounded stopping times {Tn}:=1 in (2.32). In the notation of (2.33), each )0") is in 2*(M(")) and therefore can be approximated by a sequence of simple processes PC"'''));,°_, g z in the sense T
lim E k-co
j.
- )0")I2 d<M>, = 0, V T < co
o
(Proposition 2.8). Let us fix a positive number T < co and consider n > T; we have P 01, I )0") - X,I2 d<M>, > 0] = > 0] n1 P[T, < T] = P[S < T or 11-,X,2 d<M>, > N], and the last quantity converges to zero as n -0 oo, by assumption. Now, given any E > 0, we have
PH. T In") - X,I2 d<M>, > Ei
P[f
T
In") - n)I2 d<M>, > -8] 2
o P[fT
IX} ")- X,I2d <M >,
+
>0
< -2 E j. T I Xl"'k) - n')I2 d <M > t + P[T < T] E
by the ebygev inequality.
0
3.8. Solutions to Selected Problems
n
229
For any given ö > 0 we can select 115> T so that P[T,, < T] < 312 for every n0, and we can find an integer kr,, 1 so that E J.T XP4.k..4) -)0 )12 d I
ei5
, < 4
o
It follows that for e = S = j; j > 1, there exist integers n, and k" such that, with Yul A X('0"), we have
- X,I2 d <Mx >
P[f
j
Then according o Proposition 2.26, both sequences of random variables T
YtLI) - X,12
sup 11,(YllI) - /,(X)I
d<M>,, 0
converge to zero in probability, as j
oo, and there exists a subsequence for which the convergence takes place almost surely. Having done this construction for T fixed, we use a diagonalization argument as in the first paragraph of the proof of Lemma 2.4 to obtain a sequence which works for all T
In the case that M is Brownian motion, we use Proposition 2.6 rather than Proposition 2.8 in this construction. 2.29. We have 1 m-1
s,(11) = i0 E
"' - 14c2) +
-
i=0
"i -Hit )2
1) "
1
= 2147,2 + E - -
2
)2. Ei=0(wt"1 - W,)2.
Recalling the discussion preceding Lemma 1.5.9, we may write 2
m-1
E[ i=0 E ovi, -
wo2 -
=E E [(141,+, - 111,, )2 -
ti)]
(t1 +1
1=0
= i-0 E EUK.i -14/02 -of+, -
12
032
.-1
ER,, - W04 - Ei=0(ti+, -
t1)2
m-1
02 i=o
5_ c2t1lnii,
where we have used Problem 2.2.10. This proves (2.37). To see that E = 0 corresponds to the Ito integral, consider the (piecewise constant) process m-1
i=o
in ..r,*(W), for which
230
3. Stochastic Integration Cl
E
..-1
ti+,
Inl - Ws12 ds = E
Eiwi, -we dt
1=0
o
11
ti
m-1
=E
1=0
1
'n-1
2
1=0
- ti) dt = - E 41+1- 02-40
(t
ti
0. By definition, the HO integral pow; dWs is the L2-limit of Se(II) =
as Mill Ito Xr dWk.
3.7. The proof is much like that of Theorem 3.3. The Taylor expansion in Step 2 of that proof is replaced by
f(tk, Xtd -
Xik_,)
(tk_l, x,_,)]
X,k)] + [ffik-1, Xid
= [ffik, Xik)
a
- 4_1) + E
i=1 OX i
d
1
d
32
+ E j=i
A
x,ox;
fitk_i , love - xe,)(xe - xed,
where tk_i < srk < tk and tik is as before.
3.14. Let Y(i) = af(X,)/axi, so according to Ito's rule (Theorem 3.6), V() =
f
+ J=1 1
°
32
axiax;
fixs)dmo+
33
°
s)d
2 k=1 j=1 aXiaXjaXk
32, focs)dBe OXiOXi
j=1
0) M(k)>s.
It follows from (3.9) that 1
vo.
= .10 vodxv + -E
2 J.1
32
t
aXiax;
f(Xs) d <Ma), Mo)>s,
and now (3.10) reduces to the HO rule applied to f(X i).
3.15. Let X and Y have the decomposition (3.7). The sum in question is m-1
-
E
1=o
The first term is ro
1 'n-1 X11)
dMs +
+
E (}
2 1=0
- i,)(x+, - X ).
dBs, where
rn -1 1 =0
and the continuity of Y implies the convergence in probability of Po (Ys' Y)2 d<M> s and Po [Ysn - Ys] dBs to zero. It follows from Proposition 2.26 that rn-1
E
i=0
The other term is
- x)
rsdxs. o
3.8. Solutions to Selected Problems
E
L i=c1
(M1.
- MOlArt,,
231 m-i
-
+-E
"
2 i=o
m-i
-
1 m-i
+ -2 Xo (N,, -
- Be.) +
(B4.
'
)(C.,
"
- C,)
-
-
which converges in probability to l<M, N>, because of Problem 1.5.14 and the bounded variation of B and C on [0, t].
3.29. For x; > 0, i = 1,
xr +
d, we have
d(x +
+
dm+1(xr +
+ xd)m
+ x,r).
Therefore 104,112m
(8.2)
[E (VT]
dm E 00)12,. i=i
and
E 014(i)>7 < d(E
(8.3)
i=1
<00>T)
= dAT.
Taking maxima in (8.2), expectations in the resulting inequality and in (8.3), and applying the right-hand side of (3.37) to each M('), we obtain EOM 114;12 m < dm E Eumolti2m
dm E KmEum(oyn
i=i
Kmdm+1E(A7)
1=1
A similar proof can be given for the lower bound on EU M 4.5.
(i) The nondecreasing character of T is obvious. Thus, for right-continuity, we need only show that lim045 T(0) < T(s), for 0 s < S. Set t = T(s). The definition of T(s) implies that for each E > 0, we have A(t + c) > s, and for s < 0 < A(t + E), we have T(0) < t + E. Therefore, lim,4s T(9) < t. (ii) The identity is trivial for s > S; if s < S, set t = T(s) and choose E > 0. We
have A(t + e) > s, and letting e l 0, we see from the continuity of A that A(T(s)) > s. If t = T(s) = 0, we are done. If t > 0, then for 0 < e < t, the definition of T(s) implies A(t - e) < s. Letting E l 0, we obtain A(T(s)) < s. (iii) This follows immediately from the definition of T(). (iv) By (iii), T(A(t)) = t if and only if A(T) = A(t), in which case 49(t) = q9(t). Note
that if S < cc and A(t) = A(cc) for some t < co, then T(A(t)) = co and (i9 is constant and equal to 41(t) on [t, co); hence, (p(co) = lira, (Mu) exists and equals (p(t).
(v) This is a direct consequence of the definition of T and the continuity of A. (vi) For a 5 t1 < t2 5 b, let G(t) = 11,,12,(t). According to (v), t1 5 T(s) < t2 if and only if A(t,) s < A(t2), so A(b)
G(t)dA(t) = A(t2) - A(t1) =
G(T(s))ds. A(a)
The linearity of the integral and the monotone convergence theorem imply that the collection of sets C e .41([a,b]) for which (8.4)
1,(t) dA(t)
f
A(b)
A(o)
1,(T(s))ds
232
3. Stochastic Integration forms a Dynkin system. Since it contains all intervals of the form [t1, t2) c [a, b], and these are closed under finite intersection and generate 4([a, b]), we have (8.4) for every C e M([a, b]) (Dynkin System Theorem 2.1.3). The proof of (vi) is now straightforward.
4.7. Again as before, every <M>, (resp., T(s)) is a stopping time of {s} (resp., {F,}), and the same is true of S <M>, (Lemma 1.2.11). The local martingale /II has quadratic variation CAI>, < <M>,(s2)= S n s2 < s2 < co (Problem 4.5 (ii)),
so again both a, a2 - are uniformly integrable martingales, and by optional sampling: E [AlT(s2)
MT(Si)1 T(si )] = 0,
EP-4($2) - ICIT(.0)21,77(.)] = EE<M>T(s2) - <M>T001,71501; a.s. P.
It follows that MoT A {MTN, Ws; 0 < s < co} is a martingale with <M 0 T >s =
<M>T(s), and by Problem 4.5 (iv), M 0 T has continuous paths. Now if { Ws, Ws; 0 < s < co} is an independent Brownian motion, the process
BsAWs-W s A S +M- T(s),4- s, 0_s s = s - (s A S) + <M>T(s) = s, i.e., a Brownian motion. For this process, (4.17) is established by using Problem 4.5 (iv). 4.11. Let (p be a deterministic, strictly increasing function mapping [0, co) onto [0,1), and define M E ir 1°' by M, --4
f
ow Xs dWs,
so <M >1 =
fow XS ds;
o
0 < t < co,
o
and lim, <M>, = co, a.s. on E. According to Problem 4.7, there is a Brownian motion B such that Jt
X s dWs = Bp(,),
where p(t) A <M>,-i(i).
We have limgti p(t) = oo a.s. on E, and so, by the law of the iterated logarithm for Brownian motion,
lm B,1) = -lim Bp(1) = +co, a.s. on E.
it
it 1
4.12. (Adapted from Watanabe (1984).) From Problem 4.7 we have the representation (4.17) for a suitable Brownian motion B. Taking n > 3 max(' xi, pT) and denoting R = inf ft _>.: 0;141 n/31, we have the inclusions
{max I X,' (:)1,5T
n}
g { max 141 051.sT
.1= {<M>T. 3
R,,}
{PT
Rn},
which lead, via (2.6.1), (2.6.2), and (2.9.20), to m ax P [CIT
I Xr1
n]
P[R 5 pT]
2R J
The conclusion follows.
2P[T13 '- PT] = 4P BpT dz <
12 jp n
2R
exp
3
18pT
3.8. Solutions to Selected Problems
233
6.12. Let h have support in [0, b], and consider the sequence of partitions 2"};
D"=
n
>1
of this interval. Choosing a modification of Po 1()( Ws) dWs which is continuous in a (cf. (6.21)), we see that the Lebesgue (and Riemann) integral on the left-hand side of (6.24) is approximated by the sum
2"-I b
E -h(b17))( f i(bno")dw) =
k=0 2"
F(Ws)dWs,
where the uniformly bounded sequence of functions
2"-I b
E -hoe)lor,,,,(x);
F(x)
k=0 2n
n
1
converges uniformly, as n -* co, to the Lebesgue (and Riemann) integral F(x)
Therefore, the sequence of stochastic integrals {it F(Ws)dWs}`°_i converges in L2 to the stochastic integral Po F(Ws)dWs, which is the right-hand side of (6.24). 6.13.
(i) Under any P, B(a) is a continuous, square-integrable martingale with quadratic variation process , =
f [sgn(Ws - a)]2 ds = t;
0 5 t < co, a.s. P .
o
According to Theorem 3.16, B(a) is a Brownian motion. (ii) For w in the set Q* of Definition 6.3, we have (6.2) (Remark 6.5), and from this we see immediately that L o(a, w) = 0 and L,(a, a)) is nondecreasing in t. For each z E R, there is a set n e.57" with p(n) = 1 such that 2°,(a) is closed for all CO E n. For wenn Ce, the complement of .2°,0(a) is the countable
union of open intervals U. NI I. To prove (6.26), it suffices to show that Si. dL,(co) = 0 for each a e NI. Fix an index a and let 1 = (u, v). Since W(w) -a
has no zero in (u, v), we know that I W(a)) - al is bounded away from the origin on [u + (1/n), v - (1/n)], where n > 2/(v - u). Thus, for all sufficiently small e > 0, 1
meas { 0< s < u + -; I Ws - al 5 e} = meas {0 < s
v
--1; I Ws - aI 5 E},
co). It follows that jr,,,(I,,,),-(1/0] dL,(a,w) = whence 4,(1/)(a, co) = 0, and letting n oo we obtain the desired result. = -B,(0) + 24(0); 0 .5 t < oo, a.s. P°. (iii) Set z = a = 0 in (6.25) to obtain I The left-hand side of this relation is nonnegative; B,(0) changes sign infinitely often in any interval [0, E], e > 0 (Problem 2.7.18). It follows that L,(0) cannot remain zero in any such interval.
(iv) It suffices to show that for any two rational numbers 0 5 q < r < co, if < W,(co) = a for some t e (q, r) then Lq(a, w), P-a.e. w. Let T(w) inf.{ t q; W(w) = a}. Applying (iii) to the Brownian motion {Ws ., - a; 0 5 s < co} we conclude that
3. Stochastic Integration
234 Li-00(a, a)) < LT(0,),(a, w)
for all s > 0, Pz-a.e. w,
by the additive functional property of local time (Definition 6.1 and Remark 6.5). For every we T < r} we may take s = r - T(w) above, and this yields Lq(a, w) < L,(a, a)).
6.19. From (6.38) we obtain lim),4f (y) 5 f(x), limyt f(y) 5 f(z) and f(y) 5 lim,ty f(x), f(y) 5_ lin- ir.ty f(z). This establishes the continuity of f on R. For E l fixed and 0 < h, < h2, we have from (6.38), with x = y = u + h1,
z= + h2:
of(; h1) A.N;h2) On the other hand, applying (6.38) with x = - h2, y = - hi, and z = yields
(8.5)
AN; -h2) 5 4f(; -hi).
(8.6)
Finally, with x = - e, y =
z = + 6, we have
Af(; -E)
(8.7)
AN; 6); E, 6 > a
Relations (8.5)-(8.7) establish the requisite monotonicity in h of the difference quotient (6.39), and hence the existence and finiteness of the limits in (6.40). In particular, (8.7) gives D- f(x) 5 Liff (x) upon letting E 10, 610, which establishes the second inequality in (6.41). On the other hand, we obtain easily from (8.5) and (8.6) the bounds (8.8)
(y - x) to+f(x) 5 f(y) - f(x) 5 (y - x) Df(y); x < y,
which establish (6.41).
For the right-continuity of the function Dtf(), we begin by observing the inequality D+ f (x) 5 limy 4,, D+f(y); x e R, which is a consequence of (6.41). In the
opposite direction, we employ the continuity of f, as well as (8.8), to obtain for x < z:
f(z) - f(x) = lim f (z) -f (y)
z -X
y ,i,
z -y
lim Go+ f (y). y 4,.
Upon letting z 1 x, we obtain D+ f(x)_ limy 4 x D+f(y). The left-continuity of IT f(-) is proved similarly. From (8.8) we observe that, for any function (p: R - l satisfying D- f(x) < (p(x) S D+f(x);
(8.9)
x E ER,
we have for fixed y E R, (8.10)
f(x)
Gy(x)
f(y) + (x - y)(p(y);
x e R.
The function Gy) is called a line of support for the convex function fe). It is immediate from (8.10) that f(x) = supyERGy(x); the point of (6.42) is that f() can be expressed as the supremum of countably many lines of support. Indeed, let E be a countable, dense subset of R. For any XE R, take a sequence { y }'.1 of numbers in E, converging to x. Because this sequence is bounded, so are the sequences {D±f(yO} 1 (by monotonicity and finiteness of the functions D±)) and {q)(y)},T=1 (by (8.9)). Therefore, lim, Gyn(x) = f(x), which implies that f(x) = supy e EGy(x).
3.8. Solutions to Selected Problems
235
6.20. (iii) For any x < y < z, we have (8.11)
(1)(y) - (1)(x)
1
y-x
y-x
1
z
y
f
co(u)du
9(y)
co(u)du -(1)(z) - 4)(Y) y
9(z)
x
This gives (D(Y)
z -y y -x 4)(x) + z -x z -x 4:0(z),
which verifies convexity in the form (6.38). Now let x j y, z 1 y in (8.11), to obtain 9-(y) < D (D(Y) < 9(.1) < D+O(Y) 5- 9+(y);
Ye
At every continuity point x of 9, we have 9±(x) = 9(x) = D± (1)(x). The left- (respectively, right-) continuity of 9_ and D- co (respectively, and WO) implies tp__(y) = D -1(y) (respectively, 9.,(y) = D+4)(y)) for all ye R. (iv) Letting x 1 y (respectively, x T y) in (6.44), we obtain Ir.f(Y) < (PAY) 5 9(Y) 5 9+(Y) 5- D+.1(Y);
Y e R.
But now from (6.41) one gets 50+(x)
D+ f(x) 5- D.1(Y) -5 9-01 5 OA x <
and letting y 1 x we conclude 9+(x) = 117(x); x e R. Similarly, we conclude 9_(x) = D-f(x); xe R. Now consider the function G A f - 410, and simply notice the consequences D±G(x) = (x) - D±(1)(x) = 0; x E P., of the preceding discussion; in other words, G is differentiable on P. with derivative which is identically zero. It follows that G is identically constant. 6.29. ( iv )
( vi): Let t e(0, co) be such that P°[1z, f(Ws) ds < co] = 1. For x = 0, just take S = t. For x # 0, consider the first passage time 7; and notice that P°[0 < Tx < oo] = 1, P°[27; < t] > 0, and that {B, 4=' Ws+t, - x, 0 < s < co} is a Brownian motion under P°. Now, for every we {2T < t}: 2T (m)
f(x + 13,(a)))ds o
LA.)
f(W(a)))du
f f(Wx(co))du < oo,
t} g {fp f(x + BOds < co}, a.s. P°. We conclude that this latter event has positive probability under P°, and (vi) whence {2 Tx
follows upon taking S = Tx. Lemma 6.26 gives, for each x e K, the existence of an open neighborhood U(x) of x with fu(x)f(y)dy < co. Now (iii) follows from the compactness of K. For fixed x e R, define gx(y) = f(x + y) and apply the known implication (iii) (ii) to the function gx. 7.3. We may write f as the difference of the convex functions
3. Stochastic Integration
236
f,(x)
f(0) + xf'(0) +
[f"(z)]+ dz dy, o
f2(x) -4
o
f1 x
)
o
0
[f"(z)]- dz dy,
and apply (7.4). In this case, Adx) = f "(x) dx, and (7.3) shows that I% A,(a)m(da) =
ifof "(Xs)d <M>s.
7.6. Let 1%,(w) denote the total variation of V(w) on [0, t]. For P-a.e. we a we have I7,(w) < cc; 0 < t < co. Consequently, for a < b, 131(a) - .1(b)1
-
+ IJ,(a) - Jr(b)I < I V - VzI +
and these last expressions converge to zero a.s. as r
1(am(XS)dVs, 0
t and b La. Furthermore,
the exceptional set of CO E S2 for which convergence fails does not depend on t or a. Relation (7.20) is proved similarly.
7.7. The solution is a slight modification of Solution 6.12, where now we use Lemma 7.5 to establish the continuity in a of the integrand on the left-hand side.
3.9. Notes Section 3.2: The concept of the stochastic integral with respect to Brownian motion was introduced by Paley, Wiener & Zygmund (1933) for nonrandom integrands, and by K. Ito (1942a, 1944) in the generality of the present section. ItO's motivation was to achieve a rigorous treatment of the stochastic differential equation which governs the diffusion processes of A. N. Kolmogorov (1931). Doob (1953) was the first to study the stochastic integral as a martingale, and to suggest a unified treatment of stochastic integration as a chapter of martingale theory. This task was accomplished by Courrege (1962/1963), Fisk (1963), Kunita & Watanabe (1967), Meyer (1967), Millar (1968), DoleansDade & Meyer (1970). Much of this theory has become standard and has
received monograph treatment; we mention in this respect the books by McKean (1969), Gihman & Skorohod (1972), Arnold (1973), Friedman (1975), Liptser & Shiryaev (1977), Stroock & Varadhan (1979), and Ikeda & Watanabe (1981) and the monographs by Skorohod (1965) and Chung & Williams (1983).
Our presentation draws on most of these sources, but is closest in spirit to Ikeda & Watanabe (1981) and Liptser & Shiryaev (1977). The approach suggested by Lemma 2.4 and Problem 2.5 is due to Doob (1953). A major recent development has been the extension of this theory by the "French school" to include integration of left-continuous, or more generally, "predictable," processes with respect to discontinuous martingales. The fundamental reference for this material is Meyer (1976), supplemented by Dellacherie & Meyer (1975/1980); other accounts can be found in Metivier & Pellaumail (1980), Metivier (1982), Kopp (1984), Kussmaul (1977), and Elliott (1982). Section 3.3: Theorem 3.16 was discovered by P. Levy (1948: p. 78); a different
3.9. Notes
237
proof appears on p. 384 of Doob (1953). Theorem 3.28 extends the Burkholder-
Davis-Gundy inequalities of discrete-parameter martingale theory; see the excellent expository article by Burkholder (1973). The approach that we follow
was suggested by M. Yor (personal communication). For more information on the approximations of stochastic integrals as in Problem 3.15, see Yor (1977).
Section 3.4: The idea of extending the probability space in order to accommodate the Brownian motion W in the representation of Theorem 4.2 is due to Doob (1953; pp. 449-451) for the case d = 1. Problem 4.11 is essentially from McKean (1969; p. 31). Chapters II of Ikeda & Watanabe (1981) and XII of Elliott (1982) are good sources for further reading on the subject matter of Sections 3.3 and 3.4. For a different proof and further extensions of the F. B.
Knight theorem, see Cocozza & Yor (1980) and Pitman & Yor (1986) (Theorems B.2, B.4), respectively. Section 3.5: The celebrated Theorem 5.1 was proved by Cameron & Martin
(1944) for nonrandom integrands X, and by Girsanov (1960) in the present generality. Our treatment was inspired by the lecture notes of S. Orey (1974). Girsanov's work was presaged by that of Maruyama (1954), (1955). Kazamaki (1977) (see also Kazamaki & Sekiguchi (1979)) provides a condition different
from the Novikov condition (5.18): if exp(14) is a submartingale, then 4 = exp(M, - l<M>1) is a martingale. The same is true if E[exp(1111,)] < co (Kazamaki (1978)). Proposition 5.4 is due to Van Schuppen & Wong (1974). Section 3.6: Brownian local time is the creation of P. Levy (1948), although
the first rigorous proof of its existence was given by Trotter (1958). Our approach to Theorem 6.11 follows that of Ikeda & Watanabe (1981) and McKean (1969). One can study the local time of a nonrandom function divorced from probability theory, and the general pattern that develops is that regular local times correspond to irregular functions; for instance, for the highly irregular Brownian paths we obtained Holder-continuous local times (relation (6.22)). See Geman & Horowitz (1980) for more information on this topic. On the other hand, Yor (1986) shows directly that the occupation time B
r; (B, co) of (6.6) has a density.
The Skorohod problem of Lemma 6.14, for RCLL trajectories y, was treated by Chaleyat-Maurel, El Karoui & Marchal (1980). The generalized Ito rule (Theorem 6.22) is due to Meyer (1976) and Wang (1977). There is a converse to Corollary 6.23: if f( W) is a continuous semimartingale, then f is the difference of convex functions (Wang (1977), cinlar, Jacod, Protter & Sharpe (1980)). A multidimensional version of Theorem 6.22, in which convex functions are replaced by potentials, has been proved by Brosamler (1970). Tanaka's formula (6.11) provides a representation of the form f(147,) f(Wo) + f o g(W)dWs for the continuous additive functional L,(a), with a e fixed. In fact, any continuous additive functional has such a representation, where f may be chosen to be continuous; see Ventsel (1962), Tanaka (1963).
238
3. Stochastic Integration
We follow Ikeda & Watanabe (1981) in our exposition of Theorem 6.17. For more information on the subject matter of Problem 6.30, the reader is referred to Papanicolaou, Stroock & Varadhan (1977). Section 3.7: Local time for semimartingales is discussed in the volume edited by Azema & Yor (1978); see in particular the articles by Azema & Yor (pp. 3-16) and Yor (pp. 23-36). Local time for Markov processes is treated by Blumenthal & Getoor (1968). Yor (1978) proved that local time AM for
a continuous semimartingale is jointly continuous in t and RCLL in a. His proof assumes the existence of local time, whereas ours is a step in the proof of existence. Exercise 7.13 comes from Azema & Yor (1979); see also Jeulin & Yor (1980)
for applications of these martingales in the study of distributions of random variables associated with local time. Exercise 7.14 is taken from Yor (1979).
CHAPTER 4
Brownian Motion and Partial Differential Equations
4.1. Introduction There is a rich interplay between probability theory and analysis, the study of which goes back at least to Kolmogorov (1931). It is not possible in a few sections to develop this subject systematically; we instead confine our attention to a few illustrative cases of this interplay. Recent monographs on this subject are those of Doob (1984) and Durrett (1984). The solutions to many problems of elliptic and parabolic partial differential equations can be represented as expectations of stochastic functionals. Such representations allow one to infer properties of these solutions and, conversely, to determine the distributions of various functionals of stochastic processes by solving related partial differential equation problems. In the next section, we treat the Dirichlet problem of finding a function which is harmonic in a given region and assumes specified boundary values. One can use Brownian motion to characterize those Dirichlet problems for which a solution exists, to construct a solution, and to prove uniqueness. We shall also derive Poisson integral formulas and see how they are related to exit distributions for Brownian motion. The Laplacian appearing in the Dirichlet problem is the simplest elliptic operator; the simplest parabolic operator is that appearing in the heat equation. Section 3 is devoted to a study of the connections between Brownian motion and the one-dimensional heat equation, and, again, we give probabilistic proofs of existence and uniqueness theorems and probabilistic interpretations of solutions. Exploiting the connections in the opposite direction, we show how solutions to the heat equation enable us to compute boundary crossing probabilities for Brownian motion. Section 4 takes up the study of more complicated elliptic and parabolic
4. Brownian Motion and Partial Differential Equations
240
equations based on the Laplacian. Here we develop formulas necessary for the treatment of Brownian functionals which are more complex than those appearing in Section 2.8. The connections established in this chapter between Brownian motion and elliptic and parabolic differential equations based on the Laplacian are a foreshadowing of a more general relationship between diffusion processes and second-order elliptic and parabolic differential equations. A good deal of the more general theory appears in Section 5.7, but it is never so elegant and surprisingly powerful as in the simple cases of the Laplace and heat equations developed here. In particular, in the more general setting, one must rely on existence theorems from the theory of partial differential equations, whereas in this chapter we can give probabilistic proofs of the existence of solutions to the relevant partial differential equations.
4.2. Harmonic Functions and the Dirichlet Problem The connection between Brownian motion and harmonic functions is profound, yet simply explained. For this reason, we take this connection as our first illustration of the interplay between probability theory and analysis. Recall that a function u mapping an open subset D of Rd into R is called harmonic in D if u is of class C2 and Au (52 u/a4) = 0 in D. As we shall prove shortly, a harmonic function is necessarily of class C' and has the mean-value property. It is this mean-value property which introduces Brownian motion in a natural way into the study of harmonic functions. Throughout this section, {147 A; 0 < t < co}, (S2, {F"}ER, is a ddimensional Brownian family and {A} satisfies the usual conditions. We denote by D an open set in Rd and introduce the stopping time (Problem 1.2.7) (2.1)
TD
inf It
0; W, e De 1,
the time of first exit from D. The boundary of D will be denoted by ap, and D = D u ap is the closure of D. Recall (Theorem 2.9.23) that each component of W is almost surely unbounded, so (2.2)
Px[TD < co] = 1; V x eD c Rd, D bounded.
Let Br A {x E Rd; Ilxll < r} be the open ball of radius r centered at the origin. The volume of this ball is 2r d d/2
(2.3)
d F (d)' 2
and its surface area is (2.4)
2rd-1 nal2 Sr
F(d/2)
- dr
4.2. Harmonic Functions and the Dirichlet Problem
241
We define a probability measure t, on 8B, by
r > 0.
po(dx) = P°[W,Bre dx];
(2.5)
A. The Mean-Value Property Because of the rotational invariance of Brownian motion (Problem 3.3.18), the measure u, is also rotationally invariant and thus proportional to surface measure on BB,. In particular, the Lebesgue integral of a function f over B, can be written in iterated form as
f(x)dx =
(2.6)
Sp
f(x))ip(dx)dp.
.1 aeo
eo
2.1 Definition. We say that the function u: D -4 IR has the mean-value property
if, for every a e D and 0 < r < oo such that a + B, c D, we have
u(a) = J
u(a + x)Eir(dx). aBr
With the help of (2.6) one can derive the consequence
u(a) = -1 Vr
u(a
x)dx
Br
of the mean-value property, which asserts that the mean integral value of u over a ball is equal to the value at the center. Using the divergence theorem one can prove analytically (cf. Gilbarg & Trudinger (1977), p. 14) that a harmonic function possesses the mean-value property. A very simple probabilistic proof can be based on It6's rule. 2.2 Proposition. If u is harmonic in D, then it has the mean-value property there.
PROOF. With a e D and 0 < r < oo such that a + B, c D, we have from Ito's rule d
U(Wt
E i=1
) = U(WO)
0
uxi
(Ws)dKi)
A to*,
1
+-
2
to ra,Br
f
Au(W.$)ds;
0 < t < oo.
0
Because u is harmonic, the last (Lebesgue) integral vanishes, and since (8u /ax;);
1 < i < d, are bounded functions on a + B the expectations under P° of the stochastic integrals are all equal to zero. After taking these expectations on both sides and letting t oo, we use (2.2) to obtain u(a) = E° u(W,..r ) =
f ()Br
u(a + x)yr(dx).
242
4. Brownian Motion and Partial Differential Equations
2.3 Corollary (Maximum Principle). Suppose that u is harmonic in the open, connected domain D. If u achieves its supremum over D at some point in D, then u is identically constant.
PROOF. Let M = supxe, u(x), and let DM = {x e D; u(x) = M }. We assume that DM is nonempty and show that DM = D. Since u is continuous, DM is closed relative to D. But for a e Dm and 0 < r < co such that a + Br D, we have
the mean value property:
M = u(a) =
1
V,
u(a + x)dx,
J Br
which shows that u = M on a + Br. Therefore, DM is open. Because D is connected, either DM or D \DM must be empty.
2.4 Exercise. Suppose D is bounded and connected, u is defined and continuous on D, and u is harmonic in D. Then u attains its maximum over D on D. If v is another function, harmonic in D and continuous on D, and v = u on (3D, then v = u on D as well.
For the sake of completeness, we state and prove the converse of Proposition 2.2. Our proof, which uses no probability, is taken from Dynkin & Yushkevich (1969).
2.5 Proposition. If u maps D into IR and has the mean value property, then u is of class C° and harmonic.
PROOF. We first prove that u is of class C. For e > 0, let ge:
[0, co) be
the C' function c(e)exp [11x112 g e(x) =
- 2 ];
0;
114 < 114
E,
where c(e) is chosen so that (because of (2.6))
g,(x)dx = c(e)
(2.7) 18,
f 0
s exp (p 2
1
E2
dp
= 1.
For e > 0 and a e D such that a + Be c D, define u(a + x)ge(x)dx =
ue(a) A J B,
f
u(y)g e(y - a) dy.
ad
From the second representation, it is clear that u, is of class C' on the open subset of D where it is defined. Furthermore, for every a e D there exists e > 0 so that a + Be D; from (2.6), (2.7), and the mean-value property of u, we may then write
4.2. Harmonic Functions and the Dirichlet Problem
243
u,(a) = J u(a + x)g,(x)dx
f
= c(e)
j'
= c(E)
u(a + x)exp(
Sp,
P
Jae
'
1
2
-
2)1.2,(dx)dp E
1
Spu(a)exp(
P, - E, dP = u(a), and conclude that u is also of class C. o
In order to show that Au = 0 in D, we choose a ED and expand a la Taylor
in the neighborhood a + B
au
d
(2.8)
u(a + y) = u(a) + E 3,1
()n
1
d
02
d
(a) + E E yiy; Xi i=1 J=1 2
(a) WC;UXJ
YEk,
+ 0(II312);
where again e > 0 is chosen so that a + BE c D. Odd symmetry gives us Jas,
YtiMdY) = 0,
Jas,
Yiblit(dY) = 0;
i
j,
so upon integrating in (2.8) over OA and using the mean-value property we obtain (2.9)
u(a) =
fas, u(a + y)ue(dy) 1
= u(a) + 2
d
02
E 04 -(a)
pc(dy) + o(0) as,
But E2
= 711
as,
as,
Y lit(dY) =
d
and so (2.9) becomes p2
2d
+ o(e2) = 0.
Dividing by E2 and letting e 0, we see that Au(a) = 0.
B. The Dirichlet Problem We take up now the Dirichlet problem (D,f): with D an open subset of Rd and f: OD R a given continuous function, find a continuous function u: such that u is harmonic in D and takes on boundary values specified by f; i.e., u is of class C2 (D) and
244
4. Brownian Motion and Partial Differential Equations
(2.10)
Au = 0;
(2.11)
u = f; on aD.
in D,
Such a function, when it exists, will be called a solution to the Dirichlet problem
(D, f). One may interpret u(x) as the steady-state temperature at x e D when the boundary temperatures of D are specified by f. The power of the probabilistic method is demonstrated by the fact that we can immediately write down a very likely solution to (D, f), namely x
u(x) A Exf(W,.);
(2.12)
provided of course that Exlf(W,D)1 < co;
(2.13)
V x e D.
By the definition of TD, u satisfies (2.11). Furthermore, for a a D and Br chosen
so that a + Br c D, we have from the strong Markov property: u(a) = Eaf(N)) = Ea {Ea ER wz,,) 1,,,*)}
= Ea lu(W,± )1 =
f
u(a + x)y,.(dx).
aar
Therefore, u has the mean-value property, and so it must satisfy (2.10). The only unresolved issue is whether u is continuous up to and including OD. It turns out that this depends on the regularity of aD, as we shall see later. We summarize our discussion so far and establish a uniqueness result for (D,f) which strengthens Exercise 2.4. 2.6 Proposition. If (2.13) holds, then u defined by (2.12) is harmonic in D.
2.7 Proposition. If f is bounded and
P`Tri, < cc] = 1; V a eD,
(2.14)
then any bounded solution to (D, f) has the representation (2.12).
PROOF. Let u be any bounded solution to (D,f), and let D A Ix e D; infy cap fix
- yll > 1 /n }. From ItO's rule we have d
u(WtAnAtDn) = u(Wo) + E
i=i
-(WO dWe; 0 < t < oo, n > 1. ax,
Since (au/axi) is bounded in B n D, we may take expectations and conclude that u(a) = Ea 14(Wi A TB., TD);
As t
co, n
0 < t < co,
n
1,
a e D.
cc, (2.14) implies that u(Wt A 28 A 28 ) converges to f(W,D), a.s. P°.
The representation (2.12) follows from the bOunCled convergence theorem.
4.2. Harmonic Functio.is and the Dirichlet Problem
245
2.8 Exercise. With D = {(x,, x2); x2 > 0} and f(x,, 0) = 0; x, e R, show by example that (D, f) can have unbounded solutions not given by (2.12). In the light of Propositions 2.6 and 2.7, the existence of a solution to the Dirichlet problem boils down to the question of the continuity of u defined by (2.12) at the boundary of D. We therefore undertake to characterize those points a e aD for which (2.15)
lim Exf(W,D) = f(a) x-a xeD
holds for every bounded, measurable function f: aD -9 at the point a.
which is continuous
2.9 Definition. Consider the stopping time of the right-continuous filtration {,,,} given by aD 9 inf ft > 0; 14/,e Del (contrast with the definition of TD in (2.1)). We say that a point a e aD is regular for D if Pa[crp = 0] = 1; i.e., a Brownian path started at a does not immediately return to D and remain there for a nonempty time interval.
2.10 Remark. A point a e aD is called irregular if /3/up = 0] < 1; however, the event {o- = 0} belongs to Fo","_, and so the Blumenthal zero-one law (Theorem 2.7.17) gives for an irregular point a: P °[aD = 0] = 0. 2.11 Remark. It is evident that regularity is a local condition; i.e., a e aD is regular for D if and only if a is regular for (a + 13,) n D, for some r > 0. In the one-dimensional case every point of aD is regular (Problem 2.7.18) and the Dirichlet problem is always solvable, the solution being piecewiselinear. When d > 2, more interesting behavior can occur. In particular, if D = {x e Rd; 0 < jlx II < 1} is a punctured ball, then for any x e D the Brownian
motion starting at x exits from D on its outer boundary, not at the origin (Proposition 3.3.22). This means that u defined by (2.12) is determined solely
by the values off along the outer boundary of D and, except at the origin, this u will agree with the harmonic function a(x) o Exf(141,B) =-- Exf(W,D);
x e B,.
Now u(0) f(0), so u is continuous at the origin if and only if f(0) = When d > 3, it is even possible for aD to be connected but contain irregular points (Example 2.17). 2.12 Theorem. Assume that d > 2 and fix a e aD. The following are equivalent:
(i) equation (2.15) holds for every bounded, measurable function f: aD which is continuous at a; (ii) a is regular for D;
246
4. Brownian Motion and Partial Differential Equations
(iii) for all e > 0, we have lim Px[TD > e] = 0.
(2.16)
x-qi
xeD
PROOF. We assume without loss of generality that a = 0, and begin by proving
the implication (i)
(ii) by contradiction. If the origin is irregular, then
P ° [6D = 0] = 0 (Remark 2.10). Since a Brownian motion of dimension d > 2 never returns to its starting point (Proposition 3.3.22), we have
lim r[WD e Br] = P° [W,D = 0] = 0.
40
Fix r > 0 for which P°[W,,, e B,] < (1/4), and choose a sequence On In°., for which 0 < 6 < r for all n and ön .1. 0. With ; A inf {t z 0; II Wtli b.}, we have P° [Tni. 0] = 1, and thus, limn, P° ['s . < CD] = 1. Furthermore, on the event Itn < aDI we have Wz. e D. For n large enough so that r [rn < o-D] > (1/2), we may write
-1
4
> r[W,DeB,] > PqW,, e B Tn < api = E°(1{,<,,,,}P°[Wane 41.9c]) Px [Hitt, e Br] P° Lin < Cfp, Hit. edx]
=f
DnBon
1
inf
z2 xeDnBon
Px[W,D e BA,
from which we conclude that Px"[W,D e B,] < (1/2) for some xeD nBon. Now choose a bounded, continuous function f: OD IR such that f = 0 outside B
f < 1 inside B and f(0) = 1. For such a function we have
and (i) fails.
lim Ex.f(W,)
lim Px^ [Wt. e B,] <
n-co
n-,co
-1
2
< f(0),
4.2. Harmonic Functions and the Dirichlet Problem
We next show that (ii)
247
(iii). Observe first of all that for 0 < ö < c, the
function gb(x) A Px[Wse D; .5 < s < e] = Ex(PwITD > E
R.
- 6])
PY ['CD > E- 6] Px[WaedY]
is continuous in x. But go(x)1 g(x) A I' [147, e D; 0 < s <
= Px [o-D > c]
as 510, so g is upper semicontinuous. From this fact and the inequality tD G o-D, we conclude that lim,, Px[rD > E] < lim,, g(x) < g(0) = 0, by (ii). xeD
(1). We know that for each r > 0, II Wt -Wo ll < r] does not depend on x and approaches one as
Finally, we establish (iii) Px [max° < t <
E 10. But then
Px[liVK - Woll < 11
Px [{ max II W, - Woll < r }n {To 0
[ max II Wtil < P° 0
El
rl - Px[rD > E].
Letting x -4 0 (x e D) and e 10, successively, we obtain from (iii)
lim Px[II W. -x II < /1 = 1; 0 < r < co. x-HD
xeD
The continuity off at the origin and its boundedness on ap give us (2.15). El
C. Conditions for Regularity For many open sets D and boundary points a e ap, we can convince ourselves intuitively that a Brownian motion originating at a will exit from D immediately; i.e., a is regular. We formalize this intuition with a careful discussion of regularity.
We have already seen that when d = 2, the center of a punctured disc is an irregular boundary point. The following development, culminating with Problem 2.16, shows that, in 112, any irregular boundary point of D must be "isolated" in the sense that it cannot be connected to any other point outside D by a simple arc lying outside D. 2.13 Definition. Let D c Rd be open and an D. A barrier at a is a continuous function v: D R which is harmonic in D, positive on D \ {a}, and equal to zero at a.
4. Brownian Motion and Partial Differential Equations
248
2.14 Example. Let D c Br. c R2 be open, where 0 < r < 1, and assume (0, 0) e aD. If a single-valued, analytic branch of log(x1 + ix2) can be defined in D \(0, 0), then Re
v(x,, x2) A
log Jxi + xi
1
log(x + ix2)
llog(x, + ix2)12'
(x1, x2)eD\(0,0), (x1, x2) = (0,0),
0;
is a barrier at (0,0). Indeed, being the real part of an analytic function, v is harmonic in D, and because 0 < x? + xi < r < 1 in D\(0, 0), v is positive on this set.
2.15 Proposition. Let D be bounded and a E aD. If there exists a barrier at a, then a is regular. PROOF. Let v be a barrier at a. We establish condition (i) of Theorem 2.12.
With f: ap R bounded and continuous at a, define M = supxeaD Choose a> 0 and let 8 > 0 be such that I f(x) - f(a)1 < a if xeaD and Ilx - all < 6. Choose k so that kv(x) > 2M for x e D and 'Ix - all L 6. We then have l f(x) - f(a)l < E + kv(x); x e aD, so 1 Ex.i. (WD) - f(a)l < E
k Ex v(14/, D) = E
k v(x); x E D
by Proposition 2.7. But v is continuous and v(a) = 0, so
lim IP.RWt,)- f(a)I < E. x-oa
se D
0
Finally, we let a 1.0 to obtain (2.15).
2.16 Problem. Let D c 1R2 be open, and suppose that a e al) has the property that there exists a point b 0 a in 1R2 \ D, and a simple arc in R2 \ D connecting a to b. Show that a is regular.
D
D
In three or more dimensions, it is possible to create a cusped region D so that the boundary point at the end of the cusp is irregular. We illustrate this situation in R'. In particular, we will construct a region as demarcated in the following figure by the broken line, so that, when this region is rotated about the xraxis, the resulting solid has an irregular boundary point at the
4.2. Harmonic Functions and the Dirichlet Problem
249
origin. It is a simple matter to replace this solid by an even larger one, having a smooth boundary except for a cusp at the origin (dotted curve). It is a direct consequence of Definition 2.9 that the origin is also an irregular boundary point for this larger solid. 2.17 Example (Lebesgue's Thorn). With d = 3 and {e}°1_, a sequence of positive numbers decreasing to zero, define
E = {(x1, x2, x3); -1 < x1 < 1, xi + xi < 1},
F = { (xi, x2, x3); 2-n < x1 < 2 -n +1 x2 + xi < s D=
,
E\(U F). n=1
Now P° [(W(2), Wi(3)) = (0, 0), for some t > 0] = 0 (Proposition 3.3.22), so the
P°-probability that W = (Ww, W(2), W3) ever hits the compact set K 4 {(x x2, x3); 2- < x1 < 2-n+1, x2 = x3 = 0} is zero. According to Problem 3.3.24, lime = x a.s. P°, so for P°-a.e. co ell, the path t 1-- Wi(u)) remains bounded away from K. Thus, if s is chosen sufficiently small, we can
ensure that P°[W,e F, for some t > 0] < 3-". If W, beginning at the origin, does not return to D immediately, it must avoid D by entering U',., F. In other words,
Lebesgue's Thorn p0 [0.D
co
0] < p0[VV, .., e F, for some t > 0 and n > 1] < E 3-11 < 1. n=1
If cusplike behavior is avoided, then the boundary points of D are regular, regardless of dimension. To make this statement precise, let us define for
250 ye
4. Brownian Motion and Partial Differential Equations
\ {0} and 0 5 8 5
1C,
the cone C(y, 0) with direction y and aperture 0 by
C(y, 0) = {x e Rd; (x,
cos 0}.
II
2.18 Definition. We say that the point a e OD satisfies Zaremba's cone con-
0 and 0 < 0 < it such that the translated cone
dition if there exists y
a + C(y, 0) is contained in Rd \ D.
2.19 Theorem. If a point a E ap satisfies Zaremba's cone condition, then it is regular.
PROOF. We assume without loss of generality that a is the origin and C(y, 0) c d \ D, where y 0 and 0 < 0 < n. Because the change of variable z = (x/.1i) maps C(y, 0) onto itself, we have for any t > 0,
[ = c(m) (2ittr2 exp 1
P°[ Wt CU,
=
f
1
exp [
11x1121dx
2t
I 21 dz 2
q > 0,
where q is independent of t. Now P° [6D < t] > P° [W, E C(y, 0)] = q, and letting t j 0 we conclude that Pqo-D = 0] > 0. Regularity follows from the Blumenthal zero-one law (Remark 2.10).
2.20 Remark. If, for a e ar) and some r > 0, the point a satisfies Zaremba's cone condition for the set (a + 13,) r D, then a is regular for D (Remark 2.11).
4.2. Harmonic Functions and the Dirichlet Problem
251
D. Integral Formulas of Poissont We now have a complete solution to the Dirichlet problem for a large class of open sets D and bounded, continuous boundary data functions f: OD R. Indeed, if every boundary point of D is regular and D satisfies (2.14), then the unique bounded solution to (D,f) is given by (2.12) (Propositions 2.6, 2.7 and Theorem 2.12). In some cases, we can actually compute the right-hand side of (2.12) and thereby obtain Poisson integral formulas. 2.21 Theorem (Poisson Integral Formula for a Half-Space). With d 2, D = {(x1,..., x); x, > 0} and f: OD bounded and continuous, the unique bounded solution to the Dirichlet problem (D,f) is given by O
u(x) =
(2.17)
f(/2) d
xdf (Y)
ird/2
d
dy;
xeD.
2.22 Problem. Prove Theorem 2.21.
The Poisson integral formula for a d-dimensional sphere can be obtained from Theorem 2.21 via the Kelvin transformation. Let go: Rd VOI
Rd \ {0} be
defined by q;o(x) = (x/11.42). Note that cp is its own inverse. We simplify notation by writing x* instead of co(x). For r > 0, let B = {x e Rd: llx - Cul < r}, where c = red and e1 is the unit vector with a one in the i-th position. Suppose f: aB R is continuous (and hence bounded), so there exists a unique function u which solves the Dirichlet problem (B, f). The reader may easily verify that cp(B) = H
fx* e IRd; (x*, c) > 11
and (p(OB \ {O}) = aH = {x* e Rd; (x*, c) = -I}. We define u*: H -+ R, the Kelvin transform of u, by 1
u*(x*) = oc*Ild-2 u(x).
(2.18)
A tedious but straightforward calculation shows that Au*(x*) = Au(x), so u* is a bounded solution to the Dirichlet problem (H, f*) where (2.19)
f *(x*) =
x*II d -2 fix);
Because H = (1/2r)ed + D, where D is as in Theorem 2.21, we may apply (2.17)
to obtain The results of this subsection will not be used later in the text.
4. Brownian Motion and Partial Differential Equations
252
1
(2.20)
u*(x*) =
(d/2) f
xd
lid/2
r)f*(Y*) Y* - x*
dy*;
x* e H.
Formulas (2.18)-(2.20) provide us with the unique solution to the Dirichlet problem (B, f). These formulas are, however, a bit unwieldy, a problem which can be remedied by the change of variable y = q)(y*) in the integral of (2.20). This change maps the hyperplane OH into the sphere 5B. The surface element on aB is Sra,(dy - c) (recall (2.4), (2.5)). A little bit of algebra and calculus on manifolds (Spivak (1965), p. 126) shows that the proposed change of variable in (2.20) involves
dy* =
(2.21)
Srar(dy
- c)
1131112(")
(The reader familiar with calculus on manifolds may wish to verify (2.21) first for the case y* = e, + (1/2r)ed and then observe that the general case may be reduced to this one by a rotation. The reader unfamiliar with calculus on manifolds can content himself with the verification when d = 2, or can refer to Gilbarg & Trudinger (1977), p. 20, for a proof of Theorem 2.23 which uses the divergence theorem but avoids formula (2.21).)
On the other hand, (2.22)
(2.23)
ILV* - X*112 =
Ilx - y112 1142
r2 - Ilx - cii 2 = 114 2[2(c, x*) - 1] = 2r Ike (x:
- 2r). 1
Using (2.18), (2.19), and (2.21)-(2.23) to change the variable of integration in (2.20), we obtain (2.24)
u(x) = rd-2(r -
cii 2)
I f011tr(dY as
x e B.
I1Y -
Translating this formula to a sphere centered at the origin, we obtain the following classical result.
2.23 Theorem (Poisson Integral Formula for a Sphere). With d > 2, Br = {xe 141d; 114 < r}, and f: aB, continuous, the unique solution to the Dirichlet problem (B,, f) is given by f(Y)fc(dY)
(2.25)
1142) foe,
Y-
;
x e B,..
2.24 Exercise. Show that for x E Br, we have the exit distribution (2.26)
rd-2(r2 Px[WtBe dY1 =
11
x
X112"1
-
Y Ild
(dY);
= r-
4.2. Harmonic Functions and the Dirichlet Problem
253
E. Supplementary Exercises 2.25 Problem. Consider as given an open, bounded subset D of Rd and the bounded, continuous functions g: D -, R and f: 8D -, R. Assume that u: D -gcR is continuous, of class OD), and solves the Poisson equation 1
2Au = -g;
in D
subject to the boundary condition u=f;
on aD.
Then establish the representation ,1)
(2.27)
u(x) = Ex[ f(W,D) + .1
g(W) dt];
x e D.
o
In particular, the expected exit time from a ball is given by r2
(2.28)
Ex TB,.
=
i
ixil 2
d
;
X E Br.
(Hint: Show that the process {M, gL- u(W, AO + Po^ `D g(W) ds, Ft; 0 < t < col
is a uniformly integrable martingale.) 2.26 Exercise. Suppose we remove condition (2.14) in Proposition 2.7. Show
that v(x) A Px [-up = co] is harmonic in D, and if aeaD is regular, then limx.a v(x) = 0. In particular, if every point of al) is regular, then with xe D
u(x) = Ex[f(41/01{,D,)}], the function u + Av is a bounded solution to the Dirichlet problem (D, f) for any A e R. (It is possible to show that every bounded solution to (D, f) is of this form; see Port & Stone (1978), Theorem 4.2.12.)
2.27 Exercise. Let D be bounded with every boundary point regular. Prove that every boundary point has a barrier. 2.28 Exercise. A complex-valued Brownian motion is defined to be a process W = {IV) + ilgt(2), .Ft; 0 < t < co}, where W = {(W('0, W(2)), A; 0 < t < co} is a two-dimensional Brownian motion and i = .\/- 1:
(i) Use Theorem 3.4.13 to show that if W is a complex-valued Brownian motion and f: C --0 C is analytic and nonconstant, then (under an appropriate condition) f(W) is a complex-valued Brownian motion with a random time-change (P. Levy (1948)).
(ii) With e C \ {0}, show that M, A e''',, 0 < t < co is a time-changed, complex-valued Brownian motion. (Hint: Use Problem 3.6.30.) (iii) Use the result in (ii) to provide a new proof of Proposition 3.3.22. For additional information see B. Davis (1979).
254
4. Brownian Motion and Partial Differential Equations
4.3. The One-Dimensional Heat Equation In this section we establish stochastic representations for the temperatures in infinite, semi-infinite, and finite rods. We then show how such representations allow one to compute boundary-crossing probabilities for Brownian motion. Consider an infinite rod, insulated and extended along the x-axis of the (t, x) plane, and let f(x) denote the temperature of the rod at time t = 0 and location x. If u(t, x) is the temperature of the rod at time t > 0 and position x e R, then, with appropriate choice of units, u will satisfy the heat equation 314
(3.1)
102u
at = 2 0x2'
with initial condition u(0, x) = f(x); x E R. The starting point of our probabilistic treatment of (3.1) is furnished by the observation that the transition density p(t; x, y)
1
y
Px[W,edy] =
1
e-(x-y)212t;
t > 0,
x, ye R,
of the one-dimensional Brownian family satisfies the partial differential equation (3.2)
Op
102p
at
2 0x2
Suppose then that f: R -+ R is a Borel-measurable function satisfying the condition
f_. e-`"2 I f (x)I dx < co
(3.3)
for some a > 0. It is well known (see Problem 3.1) that u(t, x) A Exf(Wi) =
(3.4)
f(y)p(t; x, y) dy
is defined for 0 < t < (1/2a) and x E R, has derivatives of all orders, and satisfies the heat equation (3.1).
3.1 Problem. Show that for any nonnegative integers n and m, under the assumption (3.3), we have ani-m
(3.5)
Otnaxm
ani-m
oo
u(t, x) =
f(y)
OtnOxm
p(t; x, y) dy;
0
x e R. 1
Iff is bounded and continuous, then rewriting (3.4) as u(t, x) = E °f(x + 147d, we can use the bounded convergence theorem to conclude
4.3. The One-Dimensional Heat Equation
255
f(x) = lim u(t, y),
(3.6)
V x e IR.
viro y -.x
In fact, we have the stronger result contained in the following problem.
3.2 Problem. If f: is a Borel-measurable function satisfying (3.3) and f is continuous at x, then (3.6) holds.
A. The Tychonoff Uniqueness Theorem We shall call p(t; x, y) a fundamental solution to the problem of finding a function u which satisfies (3.1) and agrees with the specified function f at time
t = 0. We shall say that a function u: has continuous derivatives up to a certain order on a set G, if these derivatives exist and are continuous in the interior of G, and have continuous extensions to that part of the boundary aG which is included in G. With this convention, we can state the following uniqueness theorem. For nonnegative functions, a substantially stronger result is given in Exercise 3.8.
3.3 Theorem (Tychonoff (1935)). Suppose that the function u is strip (0, T] x FE and satisfies (3.1) there, as well as the conditions lim u(t, y) = 0;
(3.7)
0,2
on the
xe
y -.x
(3.8)
sup I u(t, x)I < Keax2;
0
xe
for some positive constants K and a. Then u = 0 on (0, T] x 3.4 Remark. If u, and u2 satisfy (3.1), (3.8) and lim u (t, y) = lim u2(t, t4.0
then Theorem 3.3 applied to u, - u2 asserts that u, = u2 on (0, T) x R. 3.5 Remark. Any probabilistic treatment of the heat equation involves a timereversal. This is already suggested by the representation (3.4), in which the initial temperature function f is evaluated at W, rather than K. We shall see this time-reversal many times in this section, beginning with the following probabilistic proof of Theorem 3.3. PROOF OF THEOREM 3.3. Let Ty be the passage time of W to y as in (2.6.1). Fix
x e R, choose n > Ix!, and let R = 7' n T_. With t e [0, T) fixed and v(0, x)
u(T - t - 0, x); 0 < 0 < T - t,
256
4. Brownian Motion and Partial Differential Equations
we have from Ito's rule, for 0 5 s < T - t, u(T - t, x) = v(0, x) = Ex v(s
(3.9)
=
R, Ws
R,,)
[v(s, Ws)1{,
J].
Now I v(s, Ws)11{,
max
< s
I u(T - t -
Ke"2,
IA
and v(s, Ws) converges Px-a.s. to zero as s j T - t, thanks to (3.7). Likewise, I v(R, WR.)11{, R,,} is dominated by Ke°"2. Letting sIT-t in (3.9), we obtain from the bounded convergence theorem:
u(T- t, x) = Ex [v(R, WR,,)1{R
Therefore, with 0 < t < T,
I u(T - t, x)I < Kew° Px [R < T - t]
Ke°"2 (P° [T, < 2
< Kea 2
where we have used
+ P° [Tn+x < T])
f(n-x)/Ii
(2.6.2).
e- '212 dz
x2I2 dz +
But from
,
(n+x)hiT
(2.9.20) it
is
evident that
e-z212 dz = 0, provided a < 1/2T
en2
Having proved the theorem for a < (1/2T), we can easily extend it to the < case where this inequality does not hold by choosing To = 0 < T, < T,, = T such that a < (1/2(Ti i = 1, n, and then showing successively that u = 0 in each of the strips (T,_,, Ti]; i = 1,
n.
It is instructive to note that the function (3.10)
h(t, x)
-x p(t;
a x, 0) = a-p(t; x, 0); x
t > 0, x E R,
solves the heat equation (3.1) on every strip of the form (0, x R; furthermore, it satisfies condition (3.8) for every 0 < a < (1/2T), as well as (3.7) for every x 0. However, the limit in (3.7) fails to exist for x = 0, although we do have lima° h(t, 0) = 0.
B. Nonnegative Solutions of the Heat Equation If the initial temperature f is nonnegative, as it always is if measured on the absolute scale, then the temperature should remain nonnegative for all t > 0; this is evident from the representation (3.4). Is it possible to characterize the nonnegative solutions of the heat equation? This was done by Widder (1944),
4.3. The One-Dimensicnal Heat Equation
257
who showed that such functions u have a representation CO
u(t, x) = J
p(t; x, y) dF(y); x e
where F: is nondecreasing. Corollary 3.7 (i)', (ii)' is a precise statement of Widder's result. We extend Widder's work by providing probabilistic characterizations of nonnegative solutions to the heat equation; these appear as Corollary 3.7 (iii)', (iv)'. 3.6 Theorem. Let v(t, x) be a nonnegative function defined on a strip (0, T) x where 0 < T < cc. The following four conditions are equivalent:
(i) for some nondecreasing function F: CO
(3.11)
p(T - t; x, y)dF(y); 0 < t < T,
v(t, x) = J
X E fR;
-co
(ii) v is of class C1'2 on (0, T) x IR and satisfies the "backward" heat equation
av
-at + -21 02axev = 0
(3.12)
on this strip; (iii) for a Brownian family {W. ..,;; 0 < s < op}, (S2,
ikPxxe
fixed t e (0, T), x e 0:2, the process {v(t + s, Ws), .mss; 0 tingale on (SI, Px);
(iv) for a Brownian family {Ws, .; 0 (3.13)
v(t, x)
s
s < co},
aindeaacrh_
T-}
{Px}xefra we have
0 < t < t + s < T, xeEJ.
Ex v(t + s,VV,);
PROOF. Since (a/at)p(T - t; x, y) + (1/2)(02/ex2)p(T - t; x, y) = 0, the impli-
cation (i)
(ii) can be proved by showing that the partial derivatives of
v can be computed by differentiating under the integral in (3.11). For a > 1/2T we have
dF(y) = j_ooe-aY2
-v (T -
< cc.
This condition is analogous to (3.3) and allows us to proceed as in Solution 3.1.
For the implications (ii)(iii) and (ii)
(iv), we begin by applying Ito's
rule to v(t + s, Ws); 0 < s < T - t. With a < x < b, we consider the passage times To and Tb as in (2.6.1) and obtain: v(t +(s n Ta n Tb), Ws A T. A
= V(t, Wo)
sAToTb
sATanTh a -04 ± 0-, Wcy)dHfc,
ax
a
1a2
-2
x2
±
W(,) do-.
258
4. Brownian Motion and Partial Differential Equations
Under assumption (ii) the Lebesgue integral vanishes, as does the expectation of the stochastic integral because of the boundedness of (0/ax)v(t + o, y) when a < y < b and 0 < a <s
v(t, x) = Exv(t + (s
A T:, A 70, Ws A T. A TO'
Now let a I -co, b i Go and rely on the nonnegativity of v and Fatou's lemma to obtain (3.15)
Exv(t+s, Ws);
v(t, x)
0
x e R.
Inequality (3.15) implies that for fixed t e (0, T) and x e R, the process Ws), ,Fs; 0 < s < T - t} is a supermartingale on (0,.r,, , Px). Indeed, for 0 < s1 < s2 < T - the Markov property yields {v(t + s,
t,
(3.16)
Ex [v(t +
s2, Ws2)1A,]((a) = f(Ws i(w)),for Px-a.e. w e
0,
where (3.17)
f(y) A EY v(t +
s2, W,2_,i)
(see Proposition 2.5.13). From (3.15), we have EY v(t +
s2, W,,)
and so for 0 <
t < t + si < t +
(3.18)
v(t +
s1, WO
s2 < T,
> Ex[v(t +
v(t + si, y),
xe R: a.s. Px.
s2, 14/s2)1,,,],
It is clear from this argument that if equality holds in (3.15), then {v(t + s, iv, ( iii) and t} is a martingale. To complete our proof of ( ii) (ii) (iv), we must establish the reverse of inequality (3.15). Returning to (3.14), we may write
.; 0 < s < T -
v(t, x) = Ex[v(t + s,
Ws)i{s< T A To] + Ex[v(t + T, a)1{1.<., A TO]
+ Ex[v(t + Tb,b)1{Tb<s A T.}] < Exv(t + s, Ws) + Ex[v(t + T, a)l{r<s}i
+ Ex[v(t + Tb, b)1{Tb<s}i.
We will have established (3.13) as soon as we prove lim
(3.19)
Ex[v(t +
'Tb,b)1{Tb<s}] = 0
b-..co
(a dual argument then shows that lima__. Ex[v(t
+
'T,a)1,-.<4] = 0). For
(3.19), it suffices to show that with B > 0 large enough, we have J : Ex[v(t + Tb,b)1{Tb<s}] db < o o .
We choose x e R, 0 (3.10) we have
< t < T,
and 0 < s <
t so
Px[Tb Edo] = h(o-; b - x)da; i
that s + t
<
T From (2.6.3) and
b > x, a > 0.
4.3. The One-Dimensional Heat Equation
259
For B _.- x sufficiently large, h(a; b - x) is an increasing function of a e (0,$), provided b > B. Furthermore, for r e (s, t) and B perhaps larger, we have p(r; x, b);
h(s,b - x) ._-_
b _>._ B.
It follows that Ex[v(t + T6, b)1{Tb<s}] db = .133 fs
v(t + a,b)h(a,b - x)da db
0
B
s
ly
fs
J o
v(t + o-,b)p(r; x, b) db doB
-113- I Ex v(t + a,W,)doo
r is 0
where the next to last inequality is a consequence of (3.15). This proves (3.13)
for x e fR, 0
0
(3.20)
then v(t, x) = Ex v(t + s, Ws);
(3.21)
x e R.
This will yield (3.13) for the range of values indicated there. We have just established that (3.20) implies (3.21) when k = 1. Assume this implication for some k > 1, so {v(t + s, Ws), Fs; 0 < s < kt} is a martingale. Choose s2 e [kt,(k + 1)t) and si e [0, kt) so that 0 < s2 - s, < t. Then Ex v(t + s2, Ws) = Ex {Ex[v(t + S2, W011]) = Ex v(t + si, WO = v(t, x), where we have used (3.16), (3.17), and the induction hypothesis in the form E" v(t + s2, 147,2_0 = v(t + s 1, y).
Finally, we take up the implication (iv) =(i). For 0 <s < (T/4), (T/2) < t < T, (3.13) gives
v(t - s, x) .-- Exv(T - e, Wr-t) =
i' p(T - t; x, y)
i..
dF,(y),
P (- -27= ; 0 , Y)
where F, is the nondecreasing function FE(x) -4-
; 0, y)v(T - s, y)dy; fx_,, p ET 2
x e IR.
260
4. Brownian Motion and Partial Differential Equations
Again from (3.13), FE(co) = ev( T sup
e, WT /2) =
v(s,0) < co.
max
FE(co) <
v(( T/2) - e, 0), and thus
(T14),s(T12)
0 <e<(T14)
> By Helly's theorem (Ash (1972), p. 329), there exists a sequences, > E2 > 10 and a nondecreasing function F*: R -, [0, ao), such that lim, F,k(x) =
Ek
F*(x) for every x at which F* is continuous. Because for fixed x e R and t e((T/2), T) the ratio (p(T - t; x, y)/p((T/2); 0, y)) is a bounded, continuous function of y, converging to zero as IyI -, co, we have v(t, x) = lim v(t - Ek, x) =
iT 2;(")
-o0
k-oco
1)
dF*(y)
by the extended Helly-Bray lemma (Loeve (1977), p. 183). Defining F(x) = If:, (dF*(y)/p((T/2); 0, y)), we have (3.11) for (T/2) < t < T, x e R. If 0 < t _<_ (T/2), we choose t, e((T /2), T) and use (3.13) to write v(t, x) = J
-0
p(t, - t; x, y)v(t y)dy
=
p(t, - t; x, y)p(T - t,; y, z)dy dF(z)
1. =
p(T - t; x, z)dF (z).
CI
3.7 Corollary. Let u(t, x) be a nonnegative function defined on a strip (0, T) x R, where 0 < T < co. The following four conditions are equivalent: (i)' for some nondecreasing function F: R - R, CO
(3.22)
u(t, x) = J
p(t; x, y) dF(y);
0 < t < T, x e R;
(ii)' u is of class C'.2 on (0, T) x R and satisfies the heat equation (3.1) there; (iii)' for a Brownian family {Ws, Fs; 0 -_ s < co}, (SI, F), {PlxeR and each fixed t e (0, T), xeR, the process {u(t - s, Ws), F; 0 s < t} is a martingale on (0,F , Px);
(iv)' for a Brownian family {Ws, Fs; 0 < s < op}, (0, F), {Px}x. R we have (3.23)
u(t, x) = Exu(t - s, Ws);
0
<s
PROOF. If T is finite, we obtain this corollary by defining v(t, x) = u(T - t, x) and appealing to Theorem 3.6. If T = co, then for each integer n 1 we set v(t, x) = u(n - t, x); 0 < t < n, x e R. Applying Theorem 3.6 to each v we see that conditions (ii)', (iii)', and (iv)' are equivalent, they are implied by (i)', and
they imply the existence, for any fixed n > 1, of a nondecreasing function F: R -, R such that (3.22) holds on (0, n) x R. For t > n, we have from (3.23): L
4.3. The One-Dimensional Heat Equation
u(t, x) = Exu(--n2,
-/2)) =
261
cc)
01
z
c0
CO
PG; z,y)p (t =
f
14
Z)p(t -12
2'
- 2; x,
dz
x,
z) dz
(y)
p(t; x, y) dF(y).
3.8 Exercise (Widder's Uniqueness Theorem).
defined on the strip (i) Let u(t, x) be a nonnegative function of class (0, T) x R, where 0 < T < co, and assume that u satisfies (3.1) on this strip and lim u(t, y) = 0;
x e R.
110
Show that u = 0 on (0, T) x R. (Hint: Establish the uniform integrability of the martingale u(t - s, Ws); 0 5 s < t.) (ii) Let u be as in (i), except now assume that lima° u(t, y) = f(x); x e R. Show that CO
u(t, x) =
f-co p(t; x, y)f(y)dy;
0 < t < T, x e R.
Can we represent nonnegative solutions v(t, x) of the backward heat equation (3.12) on the entire half-plane (0, co) x R, just as we did in Corollary 3.7
for nonnegative solutions u(t, x) of the heat equation (3.1)? Certainly this cannot be achieved by a simple time-reversal on the results of Corollary 3.7. Instead, we can relate the functions u and v by the formula (3.24)
v(t, x) =
j2ir exp (x2)u(1- x
0 < t < co , x e R. "
The reader can readily verify that v satisfies (3.12) on (0, co) x R if and only if u satisfies (3.1) there. The change of variables implicit in (3.24) allows us to deduce the following proposition from Corollary 3.7.
3.9 Proposition (Robbins & Siegmund (1973)). Let v(t,x) be a nonnegative function defined on the half-plane (0, co) x R. With T= co, conditions (ii), (iii), and (iv) of Theorem 3.6 are equivalent to one another, and to (i) ":
(i)" for some nondecreasing function F: R (3.25)
v(t, x) = J
exp
1
R,
yx - y2 t) dF(y);
0 < t < oo, x e R.
4. Brownian Motion and Partial Differential Equations
262
PROOF. The equivalence of (ii), (iii), and (iv) for T = co follows from their equivalence for all finite T If v is given by (3.25), then differentiation under the integral can be justified as in Theorem 3.6, and it results in (3.12). If v satisfies (ii), then u given by (3.24) satisfies (ii)', and hence (i)', of Corollary 3.7. But (3.24) and (3.22) reduce to (3.25). 0
C. Boundary-Crossing Probabilities for Brownian Motion The representation (3.25) has rather unexpected consequences in the computation of boundary-crossing probabilities for Brownian motion. Let us consider a positive function v(t, x) which is defined and of class C`2 on (0, co) x IR, and satisfies the backward heat equation. Then v admits the representation (3.25) for some F, and differentiating under the integral we see that
- v(t,x)< 0; a
(3.26)
0 < t < oo, x E R
at
and that v(t, ) is convex for each t > 0. In particular, lim,4,0 v(t, 0) exists. We assume that this limit is finite, and, without loss of generality (by scaling, if necessary), that lim v(t, 0) = 1.
(3.27)
ti.o
We also assume that lim v(t, 0) = 0,
(3.28)
r00
lim v(t, x) = co;
(3.29)
0 < t < co,
x-co
lim v(t, x) = 0,
(3.30)
0 < t < co.
It is easily seen that (3.27)-(3.30) are satisfied if and only if F is a probability distribution function with F(0+) = 0. We impose this condition, so that (3.25) becomes (3.31)
v(t, x) =
1
c°
1
exp yx - 2y2 t)dF(y); 0 < t < oo, X E R,
0+
where F(co) = 1, F 0 + ) = 0. This representation shows that v(t, .) is strictly increasing, so for each t > 0 and b > 0 there is a unique number A(t, b) such that (3.32)
v(t, A(t, b)) = b.
It is not hard to verify that the function A(- , b) is continuous and strictly increasing (cf. (3.26)). We may define A(0, b) = lima° A(t, b). We shall show how one can compute the probability that a Brownian path W, starting at the origin, will eventually cross the curve A(.,b). The problem of
4.3. The One-Dimensional Heat Equation
263
computing the probability that a Brownian motion crosses a given, timedependent continuous boundary {/i(t); 0 < t < co} is thereby reduced to finding a solution v to the backward heat equation which also satisfies (3.27)-(3.30) and v(t, 0)) = b; 0 < t < co, for some b > 0. In this generality both problems are quite difficult; our point is that the probabilistic problem can be traded for a partial differential equation problem. We shall provide an explicit solution to both of them when the boundary is linear. Let { W gt; 0 < t < co}, (CI, .x), {Px}. pl be a Brownian family, and define Z, = v(t, Wt);
0 < t < oo.
For 0 < s < t, we have from the Markov property and condition (iv) of Proposition 3.9:
EqZ,1";] = f(14/5) = v(s, Ws) = Z a.s. P °, where f(y) A EY v(t, Wi_s). In other words, {Z Ft; 0 < t < co} is a continuous,
nonnegative martingale on (0, P°). Let {tn} be a sequence of positive numbers with tn.j. 0, and set Z0 = Z. This limit exists, P ° -a.s., and is independent of the particular sequence {t.} chosen; see the proof of Proposition 1.3.14(i). Being "O",1-measurable, Zo must be a.s. constant (Theorem 2.7.17).
3.10 Lemma. The extended process Z A {Z gi; 0 < t < co} is a continuous, nonnegative martingale under P° and satisfies Z0 = 1, 4 = 0, P°-a.s. PROOF. Let {t.} be a sequence of positive numbers with t l O. The sequence {4},D=1 is uniformly integrable (Problem 1.3.11, Remark 1.3.12), so by the Markov property for W, we have for all t > 0:
E°[Z,IF0] = E°Z, = lim E°Zi. = E°Z0 = Z0. n-,co
This establishes that {Z .a.07,; 0 < t < co} is a martingale. Zi exists P°-a.s. (Problem 1.3.16), as does Z0 Since
lima° 4,
it suffices to show that lima° 4 = 1 and lim, Z, = 0 in P °- probability. For every finite c > 0, we shall show that lim sup
(3.33)
1v(t,x) - 11 = 0.
Indeed, for t > 0, 1x1
1
exp - yc.,/ t - - y2 t)dF(y)
J o+
v(t, x)
2
CO
<
exp (yc
1
- -2y2 t)dF (y).
Because ± yc.,/t - y2 t/2 5 c2 /2; V y > 0, the bounded convergence theorem implies that both integrals in (3.34) converge to 1, as ti 0, and (3.33) follows. Thus, for any a > 0, we can find tc, depending on c and a, such that
4. Brownian Motion and Partial Differential Equations
264
l - E < v(t, x) < 1 ± E;
I XI < C-s/t,
0 < t < tc,,
Consequently, for 0 < t < tc, P°[14 - 11> e] = P°[ 1v((, Wt) - 11> E]
P°[1Wt1> c-s/i] = 2[1 - (I)(e)],
where (1)(x) 1=--`
i' x
,1
-.
N/27r
e-2212 dz.
Letting first t 10 and then c cc, we conclude that Z, t 10. A similar argument shows that
1 in probability as
lim sup v(t, x) = 0,
(3.35)
too ixi .c.,./i
and, using (3.35) instead of (3.33), one can also show that Z, -0 in probability
as t - co.
111
It is now a fairly straightforward matter to apply Problem 1.3.28 to the martingale Z and obtain the probability that the Brownian path {W,(w); 0 < t < co} ever crosses the boundary {A(t, b); 0 < t < co}. 3.11 Problem. Suppose that v: (0, cc) x FR - (0, cc) is of class C' '2 and satisfies
(3.12) and (3.27)-(3.30). For fixed b > 0, let A( , b): [0, co) -, R be the continuous function satisfying (3.32). Then, for any s > 0 and Lebesgue-almost every a e R with v(s, a) < b, we have (3.36)
(3.37)
P° [Wt > A(t, b), for some t > sl Ws = a] =
P° [W
v(s, a)
b'
A(t, b), for some t > s]
= 1 -(I)(
A(s,b)) .,/s
+
1 f (A(s,b) (I) u
o+
Is
y Nfi)dF (y),
where F is the probability distribution function in (3.31).
3.12 Example. With it > 0, let v(t, x) = exp(px - p2 t /2), so A(t,b) = fit + y, where 13 = (it/2), y = (1/12)log b. Then F(y) = 11,,,00(y), and so for any s > 0, 13 > 0, y e R, and Lebesgue-almost every a < y + 13s: (3.38)
P° [W, > fit + y, for some t > sI W., = a] = e-"('''' I's),
and for any s > 0, iq > 0, and y e R: (3.39)
P°[W,
fit + y, for some t > s] = 1 -(I) (
+ /3is -s
+
e-211Y(13(
,
.,/s
I3A.
4.3. The One-Dimensional Heat Equation
265
The observation that the time-inverted process Y of Lemma 2.9.4 is a Brownian motion allows one to cast (3.38) with y = 0 into the following formula for the maximum of the so-called "tied-down" Brownian motion or "Brownian bridge": (3.40)
P° [ max W > /3 (:1
]
1,177.
e-2p(p-a)/T
(57.
for T > 0, > 0, a.e. a < 13, and (3.39) into a boundary-crossing probability on the bounded interval [0, T]: (3.41)
P °[W > /3 + yt, for some t e [0, Ti]
= 1 - (1)(7.,j+ 13,) + e-2137,1)(VT
,/ T
.\/ T
> 0,y e R.
3.13 Exercise. Show that P° [W, > ft + y, for some t > 0] = e-211Y, for 13 > 0 and y > 0 (recall Exercise 3.5.9).
D. Mixed Initial/Boundary Value Problems We now discuss briefly the concept of temperatures in a semi-infinite rod and the relation of this concept to Brownian motion absorbed at the origin. Suppose that f: (0, oo) R is a Borel-measurable function satisfying co° e-"21f(x)Idx < oo
(3.42)
for some a > 0. We define (3.43)
0
ui(t, x) -4 Ex[f(Wf)1(ro>t)];
x > 0.
The reflection principle gives us the formula (2.8.9)
Px [ W, e dy, To > t] = p At; x, y)dy A [p(t; x, y) - p(t; x, -y)] dy
for t > 0, x > 0, y > 0, and so o
(3.44)
f(y)p(t; x, y)dy -
ui(t, x) = I 0
j
AP(t; x, AdY,
which gives us a definition for t. valid on the whole strip (0, 1/2a) x R. This representation is of the form 9.4), where the initial datum f satisfies f(y) = -f( -y); y > 0. It is clear then that u, has derivatives of all orders, satisfies the heat equation (3.1), satisfies (3.6) at all continuity points of f, and lim ui (t, 0) = 0; x10
1
0 < t < -2a .
4. Brownian Motion and Partial Differential Equations
266
We may regard u (t, x); 0 < t < (1/2a), x > 0, as the temperature in a semiinfinite rod along the nonnegative x-axis, when the end x = 0 is held at a constant temperature (equal to zero) and the initial temperature at y > 0 is
f(v)
Suppose now that the initial temperature in a semi-infinite rod is identically
zero, but the temperature at the endpoint x = 0 at time t is g(t), where g: (0, 1/2a)
R is bounded and continuous. The Abel transform of g, namely
(3.45)
u2(t, x) -4 Ex [g(t -To)1{,0,}]
=
J
=
.1
r
g(t - r)h(r, x) dr g(s)h(t - s, x)ds;
0 < t < -1 , x > 2a
o
0
with h given by (3.10), is a solution to (3.1) because h is, and h(0, x) = 0 for x > 0. We may rewrite this formula as
u2(t, x) = E° [g(t -7'x)1{,,,,}]; 0 < t <
1
, x > 0,
and then the bounded convergence theorem shows that lim u2(s, x) = g(t); s-t lim u2(t, y) = 0;
tlo
0 < t < -1 , 2a
0 < x < co.
y -.x
We may add ul and u2 to obtain a solution to the problem with initial datum f and time-dependent boundary condition g(t) at x = 0.
3.14 Exercise (Neumann Boundary Condition). Suppose that f: (0, co) is a Borel-measurable function satisfying (3.42), and define u(t, x)
R
1
Exf(11411);
0 < t < - , x > 0. 2a
Show that u is of class C1'2, satisfies (3.1) on (0, 1/2a) x (0, co) and (3.6) at all continuity points of f, as well as lim
0
s_.t uX
u(s, x) = 0;
0
-. 2a 1
x4,0
3.15 Exercise (Finite Rod). Suppose that g, k are bounded, continuous functions from (0, co) into R, and f is a bounded, continuous function from (0, b) into R. We seek a function u which is of class 0.2 on (0, co) x (0, b) and which has a continuous extension to the boundaries {0} x (0, b), (0, co) x {0}, and
4.4. The Formulas of Feynman and Kac
267
(0, co) x {b}, such that 814
102u
at
2 0 x2
on (0, co) x (0, b),
u(0, x) = f(x);
0 < x < b,
u(t, 0) = g(t);
0 < t < oo,
u(t, b) = k(t);
0 < t < co .
Show that the unique bounded solution to this problem is given by the expression (3.46)
u(t, x) = -x F r
tiw)i-{t
TO
+ k(t - Tb)1{Tb
g(t
TO)1{7.0: A TO
To}l;
0 < t < oo , 0 < x < b.
(Hint: Use Proposition 2.8.10 and Formulas (2.8.25), (2.8.26).)
4.4. The Formulas of Feynman and Kac We continue our program of obtaining stochastic representations for solutions of partial differential equations. In the first subsection, we introduce the Feynman-Kac formula, which provides such a representation for the solution of the parabolic equation (4.1)
au
1
+ ku = - Au + g; (t, x) e (0, oo) x Rd 2
subject to the initial condition u(0, x) = f(x);
(4.2)
x e Rd
for suitable functions k: Rd [0, co), g: (0, co) x R4 IR and f: Rd R. In the special case of g = 0, we may define the Laplace transform
z(x) A
f
e'u(t, x)dt;
x e Rd,
and using (4.1), (4.2), integration by parts, and the assumption that lim, e'u(t, x) = 0; a > 0, x e Rd, we may compute formally (4.3)
1 2
-1 f° Cat Au dt = (a + k)z,, -f. 2
0
The stochastic representation for the solution z of the elliptic equation (4.3) is known as the Kac formula; in the second subsection we illustrate its use when d = 1 by computing the distributions of occupation times for Brownian motion. The second subsection may be read independently of the first one.
4. Brownian Motion and Partial Differential Equations
268
Throughout this section, {W, .; 0 s t <
{-Px}xeRd is a d-
dimensional Brownian family.
A. The Multidimensional Formula R, k: Rd 4.1 Definition. Consider continuous functions f : [0, co), and g: [0, T] x Rd R. Suppose that v is a continuous, real-valued function on
[0, T] x Rd, of class C" on [0, T) x Rd (see the explanation preceding Theorem 3.3), and satisfies av
1
--at + kv = Av + g; on [0, T) x Rd,
(4.4)
v(T, x) = f(x);
(4.5)
x e Rd.
Then the function v is said to be a solution of the Cauchy problem for the backward heat equation (4.4) with potential k and Lagrangian g, subject to the terminal condition (4.5). 4.2 Theorem (Feynman (1948), Kac (1949)). Let v be as in Definition 4.1 and assume that max I v(t, x)I + max Ig(t, x)I < Kell x112; 05t5T 0
(4.6)
V x e Rd,
for some constants K > 0 and 0 < a < 1/2Td. Then v admits the stochastic representation T-t (4.7)
v(t,
= EX J.( Wr-i) exP - J
k(147s)ds}
T-:
k(llis) ds}
g(t + 0, W0) exp -
di 05t5T,xERd.
In particular, such a solution is unique.
43 Remark. If g > 0 on [0, T] x Rd, then condition (4.6) may be replaced by (4.8)
max I v(t, x)I < Ke°14112;
V x e Rd.
This leads to the following maximum principle for the Cauchy problem: if the
continuous function v: [0, T] x Rd
R is of class C1'2 on [0, T) x Rd and
satisfies the growth condition (4.8), as well as the differential inequality av
at
1
+ kv > - Av on [0, T) x Rd
-2
with a continuous potential k: Rd v > 0 on [0, T] x Rd.
[0, oo), then v > 0 on {T} x Rd implies
4.4. The Formulas of Feynman and Kac
269
4.4 Remark. If we do not assume the existence of a 0.2 solution to the Cauchy problem (4.4), (4.5), then the function defined by the right-hand side of (4.7)
need not be C1.2. The reader is referred to Friedman (1964), Chapter 1, Friedman (1975), p. 147, or Dynkin (1965), Theorem 13.16, for conditions under which the Cauchy problem of Definition 4.1 admits a solution. This is the case, for example, if k is bounded and uniformly Holder-continuous on compact subsets of Rd, g is continuous on [0, T] x l and Holder-continuous in x uniformly with respect to (t, x) e [0, T] x Rd, f is continuous, and for some constants L and v > 0 we have max I g(t, x)I + I f(x)I _< L(1 + 11x11");
X E Rd.
OS.t
PROOF OF THEOREM 4.2. We obtain from Ito's rule, in conjunction with (4.4):
d[v(t + 0, Wo)exp
= exp
-f
k(W) ds}] d
k(W)ds}[- g(t + 0, W0)d0 + E
0
ax t
v(t + 0, Wo)dWoal
Let S = inf {t 0; II 14/,11 > n,/d}; n > 1. We choose 0 < r < T - t and integrate on [0, r n Se]; the resulting stochastic integrals have expectation zero, so
rAs,,
g(t + 0, Wo)exp {- f k(W)ds} d0
v(t, x) = o
+ Ex [v(t [ + Sn, Ws.) exp
-f
s" k(Ws) ds}
1{sr}]
o
+ Ex[v(t + r, W.) exp
k(W)ds} 1 {sn,r}].
The first term on the right-hand side converges to T-t
Ex J
g(t + 0, Wo) exp
k(VV,)ds} dB
oo and r j T - t, either by monotone convergence (if g > 0) or by dominated convergence (it is bounded in absolute value by f o -`fg(t + 0, W0)Id0, which has finite expectation by virtue of (4.6)). The second term is dominated as n
by
Ex [Iv(t + S, Wsn)11(snr--1)] < Kea"' Px[S < T] < Keadn2 E px [ j=1
max IV! >
0.t5T
d
2Kead"2 E {px[wp) > n]
+ Px[ -wp)
n]
4. Brownian Motion and Partial Differential Equations
270
where we have used (2.6.2). But by (2.9.20), eadn, px[+ W- ) > n]
T eadn2
1
e(n -T- x()))212T
2n n -T
which converges to zero as n oo, because 0 < a < (1/2Td). Again by the dominated convergence theorem, the third term is shown to converge to Ex [v(T, W -i) exP -g-k(Ws)dsl] as n co and rT T - t. The FeynmanKac formula (4.7) follows. 4.5 Corollary. Assume that f: Rd -> R, k: Rd -> [0, oo), and g: [0, oo) x Rd -> R are continuous, and that the continuous function u: [0, co) x Rd -R is of class C''2 on (0, co) x Rd and satisfies (4.1) and (4.2). If for each finite T > 0 there
exist constants K > 0 and 0 < a < (1/2Td) such that max lu(t,x)1 + max Ig(t,x)I < Kea11x112;
OStST
V x e R,
0
then u admits the stochastic representation (4.9)
u(t, x) = Ex[f(140exp j -f t k(Ws)ds}
+ j'i g(t - 0, Wo) exp -
k(Ws)ds} Al
0
t < co, x e Rd.
In the case g = 0 we can think of u(t, x) in (4.1) as the temperature at time t > 0 at the point x e Rd of a medium which is not a perfect heat conductor,
but instead dissipates heat locally at the rate k (heat flow with cooling). The Feynman-Kac formula (4.9) suggests that this situation is equivalent to Brownian motion with annihilation (killing) of particles at the same rate k: the probability that the particle survives up to time t, conditional on the path { Ws; 0 < s < t }, is then exp{ -f o k(Ws) ds}
4.6 Exercise. Consider the Cauchy problem for the "quasilinear" parabolic equation (4.10)
av
= AV
1
ilVVII2
V(0, x) = 0;
(4.11)
k;
in (0, co) x Rd,
x e Rd
(linear in (0 V/0t) and the Laplacian AV, nonlinear in the gradient VV), where k: [0, co) is a continuous function. Show that the only solution V: [0, co) x fed -> R which is continuous on its domain, of class CI.2 on (0, 0o) x Rd, and satisfies the quadratic growth condition for every T > 0:
- V(t, x)
C ± a Ilx112 ;
(t, x)e [0,
x Rd
4.4. The Formulas of Fe; nman and Kac
271
where T > 0 is arbitrary and 0 < a < 1/2Td, is given by (4.12)
V(t, x) = - log Elexp j
-f k(147,)dsfl;
0 < t < co, x e Rd.
0
We turn now our attention to equation (4.3). The discussion at the beginning of this section and equation (4.9) suggest that any solution z to the equation
(a + k)z = 2-Az +
(4.13)
f;
on Rd
should be represented as
z(x) = Ex
(4.14)
f
f (WE) exp {
- at
-f k(Ws)ds} dt.
4.7 Exercise. Let f: Rd -R and k: Rd - [0, oo) be continuous, with (4.15)
Ex
f
.0
if(NI exp {-
0
at -f k(Ws)d.s} dt < co; VxeRd, 0
for some constant a > 0. Suppose that 1//: Rd -+ R is a solution of class C2 to (4.13), and let z be defined by (4.14). If ti/ is bounded, then tai = z; if tk is nonnegative, then ti/ > (Hint: Use Problem 2.25).
B. The One-Dimensional Formula In the one-dimensional case, the stochastic representation (4.14) has the remarkable feature that it defines a function of class C2 when and k are continuous. Contrast this, for example, to Remark 4.4. We prove here a slightly more general result.
f
4.8 Definition. A Borel-measurable function f: R -R is called piecewisecontinuous if it admits left- and right-hand limits everywhere on R and it has only finitely many points of discontinuity in every bounded interval. We denote by Df the set of discontinuity points of f. A continuous function R R is called piecewise Ci, j > 1, if its derivatives f"), 1 < i < j - 1 are
f
continuous, and the derivative f4-1) is piecewise-continuous.
4.9 Theorem (Kac (1951)). Let f : R -0 R and k: R - [0, co) be piecewisecontinuous functions with (4.16)
J
W
I
f(x + y)le-lYkrix dy < co;
V x E R,
for some fixed constant a > 0. Then the function z defined by (4.14) is piecewise C2 and satisfies
272
4. Brownian Motion and Partial Differential Equations 1
(a + k)z = -2z" + f;
(4.17)
on 01\(Df u Dk).
4.10 Remark. The Laplace transform computation T00
e' /2nt e-42/2` dt = 1
1
e-I41; a>
,
o
0,
eR
/ 2a enables us to replace (4.16) by the equivalent condition OD
(4.16)'
f e-alf1WtHdt < Vco, x e R.
Ex
PROOF OF THEOREM 4.9. For piecewise-continuous functions g which satisfy condition (4.16), we introduce the resolvent operator G. given by OD
e.' a g(W)dt =
(G.g)(x) A Ex
1 Jo
=
1
-03
x
1
03
+I
e(Y-x)
.,./Tot
' g(y) dy
e-Iv -xl
I
.,./2a
-cc
g(y) dy1;
e(x-Y
x e R.
x
Differentiating, we obtain (Gag)'(x) = (4.18)
J
e(x-v1 Nr2xg(y)dy - J
(Gag)" (x) = - 2g(x) + 2a(Gag)(x);
x
(y)dy;
e(Y-x1
xe
x e R,
Dg.
It will be shown later that
G7(kz) = Gaf -z
(4.19)
and G2(1kz1)(x) < co;
(4.20)
V x e R.
If we then write (4.18) successively with g = f and g = kz and subtract, we obtain the desired equation (4.17) for x e R\(Df u DkZ), thanks to (4.19). One can easily check via the dominated convergence theorem that z is continuous, so DkZ g Dk. Integration of (4.17) yields the continuity of z'. In order to verify (4.19), we start with the observation
0<
k(Ws) exp - J k(WJdul ds = 1 - exp - J
k(W,,) du} < 1;
and so by Fubini's theorem and the Markov property:
(Gaf - z)(x) = Ex
e't(1 - e-ftok(ws)ds)f(W,)dt Jo
=
f e-"f(147,) o
fo k(Ws)exp
-f k(W) du} dsdt
t > 0,
4.4. The Formulas of Feynman and Kac
exp -
=
Ex [k(Ws)
=
Cc's Ex[ t k(W.,)
ca
k(VV.)
-f
du} f(W,) dtids
k(Ws,..) du}
f(Ws,)dtids
e'k(Ws) Ex[f exp - at - J k(Ws,) du} f(W54.,)dt
= Ex
= Ex
exp
273
f
x
k(Ws)z(WS)ds = (G(kz))(x);
ds
fR,
which gives us (4.19). We may replace fin (4.14) by If I to obtain a nonnegative
function > Izi, and just as earlier we have (G,z(k1))(x) = (G(I f I) - i)(x) < oo;
G,r(ikz1)(x)
XE
Relation (4.20) follows.
Here are some applications of Theorem 4.9. 4.11 Proposition (P. Levy's Arc-Sine Law for the Occupation Time of (0, co)). Let 1"+(t)
(4.21)
rol(0,03)(W.$)ds. Then
P°[F,(t) lc 0] =
el'
fo
ds
r,/s(1 - s)
=
2
< < t.
arc sin
PROOF. For a > 0, fl > 0 the function CO
z(x) = Ex J
f
exp - at -
1(0,03)(Ws)
ds) dt
(with potential k = # 1(0,,) and Lagrangian f = 1) satisfies, according to Theorem 4.9, the equation acz(x) = +z "(x) - #2(x) + 1;
x > 0,
az(x) = Zz "(x) + 1;
x < 0,
and the conditions
z(0 +) = z(0 -); z'(0 +) = z'(0 -). The unique bounded solution to the preceding equation has the form
+ 1
Bex/ i-`" + -; a
1
a+
x>0 "
x < 0.
The continuity of z( ) and z'() at x = 0 allows us to solve for A = (.\ / + # - .1,-2)/(cc SO, so
274
4. Brownian Motion and Partial Differential Equations fco
e'
E° e-"i)dt = z(0) =
o
1
a > 0,
;
,/a(a + /3)
f > 0.
We have the related computation
1o
Ca'
CI" ' CI' " e-at dt dB d0 dt = 1o ik/0(t - 0) fo 7r/B 8 ./t -0 c°e-(24-4" ..1°D e-.
1
=
It
/13
o
ds d0
o
1
since e-7`
(4.22)
dt =
y; y > 0.
Jo
The uniqueness of Laplace transforms implies (4.21).
4.12 Proposition (Occupation Time of (0, co) until First Hitting b > 0). For > 0, > 0, we have exp [ - 131-_,(Tb)] A E °exp
(4.23)
[-
Tb
1(0,.)(Ws) (Li
1
cosh PROOF. With 1-6(t) A Po 1(b,c0)(Ws)ds, positive numbers cc, 13, y and
z(x)
EX J
1(0,00)(W,)exp(- at - $F (t) - yl-b(t))dt,
we have Z(0) = E°
exp(- at - #1-+(t))dr+(t)
+
exp( -ca - 13r+(t) - yrb(t))dr+(t) Ty
Since Tb(t) > 0 a.s. on { Tb < t} (Problem 2.7.19), we have Tb
lim z(0) = E° J
exp(- at - 131-+(t)) dr+(t)
rt co ('o
(4.24)
lim lim z(0) = E° J b exp(-I3F+(t))dr+(t) rt cc) 1
= -[1 - E° exp(- #1-+(Tb))]
4.5. Solutions to Selected Problems
275
According to Theorem 4.9, the function z() is piecewise C2 on R and satisfies the equation (with a = a + p3):
x < 0,
az(x) = z" (x);
crz(x) = lz"(x) + 1; 0 < x < b,
x > b.
(a- + y)z(x) = lz"(x) + 1;
The unique bounded solution is of the form
x < 0,
Aex12;;
z(x) =
Bex zQ + Ce-xN/ia +
+
0 < x < b,
a
1
a
x > b.
;
y
Matching the values of z() and z'() across the points x = 0 and x = b, we obtain the values of the four constants A, B, C, and D. In particular, z(0) = A is given by
sinh b.,/2a
a
2
-1
a + y [-cosh b
,/2a
L
(,./2ot + \/2(a + y)) cosh 62a + (.12ot
1
,/2(a + y) ± Y) +
sinh bN/2a
whence 2
lim z(0) =
yt
(cosh b,./2(ot +
- 1)
N/2(oz + 13) cosh b.,/2(a + /3) + ,./2a sinh b.\/2(« + fl)
and lim lim z(0) = a4,0 ytco
1
1
/3
cosh b
-[1
The result (4.23) now follows from (4.24).
4.5. Solutions to Selected Problems 2.16. Assume without loss of generality that a = O. Choose 0 < r < fl b II A 1. It suffices to show that a is regular for Br c D (Remark 2.11). But there is a simple arc C in
Old\D connecting a = 0 to 6, and in 13,\C, a single-valued, analytic branch of log(x, + ix2) can be defined because winding about the origin is not possible. Regularity of a = 0 is an immediate consequence of Example 2.14 and Proposition 2.15.
276
4. Brownian Motion and Partial Differential Equations
2.22. Every boundary point of D satisfies Zaremba's cone condition, and so is regular. It remains only to evaluate (2.12). Whenever Y and Z are independent random variables taking values in measurable spaces (G, W) and (H, le), respectively, and f: G x H R is bounded and measurable, then
Ef(Y,Z) =
fHfG f(y,z)P[Y dy]P[Z dz].
We apply this identity to the independent random variables TD = inf {t
0;
Wt(d) = 01 and {(W,"), W(d-1)); 0 < t < oo 1, the latter taking values in C[0, cold'. This results in the evaluation (see (2.8.5)) cc
E'fiwt,,) =
J la
ad-t
/(Y1,
, Yd-1, 0)P[(14/t(1),
,
Wt(d-1))E (dY I,
,dYd-i)] P"[TDE dt]
ao
xdfol
- x112]dt t(2rztr2exP
2t
dy,
and it remains only to verify that 2,42 r (do)
Az
1
(5.1)
ed+2)/2 exP
[
A > 0.
2t dt =
Ad
For d = 1, equation (5.1) follows from Remark 2.8.3. For d = 2, (5.1) can be verified by direct integration. The cases of d > 3 can be reduced to one of these two cases by successive application of the integration by parts identity
r" Jo
e-''212` dt =
r1+
2(ot
1)
"1
dt, a > 1.
ta e
Az
1
2.25. Consider an increasing sequence {D};,°_, of open sets with /5 c D; V n 1 and U,T=, D = D, so that the stopping times zn = inf {t > 0; IV, D} satisfy limn-. tn = TD, a.s. Px. It is seen from ItO's rule that
tnr Mr)
0
g(Ws)ds,
is a P"-martingale for every n > 1, X ED; also, both I Mg(w)1, I M,")(0)1 are bounded
above by max lul + (t A TD(co)) max WI, for Px-a.e. w E
By letting n cc and using the bounded convergence theorem, we obtain the martingale property of the process M; its uniform integrability will follow as soon as we establish (5.2)
E'rD < cc, V x ED.
Then the limiting random variable Mm = martingale M is identified as
Mt of the uniformly integrable
4.5. Solutions to Selected Problems
277
M = u(W,D) +
a.s. P' (Problem 1.3.20)
g(141,)dt,
and the identity E"Mo = PA/co yields the representation (2.27).
As for (5.2), it only has to be verified for D = B,. But then the function v(x) = (r2 - 1142)/d of (2.28) satisfies (1/2)Av = -1 in B, and v = 0 on OB,, and by what we have already shown, {v(Wt,,,) + (t A TO, Ft; 0 t < op) is a P"-martingale. So v(x) = Eqv(W,.,) + (t -ra)] ET A TB), and upon letting t
co we obtain Ex TB
v(x) < GC .
3.1. (Copson (1975)): Fix fi > 0, E > 0, 0 < to < t, s (1/2(a + E)), and set B = {(t, x); to < t < tI,Ixj < fi}. For (t, JOE B, ye R, we have ay
(x
Y)2
2t
1
ay2 2t- -, (Ix1 - ly1)2 aye --2ty2 + - IxIlyI , t, 1
1
E
R2
#
E
Y2 - 2- Y2 + -t,IYI 2
+
2
2Et?
For any nonnegative integers n and m, there is a constant C(n, m) such that on+m
Ot"Ox 'n
p(t; x, y)
(x - y)2} C(n, m)(1 + 1Y12"-"") exP
2t
;
(t, x)e B, ye
and so an+m
(5.3)
et"Ox'"
5- IBA
p(t; x, y)
m)(1 + ly12"-"")expl-(a + -2)3/2 + -2SE2t1}
D(n,m)If(y)le-aY2; (t, x)e B, y E R,
where D(n, m) is a constant independent of t, x, and y. It follows from (3.3) that thA integral in (3.5) converges uniformly for (t, x) e B, and is thus a continuous function of (t, x) on (0,1/2a) x We prove (3.5) for the case n = 0, m = 1; the general case is easily established by induction. For (t, x) E B, (t, x + h) e B, we have 1
-[u(t, x + h) - u(t, x)] = h
J
,
-[p(t; x + h, y) - p(t, x, yflf(y)dy h
a
=
-pct; oho, 31, YV(Y)dy, Ox
where, according to the mean-value theorem, O(t, y) lies between x and x + h. We now let h 0, using the bound (5.3) and the dominated convergence theorem, to obtain (3.5).
3.2. (Widder (1944)): We suppose f is continuous at xo and assume without loss of generality that f(xo) = 0. For each E > 0, there exists (5 > 0 such that If(y)I < E for Iy - xo 15 S. We have for X E [x0 - (5/2), -X0
(6/2)],
4. Brownian Motion and Partial Differential Equations
278
(5.4)
I u(t, x)I
f
so+6
so -6
I fly)1P(t; x, y)dy
I f( APO; -; A dy +
so-6
fco
+ ..+6
I f(Y)IP(t; x, Ady.
The middle integral is bounded above by e; we show that the other two converge to zero, as t 10. For the third integral, we have the upper bound
(5.5)
r
(y - xo by1
z
e'Y If(y)1 exp ay 2
2t
27cJt J so+6
dy.
For t sufficiently small, exp[ay2 - (y - xo - 6/2)2 /2t] is a decreasing function of y for y xo + 6 (it has its maximum at y = (x0 + 6/2)/(1 - 2at)). Therefore, the expression in (5.5) is bounded above by 6
p t; 0, -2) exp[a(x,
.,
+ 6)2] j.
e "2 If(Al dY
xo+ 6
which approaches zero as t10. The first integral in (5.4) is treated similarly.
4.6. Notes Section 4.2: The Dirichlet problem has a long and venerable history (see, e.g., Poincare (1899) and Kellogg (1929)). Zaremba (1911) was the first to observe that the problem was not always solvable, citing the example of a punctured region. Lebesgue (1924) subsequently pointed out that in three or more dimensions, if D has a sufficiently sharp, inward-pointing cusp, then the problem can fail to have a solution (our Example 2.17). Poincare (1899) used barriers to show that if every point on ap lies on the surface of a sphere which does not otherwise intersect D, then the Dirichlet problem can be solved in D. Zaremba (1909) replaced the sphere in Poincare's sufficient condition by a cone. Wiener (1924b) has given a necessary and sufficient condition involving the capacity of a set.
The beautiful connection between the Dirichlet problem and Brownian motion was made by Kakutani (1944a, b), and his pioneering work laid the foundation for the probabilistic exposition we have given here. Hunt (1957/1958) studied the links between potential theory and a large class of transient Markov processes. These matters are explored in greater depth in Ito & McKean (1974), Sections 7.10-7.12; Port & Stone (1978); and Doob (1984).
Section 4.3: The representation (3.4) for the solution of the heat equation is usually attributed to Poisson (1835, p. 140), although it was known to both Fourier (1822, p. 454) and Laplace (1809, p. 241). The heat equation for the semi-infinite rod was studied by Widder (1953), who established uniqueness
4.6. Notes
279
and representation results similar to Theorems 3.3 and 3.6. Hartman & Wintner (1950) considered the rod of finite length. For more examples and further information on the subject matter of Subsection C, including applications to the theory of statistical tests of power one, the reader is referred to Robbins & Siegmund (1973), Novikov (1981), and the references therein. Section 4.4: Theorem 4.2 was first established by M. Kac (1949) for d = 1; his work was influenced by the derivation of the Schrodinger equation achieved by R. P. Feynman in his doctoral dissertation. Kac's results were strengthened and extended to the multidimensional case by M. Rosenblatt (1951), who also provided Holder continuity conditions on the potential k in order to guarantee a C1.2 solution. Proposition 4.12 is taken from ItO & McKean (1974).
k(x) = co; then the
Let k: l -4 [0, co) be continuous and satisfy eigenvalue problem (k(x) - A)tli(x) = - tr(x); 2
xe
with e L2(R), has a discrete spectrum Al < A2 < ... and corresponding eigenfunctions ti/J;17_1 g L2(R). Kac (1951) derived the stochastic representation 1
(6.1)
k(14(,) ds}]
- lirn - log Ex[exp
Al
,_) t
for the principal eigenvalue, by combining the Feynman-Kac expression
u(t,x) = Ex[exp { -Jo k(141s)ds}] for the solution of the Cauchy problem Ou
ot
(6.2)
1
+ k u = -Au;
(0, c3o) x
u(0, x)
xel
2
1;
(Corollary 4.5), with the formal eigenfunction expansion
u(t,x)
00
E cie-Aittp,(x) .i=1
for the solution of (6.2). Recall also Exercise 4.6, and see Karatzas (1980) for a control-theoretic interpretation (and derivation) of this result. Sweeping generalizations of (6.1), as well as an explanation of its connection with the classical variational expression (6.3)
A, =
inf
k(x)tly 2 (X) dx + 2
J.
(
/(x))2dx } ,
_co
if°, 14,2(x) dx=1.
are provided in the context of the theory of large deviations of Donsker &
280
4. Brownian Motion and Partial Differential Equations
Varadhan (1975), (1976). This theory constitutes an important recent development in probability theory, and is overviewed succinctly in the monographs by Stroock (1984) and Varadhan (1984).
The reader interested in the relations of the results in this section with quantum physics is referred to Simon (1979).
Alternative approaches to the arc-sine law for fi.(t) can be found in Exercise 6.3.8 and Remark 6.3.12.
CHAPTER 5
Stochastic Differential Equations
5.1. Introduction We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator. In order to fix ideas, let us consider a d-dimensional Markov family X =
{X A; 0 < t < co}, (52, ,F), Px 1 xe Rd, and assume that X has continuous paths. We suppose, further, that the relation (1.1)
1
- [ef(X,) -f (x)] = (cif )(x); dxePd rlo t lim
holds for every f in a suitable subclass of the space C2(Rd) of real-valued, twice continuously differentiable functions on Rd; the operator s/f in (1.1) is given by I
(1.2)
(sif )(x)
d
d
.E1 kE1 a ik(x)
a2f( x)
d
ofi x)
+
2
for suitable Borel-measurable functions aik: Rd R, 1 5 i, k S d. The lefthand side of (1.1) is the infinitesimal generator of the Markov family, applied to the test function f. On the other hand, the operator in (1.2) is called the secondorder differential operator associated with the drift vector b = (1)1, . , bd) and the diffusion matrix a = {aik}
282
5. Stochastic Differential Equations
y e Rd. Assuming that (1.1) holds for these test functions, we obtain (1.3)
Ex[Xls) - xi] = tb,(x) + o(t)
(1.4)
Ex [(XiI) - xj)()Ok) - xk)] = ta,k(x) + o(t)
as t J, 0, for 1 < i, k < d. In other words, the drift vector b(x) measures locally the mean velocity of the random motion modeled by X, and a(x) approximates
the rate of change in the covariance matrix of the vector X, - x, for small values of t > 0. The monograph by Nelson (1967) can be consulted for a detailed study of the kinematics and dynamics of such random motions.
1.1 Definition. Let X = IX ,57;,; 0 < t < co }, dimensional Markov family, such that
(12, g;),
{1'1. Esd
be a d-
(i) X has continuous sample paths; (ii) relation (1.1) holds for every f e C2(Rd) which is bounded and has bounded first- and second-order derivatives; (iii) relations (1.3), (1.4) hold for every x e Rd; and
(iv) the tenets (a)-(d) of Definition 2.6.3 are satisfied, but only for stopping times S.
Then X is called a (Kolmogorov-Feller) diffusion process.
There are several approaches to the study of diffusions, ranging from the purely analytical to the purely probabilistic. In order to illustrate the traditional analytical approach, let us suppose that the Markov family of Definition 1.1 has a transition probability density function Px [X, e dy] = F (t; x, y)dy;
(1.5)
t > 0.
Vxe
Various heuristic arguments, with (1.1) as their starting point, can then be employed to suggest that r(t; x, y) should satisfy the forward Kolmogorov equation, for every fixed x e Rd: (1.6)
0
Ot
f(t; x, y) = ..il*F(t; x, y);
(t, y)E(0, 00) x Rd,
and the backward Kolmogorov equation, for every fixed y e Rd: (1.7)
- I-(t; x, y) = sir(t; x, y); 0
at
(t, x) e (0, co) x Rd.
The operator d* in (1.6) is given by (1.8)
(.91*.f)(Y)
i=i k=1
uy (4, [aik(Y)f(Y)]
1
=1
Ebt(Y)f(Y)1,
the formal adjoint of .4 in (1.2), provided of course that the coefficients k, ask
possess the smoothness requisite in (1.8). The early work of Kolmogorov (1931) and Feller (1936) used tools from the theory of partial differential
5.1. Introduction
283
equations to establish, under suitable and rather restrictive conditions, the existence of a solution C(t; x,y) to (1.6), (1.7). Existence of a continuous Markov process X satisfying (1.5) can then be shown via the consistency Theorem 2.2.2 and the entsov-Kolmogorov Theorem 2.2.8, very much in the spirit of our approach in Section 2.2. A modern account of this methodology is contained in Chapters 2 and 3 of Stroock & Varadhan (1979).
The methodology of stochastic differential equations was suggested by P. Levy as an "alternative," probabilistic approach to diffusions and was carried out in a masterly way by K. ItO (1942a, 1946, 1951). Suppose that we have a continuous, adapted d-dimensional process X = {X 3,7,; 0 < t < cc) which satisfies, for every x e Rd, the stochastic integral equation (1.9)
AT)
= xi +
t
r
't
o
J =1
0
f bi(Xs)ds + E
on a probability space KZ
0-1;(xjawch; 0 < t < co, I < i < d
Px), where W = { W ..F,; 0 < t < col is a
Brownian motion in 01' and the coefficients 6,, au: Rd -> R; 1 < i < d, 1 < j < r
are Borel-measurable. Then it is reasonable to expect that, under certain conditions, (1.1)-(1.4) will indeed be valid, with (1.10)
au,(x)
Er aii(90"ki(X). i=1
We leave the verification of this fact as an exercise for the reader. 1.2 Problem. Assume that the coefficients bi, au are bounded and continuous, and the Or-valued process X satisfies (1.9). Show that (1.3), (1.4) hold for every x e Or, and that (1.1) holds for every f e C2(Rd) which is bounded and has bounded first- and second-order derivatives.
Ito's theory is developed in Section 2 under the rubric of strong solutions. A strong solution of (1.9) is constructed on a given probability space, with respect to a given filtration and a given Brownian motion W. In Section 3 we take up the idea of weak solutions, a notion in which the probability space, the filtration, and the driving Brownian motion are part of the solution rather than the statement of the problem. The reformulation of a stochastic differential
equation as a martingale problem is presented in Section 4. The solution of this problem is equivalent to constructing a weak solution. Employing martingale methods, we establish a version of the strong Markov propertycorresponding to (iv) of Definition 1.1-for these solutions; they thereby earn the right to be called diffusions.
The stochastic differential equation approach to diffusions provides a powerful methodology and the useful representation (1.9) for a very large class of such processes. Indeed, the only important strong Markov processes with continuous sample paths which are not directly included in such a development are those which exhibit "anomalous" boundary behavior (e.g., reflection, absorption, or killing on a boundary).
284
5. Stochastic Differential Equations
Certain aspects of the one-dimensional case are discussed at some length in Section 5; a state-space transformation leads from the general equation
to one without drift, and the latter is studied by the method of random time-change. The notion and properties of local time from Sections 3.6, 3.7 play an important role here, as do the new concepts of scale function, speed measure, and explosions. Section 6 studies linear equations; Section 7 takes up the connections with partial differential equations, in the spirit of Chapter 4 but not in the same detail. We devote Section 8 to applications of stochastic calculus and differential equations in mathematical economics. The related option pricing and consumption/investment problems are discussed in some detail, providing concrete illustrations of the power and usefulness of our methodology. In particular, the second of these problems echoes the more general themes of stochastic control theory. The field of stochastic differential equations is now vast, both in theory and in applications; we attempt in the notes (Section 10) a brief survey, but we abandon any claim to completeness.
5.2. Strong Solutions In this section we introduce the concept of a stochastic differential equation with respect to Brownian motion and its solution in the so-called strong sense. We discuss the questions of existence and uniqueness of such solutions, as well as some of their elementary properties. Let us start with Borel-measurable functions Mt, x), x); 1 < i < d, 1 < j < r, from [0, co) x Rd into R, and define the (d x 1) drift vector b(t, x) = {13,(t, x)} and the (d x r) dispersion matrix o-(t, x) = {6;;(t, x) }, The intent is to assign a meaning to the stochastic differential equation (2.1)
sjSr
dX, = b(t, X ,) dt + o(t, X,)dW
written componentwise as (2.1)'
= bat, X,)dt + E
xt) dwto;
1 < i < d,
.i=i
where W = {W; 0 < t < oo } is an r-dimensional Brownian motion and X = {X,; 0 < t < co} is a suitable stochastic process with continuous sample paths and values in Rd, the "solution" of the equation. The drift vector b(t, x) and the dispersion matrix o-(t, x) are the coefficients of this equation; the (d x d) matrix a(t, x) A x)o-T(t, x) with elements (2.2)
aik(t, x) A E 0-,;(t,x)o-kp,x); 1=1
will be called the diffusion matrix.
1 < i, k < d
5.2. Strong Solutions
285
A. Definitions In order to develop the concept of strong solution, we choose a probability space (0, P) as well as an r-dimensional Brownian motion W = .317,w; 0 < t < co} on it. We assume also that this space is rich enough to accommodate a random vector taking values in Rd, independent of 'cow, and with given distribution
ell; re ARd)-
u(r) =
We consider the left-continuous filtration
a (0 v
= cr(, Ws; 0
s
t);
0 < t < op,
as well as the collection of null sets P-4 {Ns O.; 3 G e W., with N
G and P(G)= 0},
and create the augmented filtration a (g, u .41),
(2.3)
0 < t < oo;
(u t
0
by analogy with the construction of Definition 2.7.2. Obviously, { W A; o < t < co} is an r-dimensional Brownian motion, and then so is { W
0 < t < co} (cf. Theorem 2.7.9). It follows also, just as in the proof of Proposition 2.7.7, that the filtration {",} satisfies the usual conditions. 2.1 Definition. A strong solution of the stochastic differential equation (2.1), on the given probability space (S2, F, P) and with respect to the fixed Brownian motion W and initial condition is a process X = {X,; 0 .5 t < oo} with continuous sample paths and with the following properties: (i) X is adapted to the filtration {..F,} of (2.3),
(ii) P[X0 = ] = 1, Xs)1 + Xs)} ds < co] = 1 holds for every 1 < i < d, r and 0 t < co, and (iv) the integral version of (2.1)
(iii) P[Po 1
(2.4)
j
X, = X0 +
J b(s, X s) ds +
ft o-(s, Xs) dWs;
0
,
a
or equivalently, (2.4)'
)01) = Xg) +
J
bi(s, X 5) ds + E I 0-,;(s, X s)dWP); J=1 Jo
0 < t < co, 1 < i
d,
holds almost surely.
2.2 Remark. The crucial requirement of this definition is captured in condition (i); it corresponds to our intuitive understanding of X as the "output"
5. Stochastic Differential Equations
286
of a dynamical system described by the pair of coefficients (b, Q), whose "input"
is W and which is also fed by the initial datum
1
input W
b, a
X output
The principle of causality for dynamical systems requires that the output X, at time t depend only on c and the values of the input Ws; 0 < s < t} up to that time. This principle finds its mathematical expression in (i). Furthermore, when both c and { W; 0 < t < ool are given, their specification should determine the output {Xt; 0 < t < oo} in an unambiguous way. We are thus led to expect the following form of uniqueness. 2.3 Definition. Let the drift vector b(t, x) and dispersion matrix °It, x) be given.
Suppose that, whenever W is an r-dimensional Brownian motion on some P), c is an independent, d-dimensional random vector, {A} is given (0, by (2.3), and X, 31 are two strong solutions of (2.1) relative to W with initial condition then P[X, = ; 0 < t < oo] = 1. Under these conditions, we say that strong uniqueness holds for the pair (b, Q). We sometimes abuse the terminology by saying that strong uniqueness holds for equation (2.1), even though the condition of strong uniqueness requires us
to consider every r-dimensional Brownian motion, not just a particular one. 2.4 Example. Consider the one-dimensional equation dX, = b(t, X,) dt + dW, where b: [0, co) x IR
E is bounded, Borel-measurable, and nonincreasing in
the space variable; i.e., b(t, x) < b(t, y) for all 0 < t < oo, -oo < y <x < oo. For this equation, strong uniqueness holds. Indeed, for any two processes X(1), X(2) satisfying P-a.s.
Xi) = X0 + 1 t b(s, XV) ds + W; 0 < t < oo and i = 1, 2, o
we may define the continuous process A, = XP) - XP) and observe that
,6, = 2 1 t (X11) - V))[b(s, XP)) - b(s, X!2))] ds < 0; 0 < t < oo, a.s. P. o
B. The ItO Theory If the dispersion matrix °(t, x) is identically equal to zero, (2.4) reduces to the ordinary (nonstochastic, except possibly in the initial condition) integral
5.2. Strong Solutions
287
equation
X, = X, +
(2.5)
ft b(s,
In the theory for such equations (e.g., Hale (1969), Theorem 1.5.3), it is customary to impose the assumption that the vector field b(t, x) satisfies a local Lipschitz condition in the space variable x and is bounded on compact subsets of [0, op) x Rd. These conditions ensure that for sufficiently small t > 0, the Picard-Lindelof iterations (2.6)
Xr)
Xo;
X:"+" = X0 +
b(s, Xl"))ds,
n > 0,
0
converge to a solution of (2.5), and that this solution is unique. In the absence
of such conditions the equation might fail to be solvable or might have a continuum of solutions. For instance, the one-dimensional equation Xt = f t !Xs!' ds
(2.7)
has only one solution for a > 1, namely Xt -.E. 0; however, for 0 < a < 1, all functions of the form
0 < t < s, s < t < oo,
;
with # = 1/(1 - a) and arbitrary 0 < s < co, solve (2.7). It seems then reasonable to attempt developing a theory for stochastic differential equations by imposing Lipschitz-type conditions, and investigating what kind of existence and/or uniqueness results one can obtain this way. Such a program was first carried out by K. ItO (1942a, 1946). 25 Theorem. Suppose that the coefficients b(t, x), o-(t, x) are locally Lipschitzcontinuous in the space variable; i.e., for every integer n 1 there exists a
constant K > 0 such that for every t (2.8)
b(t,
- Mt,
0, 114
+ 110'(t, x) -
n and Ilyll
n:
Y)11 < KnIlx - .3111.
Then strong uniqueness holds for equation (2.1).
2.6 Remark on Notation. For every (d x r) matrix a, we write d
(2.9)
r
116112 -4 E E 6. i =1 j =1
Before proceeding with the proof, let us recall the useful Gronwall inequality.
2.7 Problem. Suppose that the continuous function g(t) satisfies
288
5. Stochastic Differential Equations
0 < g(t) < a(t) + # J r g(s) ds; 0 < t < T,
(2.10)
0
with fl > 0 and a: [0, T]
integrable. Then
f
g(t) < a(t) + 13
(2.11)
a(s)e"-s) ds; 0 < t < T
0
PROOF OF THEOREM 2.5. Let us suppose that X and 2 are both strong solutions,
defined for all t > 0, of (2.1) relative to the same Brownian motion W and the same initial condition on some (SI, P). We define the stopping times rn = inf ft > 0; IX, II > n} for n > 1, as well as their tilded counterparts, and
we set S T A in. Clearly lim, S = co, a.s. P, and t A Su
Xt A S
Xt A S
b(u, gu)} du
{NU, Xu)
+
f
tAsu
Using the vector inequality Ilv, + + vkIIZ k2(Ilvi 112 + 1114112), the Holder inequality for Lebesgue integrals, the basic property (3.2.27) of stochastic integrals, and (2.8), we may write for 0 < t E 11X
A
- jet A sull2
[f
T:
tAS,,
Ilb(u, X) - b(u, gu)II du12
o
r
[E f
+ 4E
tAS
Xu) -
u)) d W utn 12
o
i=1 fI A
Ilb(u, Xu) - b(u, g)112 du
4tE
0 t A Su
+ 4E .1 4(T + 1)K,;
Xu) - a(u, 5tu)Il2 du
ft E X
uA
- luAsull2 du.
We now apply Problem 2.7 with g(t) A E II X t A s, 21A 5112 to conclude that {X, A S; 0 < t < 00 } and {X n 5; 0 < t < co} are modifications of one another, oo, we see that the same is true for and thus are indistinguishable. Letting n t < co }. {X,; 0 _< t < co } and {)-1,; 0
2.8 Remark. It is worth noting that even for ordinary differential equations, a local Lipschitz condition is not sufficient to guarantee global existence of a solution. For example, the unique (because of Theorem 2.5) solution to the equation
5.2. Strong Solutions
289
X, = 1 + f t X1 ds o
is X, = 1/(1 - t), which "explodes" as t T 1. We thus impose stronger conditions in order to obtain an existence result.
2.9 Theorem. Suppose that the coefficients b(t, x), a(t, x) satisfy the global Lipschitz and linear growth conditions 116(t, x) - b(t, y)II + lio(t, x) - alt, AII Lc. K 11x - yil,
(2.12)
(2.13)
ilb(t, x)O2 + ki(t, X)Il 2 < K2(1 + 0x112),
for every 0 < t < oo, x e Rd, y e Rd, where K is a positive constant. On some probability space (0, ..F, P), let be an Rd-valued random vector, independent of the r-dimensional Brownian motion W = {Hit, ,F,w ; 0 < t < co}, and with finite second moment: (2.14)
Ell92 < 00.
Let {A} be as in (2.3). Then there exists a continuous, adapted process X = {X .57;,; 0 < t < col which is a strong solution of equation (2.1) relative to W, with initial condition . Moreover, this process is square-integrable: for every T > 0, there exists a constant C, depending only on K and T, such that
E Vie
(2.15)
C(1 + E g 112 )ect;
0
t
T
The idea of the proof is to mimic the deterministic situation and to construct recursively, by analogy with (2.6), a sequence of successive approximations by setting Xr --- and
Xr1) A-
(2.16)
+
f
t
t
b(s, XP)ds +
o
5
o(s, XP)dWs; 0 < t < co,
o
for k > 0. These processes are obviously continuous and adapted to the filtration {,,}. The hope is that the sequence {X(k)}1°_, will converge to a solution of equation (2.1). Let us start with the observation which will ultimately lead to (2.15).
2.10 Problem. For every T > 0, there exists a positive constant C depending only on K and T, such that for the iterations in (2.16) we have E Il V)II2
(2.17)
C (1 + E g 112)ect; OT,k 0.
PROOF' OF THEOREM 2.9. We have Xr1) - V) = B, + M, from (2.16), where B, -4
t
o
{b(s V)) - b(s, Xlk-1))1 ds,
M, A 5 z fa(s, X!k)) - Os, Xlk-1))1 dWs. o
Thanks to the inequalities (2.13) and (2.17), the process {M, = (MP ), ... , Mr),
5. Stochastic Differential Equations
290
..97t; 0 < t < co
is seen to be a vector of square-integrable martingales, for
which Problem 3.3.29 and Remark 3.3.30 give
E[max
0 <s
X?)) - 0-(s, Xr1))11 2 ds
A, E
ilMs112
0 t
< AIK2E 1 IV) - Xr1)112ds. o
On the other hand, we have E 1113t II 2 < K2 t f o Ell X?) - X.!k-1) 2 ds, and therethere11
fore, with L = 4K2(A1 + T), (2.18)
E[ max
linc+1)
nc). 2
o<s
J t
E X?) - Xr1)112 ds; 0 < t < T.
Inequality (2.18) can be iterated to yield the successive upper bounds (2.19)
noil2 < c* (Lt)k
E[ max xr-i)
-
o<s
where C* = maxo
-
1
.
0
t
T,
k!
a finite quantity because of (2.17).
2,
Relation (2.19) and the Cebygev inequality now give (2.20)
P[ max
O
xr-1)
X(k) ,.
1
II > 2 k +
1
4C*
(4L T)k ;
k!
k = 1, 2, ... ,
and this upper bound is the general term in a convergent series. From the Borel-Cantelli lemma, we conclude that there exists an event Kr e .F with P(S1*) = 1 and an integer-valued random variable N(w) such that for every xr-i)(0 xpo(co) < 2 -(k +1) V k > N(w). Consequently, co en*: maxo
max II Xrn)(w) - )0k)(0))11 < 0
V m > 1, k > N(w).
We see then that the sequence of sample paths 00k)(w); 0 < t < T1 ic°_, is convergent in the supremum norm on continuous functions, from which follows the existence of a continuous limit IXt(w); 0 < t < T} for all w e Since T is arbitrary, we have the existence of a continuous process X = {X t; 0 < t < col with the property that for P-a.e. w, the sample paths Ir)(w)},`,°_, converge to X Jo)), uniformly on compact subsets of [0, co). Inequality (2.15) is a consequence of (2.17) and Fatou's lemma. From (2.15) and (2.13) we have condition (iii) of Definition 2.1. Conditions (i) and (ii) are also clearly satisfied by X. The following problem concludes the proof. 2.11 Problem. Show that the just constructed process (2.22)
Xt P-- lim V); 0 < t < oo k-.co
satisfies requirement (iv) of Definition 2.1.
5.2. Strong Solutions
291
2.12 Problem. With the exception of (2.15) and the square-integrability of X, the assertions of Theorem 2.9 remain valid if the assumption (2.14) is removed.
C. Comparison Results and Other Refinements In the one-dimensional case, the Lipschitz condition on the dispersion coefficient can be relaxed considerably. 2.13 Proposition (Yamada & Watanabe (1971)). Let us suppose that the coefficients of the one-dimensional equation (d = r = 1)
dX, = b(t, Xddt + o(t, XddW,
(2.1)
satisfy the conditions (2.23) (2.24)
ib(t, x) - b(t, y)i
K ix - yi,
-
h(lx - .Y1),
I
for every 0 < t < oo and x eR, yeR, where K is a positive constant and h: [0, oo)
[0, co) is a strictly increasing function with h(0) = 0 and h-2(u) du = oo; V s > 0.
(2.25)
Then strong uniqueness holds for the equation (2.1).
2.14 Example. One can take the function h in this proposition to be h(u) = (1/2).
PROOF OF PROPOSITION 2.13. Because of the conditions imposed on the function
h, there exists a strictly decreasing sequence {a},T=0 g (0, 1] with ao = 1,
lim, a = 0 and fa:-' h-2(u)du = n, for every n > 1. For each n > 1, there exists a continuous function p on fJ with support in (a, a_,) so that 0 < p(x) < (2/nh2(x)) holds for every x > 0, and 117-' p(x) dx = 1. Then the function p,,(u)dudy;
4 /(x)
(2.26)
o
is
xeR
o
even and twice continuous y differentiable, with tir,;(x)1 < 1 and 0(x) = Ixi for x e rt. Furthermore, the sequence {tp}-_, is
nondecreasing. Now let us suppose that there are two strong solutions x(') and X(2) of (2.1)
with n) = Xo a.s. It suffices to prove the indistinguishability of xo) and X(2) under the assumption (2.27)
E
XV)12 ds < oo;
0
t < co, i = 1, 2;
292
5. Stochastic Differential Equations
otherwise, we may use condition (iii) of Definition 2.1 and a localization argument to reduce the situation to one in which (2.27) holds. We have
A, A- XP ) - V ) = ft {b(s, X.P)) - b(s, V))} ds o
ft
+
{Os, X11)) - Os, V))1 dWs,
o
and by the ItO rule, (2.28)
*At) = fo tk(As)[b(s, XS')) - b(s, X12))] ds + -1 I t ti,;(As)[a(s, Vs1)) - o-(s, X12))]2 ds 2
+
0
it0 IMA.,)[a(s, X,P)) - o(s, X,12))] dWs.
The expectation of the stochastic integral in (2.28) is zero because of assumption
(2.27), whereas the expectation of the second integral in (2.28) is bounded above by ES ip"(As)h2(14,1) ds < 2t/n. We conclude that (2.29)
Eik(A,)
E
ft ii/;,(A.,)[b(s, XV)) - b(s, X52))] ds + -tn o
_.K
o
t
.1EI.A.,lds +-t ; n
0, n
1.
A passage to the limit as n -> co yields EIA,1 < K Po E IN ds; t z 0, and the conclusion now follows from the Gronwall inequality and sample path
0
continuity.
2.15 Example (Girsanov (1962)). From what we have just proved, it follows that strong uniqueness holds for the one-dimensional stochastic equation (2.30)
X,=
1 t I Xsix dWs;
0
t < co,
o
as long as a > (1/2), and it is obvious that the unique solution is the trivial one X, -a 0. This is also a solution when 0 < a < (1/2), but it is no longer the only solution. We shall in fact see in Remark 5.6 that not only does strong uniqueness fail when 0 < a < (1/2), but we do not even have uniqueness in the weaker sense developed in the next section. 2.16 Remark. Yamada & Watanabe (1971) actually establish Proposition 2.13 under a condition on b(t, x) weaker than (2.23), namely, (2.23y
I b(t, x) - b(t, Al
K(1 x - YI);
0
t < co, x e R, y e R,
5.2. Strong Solutions
293
where K: [0, co) -+ [0, oo) is strictly increasing and concave with K(0) = 0 and f(0.0(du/K(u)) = co for every s > 0.
2.17 Exercise (Ito & Watanabe (1978)). The stochastic equation
X, = 3 fX:I3 t
ds + 3
dW,
has uncountably many strong solutions of the form
:") =
t<
0 (47,3;
fle < t < co ,
where 0 < O 5 oo and Q® inf {s > W, = 01. Note that the function o(x) = 3x213 satisfies condition (2.24), but the function b(x) = 3x" fails to satisfy the condition of Remark 2.16.
The methodology employed in the proof of Proposition 2.13 can be used to great advantage in establishing comparison results for solutions of onedimensional stochastic differential equations. Such results amount to a certain kind of "monotonicity" of the solution process X with respect to the drift coefficient b(t, x), and they are useful in a variety of situations, including the study of certain simple stochastic control problems. We develop some comparison results in the following proposition and problem. 2.18 Proposition. Suppose that on a certain probability space (1-2, .F, P) equipped
with a filtration {,,} which satisfies the usual conditions, we have a standard, one-dimensional Brownian motion {W 37;,; 0 < t < oo} and two continuous, adapted processes Xth; j = 1, 2, such that (2.31)
Xli) = XVI +
ft bf(s, Xn ds + o
i
o- (s, XV)) dW.,;
0 < t < co
o
holds a.s. for j = 1, 2. We assume that
(i) the coefficients o-(t, x), k(t,x) are continuous, real-valued functions on [0, co) x R, (ii) the dispersion matrix o-(t, x) satisfies condition (2.24), where h is as described in Proposition 2.13,
(iii) XV) < XV) a.s., (iv) bi(t, x) < b2(t, x), V 0 < t < oo, x e R, and (v) either b1(t, x) or b2(t, x) satisfies condition (2.23). Then (2.32)
P[XP) < XP), V 0 < t < co] = 1.
PROOF. For concreteness, let us suppose that (2.23) is satisfied by b1(t,x). Proceeding as in the proof of Proposition 2.13, we assume without loss of
294
5. Stochastic Differential Equations
generality that (2.27) holds. We recall the functions kfr(x) of (2.26) and create a new sequence of auxiliary functions by setting ton(x) = tfr(x) 1,0,)(x); x e R,
n > 1. With A, = XP) - V), the analogue of relation (2.29) is
Eq),,(60 -n 5 E .1
(p'(As)[bi(s, x!") - b2(s, X12)11 ds
o
=E
ii co;,(As)[bi(s, X11)) - bi(s, X?))] ds 0
it
+E
cp',,(As)[bi(s, X?)) - b2(s, V))] ds < K ft WO ds,
o
o
by virtue of (iv) and (2.23) for 61(t, x). Now we can let n -' co to obtain E(A,+) < K Po E(45-1-)ds; 0 < t < co, and by the Gronwall inequality (Problem 2.7), we have E(A,+) = 0; i.e., XP) < XP) a.s. P.
2.19 Exercise. Suppose that in Proposition 2.18 we drop condition (v) but strengthen condition (iv) to (iv)'
bi(t, x) < b2(t, x);
0 < t < co, x e 41.
Then the conclusion (2.32) still holds. (Hint: For each integer m > 3, construct a Lipschitz-continuous function b,,,(t, x) such that (2.33)
bi (t, x) 5 b,(t, x) < b2(t, x);
0 < ( < m, ixi < m).
It should be noted that for the equation (2.34)
X,= + f i b(s, X s) ds + Wr;
0 5 t < co,
/3
with unit dispersion coefficient and drift b(t, x) satisfying the conditions of Theorem 2.9, the proof of that theorem can be simplified considerably. Indeed, since there is no stochastic integral in (2.34), we may fix an arbitrary w e SI
and regard (2.34) as a deterministic integral equation with forcing function { W,(co); 0 5 t < co}. For the iterations defined by (2.16), and with k = 1, 2, ..., Er)(w) A max IIV)(w) - k-i
osst
)(OE
we have the bound ak)(co) < K Pc, Dr1)(co)ds; 0 < t < co, valid for every w e e. The latter can be iterated to prove convergence of the scheme (2.16), path by path, to a continuous, adapted process X which obeys (2.34) surely. This is the standard Picard-Lindeliif proof from ordinary differential equations and makes no use of probabilistic tools such as the martingale inequality or the Borel-Cantelli lemma.
Lamperti (1964) has observed that, under appropriate conditions on the coefficients b and o-, the general, one-dimensional integral equation
5.2. Strong Solutions
(2.4)"
295
X, = + f
b(Xs)ds +
f
6(X0dWs;
0 :5_ t < cc,
can be reduced by a change of scale to one of the form (2.34); see the following exercise.
2.20 Exercise. Suppose that the coefficients a: IR - (0, oo) and b: R R are of class C2 and C1, respectively; that b' - (1/2)66" - (bof/o-) is bounded; and that (1/a) is not integrable at either +oo. Then (2.4)" has a unique, strong solution
X. (Hint: Consider the function f(x) = 1,I(du/o-(u)) and apply Ito's rule to
f(x,)) A second important class of equations that can be solved by first fixing the
Brownian path and then solving a deterministic differential equation was discovered by Doss (1977); see Proposition 2.21.
D. Approximations of Stochastic Differential Equations Stochastic differential equations have been widely applied to the study of the
effect of adding random perturbations (noise) to deterministic differential systems. Brownian motion offers an idealized model for this noise, but in many applications the actual noise process is of bounded variation and non-Markov. Then the following modeling issue arises.
Suppose that { V}°°_, is such a sequence of stochastic processes which converges, in an appropriate strong sense, to the Brownian motion W = {W .57;,; 0 < t < co}. Suppose, furthermore, that {X},7_, is a corresponding sequence of solutions to the stochastic integral equations (2.35)
)0") = +
J
b(Xr))ds +
ft o-(X!"))dVs(");
n > 1,
where the second integral is to be understood in the Lebesgue-Stieltjes sense. As n -- co, will {X(")},7_, converge to a process X, and if so, what kind of integral equation will X satisfy? It turns out that under fairly general conditions, the proper equation for X is (2.36)
X, = + f
b(Xs)ds +
f
o-(X )odWs,
where the second integral is in the Fisk-Stratonovich sense. Our proof of this depends on the following result by Doss (1977).
2.21 Proposition. Suppose that a is of class C2(R) with bounded first and second derivatives, and that b is Lipschitz-continuous. Then the one-dimensional stochastic differential equation
296
(2.36)'
5. Stochastic Differential Equations
2"
X, = + f fb(Xs) + -o-(X )cr'(X )} ds + f 1
cr(Xs)dly,
has a unique, strong solution; this can be written in the form X t(w) = u(W(w), Yt(co));
0 < t < oo, co E
R and a process Y which solves an for a suitable, continuous function u: R2 ordinary differential equation, for every w el/
2.22 Remark. Under the conditions of Proposition 2.21, the process {cr(X,), .97E; 0 < t < co} is a continuous semimartingale with decomposition 1
o-(X,) = cr(0 +
+
f
J
1
[b(Xs)er'(Xs) + 2cr(X5)(cr'(X5))2 + - o-"(Xs)cr2(X5)1ds 2
a(X3)0=(X,)dll7s,
r
and so, according to Definition 3.3.13,
fo cr(XJ0 dW, = -1 ft o-(Xs)o-'(Xs)ds + 2
0
f
o-(Xs)dW.s.
In other words, equations (2.36) and (2.36)' are equivalent. PROOF OF PROPOSITION 2.21. Let u(x, y): R2 - R be the solution of the ordinary
differential equation Ou
-= o-(u),
(2.37)
u(0, y) = y;
Ox
such a solution exists globally, thanks to our assumptions. We have then
-
02
2
(2.38)
Ox2
= a(u)o-'(u),
x0y
aU
= ce(u)-0y,
a y) = 1, yu(0,
which give (2.39)
0
-oyu(x, y) = exp
1
cr'(u(z, y)) dz} A
p(x, Y).
Let A > 0 be a bound on o' and a". Then e-Alx1 < p(x, y) < eAlx1, and (2.39) implies the Lipschitz condition Iu(x,y1) - ti(x,Y2)1 < eAl'IlYi - y21.
If L is a Lipschitz constant for b, then lb(u(x, Yi)) - b(u(x,Y2))1 < LeAlx1
1y1
Y21
and consequently, for fixed x, b(u(x, y)) is Lipschitz-continuous and exhibits linear growth in y. Using the inequality le l - ez21 < (ez' v ez2)lz - z21, we
5.2. Strong Solutions
297
may write lx1
I p(x, yi) - p(x, y2)I
[P(x, Yi) v P(x,Y 2)] is
Id(u(z, Y in - a' (u(z, Y2))1 dz
0
eAlx1 f 'xi
A lu(ztY 1) - u(zt Y Adz
03
< Alx1e2A1'11Yi- Y21.
For fixed x, p(x, y) is thus Lipschitz-continuous and bounded in y. It follows that the product f(x, y) A p(x, y) b(u(x, y)) satisfies Lipschitz and growth conditions of the form (2.40)
I f (x, y 1) -f (x, Y 2)1
(2.41)
Ifix, Al
K1 + KklYI;
Ix'
-k
x, Yi, y2
k, y E R,
where the constants Lk and Kk depend on k. We may fix co E I/ and let Yi(co) be the solution to the ordinary differential equation (2.42)
d
-dt Y(w) = f (NO, Y(w)) with Y0(w) =
((.0)
Such a solution exists globally and is unique because of (2.40) and (2.41). The resulting process Y is adapted to {,t}, and the same is true of Xt(w) A u(W,((o), Y(o))). An application of Ito's rule shows that X satisfies (2.36)'.
0
We turn our attention to the integral equation (2.35).
2.23 Lemma. Let P-a.e. path V.(co) of the process V = {11,; 0 < t < co} be continuous and have finite total variation V;((o) on compact intervals of the form
[0, t]. If b, a: IR --, R are Lipschitz-continuous, then the equation (2.35) with V"
V possesses a unique solution.
< co and define recursively for k > 1:
PROOF. Set )0°) = c; 0 (2.43)
Xr1) -A
+ f t b(XP) ds +
ft o-(X.1k))dVs;
0 < t < co.
o
0
Then Mk+1).-- maxo<sst IXr"- xyol satisfies dpo, mk-1-1) < L f t Levis 0
where Lisa Lipschitz constant for b and o. Iteration of this inequality leads to mk-1-1) < m.) Lk(t ± Vt)k,,
0v ...-
t < 00, k
0.
For 0 < t < co fixed, the right-hand side of this last inequality is summable
298
5. Stochastic Differential Equations
over k, so Inc); 0 < s 5 tIp_o is Cauchy in the supremum norm on C[0, a Since t is arbitrary, X(k) must converge to a continuous process X, and the convergence is uniform on bounded intervals. Letting k oo in (2.43), we see that X solves (2.35). If Y is another solution to (2.35), then Dt A maxo <s
Lk(t + lt)k ,
k!
0 < t < oo, k > 0,
which implies that D = 0. 2.24 Proposition. Suppose that o- is of class C2 (R)with bounded first and second derivatives, and that b is Lipschitz-continuous. Let {Vt("); 0 < t < Inc°_1 be a sequence of processes satisfying the same conditions as V in Lemma 2.23, and let {W, ,Ft; 0 < t < I be a Brownian motion with lim sup IV?) -W = 0, a.s., rm O<s
unique solution X of equation (2.36).
PROOF. Let u and f be as in the proof of Proposition 2.21; let y(n)(w) be the solution to the ordinary differential equation (cf. (2.42)): dt
Yt(n)1(0) = f117i(n)1(0), Yi(n)1(0»;
Ye%) =
( (o),
and define )qn)(w) A u(Vt(n)(w), y(")(w)). Ordinary calculus shows that X(") is the (unique) solution to (2.35). It remains only to show that with Y defined by (2.42), we have for every 0 < t < oo: (2.44)
lim n--.a
sup
O<s
Ys(") - ysl = 0,
a.s.
Let us fix w e SZ, 0 < t < oo, and a positive integer k. With Lk as in (2.40), we may choose c > 0 satisfying c < e Lkt. Let rk(w) = t n inf {0 < s < t; I Ys(w)1 k - 1 or 1Ws(w)1 > k - 1 }, Tin%) = t n inf {0 < s < t; I Ys(n)(0)1 > k }. For fixed w e [R, we may choose n sufficiently large, so that If (Vs' q(0), Ys(w)) -f (ws(w), Ys(w))I < E2 and I vs(n)(0))1 < k hold for every se [0, rk(w) rk")(con, and thus ds 1Y(n)(w) - Ys(w))
If (vsn(0), Ys' (a))) -f (vsn(0), Ys((0))1 + If (Vs' q(0), Ys(w)) -f (w.s(w), Y.s(w))I Lk 1177 qa))
Ys(w)1
E2.
An application of Gronwall's inequality (Problem 2.7) yields Ys(n)(0) - Y.s(01 < E2eLkt < c;
0 < s < rk(w) A 1)(W).
5.2. Strong Solutions
299
This last inequality shows that tk(w) < tin)(w), so we may let first n -> co and then e 10 to conclude that
sup
lim
11;( ")(co) - l's(co)1 = 0.
n-co 0 5s5rk(c))
For k sufficiently large we have Tk(w) = t, and (2.44) follows.
El
2.25 Remark. The proof of Proposition 2.24 works only for one-dimensional stochastic differential equations with time-independent coefficients. In higher dimensions there may not be a counterpart of the function u satisfying (2.37) (see, however, Exercise 2.28 (i)), and consequently Proposition 2.24 does not hold in the same generality as in one dimension. However, if the continuous processes { V(")}'_, of bounded variation are obtained by mollification or piecewise-linear interpolation of the paths of a Brownian motion W, then the multidimensional version of the Fisk-Stratonovich stochastic differential equation (2.36) is still the correct limit of the Lebesgue-Stieltjes differential equations (2.35). The reader is referred to Ikeda & Watanabe (1981), Chapter VI, Section 7, for a full discussion.
E. Supplementary Exercises 2.26 Exercise (The Kramers-Smoluchowski Approximation; Nelson (1967)). Let b(t, x): [0, co) x R -÷ R be a continuous, bounded function which satisfies the Lipschitz condition (2.23), and for every finite a > 0 consider the stochastic differential system
XV =
d.)02) = V') dt;
dY,(1) = 016(t, X``))dt - ca,(') dt + a dW; ye = n, where , ri are a.s. finite random variables, jointly independent of the Brownian motion W.
(i) This system admits a unique, strong solution for every value of a E (0, CO). (ii) For every fixed, finite T > 0, we have lim
sup 1,02) - XtI = 0, a.s.,
ago 0.T
where X is the unique, strong solution to (2.34). 2.27 Exercise. Solve explicitly the one-dimensional equation
dX, = (,/1 + X,2 + PC,)dt + .,/1 + X, MT,. 2.28 Exercise. (i) Suppose that there exists an Rd-valued function u(t, y) = (1410,11,
of class C "2([0, co) x Rd), such that
,ud(t,17))
300
5. Stochastic Differential Equations
au;
at
(t, y) = bi(t,u(t, y)),
aui y,
(t, y) = o- ;At, u(t, y));
1
j
d
hold on [0, co) x Rd, where each bi(t, x) is continuous and each o-,;(t, x) is of class C1'2 on [0, co) x Rd. Show then that the process X,
u(t, W);
0 < t < oo,
where W is a d-dimensional Brownian motion, solves the Fisk-Stratonovich
equation (2.36)"
dX, = b(t, X,) dt + cr(t, X,) 0 dW,.
(ii) Use the above result to find the unique, strong solution of the onedimensional Ito equation d X --
[ 1 2+ t X,
a(1 + t)21dt + a(1 + t)2 dW; 0 < t < oo.
5.3. Weak Solutions Our intent in this section is to discuss a notion of solvability for the stochastic differential equation (2.1) which, although weaker than the one introduced in the preceding section, is yet extremely useful and fruitful in both theory and applications. In particular, one can prove existence of solutions under assumptions on the drift term b(t, x) much weaker than those of the previous section, and the notion of uniqueness attached to this new mode of solvability will lead naturally to the strong Markov property of the solution process (Theorem 4.20).
3.1 Definition. A weak solution of equation (2.1) is a triple (X, W), {A}, where
P),
P) is a probability space, and {F,} is a filtration of sub-a-fields of .97; satisfying the usual conditions,
(i) (f2,
(ii) X = IX A; 0 < t < col is a continuous, adapted Rd-valued process, W={W 0 < t < oo} is an r-dimensional Brownian motion, and (iii), (iv) of Definition 2.1 are satisfied.
The probability measure p(T) A P[X0 e 11, F e 4(1?) is called the initial distribution of the solution. The filtration {.57,} in Definition 3.1 is not necessarily the augmentation of the filtration = o-() v 0 < t < oo, generated by the "driving Brownian motion" and by the "initial condition" = X0. Thus, the value of the solution X,(w) at time t is not necessarily given by a measurable functional of the Brownian path { Ws(w); 0 < s < t} and the initial condition (w). On the other
5.3. Weak Solutions
301
hand, because W is a Brownian motion relative to {A}, the solution X,(w) at time t cannot anticipate the future of the Brownian motion; besides { Ws(a)); 0 < s < t} and (co), whatever extra information is required to compute Xt(w) must be independent of { Wo(w) - W(w); t < 0 < col.
One consequence of this arrangement is that the existence of a weak solution (X, W), (fl, P), } does not guarantee, for a given Brownian motion { l , A; 0 < t < cc } on a (possibly different) probability space (n,,,P), the existence of a process )7 such that the triple (51, 1,P), (n, F),
{A} is again a weak solution. It is clear, however, that strong solvability implies weak solvability.
A. Two Notions of Uniqueness There are two reasonable concepts of uniqueness which can be associated with weak solutions. The first is a straightforward generalization of strong uniqueness as set forth in Definition 2.3; the second, uniqueness in distribution, is better suited to the concept of weak solutions.
3.2 Definition. Suppose that whenever (X, W), .F, P), {A}, and (X, W), (SI, y, P), {A}, are weak solutions to (2.1) with common Brownian motion W (relative to possibly different filtrations) on a common probability space P) and with common initial value, i.e., P[X0 = 5C0] = 1, the two processes X and X are indistinguishable: P[X, = V0 < t < co] = 1. We say then that pathwise uniqueness holds for equation (2.1).
3.3 Remark. All the strong uniqueness results of Section 2 are also valid for pathwise uniqueness; indeed, none of the proofs given there takes advantage of the special form of the filtration for a strong solution. 3.4 Definition. We say that uniqueness in the sense of probability law holds for equation (2.1) if, for any two weak solutions (X, W), P), {A}, and (g, 11), c, F), with the same initial distribution, i.e., P[X 0 e
= i3 [g0 e 1]; V I- e R(Rd),
the two processes X, g have the same law.
Existence of a weak solution does not imply that of a strong solution, and uniqueness in the sense of probability law does not imply pathwise uniqueness. The following example illustrates these points amply. However, pathwise uniqueness does imply uniqueness in the sense of probability law (see Proposition 3.20).
35 Example. (H. Tanaka (e.g., Zvonkin (1974))). Consider the one-dimensional equation
302
5. Stochastic Differential Equations
X, =
(3.1)
f sgn(Xs) d Ws;
0 < t < co,
where
sgn(x) =
x > 0,
1;
1 -1; x <0.
If (X, W), P), {A} is a weak solution, then the process X = {X .F,; 0 < t < co} is a continuous, square-integrable martingale with quadratic variation process <X>, = jo sgn2(Xs) ds = t. Therefore, X is a Brownian
motion (Theorem 3.3.16), and uniqueness in the sense of probability law holds. P), {A} is also a weak solution, so once On the other hand, (- X, W), (SI,
we establish existence of a weak solution, we shall also have shown that pathwise uniqueness cannot hold for equation (3.1).
Now start with a probability space (0, .
P) and a one-dimensional
Brownian motion X = {X .519;ix; 0 < t < co} on it; we assume P[X0 = 0] = 1 and denote by IR the augmentation of the filtration } under P. The same argument as before shows that sgn(Xs)dXs;
0 < t < co
0
is a Brownian motion adapted to {.3,' }. Corollary 3.2.20 shows that (X, W), (1, P), { x} is a weak solution to (3.1). With {#,w} denoting the augmen-
tation of {Fr}, this construction gives
Ax, which is the opposite
inclusion from that required for a strong solution! Let us now show that equation (3.1) does not admit a strong solution. Assume the contrary; i.e., let X satisfy (3.1) on a given (0, P) with respect to a given Brownian motion W, and assume ..9%x Ary for every t > 0. Then X is
necessarily a Brownian motion, and from Tanaka's formula (3.6.13) with a = 0, we have sgn(Xs)dXs = I X ,I - 2Lx, (0)
W, = 0
=
1
- lim - meas{0 40 2e
s
t;
I Xsj
e},
0 < t < co , a.s. P,
where a(0) is the local time at the origin for X. Consequently, Aw c and thus also Ax Axl holds for every t > 0. But this last inclusion is absurd.
B. Weak Solutions by Means of the Girsanov Theorem The principal method for creating weak solutions to stochastic differential equations is transformation of drift via the Girsanov theorem. The proof of the next proposition illustrates this approach.
5.3. Weak Solutions
303
3.6 Proposition. Consider the stochastic differential equation (3.2)
d X, = b(t, X,) dt + dW,;
0 < t < T,
where T is a fixed positive number, W is a d-dimensional Brownian motion, and b(t, x) is a Borel-measurable, Rd-valued function on [0, T] x O which satisfies (3.3)
K(1 + DM; 0 .t,T,xe9;Rd
ilb(t,x)11
for some positive constant K. For any probability measure u on (Rd,.4(Rd)), equation (3.2) has a weak solution with initial distribution II.
PROOF. We begin with a d-dimensional Brownian family X = {X
{P}xE Rel. According to Corollary 3.5.16,
4 A- exp E E J=1
f
t
ys, xs)dxv) -
I ft 2-
o
II b(s, X0112
ds}
o
is a martingale under each measure 13', so the Girsanov Theorem 3.5.1 implies that, under Qx given by (dQxldr) = ZT, the process (3.4)
W, A X, - X0 -
b(s, X ,)ds;
0
T
is a Brownian with Qx[W0 = 0] = 1, Vx e Rd. Simply rewriting (3.4) as
X, = Xo + i't b(s, Xs)ds + Wt; 0
t < 'T,
we see that, with Q"(A) fad Qx(A)ti(dx), the triple (X ,W), is a weak solution to (3.2).
Q"), {Ft}
3.7 Remark. If we seek a solution to (3.2) defined for all 0 < t < co, we can repeat the preceding argument using the filtration {,Fix } instead of {,57;,} and citing Corollary 3.5.2 rather than Theorem 3.5.1. Whereas {,57;,} in Proposition 3.6 can be chosen to satisfy the usual conditions, {....F,x} does not have this
property. Thus, as a last step in this construction, we take
to be the
collection of null sets of (0, ,F,, QP), set A = o(.Ftx u ./11) and W, = A+. The filtration {/,} satisfies the usual conditions.
3.8 Remark. It is apparent from Corollary 3.5.16 that Proposition 3.6 can be extended to include the case (3.5)
X, = X0 +
ft b(s, X)ds + Wt;
0 < t < T,
where b(t, x) is a vector of progressively measurable functionals on C[0, co)d; see Definition 3.14.
3.9 Remark. Even when the initial distribution /..t degenerates to unit point mass at some x e Rd, the filtration {,Fr} of the driving Brownian motion in
5. Stochastic Differential Equations
304
(3.5) may be strictly smaller than the filtration {,F Z` } of the solution process (see the discussion following Definition 3.1). This is shown by the celebrated
example of Cirel'son (1975). Nice accounts of Cirel'son's result appear in Liptser & Shiryaev (1977), pp. 150-151, and Kallianpur (1980), pp. 189-191.
The Girsanov theorem is also helpful in the study of uniqueness in law of weak solutions. We use it to establish a companion to Proposition 3.6. 3.10 Proposition. Assume that (X(1), W(l)), (Sr), .5"), P(i)), {.F(i)}; i = 1, 2, are weak solutions to (3.2) with the same initial distribution. If P")
(3.6)
[1
11 b(t, X:1))112 dt <
oo] = 1; i = 1, 2,
then (X"), Wu)) and (X(2), W2)) have the same law under their respective probability measures. PROOF. For each k > 1, let (3.7)
41)
inf {0 < t < T;
T
ft
Ms, XV)112 ds = k} .
03
According to Novikov's condition (Corollary 3.5.13), (3.8)
1`)(X(I)) -A exp
{-
tA tip
io
1
(b(s, r), d1V)) -
116(s, )0112 ds} 0
is a martingale, so we may define probability measures Pr on ,F19, i = 1, 2, according to the prescription (dPe/dP(i)) = VIc)(X")). The Girsanov Theorem 3.5.1 states that, under Pe), the process ( 3.9)
Xft tip = Xg) + J
b(s, XV) ds + W,(i,,) tip;
0
o
is a d-dimensional Brownian motion with initial distribution it, stopped at time TV). But TV), {IV); 0 < t < 4)}, and Vc)(X(`)) can all be defined in terms of the process in (3.9) (see Problem 3.5.6). Therefore, for 0 = to < t1 < < t = T and F e,l(R2d("+1)) we have XLI. )5 Wt(1 )) tv.) (3.10) pol[poi), W(0l )
Si ry
(x(i))1{0410),),..., xv.), wocv4)=
di3,(c1)
L2),,)(x(2)) ,,(x,20),w,02),...,x,,2))c,;,,)=T}d,2) p(2)[(x102), W(02),
X: 2), W(2))
r; 42)
T].
5.3. Weak Solutions
305
By assumption (3.6), limk 13(l)[4,4 = T] = 1; i = 1, 2, so passage to the limit as k co in (3.10) gives the desired conclusion.
3.11 Corollary. If the drift term b(t, x) in (3.2) is uniformly bounded, then uniqueness in the sense of probability law holds for equation (3.2). Furthermore,
with 0 5 t1 < t, <
< t < T and with the notation developed in the proof of
Proposition 3.6, we have then
X,n)er] = J
(3.11)
Ex[1{(x, Qgd
F e R(Rd").
.x, )E ,--}ZT]p(dx); I
3.12 Exercise. According to Proposition 3.6 and Corollary 3.11, the onedimensional stochastic differential equation
dX, = -sgn(X,)dt + dW X0 = 0
(3.12)
possesses a weak solution which is unique in the sense of probability law. Show that if (X, W), P), {",} is such a solution and L,(0) is the local time at the origin for the Brownian family 1X,, 1, (n,"), {px}ER, then (3.13)
Q[X,ef] = e-1/2 E°[1{,G exP( - I Xt1
2L,(0))];
e *IR).
(An explicit formula for the right-hand side of (3.13) can be derived from Theorem 3.6.17 and relation (2.8.2). See Problem 6.3.4 and Exercise 6.3.5.)
3.13 Problem. Consider the stochastic differential equation (2.1) with a(t, x) a (d x d) nonsingular matrix for every t z 0 and x e Rd. Assume that b(t, x) is uniformly bounded, the smallest eigenvalue of a(t, x) is uniformly bounded away from zero, and the equation dX, = o-(t, X,)dW,;
(3.14)
0
has a weak solution with initial distribution y. Show that (2.1) also has a weak solution for 0 < t < T with initial distribution p. (We shall have more to say about existence and uniqueness of solutions to (3.14) in Sections 4 and 5.)
3.14 Definition. Let b,(t, y) and o-ti(t, y); 1 < i < d, 1 < j < r, be progressively measurable functionals from [0, co) x C[0, co )d into ER (Definition 3.5.15). A weak solution to the functional stochastic differential equation
0 < t < co,
dX, = b(t, X) dt + a(t, X) dW,;
(3.15)
is a triple (X, W), (12, .9,, P), {.9,} satisfying (i), (ii) of Definition 3.1, as well as X)I + cr,21(s, X)} ds < co;
(iii)
1
i
d,
15j
r,
t > 0,
Jr0
(iv)
X, = X0 + almost surely.
b(s, X)ds +
a(s, X)dWs;
0
t < co,
5. Stochastic Differential Equations
306
3.15 Problem. Suppose bi(t, y) and o-(t, y); 1 < i < d, 1 5 j
r, are progres-
sively measurable functionals from [0, co) x C[0, oo)d into ER satisfying (3.16)
II b(t,Y)II2 + 110(t,Y)112
K (1 + max II Yls/112);
o<st
V 0 5 t< co, y e C[0, °o r, where K is a positive constant. If (X, W), (Si, F, P), {A} is a weak solution to (3.15) with E II X oil' < co for some m > 1, then for any finite T > 0, we have (3.17)
E ( max 0<s
(3.18)
E 11X, - Xs1121"
11Xs1I2m)
< co + EllX0112m)ect; 0 < t < T,
C(1 + E 11X01121")(t - s)m;
0 _s
where C is a positive constant depending only on m, T, K, and d.
C. A Digression on Regular Conditional Probabilities We know that indistinguishable processes have the same finite-dimensional distributions, and this causes us to suspect that pathwise uniqueness implies uniqueness in the sense of probability law. The remainder of this section is devoted to the confirmation of this conjecture. As preparation, we need to state certain results about regular conditional probabilities, in the spirit of Definition 2.6.12, but in a form better suited to our present needs. We refer
the reader to Parthasarathy (1967), pp. 131-150, and Ikeda & Watanabe (1981), pp. 12-16, for proofs and further information. 3.16 Definition. Let (S1, F, P) be a probability space and a sub-a-field of F. A function Q(co; A): 0 x F -> [0,1] is called a regular conditional probability
for F given 4 if (i) for each w en, Q(w; -) is a probability measure on (0.,,F), (ii) for each A e , F , the mapping wi- Q(w; A) is 4-measurable, and (iii) for each A e 9; , Q(w; A) = P[A14](co); P-a.e. w e e. Suppose that, whenever Q' (co; A) is another function with these properties, there exists a null set N e ,y such that Q(w; A) = Q' (co; A) for all A e .97 and w e n\N. We then say that the regular conditional probability for F given 4 is unique.
Note that if X in Definition 2.6.12 is the identity mapping, then conditions (i)-(iii) of that definition coincide with those of Definition 3.16. 3.17 Definition. Let (Si, F) be a measurable space. We say that F is countably
determined if there exists a countable collection of sets di F such that, whenever two probability measures agree on dt, they also agree on F. We
5.3. Weak Solutions
307
say that .9" is countably generated if there exists a countable collection of sets (6' g such that 6((6).
In the space C[0, oor, for an arbitrary integer m > 1, let us introduce the o--fields
(3.19)
ajcp, con A o(z(s); 0 < s < t) =
(a(cp, co)'"))
for 0 < t < cc, where tpt: C[0, oor is the truncation mapping (x))'" (ptz)(s) A z(t A s); z e C[0, oor, 0 5 s < co. As in Problem 2.4.2, it is shown that A(C[0, corn) = oM), where (6, is the countable collection of all finitedimensional cylinder sets of the form
C = {ze C[0, co)'"; (z(ti),..., z(t,,))e AI
with n > 1, tie [0, t] n Q, for 1 5 i < n, and A e AR") equal to the product of intervals with rational endpoints. The continuity of ze C[0, corn allows us to conclude that a set of the form C is in a(%), even if the points t, are not necessarily rational. It follows that a t(c [0, cor) is countably generated. On the other hand, the generating class (6; is closed under finite intersections, so any two probability measures on A(C[0, con which agree on (6, must also agree on ai(c 0, con,
by the Dynkin System Theorem 2.1.3. It follows that A(c[o, con is also countably determined.
More generally, Theorem 2.1.3 shows that if a 6 -field .97 is generated by a countable collection of sets (6' which happens to be closed under pairwise intersection, then ,9-", is also countably determined. In a topological space with a countable base (e.g., a separable metric space), we may take this to be the collection of all finite intersections of complements of these basic open sets. 3.18 Theorem. Suppose that SI is a complete, separable metric space, and denote
the Borel 6 field y = WI). Let P be a probability measure on (0,,), and let be a sub-o--field of F. Then a regular conditional probability Q for F given exists and is unique. Furthermore, if dr is a countably determined sub-o--field of then there exists a null set N e such that (iv)
Q(o); A) = Wu)); A e
, w e SI\N.
In particular, if X is a -measurable random variable taking values in another complete, separable metric space, then with ° denoting the a-field generated by X, (iv) implies (iv)'
Q(w; {w' e
X(&) = X(01) = 1; P-a.e.
When the a-field is generated by a random variable, we may recast the assertions of Theorem 3.18 as follows. P) be as in Theorem 3.18, and let X be a measurable 3.19 Theorem. Let (SI, mapping from this space into a measurable space (S, 91, on which it induces the
5. Stochastic Differential Equations
308
distribution P X-1 (B) -A- P [w e O.; X (co) e B], B e 9'. There exists then a function Q(x; A): S x 07", [0, 1], called a regular conditional probability for y given
X, such that
(i) for each xeS, Q(x; ) is a probability measure on (52, g"), (ii) for each A e ", the mapping xi- Q(x; A) is Y-measurable, and (iii) f o r each A e 9-;, Q(x; A) = P[Al X = x], P X -1-a.e. xeS.
If Q'(x; A) is another function with these properties, then there exists a set Neff with PX-1(N) = 0 such that Q(x; A) = Q'(x; A) for all A e .f; and xe S\N. Furthermore, if S is also a complete, separable metric space and = M(S), then N can be chosen so that we have the additional property: B e ff, x e S\N.
Q(x; Ico e O.; X (co)e BI) = 1,(x);
(iv)
In particular, Q(x; 1w e O.; X(w) = x }) = 1;
(iv)'
PX-1-a.e. xeS.
D. Results of Yamada and Watanabe on Weak and Strong Solutions Returning to our initial question about the relation between pathwise uniqueness and uniqueness in the sense of probability law, let us consider two weak }; j = 1, 2, of equation (2.1) with solutions (X-', W-'), (51i, vi), y(B) A vi[n) e 13] = v2[X,1321e 13];
(3.20)
We set Ic(f) =
B e M(f?).
- X,(3j); 0 < t < co, and we regard the j-th solution as
consisting of three parts: Xe, W(i), and 17"). This triple induces a measure Pi on
(0, .4(0)) A Old x C[0, cc)' x C[0, co)d, M(Rd) © .4(C[0, cor) 0 .4(C [0, co)d))
according to the prescription (3.21)
pi(A) A vi[(XW), W"), Yu) e A];
A e .4(13), j = 1, 2.
We denote by B = (x, w, y) the generic element of O. The marginal of each Pi on the x-coordinate of B is the marginal on the w-coordinate is Wiener
measure P,, and the distribution of the (x, w) pair is the product measure x P1 because X,(3-(*) is .F1)-measurable and W") is independent of .9-1) (Problem 2.5.5). Furthermore, under Pi, the initial value of the y-coordinate is zero, almost surely. The two weak solutions (X"), W")) and (X(2), W(2)) are defined on (possibly) different sample spaces. Our first task is to bring them together on the same, canonical space, while preserving their joint distributions. Toward this end, we note that on (0, .4(0), Pi) there exists a regular conditional probability for .4(0) given (x, w). We shall be interested only in conditional probabilities of
5.3. Weak Solutions
309
sets in MO) of the form Rd x C[0, coy x F, where F e 4(C[0, °or). Thus, with a slight abuse of terminology, we speak of Qi(x, w; F):
x C[0, coy x .4(C[0, cor)
[0,1]
as the regular conditional probability for M(C[0, oor) given (x, w). According
to Theorem 3.19, this regular conditional probability enjoys the following properties: (3.22) (i) for each x e Rd, w e Cr[0, co), Q3(x, w; ) is a probability measure on
(C[0, cc)", MC [0, cor)), (3.22) (ii) for each F e [0, cor), the mapping (x,w)i--. Q;(x,w; F) is MO) AC [0, con-measurable, and (3.22) (iii) PJ(G x F) = $G Qi(X w; F)p.(dx)P,,,(dw); F e M(C [0, cor), G e .4(01d) ®M(C[0, coy).
Finally, we consider the measurable space (0,F), where SI = O x C[0, co)d and F is the completion of the o--field Af(0) 0 AC[0, cor) by the collection ..)1( of null sets under the probability measure P(dco) A Qi(x, w; dyi )Q2(x, w;dy2)p(dx)P,(dw).
(3.23)
We have denoted by w = (x, w4142) a generic element of a In order to endow (SI,
P) with a filtration that satisfies the usual conditions, we take A -4 Q { (x, w(s), y (s), y2(s)); 0 < s < t} ,
,A
,+;
u
0 < t < co.
It is evident from (3.21), (3.22) (iii), and (3.23) that (3.21)'
P [co e SI; (x, w, yj) e
and so the distribution of (x
= vi[(4-1), Wu), ri))e A]; A G.4(:3), j = 1, 2, w) under P is the same as the distribution of
(Xu), WLD) under vi. In particular, the w-coordinate process {w(t), A; 0 < t < co}
P), and it is then not difficult is an r-dimensional Brownian motion on (SI, to see that the same is true for {w(t), Ft; 0 < t < co}. 3.20 Proposition (Yamada & Watanabe (1971)). Pathwise uniqueness implies uniqueness in the sense of probability law. {..FP)}; PROOF. We started with two weak solutions (Xth, Wu)), (C/J, j = 1, 2, of equation (2.1), with (3.20) satisfied. We have created two weak solutions (x + yj, w), j = 1, 2, on a single probability space (II, F, P), {",}, such that (X), WU)) under v.; has the same law as (x + yj, w) under P. Pathwise uniqueness implies P[x + y1 (t) = x + y2(t), V 0 < t < co] = 1, or equivalently,
(3.24)
no) = (x5w41,Y2)E CI; yi = y2] = 1.
It develops from (3.21)', (3.24) that
5. Stochastic Differential Equations
310
viun),14/0), Y(1))e A] = P[co ei); (x, w, yi )e A] = P[to e 0; (x, w, y2) e in = v2[(X,(32), W(2), Y(2)) e A];
A e R(D),
and this is uniqueness in the sense of probability law.
CI
Proposition 3.20 has the remarkable corollary that weak existence and pathwise uniqueness imply strong existence. We develop this result.
3.21 Problem. For every fixed t > 0 and F e .4,(C[0, co)d), the mapping (x, w)i- Qi(x, w; F) is 4,-measurable, where IA 1 is the augmentation of the filtration 1.4(Rd) ® at(c [0, con} by the null sets of p(dx)P,(dw).
(Hint: Consider the regular conditional probabilities Z(x, w; F): Rd x C[0, co)' x .4,(C[0, co)d) -* [0, 1] for RAC [0, co)d), given (x, cp,w). These
enjoy properties analogous to those of Qi(x,w; F); in particular, for every F e .4,(C[0, co)d), the mapping (x, w) r- Ql(x, w; F) is .4(Rd) 0 .4,(C [0, conmeasurable, and
Pj(G x F) = 1 Vi(x,w; F) p(dx)P,(dw)
(3.25)
G
for every G e .4(Rd) 0 .4,(C[0, con. If (3.25) is shown to be valid for every G e .4(Rd) 0 .4(C[0, con, then comparison of (3.25) with (3.22) (iii) shows that Qi(x, w; F) = Q;(x,w; F) for it x P,-a.e. (x, w), and the conclusion follows. Establish (3.25), first for sets of the form (3.26)
G = G1 x (pt.' G2 n o-,-1 G3);
G1 e .4(Rd),
G2, G, e .4(C [0, co)'),
where (o-,w)(s) A w(t + s) - w(t); s > 0, and then in the generality required.)
3.22 Problem. In the context of Proposition 3.20, there exists a function k: Rd x C[0, co)'
C[0, co)d such that, for it x P,-a.e. (x, w) e Rd x C[0, co)",
we have (3.27)
Q 1 (x, w; {k(x, w)} ) = Q2(x, w; {k(x, w)} ) = 1.
This function k is .4(Rd) 0 .4(C [0, con/ .4(C [0, co)d)-measurable and, for each
0 < t < co, it is also A/.4,(C[0, co)d)-measurable (see Problem 3.21 for the definition of A). We have, in addition, (3.28)
P[o) = (x, w, yi, Y2)6 n; Y1 = Y2 = k(X, W)] = 1.
3.23 Corollary. Suppose that the stochastic differential equation (2.1) has a weak
solution (X, W), (0, .F, P), 1.0 with initial distribution 12, and suppose that pathwise uniqueness holds for (2.1). Then there exists a .4(Rd) 0 .4(C [0, con/ .4(C[0, co)d)-measurable function h: Rd x C[0, co)' C[0, co)d, which is also A/at(c [o, oo)d)-measurable for every fixed 0 < t < co, such that (3.29)
X. = h(X0, W), a.s. P.
5.4. The Martingale Problem of Stroock and Varadhan
311
Moreover, given any probability space P) rich enough to support an Rd-valued random variable with distribution p and an independent Brownian
motion {ll, (3.30)
; 0 < t < ool, the process X.
h(, W.)
is a strong solution of equation (2.1) with initial condition
PROOF. Let h(x, w) = x + k(x, w), where k is as in Problem 3.22. From (3.28) and (3.21)' we see that (3.29) holds. For c and 14, as described, both (X0, W) and ITO induce the same measure p x P,, on Rd x [0, con, and since W) satisfies (2.1), so does (X. = Pr.). The process )7 is (X. = h(X 0, adapted to {A} given by (2.3), because h is / MAC [0, co)d)- measurable. The functional relations (3.29), (3.30) provide a very satisfactory formulation of the principle of causality articulated in Remark 2.2.
5.4. The Martingale Problem of Stroock and Varadhan We have seen that when the drift and dispersion coefficients of a stochastic differential equation satisfy the Lipschitz and linear growth conditions of Theorem 2.9, then the equation possesses a unique strong solution. For more general coefficients, though, a strong solution to the stochastic differential equation might not exist (Example 3.5); then the questions of existence and uniqueness, as well as the properties of a solution, have to be discussed in a different setting. One possibility is indicated by Definitions 3.1 and 3.4: one attempts to solve the stochastic differential equation in the "weak" sense of finding a process with the right law (finite-dimensional distributions), and to do so uniquely. A variation on this approach, developed by Stroock & Varadhan (1969), formulates the search for the law of a diffusion process with given drift and dispersion coefficients in terms of a martingale problem. The latter is equivalent to solving the related stochastic differential equation in the weak sense, but does not involve the equation explicitly. This formulation has the advantage of being particularly well suited for the continuity and weak convergence arguments which yield existence results (Theorem 4.22) and "invariance principles", i.e., the convergence of Markov chains to diffusion processes (Stroock & Varadhan (1969), Section 10). Furthermore, it casts the
question of uniqueness in terms of the solvability of a certain parabolic equation (Theorem 4.28), for which sufficient conditions are well known. This section is organized as follows. First, the martingale problem is formulated and its equivalence with the problem of finding a weak solution to the corresponding stochastic differential equation is established. Using this martingale formulation and the optional sampling theorem, we next establish the strong Markov property for these solution processes. Finally, conditions
312
5. Stochastic Differential Equations
for existence and uniqueness of solutions to the martingale problem are provided. These conditions are different from, and not comparable to, those given in the previous section for existence and uniqueness of weak solutions to stochastic differential equations.
4.1 Remark on Notation. We shall follow the accepted practice of denoting by Ck(E) the collection of all continuous functions f: E gl which have continuous derivatives of every order up to k; here, E is an open subset of some
gl is a continuous function, we
Euclidean space Rd. If f(t, x): [0, T) x E
write f e C([0, T) x E), and if the partial derivatives (afia(),(afiaxi),(a2fiaxiaxi); 1 < i,j < d, exist and are continuous on (0, T) x E, we write f e C"2((0, T) x E). The notation f e C" 2 ( [0, T) x E) means that f e C" 2 ((0, T) x E) and the
indicated partial derivatives have continuous extensions to [0, T) x E. We shall denote by Ct(E), Ct(E), the subsets of Ck(E) of bounded functions and functions having compact support, respectively. In particular, a function in Ct(E) has bounded partial derivatives up to order k; this might not be true for a function in ct(E).
A. Some Fundamental Martingales In order to provide motivation for the martingale problem, let us suppose P), {.??7,} is a weak solution to the stochastic differential that (X, W), equation (2.1). For every t > 0, we introduce the second-order differential operator a2foo 1 d d af(x) (4.1)
(.91, f )(x) A
-
2 i=i k=1
+
atk(t, x)
NOXi 10Xk
b.(t, x)
;
f e C2 (Rd),
a Xi
1=1
where aik(t, x) are the components of the diffusion matrix (2.2). If, as in the next proposition, f is a function of t e [0, oo) and x e gld, then (.91,f)(t, x), is obtained by applying sr', to f(t, -). 4.2 Proposition. For every continuous function f(t, x): [0, oo) x gld -, gl which belongs to C"2((0, oo) x Rd), the process Mf = {Mir , .F;; 0 < t < op} given by (4.2)
Ng A f(t, Xt) - f(0, X0) -
t (a-f + as
s)ds
sisf)(s, X
Jo
is a continuous, local martingale; i.e., Mf je, loc. If g is another member of C([0, co) x Rd) C. , 2 ( (05 oo) x glg), then Mg e .41'''" and d d 't a a (4.3)
<Mf, Mg>, = 1 1
i=1 k=1
afk(s, Xs)
0
ax;
f(s, Xs)
ax,,
g(s, XS) ds.
Furthermore, if f e C0([0, oo) x Rd) and the coefficients r; 1 5 i s d, 1 s j 5 r, are bounded on the support of f, then Mf e JP',
5.4. The Martingale Problem of Stroock and Varadhan
313
PROOF. The Ito rule expresses M1 as a sum of stochastic integrals: r
d
(4.4)
Mir = E E gi,J), with MI" n r aii(s, Xs)f(s, X s)dWo. i=1 j=1
ax,
Introducing the stopping times
S
inf
> 0; X,11 > n or
a5(s, Xs)
n for some (i,j)}
and recalling that a weak solution must satisfy condition (iii) of Definition 2.1,
we see that limn, S = co a.s. The processes Ali f(n)
tAS
r
d
(4.5)
AlfAs = E E
i=1 j=1
J
0 a ii(S, X s)- f(s XS) d Wsti); Oxi
n > 1,
are continuous martingales, and so M1 e dr'. The cross-variation in (4.3) follows readily from (4.5). If f has compact support on which each au is bounded, then the integrand in the expression for M" in (4.4) is bounded, so Mf With the exception of the last assertion, a completely analogous result is valid for functional stochastic differential equations (Definition 3.14). We elaborate in the following problem.
4.3 Problem. Let bi(t, y) and c(t, y); 1 < i < d, 1 < j < r, be progressively measurable functionals from [0, co) x C[0, oo)d into R. By analogy with (2.2), we define the diffusion matrix a(t, y) with components (4.6)
ak(t, y) A E aiy,y)o-,;(t,y); 0 < t < co, y e C[0, co )d. .J=1
{;}, is a weak solution to the functional Suppose that (X, W), stochastic differential equation (3.15), and set 1
(sift'u)(Y) =
d
Ou(y(t)) 02u(y(t)) + bi(t, y) i.1 oxi Oxiax,
d
E E aik1t, 2 i=i k=1
0 < t < oo, u e C2(Rd), y e C[0, oo)d. Then, for any functions f, g e C([°, co) x Rd) cr. ,2(0, co. x Rd), the process (4.2)'
mif 4 f(t, Xt) - f(0, X0) -
ft o
+ .4.11(s, X) ds, Ft; 0 < t < 00 os
is in .11`.1", and d
(4.3)'
d
t
<M1, M9 >, = E E
i=1 k=1
aik(s, X) o
a
oXi
f(s, X,)-g(s, XS) ds. Oyck
Furthermore, if f e Co([0, co) x Rd) and for each 0 < T < co we have
314 (4.7)
5. Stochastic Differential Equations KT,
Haft,
t
0
T,
}le CLO, 00)d,
where K T is a constant depending on T, then f e ./11`2.
The simplest case in Proposition 4.2 is that of a d-dimensional Brownian motion, which corresponds to k(t, x) 0 and o-u(t, x) Su; 1 j < d. Then the operator in (4.1) becomes d
82f
1
1
2
2 i=1 ax?;
sif = -Af = E
fe C2(g1°).
4.4 Problem. A continuous, adapted process W = {W, ,Ft; 0 < t < co} is a d-dimensional Brownian motion if and only if
f(w) - fiwo) - -2
0
Ai(Ws) ds,
0 < t < co,
is in .1r1" for every f e C2(01").
B. Weak Solutions and Martingale Problems Problem 4.4 provides a novel martingale characterization of Brownian motion. The basic idea in the theory of Stroock & Varadhan is to employ AV of (4.2) in a similar fashion to characterize diffusions with general drift and dispersion coefficients.
To explain how this characterization works, we shall find it convenient
to deal temporarily with progressively measurable functionals bi(t, y), o-ii(t, y): [0, co) x C[0, oo)d R, 1 < i < d, 1 < j < r. We recall the family of operators I.go71 of (4.1)'.
4.5 Definition. A probability measure P on (C[0, co)°, .R(C [0, cor)), under which (4.8)
MI = f (y(t)) -f (y(0)) -
J t
(s4f)(y)ds, ..Ft; 0 < t < co,
is a continuous, local martingale for every f e C2(l . "), is called a solution to the local martingale problem associated with {,207}. Here
According to Problem 4.3, a weak solution to the functional stochastic differential equation (3.15) induces on (C[0, oo)°,.4(C[0, cor)) a probability measure P which solves the local martingale problem associated with {Al. The converse of this assertion is also true, as we now show.
5.4. The Martingale Problem of Stroock and Varadhan
315
4.6 Proposition. Let P be a probability measure on (C[0, oor,.4(C[0, oor)) under which the process M1 of (4.8) is a continuous, local martingale for the choices f(x) = x, and f(x) = xixk; 1 < i, k < d. Then there is an r-dimensional Brownian motion W = {W, .; 0 < t < oo} defined on an extension (n, P) of (C[0, oo)d, -.4(C[0, oo)d), P), such that (X, A y(t), W,),
P), {A}, is a weak
solution to equation (3.15). PROOF. By assumption, MP) =°
XP) - n) _ f ' b,(s, X) ds, ..Ft;
t < oo
0
o
is a continuous, local martingale under P. In particular,
P[f
(4.9)
t
lbas, X)I ds < cx); 0 ._ t < op] = 1;
1 < i < d.
o
With f(x) = xixk, we see that
MP.° '=1- XP)r) - Xg)n)
-f
[XP bk(s, X) + XY`)bi(s, X) + aik(s, X)] ds
o
is also a continuous, local martingale. But one can express
MP)/e) -
(4.10)
a,k(s, X) ds
J
as the sum of the continuous local martingale mi,k) - xg)A4:0 - nogi) and the process (4.11)
fo
(X()
X)ds +
X(`))b (s
s
k
()CY') - XP'))bi(s, X) ds
'
+ f t k(s, X)ds o
=
ft (Mr - MP)bk(s, X) ds +
t
bk(s, X)ds
o
ft (MY') - M1k))bi(s, X) ds
o
=-
f
o
ft Lis bk(u, X) du &lir o
ot
os
Mu, X) du d le).
o
The last identity may be verified by applying ItO's rule to both processes claimed to be equal. We see then that the process of (4.11) is a continuous, local martingale of bounded variation; it follows from Exercise 1.5.21 that it is identically equal to zero. Therefore, the process of (4.10) is in Jr)", and (4.12)
M(k)>, = J aik(s, X)ds;
0 < t < co, a.s.
316
5. Stochastic Differential Equations
We may now invoke Theorem 3.4.2 to conclude the existence of a d-dimensional Brownian motion {1,V A; 0 < t < co} on an extension (n, g, P) of (CEO, oo)d, .-1(C[0, ao)d), P) endowed with a filtration which satisfies the usual condi-
tions, as well as the existence of a matrix p = pp), A; 0 5 t <
<1.i
of measurable, adapted processes with
P[ ft pl(s)ds < ad=
(4.13)
1s j
1;
d, 0
t < oo
such that d
=E
(4.14)
j
=1
1 < i < d, 0 < t < oo
pu(s)daisti); 0
holds a.s. P. This last equation can be rewritten as (4.15)
X, = X0 +
b(s, X) ds +
J
p(s)dais;
0 5_ t < oo.
In order to complete the proof, we need to establish the existence of an r-dimensional Brownian motion W = {W A; 0 < t < co} on (n, P), such that p(s) dais =
(4.16)
ft a(s, X) d4; 0 < t < co
fo holds P-almost surely. From (4.12), (4.14) and with the notation (4.6), it will then be clear that d
P E pii(t)pki(t) = aik(t, X), for a.e. t > 0 = 1;
1 < i, k < d
i=
and (4.13) will imply (4.17)
P[i.
ots, X) ds <
1;
1 < i < d, 1 < j < r, OSt
o
{A} as a
The relations (4.9), (4.15)-(4.17) will then yield (X, W), weak solution to (3.15).
It suffices to construct W satisfying (4.16) under the assumption r = d. Indeed, if r > d, we may augment X, b, and a by setting ,qi) = 61(t5 y) = o-ii(t, y) = 0; d + 1 < i < r, 1 < j < r. This r-dimensional process X satisfies an appropriately modified version of (4.8), and we may proceed as before except now we shall obtain a matrix p which, like a, will be of dimension (r x r). On the other hand, if r < d, we need only augment a by setting y) = 0; 1 < i < d, r + 1 < j < d, and nothing else is affected. Both p and a are then (d x d) matrices. According to Problem 4.7 following this proof, there exists a Borelmeasurable, (d x d)-matrix-valued function R( p, a) defined on the set (4.18)
D A {(p,o-); p and a are (d x d) matrices with pT
aT
5.4. The Martingale Problem of Stroock and Varadhan
317
such that a = pR(p, a) and R(p, a) RT (p, a) = I, the (d x d) identity matrix. We set WA
f
a(s, X))dit 0 < t <
Then W") e dr)"; 1 < i < d, and <10), Wo>, = Mu;
1 < i,j < d, 0 < t < co.
It follows from Levy's Theorem 3.3.16 that {W,, .P; 0 < t < co} is a d-dimensional Brownian motion. Relation (4.16) is apparent. 4.7 Problem. Show that there exists a Borel-measurable, (d x d)-matrix-valued function R(p, a) defined on the set D of (4.18) and such that a = pR(p, o-),
R(p, a)RT (p, a) = I;
(p, a) e D.
(Hint: Diagonalize ppT = ao-T and study the effect of the diagonalization transformation on p and a.) 4.8 Corollary. The existence of a solution P to the local martingale problem associated with {sit'} is equivalent to the existence of a weak solution (X, W), (5,., P), {A} to the equation (3.15). The two solutions are related by P = Px-i; i.e., X induces the measure P on (C[0, cor,,It(C[0, oo)°)). 4.9 Corollary. The uniqueness of the solution P to the local martingale problem with fixed but arbitrary initial distribution
P[y e C[0, °or; y(0) ell = p(T); F e M(Rd) is equivalent to uniqueness in the sense of probability law for the equation (3.15).
Because of the difficulty in computing expectations for local martingales, it is helpful to introduce the following modification of Definition 4.5.
4.10 Definition (Martingale Problem of Stroock & Varadhan (1969)). A probability measure P on (C[0, oo)°,4(C[0, °o)°)) under which M1 in (4.8) is a continuous martingale for every f e CROV) is called a solution to the martingale problem associated with {34}.
Given progressively measurable functionals bi(t,y), o-u(t, y): [0, oo) x R; 1 < i < d, 1 < j < r, the associated family of operators {s47},
C[0, co)"
and a probability measure p on .4(Rd), we can consider the following three conditions:
(A) There exists a weak solution to the functional stochastic differential equation (3.15) with initial distribution p. (B) There exists a solution P to the local martingale problem associated with with P[y(0) e n = p(r); r e M(01°).
5. Stochastic Differential Equations
318
(C) There exists a solution P to the martingale problem associated with = it(F); F e .4(Fe). with P [y(0) e 4.11 Proposition. Conditions (A) and (B) are equivalent and are implied by (C). Furthermore, (A) implies (C) under either of the additional assumptions:
(A.1) For each 0 < T < co, condition (4.7) holds. (A.2) Each o-i(t, y) is of the form o-ii(t, y) = y(t)), where the Borelmeasurable functions d: [0, co) x are bounded on compact sets.
PROOF. We have already established the equivalence of (A) and (B.) If P is a solution to the martingale problem and f e C2(Rd) does not necessarily have compact support, we can define for every integer k > 1 the stopping time (4.19)
Sk A inf.{ t
0: Y(011 > k }.
Let gke CAW) agree with f on Ix e Rd; 11x11 < k }. Under P, each Mrk is a martingale which agrees with Mir for t < Sk. It follows that Mf e dff'"; thus (C)
(B). Under (A.1), Problem 4.3 shows (A)
(C); under (A.2), the argument
for this implication is given in Proposition 4.2.
4.12 Remark. It is not always necessary to verify the martingale property of M-1. under P for every f e CARd), in order to conclude that P solves the martingale problem. To wit, consider fi(x) A xi and fii(x) A xixi for 1 j < d, and choose sequences I 10,5)11°_, of functions in Co(Rd) such that g1 (x) = fi(x), g1 (x) = fy(x) for 114 k. If Me) and /1/19`,k; are martingales for 1 < i, j < d and k > 1, then M-1; and M-fq are local martingales. According to Proposition 4.6, there is a weak solution to (3.15), and Corollary 4.8 now implies that P solves the local martingale problem. Under either of the assumptions (A.1) or (A.2), P must also solve the martingale problem. It is also instructive to note that in the Definitions 3.1 and 3.14 of weak solution to a stochastic differential equation, the Brownian motion W appears only as a "nuisance parameter." The Stroock & Varadhan formulation eliminates this "parametric" process completely. Indeed, the essence of the implication (B) (A) proved in Proposition 4.6 is the construction of this process. In keeping with our practice of working with filtrations which satisfy the usual conditions, we have constructed from ,4,(C[0, oo )d) such a filtration {A} for the Definitions 4.5 and 4.10. Later in this section, we shall instead want to deal with {A} itself, because this filtration does not depend on the probability measure under consideration and each is countably determined. Toward this end, we need the following result. 4.13 Problem. Assume either (A.1)' KT; 0 5 t < T, y e C[0, 05)d, for every b (t, + 0 < T < co, where KT is a constant depending on T, or else
5.4. The Martingale Problem of Stroock and Varadhan
319
b,(t, y) and 0-0(t, y) are of the form b,(t, y) = (t, y(t)), y) = er(t,y(t)), where the Borel-measurable functions bi, au: [0, oc) x Fl are bounded on compact sets.
(A.2)'
Let P be a probability measure on (C[0, oo)d, AC[0, °or)); {.f,} be as in Definition 4.5; and f E ce(ll 0). Show that if {Mif, A; 0 < t < oo} is a martingale, then so is {Mir, ,97,; 0 < t < co}.
C. Well-Posedness and the Strong Markov Property We pause here in our development of the martingale problem to discuss the strong Markov property for solutions of stochastic differential equations. Consistent with our discussion of Markov families in Chapter 2, we shed the trappings of time-dependence. Thus, we have time-homogeneous and Borel-measurable drift and dispersion coefficients bi: Rd R, R; 1 < i < d,1 < j < r, and we shall study the time-homogeneous version of (2.1), written here in integral form as (4.20)
X,
=x+
f b(Xs) ds + f
a(X5) diVs;
0 < t < co.
This model actually does allow for time-dependence because time can be
appended to the state variable; i.e., we may take Xr1) = t, b, (x) = 1, 1 :54 :5_ r. We adopt the time-homogeneous assumption primarily for simplicity of exposition. Note, however, that some results (e.g., crd+i,J(x) =
Remark 4.30 and Refinements 4.32) require the nondegeneracy of the diffusion coefficient, which is not valid in such an augmented model.
4.14 Definition. The stochastic integral equation (4.20) is said to be well posed if, for every initial condition X E Rd, it admits a weak solution which is unique in the sense of probability law.
We know, for instance, that (4.20) is well posed if b and a satisfy Lipschitz and linear growth conditions (Theorems 2.5, 2.9). If a is the (d x d) identity matrix and b is uniformly bounded, (4.20) is again well posed (Proposition 3.6 and Corollary 3.11). Later in this section, we shall obtain well-posedness under even less restrictive conditions on a (Theorems 4.22 and 4.28, Corollary 4.29).
If (4.20) is well posed, then the solution X with initial condition X0 = x induces a measure Fix on (C[0, co)", a(c[o,cor)). One can then ask whether the coordinate mapping process on this canonical space, the filtration {M, }, and the family of probability measures {Px},ce R. constitute a strong Markov family. We shall see that if b and a are bounded on compact subsets of Rd, the answer to this question is essentially affirmative. Our analysis proceeds via the martingale problem, which we now specialize to the case at hand. We denote by
5. Stochastic Differential Equations
320 1
d
02f(x)
d
d
E E a,(x) (df)(x) = -2 1=1 CX OXk k=1
(1.2)
Of(x)
E
OX/
1=1
the time-homogeneous version of the operator in (4.1).
4.15 Definition. Assume that R and oii: Rd R; 1 < i < d, 1 < j 5 r are bounded on compact subsets of Rd. A probability measure P on (Q, a.) A (C[0, oo)d, (4.21)
((C[0, oo)d)) under which
E[f(y(t)) - f(y(s)) -
(szif)(y(u))du Ad=
0,
a.s. P
holds for every 0 < s < t < oo, f e CARd), is called a solution to the timehomogeneous martingale problem. We denote by 13' any solution for which 13' [y e C[0, oo)d; y(0) = x] = 1.
(4.22)
We say that the time-homogeneous martingale problem is well posed if, for every x e Rd, there is exactly one such measure Px. 4.16 Remark. The replacement of ..97, by Ms in (4.21) is justified by Problem 4.13.
4.17 Remark. Under the conditions of Definition 4.15, well-posedness of the time-homogeneous martingale problem is equivalent to well-posedness of the stochastic integral equation (4.20) (Corollaries 4.8 and 4.9 and Proposition 4.11).
4.18 Lemma. For every bounded stopping time T of the filtration {A}, we have = (431(t A T); 0
t < oo).
The MT-measurability of each y(t n T) follows directly from Definition 1.2.12 and Proposition 1.2.18. It remains to show MT g o-(y(t n T); 0 < t < oo). C[0, oo)d be given by (cpty)(s) = y(t n s); 0 < s < oo, for Let cpt: C[0, oo)d arbitrary t > 0. Problem 1.2.2 shows that T(y) = T(pT(y)(y)) holds for every y e C[0, oo)d and so, with A e ,IT and t A T(y), we have PROOF.
ye A
ye[A n
5 t1]-4* cpty e [A n
5 tn.:* (Aye A.
The second of these equivalences is a consequence of the facts An{ T< t} e M and y(s) = (cp,(y))(s); 0 < s < t. We conclude that A = {ye C[0, oo)d; ch(y)(y)e Al =
e C[0, oo)d; y( n T)e Al.
For the next lemma, we recall the discussion of regular conditional probabilities in Subsection 3.C, as well as the formula (2.5.15) for the shift operators: (Osy)(t) = y(s + t); 0 < t < oo for s > 0 and y e C[0, oo)d.
5.4. The Martingale Problem of Stroock and Varadhan
321
4.19 Lemma. Let T be a bounded stopping time of {.4,} and 5 a countably determined sub-a-field of MT such that y(T) is 5-measurable. Suppose that b and Q are bounded on compact subsets of Rd, and that the probability measure P on (SI, = (C[0, co)°, .4(C [0, cx)r), solves the time-homogeneous martingale problem of Definition 4.15. We denote by Q,(F) = Q(a); F): S2 x P4 -+ [0,1] the regular conditional probability for .4 given 1. There exists then a P-null event N e 5 such that, for every w N, the probability measure P. 4 Q,° 13i.1 solves the martingale problem (4.21), (4.22) with x = w(T(w)).
PROOF. We notice first that, thanks to the assumptions imposed on /, Theorem 3.18 (iv)' implies the existence of a P-null event N e 5, such that Q(0); {yen; Y(T(Y)) = w(T(w)) }) = 1,
and therefore also PAY e f); .Y(0) = (0(7.(0))] = Q.Ey e n; Y(T(Y)) = w(T(w))] = 1 hold for every w 0 N. Thus (4.22) is satisfied with x = w(T(w)).
In order to establish (4.21), we choose 0 < s < t < co, Gel, Fe.4 f e q(Rd); define
f(Y(t)) - f(Y(s)) - I (sin1Y1un du; ye C[0, co)d,
Z(Y)
and observe that (4.23)
Z(y)1F(Y)13.(dY) =
Z(07-0).01F(07.61Y)0(0;dY) 1-1
= E[loiF(Z 0 OT)ig](a) = E[E(Z 0 oTiaT+3). lei.,F11] (03)
= 0,
P-a.e. co.
We have used in the last step the martingale property (4.21) for P and the optional sampling theorem (Problem 1.3.23 (i)).
Let us observe that, because of our assumptions, the random variable Z is bounded; relation (4.23) shows that the /-measurable random variable w
Z(y)P,(dy) is zero except on a P-null event depending on s, t, f, and F.
Consider a countable subcollection g of as and a P-null event N(s, t, f) e , such that Z(y)Pc,(dy) = 0;
V co
N(s, t, f),
V F e g.
5. Stochastic Differential Equations
322
agree on 6', and since is countably determined, the subcollection g can be chosen so as to permit the conclusion Then the finite measures v,±-(F) A IF Z± (y)P,(dy); F e
f,
Z(y)P,(dy) = 0;
We may set now N(f) = lj Li e Q 0
V ru
N(s,t, f),
VFe
N(s, t, f), and use the boundedness and
s
continuity (in s, t) of Z to conclude that
f(y(t)) - f(y(0)) -
Mir
ft (sif )(y(u)) du, .4t;
0 < t < co
is a martingale under P., for every w N(f). Finally, we see that there exists a P-null event N a under which M-r is a P.; martingale for all w N and countably many f e CROld); because of Remark 4.12, Pc, solves the timehomogeneous martingale problem for all a) 0 N. 4.20 Theorem. Suppose that the coefficients b, a are bounded on compact subsets
of Rd, and that the time-homogeneous martingale problem of Definition 4.15 (or equivalently, the stochastic integral equation (4.20)) is well posed. Then for every stopping time T of {A}, F e .4(C [0, co )d) and x e , we have the strong Markov property (4.24)
Px[0,-1 F1.4,]((o) = P'm[F],
Px-a.s. on {T < col.
PROOF. If the stopping time T is bounded, we let
in Lemma 4.19 be .4T, which
is countably determined by Lemma 4.18. Using the notation of Lemma 4.19, we may write then, for every F e a/(C[0, co)d): Px
F I .431,] (w) = Q(o);OV F) = P,(F) = Pc'm"[F],
for Px-a.e. w en. The last identity is a consequence of the uniqueness of solution to the time-homogeneous martingale problem with initial condition x = co(T(w)).
Unbounded stopping times are handled as in Problem 2.6.9 (iii). 4.21 Remark. The strong Markov property of (4.24) is the same as condition (e") of Theorem 2.6.10, except that we have succeeded in proving it only for stopping (rather than optional) times.
Condition (a) of Definition 2.6.3 is satisfied under the assumptions of Theorem 4.20. Indeed, well-posedness implies that the mapping x H Px(F) is Borel-measurable for every F e .4(C[0, co )d), but the proof of this statement requires a rather extensive set-theoretic development (see Stroock & Varadhan (1979), Exercise 6.7.4, and Parthasarathy (1967), Corollary 3.3, p. 22). This result is of rather limited interest, however, because when a proof of well-posedness is given, it typically provides a constructive demonstration of the measurability of the mapping x Px(F).
5.4. The Martingale Problem of Stroock and Varadhan
323
D. Questions of Existence It is time now to use the martingale problem in order to establish the fundamental existence result for weak solutions of stochastic differential equations with bounded, continuous coefficients. 4.22 Theorem (Skorohod (1965), Stroock & Varadhan (1969)). Consider the stochastic differential equation d X , = b(X,) dt + o- (X ,) dW
(4.25)
where the coefficients bi, are bounded and continuous functions. Corresponding to every initial distribution kt on 4(91°) with IIx1I2mµ(dx) < co, for some m > 1,
fRa
there exists a weak solution of (4.25).
PROOF. For integers j > 0, n > 1 we consider the dyadic rationals 4") = jr" and introduce the functions i(i(t) = t11); t e [tj"), 01_1). We define the new coefficients (4.26)
b(n)(t,
b(Y(0.(t))),
(7(")(t, Y) g 6(Y(IP.(0);
0 < t < co, y e C[0, oo)d,
which are progressively measurable functionals. Now let us consider on some probability space (0,F, P) an r-dimensional
Brownian motion W = {W, Fr; 0 < t < col and an independent random vector with the given initial distribution
and let us construct the filtration
1,,1 as in (2.3). For each n > 1, we define the continuous process V") = Wm),
0 < t < co} by setting X,(,,") = and then recursively:
X") = 41!) + b(X(n)))(t - t(p) + o-PC:!))(1,V, - K.));
j
0, tj") < t <
.
Then V") solves the functional stochastic integral equation (4.27)
X:") = +
f b(")(s, X("))ds + ft
o-(")(s, X("))dWs;
0 < t < co.
Fix 0 < T < co. From Problem 3.15 we obtain sup E II X:") - X!")1121" n>1
C(1 +FII
112"1)(t - s)m;
0 ,s
where C is a constant depending only on m, T, the dimension d, and the bound on 11b112 + 110112. Let P(") c P(X("))-1; n > 1 be the sequence of probability measures induced on (C[0, oo)d,a(C[0, oo)d)) by these processes; it follows from Problem 2.4.11 and Remark 2.4.13 that this sequence is tight. We may
then assert by the Prohorov Theorem 2.4.7, relabeling indices if necessary,
324
5. Stochastic Differential Equations
that the sequence 1/34")};°_, converges weakly to a probability measure P* on this canonical space. According to Proposition 4.11 and Problem 4.13, it suffices to show (4.28) (4.29)
P*[y e C[0, cot y(0) e I-1 = p(F); I- e A(Rd), E*C./(Y(0) - .AY(s))
-1
t
a.s. P*,
(sif)(Y (14)) du IA] = 0,
s
for every 0 < s < t < oo, fe Cj,(Rd). For every fE Cb(Old), the weak convergence of {/34")},°,'_, to P* gives
i
E*f(y(0)) = lim E(n)f(y(0)) = n-.co
f(x)p(dx),
Rd
and (4.28) follows. In order to establish (4.29), let us recall from (4.27) and Proposition 4.11 that for every f e oRd) and n > 1,
f(y(t)) -f (y(0))
-f
(s1,;")f )(y) du, ,4,;
0
t < co
o
is a martingale under P("), where (sit(n).n(Y) A
i bln)(t, y) af (At)) + axi
i=1
is i 0-1;*, y)011)(t, y)a2f (Y(t)).
1
2 1=1 k=l J=I
oXiaKk
Therefore, with 0 < s < t < co and g: C[0, co)d -> II a bounded, continuous, AS measurable function, we have (4.30)
E4")[{f(y(t)) - f(y(s))
-f 1 (sii;")f )(y) du} g(y)1= 0.
We shall show that for fixed 0 < s < t < co, the expression Fn(Y) A f(Y(t)) - f(y(s))
f
(clin)f)(Y) du
s
converges uniformly on compact subsets of C[0, co)d to
F(Y) A f(Y(t)) - f(y(s))
f
(slf )(Y (u)) du.
Then Problem 2.4.12 and Remark 2.4.13 will imply that we may let n -> co in (4.30) to obtain E*[F(y)g(y)] = 0 for any function g as described, and (4.29) will follow. Let K c C[0, co)d be compact, so that (Theorem 2.4.9 and (2.4.3)):
M A sup yEK
11)7(41 < GC,
o5u5t
lim sup trit(y,2-") = 0.
n-co yeX
Because b and a are uniformly continuous on {x E Rd; II x II < M }, we can find
for every s > 0 an integer n(s) such that
5.4. The Martingale Problem of Stroock and Varadhan
325
sup (11b(")(s, y) - b(Y(s))II + II a(")(s,y) - cr(Y(s))11) 5 v;
0<st yeK
n > n(E).
The uniform convergence on K of Fn to F follows.
4.23 Remark. It is not difficult to modify the proof of Theorem 4.22 to allow b and a to be bounded, continuous functions of (t, x) e [0, oo) x Rd, or even to allow them to be bounded, continuous, progressively measurable functionals.
E. Questions of Uniqueness Finally, we take up the issue of uniqueness in the martingale problem. 4.24 Definition. A collection 9 of Borel-measurable functions cc,: Rd -4 R is called a determining class on Rd if, for any two finite measures it, and it2 on R(Rd), the identity
P(x)Iii(dx) = Qga
49(x)112(dx),
V9
J 68a
implies it = 4.25 Problem. The collection C83(Rd) is a determining class on Rd. 4.26 Lemma. Suppose that for every f e Cg)(Rd), the Cauchy problem (4.31)
au
at
= siu; in (0, oo) x Rd, u(0,
(4.32)
) = f;
in Rd,
has a solution uf e C([0, oo) x Rd) n C12((0,00) x Rd) which is bounded on each strip of the form [0, T] x Rd. Then, if Px and 17:" are any two solutions of the time-homogeneous martingale problem with initial condition x E Rd, their one-dimensional marginal distributions agree; i.e., for every 0 < t < co: (4.33)
Px[y(t) e V] = Px[y(t) e
v r e R(Rd).
PROOF. For a fixed finite T > 0, the function g(t, x) A uf(T - t, x); 0 < t < x Rd) and satisfies x e W is of class Cb([0, T] x Rd) r C1 ,2(0, -ag
at
+
= 0;
g(T, ) = f;
T,
in (0, T) x Rd, in Rd.
Under either F" or /3', the coordinate mapping process X, = y(t) on C[0, 0o)d is a solution to the stochastic integral equation (4.20) (with respect to some
5. Stochastic Differential Equations
326
Brownian motion, on some possibly extended probability space; see Proposition 4.11). According to Proposition 4.2, the process {g(t, y(t)), .1t; 0 5 t 5 T} is a local martingale under both Px and Px; being bounded and continuous, the process is in fact a martingale. Therefore,
Exf(y(T)) = Ex g(T, y(T)) = g(0, x) = Exg(T,y(T)) = Exf(y(T)) Since f can be any function in the determining class Co (Rd), we conclude that (4.33) holds.
We are witnessing here a remarkable duality: the existence of a solution to the Cauchy problem (4.31), (4.32) implies the uniqueness, at least for onedimensional marginal distributions, of solutions to the martingale problem. In order to proceed beyond the uniqueness of the one-dimensional marginals
to that of all finite-dimensional distributions, we utilize the Markov-like property obtained in Lemma 4.19. (We cannot, of course, use the Markov property contained in Theorem 4.20, since uniqueness is assumed in that result.)
4.27 Proposition. Suppose that for every x e Rd, any two solutions Px and Px of the time-homogeneous martingale problem with initial condition x have the same one-dimensional marginal distributions; i.e., (4.33) holds. Then, for every xe Rd, the solution to the time-homogeneous martingale problem with initial condition x is unique.
PROOF. Since a measure on C[0, corl is determined by its finite-dimensional
distributions, it suffices to fix 0 < t, < t2 < < t < a) and show that Px and Px agree on A o(y(t,),..., y(t)). We proceed by induction on n. We have assumed the truth of this assertion for n = 1. Suppose now that Px and Px agree on q(t 11. t, _1), and let Qy be a regular conditional probability for M(C[0, oc)°) given W(tI, tn_i), corresponding to Px. According to Lemma 4.19, there exists a Px-null set N e q(tI, to _1) such that for y 0 N, the measure Py A Qy o et lI solves the time-homogeneous martingale problem with initial condition y(tn_i). Likewise, there is a regular conditional probability Oy corresponding to Px and a Px-null set e 5(t , , t_,), such that PY 0 Y 0-1 In-I solves this martingale problem whenever y R. For y not in the
null (under both Px, P) set N u R, we know that Py and Py have the same one-dimensional marginals. Thus, with A e .4(Rd(n-1)) and B e .4(Rd), we have
Px[(y(t,),... , y(t,,_,))e A, y(t) e B] = Ex[110,40,...,yon_inE A}Pylco
co(tn - tn_i)EB }]
= Ex [1 {(Ati),
(.0(tn -
IDE Al PY {W
= x[ 1(0,00,...,yon ,EA}Pylw e D
}]
tn_i)EB w(tn - tn_i )e B 1]
= Px[(y(ti),..., y(tn_i))e A, y(tn)e 13],
5.4. The Martingale Problem of Stroock and Varadhan
327
where we have used not only the equality of Py {co ef2; co(t. - tn_i )e B} and Py{w e S2; w(t - t_, )e B}, but also their W(t1,... , t_1)-measurability. It is
now clear that P and Pr agree on g(t1,..., t). We can now put the various results together.
4.28 Theorem (Stroock & Varadhan (1969)). Suppose that the coefficients b(x) and a(x) in Definition 4.15 are such that, for every f e Co (RI), the Cauchy problem (4.31), (4.32) has a solution uf e C([0, oo) x Rd) n Cl '2((0, co) x lild) which is bounded on each strip of the form [0, T] x g8 °. Then, for every x ell?, there exists at most one solution to the time-homogeneous martingale problem.
4.29 Corollary. Let the coefficients b(x), a(x) be bounded and continuous and satisfy the assumptions of Theorem 4.28. Then the time-homogeneous martingale problem is well posed.
4.30 Remark. A sufficient condition for the solvability of the Cauchy problem (4.31), (4.32) in the way required by Theorem 4.28 is that the coefficients bi(x), aik(x); 1 < j, k < d be bounded and Holder-continuous on Rd, and the matrix a(x) be uniformly positive definite; i.e., d
(4.34)
d
E E a ik(x)M >_ A11112; V x, e Rd
and some A > 0.
i=1 k=1
We refer to Friedman (1964), Chapter 1, and Friedman (1975), §6.4, §6.5, or Stroock & Varadhan (1979), Theorem 3.2.1, for such results; see also Remark 7.8 later in this chapter. 4.31 Remark. If aike C2(Rd) for 1 < i, k < d, then a(x) has a locally Lipschitz-
continuous square root,
i.e.,
a (d x d)-matrix-valued function a' (x) =
{iiii(x)} 1 , i, j, such that a ik(x) = El=iiiii(x)a,;(x); xe Rd (Friedman (1975), Theorem 6.1.2). We do not necessarily have a(x) = Q(x) (indeed, one interest-
ing case is the one-dimensional problem with a(x) = sgn(x) and a(x) = 1; x e R). If, in addition, bi e C 1 (Rd); 1 < i 5 d, then according to Theorem 2.5, Remark 3.3, Proposition 3.20, Corollary 4.9, and Proposition 4.11, for every x e I there exists at most one solution to the time-homogeneous martingale
problem. This result imposes no condition analogous to (4.34) and is thus especially helpful in the study of degenerate diffusions. 4.32 Refinements. It can be shown that if the coefficients b(x), a(x) are bounded
and Borel-measurable, and the matrix a(x) is uniformly positive definite on compact subsets of Rd, then the time-homogeneous martingale problem admits a solution. In the cases d = 1 and d = 2, this solution is unique. See Stroock & Varadhan (1979), Exercises 7.3.2-7.3.4, and Krylov (1969), (1974).
5. Stochastic Differential Equations
328
F. Supplementary Exercises 4.33 Exercise. Assume that the coefficients bi: IR, o-ij: IRd IR; 1 < i < d, 1 < j < r are measurable and bounded on compact subsets of Rd, and let al be the associated operator (1.2). Let X = { X .f.,; 0 < t < co} be a continuous process on some probability space P) and assume that {.F,} satisfies the usual conditions. With e C2(Rd) and a e IR, introduce the processes ('f Mt g f (X,) - f(X0) - J df(Xs)ds, .F,; 0 < t < co,
A, -A e-x`f(X,) - f(X0) +
J
and show M e dr' a A
e-"(ocf(Xs) - alf(X))ds,
0 < t < oo,
loc. If f is bounded away from zero on compact
sets and
df(X,)
N, g f(X,)exp o
f (x
.f(XS) ds
.97,;
0
< 00,
then these two conditions are also equivalent to: N e .1r1". (Hint: Recall from
Problem 3.3.12 that if M e dr' and C is a continuous process of bounded variation, then C,M, - f o MS dC, = Ito CS dM, is in dr 1".)
P), 1,9;1 be a weak solution to the functional 4.34 Exercise. Let (X, W), stochastic differential equation (3.15), where condition (A.1)' of Problem 4.13 holds. For any continuous function f: [0, co) x IRd FR of class C'2((0, co) x Rd) and any progressively measurable process Ik .97,; 0 < t < col, show that ,
At
+ als'f - ksf)e-P'ku" ds, Ft;
f(t, X,)e-fr'ku" - f(0, X0) o
as
0 < t < oo
is in dr'. If, furthermore, f and its indicated derivatives are bounded and k is bounded from below, then A is a martingale.
4.35 Exercise. Let the coefficients b, r be bounded on compact subsets of Fe, and assume that for each x e Fe, the time-homogeneous martingale problem of Definition 4.15 has a solution Px satisfying (4.22). Suppose that there exists a function f: [0, oc) of class C2(11?) such that
df(x) + Af(x) < c, holds for some A > 0, c > 0. Then Exf(y(t))
f(x)e-At + ;.(1 - e-1`); 0 < t < co, x e
329
5.5. A Study of the One-Dimensional Case
5.5. A Study of the One-Dimensional Case This section presents the definitive results of Engelbert and Schmidt concerning weak solutions of the one-dimensional, time-homogeneous stochastic differential equation
dX, = b(Xi)dt + a(X,) dW,
(5.1)
with Borel-measurable coefficients b: R R and a: R -+ R. These authors provide simple necessary and sufficient conditions for existence and uniqueness when b --- 0, and sharp sufficient conditions in the case of general drift coefficients. The principal tools required for this analysis are local time, the generalized Ito rule, the Engelbert-Schmidt zero-one law (see Subsection 3.6.E),
and the notion of time-change. Solutions of equation (5.1) may not exist globally, but rather only up to an "explosion time" S; see, for example, Remark 2.8. We formalize the idea of explosion. 5.1 Definition. A weak solution up to an explosion time of equation (5.1) is a triple (X, W), (S2, F, 11, {,;}, where (i)
(SI, ,F, P) is a probability space, and {..F,} is a filtration of sub-a-fields of Y., satisfying the usual conditions,
(ii) X = IX ,f",,; 0 < t < col is a continuous, adapted, 11 u 1 + co 1-valued process with 1X01 < co a.s., and { 147 gt; 0 < t < co} is a standard, onedimensional Brownian motion, (iii) with (5.2)
Sn A- inf{t _>._ 0; 1X,1
n},
we have
,AS
(5.3)
P[f {1b(Xs)I + e(Xs)} ds < col = 1; V0
and (iv)
(5A)
PEX,s. = X0 +
b(X01{,s.}ds o
J0-(X.011ssId3; V 0 +
o
valid for every n > 1. We refer to (5.5)
S = liM S n->co
t < co] = 1
5. Stochastic Differential Equations
330
as the explosion time for X. The assumption of continuity of X in the extended real numbers implies that (5.6)
and
S = inf{t > 0: X, st RI
Xs = +co a.s. on {S < co}.
We stipulate that X, = Xs; S < t < co. The assumption of finiteness of X0 gives P[S > 0] = 1. We do not assume that lim, X, exists on {S = col, so Xs may not be defined on this event. If P[S = co] = 1, then Definition 5.1 reduces to Definition 3.1 for a weak solution to (5.1). We begin with a discussion of the time-change which will be employed in both the existence and uniqueness proofs.
A. The Method of Time-Change Suppose we have defined on a probability space a standard, one-dimensional
Brownian motion B = {Bs, FB; 0 < s < co} and an independent random variable with distribution p. Let {s) be a filtration satisfying the usual conditions, relative to which B is still a Brownian motion, and such that is q0-measurable (a filtration with these properties was constructed for Definition 2.1). We introduce TA
(5.7)
du 0 U 2g fs+
Bu);
< < CC,
a nondecreasing, extended real-valued process which is continuous in the topology of [0, co] except for a possible jump to infinity at a finite time s. From Problem 3.6.30 we have To A lim T, = co
(5.8)
a.s.
s t co
We define the "inverse" of (5.9)
Ts
by
0 < t < co
A, -4 inf{s > 0; T > t};
and
A3,
lim A,.
Whether or not Ts reaches infinity in finite time, we have a.s. (5.10)
Ao = 0,
A, < co; 0
t < co
and
Acc, = inf {s
0: T, = co}.
Because Ts is continuous and strictly increasing on [0, IL), A, is also continuous on [0, co) and strictly increasing on [0, TA,,_). Note, however, that if T, jumps
to infinity (at s = A), then (5.11)
A, = Az°, V t > TA._ ;
if not, then TA._ = TA.= co, and (5.11) is vacuously valid. The identities (5.12)
TA, = t;
0 5 t < T/c_,
5.5. A Study of the One-Dimensional Case (5.13)
331
0 < s < Aco,
AT= = s;
and
hold almost surely. From these considerations we deduce that 0; A, > sl;
'Ts = inflt
(5.14)
0 < s < oo, a.s.
In other words, A, and Ts are related as in Problem 3.4.5. Let us consider now the closed set dy
E
/(a) = { x E R; f_e
(5.15)
= co, V e > 01,
define
R A inf{s z 0;
(5.16)
+ Bs e 40)}.
5.2 Lemma. We have R = Acc, a.s. In particular, c R+
(5.17)
du
v2( + B.)
= co,
a.s.
PROOF. Define a sequence of stopping times
0; p( + B 1(a))
R,, A inf s
1
-1; n > 1,
where
p(x, 1(a)) A inf{ x - yl; ye/(a)}.
(5.18)
Because 1(a) is closed, we have limn, R = R, a.s. (recall Solution 1.2.7). For n > 1, set (x);
p(x,1(a)) z
a(x) = p(x, 1(a)) <
1
n, 1
We have cr,;-2(x) dx < co for each compact set K c R, and the EngelbertSchmidt zero-one law (Proposition 3.6.27 (iii) Problem 3.6.29 (v)) gives a.s.
[R." J0
du
a ff + R 1
R^^s 0
du + Bu )
s
du
± B.) < co; 0
For P-a.e. w e f and s chosen to satisfy s < R (co), we can let n cs
s < oo.
oo to conclude
du
cr2g(w) + B((0)) < c). It follows that Ts < co on {s < R}, and thus A. 5 R a.s. (see (5.10)). For the opposite inequality, observe first that 1(a) = 0 implies R = and in this case there is nothing to prove. If 1(a) 0 0 then R < co a.s. and o
5. Stochastic Differential Equations
332
for every s > 0, R+s
s
du
>
du
cr2( ± B.) fo Q2( + BR + Wu)' where W --4- B - R+u -BR is a standard Brownian motion, independent of BR Jo
(Theorem 2.6.16). Because + BR e 1(a), Lemma 3.6.26 shows that the latter integral is infinite. It follows that TR = CO, so from (5.10), fl, < R a.s.
We take up first the stochastic differential equation (5.1) when the drift is identically zero. One important feature of the resulting equation dX, = a(X,)dW,
(5.19)
is that solutions cannot explode. We leave the verification of this claim to the reader.
5.3 Problem. Suppose (X, W), (0,F,P), {,,}
is
a weak solution of
equation (5.19) up to an explosion time S. Show that S = co, a.s. (Hint: Recall Problem 3.4.11.)
We also need to introduce the set Z(a) = Ix E R; a(x) = 0 }.
(5.20)
The fundamental existence result for the stochastic differential equation (5.19) is the following.
5.4 Theorem (Engelbert & Schmidt (1984)). Equation (5.19) has a nonexploding weak solution for every initial distribution a if and only if
1(6) g Z(6);
(E)
i.e., if re
dy
j_, a2(x + y)=
co'
Ve>0
Cr(X) = O.
5.5 Remark. Every continuous function a satisfies (E), but so do many discontinuous functions, e.g., a(x) = sgn(x). The function o(x) = 1{0}(x) does not satisfy (E). PROOF OF THEOREM 5.4. Let us assume first that (E) holds and let {
+ B A;
0 < s < oo 1 be a Brownian motion with initial distribution ii, as described at the beginning of this subsection. Using the notation of (5.7)-(5.16), we can verify from Problem 3.4.5 (v) that each A, is a stopping time for {A}. We set (5.21)
M, = B, X, = + M g" =WA; 0 5 t < oo.
Because A is continuous, I.F,1 inherits the usual conditions from {
'.,}
(cf.
Problem 1.2.23). From the optional sampling theorem (Problem 1.3.23 (i)) and
5.5. A Study of the One-Dimensional Case
333
the identity A, T, = A, n s (Problem 3.4.5 (ii), (v)), we have for 0 < t1 < t2 < co
and n > 1: E[Mt, A
= E[BA, A nIA, ] = BA, An =
Since lim, T = co Mt2A
Mt! A Tn,
a.s.
we conclude that M e Jr'. Furthermore,
a.s.,
nn -(A, A n) is in ,fic for each n > 1, so <M>, = A,; 0 < t < co, a.s.
T - At A T =
(5.22)
that the process of (5.9) is given by
As the next step, we
A, =
(5.23)
t
62 (X )dv;
0 < t < co, a.s.
0
Toward this end, fix w e {R = 4,0}. For s < A,(w), (5.7) and (5.10) show that the function 7'(w) restricted to u e [0, s] is absolutely continuous. The change of variable v = 7;,(w) is equivalent to Ay(w) = u (see (5.12), (5.13)) and leads to the formula At(m)
(5.24)
A,(w) =
62(((o) + B.(0)dr,,((o) =
a2 (X (w)) dv,
valid as long as A,(w) < 21,0(w), i.e., t < T(w), where (5.25)
T
TA._ = inf { u > 0; A = A. }.
If t(w) = co, we are done. If not, letting t T t(w) in (5.24) and using the continuity of A(w), we obtain (5.23) for 0 < t < t(o)). On the interval [-r(w), co],
A00(co) = A.(w) = R(w). If R(w) < co, then
A,(w)
c(w) + BR(W)(w) E 1(a)
A',40(co)
Z(o-),
and so o-(X,(w)) = o-(X too(w)) = 0;
t(w) < t < co.
Thus, for t > t(co), equation (5.23) holds with both sides equal to 240.)(co). From (5.22), (5.23), and the finiteness of A, (see (5.10)), we conclude that
<M> is a.s. absolutely continuous. Theorem 3.4.2 asserts the existence of a Brownian motion 4V = 1147 0 < t < col and a measurable, adapted
process p = Ip At; 0 < t < col on a possibly extended probability space F), such that M, =
f
dITT,
<M>, =
f
py2 dv;
0
t < co , P a.s.
In particular, P[pi2 = 62(Xj) for Lebesgue a.e. t > 0] = 1. We may set
W=
ft
0
sgn(p09-(XJ)dIR;
0 < t < co;
5. Stochastic Differential Equations
334
observe that W is itself a Brownian motion (Theorem 3.3.16); and write
X,= + M, = + f t a(X)dW; 0 < t < co, P-a.s. 0
Thus, (X, W) is a weak solution to (5.19) with initial distribution ft. To prove the necessity of (E), we suppose that for every x e 64E, (5.19) has
a nonexploding weak solution (X, W) with X0 = x a.s. Here W = {W, ",; 0 < t < co} is a Brownian motion with W0 = 0 a.s. Then
0 < t < co
M, = X, - X0, ..17;,;
(5.26)
is in de.'" and <M>, =
(5.27)
I: a2(Xv)dv < co;
0 < t < oo, a.s.
o
According to Problem 3.4.7, there is a Brownian motion B = 14, W's; 0 s < co} on a possibly extended probability space, such that
M, = Boot; 0 < t < co, a.s.
(5.28)
Let Ts = inf{t _. 0; <M>, > s }. Then s A <M>co = <M> T., (Problem 3.4.5 (ii)),
so using the change of variable u = <M> (ibid. (vi)) and the fact that d<M> assigns zero measure to the set {v > 0; a2(X) = 0 }, we may write (5.29)
du
5 A <MX°
1<m)T.
j0
a2(X0 + B.)
=
f
du
a2(X0 + B.)
T,
ii; d<mx, j0 a2(X,,)
d<M>v l {,,,2(x d>, 0}
(x
T,
=
J.0
1 {2(x 0") dv _<_ T.
Let us choose the initial condition X0 = x in Z(o)`. Then a(x) 0 0, so a solution to (5.19) with such an initial condition cannot be almost surely constant. Moreover P[<M>co > 0] > 0, and thus for sufficiently small positive s, P[T, < co, <MX° > 0] > 0. Now apply Lemma 3.6.26 with T(w
A <M>.(w); s;
if Ts(co) < oo, <M>.(w) > 0, otherwise.
We conclude from this lemma and (5.29) that x e 1 (o)`. It follows that /(u) g Z(a). 5.6 Remark. The solution (5.21) constructed in the proof of Theorem 5.4 is nonconstant up until the time (5.30)
t A inf.{ t > 0; X, e /(a)}.
(This definition agrees with (5.25).) In particular, if X0 = x e 1(o)` , then X is
5.5. A Study of the One-Dimensional Case
335
not identically constant. On the other hand, if x e Z(a), then Y = x also solves (5.19). Thus there can be no uniqueness in the sense of probability law if Z(a) is strictly larger than 1(a). This is the case in the Girsanov Example 2.15;
if a(x) = jxIa and 0 < a < (1/2), then 1(a) = 0 and Z(a) = {0}. Engelbert & Schmidt (1985) show that if a: IR -* IR is any Borel-measurable function satisfying 1(a) = 0, then all solutions of (5.19) can be obtained from the one constructed in Theorem 5.4 by "delaying" it when it is in Z(a). An
identically constant solution corresponds to infinite delay, but other, less drastic, delays are also possible.
5.7 Theorem. (Engelbert & Schmidt (1984)). For every initial distribution p, the stochastic differential equation (5.19) has a solution which is unique in the sense of probability law if and only if
(E + U)
1(a) = Z(cr).
PROOF. The inclusion 1(a) g Z(a), i.e., condition (E), is necessary for existence
(Theorem 5.4); in the presence of (E), the reverse inclusion is necessary for uniqueness (Remark 5.6). Condition (E) is also sufficient for existence (Theorem 5.4), so it remains only to show that the equality 1(a) = Z(a) is sufficient for uniqueness. We shall show, in fact, that the inclusion 1(a) Z(0) is sufficient for uniqueness. Assume 1(a) Z(a) and let (X, W), (n,F,p), {,,} be a weak solution
of (5.19) with initial distribution It. We define M as in (5.26) and obtain (5.27)-(5.29), where B is a standard Brownian motion. We also introduce via (5.30). By assumption,
a2(X,) > 0;
(5.31)
0
t < T, a.s.,
so <M> is strictly increasing on [0, t]. In particular, (5.32)
0; X0 + Bse 1(a)} = <M>t,
R A inf {s
a.s.
We set (5.33)
T, A int.{ t
0; <M>, >
so T is nondecreasing, right-continuous, and (5.34)
{TS
<
= Is < <M>,}.
We claim that (5.35)
du JS
a2 (X0 + Bu)
-Ts ;
0 < s < <MX°, a.s.
In light of (5.29), (5.31), to verify this claim it suffices to show (5.36)
R
<M>., a.s.
Indeed, on the set {R < <M>,,,} we have by the definition (5.33) that TR < CO;
336
5. Stochastic Differential Equations
letting sIR in (5.29), we obtain then TR+ o
du
6.2(X0 + B)
on {R < <M> 1.
Comparing this with (5.17), we see that PER < <M>] = 0, and (5.35) is established. We next argue that s+
(5.37)
Ts =
.1.0
du
6 (X0 + B)
;
0 < s < co, a.s.
For s < <M>, we have Ts < co and so (5.37) follows immediately from (5.35). For s = <M> = R, it is clear from (5.33) that Ts = oo, and (5.37) follows from (5.17). It is then apparent that (5.37) also holds for s > <M> with both sides equal to infinity.
Comparing (5.37) and (5.7), we conclude that the discussion preceding Lemma 5.2 can be brought to bear. The process A defined in (5.9) coincides then with <M>, which is thus seen to be continuous and real-valued and to satisfy s+
<M>, = inf Is
du
> t } ; 0 t < co. fo u2(X0 + B) We now prove uniqueness in the sense of probability law by showing that the distribution induced by the solution process X on the canonical space (C[0, co), ,l(C[0, co))) is completely determined by the distribution it of Xo. The distribution induced on this space by the process X0 + B is completely (5.38)
0;
determined by it because B is a standard Brownian motion independent of X0 (Remark 3.4.10). For CO E C [0, Go), define 9.(w) to be the right-continuous process tp,(w) A inf Is
0;
iso+
du o-2(w(u))
> t}; 0 < t < co
with values in [0, co]. Because of its right-continuity, 9 is 4( [0, co)) ® .4(C[0, co))-measurable (Remark 1.1.14). Let 7C: [0, CO) x C [0, CO) -+ R be the measurable projection mapping tr,(co) = w(t). Consider the P. co)) ® M(C [0, co))-measurable process Co)) A co(0) + tr ,,,,(,,)(a)). We have (5.39)
X, = X0 + Boo, = 11/,(X0 + B.),
0
t < co,
and so the law of X is completely determined by that of X0 + B..
5.8 Remark. It is apparent from the proof of Theorem 5.7 that under the assumption 1(a) 2 Z(r), any solution to (5.19) remains constant after arriving in 1(o). See, in particular, the representation (5.39) and recall that <M>, is constant for t < t < CO.
5.5. A Study of the One-Dimensional Case
337
Using local time for continuous semimartingales, it is possible to give a simple but powerful sufficient condition for pathwise uniqueness of the solution to equation (5.19).
5.9 Theorem. Suppose that there exist functions f: L -> [0, co) and h: [0, co) -> [0, co) such that: (i)
at every x el(a)t, the quotient (f/a)2 is locally integrable; i.e., there exists e > 0 (depending on x) such that
(f(Y))2 dy
(ii) the function h is strictly increasing and satisfies h(0) = 0 and (2.25); (iii) there exists a constant a > 0 such that
+ y) - a(x)I
f(x)klYD;
Vxe
y e [ - a, a].
Then pathwise uniqueness holds for the equation (5.19).
PROOF. Observe first of all that if a(x) = 0, then (ii) and (iii) imply that fe
dy
h'(y)dy = oo,
> 2f -2(x)
cf2(x +
V e > O.
0
Thus, Z(a) 1(a). Suppose now that X(`) = {XI°, .F.,; 0 < t < co}; i = 1, 2, are solutions to
0 < t < co} on (5.19) relative to the same Brownian motion W = {W, some probability space (SI, .F, P), with P[XP = XVI = 1. Defining -c(l) = inf{t > 0; XP) e /(a)}, we recall from Remark 5.8 that XP) = 'CPL.; 0 < t < co, a.s., so it suffices to prove that xp) )02); 0 < t < T(1), (5.40)
For each integer k > (1/a), there exists sk > 0 We set A, = XF) ) d = k. Using the continuity of the local time for A
for which 1,11c h-2
(Remark 3.7.8), (3.7.3), and assumption (iii), we may write for every random time r:
1 fs 1
(5.41)
lik
A°(0) = lim k
a0
h-2(x)M(x)dx
k
= lim k
t
0
1 ft
< lim -,, k-,,,, PC
0
h-2(A)1
f 2 (XV)) ds,
(As )(a(X(2))
- a(XP)))2 ds
a.s. P.
Now M, A XI" - 4 ) is in jr.loc, and so M admits the representation M, = B<m>, where B is a Brownian motion (Problem 3.4.7). Furthermore,
5. Stochastic Differential Equations
338
<M>, = focr2(X!,1))dv, and 62(X,")) > 0 for 0 < v < -r"), so a change of variable results in the relation <m>,
f
t
f 2 (XV)) dv =
o
f2041)
B.)0.-2(A-,i) o+
B,Jdu; 0 < t < -r").
o
We take T. A inf It > 0; p(X:1),I(o-))
-1
n},
R A <M>,. = inf { s > 0; p(X0 + B5,1(0)) < ;11}, where p is given by (5.18). We alter f and a near 1(6) by setting 1
f(x) = o-(x) = 1;
if p(x, /(o-)) <
f(x) = f(x), o-(x) = o-(x);
if p(x,l(u)) .._. 1, n
so f20-;2 is locally integrable at every point in 11. From the implication (iii) =. (v) in Proposition 3.6.27 and Problem 3.6.29, it follows that
r 1o
R,,
f 2 (XP)) dv = 1.
f 2 (XW ) + B,Jo-- 2 (X W ) + B) du
o
TR fn2(xW) + B.)0;2,klio ,I1) + Buidu < co,
a.s.
0
From (5.41) we see now that Azt(0) = 0; n > 1, a.s. Since t T -c" ) as n co, we conclude from the continuity of ANO) that A'.:`0)(0) = 0, a.s. P. Remark 3.7.8 shows that E I X).,,i, - X:1,),i)1 = 0; 0 < t < oo, and (5.40) follows.
We recall from Corollary 3.23 that the existence of a weak solution and pathwise uniqueness imply strong existence. This leads to the following result.
5.10 Corollary. Under condition (E) of Theorem 5.4 and conditions (i)-(iii) of Theorem 5.9, the equation (5.19) possesses a unique strong solution for every initial distribution p. 5.11 Remark. If the function f is locally bounded, condition (i) of Theorem 5.9 follows directly from the definition of 1(6). We may take h(y) = y"; a > (1/2),
and (iii) becomes the condition that a be locally Holder-continuous with exponent at least (1/2); see Examples 2.14 and 2.15.
5.5. A Study of the One-Dimensional Case
339
B. The Method of Removal of Drift It is time to return to the stochastic differential equation (5.1), in which a drift term appears. We transform this equation so as to remove the drift and thereby reduce it to the case already studied. This reduction requires assumptions of nondegeneracy and local integrability: (ND) (LI)
a2 (x) > 0;
Vx
> 0 such that
V x e R, 3
e R,
x+(AI ' b dy
< co.
Under these assumptions, we fix a number c e R and define the scale function
exp { -2 14 b() 61} g x e R. a2g) The function p has a continuous, strictly positive derivative, and p" exists (5.42)
P(x)
x
almost everywhere and satisfies (5.43)
p"(x) -
2b(x)
0.2(x) p'(x).
Henceforth, whenever we write p", we shall mean the (locally integrable) function defined by (5.43) on the entire of R; this definition is possible because of (ND). The function p maps R onto (p( -co), p(co)) and has a continuously differ-
entiable inverse q: (p( -co), p(oo)) -> R. This latter function has derivative qf(y)= (1/ p'(q(y))), which is actually absolutely continuous, with (5.44)
P(q(Y)) "
y))2
p'(q(Y))
2b(q(y)) a2(q(y))
1
(q( y))) 2
a.e. in (p( -co), p(oo)).
We extend p to [-co, co] and q to [p(- co), p(co)] so that the resulting functions are continuous in the topology on the extended real number system.
5.12 Problem. Although not explicitly indicated by the notation, p(x) defined by (5.42) depends on the number c e R. Let us for the moment display this dependence by writing pc(x). Show that (5.45)
Pa(x) = pa(c) + PL(c)Pc(x).
In particular, the finiteness (or nonfiniteness) of pc(+ co) does not depend on the choice of c.
5.13 Proposition. Assume (ND) and (LI). A process X = {X Ft; 0 < t < oo} is a weak (or strong) solution of equation (5.1) if and only if the process
340
5. Stochastic Differential Equations
Y=Y
p(X,), r#,; 0 < t < col is a weak (or strong) solution of
=
(5.46)
+
f
0 < t < oo,
d(Ys)dWs;
where
p(-00) < Yo < p(00) a.s.,
(5.47) (5.48)
(y) =
p( -co) < y < p(co),
{p'(q(y))6.(q(y));
otherwise.
0;
The process X may explode in finite time, but the process Y does not.
PROOF. Let X satisfy (5.1) up to the explosion time S, define Y, = p(X,), and recall S from (5.2); we obtain from the generalized Ito rule (Problem 3.7.3) and (5.43): t
5,,
LAS = P(X0)
(5.49)
0 < t < co, n > 1.
6(Ys)dWs;
Because X is a continuous, extended-real-valued process defined for all 0 < t < co (Definition 5.1) and p: [-co, co] [p(-oo),p(co)] is continuous, Y is also continuous for all 0 < t < co. Hence, as n co, the left-hand side of (5.49) converges to Y n s, and the right-hand side must also have a limit. In light of Problem 3.4.11, this means that st0A s d2(Ys) ds < co; 0 < t < co, a.s. Because Xs, and hence Ys, is constant for S < s < co, we must in fact have Sto ei2(Ys)ds < co; 0 Lc. t < co, a.s. It follows that Po 6(Ys)d147, is defined for all
0 < t < cc and is a continuous process. From (5.49) we see that (5.46) holds for 0 _5 t < S. This equality must also hold for 0 < t S on the set {S < co}, because of continuity. The integrand a(Ys) vanishes for S < s < co, and so (5.46) in fact holds for all 0 < t < co. The solution of this equation cannot explode in finite time; see Problem 5.3. For the converse, let us begin with a process Y satisfying (5.46), (5.47) which
takes values in [p( -co), p(oo)]. We can always arrange for Y to be constant after reaching one of the endpoints of this interval, because the integrand in (5.46) vanishes when Ys (p( -co), p(oo)). Define X, A q(Y); 0 5 t < co, and let S be given by (5.2). Then the generalized Ito rule of Problem 3.7.3 gives, in conjunction with the properties of the function q (in particular, (5.44)) and (5.48), 1
Xt
S,,
= X0 ±
(107,$)dY.s ±
(1"117.0ds o
tAS
= X°
1
ei(Ys)dW, +
+
PVCs)
f
o
s- b(Xs)
1
472(Xs) (P'1Xs02
tAs,,
tAs,,
= X0 +
tA
b(Xs) ds +
almost surely, for every n
1.
cr(Xs)dWs; 0
ii2(Y )ds 3
5.5. A Study of the One-Dimensional Case
341
5.14 Exercise. Show by example that if (LI) holds but (ND) fails, then (5.46) can have a solution Y which is not of the form p(X), for some solution X of (5.1 )
5.15 Theorem. Assume that a-2 is locally integrable at every point in R, and conditions (ND) and (LI) hold. Then for every initial distribution II, the equation (5.1) has a weak solution up to an explosion time, and this solution is unique in the sense of probability law.
PROOF. Let 6 be defined by (5.48). According to Theorem 5.7 and Proposition 5.13, it suffices to prove that 1(6) = 43). Now Z(6) = (p( -co), p(+ co))`, and
1(6) contains this set. We must show that 6 -2 is locally integrable at every point yo e (p(-co), p(co)). At such a point, choose E > 0 so that P( -Do) < Yo - E < yo + 6
and write
Yo' dy Yo
f
9(Y°+')
dx
6 2(Y) - q(yo-e) P'(072(X)
The second integral is finite, because p' is bounded away from zero on finite intervals and a-2 is locally integrable. 5.16 Corollary. Assume that a-2 is locally integrable at every point in R, and that conditions (ND), (LI), and
I b(x) - b(y)I < K ix - yl;
(x, y) e 082
I o-(x) - a(y)I -. h( I x - ye);
(x, y) e R2
(5.50)
hold, where K is a positive constant and h: [0, co) - [0, co) is a strictly increasing
function for which h(0) = 0 and (2.25) hold. Then, for every initial condition independent of the driving Brownian motion W = {W, .Ft; 0 < t < co}, the equation (5.1) has a unique strong solution (possibly up to an explosion time). PROOF. Weak existence (Theorem 5.15) and pathwise uniqueness (Proposition
2.13 and Remark 3.3) imply strong existence (Corollary 3.23). These results
can easily be localized to deal with the case of possible explosion of the solution.
5.17 Proposition. Assume that b: R R is bounded and a: R -- R is Lipschitzcontinuous with a2 bounded away from zero on every compact subset of 01. Then, for every initial condition independent of the driving Brownian motion
W = {Hi, ,,; 0 5 t < co}, equation (5.1) has a nonexploding, unique strong solution.
PROOF. We show first that the boundedness of b prevents the explosion of any solution X to (5.1). We fix t e (0, co) and let n -, co in (5.4); the Lebesgue integral
342
5. Stochastic Differential Equations
Ito b(XN)1{,,s.} ds converges to .1:)^ s b(Xs)ds, a finite expression because b is bounded. On the other hand, the stochastic integral ro a(Xs)1{s<s} dWs converges to ro^ s a(X.,)d W., on the event A A ,i j bA S a 2 (Xs)ds < col, and does
not have a limit on A`; cf. Problem 3.4.11. It develops that the limit of the right-hand side of (5.4) exists and is finite a.s. on A, and does not exist on /1`. On the left-hand side of (5.4) we have lim, X, , s. = X, , s, which is defined
a.s. and is equal to +co on {S < t }. It follows that P[S < t] = 0 holds for every t > 0, so P[S = co] = 1. We turn now to the questions of existence of a weak solution, and of pathwise uniqueness, for equation (5.1). The assumptions on b and a imply (ND), (LI), and the local integrability of a', so weak existence follows from Theorem 5.15. According to Proposition 5.13, the pathwise uniqueness of (5.1) -(2b/o-2)p' is locally bounded, is equivalent to that of (5.46). Because p" p' is locally Lipschitz. It follows that ii(y) defined by (5.48) is locally Lipschitz at every point y E(p( -oo), p(oo)). We have shown that any solution X to (5.1) does not explode, so any solution Y to (5.46), (5.47) must remain in the interval (p( -co), p(co)). Under these conditions, the proof given for Theorem 2.5 shows that pathwise uniqueness holds for (5.46). We appeal to Corollary 3.23 in order to conclude the argument.
Proposition 5.13 raises the interesting issue of determining necessary and sufficient conditions for explosion of the solution X to (5.1). Since Y given
by (5.46) does not explode, and Y, = p(X,), it is clear that the condition p(+cc) = +oo guarantees that X is also nonexploding; however, this sufficient condition is unfortunately not necessary (see Remark 5.18). We develop the necessary and sufficient condition known as Feller's test for explosions in Theorem 5.29.
5.18 Remark. Consider the case of b(x) = sgn(x), a(x)
a > 0. The scale
function p of (5.42) is bounded, and according to Proposition 5.17 the equation (5.1) has a nonexploding, unique strong solution for any initial distribution.
5.19 Remark. The linear growth condition I b(x)I + I a(x)I < K(1 + Ix I );
VX E R
is sufficient for P[S = co] = 1; cf. Problem 3.15.
C. Feller's Test for Explosions We begin here a systematic discussion of explosions. Rather than working exclusively with processes taking values on the entire real line, we start with an interval 1 = (1",r);
-oo < i < r < co
5.5. A Study of the One-Dimensional Case
and assume that the coefficients a: I (ND)'
343
R, b: I -4 F satisfy
Q2(x) > 0;
V x e /, 3 e> 0
(LI)'
such that
V x e l,
ix", + I b(y)I
J.-.
dy < co.
a2(Y)
We define the scale function p by (5.42), where now the number c must be in I. We also introduce the speed measure m(dx) 4
(5.51)
2 dx p'(x)a 2 (X) '
xe I
and the Green's function (5.52)
Ga,b(x, .Y) =
(p(x A y) - p(a))(p(b) - p(x v y)) ; x, y E [a, b] g_ I. p(b) - p(a)
In terms of these two objects, a solution to the equation (5.53)
b(x)M'(x) + lo-2(x)M"(x) = - 1; a < x < b,
(5.54)
M(a) = M(b) = 0
is given by
i G,,,(x, y)m(dy) to
(5.55) M,(x) A-
a -1.
=
(P(x) - P(.0)m(dY) +P(x)
P(a) f b (PO) - P(A)m(dY)
p(b) - p(a)
a
a
As was the case with p", the second derivative Ma",6 exists except possibly on a set of Lebesgue measure zero; we define M,". at every point in (a, b) by using (5.53). Note that G., be , ), and hence Ma.b(), is nonnegative.
5.20 Definition. A weak solution in the interval I = (1', r) of equation (5.1) is a triple (X, W), (Si, ,F, P), {..-,}, where
(i)' condition (i) of Definition 5.1 holds, (ii)' X = {X ..*--,; 0 < t < co} is a continuous, adapted, [t, r]-valued process with X0 e / a.s., and { W A; 0 < t < col is a standard, one-dimensional Brownian motion, (iii)' with {e}n°°_,, and {r},7,, strictly monotone sequences satisfying t < to < r < r, lim,,,, f = t, iim 00 rn = r, and (5.56)
S A inf{t z 0: X, it (e, r)}; n > 1,
the equations (5.3) and (5.4) hold. We refer to (5.57)
S = inf { t
__
0: X, 0 (e , 0} = tun s n-co
344
5. Stochastic Differential Equations
as the exit time from I. The assumption X0 E I a.s. guarantees that P[S > 0] = 1. If = -co and r = +co, Definition 5.20 reduces to Definition 5.1, once we
stipulate that Xi = Xs; S < t < cc. Let (X, W) be a weak solution in 1 of equation (5.1) with X0 = X E (a, b) and set
t = inf { t
_
.
o-2(Xs)ds> n
0:
;
I,
n = 1, 2, ...,
o
Tab= inf {t >0;Xt (a, b)};
e < a < < r.
We may apply the generalized Ito rule (Problem 3.7.3) to M,,,b(X,) and obtain t A T A T.,b
Ma,b(Xt A T A T,,,b) = Ma,b(X)
(t A Tr, A 1:7,0 ± J
Nra,b(X.00-(Xs)d o
co, we see that
Taking expectations and then letting n (5.58)
E(t A Tad)) = Ma,b(X)
Eltla,b(XtA T b) 5 M a b(X) < 00,,
and then letting t co we obtain ET,b < Ma.b(x) < co. In other words, X exits from every compact subinterval of ((,r) in finite expected time. Armed with this observation, we may return to (5.58), observe from (5.54) that limi, EM,,,b(Xt A Tab) = 0, and conclude ET,b = Ma,b(x);
(5.59)
a < x < b.
On the other hand, the generalized Ito rule applied in the same way to p(X,) gives p(x) = Ep(X, A Tab), whence (5.60)
p(x) = Ep(XT.,,b)= p(a)P[XTo.b = a] + p(b)P[XT..b = b],
co. The two probabilities in (5.60) add up to one, and thus
upon letting t (5.61)
P[XT = a] = a.13
p(b) - p(x)
P[XT = b] =
p(b)- p(a)'
p(x)- p(a) p(b) - p(a)
These expressions will help us obtain information about the behavior of X near the endpoints of the interval (t,r) from the corresponding behavior of the scale function. Problem 5.12 shows that the expressions on the right-hand sides of the relations in (5.61) do not depend on the choice of c in the definition of p.
5.21 Remark. For Brownian motion Won I = (-co, co), we have (with c = 0 in (5.42)) p(x) = x, m(dx) = 2 dx. For a process Y satisfying
=+
d(K)diVs, 0
we again have p(R) = g, but m(dx) = (2 dife12(50). Now Y is a Brownian motion run "according to a different clock" (Theorem 3.4.6), and the speed measure simply records how this change of clock affects the expected value of
5.5. A Study of the One-Dimensional Case
345
exit times: (5.62)
- (5-c v y)) 2 di)
ET'a,s(5e) = f b ((x A
b-a
(5:2
Once drift is introduced, the formulas become a bit more complicated, but the idea remains the same. Indeed, suppose that we begin with X satisfying (5.1), compute the scale function p and speed measure m for X by (5.42) and (5.51), and adopt the notation g = p(x), y = p(y), a = p(a), b = p(b); then (5.55), (5.59) show that ET,,,b(x) is still given by the right-hand side of (5.62), where now if is the dispersion coefficient (5.48) of the process Y, A p(Xi). We say Y, = p(X,)
is in the natural scale because it satisfies a stochastic differential equation without drift and thus has the identity as its scale function. 5.22 Proposition. Assume that (ND)', (LI)' hold, and let X be a weak solution of (5.1) in I, with nonrandom initial condition X0 = x e I. Let p be given by (5.42) and S by (5.57). We distinguish four cases:
(a) pv +) = -co, p(r -) = co. Then
P[S = co] =
P[ sup X, =
=
P[
inf X, =
ot
el= t.
In particular, the process X is recurrent: for every ye I, we have
P[X, = y; some 0
t < co] = 1.
(b) pv +)> -co, p(r -) = co. Then
P[ sup X, < 7.1= its (c) p(1 +) = -co, p(r-) < co. Then P[lim X, =
P[lim X, =
P[
inf X, >
1.
0 -co, p(r -) < co. Then
P[lim X =
its
1.
(1=1 - P[lim X = r] = p(r -) - p(x)
P(r -) - p(e +).
its
5.23 Remark. In cases (b), (c), and (d), we make no claim concerning the finiteness of S. Even in case (d), we may have P[S = co] = 1, as demonstrated in Remark 5.18. PROOF OF PROPOSITION 5.22. For case (a), we have from (5.61) for 1' < a <
x < b < r: (5.63)
p[
inf o5.t<s
X1
P[XT,, =
=
1 - (p(x) /p(b))
346
5. Stochastic Differential Equations
Letting b T r, we obtain P [info <,,s X, < a] = 1 for every a e /. Now we let a J e to get P[info ,,,s X, = = 1. A dual argument shows that P[supo <,<s X, =
r] = 1. Suppose now that P[S < co] > 0; then the event {limas X, exists and is equal to e or r} has positive probability, and so {supo <,<5 X, = r} and {info <,<s X, = el cannot both have probability one. This contradiction shows
that P[S < oo] = 0. For case (b), we first observe that (5.63) still implies P [info <,<5 X, = = 1. If, however, we recall P[X7...b = b] = (p(x) - p(a))/(p(b) - p(a)) from (5.61) and let a le, we see that
P[X, = b; some 0 < t < S] = P(x) - P(e+) P(b) - p(e +) Letting now b T r, we conclude that P[suPo
P[ inf X , . =-el=
ot<s
P[
sup Xt <
0
ri = 1.
It remains only to show that limits X, = info <,<5 X and for this it suffices to establish that the limit exists, almost surely. With S as in (5.56), the process Yt(n) A P(XtAs,,) -
+); 0 < t < co is for each n > 1 a nonnegative local
martingale (see (5.49)); letting n oo and using Fatou's lemma, we see that Y A p(X,,$) - p(e +); 0 < t < oo is a nonnegative supermartingale. As such, it converges almost surely as t oo (Problem 1.3.16). Because p: [e ,r) has a continuous inverse, lim,, X, AAS must exist. Case (c) is dual to (b), and case (d) is obtained easily, by taking limits in (5.61).
0
5.24 Example. For the Brownian motion X, = ut + trig, on / = ( -co, co) with drift u > 0 and variance a2 > 0, we have (setting c = 0 in (5.42)) that p(x) = (1 - e-Px)/13 and m(dx) = (2efix /0-2) dx, where /3 = 41/(72. We are in case (c). Compare this result with Exercise 3.5.9.
5.25 Example. For the Bessel process with dimension d > 2 (Proposition 3.3.21), we have I = (0, co), b(x) = (d - 1)/2x, and a2(x) = 1. With c = 1, we obtain (i) for d = 2: p(x) = log x, m(dx) = 2x dx (case (a)),
(ii) for d > 3: p(x) = (1 - x')/(d - 2), m(dx) = 2x" dx (case (c)). Compare these results with Problem 3.3.23.
Proposition 5.17, Remark 5.19, and part (a) of Proposition 5.22 provide sufficient conditions for nonexplosion of the process X in (5.1), i.e., for P[S = co] = 1. In our search for conditions which are both necessary and sufficient, we shall need the following result about an ordinary differential equation.
5.5. A Study of the One-Dimensional Case
347
We define, by recursion, the sequence fun }`°_,) of real-valued functions on /,
by setting u, ar 1 and x
i u_1(z)m(dz)dy; Y
u,,(x) = fc p'(y)
(5.64)
X E 1, n > 1
c
where, as before, c is a fixed number in I. In particular we set for x e /: (5.65)
v(x) A u1 (x) =
.1 x
p'(y)
x
2 dz
TY
, p ( z)o- (z)
f,
dy =
(p(x) - p(y))m(dy)
5.26 Lemma. Assume that (ND)' and (LI)' hold. The series CO
u(x) = > un(x); x e I
(5.66)
n=0
converges uniformly on compact subsets of I and defines a differentiable function with absolutely continuous derivative on I. Furthermore, u is strictly increasing (decreasing) in the interval (c, r) (respectively, (e, c)) and satisfies (5.67)
Icr2(x)u"(x) + b(x)u'(x) = u(x);
(5.68)
u(c) = 1, u'(c) = 0,
a.e. x el,
as well as
1 + v(x) < u(x) < ev(x); x e /.
(5.69)
PROOF. It is verified easily that the functions funI,T_I in (5.64) are nonnegative, are strictly increasing (decreasing) on (c, r) (respectively, v, 0 , and satisfy
a.e. xe I.
lo-2(x)14(x) + b(x)u'(x) = un_1(x),
(5.70)
We show by induction that
u(x) -_
(5.71)
v"(x)
n = 0, 1, 2, ...
;
n!
Indeed, (5.71) is valid for n = 0; assuming it is true for n = k - 1 and noting that uk(x) = p'(x)
(5.72)
f
x
x e I,
uk_1(z)m(dz);
c
we obtain for c < x < r:
i
Uk(X)
=
y vk-1 2)
x 13
'01) .1.x
1
(k
-
(k -
1)!
,
1
M(dZ)dy
0 (y)dv(y) =
(k - 1)!
fx ,
p'(y)vk-1(y)m((c, Ody
vk(kx) . !
A similar inequality holds for e < x 5 c. This proves (5.71), and from (5.72)
348
5. Stochastic Differential Equations
we have also v"-1(x)
lu.(x)1
;
(n - 1)!
n = 1, 2, ... .
It follows that the series in (5.66), as well as I,T=c, u,,'(x), converges absolutely
on I, uniformly on compact subsets. Solving (5.70) for u;; (x), we see that EnTh u;; (x) also converges absolutely, at each point x e /, to an integrable function. Term-by-term integration of this sum shows that E`°_, u;; (x) is almost everywhere the second derivative of u in (5.66), and that u'(x) = u'(x) holds for every x e I. The other claims follow readily. 5.27 Problem. Prove the implications
p(r -) = co
(5.73)
v(r -) = co,
p(( +) = -co .v(e+) = co.
(5.74)
5.28 Problem. In the spirit of Problem 5.12, we could display the dependence of v(x) on c by writing v
x
(5.75)
yc(x) -4'
PjY)
2 dz
dy.
P:(z)o- 2 (Z)
Show that for a, c e 1: (5.76)
va(x) = v(c) + vL(c)p(x) + vc(x);
x e I.
In particular, the finiteness or nonfiniteness of vc(r -), vc(i + ) does not depend on c.
5.29 Theorem (Feller's (1952) Test for Explosions). Assume that (ND)' and be a weak solution in I = (e,r) of (LI)' hold, and let (X, W),
(5.1) with nonrandom initial condition X0 = zeI. Then P[S = co] = 1 or P[S = co] < 1, according to whether
v(( +) = v(r-) = co
(5.77)
or not.
PROOF. Set r A inf {t > 0: yo o-2(Xs)ds> n} and Z") A u(X, A sn n). According
to the generalized Ito rule (Problem 3.7.3) and relation (5.67), Z(") has the representation u (X0ds
4") = Z,(3") +
+
u'(Xs)o-(X0dWs.
To
Consequently, MP) A e-t A S" 2" 4") has the representation tA5, Ar,,
Min) = Mg
e-su'(Xs)o-(Xs)dIgs 0
5.5. A Study of the One-Dimensional Case
349
as a nonnegative local martingale. Fatou's lemma shows that any nonnegative local martingale is a supermartingale, and that this property is also enjoyed
by the process M, A lim, MP) = e-tAsu(X,); 0 5 t < cc. Therefore, Mx = lim e-t^su(X, As)
(5.78)
exists and is finite, almost surely (Problem 1.3.16).
Let us now suppose that (5.77) holds. From (5.69) we see that u(f + ) = u(r-) = co, and (5.78) shows that Mx = oe a.s. on the event {S < }. It follows that P[S < oo] = 0. For the converse, assume that (5.77) fails; for instance, suppose that v(r -) < cc. Then (5.69) yields u(r-) < oo. In light of Problem 5.28, we may assume without loss of generality that c < x < r, and set = inf{t > 0; X, = c }. The continuous process ME
A
T.,. =
e-(t A A Tc)I4(X
t ASA Tc);
GO
is a bounded local martingale, hence a bounded martingale, which therefore converges almost surely (and in 1,1) as t co; cf. Problem 1.3.20. It develops that
- 1 {T<s}. U(X) = Ee-S A T`U(XS AT) = u(r-)Ee- s 1 {s
If P[S = co] = 1, the preceding identity gives u(x) = u(c)Ee-T` < u(c), contradicting the fact that u is strictly increasing on [c, x]. It follows that P[S co] < 1.
5.30 Example. For Brownian motion W on 1 = (-oo, co), we have already computed p(x) = x, m(dx) = 2 dx (with c = 0). Consequently, v(x) = x2 and v(+oo) = oo. 5.24 Example (continued). For Brownian motion with drift it > 0 and variance 62 > 0, it turns out that v(x) = 2(fix - 1 + e- tot)332a2, with /3 _ (2/1/62) and c = 0. Again, v(+oo) = co.
5.25 Example (continued). For the Bessel process with dimension d > 2 and c = 1, we have v(x) = [x2 - 1 - 2p(x)]/d; 0 < x < co, and v(0 + ) = v(oo) = oo.
5.31 Exercise. Consider the geometric Brownian motion X, which satisfies the stochastic integral equation (5.79)
X,=x+ttf
Xsds + v
Xsdl/K,
where x > 0. Use Theorem 5.29 and Proposition 5.22 to show that X, e (0, oo) for all t, and (i) (ii) (iii)
if it < v2/2, then X, = 0, suPo < X, < oo, a.s.; if it > v2/2, then info.<,,, X, > 0, X, = co, a.s.; if it= v2/2, then info <,<, X, = 0, supo
5. Stochastic Differential Equations
350
(The solution to (5.79) is given by
X, = xexpt(it - iv2)t + vW,I, and so (i)-(iii) can also be deduced from the properties of Brownian motion with drift (see, e.g., Problem 2.9.3 and Proposition 2.9.23).)
Let us suppose now that (5.77) is violated, so that P[S < co] is positive. Under what additional conditions can we guarantee that this probability is actually equal to one, i.e., that explosion occurs almost surely? 5.32 Proposition. Assume that (ND)' and (LI)' hold. We have P[S < co] = 1 if and only if one of the following conditions holds: (i) v(r -) < co and v(e +) < co, (ii) v(r -) < co and p(e +) = -co, or (iii) vv +) < co and p(r-) = co.
In the first case, we actually have ES < co.
5.33 Remark. If (i) prevails, then we also have finiteness of p(r-) and p(e +) (Problem 5.27), and we can define by analogy with (5.52) the Green's function G(x, y) =
[P1x A y) -
p(e +)][P(r -) - P1x v .0] ; (x, y)e I 2
I* -) - XI' +)
for the entire interval I = (C, r). We also define the counterpart of (5.55):
M(x) =
f
, G(x, y)m(dy);
x e(e,r).
e
This function satisfies M(e+) = M(r-) = 0, has an absolutely continuous derivative on I, and satisfies the equation (5.53) there. The same procedure
that led to (5.59) now gives ES = M(x) < co, justifying the last claim in Proposition 5.32. PROOF OF SUFFICIENCY IN PROPOSITION 5.32. We just dealt with (i). Suppose
that (ii) holds; then with {e},T=, and {r};,°=, as in Definition 5.20 (iii)', we introduce the stopping times
R g inftt 0; X, = enl; T, . g inf tt
0; X, = /1,
n
1,
'Te -A inftt
0; X, = el = iim Rn. n-,co
Because v(r -) < oo, v( e ) < co, we obtain as in Remark 5.33 that E(R,, A Tr) < co; n 1. On the other hand, (5.73) gives p(r -) < co, so we are in the case (c) of Proposition 5.22. Therefore, for P-a.e. o.) e D, R(a)) = co for sufficiently large n (depending on w), and thus S(w) = 'T,.(a.)) < co. Condition (iii) is dual to (ii). 0
5.5. A Study of the One-Dimensional Case
351
PROOF OF NECESSITY IN PROPOSITION 5.32. Assume that P[S < co] = 1; from
Theorem 5.29 we conclude that either v(( +) < co or v(r -) < co. Let us suppose v(( + ) < co, and that none of (i), (ii), (iii) holds. Then necessarily p(r -) < oo = v(r -) and p(( + ) > -co (remember (5.74)), so we are in case (d) of Proposition 5.22, and thus A, g {limo., X, = r} has positive probability.
But now we recall from the proof of Theorem 5.29 that M, = e-` ^su(X,) is a nonnegative supermartingale with M° of (5.78) an almost surely finite random variable. According to (5.69), u(r -) = co, and so S = co on A,. This shows that P[S < co] < 1, contradicting our initial assumption. It follows that at least one of (i), (ii), (iii) must hold, if P[S < co] = 1 does.
1=1
D. Supplementary Exercises 5.34 Exercise. Take o(x)
1 and I = (-co, co).
(a) If b(x) = lx2, show that P[S < co] = 1. (b) If b(x) = 2x3, show that ES < oo. 5.35 Exercise. Take I = (0, oo) and b(x) = k, o(x) = positive constants.
where k and e are
(i) Show that we are in case (a), (b), or (c) of Proposition 5.22 according as it A (2k/e) is equal to, less than, or greater than one, respectively. (ii) In the first and third cases, the solution is nonexploding; in the second case the origin is reached in finite time with probability one. (iii) The cases e = 2, k = 2, 3, ... should be familiar; can you relate the solution to a known process? (iv) Solve the stochastic differential equation explicitly in the case e2 = 4k. (Hint: Recall Proposition 2.21.)
5.36 Exercise. Show that the solution of the equation dX, = (1 + X,)(1 + X,2)dt + (1 + X,2)dW,;
7I
X0e(- -2' )'
explodes (to +co) in finite expected time. 5.37 Exercise. Discuss the possibility of explosion in the cases: (i) b(x) - x, o(x) = + x2), I = R; (ii) b(x) = o-(x) = Sx, / = (0, co), and e R, > 0; o-(x) = 6x, I = (0, oo), and ye R, S > 0. (iii) b(x) = 1 + (Hint for (iii): Recall Exercise 3.3.33.)
R be locally square-integrable, i.e., V x e R, 3e > 0 such that P,`,!:b2(y)dy < co.
5.38 Exercise. Let the function b: R
352
5. Stochastic Differential Equations
(0 Show that, corresponding to every initial distribution /4. on (FR, R(R)), the
equation dX, = b(X,) dt + dW, has a weak solution up to an explosion time S, and this solution is unique in the sense of probability law.
(ii) Prove that for every finite T > 0, this solution obeys the generalized Girsanov formula: T
P[X e
S > T] = E[exp (f b(W) digs --
2
b2(1Vds)- 1{w.e r}1
for every I" earT(c[o, oon, the a-field of (3.19) with m = 1. Here, W is a one-dimensional Brownian motion with P(Wo e B) = p(B); B ER(R).
(Hint: Try to emulate the proof of Proposition 3.10, and recall the Engelbert-Schmidt zero-one law of Section 3.6.)
(iii) In particular, conclude that with W as in (ii), the nonnegative supermartingale
Z, = exp
b(Ws) dills
--1 f b2(W)ds}; 2
0
t < co,
0
is a martingale if and only if P[S = co] = 1; this is equivalent to cp( + co) = co, where cp(x) = fx 1 Y exp { o
-2 I
Y
b(u) du} dz dy.
o
5.39 Exercise. In the setting of Subsection C, show that for every bounded, piecewise continuous function h: I -> [0, co) and x e (a, b) 1, we have To. I,
h(X,) dt =
Ex o
Ga,b(x, y)h(y)m(dy) a
Here and in the following exercises we denote by a superscript x on probabilities
and/or expectations the initial condition X0 = x. (Hint: Proceed by analogy with (5.55), (5.59).)
5.40 Exercise (Pollack & Siegmund (1985)). In the setting of Subsection C with b and a bounded on compact subintervals of I and with the scale function p(x) and the speed measure m(dx) satisfying
p(e+) = -co, p(r-) = oo, m(I) < co, the solution X to (5.1) starting at x e I never exits I (Proposition 5.22). We assume that (5.80)
Px(X, = z) = 0; V x, z e /, t > 0.
Show in the following steps that the normalized speed measure (m(dx)/m(1)) is the limiting distribution of X:
5.5. A Study of the One-Dimensional Case
lirn Px(X, < z) =
(5.81) (i)
353
m ((i, z ))
m(/)
;
x, z e I.
Introduce the stopping times T, = inf {t 0; X, = a}, Sod, = inf { t Ta; X, = b} for a, b e I. Deduce from the assumptions and (5.59) the positive
recurrence properties, with e
-f
(p(x) - p(y))m(dy) + (p(x) - p(a))- m((a, r)) <
a
Ex Tb =
(P(Y)
P(x))m(dY) + (P(b) - P(x))' m(((, b)) < co,
x
ExS,x = Ex T + E"Tx = (p(u) - p(x))- m(I) < co. (ii) Show that the same methodology as in (i) allows us to conclude, with the help of Exercise 5.39:
s,
Ex 1
1 (e, z)(X f)dt = (p(u) - p(x)) m((e , z));
V z e I.
Jo
(iii) With the aid of assumption (5.80), show that for e<x
x, the function a(t)
Px(X, < z, S,x > t); 0
t < co,
is continuous and D,°-1 max.-1 s ,< a(t) < co. This last condition implies
direct Riemann integrability of a() (see Feller (1971), Chapter XI, Section 1).
(iv) Establish a renewal-type equation
F(t) = a(t) +
ft F(t - s)v(ds);
0 < t < co
for the function F(t) s Px(X, < z), with appropriate measure v(dt) on [0, co). Show that the renewal theorem (Feller (1971), Chapter XI) implies (5.81).
5.41 Exercise (Le Gall (1983)). Provide a proof of Proposition 2.13 based on semimartingale local time. (Hint: Apply the Tanaka-Meyer formula (3.7.9) to the difference X ±4. X(1) X(2' between two solutions X(1), X(2) of (2.1). Use Exercise 3.7.12 to show that
the local time at the origin for X is identically zero, almost surely.)
5.42 Exercise (Le Gall (1983)). Suppose that uniqueness in the sense of probability law holds for (2.1), and that for any two solutions X(1), X(2) on the
same probability space and with respect to the same Brownian motion and initial condition, the local time of X = X(1) - X(2) at the origin is identically equal to zero, almost surely. Then pathwise uniqueness holds for (2.1). (Hint: Use a Tanaka-Meyer formula to show that X(1) v X(2) also solves (2.1).)
354
5. Stochastic Differential Equations
5.6. Linear Equations In this section we consider d-dimensional stochastic differential equations in which the solution process enters linearly. Such processes arise in estimation and control of linear systems, in economics (see Section 5.8), and in various other fields. As we shall see, one can provide a fairly explicit representation for the solution of a linear stochastic differential equation. For most of this section, we study the equation (6.1)
dX, = [A(t)X, + a(t)] dt + a(t)dW, 0 < t < oo,
X, = where W is an r-dimensional Brownian motion independent of the ddimensional initial vector and the (d x d), (d x 1) and (d x r) matrices A(t), a(t), and a(t) are nonrandom, measurable, and locally bounded. In Problem 6.15 we generalize (6.1) for one-dimensional equations by allowing the solution X also to appear in the dispersion coefficient. The deterministic equation corresponding to (6.1) is (6.2)
(t) = A(t)l;(t) + a(t);
(0) =
Standard existence and uniqueness results (Hale (1969), Section 1.5) imply that for every initial condition E Rd, (6.2) has an absolutely continuous solution W) defined for 0 < t < oo. Likewise, the matrix differential equation (6.3)
4(t) = A(t)1(t), 0(0) = I
has a unique (absolutely continuous) solution defined for 0 < t < co. (Here / is the (d x d) identity matrix.) This matrix function b is called the fundamental solution to the homogeneous equation (t) = A(t)c(t).
(6.4)
For each t > 0, the matrix C(t) is nonsingular, for otherwise there would be a to > 0 and a nonzero vector Ae Old such that (1)(t0)). = 0. But t(t)A is a solution to (6.4), and since the identically zero function is the unique solution which vanishes at to, we must have 4:I)(t)). = 0 for all t. This would contradict
the initial condition OM = I. In terms of (I), the solution of the deterministic equation (6.2) is simply (6.5)
(t) = (1)(t)
(0) +
(I)-1 (s)a(s)ds].
J 0
(See Hale (1969), Chapter 3, for additional information.) A pleasant fact is that the solution of (6.1) has a representation similar to (6.5). Indeed, it is easily verified by Ito's rule that 7 (6.6)
X, A 4)(t) [X, + Ij''
4
(s)a(s) ds +
(V' (s)6(s)d141,1; 0 < t < co
solves (6.1). Pathwise uniqueness for equation (6.1) follows from Theorem 2.5.
5.6. Linear Equations
355
6.1 Problem. Suppose that EIRC0112 < co, and introduce the mean vector and covariance matrix functions
m(t) A EX
(6.7)
p(s, t) A- E[(X, - m(s))(X, - m(t))T],
(6.8)
(6.9)
V(t) A- p(t, t).
Show that (6.10)
m(t) = OW [m(0) + J i (1)-1(s)a(s)dsi, o
(6.11)
p(s, t) =
col- V(0) + L
SAt
01)'(u)o-(u)(0-1(u)o-(u))T du]f:DT (t), w0
hold for every 0 < s, t < co. In particular, m(t) and V(t) solve the linear equations (6.12)
rh(t) = A(t)m(t) + a(t),
(6.13)
V(t) = A(t)V(t) + V(t)AT(t) + a(t)aT(t).
6.2 Problem. Show that if X0 has a d-variate normal distribution, then X is a Gaussian process (Definition 2.9.1).
A. Gauss-Markov Processes If X0 is normally distributed, then the finite-dimensional distributions of the
Gaussian process X in (6.6) are completely determined by the mean and covariance functions. In this case, we would like to know under what additional conditions we can guarantee the nondegeneracy of the distribution of X i.e., the positive definiteness of the matrix (6.14)
V(t) = OW [V(0) + f t (I)-1(u)o-(u)(4)-1(u)a(u))T dui
(DT (t),
o
for every t > 0. In order to settle this question, we shall introduce the concept of controllability from linear system theory.
6.3 Definition. The pair of matrix-valued, measurable, locally bounded functions (A, a) is called controllable on [0, T] if for every pair x, y E Rd, there exists a measurable, bounded function v: [0, T] -> Rr, such that (6.15)
Y(t) = x +
ft A(s)Y (s) ds + ..1: cr(s)v(s) ds; o
0
t < T,
o
satisfies Y(T) = y. In other words, for every pair x, y e Rd, there exists a control function v) which steers the linear system (6.15) from Y(0) = x to Y(T) = y.
5. Stochastic Differential Equations
356
6.4 Proposition. The pair of functions (A, a) is controllable on [0, T] if and only if the matrix (6.16)
M(T) A
f
41)-'(t)o-(t)(0-1(t)o-(t))T dt
is nonsingular. PROOF. Let G(t) = (1)-1(t)o-(t). From (6.5), the solution to (6.15) is
Y(t) = (I)(t)[x + J G(s)v(s)dsi, I)
and because of the nonsingularity of (1)(T), controllability is equivalent to the condition that G(s) v(s) ds range over all of OrI as v ranges over the bounded, measurable functions from [0, T] to Or.
Choose an arbitrary z e Old. Under the assumption of nonsingularity of M(T), we may set v(s) = GT (s)M-1 (T)z, and then we have z = G(s)v(s) ds. On the other hand, if M(T) is singular, then there exists a nonzero z EIR° such that zT M(T)z = 0, i.e., SI; z TG(s)GT(s)z ds = 0, which shows that zT G(s) = 0 for Lebesgue-almost every se [0, T]. Consequently, 2T g G(s)v(s) ds = 0 for any bounded, measurable v, which contradicts controllability.
We see from its definition that V(0) is positive semidefinite, so the nonsingularity (and hence the positive-definiteness) of M(T) implies the same property for V(T). We obtain thereby the following result: if the pair (A, a) is controllable on [0, T], then the matrix V(T) of (6.14) is nonsingular. A bit of reflection shows that the two conditions are actually equivalent, provided V(0) = 0, i.e., X0 in (6.1) is nonrandom. When the matrices A and a appearing in (6.1) are constant, this result takes a more explicit form. In this case, the fundamental solution to (6.4) is (6.17)
(Dm
etA A E
tn
An
n=0 n!
and controllability reduces to the following rank condition.
6.5 Proposition. The pair of constant matrices (A, a) is controllable (on any interval [0, T]) if and only if the (d x d) controllability matrix C A [a, Ao-, A2 Ad-10] has rank d.
PROOF. Let us first assume that rank(C) < d, and show that M(T) given by (6.16) is singular for every 0 < T < co. The assumption rank(C) < d is equivalent to the existence of a nonzero vector z a l for which (6.18)
ZT = ZT Acr = ZT A20- =
= Z T Ad-1 Cr = O.
By the Hamilton-Cayley theorem, A satisfies its characteristic equation; thus,
5.6. Linear Equations
357
every positive, integral power of A can be written as a linear combination of /, A, It follows from (6.18) that z TAna = 0; n > 0, and thereby z7-0-1
E (-trzTA"o- = 0; t > 0.
zT e-tA
n!
n=0
The singularity of M(T) follows from zT M(T)z = 0. Let us now assume that for some T > 0, M(T) is singular and show that rank (C) < d. The singularity of M( T)enables us to find a nonzero vector z E Rd such that
0 = zT M(T)z =
From this we see that f(t)
f
lIzTe-mo-112 dt.
is identically zero on [0, T], and
zT e-"`a
evaluating f(0), f'(0),...,f(d-1)(0) we obtain (6.18). When A and a are constant, equations (6.13) and (6.14) take the simplified form V(t) = AV(t) + V(t)AT
(6.13)' (6.14)'
V(t) =
e" [V(0) +
aaT,
--sAcro.re-sAT ds
ll
etAT.
One could hope that by proper choice of V(0), it would be possible to obtain
a constant solution to these equations. Under the assumption that all the eigenvalues of A have negative real parts, so that the integral VA
(6.19)
esito.o.resAT ds
converges, one can verify that V(t) V does indeed solve (6.13)', (6.14)'. We leave this verification as a problem for the reader. 6.6 Problem. Show that if V(0) in (6.14)' is given by (6.19), then V(t) = V(0). In particular, V of (6.19) satisfies the algebraic matrix equation AV + VAT = --o-o-T.
(6.20)
We have established the following result.
6.7 Theorem. Suppose in the stochastic differential equation (6.1) that a(t) a a, a(t) 0, all the eigenvalues of A(t) A have negative real parts, and the initial random vector = X0 has a d-variate normal distribution with mean m(0) = 0 and covariance V = E(X0.11) as in (6.19). Then the solution X is a stationary, zero-mean Gaussian process, with covariance function (6.21)
p(s,
=
Ve(`-"T; (e("-"V;
0 < t < s < op 0 5 s < t < co.
358
5. Stochastic Differential Equations
PROOF. We have already seen that V(t) = V; (6.21) follows from (6.14)' and (6.11).
6.8 Example (The Ornstein-Uhlenbeck Process). In the case d = r = 1, a(t) .m0, A(t) = -a < 0, and a(t) a > 0, (6.1) gives the oldest example
dX, = -aXidt + a dW,
(6.22)
of a stochastic differential equation (Uhlenbeck & Ornstein (1930), Doob (1942), Wang & Uhlenbeck (1945)). This corresponds to the Langevin (1908)
equation for the Brownian motion of a particle with friction. According to (6.6), the solution of this equation is X, = X 0e-Ga + o-
ft e-2"-s) dWs;
0 < t < co.
o
If EX,!, < co, the expectation, variance, and covariance functions in (6.7)-(6.9) become
m(t) A- EX, = m(0)e', 0.2
0.2
+ (V(0)
V(t) A Var(X,) =
- -2a) e-2', 0.2
p(s,t) A Cov(X Xt) = [V(0) + - (e2'(` ^ s) - Me-2(f'). 2a
If the initial random variable X0 has a normal distribution with mean zero and variance (a2/2a), then X is a stationary, zero-mean Gaussian process with covariance function p(s, t) = (a2/2a)e-'1".
B. Brownian Bridge Let us consider now the one-dimensional equation (6.23)
dX, =
TT
X, t
dt + dWi;
0 < t < T, and Xo = a,
for given real numbers a, b and T > 0. This is of the form (6.1) with A(t) = -11(T - t), a(t) = b /(T - t) and a(t) ._ 1, whence OW = 1 - (t /T). From (6.6) we have
X,=a(1-
-t) t)
t
+b-T +(T
dig -Of Ts; 0
6.9 Lemma. The process (6.24)
X=
{(T - t) 0;
f
' digs
T
s;
Cl-t
'T.
5.6. Linear Equations
359
is continuous, zero-mean, and Gaussian, with covariance function
0 < S, t < T
p(s, t) = (s A t)
(6.25)
f o (T - s)-1 dWs; 0 < t < T is a continuous, squareintegrable martingale with quadratic variation PROOF. The process M,
ds
o
<M>t
0 (T - s)2
1
1
T-t
T
According to Theorem 3.4.6, there exists a standard, one-dimensional Brownian motion B such that M, = Boot; 0 < t < T It follows from the strong law of large numbers for Brownian motion (Problem 2.9.3) that Y, -4 B0011(<M>, + T-1) converges almost surely to zero as t T T The process Y of (6.24) is thus seen to be almost surely continuous on [0, T], and to be a zero-mean Gaussian process with tAs
E(Y,Y,) = (T - t)(T - s) f o
du
(T - u)2
st
t)--T,
= (s
provided 0 < s,t<TIfsv t = T, the preceding expectation is trivially zero, in agreement with (6.25).
6.10 Corollary. The process
(I
(6.26)
- -tT ) + b-tT + (T - t) fo' dW T -s s
0-
b;
is Gaussian with a.s. continuous paths, expectation function (6.27)
m(t) A E(X,) = a
1 - -t + b -t 0 < t < T T'
and covariance function p(s, t) given by (6.25). This process is the pathwise unique solution of equation (6.23) on [0, T).
6.11 Problem. Show that the finite-dimensional distributions for the process X in (6.26) are given by (6.28)
P[X,e dx,, =
,
X,. e dxn]
p(ti - ti-i; x,- xi) t=i
p(T - t; xn, b) p(T; a, b)
dx,
dx,,,
where 0 = to < t, < < t < T, x, = a, (x,, , xn)e ER", and p(t; x, y) is the Gaussian kernel (2.2.6). (Hint: The normal random variables Zr Xi,/
(T - ti) - X, i/(T - 4_1); i = 1, . . ,n, are independent.) .
5. Stochastic Differential Equations
360
6.12 Definition. A Brownian bridge from a to b on [0, T] is any almost surely continuous process defined on [0, T], with finite dimensional distributions specified by (6.28).
It is apparent that any continuous, Gaussian process on [0, T] with mean and covariance functions specified by (6.27) and (6.25), respectively, is a Brownian bridge from a to b. Besides the representation of Brownian bridge as the solution to (6.23), there are two other ways of thinking about this process; they appear in the next two problems. 6.13 Problem. Let {l4',.; 0 < t < cm}, (S2,,), {Pa}ac n be a one-dimensional
Brownian family. Show that for 0 = to < t, <
< t,, < T, x0 = a, and
(x ... , x.) e IV% the conditional finite-dimensional distributions I [W,, e dx"..., W. e dx,,IWT = b] are given by the right-hand side of (6.28), for Lebesgue-almost every be P. In other words, Brownian bridge from a to b on [0, T] is Brownian motion started at a and conditioned to arrive at b at time T.
6.14 Problem. Let W be a standard, one-dimensional Brownian motion and define (6.29)
t
Brb A a ( 1
t
t
0 < t < 'T.
Then Barb is a Brownian bridge from a to b on [0, T].
C. The General, One-Dimensional Linear Equation Let us consider the one-dimensional (d = 1, r > 1) stochastic differential equation (6.30)
dX, =
(t) X, + a(t)] dt +
[S;(t)X, + o-j(t)] i =i
, W,(6), 0 < t < co} is an r-dimensional Brownian motion, and the coefficients A, a, Si, oi are measurable, {. }-
where W = {141, = (Him,
adapted, almost surely locally bounded processes. We set 1
ift
ct
J=1
(6.31)
Zt 4- exp
o
S(u)dW°) "
r
- -2 j=1
S.12(u)du, To
[ft A(u) du +
6.15 Problem. Show that the unique solution of equation (6.30) is
5.6. Linear Equations
(6.32)
361
X, = Z, [X0 +
f
1
- la(u) 0
S;(u)o-,(u)} du + 1=1
.1=1
Wol f af(u)d Zu "
In particular, the solution of the equation
dX, = A(t)X,dt +
(6.33)
S;(t)X,dW,u) J=1
is given by (6.34)
X, = Xoexp[f {A(u) -
,
1
2,
Si (u)} au
.1=
f
Si(u) dW,,u)].
0
In the case of constant coefficients A(t) = A, S;(t) S; with 2A < E;=, s.,2 in (6.34), show that lim X, = 0 a.s., for arbitrary initial condition Xo.
D. Supplementary Exercises 6.16 Exercise. Write down the stochastic differential equation satisfied by Y = X1`; 0 < t < co, with k > 1 arbitrary but fixed and X the solution of equation (5.79). Use your equation to compute E(X,k). 6.17 Exercise. Define the d-dimensional Brownian bridge from a to b on [0, T] (a, b e Rd) to be any almost surely continuous process defined on [0, T], with finite dimensional distributions specified by (6.28), where now iix
p(t; x, y) = (27rt)-(d/2) exp
Yil 2
2t
J
x, y e
t > O.
(i) Prove that the processes X given by (6.26) and BO given by (6.29), where W is a d-dimensional Brownian motion with Wo = 0 a.s., are d-dimensional Brownian bridges from a to b on [0, T]. (ii) Prove that the d-dimensional processes fig'; 0 < t < T} and { Br: ; 0 < t < T} have the same law. (iii) Show that for any bounded, measurable function F: C[0, T]' R, we have (6.35)
EF(a + W.) =
f
EF(B!')p(T; a, b)db.
ud
6.18 Exercise. Let (I): R be of class C2 with bounded second partial derivatives and bounded gradient V(I), and consider the Smoluchowski equation (6.36)
dX, = V(1)(X,)dt + dWi;
0 < t < co,
362
5. Stochastic Differential Equations
where W is a standard, Rd-valued Brownian motion. According to Theorems 2.5, 2.9 and Problem 2.12, this equation admits a unique strong solution for every initial distribution on Xo. Show that the measure u(dx) = e2G(x)dx
(6.37)
on a(Rd)
is invariant for (6.36); i.e., if X(') is the unique strong solution of (6.36) with initial condition Xr = a e Rd, then (6.38)
µ(A) =
f
P(X:")e A) p(da);
V A e M(0')
holds for every 0 < t < (Hint: From Corollary 3.11 and the Ito rule, we have (6.39)
Ef Ma)) = E[f(a +
exp Istro(a +
- (1)(a)
- 1 f (AO + IIV(1)112)(a +
ds}1
for every f e C43(01°). Now use Exercise 6.17 (ii), (iii) and Problem 4.25.)
6.19 Remark. If d = 1 in Exercise 6.18, the speed measure of the process X is
given by m(dx) = 2 exp{ - 20(c)}tt(dx) and is therefore invariant. Recall Exercise 5.40.
6.20 Exercise (The Brownian Oscillator). Consider the Langevin system d X, = Y, dt
dY, = - X, dt - aY, dt + o- dW where W is a standard, one-dimensional Brownian motion and /3, a, and a are positive constants. (i) Solve this system explicitly.
(ii) Show that if (X0, Yo) has an appropriate Gaussian distribution, then (X 171) is a stationary Gaussian process. (iii) Compute the covariance function of this stationary Gaussian process.
6.21 Exercise. Consider the one-dimensional equation (6.1) with a(t) = 0, a(t) a > 0, A(t) < -a < 0, V 0 t < cc, and = x e R. Show that 0.2
E X,2 <
2a
x2
0.2
2a
e-2t;
Vt > O.
= (W,(1), Wt(r)),,t; 0 < t < co} be an rdimensional Brownian motion, and let A(t), S(P)(t); p = 1, . . . , r, be adapted, bounded, (d x d) matrix-valued processes on [0, T]. Then the matrix stochas-
6.22 Exercise. Let W =
5.7. Connections with Partial Differential Equations
363
tic integral equation (6.40)
X(t) = I +
f A(s) X (s)ds +
J p=1
t
S(P)(s)X (s)dW,(P)
o
has a unique, strong solution (Theorems 2.5, 2.9). The componentwise formulation of (6.40) is d
Xki(t) = oki + E
t
d
snsp,;(s)dws(P).
A ke(s)X o(s) ds + E E p=11=1
C =1
0
Show that X(t) has an inverse, which satisfies (6.41)
X-1(t) = I + I X -1(s)[
-E P =1
(S(P)(s))2 - A(s)ids P -1
o
f x-i(s)s(P)(s) d HIP) 0
5.7. Connections with Partial Differential Equations The connections between Brownian motion on one hand, and the Dirichlet and Cauchy problems (for the Poisson and heat equations, respectively) on the other, were explored at some length in Chapter 4. In this section we document analogous connections between solutions of stochastic differential equations, and the Dirichlet and Cauchy problems for the associated, more general elliptic and parabolic equations. Such connections have already been presaged in Section 4 of this chapter, in the prominent role played there by the differential operators a', and (a /at) + si as well as in the relevance of the Cauchy problem to the question of uniqueness in the martingale problem (Theorem 4.28).
In Chapter 4 we employed probabilistic arguments to establish the existence and uniqueness of solutions to the Dirichlet and Cauchy problems considered there. The stochastic representations of solutions, which were so useful for uniqueness, will carry over to the generality of this section. As far as existence is concerned, however, the mean-value property for harmonic
functions and the explicit form of the fundamental solution for the heat equation will no longer be available to us. We shall content ourselves, therefore, with representation and uniqueness results, and fall back on standard references in the theory of partial differential equations when an existence result is needed. The reader is referred to the notes for a brief discussion of probabilistic methods for proving existence. Throughout this section, we shall be considering a solution to the stochastic integral equation
364
5. Stochastic Differential Equations s
(7.1)
X!") = x+ f 6(0, XV))d0 +
o-(0, Xl,"))dW,;
t s s<
cc
,
i
under the standing assumptions that (7.2)
(7.3) (7.4)
the coefficients bi(t, x), 6,,(t, x): [0, co) x Rd R are continuous and satisfy the linear growth condition (2.13);
the equation (7.1) has a weak solution (X("), W), (SI, .F, P), {cr's} for every pair (t, x); and
this solution is unique in the sense of probability law.
We frequently suppress the superscripts (t, x) in X(), and write E" to indicate the expectation computed under these initial conditions. Associated with (7.1) is the second-order differential operator At, of (4.1). When b and o- do not depend on t, we write Al (as in (1.2)) instead of si, and EX instead of E".
A. The Dirichlet Problem Let D be an open subset of Rd, and assume that b and a do not depend on t. 7.1 Definition. The operator .sal of (1.2) is called elliptic at the point x e l if d
d
aik(xgi4 > 0; EE 1=1 k=1
;/ E Old \ {0}.
If Ai is elliptic at every point of D, we say that .91 is elliptic in D; if there exists
a number 6 > 0 such that d
d
E E aik(xgik 1=1 k=1
611a2; VxeD,eRd,
we say that sal is uniformly elliptic in D.
Let .sal be elliptic in the open, bounded domain D, and consider the continuous functions k: D [0, co), g: D R, and f: ap R. The Dirichlet P such that u is of class C2(D)
problem is to find a continuous function u: D and satisfies the elliptic equation (7.5)
- ku = -g;
in D
as well as the boundary condition (7.6)
u = f; on ap.
7.2 Proposition. Let u be a solution of the Dirichlet problem (7.5), (7.6) in the open, bounded domain D, and let tD -4 inf It > 0; X,It DI. If (7.7)
Ex D < co;
VxeD,
5.7. Connections with Partial Differential Equations
365
then under the assumptions (7.2)-(7.4) we have (7.8)
U(X) = Ex[f(X,D)exp {
-i
rift
f
k(Xs)ds}+
o
tb
g(X,)exp { -
J k(Xs)ds} dt] o
o
for every x ED.
7.3 Problem. Prove Proposition 7.2. When should the condition (7.7) be expected to hold? Intuitively speaking, if
there is enough "diffusion" to guarantee that, in at least one component, X behaves like a Brownian motion, then (7.7) is valid. We render this idea precise in the following lemma.
7.4 Lemma. Suppose that for the open, bounded domain D, we have for some 1 < 1 < d: min ciee(x) > 0.
(7.9)
xeD
Then (7.7) holds.
PROOF (Friedman (1975), p. 145). With b A mazE 51b(x)I, a A minx 5 ciee(x),
A mine 5 Xe, and v > (2b/a), we consider the function h(x) = -fiexp(vxj); x = (x1,...,xd)eD, where the constant ti > 0 will be determined later. This function is of class C' (D) and satisfies q
-(.91h)(x) = ye"i{iv2aei(x) + vbj(x)}
-2 'Iva e" v -2ba- ; 1
xeD.
Choosing p sufficiently large, we can guarantee that .cdh < -1 holds in D. Now the function h and its derivatives are bounded on D, so by Ito's rule we have for every x ED, t > 0:
ET A ID) _.._ h(x) -
Ex h(X, A,D) s 2 max ih(y)1 < 00.
yD
Let t -> c0 to obtain (7.7).
El
75 Remark. Condition (7.9) is stronger than ellipticity but weaker than uniform ellipticity in D. Now suppose that in the open bounded domain D, we have that (i) sad is uniformly elliptic, (ii) the coefficients au, b,, k, g are Holder-continuous, and (iii) every point a e OD has the exterior sphere property; i.e., there exists a ball B(a) such that B(a) n D = 0, B(a) n 8D = {a}.
We also retain the assumption that
f is continuous on
OD.
Then there exists
5. Stochastic Differential Equations
366
a function u of class C(D) n CZ(D) (in fact, with Holder-continuous second partial derivatives in D), which solves the Dirichlet problem (7.5), (7.6); see Gilbarg & Trudinger (1977), p. 101, Friedman (1964), p. 87, or Friedman (1975), p. 134. By virtue of Proposition 7.2, such a function is unique and is given by (7.8).
B. The Cauchy Problem and a Feynman-Kac Representation With an arbitrary but fixed T > 0 and appropriate constants L > 0, 2 > 1, we consider functions f (x): Rd -, R, g(t, x): [0, T] x08° -, 01 and k(t, x): [0, T] x08° -.10, co) which are continuous and satisfy (7.10)
(i)
1./(x)1 < L(1 + 11421)
(ii) f(x) > 0; V xe Rd
Or
as well as (7.11)
L(1 + 11421)
(i) Ig(t,x)I
or (ii) g(t, x) > 0;
V 0 < t < T, x e Rd.
We recall also the operator .21, of (4.1), and formulate the analogue of the Feynman-Kac Theorem 4.4.2:
7.6 Theorem. Under the preceding assumptions and (7.2)-(7.4), suppose that v(t, x): [0, T] x08° - 08° is continuous, is of class CL2([0,T) x08°) (Remark 4.1), and satisfies the Cauchy problem av
(7.12)
--at + kv = At) + g; in [0,T) x08°,
(7.13)
v(T, x) . f(x); x e01°,
as well as the polynomial growth condition (7.14)
max Iv(t, x)1 0
M(1 + II x112P);
x e Rd,
for some M > 0, p _. 1. Then v(t, x) admits the stochastic representation (7.15)
v(t, x) =
E"[f (X 7.) exp { -i T
+ 1 g(s, Xs) exp {
T
k(0, XOrli s
-i k(0,X0)0} ds]
on [0, T] x Rd; in particular, such a solution is unique. PROOF. Proceeding as in the proof of Theorem 4.4.2, we apply the Ito rule to
the process v(s, Xs)exp{ -If k(0, X 8)0}; s e [t, T], and obtain, with Ti, '' inf{s > t; IIX sll
n},
5.7. Connections with Partial Differential Equations
(7.16)
v(t, x) =
E"[J.
367
T
g(s, XS) exp -J k(0, X8) dO} ds]
+ E" [v(t,
exp
+ E'x[f (X T) exp
{-
{-
k(0, X0) dei
k(0, X0) de}
ilt
Let us recall from (3.17) the estimate (7.17)
Et' max 11X9112m
C(1
0012m)eC(s-t);
t < < T,
i
valid for every m > 1 and some C = C(m, K, T, d) > 0. Now the first term on the right-hand side of (7.16) converges as n -p cc to J'
k(0, X0)0} ds,
g(s, Xs) exp -
by either the dominated convergence theorem (thanks to (7.11) (i) and (7.17)) or the monotone convergence theorem (if (7.11) (ii) prevails). The second term is bounded in absolute value by
E"[I ver,
(7.18)
M(1 + n2")Pfx[T,, < T].
However, this last probability can be written as max r x[r
pt,x[tn
n
ri-2'"E" max
II X 0112in
cn-2m11
by virtue of (7.17) and the Ceby§ev inequality. Simply selecting m > u, we see that the right-hand side of (7.18) converges to zero as n cc. Finally, the last term in (7.16) converges to Et.
x[f (X T) exp j -f
k(0, X0) d0}1,
either by the dominated or by the monotone convergence theorem. 7.7 Problem. In the case of bounded coefficients, i.e., (7.19)
I b,(t,
x) I + E 0-4(t,x) < p; 0 < t < co, x
1
d,
J=1
the polynomial growth condition (7.14) in Theorem 7.6 may be replaced by (7.20)
max I v(t, x)I < Me'llx1I2; (:)
x e Rd
T
for some M > 0 and 0 < u < (1/18p Td). (Hint: Use Problem 3.4.12.)
368
5. Stochastic Differential Equations
7.8 Remark. For conditions under which the Cauchy problem (7.12), (7.13) has a solution satisfying the exponential growth condition (7.20), one should consult Friedman (1964), Chapter I. A set of conditions sufficient for the existence of a solution v satisfying the polynomial growth condition (7.14) is: (i)
Uniform ellipticity: There exists a positive constant S such that d
d
E E a,(t, x)M ,_. 1=1 k=1
451a2
holds for every e Rd and (t, x) e [0, co) x Rd; (ii) Boundedness: The functions aik(t, x), bi(t, x), k(t, x) are bounded in [0, 7] x Rd; (iii) Holder continuity: The functions aik(t, x), bi(t, x), k(t, x), and g(t, x) are uniformly Holder-continuous in [0, T] x Rd; (iv) Polynomial growth: The functions f(x) and g(t, x) satisfy (7.10) (i) and (7.11) (i), respectively.
Conditions (i), (ii), and (iii) can be relaxed somewhat by making them local
requirements. We refer the reader to Friedman (1975), p. 147, for precise formulations. 7.9 Definition. A fundamental solution of the second-order partial differential equation au (7.21) + ku = .21,u
-t
is a nonnegative function G(t, x; t, defined for 0 < t < r < T, x e Rd, e Rd, with the property that for every f e Co(Rd), t e (0, T], the function (7.22)
G(t, X; T,
u(t, x) 4=='-
(0 cg; 0 < t < T, x e
Rid
is bounded, of class C1'2, satisfies (7.21), and (7.23)
lim u(t, x) = f(x);
ttt
x e Rd.
Under conditions (i)-(iii) of Remark 7.8 imposed on the coefficients bi(t, x), aik(t, x), and k(t, x), a fundamental solution of (7.21) exists (see Friedman (1975), pp. 141, 148 and Friedman (1964), Chapter I). For fixed (r, e (0, T] x Rd, the
function co(t, x) g G(t, x; t,
is of class C1'2([0, r) x Rd) and satisfies the backward Kolmogorov equation (7.21) in the backward variables (t, x). If, in addition, the functions (a/ext)bat, x), (0/axi)aik(t, x), (a2 /ax; axk)aik(t, x) are bounded and Holder-continuous, then for fixed (t, x) e [0, T) x l the function
thr,
-4 G(t, x; t,
5.7. Connections with Partial Differential Equations
369
is of class C12((t, T] x Rd) and satisfies the forward Kolmogorov equation (7.24)
0
ar
d
32
d
=2 E E 1=1 k=1 1
t/(T,
-
i=1
[a.k(T, )tl(T,
[bAT , 00-r, 01 - k(-r, 00T, 0
in the forward variables (r, Returning to the Cauchy problem (7.21), (7.23) with k = 0, we recall from Theorem 7.6 that its solution is given by
u(t, x) = Ef(n.x)); f e Co(Rd).
(7.25)
A comparison of (7.22), (7.25), in conjunction with Problem 4.25, leads to the conclusion that any fundamental solution G(t, x; r, is also the transition probability density for the process )0) determined by (7.1); i.e.,
(7.26) P[Xx) e A] =
G(t, x; r,
g; A e .4
0
t<
T
A
In particular, under the conditions (7.2)-(7.4), this fundamental solution is unique, and the representation (7.15) of the solution to the Cauchy problem av
--at = sit!) +
(7.27)
in [0, T) x Rd,
g;
v(T, x) = f(x);
(7.28)
x e Rd,
now takes the form T
(7.29)
v(t, x) =
G(t, x; T, Of(0 do +
J Rd
J
G(t, x; T,
J t
g(r, )(g dr.
Rd
C. Supplementary Exercises 7.10 Exercise. The Cauchy problem (7.27), (7.28) does not include the potential term kv appearing in (7.12). The case of nonzero k corresponds to a diffusion
with killing. In particular, let X() be the solution to (7.1); let Y be an independent, exponentially distributed random variable with mean 1; and define the lifetime p(t.X)
inf
> t;
k(9, XS")) d0 >
The killed diffusion process is (i.x) A
Txp -); t < s < p("), A;
s y po-x),
370
5. Stochastic Differential Equations
where A is a cemetery state isolated from Rd. Assume the conditions of Theorem 7.6 and let G(t, x; r, have
be a fundamental solution of (7.21). Then we
(7.30) P[X!" e A] = J G(t, x; r, g; A e .4(GV), 0 < t < r < T, A
and the solution (7.15) of the Cauchy problem (7.12), (7.13) takes the form (7.29).
7.11 Exercise. Suppose that bi(t, x); 1 < i < d, are uniformly bounded on [0, T] x R' and that f(x) and g(t, x) satisfy (7.10) and (7.11), respectively. If v(t, x) is a solution to the Cauchy problem at1
= Av + (b, Vv) + g; in [0, T) x Rd
-
v(T, x) = f(x);
xe
and (7.20) holds, then T-t
v(t, x) -= Ex[f(W7-_,) exp 1.
(b(t + 0, WO, dWo) o
+ 0, W0)112
t
d0}
o
T-t
g(t + s, Ws) exp
-2 -1
where {W .F,; 0 < t < T },
f
Il b(t 0
8
(b(t + 0, Wo), d Wo)
+
Wo)112
d0} ds],
Px 1.E Re is a d-dimensional Brownian
(CI,
family.
7.12 Exercise. Write down the Kolmogorov forward and backward equations with k 0 for one-dimensional Brownian motion with constant drift it, and
verify that the transition probability density of this process satisfies these equations in the appropriate variables. 7.13 Exercise. Let the coefficients b, a in (7.1) be independent of t, and assume that condition (7.9) holds for every open, bounded domain D c Rd. Suppose also that there exists a function f : R° \ {0} R of class C2, which satisfies (7.31)
and is such that F(r)
df(x) 5 0
on Rd\ {0}
is strictly increasing with
(i) Show that we have the recurrence property
F(r) = co.
5.8. Applications to Economics (7.32)
371
Px(T, < co) = 1;
for every r > 0, where B, = {X G Rd; (ii) Verify that (7.32) holds in the case (7.33)
(x, b(x)) +
1>
aii(x)
II
\B,
Vxe
X II
< r} and T,. = inf It
(x, a(x)x);
0; X, E B, 1.
V x E Rd\{0}.
1x112
(iii)
If (7.31) is strengthened to sif(x) < -1; V x e Rd\ {0}, then we have the positive recurrence property (7.34)
ExT, < 00
VXE
\ fir.
5.8. Applications to Economics In this section we apply the theory of stochastic calculus and differential equations to two related problems in financial economics. The first of these is option pricing, where we derive the celebrated Black & Scholes (1973) option pricing formula. The second application is the optimal consumption/investment
problem formulated by Merton (1971). These problems are unified by their reliance on the theory of stochastic differential equations to model the trading of risky securities in continuous time. In the second problem, this theory allows us to characterize the value function and optimal consumption process in a context more general than considered heretofore. We subsequently specialize the model to the case of constant coefficients, so as to illustrate the use of the Hamilton-Jacobi-Bellman equation in stochastic control.
A. Portfolio and Consumption Processes Let us consider a market in which d + 1 assets (or "securities") are traded continuously. We assume throughout this section that there is a fixed time horizon 0 < T < oc. One of the assets, called the bond, has a price Po(t) which evolves according to the differential equation (8.1)
dPo(t) = r(t)P0(t)dt, P0(0) = po;
0 < t < T.
The remaining d assets, called stocks, are "risky"; their prices are modeled by the linear stochastic differential equation for i = 1, .. , d: (8.2)
dPi(t) = bi(t)Pi(t)dt + PM)
Pi(0) = pi;
0
J =1
The process W = {W = (W,"),..., W,49)T,,F,; 0 < t < T} is a d-dimensional Brownian motion on a probability space (f2, P), and the filtration 1.F, is the augmentation under P of the filtration {",""} generated by W. The interest rate process {r(t), .9v,; 0 5 t < T}, as well as the vector of mean rates of return
372
5. Stochastic Differential Equations
{b(t) = (131(t),...,bd(t))T,,,: 0 S t 5 T} and the dispersion matrix {O(t) = (a1 i(t))1< i,j 0, (8.3)
.ra(t)
Eg112;
e[Fad,
0 < t < T,a.s.
8.1 Problem. Under assumption (8.3), a T(t) has an inverse and
- , - a; V e Rd, 0 < t < T,
(8.4)
a.s.
11(aT(01-1
Moreover, with d(t) A 6T(t)a(t), we have (8.5)
Ta(cg
a.s.
gl12; v
so a(t) also has an inverse and (8.6)
11(010)-111
II;
Ve
0_
a.s.
We imagine now an investor who starts with some initial endowment x > 0
and invests it in the d + 1 assets described previously. Let NM denote the number of shares of asset i owned by the investor at time t. Then X0 a x = El=c, N,(0)p,, and the investor's wealth at time t is
X, = E Ni(t)P,(t).
(8.7)
i=o
If trading of shares takes place at discrete time points, say at t and t + h, and there is no infusion or withdrawal of funds, then (8.8)
Xt+h - X, =
INI,(t)[P,(t + h) - PM]. i=0
If, on the other hand, the investor chooses at time t + h to consume an amount hCH.,, and reduce the wealth accordingly, then (8.8) should be replaced by d
(8.9)
Ni(t)[P,(t + h) - P,(t)] - hC,+h.
Xt+h - Xt = i=o
The continuous-time analogue of (8.9) is
Ni(t)d13,(t) - cdt.
dX, = i=o
Taking (8.1), (8.2), (8.7) into account and denoting by tr,(t) A N,(t)P,(t) the amount invested in the i-th stock, 1 < i < d, we may write this as (8.10) d
d
d
dX, = (r(t)X, - C,)dt + E (kW - r(t))n,(t)dt + E E ni(t)a,i(t)dW,(11. 1=1
1=1 j =1
5.8. Applications to Economics
373
8.2 Definition. A portfolio process it = {n(t) = (n 1(t), is a measurable, adapted process for which T
d
E
(8.11)
,nd(t))T,..97,; 0 5 t
ir(t)dt < co,
i=1
a.s.
o
A consumption process C = {C,Ft; 0 < t < T} is a measurable, adapted process with values in [0, co) and C, dt < co, a.s.
(8.12)
fo
83 Remark. We note that any component of n(t) may become negative, which is to be interpreted as short-selling a stock. The amount invested in the bond, d
no(t)
X t - E nt(t), i=1
may also be negative, and this amounts to borrowing at the interest rate r(t).
8.4 Remark. Conditions (8.11), (8.12) ensure that the stochastic differential equation (8.10) has a unique strong solution. Indeed, formal application of Problem 6.15 leads to the formula (8.13)
X, = e.V. r(s) as
{x +
to e-P,roo du ur(s)T(b(s)
- r(s)1) - C5] ds
0 < t < T, ft CS° r(u)d" irT (s)o-(s)dWs}; + where 1 is the d-dimensional vector with every component equal to 1. All vectors are column vectors, and transposition is denoted by the superscript T The verification that under (8.11), (8.12), the process X given by (8.13) solves (8.10) is straightforward.
8.5 Definition. A pair (7c, C) of portfolio and consumption processes is said to
be admissible for the initial endowment x > 0 if the wealth process (8.13) satisfies (8.14)
X > 0; 0 < t < T, a.s.
If b(t) = r(t)1 for 0 S t < T, then the discount factor CP. os) ds exactly offsets the rate of growth of all assets and (8.13) shows that (8.15)
M, -A X te-
r(s) ds
x+
r(u) du Cs ds
is a stochastic integral. In other words, the process consisting of current wealth plus cumulative consumption, both properly discounted, is a local martingale. When b(t) r(t)1, (8.15) is no longer a local martingale under P, but becomes
5. Stochastic Differential Equations
374
one under a new measure P which removes the drift term n(t)T(b(t) - r(t)1) in (8.10). More specifically, recall from Problem 8.1 that the process 0(t)
(8.16)
(6(t))-1(b(t) - r(t)1)
is bounded, and set (8.17)
Z = exp {- E
0(s)dWP1 2
i=1
Then Z = {Z
f 0
110(012 ds}.
0 < t < T} is a martingale Corollary 3.5.13); the new
probability measure (8.18)
P(A) A E[ZT1A]; A E .FT
is such that P and P are mutually absolutely continuous on .FT, and the process
PfrAW,+fi 0(s)ds; 0< t< T,
(8.19)
is a d-dimensional Brownian motion under P (Theorem 3.5.1). In terms of IV, (8.13) may be rewritten as
ds = x + Ir e-p,r(u)du ITT (00.(s) dais;
e-po r(u) du
(8.20) Xie-f; r(s)ds
JO
0
0 < t < T, a.s. For an admissible pair (it, C) the left-hand side of (8.20) is nonnegative and the right-hand side is a P-local martingale. It follows that the left-hand side, and hence also Xie- A r(s)ds is a nonnegative supermartingale under P (Problem 1.5.19). Let (8.21)
To = T A inf It e [0, T]; X, = 0 }.
According to Problem 1.3.29,
X, = 0; To < t < T holds a.s. on {To < If To < T, we say that bankruptcy occurs at time To. From the supermartingale property in (8.20) we obtain (8.22)
E[XTe-gr(s)ds
Cte- r(s)ds dt < x,
whence the following necessary condition for admissibility: (8.23)
E
e-Jo r(s)ds C, dt < x.
This condition is also sufficient for admissibility, in the sense of the following proposition.
5.8. Applications to Economics
375
8.6 Proposition. Suppose x > 0 and a consumption process C are given so that (8.23) is satisfied. Then there exists a portfolio process it such that the pair (it, C) is admissible for the initial endowment x. PROOF. Let D 4 g, ce-ro 'Is)" dt, and define the nonnegative process -4
E[
T
ll
Cse- Pr(u)" ds ..F; + (x - ED) el° r(s)ds,
so that (8.24)
St
e50 r(s) ds
- ft Cse-
Ix +
if, r(u) du ds
o
where
m,=° E[DI,F,] - ED =
]
E[DZTI
E(DZT)
Zi
from the Bayes rule of Lemma 3.5.3. Thanks to Theorem 1.3.13, we may assume that P-a.e. path of the martingale Ni A E(DZTIA), ..Ft;
0
is RCLL, and so, by Problem 3.4.16, there exists a measurable, {.F,}- adapted, Rd-valued process Y with fT
(8.25)
II 17 (t)112 dt < oo
and
0
(8.26)
i ft y(s)dWsu);
1%4= E(DZT) +
t---1
0
t
T,
o
valid a.s. P. Now mi = u(/V Zi) - E(DZT), where u(x, y) = (x/y), and from Ito's rule we obtain with (1)4) A (Y(t) + Alt0(t))/Zi: d
(8.27)
J.(pi(s)dtillm; ' 0
m, = j =1
o
We have used the relations dZi = -Z,OT(t)dWt and (8.19). Now define (8.28)
ir(t) A el0'")ds(aT(t))-1 (1)4),
= X. Condition (8.11) follows from (8.4), (8.25), the boundedness of 0, and the path so that (8.24) becomes (8.20) when we make the identification
continuity of Z and N (the latter being a consequence of (8.26)).
Remark. The representation (8.27) cannot be obtained from a direct application of Problem 3.4.16 to the P-martingale fm A; 0 t < T }. The reason is that the filtration {A} is the augmentation (under P or P) of {Aw}, not of {Awl.
376
5. Stochastic Differential Equations
B. Option Pricing In the context of the previous subsection, suppose that at time t = 0 we sign a contract which gives us the option to buy, at a specified time T (called maturity or expiration date), one share of stock 1 at a specified price q, called the exercise price. At maturity, if the price PP ) of stock 1 is below the exercise
price, the contract is worthless to us; on the other hand, if PP) > q, we can exercise our option (i.e., buy one share at the preassigned price q) and then sell the share immediately in the market for PP. This contract, which is called an option, is thus equivalent to a payment of (PV - q)+ dollars at maturity. Sometimes the term European option is used to describe this financial instrument, in contrast to an American option, which can be exercised at any time between t = 0 and maturity. The following definition provides a generalization of the concept of option. 8.7 Definition. A contingent claim is a financial instrument consisting of:
(i) a payoff rate g = Ig Ft; 0 < t < T }, and (ii) a terminal payoff fT at maturity. Here g is a nonnegative, measurable, and adapted process, fT is a nonnegative, 3,7T-measurable random variable, and for some pi > 1 we have T
E[fT +
(8.29)
gt dt
< oo.
8.8 Remark. An option is a special case of a contingent claim with g
0 and
fT = (PP) - qr. 8.9 Definition. Let x > 0 be given, and let (ir, C) be a portfolio/consumption process pair which is admissible for the initial endowment x. The pair (it, C) is called a hedging strategy against the contingent claim (g, fT), provided (i) C, = gt; 0 (ii) XT = fT
t
T, and
hold a.s., where X is the wealth process associated with the pair (n, C) and with the initial condition X0 = x. The concept of hedging strategy is introduced in order to allow the solution of the contingent claim valuation problem: What is a fair price to pay at time
t = 0 for a contingent claim? If there exists a hedging strategy which is admissible for an initial endowment X0 = x, then an agent who buys at time t = 0 the contingent claim (g, fT) for the price x could instead have invested the wealth in such a way as to duplicate the payoff of the contingent claim. Consequently, the price of the claim should not be greater than x. Could one begin with an initial wealth strictly smaller than x and again duplicate the
377
5.8. Applications to Economics
payoff of the contingent claim? The answer to this question may be affirmative, as shown by the following exercise.
8.10 Exercise. Consider the case r = 0, d = 1, bl = 0 and a contingent claim g
1. Let the
0 and fT = 0 be given, so obviously there exists a hedging
strategy with x = 0, C 0, and it = 0. Show that for each x > 0, there is a hedging strategy with X0 = x. The fair price for a contingent claim is the smallest number x > 0 which allows the construction of a hedging strategy with initial wealth x. We shall show that under the condition (8.3) and the assumptions preceding it, every contingent claim has a fair price; we shall also derive the explicit Black & Scholes (1973) formula for the fair price of an option.
8.11 Lemma. Let the contingent claim (g, fT) be given and define Q
(8.30)
e-p; r(u)dufT
r(u) du
J-
Y., C16*
0
Then EQ is finite and is a lower bound on the fair price of (g, fT).
PROOF. Recalling that r is uniformly bounded in t and w, we may write Q < L( fT + f o gs ds), where L is some nonrandom constant. From (8.17) we have for every v > 1: T
a
ZT = exp {-
vOi(s)dWP i=1
x exp
t
v(v - 1)
f
2
0
- -2
IlvO(s)112
ds}
0
110(s)112 ds},
and because 11011 is bounded by some constant K, it follows that EZ'4. < exp
v(v - 1) K27,} 2
With p as in (8.29) and v given by (1/v) + (1/p) = 1, the Holder inequality implies EQ
LE[ZT fT +
I gsds) 1 T
E fT +
gsds
< CC.
Now suppose that (it, C) is a hedging strategy against the contingent claim
(9,M, and the corresponding wealth process is X with initial condition X0 = x. Recalling the Definition 8.9 and (8.30), we rewrite (8.22) as EQ 5 x.
378
5. Stochastic Differential Equations
8.12 Theorem. Under condition (8.3) and the assumptions preceding it, the fair
price of a contingent claim (g, fT) is EQ. Moreover, there exists a hedging strategy with initial wealth x = EQ. PROOF. Define ey, r(s)ds[EQ
(8.31)
m,
f e-pc, r(u) du ds], 0
- EQ. Proceeding exactly as in the proof of Proposition 8.6 with D replaced by Q, we define it by (8.28) and C g, so that (8.31) where m, = E EQ1
becomes (8.20) with the identifications X = be cast as T
[
e- -1; r(u) dugs ds
X, = E e- 1 7- r(u) du fT
(8.32)
x = EQ. But then (8.31) can also
,F,i;
0 < t < T,
i
whence X, > 0; 0 < t < T and
= fT are seen to hold almost surely.
1=1
8.13 Exercise. Show that the hedging strategy constructed in the proof of Theorem 8.12 is essentially (in the sense of meas x P-a.e. equivalence) the only
hedging strategy corresponding to initial wealth x = EQ. In particular, the process X of (8.32) gives the unique wealth process corresponding to the fair price; it is called the valuation process of the contingent claim.
8.14 Example (Black & Scholes (1973) option valuation formula). In the setting of Remark 8.8 with d = 1 and constant coefficients r(t) r > 0, 611(t) a- o- > 0, the price of the bond is
Po(t) = poe"; 0 < t < 'T,
and the price of the stock obeys dP, (t) = 61 (t)P, (t) dt + o- Pi(t)(114/, = r P,(t) dt + o-
(t)
For the option to buy one share of the stock at time T at the price q, we have from (8.32) the valuation process (8.33) X, E[e-rci--0(131(T) ig) + 1 Al 0 T.
-
In order to write (8.33) in a more explicit form, let us observe that the function (8.34) v(t, x)
(T - t, x)) - qe-r(T-1)(1)(p_(T - t, x));
0 < t < T, x > 0,
t= T, x > 0
(x - q)+;
with p+ (t, x) =
1
o-.1i
[log x + t (7- +
0-2
)], 2
4:1)(x) =
f ,./27r
x
e -op
12 dz,
5.8. Applications to Economics
379
satisfies the Cauchy problem all
(8.35)
Ot
+ ry =1 a2 x2 2
02
ex
+
rxO-x v ;
x
0,
v(T, x) = (x - q)+ ;
e
on [0, T) x (0, oc,)
as well as the conditions of Theorem 7.6. We conclude from that theorem and the Markov property applied to (8.33) that (8.36)
X, = v(t,P,(t)); 0 < t
T,
a.s.
We thus have an explicit formula for the value of the option at time t in terms of the current stock price P, (t), the time-to-maturity T - t, and the exercise price q.
8.15 Exercise. In the setting of Example 8.14 but with fT = h(P, (T)), where h: [0, co) [0, co) is a convex, piecewise C2 function with h(0) = h'(0) = 0, show that the valuation process for the contingent claim (0,fT) is given by (8.37)
X, = E[cr(T-')h(P,(T))1.97,1 = i
h"(9)1,,,T(t, P, (t)) dq.
We denote here by ty,,,T(t, x) the function of (8.34).
C. Optimal Consumption and Investment (General Theory) In this subsection we pose and solve a stochastic optimal control problem for the economics model of Subsection A. Suppose that, in addition to the data
given there, we have a measurable, adapted, uniformly bounded discount process 13 = {Ns), Fs; 0 < s < T} and a strictly increasing, strictly concave, continuously differentiable utility function U: [0, co) - [0, co) for which
U(0) = 0 and U'( co) --' lime U'(c) = 0. We allow the possibility that [r(0) A limy, 0 U'(c) = co. Given an initial endowment x > 0, an investor wishes to choose an admissible pair (ir, C) of portfolio and consumption processes, so as to maximize T
V,,,e(X) A E
1 e-ilk")"U(COds. o
We define the value function for this problem to be (8.38)
V(x) = sup V,c(x), (n,c)
where the supremum is over all pairs (n, C) admissible for x. From the admissibility condition (8.23) it is clear that V(0) = 0. Recall from Proposition 8.6 that for a given consumption process C, (8.23) is satisfied if and only if there exists a portfolio it such that (n, C) is admissible for x. Let us define 2(x) to be the class of consumption processes C for which
380
5. Stochastic Differential Equations T
.1e-P.'")dsC,dt = x.
(8.39)
E
o
It turns out that in the maximization indicated in (8.38) we may ignore the portfolio process it and we need only consider C e 9(x). 8.16 Proposition. For every x > 0 we have T
V(x) = sup E C e fd(x)
e-f 0 t3(s)ds U(C,) dt. 0
PROOF. Suppose (it, C) is admissible for x > 0, and set
y '' E
i
T
e-J°'(."' Ctdt < x.
0
If y > 0, we may define C, = (x/y)C, so that 0 e g(x). There exists then a portfolio process A such that (A, 0) is admissible for x, and V,c(x) < Vi,e(x).
(8.40)
If y = 0, then: C, = 0; a.e. t e [0, T], almost surely, and we can find a constant
c > 0 such that 0, .--- c satisfies (8.39). Again, (8.40) holds for some it chosen so that (A, 0) is admissible for x.
1=1
Because U': [0, co] '''' [0, U'(0)] is strictly decreasing, it has a strictly decreasing inverse function 1: [0, U'(0)] --4onto [0, co]. We extend I by setting
1(y) = 0 for y > U'(0). Note that /(0) = co and /(x)) = 0. It is easily verified that (8.41)
U(I(y)) - yl(y)
U(c) - yc; 0 < c < op, 0 < y < oo.
Define a function X: [0, co] - [0, co] by E
(8.42)
T
e-1,. r(u) du
I (yZsef°(fl(")-1( "))")ds,
0
and assume that (8.43)
X(y) < co;
0 < y < Do.
We shall have more to say about this assumption in the next subsection, where
we specialize the model to the case of constant coefficients. Let us define y 4- sup{ y
0; .1. is strictly decreasing on [0, y]}.
8.17 Problem. Under condition (8.43), .i" is continuous and strictly decreasing
on [0, y] with X(0) = co and gly) = 0. to : [0, co] -o-4 [0, y] be the inverse of .?C. For a given initial endowment x > 0, define the processes
Let
5.8. Applications to Economics
381
(8.44)
r/: g V(x)zsePo (poo-rom du
(8.45)
C: A join.
The definition of -N implies C* eg(x). We show now that C* is an optimal consumption process.
8.18 Theorem. Let x > 0 be given and assume that (8.43) holds. Then the consumption process given by (8.45) is optimal:
V(x) = E
(8.46)
CP° P(s)ds U(Ct*)dt. 0
PROOF. It suffices to compare C* to an arbitrary C e g(x). For such a C, we have E
e-S°P(s)ds(U(Ct*)
=E
- U(Ct))dt
CS° P(s)'[(U(I(e)) - ri7 Re)) - (U(C,) - ri7Ct)]dt V(X)E
e-f`o r(s)ds (ct)I,
Ct)dt.
The first expectation on the right-hand side is nonnegative because of (8.41); the second vanishes because both C* and C are in (x).
Having thus determined the value function and the optimal consumption process, we appeal to the construction in the proof of Proposition 8.6 for the determination of a corresponding optimal portfolio process 7E*. This does not provide us with a very useful representation for n *, but one can specialize the model in various ways so as to obtain V, C* and ir* more explicitly. We do this in the next subsection.
D. Optimal Consumption and Investment (Constant Coefficients) We consider here a case somewhat more general than that originally studied
by Merton (1971) and reported succinctly by Fleming & Rishel (1975), pp. 160-161. In particular, we shall assume that U is three times continuously differentiable and that the model data are constant: (8.47)
fl(t)
13,
r(t)
r,
b(t)
b,
Q(t) = a,
where b ege and a is a nonsingular, (d x d) matrix. We introduce the linear, second-order partial differential operator given by
5. Stochastic Differential Equations
382 IAP(t,Y)
- (Pt(t, y) + i3P(t,.11) - 113 - 031(Py(t, y) - 111011 2.Y2(P(t,37),
where B = o-' (b - r1) in accordance with (8.16). Our standing assumption throughout this subsection is that B is different from zero and there exist C' '3 functions G: [0, T] x (0, co) [0, oo ) and S: [0, T] x (0, co) [0, co) such that LG(t, y) = U (1 (y));
(8.48)
y>0
G(T, y) = 0;
(8.49)
0 < t < 'T, y > 0
and
0 < t < T, y > 0
LS(t, y) = y1 (y);
(8.50)
y > O.
S('T, y) = 0;
(8.51)
Here we mean that G,(t, y), Gty(t, y), Gy(t, y), Gyy(t, y), and Gyyy(t, y) exist for all
0 < t < T, y > 0, and these functions are jointly continuous in (t, y). The same is true for S. We assume, furthermore, that G, Gy, S, and Sy all satisfy polynomial growth conditions of the form (8.52)
max H(t, y) < M(1 + y -'' + y A); 1)
y>0
T
for some M > 0 and A > O.
8.19 Problem. Let H: [0, T] x (0, co) -, [0, co) be of class C I 2 on its domain and satisfy (8.52). Let g: [0, T] x (0, co) [0, co) be continuous, and assume that H solves the Cauchy problem LH (t, y) = g(t, y);
H('T, y) = 0;
0 < t < 7; y > 0 y > O.
Then H admits the stochastic representation
I
H(t, y) = E
T
e-fl(s-`)g(s,17,(`'''))ds,
t
where, with t < s < T: (8.53) (8.54)
ys(1,y) A ye(fl -rgs-t) Zt
4 4 exp{ -0T(Ws - Wt) - 1110112(s - t)}.
(Hint: Consider the change of variable e = log y.) From Problem 8.19 we derive the stochastic representation formulas (8.55)
G(t, y) = E
i
T
e-134-`)U (1(17,("))ds,
t
(8.56)
S(t, y) = yE
f t
T
e-r(s-`)Z's I (V' Y)) ds.
5.8. Applications to Economics
383
It is useful to consider the consumption/investment problem with initial times other than zero. Thus, for 0 < t 5 T fixed and x > 0, we define the value function V(t, x) = sup E
(8.57)
0, ,c)
f
fisU (Cs) ds,
r
where (it, C) must be admissible for (t, x), which means that the wealth process determined by the equation (8.58)
(r X - C) du + E
Xs = x + d
f
(b1 - r)tri(u) du
d
+EE
dW,P);
s < T,
t
r
remains nonnegative. Corresponding to a consumption process C, a portfolio process it for which (7r, C) is admissible for (t, x) exists if and only if (cf. Proposition 8.6) E
(8.59)
f
T
e-r(s-`)Zst Cs ds < x.
$
For 0 < t < T, define a function X(t, ): [0, oo] X(t, y) 9 E
(8.60)
J T
[0, co] by
e-r(s-`)41(Y,"Y))ds.
Comparison of (8.56) and (8.60) shows that
y) = S(t, y) < co; 0 < y < oo.
(8.61)
Now y(t) g sup It > 0; X(t, ) is strictly decreasing on [0, y] = co, and we have just as in Problem 8.17 that for 0 < t < T, -) is strictly decreasing on [0, co] with .X(t, 0) = co and (t, co) = 0. We denote by [0, co] the inverse of X(t, ): (8.62)
W(t,
y)) = y;
0
y
co,
): [0, co] -°'
0
If we now define for t < s < T: (8.63)
171") 4 V(t,x)Zst e(11-04-0,
(8.64)
C: 9. /(n(st,x)),
then (8.65)
V(t, x) = E
f
CI" U (C,*)ds.
This claim is proved as in Theorem 8.18; the new feature here is that for x), one has ii(") = Y("), and consequently y=
384
5. Stochastic Differential Equations
V(t, x) = e-fl`G(t,(t,x)); 0 < t < T, x > 0.
(8.66)
Thus, if we can solve the Cauchy problems (8.48), (8.49) and (8.50), (8.51), then we can express V(t, x) in closed form.
8.20 Exercise. Let U(c) = kA
cb,
where 0 < 6 < 1. Show that if
1
1
r6
1-
2
II 0112
1-
is nonzero, then GO,
= -1(1
- e-ku-_,))(yro-t)
S(t, y) = 6G(t, y), e-k( V(t,
If
k = 0,
then
V(t,x) = e-NT -
= e-13`
-6
)5.
k
G(t, y) = (T - t)(1)°1(",
S(t, y) = 45G(t, y),
and
-6 X b.
Although we have the representation (8.66) for the value function in our consumption/investment problem, we have not as yet derived representations for the optimal consumption and portfolio processes in feedback form, i.e., as functions of the optimal wealth process. In order to obtain such representa-
tions, we introduce the Hamilton-Jacobi-Bellman (HJB) equation for this model. This nonlinear, second-order, partial differential equation offers a characterization of the value function and is the usual technique by which stochastic control problems are attacked. Because of its nonlinear nature, this equation is typically quite difficult to solve. In the present problem, we have already seen how to circumvent the HJB equation by solving instead the two linear equations (8.48) and (8.50). 8.21 Lemma (Verification Result for the HJB Equation). Suppose Q: [0, T] x [0, co) -> [0, co) is continuous, is of class C1'2([0, T) x (0, co)), and solves the HJB equation (8.67)
Qt(t,
+ max f[rx -c + (b - r1)T n] Qx(t, c>o
+
1
Mal' 702 Q.(t ,
neRd
+ e-fitU(c)} =0; 0
t
T,
0 < x < co.
Then (8.68)
V(t, x) < Q(t, x); 0 < t < T,
0 5 x < co.
5.8. Applications to Economics
385
PROOF. For any initial condition (t, x) e [0, T) x (0, co) and pair (ir, C) admis-
sible at (t, x), let {Xs; t 5 s < T} denote the wealth process determined by (8.58). With l
//
Tn A T - -
n
A id s e [t, T]; Xs > n or Xs
1
- or
1114412 du = n
,
we have E 1;"Qx(s, X On'. (s)o- OK = 0. Therefore, It6's rule implies, in conjunction with (8.58) and (8.67), 0 < EQ(T., X,,,)
= Q(t,x) + E
1
rn
(Qt(s, Xs) + [rX3 - Cs + (b - rpr ir(s)]Qx(s, Xs)
1
+ - II 0" T n (Ai 2 Qxx(s, Xs)} ds 5 Q(t, x) -E 2
f
r e-PsU(Cs)ds.
Letting n -+ co and using the monotone convergence theorem, we obtain E if CP' U(Cs) ds < Q(t, x). Maximization of the left-hand side over admissible pairs (n, C) gives the desired result.
A solution to the HJB equation may not be unique, even if we specify the boundary conditions (8.69)
Q(t,0) = 0;
0 < t < T and
Q(7; x) = 0;
0 < x < co.
This is because different rates of growth of Q(t, x) are possible as x approaches infinity. One expects the value function to satisfy the HJB equation, and, in light of (8.68), to be distinguished by being the smallest nonnegative solution of this equation.
8.22 Proposition. Under the conditions set forth at the beginning of this subsection, the value function V: [0, T] x [0, co) -* [0, co) is continuous, is of class ,,,,, T) x (0, co)), and satisfies the HJB equation (8.67) as well as the C1,2(IV boundary conditions (8.69).
PROOF. If 0 < y < U'(0), then (8.70)
U(I(y)) = U'(I(y))1'(y) = yr(y);
if y > U'(0), then 1(y) = l'(y) = 0 and (8.70) still holds. Because of our assumption that G and S are of class C, we may differentiate (8.48), (8.50)
with respect to y and observe that (pi (t, A A -YGy(t9Y) and (P2(t, A A -y2(0/0y)(S(t, y)/y) both satisfy 1.4p,(t, y) = - y2 l' (y); Sai(T, Y) = 0;
0 5 t 5 T, y > 0, y > O.
5. Stochastic Differential Equations
386
In particular, I' is continuous at y = U'(0), i.e., a necessary condition for our assumptions is U"(0) = oo. Problem 8.19 implies cp, = (p2, because both functions have the same stochastic representation. It follows that Gy(t, y) =
(8.71)
y-0
(-1
S(t, y)) = ydy(t, y)
aY
and from (8.66), (8.62) we have (8.72)
Vx(t, x) = CP' V(t, x),
(8.73)
Vt(t, gr(t, y)) = - Vx(t, lit, y)) gr,(t, y).
Finally, (8.50) and (8.61) imply that (8.74)
-1;(t, y) + r.T(t, y) - (f - r + Heir )Y .Ty(t, Y)
- 2 110112 Y2
(t, Y) = 1(y);
0 < y < oo, 0
t < 'T.
We want to check now that the function V(t, x) of (8.66) satisfies the HJB equation (8.67). With Q = V, the left-hand side of this equation becomes CP' times (8.75)
Gt(t, V(t, x)) - /3G(t, V(t, x)) + Gy(t,V(t, x)) Vt(t, x)
+ max[((rx - c) + (b - rpr n)V(t, x) +
ll o-Tn112Vx(t, x) +
U(c)].
c> 0
The maximization over c is accomplished by setting (8.76)
c = I(V(t, x)).
Because of the negativity of Vx, the maximization over it is accomplished by setting (8.77)
n = -(craT)-` (b
r1)V(t,
x)
V ,c(t, x)
Upon substitution of (8.76), (8.77) into (8.75), the latter becomes (8.78)
Gt(t,V(t, x)) - I1G(t,V(t, x)) + Gy(t,V(t, x))V,(t, x) 1
+ rx?J(t, x) - V(t, x)I(V(t, x)) - -2 1102
V2(t, x)
Vt, x)
+ U(I(Mt, x))).
We may change variables in (8.78), taking y = '(t, x) and using (8.71), (8.73), (8.48) to write this expression as
GM, y) - K(t, Y) - A(t, y) + ry.lit, Y) -Y 1(Y) - +110112y2xy(t, y) + U (I (Y)) = A Thirt(t, y) + rajt, y) - (13 - r + 110112)Y.gry(t, Y) - 1110112Y2Aryy(t, Y) - 4.01
5.9. Solutions to Selected Problems
387
which vanishes because of (8.74). This completes the proof that V satisfies the
HJB equation (8.67). The boundary conditions (8.69) are satisfied by V by virtue of its definition (8.57) and the admissibility condition (8.59) applied when x = O.
1=1
In conclusion, we have already shown that for fixed but arbitrary (t, x) e [0, T) x (0, oo), there is an optimal pair (n *, C*) of portfolio/consumption processes. Let {X:; t < s S T} denote the corresponding wealth process. If we now repeat the proof of Lemma 8.21, replacing (it, C) by (tr*, C*) and Q by V, we can derive the inequality (8.79) T
0
V(t, x) + E ft {V,(s, X:) + [rX: -C: + (b - r1)T tr*(s)] Vx(s, X:) T
1
+ -211 6 T it* (s)112 Vx(s,
X:)} ds
V(t, x) -E
e- Ps U (Cnds. t
We have used the monotone convergence theorem and the inequality (8.80)
Vi(s, X:`) + [r X: -C: + (b - r DT tr*(s)] Vx(s, X:) + Dar n*(s)112 Vxx(s, X:)
- - CP' [1 (Cn
0;
t < S < T,
which follows from the HJB equation for V. But (8.65) holds, so equality prevails in (8.79) and hence also in the first inequality of (8.80), at least for meas x P-almost every (s, w) in [t, T] x n. Equality in (8.80) occurs if and only if le: and C: maximize the expression
[rX: -c + (b - r1)T it] Vx(s, X:) + ilia TnIi2 vx.(s,xs) + e-Pt um; i.e. (cf. (8.76), (8.77)), (8.81)
C: = .1((s, X:)),
(8.82)
ir:
(0.0.T r 1 0
r1)
(s, X:) ,c(s, X:) '
where again both identities hold for meas x P-almost every (s, w) E [t, T] x a The expressions (8.81), (8.82) provide the optimal consumption and portfolio processes in feedback form.
8.23 Exercise. Show that in the context of Exercise 8.20, the optimal consumption and portfolio processes are linear functions of the wealth process X *. Solve for the latter and show that XI = 0 a.s.
5.9. Solutions to Selected Problems 2.7. We have from (2.10) -d dt
(e-flt f ' o
g(s)ds)=(g(t)-
13
f
g(s)ds) e-flt < a(t)e -flt, o
5. Stochastic Differential Equations
388
whence f u g(s) ds _5 e°' f u a(s)e - as ds. Substituting this estimate back into (2.10), we obtain (2.11).
2.10. We first check that each V) is defined for all t > 0. In particular, we must show that for k > 0,
0 < t < cc, a.s.
(II b(s, XI') )11 + 110(s, X?))112)ds < oo; Jo
In light of (2.13), this will follow from
sup EIP0k)112 < co; 0
(9.1)
T < co,
a fact which we prove by induction. For k = 0, (9.1) is obvious. Assume (9.1) for some value of k. Proceeding similarly to the proof of Theorem 2.5, we obtain the
bound for 0 5 t < T: EI1V+"112 < 9E11 112 + 9(T + 1)K2
(9.2)
(1 + E11X?)112)ds,
which gives us (9.1) for k + 1. From (9.2) we also have
f
EMk+i)112 < C(1 + E11112) + C
E11X?)112ds;
0 < t < T,
o
where C depends only on K and T Iteration of this inequality gives
(CO'
r +1)112 < C(1 + Eg112)[1 + Ct + ("2) 2
(k + 1)!
and (2.17) follows. 2.11. We will obtain (2.4) by letting k -+ co in (2.16), once we show that the two integrals
on the right-hand side of (2.16) converge to the proper quantities. With T > 0, (2.21) gives maxo,,,T1,1X,(co) - )0k)(o))11 < 2', V k > N(co). Consequently, 2
b(s, X?))ds - ft b(s, X5) ds
< K2T
f
X s12 ds 0
converges to zero a.s. for 0 < t < T, as k In order to deal with the stochastic integral, we observe from (2.19) that for fixed 0 < t < T, the sequence of random variables {.)0},T_I is Cauchy in L2(52, P), and since V) -+ X, a.s., we must have
Ell r) - X,I12 -0 as k- co. Moreover, (2.17) shows that Ell r)II 2 is uniformly bounded for 0 < t < T and k > 0, and from Fatou's lemma we conclude that E11X,112 <_ limk E11V)112 is uniformly bounded as well. From (2.12) and the bounded convergence theorem, we have E
f
cr(s, Vsk))dWs
o
-j
2 1
o-(s, Xs) dli/s
5 K2 J E II X?) - XsII2 ds -0 as E
= E .1110-(s,.V)) - a(s, X s)II2 ds 1
k --+ oo;
0 5 t 5 T.
o
2.12. For each positive integer k, define the stopping time for {,F,}: {0 Tk =
ao
if 1111 > k, if II
115 k,
5.9. Solutions to Selected Problems
389
and set 6, = 1{,4
t
b(s, V') ds +
ft o-(s, V)) dgis;
0 < t < x.
For t > k, we have that (r) - r))11, 4 <10 is equal to
j.:^ Tk {b(s, V)) - b(s, Xr)} ds + =
f
t '' Tk o
{ 0(S, V)) - CYCS, Xr)} d Ws
{b(s, V)) - b(s, Xr)}1{s< Ts} ds +
{Os, V)) - a(s, V))}1{s< Tki dWs.
By repeating the argument which led to (2.18), we obtain for 0 < t < T:
E[max
o.ct
II
- X?)(12
L
1{1141
E max 11V) - V42 1{ 0
0..s5t
4
dt,
and Gronwall's inequality (Problem 2.7) now yields max II Xsrl - X?) I1 = 0,
0<s
a.s. on {II II 5 14.
We may thus define a process {X,; 0 < t < co } by setting X Jo)) = V)(co), where k is chosen larger that II (c0)11 Then X,1{1141,
f
=
tnTk
Ns, V)) ds +
f
tnrk
Since
II 5
V)) dWs Tk
=
b(s, X s) ds +
+
EAT,
b(s, X s)ds +
cr(s, X s)dWs
ft cr(s, Xs) dWdl 1,14,1 kl.
= 1, we see that X satisfies (2.4) almost surely.
3.15. We use the inequality for p > 0; (9.3)
I ailP +
+ la IP < n(lail +
+ aI)P < nP41(la1 IP +
+ laIP).
We shall denote by C(m,d) a positive constant depending on m and d, not necessarily the same throughout the solution. From (iv) and (9.3) we have, almost surely: 2m
IIX,112m
C(m, d)[11X0112m +
2m
t
j' t b(u, X) du
(U, X) dW.
and the Holder inequality provides the bound 2m
b(u X) du
=E
b,(u, X) du)2T
fo
t2m-1
b(u, X)112'" du.
r[..1
1119(u, X)!! 2
dd
5. Stochastic Differential Equations
390
k} tend almost surely to infinity as
The stopping times Sk A inf {t 0; IIX, co, and truncating at them yields k
A Sk
max
b(u, X)112' du
II X 5112' < C(m,d) [II X 0112' + t2m-1
OSSStASk
0
fsA Sk
+ max
0(14, X) OK
a.s.
Remark 3.3.30 and Holder's inequality provide a bound for the last term, to wit, tA sk
5sA
E[ max
C(m,d) E
a(u, X)d147.
f
X)II2
o
< C(m, d)tm- E
f
I
dur
AS,
a(u, X)112'n du.
Therefore, upon taking expectations, we obtain
E(otAsk max
IIX5112m)
C(m, d)[E1IX 0112m + C(T) E
f
tnSk (111,(4, X)II 2m + 110(u, X)112m)dui
for every 0 < t < T, where C(T) is a constant depending on T, m, and d. But now we employ the linear growth condition (3.16) to obtain finally,
OM A
E(
max
UXkl12")
< C[1 + EIIXollzm +
0..5.s..5tASk
f
tpk(u) du];
0
0 < t < T, and from the Gronwall inequality (Problem 2.7): Ok(t)
C(1 + EllX0112m)ect;
0
T.
Now (3.17) follows from Fatou's lemma. On the other hand, if we fix s < t in [0, T] and start from the inequality IRC, - Xs112'" < C(m,d)(11.g b(u, X) dull2m + .R 0(u, X) dW.II2m), we may proceed much as before, to obtain Ell
- Xsem
C(t - s)"' -' f t [1 +
E( max 11X 0112m)idu
s
< C(1 + CecT)(1 + EllX0112m)(t - s)m.
3.21. We fix F .4,(C [0, oor). The class of sets G satisfying (3.25) is a Dynkin system. The collection of sets of the form (3.26) is closed under pairwise intersection and generates the a-field .4(08') .4(C[0, coy). By the Dynkin System Theorem, it suffices to prove (3.25) for G of the form (3.26). For such a G, we have
oqi(x,w;
F)/.1(dx)P,(dw)
E[.F)1.1(dx)
=
GI
,G; P{o-,-1G31.4,(C[0, CO)')}
5.9. Solutions to Selected Problems
391
Q;(x, ; F)p(dx)
= Es [J
1
,Gz] P:[al G3]
= PAGI x 9,- G2 x F] Ps[a,' G3]. The crucial observation here is that, under Wiener measure Ps, Crt-1 G3 is independent of A(C[0, con From (3.21), we have Ps[a,1 G3] = pi[(x, w, y) E 0; ajw E G3] = Vi[47, Wth E G3],
PAGI x
G2 x F] = vi[XP E G , 9,Wu) E G2, ri) E Pl.
Therefore, Q;(x, w; F)p(dx)13,(dw) = vi[XY)E G,,(p,WthE G2, Yth E F] vi[aiW(i) E G3]
T. = vi[X cji) E GI ,
W(i)E G2, CT, VV(i) E G3, y(j)
E F] = PJ[G x F],
= vi[(XV, W(J)) E G,
because (XeE G, , (p,W(i)E G2, YU) E F) Wu) E G3 is independent of it.
F]
belongs to the a-field
and
3.22. The essential assertion here is that Qi(x, w; ) assigns full measure to a singleton,
which is the same for j = 1, 2. To see this, fix (x, w) E
x C[0, Do)' and define the measure Q(x, w; dy,,dy2) A Qi(x,w; dy 1)Q2(x, w, dye) on (S, 9') A (C[0, co)" x C[0, co)", Af(C[0, co)") 0 At(C[0, co)d). We have from (3.23) (9.4)
P(G x B) =
f
Q(x,w; B)i(dx)Ps(dw);
Be
G E .gi(Rd) 0 .4(C [0, °or).
With the choice B = {(YI,Y2)ES; Y = Y2} and G = [Rd x C[0, ooY, relations .2(C[0, cor ) with (p x P.) (3.24) and (9.4) yield the existence of a set N E d(Rd) (N) = 0, such that Q(x, w; B) = 1 for all (x, w) ct N. But then from Fubini, 1 = Q(x, w; B) =
fcto..)d Qi(x,w; {y}) Q2(x,w; dy);
(x, w)0 N,
which can occur only if for some y, call it k(x, w), we have Qi(x, w; {k(x, w) } ) = 1;
j = 1, 2. This gives us (3.27). For (x, w) 0 N, and any BE AC [0, oa)d), we have k(x, w)E B Q i(x , w; B) = 1. The A WI ) 0 Af(C [0, con/I( [0, 1])-measurability of (x, w)i- Qi(x, w; B) implies the A Rd) 0 .4(C[0, co yvalc[o, co)")- measurability of k. The .4,/.4,(C[0, co) ")- measurability of k follows from a similar argument,
which makes use of Problem 3.21. Equation (3.28) is a direct consequence of (3.27) and the definition (3.23) of P. 4.7. For (p, a) E D, let A = pp' = o-o-T . Since A is symmetric and positive semidefinite, A 0 there is an orthogonal matrix Q such that QAQT = [- -1 - - , where A is a (k x k) 0
1
0
diagonal matrix whose diagonal elements are the nonzero eigenvalues of A. Since QAQT = (QP)(Q P)T = (12a)(120)T ,
r
Qp must have the form Qp =[-1- , where y, YiT = A. Likewise, Qa = Z1 0
01
392
5. Stochastic Differential Equations
where Z1ZT = A. We can compute an orthonormal basis for the (d - k)dimensional subspace orthogonal to the span of the k rows of Y1. Let Y2 be the (d - k) x d matrix consisting of the orthonormal row vectors of this basis, and
set Y=
EY-11.
A' 0
Then (Qp)yr
A0
yy,
.,.
0I0
Y2
0
where / is the (d - k) x ,
(d - k) iden ity matrix. In the same way, define Z = [Z - - - so that (Qa)ZT = Z2
(Qp)112., ZZT = YYT. Then a = pYr(ZT)', so we set R = YT(ZT)-1. It is easily verified that RRT = 1. All of the steps necessary to compute R from p and a can be accomplished by Borel-measurable transformations.
4.13. Let us fix 0 < s < t < co. For any integer n > (t - s)-' and every A e Ws_ft,/), we can find B e A+(0,) such that P(A A B) = 0 (Problem 2.7.3). The martingale property for {Mt,f, gk; 0 < u < co} implies that (9.5)
E [{ f(y(t))
- f(y(s +
I
) ) -J n
t
(s1 J) ±11,)
= E[ff ( y(t)) -f(y (s + -)) -..1 n
1A]
(sit,: f )( y) du}
,,d =0.
s+(tin)
If follows that the expectation in (9.5) is equal to zero for every A e.f.s = Ws+ . We can then let n -oo and obtain the martingale property E[(Mir - MS )1A] = 0 from the bounded convergence theorem. 4.25. Suppose jud (P(x)Pi(dx) = 1Rd 9(x)/12(dx) for every ys, Cr (Rd), and take ik e Co(111°). Let p E Co (Rd) satisfy p > 0, cRd p(x) dx = 1, and set (p(x) A juld tk(x + (y /n))p(y) dy = n IRdt1/(z)p(nz - nx) dz. Then cp e q°(Rd) and 9(x) --1//(x) for every x e Old. It follows from the bounded convergence theorem that .fRa ii/(x)111 (dx) = jRa ti/(x)/12(dx), for every 0 a Co(Rd). Now suppose G c Rd is open and bounded. Let 0(x) = 1 A infy# G nlly - xII. Then On E Co( ) for all n,
and ifr 11,. It follows from the monotone convergence theorem that it,(G) = i12(G) for every bounded open set G. The collection of sets Cif = {BE At(Rd); PI(B) = 142(B)} form a Dynkin system, and since every bounded, open set is in ', the Dynkin System Theorem 2.1.3 shows that elf = g(ge).
5.3. For t
0, let Et = {Po' a2(Xs) ds = co}. Using the method of Solution 3.4.11, we can show that lim
AS. = XO
a(Xs)dRis = o o,
lim
a.s. on E
n-.c0
AS
ct(Xs)dW = - co, as. on E,. lim XrAS = X0 + lim ft n-,co .-.0 0 But X, is continuous in the topology of the extended real numbers, so P(E,) = 0. Consequently, X, ,,s = X0 roA 0.(xs) a'His is real-valued a.s., for every t 0, so S = co a.s.
5.27. For s > 0 and c+s<x
p'(y)
2 dz
p'(z)a2(z)
dy
[p(x)
p(c + e)]
and (5.73) follows. A similar argument works for (5.74).
r
jc
dz p'(z2)a2(z)'
393
5.9. Solutions to Selected Problems
6.2. It suffices to show that if Q(t) is a locally bounded, measurable, (d x r)-matrix-
valued function of t, then l', = cc,Q(s)dW, is a Gaussian process. This is certainly true if the components of Q are simple, and the general Q can be approximated by simple matrix functions Q(") so that for each fixed t, we have Elifo [V")(s) - Q(s)] 6VA' = 0. Because the L2 limit of normal random vectors must be normal, we have the desired result. 6.11. The jointly normal random variables Z,, Z2, Z are uncorrelated, as one can see from (6.25) or by observing from (6.26) that
Z;=
b(t, - t,_,)
digs
(T - td(T - t,_,)
T-s
= 1,
n.
It is also apparent that EZ, =
b(ti
t1_,)
Var (Z,) -
(T - 4)(T - ti_1)'
Now write down the joint density of (Z,,
t,- ti_, (T - td(T - t,_,)
, Z") and make the change of variables
zi = x; /(T - ti) - x;_1 /(T - ti_1) to obtain the desired result. 6.15. Observe that dZ, = Z,[A(t)dt +
S.,(t)dWP1, J=1
and apply ItO's rule to the right-hand side of (6.32) to see that X defined by this formula satisfies (6.30). Uniqueness follows from Theorem 2.5.
In the case of constant coefficients for (6.33), the solution becomes X, = exp( Y,), where
at + aB a 'A .,./E;=, SI, p = A - a2 /2 and
l',
B, A
(1/0E'i=1 SWU) is Brownian motion (by the P. Levy characterization, Theo-
rem 3.3.16). The strong law of large numbers (Problem 2.9.3) shows that (1',/t) = p, and hence also lim X, = 0, a.s. 7.3. Proceeding as in Solution 4.2.25, we show that t A to
M, A u(X, tdexp -
k(Xdds}
A tD
g(Xs) exp
-f k(X8)(10}ds; 0 < t < op
is a uniformly integrable martingale under Px. The identity PM° = Pitt, is (7.8).
8.1. For a (d x d) matrix r, define the operator norm lin! = sup,o (11r-11/1111). We show that Ilrll = !IrTil. According to the Cauchy-Schwarz inequality, Ilt1TrT11 = II
ri111
gillIrt111 < 111-11RII 11n11.
Now set n = F% to obtain FT and thus by definition ilFT11 11F11 We obtain the opposite inequality by reversing the roles of F and FT. Now (8.3) implies that aT(t) is nonsingular, for otherwise we could find a nonzero vector which makes Ta(t) = llo-T(t)OP = 0. Letting = (aT(t))-1 we may rewrite (8.3) as
5. Stochastic Differential Equations
394
11(0 T(0)-1 /II
vqe Rd, 0 < t < T,
11,711;
a.s., which is equivalent to II(a T(0)-1 II 5 (1/fE). Because II(a(t))' II = Ho-T(0)-111, we have the equivalence of (8.3) and (8.5). We have already proved (8.4), and (8.6) is established similarly.
8.17. Because / is strictly decreasing on [0, U'(0)] and identically zero on [U'(0), oo), we see that X is nondecreasing and for 0 < yl < y, < co: (0)1
P[ min (Zief°(flo) -*)) as)
Y t) =
Yi
7'
()
in which case X(yi ) = 9C(y2) = 0. The equality 5t (y) = 0 follows. The equality 5d'(0) = limylo X(y) = co is a consequence of the monotone convergence theorem. Continuity of X on (0, y], and hence its surjectivity, follows from (8.43) and the dominated convergence theorem.
8.19. Let IN , ) = H(t, et), so Jr is defined and of class CI.2 on [0, T] x R. A bit of computation shows that if solves the Cauchy problem
-.
+
- (/3 - r --21101I2 lee --21101121fee = g(t, el); 1
fl
1
Jrcr,e)
o;
0
t
T,
CcUB
E
Condition (8.52) on H implies that Ye satisfies (7.20) for everyµ > 0 and M > 0 depending on tt. It follows from Problem 7.7 that T
iot,e) = E[f e-P('-`)g(s,e4)ds L, = where di, = (I3 -r - I10112 /2) ds - OT dWs. This is the stochastic differential equation satisfied by log Ys"), and thus
H(t, y) = "Jog y) =
E[f
T
e-13('-`)g(s,
Ys"))dsl.
5.10. Notes Section 5.1: The term stochastic differential equation was actually introduced
by S. Berngtein (1934, 1938) in the limiting study of a sequence of Markov chains arising in a stochastic difference scheme. He was only interested in the distribution of the limiting process and showed that the latter had a density satisfying the Kolmogorov equations. However, according to Gihman & Skorohod (1972), it would be an exaggeration to consider Berngtein the founder of this theory. Independently of Ito's work, I. I. Gihman (1947, 1950) developed a theory of stochastic differential equations, complete with results on existence,
5.10. Notes
395
uniqueness, smooth dependence on the initial conditions, and Kolmogorov's equations for the transition density. Since the early work of Ito and Gihman, the interest in the methodology and the mathematical theory of stochastic differential equations has enjoyed remarkable successes. The constructive and intuitive nature of the concept, as well as its strong physical appeal, have been responsible for its popularity among applied scientists; for instance, the "dW" term on the right-hand side
of (1.9) has an important interpretation as white noise in statistical communication theory. Apart from its original and continued relevance to physics (see the notes to Section 6, as well as Nelson (1967), Freidlin (1985)), the new
methodology became gradually indispensable in fields such as stochastic systems (Arnold & Kliemann (1983)) and stability theory (Friedman (1976), Khas'minskii (1980)), stochastic control and game theory (Fleming & Rishel (1975), Friedman (1976), Krylov (1980)), filtering (Liptser & Shiryaev (1977), Kallianpur (1980)), communication and dynamical systems (Wong & Hajek (1985)), mathematical economics (cf Section 5.8), mathematical biology and population genetics (cf. examples in Chapter XV of Karlin & Taylor (1981)). Stochastic partial differential equations arise in filtering (Pardoux (1979)) and neurophysiology (Walsh (1984, 1986)). On the other hand, the analytical approach to diffusions and, more generally, Markov processes, has been further developed in conjunction with the Hille-Yosida theory of semigroups; see, for instance, the lecture notes of Ito (1961a) and Chapter I of Ethier & Kurtz (1986). Section 5.2: Theorems 2.5, 2.9 are standard; they are due to ItO (1942a, 1946, 1951) and can be found in several monographs such as those by Skorohod (1965), Mc Kean (1969), Gihman & Skorohod (1972, 1979), Arnold (1973), Friedman (1975), Liptser & Shiryaev (1977), Stroock & Varadhan (1979), Kallianpur (1980), Ikeda & Watanabe (1981). These sources, as well as Friedman (1976), should be consulted for further reading on the subject matter of this chapter and some of its applications. The study of comparison results for stochastic differential equations started with Skorohod (1965). The article by Ikeda & Watanabe (1977) contains important refinements of Proposition 2.18 and Exercise 2.19, with applications to stochastic control and to tests for explosions; see also Chapter VI in Ikeda & Watanabe (1981), Yamada & Ogura (1981), Hajek (1985). Doss (1977) and Sussmann (1978) were the first authors who studied the possibility of pathwise solutions to stochastic differential equations, via an
appropriate reduction to an ordinary differential equation in the spirit of Proposition 2.21. Extension of the latter to several dimensions has Liealgebraic ramifications; see Ikeda & Watanabe (1981), pp. 107-110. The modeling issue of Subsection 5.2.D was first raised by Wong & Zakai (1965a),
who obtained Proposition 2.24 under more restrictive conditions. The onedimensional equation (2.34) was studied in detail by Chitasvili & Toronjadze (1981).
396
5. Stochastic Differential Equations
Stochastic differential equations driven by general semimartingales, instead
of just Brownian motion, have been studied by Do leans-Dade (1976) and Protter (1977, 1984) among others; see also Elliott (1982), Chapter XIV. Stochastic integral equations of the Volterra type have also been considered; see, for instance, Kleptsyna & Veretennikov (1984) and Protter (1985). Section 5.3: The methodology that leads to Proposition 3.20 and Corollary
3.23 was developed by Yamada & Watanabe (1971); we follow Ikeda & Watanabe (1981) in our exposition. Further instances of one-dimensional equations dX, = a()C,)dW, admitting no strong solution, in the spirit of Example 3.5 but with continuous dispersion coefficients o( ), have been discovered by Barlow (1982). Section 5.4: The book by Stroock & Varadhan (1979) should be consulted for further reading on martingale problems and multidimensional diffusions. For the martingale approach to more specialized questions on diffusions, such as the support theorem, boundary behavior, degeneracy, and the construction of diffusion processes on manifolds, the reader is referred to the articles by
Stroock & Varadhan (1970, 1971, 1972) and the lecture notes of Priouret (1974), respectively.
The "martingale problem" approach of this section can also be employed to characterize more general Markov processes in terms of their corresponding infinitesimal operators; a good part of Chapter IV in Ethier & Kurtz (1986) is devoted to this subject. Section 5.5: Material for the first two subsections was drawn primarily from the papers of Engelbert & Schmidt (1981, 1984, 1985). The use of local time
to prove pathwise uniqueness in Theorem 5.9 is due to Perkins (1982c), although Perkins's result has been sharpened by the use of the Engelbert and Schmidt zero-one law. Proposition 5.17 is a time-homogeneous version of a more general result due to Zvonkin (1974); according to the latter, the equation (2.1) with d = 1 has a unique strong solution, provided that (i) the drift b(t, x) is bounded and Borel-measurable,
(ii) the dispersion o-(t, x) is bounded (both above and away from the origin), is continuous in (t, x), and satisfies a Holder condition of the type (2.24) with h(u) = KO; a > (1/2).
Zvonkin's results were extended to the multidimensional case by Veretennikov (1979, 1981, 1982), who showed in particular that the equation (3.2) with
d > 1 has a unique strong solution for any bounded, Borel-measurable drift b(t, x).
In another important development, Nakao (1972) showed that pathwise uniqueness holds for the equation (5.1), provided that the coefficients b, a are bounded and Borel-measurable, and a is bounded below by a positive constant and is of bounded variation on any compact interval. For further extensions of this result (to time-dependent coefficients), see Veretennikov (1979), Nakao (1983), and Le Gall (1983). The material of Subsection C is fairly standard; we relied on sources such
5.10. Notes
397
as McKean (1969), Kallianpur (1980), and Ikeda & Watanabe (1981), particularly the latter. A generalization of the Feller test to the multi-dimensional case is due to Khas'minskii (1960) and can be found in Chapter X of Stroock & Varadhan (1979), together with more information about explosions. A complete characterization of strong Markov processes with continuous
sample paths, including the classification of their boundary behavior, is possible in one dimension; it was carried out by Feller (1952-1957) and appears in Ito & Mc Kean (1974) and Dynkin (1965), Chapters XV-XVII. See also Meleard (1986) for an approach based on stochastic calculus. The recurrence and ergodic properties of such processes were investigated by Maruyama
& Tanaka (1957); see also §18 in Gihman & Skorohod (1972), as well as Khas'minskii (1960) and Bhattacharya (1978) for the multi-dimensional case.
Section 5.6: Langevin (1908) pioneered an approach to the Brownian movement that centered around the "dynamical" equation (6.22), instead of relying on the parabolic (Fokker-Planck-Kolmogorov) equation for the transition probability density. In (6.22), X, represents the velocity of a free particle with mass m in a field consisting of a frictional and a fluctuating force, a is the coefficient of friction, and 62 = 2ockT/m, where T denotes (absolute) temperature and k the Boltzmann constant. Langevin's ideas culminated in the Ornstein-Uhlenbeck theory for Brownian motion; long considered a purely heuristic tool, unsuitable for rigorous work, this theory was placed on firm mathematical ground by Doob (1942). Chapters IX and X of Nelson (1967) contain a nice exposition of these matters, including the Smoluchowski equation for Brownian movement in a force field. Section 5.7: The monograph by Freidlin (1985) offers excellent follow-up reading on the subject matter of this section, as well as on degenerate and quasi-linear partial differential equations and their probabilistic treatment. In the setting of Theorem 7.6 with k
0, g E: 0 it is possible to verify directly,
under appropriate conditions, that the function (10.1)
u(t, x) = Ef(Xx))
on the right-hand side of (7.15) possesses the requisite smoothness and solves the Cauchy problem (7.12), (7.13). We followed such an approach in Chapter
4 for the one-dimensional heat equation. Here, the key is to establish "smoothness" of the solution V) to (7.1) in the initial conditions (t, x) so as to allow taking first and second partial derivatives in (10.1) under the expectation sign; see Friedman (1975), p. 124, for details. Questions of dependence on the initial conditions have been investigated extensively. The most celebrated of such results is the diffeomorphism theorem (Kunita (1981), Stroock (1982)), which we now outline in the context of the stochastic integral equation (4.20). Under Lipschitz and linear growth conditions as in Theorem 2.9, this equation has, for every initial position x e Rd,
a unique strong solution IX,(x); 0 "i: t < col. Consider now the (d + 1)dimensional random field d = {X t(x , w); (t, x) e [0, co) x Rd, co e ill. It can be
shown, using the Kolmogorov-entsov theorem (Problem 2.2.9) in conjunc-
398
5. Stochastic Differential Equations
tion with Problems 3.3.29 and 3.15, that there exists a modification such that: (i) (t, x) i- gt(x, w) is continuous, for a.e. w e CI; (ii) For fixed t > 0, xi-, IC,(x, w) is a homeomorphism of D
of I
into itself for a.e.
We
Furthermore, if the coefficients b, a have bounded and continuous derivatives of all orders up to k > 1, then for every t > 0, (iii) x F- fet(x, w) is a C" -diffeomorphism for a.e. w e e.
For an application of these ideas to the modeling issue of Subsection 5.2.D, see Kunita (1986).
Malliavin (1976, 1978) pioneered a probabilistic approach to the questions of existence and smoothness for the probability densities of Brownian func-
tionals, such as strong solutions of stochastic differential equations. The resulting "functional" stochastic calculus has become known as the stochastic calculus of variations, or the Malliavin calculus; it has found several exegeses and applications beyond its original conception. See, for instance, Watanabe (1984), Chapter V in Ikeda & Watanabe (1981), and the review articles of Ikeda & Watanabe (1983), Zakai (1985) and Nualart & Zakai (1986). For applications of the Malliavin calculus to partial differential equations, see Stroock (1981, 1983) and Kusuoka & Stroock (1983, 1985).
Section 5.8: The methodology of Subsection A is new, as is the resulting treatment of the option pricing and consumption/investment problems in Subsections B and C, respectively. Similar results have been obtained independently by Cox & Huang in a series of papers (e.g., (1986, 1987)). For Subsection B, the inspiration comes in part from Harrison & Pliska (1981) and Bensoussan (1984); this latter paper, as well as Karatzas (1988), should be consulted for the pricing of American options. Material for Subsection C was drawn from more general results in Karatzas, Lehoczky & Shreve (1987). The problem of Subsection D was introduced by Samuelson (1969) and Merton (1971); it has been discussed in Karatzas et al. (1986) on an infinite horizon and with very general utility functions. An application of these ideas to equilibrium analysis is presented in Lehoczky & Shreve (1986). See also Duffie (1986) and Huang (1987).
CHAPTER 6
P. Levy's Theory of Brownian Local Time
6.1. Introduction This chapter is an in-depth study of the Brownian local time first encountered
in Section 3.6. Our approach to this subject is motivated by the desire to perform computations. This is manifested by the inclusion of the conditional Laplace transform formulas of D. Williams (Subsections 6.3.B, 6.4.C), the derivation of the joint density of Brownian motion, its local time at the origin and its occupation time of the positive half-line (Subsection 6.3.C), and the computation of the transition density for Brownian motion with two-valued drift (Section 6.5). This last computation arises in the problem of controlling the drift of a Brownian motion, within prescribed bounds, so as to keep the controlled process near the origin. Underlying these computations is a beautiful theory whose origins can be traced back to Paul Levy. Levy's idea was to use Theorem 3.6.17 to replace the study of Brownian local time by the study of the running maximum (2.8.1) of a Brownian motion, whose inverse coincides with the process of first passage times (Proposition 2.8.5). This latter process is strictly increasing, but increases by jumps only, and these jumps have a Poisson distribution. A precise statement of this result requires the introduction of the concept of Poisson random measure, a notion which has wide application in the study of jump processes. Here we use it to provide characterizations of Brownian local time in terms of excursions and downcrossings (Theorems 2.21, 2.23). In Section 6.3 we take up the study of the independent, reflected Brownian motions obtained by looking separately at the positive (negative) parts of a standard Brownian motion. These independent Brownian motions are tied together by their local times at the origin, a fact which does not violate their independence. Exactly this situation was encountered in the Discussion of
6. P. Levy's Theory of Brownian Local Time
400
F. Knight's Theorem 3.4.13, where we observed that intricately connected processes could become independent if we time-change them separately and then forget the time changes. The first formula of D. Williams (Theorem 3.6) is a precise statement of what can be inferred about the time change from observing one of these reflected Brownian motions. Section 6.4 is highly computational, first developing Feynman-Kac formulas involving Brownian local time at several points, and then using these formulas to perform computations. In particular, the distribution of local time at several spatial points, when the temporal parameter is equal to a passage time, is computed and found to agree with the finite-dimensional distribution of one-half the square of a two-dimensional Bessel process. This is the RayKnight description of local time; it allows us finally to prove the DvoretzkyErdos-Kak utani Theorem 2.9.13.
6.2. Alternate Representations of Brownian Local Time In Section 3.6 we developed the concept of Brownian local time as the density
of occupation time. This is but one of several equivalent representations of Brownian local time, and in this section we present two others. We begin with a Brownian motion W = {147,, ,F,; 0 < t < co} where P[Wo = 0] = 1 and {Ft} satisfies the usual conditions, and we recall from Theorem 3.6.17 (see, in particular, (3.6.34), (3.6.35)) that (2.1)
where B, = (2.2)
= M, - B,, 2L,(0) = Mt; V 0
t < co] = 1,
sgn(Ws) AK is itself a Brownian motion,
M,= max 135; 0 < t < co, o<s
and L,(0) is the local time of W at the origin: (3.6.2)
L,(0) = lim
1 meas {0 < s < t; I147,1 < e }.
Thus, a study of the loca;ti0 time 4 of W at the origin can be reduced to a study of the more easily conceived process M. The idea of this reduction and much of what follows originated with P. Levy (1939, 1948).
A. The Process of Passage Times The process M of (2.2) is continuous, nondecreasing, and "flat" (constant) during excursions of the reflected Brownian motion I WI away from the origin. Because the Lebesgue measure of the set {0 < t < co; I W11 = 0} is zero, P-a.s.
6.2. Alternate Representations of Brownian Local Time
401
(Theorem 2.9.6), the process M has time derivative almost everywhere equal to zero; it is very much like the Cantor function. We may regard M as a new clock, which stops when I WI is away from the origin and runs at an accelerated rate when I WI is at the origin. In order to pass from this new clock to the original one, we introduce the right-continuous inverse of M, defined as in Problem 3.4.5: (2.3)
Sb
inf t
0; M, > b1 = inf {t
0; B, > b};
0
b < oo.
Each Sb is an optional (and hence also a stopping) time of the right-continuous
filtration f
Tb
inf tt
0; M, = b1 = inf t
0; B, =
0
b < co.
Regarded as processes parametrized by the spatial variable b, T = {Tb; 0 5 b < col and S = {Sb; 0 < b < co} are modifications of one another (Problem 2.7.19). The existence of two inverses for M reflects the fact that M identifies the starting and ending points of the excursions of I WI away from the origin. These excursions correspond to jumps of the process T, as we now elaborate. Recall from Remark 2.8.16 that for each t > 0, the time in se [0, for which B, = M, is almost surely unique. Thus, for each w in an event SI* of probability one, this assertion holds for every rational t. Fix w e Se, a positive number t (not necessarily rational), and define (2.5)
y,(w)
sup {s e [0, t]; Ws(co) = 0} = sup {s e [0, a Bs(co) = M,(w)1,
(2.6)
13,(w)
inf {s e [t, co); W,(w) = 0} = inf {s e [t, co); Bs(co) = M,(01.
If 14/,(co) = 0, then /3,(w) = t = y,(w). We are interested in the case W,(w) # 0, which implies (2.7)
< t < A(0). In this case, the maximum of Bs(w) over 0 < s < t is attained uniquely at s = y,(w), hence 7,,,,,(w)(w) = y,(w). Similarly Tww),(w) = Sw,)(w) = f3,(w), for otherwise there would be a rational q > 13,((o) such that Bfit(w)(w) = Bm,)(w) =
WO, a contradiction to the choice of w e Sr. We see then that for w e and t chosen so that W(w) 0 0, the size of the jump in Tb(w) at b = M,(w) is the length of the excursion interval (y,(w), Ma))) straddling t: (2.8)
TAftoo+ (0) - Tut( (0(0 =
- Y*0)
It is clear from (2.4) that Tb is strictly increasing in b, and To = 0. It is less clear that T grows only by jumps. To see this, consider the zero set of W(w), namely 22'1, A {0 < t < co; Wt(w) = 0},
which is almost surely closed, unbounded, and of Lebesgue measure zero (Theorem 2.9.6). As with any open set, £ n (0, co) can be written as a countable
6. P. Levy's Theory of Brownian Local Time
402
union of disjoint open intervals
27. n (0, co ) = U 4(0, ae A
and each of these intervals contains a number ta(w). In the notation of (2.5), (2.6), we have a G A.
la = (ye., A.);
Because meas(Aeb,) = 0 for P-a.e. w e S/, we have
=E
me A
yi.) =aeAE (Tm. -
°
°
E (Tx, - Tx); 0 < t < co,
x<M,
almost surely. Now set t = Ta to obtain
E (Tx, - Tx),
a < oo;
x
letting a T b and using the left-continuity of T, we see finally that, except on a P-null set, Tb
E (Tx+ - Tx), 0
b < oo.
x
The reverse inequality is obvious. We summarize our observations thus far.
2.1 Theorem. The processes S = {Sb; 0 < b < co} of (2.3) and T = {Tb; 0 < b < col of (2.4) are modifications of one another. Being the right(respectively, left-) continuous inverses of the process M in (2.2), they are strictly increasing. Moreover, they increase by jumps only: (2.9)
Sb = > (sx-sx_), T, = E (Tx+ - Tx); x
0
b < co,
x
a.s., where So_ A 0. These processes have stationary and independent increments, with distribution (2.10)
P[Sb -
e dt] = P[Sb_be dt] =
b -a e-(b-a)212' dt; .,./27rt3
0 < a < b, t > 0, (2.11)
Ee-asb
e-bf2;; 0 < a, b < co.
PROOF. The only new assertions are those contained in the last sentence; these follow from Proposition 2.8.5.
It is apparent from Theorem 2.1 that T must have infinitely many jumps on any interval [0, 6], b > 0, but T can have only finitely many jumps whose size exceeds a given number ( > 0. We want to know the distribution of the (random) number of such jumps. To develop a conjecture about the answer
6.2. Alternate Representations of Brownian Local Time
403
to this question, we ask another. How large must b be in order for T to have a jump on [0, b] whose size exceeds e? We designate by T, the minimal such b, and think of it as a "waiting time" (although b is a spatial, rather than temporal, parameter for the underlying Brownian motion W). Suppose we have "waited" up to "time" c and not seen a jump exceeding size e; i.e., T, > c. Conditioned on this, what is the distribution of T,? After waiting up to "time" c, we are now waiting on the reflected Brownian motion I = W.+T, WTI to undergo an excursion of length exceeding t; thus, the conditional distribution of the remaining wait is the same as the unconditional distribution of T,: (2.12)
P[T, > c + bit, > c] = P[T, > b].
This "memoryless" property identifies the distribution of T1 as exponential (Problem 2.2).
2.2 Problem. Let / = [0, co) or / = R. Show that if G: 1 -* (0, co) is monotone and (2.13)
G(b + c) = G(b)G(c);
b, c e I,
then G is of the form G(b) = e-'i6; be I, for some constant .1 e R. In particular, (2.12) implies that for some A > 0,
P[-i- > b] = e-'i6;
0 < b < co.
After T,, we may begin the wait for the next jump of T whose size exceeds e, i.e., the wait for I W+7-,1 = +7. to have another excursion of WT I
duration exceeding e. It is not difficult to show, using the strong Markov property, that the additional wait t2 is independent of Tl. Indeed, the "interarrival times" T1, t2, r3, ... are independent random variables with the same (exponential) distribution. Recalling the construction in Problem 1.3.2 of the Poisson process, we now see that for fixed b > 0, the number of jumps of the process T on [0, b], whose size exceeds e, is a Poisson random variable. To
formalize this argument and obtain the exact distributions of the random variables involved, we introduce the concept of a Poisson random measure.
B. Poisson Random Measures A Poisson random variable takes values in No A {0, 1, 2, ... and can be thought of as the number of occurrences of a particular incident of interest. Such a concept is inadequate, however, if we are interested in recording the occurrences of several different types of incidents. It is meaningless, for example, to keep track of the number of jumps in (0, b] for the process S of (2.3), because
there are infinitely many of those. It is meaningful, though, to record the number of jumps whose size exceeds a positive threshold e, but we would like
6. P. Levy's Theory of Brownian Local Time
404
to do this for all positive simultaneously, and this requires that we not only count the jumps but somehow also classify them. We can do this by letting v((0, b] x A) be the number of jumps of S in (0, b] whose size is in A e.4((0, co)),
and then extending this counting measure from sets of the form (0, b] x A to the collection .4((0, co)2). The resulting measure v on ((0, co)2, .4((0, co)2)) will
of course be random, because the number of jumps of {S.; 0 < a < b} with sizes in A is a random variable. This random measure v will be shown to be a special case of the following definition.
2.3 Definition. Let (S1,F, P) be a probability space, (H, Yr) a measurable space, and v a mapping from SI to the set of nonnegative counting measures on (H, Jr), i.e., v,,,(C)e No u {co} for every co ES./ and C e Jr. We assume that the mapping coi-+ v,,,(C) is .F/M(No u {co})-measurable; i.e., v(C) is an No u {co}-valued random variable, for each fixed Ce Ye. We say that v is a Poisson random measure if:
(i) For every C E A', either P[v(C) = co] = 1, or else A(C) -A Ev(C) < co
and v(C) is a Poisson random variable: P[v(C)
=
e--A(c)(A(C))" n!
;
neNo.
(ii) For any pairwise disjoint sets C1, ..., Cm in Jr, the random variables v(C, ),
v(Cm) are independent.
The measure A(C) = Ev(C); Cele, is called the intensity measure of v. 2.4 Theorem (Kingman (1967)). Given a o--finite measure A on (H, Jr), there exists a Poisson random measure v whose intensity measure is A. PROOF. The case A(H) < co deserves to be singled out for its simplicity. When it prevails, we can construct a sequence of independent random variables = A(C)/01(11); C E dr, as well as an . with common distribution PN1 E
independent Poisson random variable N with P[N = n] = e-411)(A(H))7n!; n e No. We can then define the counting measure v(C) A hi i=i
1c(4 C E
It remains to show that v is a Poisson random measure with intensity A. Given a collection C1, ..., Cm of pairwise disjoint sets in Y9, set Co = H Uir=i C so E,7=0 v(C,,) = N. Let no, n1, nm be nonnegative integers with n = no +
ni +
+ n,. We have P[v(C0) = no, v(C1) = n1, ..., v(Cm) = nm]
= P[N = n] P[v(C0) = no, v(Ci)
ni,
v(Cm) = n,,,IN
6.2. Alternate Representations of Brownian Local Time
=e
n!
(A(C0) yo
n!
(.1,(H))"
?T (1.(H) no! ni!... ?T.!
)
405
0.(Ci ) r 0.(Cm)r" A(H) )
A(H)
nn e_A(co(A(Ck)r" nk!
k =0
and the claim follows upon summation over no E No.
2.5 Problem. Modify the preceding argument in order to handle the case of o- finite A.
C. Subordinators 2.6 Definition. A real-valued process N = {Ng; 0 < t < co} on a probability space (1/,
S, and indeed any subordinator, can be thought of as a superposition of Poisson processes.
2.7 Theorem (Levy (1937), Hinein (1937), Ito (1942b)). The moment generating function of a subordinator N = {N1; 0 ._ t < co } on some (SI, F, P) is given by (2.14)
Ee'N, = exp [ -t {mot + J
(1 - e'e)ii(de)}]; t > 0, a > 0, (o,c0)
for a constant m
0 and a cr-finite measure /..i on (0, co) for which the integral in (2.14) is finite. Furthermore, if i7 is a Poisson random measure on (0, c0)2 (on a possibly different space (0, g", P)) with intensity measure
A(dt x de) = dt it(de),
(2.15)
then the process (2.16)
N,
ev((0,t] x de); 0 < t < co
A mt + J (0, co)
is a subordinator with the same finite-dimensional distributions as N.
2.8 Remark. The measure it in Theorem 2.7 is called the Levy measure of the
subordinator N. It tells us the kind and frequency of the jumps of N. The simplest case of a Poisson process with intensity A > 0 corresponds to m = 0
406
6. P. Levy's Theory of Brownian Local Time
andµ which assigns mass A. to the singleton { 1 }. If y does not have support on a singleton but is finite with A. = 14(0, co)), then N is a compound Poisson process; the jump times are distributed just as they would be for the usual Poisson process with intensity ;4., but the jump sizes constitute a sequence of independent, identically distributed random variables (with common distribution it(de')/),), independent of the sequence of jump times. The importance of Theorem 2.7, however, lies in the fact that it allows /.4 to be a-finite; this is exactly the device we need to handle the subordinator S = {Sb; 0 < b < which has infinitely many jumps in any finite interval (0, b]. PROOF OF THEOREM 2.7. We first establish the representation (2.14). The stationarity and independence of the increments of N imply that for a > 0, the nonincreasing function p °(t) A Ee-xl'4; 0 < t < co, satisfies the functional equation pc,(t + s) = pc,(t)p(s). It follows from Problem 2.2 that Ee'N' = e-"I1(2);
(2.17)
t > 0, a > 0,
holds for some continuous, nondecreasing function 0: [0, co) -* [0, co) with 0(0) = 0. Because the function g(x) A xe-";x > 0, is bounded for every a > 0, we may differentiate in (2.17) with respect to a under the expectation sign to obtain the existence of ti/ and the formula (a)e-"1/(2) =
1
t
1
-t
E[N,C°N1
8e-21 P[N,Ed1];
a > 0, t > 0.
Consequently, we can write
Oa) = lim kck J
(2.18)
k-'c c
(1 + [0,00)
where
(
E[1 +1N 1/, 1, p,(&) A clk
ck
1 -I-
( P[N,/, e de].
0 a.s., the theorem is trivially true, so we may assume the contrary and choose a > 0 so that c E(IVI 1{bri
(2.19)
ck >
E[
A
1
+1N i m
1
1{N
)] >
1
1+a
E[N11,1{N
k
= k(1 + a) E E[(N/k k(1 + a) We can establish now the tightness of the sequence of probability measures { pk } if= 1. Indeed,
407
6.2. Alternate Representations of Brownian Local Time
P[N, < e] > P[Nfik - Nu_of < e; j = 1,
k]
= 1P[Nlik
and thus, using (2.19), we may write
(2.20) MK cc))
1+a
k(1 + a)
kP[N,,,, >
(P[N, < tykl.
{1
= -log 0 < < 1, we can make the right-hand
Because limk k(1 -
side of (2.20) as small as we like (uniformly in k), by taking ( large. Prohorov's Theorem 2.4.7 implies that there is a subsequence {p,5}1°=, which converges
weakly to a probability measure p on ([0, co), d([0, co))). In particular, be+ e)e-e is bounded for every positive a, we must cause the function e have (1 + e)e-2? pk.,(d()=
lim
(1 + e)e-cie p(do. [0.c.)
10.c.)
Combined with (2.18), this equality shows that kick, converges to a constant c > 0, so that (2.21)
1//(a) = c
(1 + e)e-at p(cl()
J
= cp( {O})
c
(1 + e)e'ep(de); 0 < a < co.
J
op..)
Note, in particular, that i//' is continuous and decreasing on (0, co). From the fundamental theorem of calculus, the Fubini theorem, (2.21), and /i(0) = 0, we obtain now (2.22)
Oa) = acp({0}) + c f (0..)1 + e (1
e-2e)p(de);
0 < a < co.
The representation (2.14) follows by taking m = cp({0}) and (2.23)
p(de) =
c(1 +
p(de);
> 0,
the latter being a a-finite measure on (0, co). In particular, we have from (2.22): (2.24)
tp(c) = ma +
(1 - C'e')12(de) < co;
0 < a < cc.
(0.00)
We can now use Theorem 2.4 with H = (0, co)2 and lt = ge(H), to construct P) a random measure 17 with intensity given by on a probability space (n, P) via (2.16). (2.15); a nondecreasing process 13 can then be defined on (n, It is clear that Ro = 0, and because of Definition 2.3 (ii), 13 has independent increments (provided that (2.25), which follows, holds, so the increments are
6. P. Levy's Theory of Brownian Local Time
408
defined). We show that IV is a subordinator with the same finite-dimensional distributions as N. Concerning the stationarity of increments, note that for a
nonnegative simple function cp(f)=E7=iail,(l) on (0, x), where A1, ..., are pairwise disjoint Borel sets, the distribution of (o..)
t + h] x (10=
tp(t)
+ h] x Ai) i =1
is a linear combination of the independent, Poisson (or else almost surely infinite) random variables Iii((t, t + h] x Ai)}7=1 with respective expectations {h1t(Ai)}7=1. Thus, for any nonnegative, measurable co, the distribution of .1.0,00(p(I)v((t,t + h] x dl) is independent of t. Taking (Xi) = t, we have the stationarity of the increment A +h - A. In order to show that IV is a subordinator, it remains to prove that
A < co; 0
(2.25)
t < oo
and that lS is right-continuous, almost surely. Right-continuity will follow from (2.25) and the dominated convergence theorem applied in (2.16), and (2.25) will follow from the relation
Re'" = e-0(x); a > 0, t > 0,
(2.26)
= (6("1, /1")]; 1 < j < 4", we have from the monotone and bounded convergence theorems: where (I, is as in (2.24). With n > 1, and (1.) A j2-", (2.27)
E exp -
f
4^
( 17( 0, t] x de)} = lim
(0.oD)
exp - E 4!),
t] x
/.;"))} .
j= 2
n-.co
But the random variables ii((0, t] x II")); 2 < j < 4" are independent, Poisson, with expectations ty(11")); 2 < j < 4", and these quantities are finite because the integral in (2.14) is finite. The expectation on the right-hand side of (2.27) becomes 4"
Eexp{ -aer, i7((0,
x /l"))} =exp -t
! =2
4"
(1 -
)µ(/1"))
,
.1 =2
which converges to exp{ -t f(o)(1 - e'f)ki(d1)} as n co. Relation (2.26) follows and shows that for each fixed t > 0, A has the same distribution as N. The equality of finite-dimensional distributions is a consequence of the independence and stationarity of the increments of both processes. Theorem 2.7 raises two important questions:
(I) Are the constant m > 0 and the Levy measure it unique?
(II) Does the original subordinator N admit a representation of the form (2.16)?
One is eager to believe that the answer to both questions is affirmative; for
409
6.2. Alternate Representations of Brownian Local Time
the proofs of these assertions we have to introduce the space of RCLL functions, where the paths of N belong. 2.9 Definition. The Skorohod space D[0, co) is the set of all RCLL functions from [0, oo) into R. We denote by AD [0, co)) the smallest a-field containing all finite-dimensional cylinder sets of the form (2.2.1).
2.10 Remark. The space D[0, co) is metrizable by the Skorohod metric in such a way that M(D [0, co)) is the smallest a-field containing all open sets (Parthasarathy (1967), Chapter VII, Theorem 7.1). This fact will not be needed here.
2.11 Problem. Suppose that P and P are probability measures on (D[0, co), .4(D [0, co))) which agree on all finite-dimensional cylinder sets of the form
Y(WernI, < co, and Fie M(R); i = 1,
1Y GDP, GO; Ati)Gri,
where n > 1, 0 < t, < t2 <
< n. Then P and P agree on .4(D [0, co)). (Hint: Use the Dynkin System Theorem 2.1.3.) 2.12 Problem. Given a set C g_ (0, co)2, let n(; C): D[O, co) -> fro u {co} be defined by (2.28)
n(y;
A # {(t,e)E C; IY(t)
-
-)I = el,
where # denotes cardinality. In particular, n(y; (t, t + h] x ((, co)) is the number of jumps of y during (t,t + h] whose sizes exceed e. Show that n(; C) is .11(D[0, co))-measurable, for every Ce.4((0, co)2). (Hint: First show that n(; (0, t] x (i' , co)) is finite and measurable, for every t > 0, ( > 0.) Returning to the context of Theorem 2.7, we observe that the subordinator
N on (SI,,F,P) may be regarded as a measurable mapping from (12,") to (D [0, co), .4(D [0, co))). The fact that Ar defined on (5 , g, P) by (2.16) has the same finite-dimensional distributions as N implies (Problem 2.11) that N and induce the same measure on D[0, co): PEN e A] = P[KI e A]; V A e .4(D [0, oo)).
We say that N under P and under P have the same law. Consequently, for C1, C2, C, in gf((0, co)2), the distribution under P of the random vector . , n(N; Cm)) coincides with the distribution under 15 of (n(N; C1), (n(N; , n(N; Cm)). But (2.29)
n(g ; C) = ("(C);
C e *(0, 00) x
00))
is the Poisson random measure (under P) of Theorem 2.7, so (2.30)
v(C) A n(N; C) = # {(t, eC;
N,
- Nt_ = e}; C e .4((0, al')
is a Poisson random measure (under P) with intensity given by (2.15).
6. P. Levy's Theory of Brownian Local Time
410
We observe further that for t > 0, the mapping qv D[0, cc) -* [0, cc] defined by (My) A
f
in(y; (0, t] x do)
(o,o0)
is Af(D [0, oo))/.4([0, co])-measurable, and co,(N) =
f
(v((0, t] x do,
(MR) =
f
mo,t] x de).
co,c0)
co,c0)
It follows that the differences {N, - pi(N); 0 < t < col and {A - cp,(R); 0 < t < col have the same law. But A - co,(N). mt is deterministic, and thus N, - q),(N) = mt as well. We are led to the representation (2.31)
N, = mt +
i
0v((0, t] x dO);
0 < t < 00.
We summarize these remarks as two corollaries to Theorem 2.7. 2.13 Corollary. Let N = {Nt; 0 < t < co} be a subordinator with moment generating function (2.14). Then N admits the representation (2.31), where v given by (2.30) is a Poisson random measure on (0, cc)2 with intensity (2.15).
2.14 Corollary. Let N = INt; 0 < t < col be a subordinator. Then the constant m .. 0 and the a-finite Levy measure it which appear in (2.14) are uniquely determined. PROOF. According to Corollary 2.13, a(A) = Ev((0, 1] x A); A E MO, 00), where
v is given by (2.30); this shows that it is uniquely determined. We may solve (2.31) for m to see that this constant is also unique.
2.15 Definition. A subordinator N = {Nt; 0 < t < co} is called a one-sided stable process if it is not almost surely identically zero and, to each a > 0, there corresponds a constant 11(a) 0 such that {aNt; 0 < t < co } and {Now; 0 < t < col have the same law. 2.16 Problem. Show that the function Ma) of the preceding definition is continuous for 0 < a < cc and satisfies (2.32)
Nay) = Q(a)NY);
cc > 0, Y > 0
as well as (2.33)
i I t (a) = r i 3 (a);
a > 0,
where r = it/(1) is positive and 4./ is given by (2.17), or equivalently, (2.24). The unique solution to equation (2.32) is 13(a) = a', and from (2.33) we see that for a one-sided stable process N,
6.2. Alternate Representations of Brownian Local Time
411
0 < a < oo.
Ee'N. = exp{ - till(a)} = exp{ - traE};
(2.34)
The constants r, c are called the rate and the exponent, respectively, of the process. Because b is increasing and concave (cf. (2.21)), we have necessarily 0 < e < 1. The choice E = 1 leads to m = r, t = 0 in (2.14). For 0 < E < 1, we have
m = 0, m(dt) =
(2.35)
tit r(1rE E) e' +g;
e
D. The Process of Passage Times Revisited The subordinator S of Theorem 2.1 is one-sided stable with exponent E = (1/2).
Indeed, for fixed a > 0, (1/ot)Sb,A; is the first time the Brownian motion (1/.,/a) Wcu; 0 < t < oo } reaches level b; i.e.,
(Lemma 2.9.4 (i)) W* = {14/,* 1
- Sb,A,=
inf It
0; 147,* > bl.
az
Consequently, {otSb; 0 < b < co} has the same law as laSt, = Sb,";; 0 < b < col, from which we conclude that Noz) appearing in Definition 2.15 is NAi Comparison of (2.11) and (2.34) shows that the rate of S is r = .\/2, and (2.35) gives us the Levy measure
de it(de) = /2n83;; 2e3
e > 0.
Corollary 2.13 asserts then that eq(0,13] x de);
Sb
0 < b < oo,
to. co)
where, in the notation of (2.1)-(2.4), (2.30) (2.36)
v(C) = # {(b,e) e C; Sb
Sb- = el = # {(b,e) e C; WI has an excursion of duration e starting at time Tb }; CeI((0, co )2 ),
is a Poisson random measure with intensity measure (dt de'1,/27re'3). In particular, for any I eM((0, co)) and 0 < 5 < E < oo, we have (2.37)
Ev(I x [5,0) = meas(/)
dA
27c,13
= meas(/)
2(
For 0 <6<e 0, let us consider the random variables (2.38)
Ne,
v((0,13] x [6,0) = Number of jumps of size e e [6,E) suffered by
0 < a < 131,
6. P. Levy's Theory of Brownian Local Time
412
Lb'' A
(2.39)
1
tv ((0,6] x di')
[ a,
e)
= Total length of jumps of size 1 eR v) suffered by {Sa; 0 a 5 b}. We also agree to write
Nt, A N,- = v40,13] x R cc)), 14 A L'.2,'' = 1
ev((0,13] x de).
(o,e)
Np = {Np; 0 < b < op} is Poisson with intensity 02170411.A (1/,/i)); the process La- = {L:- ; 0 < b < op} is a sub-
The
process
-
ordinator which grows only by jumps (see the last paragraph of the proof of Theorem 2.7). Furthermore, rr
(2.40)
Ee-'11` = exp -b
L
de
(1 - e-'1)
1,
,./21re3 _I
e)
and these assertions concerning LV hold even if 6 = 0. The behavior of N: as 6 1 0 merits some attention. Of course, limayo N: = co a.s., and any meaningful statement will require some normalization. 2.17 Proposition. For almost every we SI, lim
(2.41)
tr6
-NI°, (o)) =- b;
V 0 5 b < 00.
al.()
PROOF. The process
Q, 4 v (0,
13]
x[12 , co));
0 5 t < op,
has nondecreasing, right-continuous paths and independent increments. For 0 s < t, the increment Qt -Qs= v((0,13] x [t-2, s-2)) is Poisson with expectation -2 S
E(Qt - Qs) = b ft -2
de
2tre3
=
2
-(t - s). it
We conclude that Q is a Poisson process, for which the strong law of large numbers of Remark 1.3.10 gives lim (Q,It)= b \/(2 /n) a.s. By definition we have NI, = Q 1/sir ,a so (2.41) holds a.s. for each fixed b. Except for w in a null set A g. Q, this relation must hold for all rational, nonnegative b. The monotonicity in b of sides of (2.41) then gives us its validity for w et-1\A, 0 < b < o o.
0 As c 10, the dominated convergence theorem shows that L 0 a.s. In other words, since Sb is finite, the total length of its jumps of size less thane must
413
6.2. Alternate Representations of Brownian Local Time
approach zero with E. We thus normalize Lb in a manner "opposite" to the normalization of (2.41).
2.18 Proposition. For almost every we SI, lim .\/1 Lb(w) = b; V 0 < b < co. elo
(2.42)
PROOF. Given e > 0 and 0 < p < 1, we have for all n > 1:
E
(2.43)
p- 2k v((0,b] x L ep2k, cp2(k -1)))
k=1 co
<E
cp2(k-1)V(0, 6] x [8p2k,cp2(k--1))).
k=1
The left-hand side of (2.43) may be written as EI pk.s/ep2k
pk+1 N/ep2(k-1)Nbep2(k-1))
k=1
which converges as el 0 (see (2.41)) to
E" (pk _pk+i)= b
b
2
pn+1)
k=1
It follows that limg10(1//04 > b.0211r)(p - pn+1), a.s., and letting first n
co and then p T 1, we obtain lim
1
2
>
Lb
a.s.
Now let 1(g) suPo<s <eiNF5Nt - b021701, so that limeio 1/(g) = 0 a.s. (Proposition 2.17). The right-hand side of (2.43) is subject to the bound
E .ip2(k-1)1#0, b]
X
[gp2k, gp2(k-1)))
k=1
Ipk-2,/,p2kNo2k_ pk-l.s/sp k=1
tpk-2 (b\fi k-=-1
b_
2
.,(8))
"
17(0(1
pk-1
(b
- ti(E)/ }
2 1
-P
It follows that lime4.0 (1/N/i)Li, < (b/P),./(2/n). Letting p T 1, we obtain 1
1im /-Lb e.Lo .s/E
b
a.s.
414
6. P. Levy's Theory of Brownian Local Time
This proves that (2.42) holds a.s. for each fixed b. We conclude as in Proposition 2.17.
2.19 Exercise. Show that for fixed, positive 6, the process X,(6) A N,6, b,/2/7r6; 0 < b < co, is a right-continuous, square-integrable martingale (with respect to the filtration 1.51;b")}). This process has stationary, independent increments and <X(6)>b =
2 biir6;
b < co.
0
Furthermore, we have the law of large numbers 1
N't), =
.\/- 2 , a.s. TEO
6* co
2.20 Exercise. Show that for fixed, positive c, the process Vb(e) A 4, 13\/(2E/n); 0 < b < co is a right-continuous, square-integrable martingale (with respect to the filtration {.F6''(e)}). This process has stationary, independent increments and 2b b = -3
(2.44)
Establish the representation
=
(2.45)
+
NI, d(;
0 < b < co, a.s.,
f(o,e)
as well as the law of large numbers lim
(2.46)
1
-14 = b
a.s.
(Hint: Obtain the moment generating function (2.47)
121 Ee'b(e) = exp ab - -b J
(1 - e-'1)
1) NI 21E1'3 _I
E. The Excursion and Downcrossing Representations of Local Time Let us now discuss the significance of Propositions 2.17, 2.18, for Brownian
local time. Returning to the context of (2.1)-(2.4) and with v the Poisson random measure given by (2.36), we may recast these propositions as (2.48)
P e
in:2 IMO, Mt] x LE, CO)) = Mt; V 0 5_ t < cd=
1,
415
6.2. Alternate Representations of Brownian Local Time
P [lim
(2.49)
40
fli f(0,0 (v((0,4] x do) = M,; d0
t < cod = 1.
LE
But, by (2.36),
v((0, M,] x [c, co)) = # {jumps of size > e suffered by IS,,; 0 < b < AI}
=#
excursion intervals away from the origin, of duration ze, made by I Wu; 0 < u < SAO
'
As observed earlier, if we have W 0 0 so that (2.7) holds, then Sm, = A is the time of conclusion of the excursion of W straddling t, and intervals away from
the origin, of duration >c, (2.50a) v((0, Mt] x [e, co)) = # {excursion
.
initiated by W before t
On the other hand, if W = 0, then intervals away from the (2.50b)
v((0,
,] x [c, oo)) = # {excursion origin, of duration .c, initiated
/14
by W at or before time t
Instead of the expressions on the right-hand side of (2.50a, b), it is perhaps easier to visualize the "number of excursion intervals away from the origin, of duration >c, completed by W at or before time t." This expression differs from v((0, Mt] x [e, co)) by at most one excursion, and such a discrepancy in counting would be of no consequence in formulas (2.48), (2.49): in the former, it would be eliminated by the factor ,,/i as c 10; in the latter, the effect on the
integral would be at most e, and even after being divided by 4, the effect would be eliminated as E 10. Recalling the identifications (2.1), we obtain the following theorem.
2.21 Theorem (P. Levy (1948)). The local time at the origin of the Brownian motion W satisfies L,(0) = lim \i-c,l.o
=
40
8
# excursion intervals away from the origin, of duration -_c, completed by {Ws; 0
s
t}
Total duration of all excursion intervals away from the origin of individual duration
,
completed by {Ws; 0 _<_ s < t}
V 0 < t < co, a.s.
2.22 Remark. The notion of local time 2L,(0) as occupation density suggests that this quantity is determined by the behavior of the Brownian path W near,
rather than at, the origin. Theorem 2.21 shows that local time is actually determined by the way in which Brownian motion spends time at the origin,
6. P. Levy's Theory of Brownian Local Time
416
for both representations in that theorem can be computed from knowledge of the zero set J1, = 10 < t < W, = 01 alone.
A third representation of local time in the spirit of Theorem 2.21, the downcrossings theorem, was conjectured by Levy (1959) and proved by Ito & McKean (1974). We offer a proof taken from Stroock (1982). Recalling the notation introduced immediately before Theorem 1.3.8, let Di(E) =
E; I WI)
be the number of downcrossings of the interval [0, a] by the Brownian motion tl. s Wsl; 0
2.23 Theorem (P. Levy's Downcrossings Representation of Local Time). The local time at the origin of the Brownian motion W satisfies
24(0) = lim ENE); 0 < t < co, a.s.
(2.51)
0
PROOF. For fixed a > 0, let us define recursively the stopping times To an
4 inf {t > tn-i; I II = e },
to
0 and
itif{t > an; III = 0}
for n > 1. With nn = an - rn -1 = rn an, we have from the strong Markov property as expressed by Theorem 2.6.16 that the pairs (n 1> are independent and identically distributed. Moreover, Problem 2.8.14 asserts that
E1 =
(2.52)
E2'
and we also have
{ j < NO} = {-5 < t} is independent of {(rh, ,)},T=J+,.
(2.53)
Suppose that t E [an, t,,) for some n > 1; then n-i -1wQ,11+ 147,I IlWtAt,} -1WtAn,11 = j=1
J =1
= -EDi(E)
e
1Wil
aDt(e). In either case,
If t U,T=1 [an, r), then ET=1{1141tAr,} -1WtAn,1} = (2.54)
W,
T
I - iWt,n,11 = -0,(a) +
WI
- a)
J =1
l[ni,)(t) J=1
On the other hand, the local time L.(0) is flat on U;,'°_1 [a, c) (cf. Problem 3.6.13 (ii)), and thus from the Tanaka formula (3.6.13) we obtain, a.s.: (2.55) J =1
{1WtA
- I WtAt7,11 = 1Wt1
From (2.54), (2.55) we conclude that
2L,(0) - E
J=1
i" tAr.,
sgn(147,)dWs.
417
6.2. Alternate Representations of Brownian Local Time
el),(e) - 2L,(0) = -I WI
(2.56)
,,
-
+E
11,,r5)(t) J=1
J=1
sgn(14/s)dWs,
a.s.
A 5_,
.i=1
2.24 Problem. Conclude from (2.56) that, for some positive constant C(t) depending only on t, we have EleD,(E) - 2L,(0)12 < C(t)s.
(2.57)
Ceby§ev's inequality and (2.57) give
P[In-2A(n-2) - 2L,(0)I > n-114] < C(t)n- 312 ,
and this, coupled with the Borel-Cantelli lemma, implies
lim n-2D,(n-2) = 2L,(0), a.s. n-oco
But for every 0 < E < 1, one can find an integer n = n(e)
1 such that
(n + 1)-2 < E < n-2, and obviously (n + 1)-2D,(n-2) < eD,(c) < n-2 D,((n + 1)-2) holds. Thus, (2.51) holds for every fixed t E [0, o0); the general statement follows
from the monotoncity in t of both sides of (2.51) and the continuity in t of 2.25 Remark. From (2.51) and (2.1), (2.2) we obtain the identity (2.58)
lim ED[o ti(0; E; M(w)
B(w)) = 114,(w);
0 < t < oo
E
for P-a.e. w E a The gist of (2.58) is the "miraculous fact," as Williams (1979)
puts it, that the maximum-to-date process M of (2.2) can be reconstructed from the paths of the reflected Brownian motion M - B, in a nonanticipative way. As Williams goes on to note, "this reconstruction will not be possible for any picture you may draw, because it depends on the violent oscillation of the Brownian path." You should also observe that (2.58) offers just one way of carrying out this reconstruction; other possibilities exist as well. For instance, we have from Theorem 2.21 that lim o
-# 2
lim
40
excursion intervals away from the origin, of duration > c, completed by {M, - 135; 0
s
total duration of all excursion intervals, away from the origin, of individual duration
hold almost surely.
M,;
V0
t < cc,
t}
= M; V 0 < t < co,
418
6. P. Levy's Theory of Brownian Local Time
6.3 Two Independent Reflected Brownian Motions Our intent in this section is to show how one can create two independent, reflected Brownian motions by piecing together the positive and negative excursions of a standard, one-dimensional Brownian motion. This result has important consequences, and we develop some of them, most notably the first formula of D. Williams (Theorem 3.6). Our basic tools will be the F. Knight Theorem 3.4.13 and the theory of Brownian local time as developed in Section 3.6; we shall retain the setting, assumptions, and notation of that section.
A. The Positive and Negative Parts of a Brownian Motion We start by examining the Tanaka formulas (3.6.11-12) a bit more closely. With a = z = 0, L(t) A L1(0), and (3.1)
f
/+(t)
100)(WO dW,,
LW A
ft
1(-00,0")d117,,
0
.3
these formulas read
W ± = -/ +(t) + L(t); 0 _5 t < oo
(3.2)
a.s. P°. The processes in (3.1) are continuous, square-integrable martingales, with quadratic variations (3.3)
<1 + >(t) = T +(t)
meas {0
s 5 t; Ws> 0}
(3.4)
<1_>(t) =
meas {0
s < t; W, 5 0}
and cross-variation equal to zero: <4, /_ > (t) =
(3.5)
1(
)(Ws) 1(,03(14/)ds = 0.
We also have
lim t +(t) = co,
(3.6)
a.s.
t-'cO
from Problem 3.6.30. On the other hand, WI are nonnegative processes which satisfy 1(0,00)(W±(s))dL(s) = 0
(3.7) J
a.s. P° (Problem 3.6.13 (ii ). It becomes evident then from (3.2), (3.7) that the pairs (L, W±-) are solutions to the Skorohod equation of Lemma 3.6.14 for the functions -1 +, respectively. From the explicit form (3.6.32) of the solution to this equation, we deduce (3.8)
L(t) = max / +(s), 147,± = max / +(s) - /±(t); 0 oss<, osst
t < co, a.s. P°.
6.3. Two Independent Reflected Brownian Motions
419
Now let us introduce the right-continuous inverses of the occupation times T+ of (3.3), (3.4), namely (T) = inf {t
(3.9)
0; I" + (t) > r};
0 < t < GC.
Theorem 3.4.13 asserts, in conjunction with (3.3)-(3.6), that the processes
±(t) A +(I-V(0); 0 5 r<
(3.10)
are independent, standard, one-dimensional Brownian motions under P°, and from Theorem 3.4.6 we also have the representations I+(t) = B ±(I ±(t));
(3.11)
0 < t < co.
Here then is the fundamental result of this section. 3.1 Theorem. The processes W+ (T) A + W-_0(,);
(3.12)
0
cc
are independent, reflected Brownian motions under P°. PROOF. We start by introducing (3.13)
L +(t) -A- L(TV(r))
and observing that because of (3.8), (3.11), and Problem 3.4.5 (ii), (iii) we have, a.s. P°: (3.14)
L +(t) =
/+(s) =
max 05s5r.V(T)
max
0
B+(1" +(s)) = max B +(u) (.114<.r
for all 0 < t < CC; in particular, L are independent, continuous nondecreasing processes. Now for each 0, I"±(T-+1 (T)) = t < 1-41"-±1(r) + (5) for all (5 > 0, and consequently 14/-L(.0 > 0, 14/1-,1(.0 0 hold a.s. P°. It follows then that (3.15)
W +(t) = Wrt,i(r) = L +(t) -B +(t) = max B ±(u) -B ±(t) O
also hold a.s. P°, first for a fixed t > 0, and then by continuity, for all t > 0 simultaneously. From Theorem 3.6.17, each of the processes W+ is a reflected Brownian motion starting at the origin; W+ are independent because B are.
3.2 Remark. Theorem 3.6.17 also yields that the pairs {(WAT), L +(T)); 0 < t < CO} have the same law as {(1W,I, 24(0)); 0 < t < col, under P°. In particular, we obtain from (3.6.36) and (3.14), (3.15): 1
(3.16)
L + (T) = lim
40 LE
meas {0 5_ o- < t; W+ (0) <
a.s. P°
for every T e [0, co). The processes L± are thus adapted to the augmentations of the filtrations Wzi- 9 a(W+(u); 0 < u < r).
6. P. Levy's Theory of Brownian Local Time
420
3.3 Remark. As suggested by the accompanying figure, the construction of W +() = W,-;,(.) amounts to discarding the negative excursions of the Brownian
path and shifting the positive ones down on the time-scale in order to close up the gaps thus created. A similar procedure, with a change of sign, gives the construction of W_() = W-74.). Theorem 3.1 asserts then, roughly speaking, that the motions of Won the two half-lines (0, co) and ( -co, 0) are independent.
wt W +(T) = w
3.4 Problem. Derive the bivariate densities (3.17)
P°[14/, e da; 24(0)e db] =
b + 1611
exp
2trt3 (3.18)
P°[
da; 24(0)e db] =2(a +
b)
exp
(b + la1)2}da db; 2t
(a + 6)2} 2t
.,./27rt3
da db;
a eR >0
for b > 0. 3.5 Exercise. Show that the right-hand side of (5.3.13), Exercise 5.3.12, is given by Sr q,(a) da, where (3.19)
qt(a) =
,/21rt
exp
(Ial + t)2} 2t
e- Thd I 'D exp. { lal
(ll
2t
dv .
6.3. Two Independent Relected Brownian Motions
421
B. The First Formula of D. Williams The processes L, of (3.13) are continuous and nondecreasing (cf. (3.14)), with right-continuous inverses (3.20)
L-,1(b)
0; max B +(u) > b} .
inf {z > 0; L +(r) > 6} = inf
0
For every fixed b e [0, co), L7,1 (b) is P°-a.s. equal to the passage time (3.21)
Tb-k A TbB±
= inf {t > 0; B +(t)_ b}
of the Brownian motion B+ to the level b; cf. Problem 2.7.19 (i).
3.6 Theorem (D. Williams (1969)). For every fixed a > 0, t > 0 we have (3.22)
E°[e'+'(')I W,(u); 0< u< co] =
.12aL,(v);
a.s. P°.
PROOF. The argument hinges on the important identity (3.23)
a.s. P°
f+-1 (r) = t + L:1 (L,(T)) = z + TL-Ao;
which expresses the inverse occupation time TV (r) as t, plus the passage time of the Brownian motion B_ to the level L,(r). But L +(t) is a random variable measurable with respect to the completion of a(W +(u); 0 < u < co), and hence independent of the Brownian motion B_ . It follows from Problem 2.7.19 (ii) that (3.24)
Lii (L+(r)) =
a.s. P°,
and this takes care of the second identity in (3.23). The first follows from the string of identities (see Problem 3.7) (3.25)
L 11 (L,(r)) = inf t
0; L_(t) > L,(T)}
= inf It > 0; L(Fil(t)) > L(r VW)} = inf It
0; 1".=1(t), FV(r)}
=1--(1"-Ti(T))=F;1(t)- r+(rV(T)) ET1(7)- t, a.s. P°. Now the independence of {B_(u); 0 < u < a)} and { W,(u); 0 < u < cc}, along with the formula (2.8.6) for the moment generating function for Brownian passage times, express the left-hand side of (3.22) as E° CiTb b= +(t)
e-cts-.1.2iLAr)
a.s. P°.
111
3.7 Problem. Establish the third and fourth identities in (3.25). Following McKean (1975), we shall refer to (3.22), or alternatively (3.23), as
the first formula of D. Williams. This formula can be cast in the equivalent
422
6. P. Levy's Theory of Brownian Local Time
forms (3.26)
PITZ1(r) < t I W+(u); 0 < < ao] =
[Tb < t -
lb-L+(t)
= 2 [1 -(I) ( Li+ (t) )1 t
rr
=
L+(t)
e -Li (r)/2(0-0
2140 - r)3
dO
a.s. P°, which follow easily from (3.23) and (2.8.4), (2.8.5). We use the notation (120(z) = e'2/2 du.
We offer the following interpretation of Williams's first formula. The reflected Brownian motion { W+(u); 0 < u < co} has been observed, and then a time r has been chosen. We wish to compute the distribution of TV (r) based on our observations. Now 147_, consists of the positive part of the original Brownian motion W, but W+ is run under a new clock which stops whenever W becomes negative. When r units of time have accumulated on this clock corresponding to W+, r +' (r) units of time have accumulated on the original clock. Obviously, F+-1(r) is the sum of T and the occupation time 1-1(1-+-1(r)). Because W_ is independent of the observed process W+, one might surmise that nothing can be inferred about r_ (t) from W. However, the independence between W+ and W_ holds only when they are run according to their respective clocks. When run in the original clock, these processes are intimately con-
nected. In particular, they accumulate local time at the origin at the same rate, a fact which is perhaps most clearly seen from the appearance of the same process L in both the plus and minus versions of (3.2). After the time changes (3.12) which transform W ± into W+, this equal rate of local time accumulation finds expression in (3.13). (From (3.16) we see that L+ is the local time of W+.) In particular, when we have observed W+ and computed its local time L+(r), and wish to know the amount of time W has spent on the negative half-line
before it accumulated r units of time on the positive half-line, we have a relevant piece of information: the time spent on the negative half-line was enough to accumulate L+(t) units of local time. Suppose L+(c) = b. How long should it take the reflected Brownian motion W- to accumulate b units of local time? Recalling from Theorem 3.6.17 that the local time process for a reflected Brownian motion has the same distribution as the maximum-to-date process of a standard Brownian motion, we see
that our question is equivalent to: How long should it take a standard Brownian motion starting at the origin to reach the level b? The time required is the passage time Tb appearing in (3.23), (3.26). Once L +(T) = b is known,
nothing else about W. is relevant to the computation of the distribution of 1--(r+-1(-0).
3.8 Exercise. Provide a new derivation of P. Levy's arc-sine law for F+ (t) (Proposition 4.4.11), using Theorem 3.6.
6.3. Two Independent Reflected Brownian Motions
423
C. The Joint Density of (W(t), L(t), r+(t)) Here is a more interesting application of the first formula of D. Williams. With T > 0 fixed, we obtain from (3.26): P°[1-_-,-i CO E dt1W+(t) = a; L+(t) = b] -
(3.27)
be-b2/2(`-') ,
- \ / 2n(t - -03
dt;
t
co,
for almost every pair (a, b) of positive numbers. Remark 3.2 and the bivariate density (3.18) give P°[W,(T)e da; L+(t) e db] =
2(a, + b)
e_(a+b)2121 da db;
,./2trr3
a > 0, b > 0,
and in conjunction with (3.27): P° [ W, (r) e da; L_F(T)e db;IT1(t)e dt] = f(a, b; t, 'Oda db dt;
(3.28)
a > 0,
b > 0, t > r
where
f(a, b; t, t) _°
(3.29)
( + b) ba
nr3/2 (t - -0312 exP 1
b2
2(t - r)
(a + 6)21 2t j .
We shall employ (3.28) in order to derive, at a given time t e (0, co), the trivariate density for the location W, of the Brownian motion; its local time L(t) = L1(0) at the origin; and its occupation time r+(t) of (0, co) as in (3.3), up to t. 3.9 Proposition. For every finite t > 0, we have (3.30)
P°[ W, e da; L(t) e db; r+(t)e dt] f(a, b; t, -r)da db dr; a > 0, b > 0, 0 < i < t, (- a, b; t, t - -c)da db (IT; a < 0, b > 0, 0 < i < t,
in the notation of (3.29).
3.10 Remark. Only the expression for a > 0 need be established; the one for
a < 0 follows from the former and from the observation that the triples (W L(t), r,(0) and ( -W L(t), t - F+(t)) are equivalent in law. Now in order to establish (3.30) for a > 0, one could write formally 1
dtr[147+(t)e 1
da; L+(r)e db; rvcoe dt]
= -PqW e da; L(t)e db;1-.,(t)e dt]; a > 0, b > 0, 0 < r < t, dt '
6. P. Levy's Theory of Brownian Local Time
424
and then appeal to (3.28). On the left-hand side of this identity, r is fixed and we have a density in (a, b, t); on the right-hand side, t is fixed and we have a density in (a, b, r). Because the two sides are uniquely determined only up to sets of Lebesgue measure zero in their respective domains, it is not clear how this identity should be interpreted. We offer now a rigorous argument along these lines; we shall need to recall the random variable fl, of (2.6), as well as the following auxiliary result. 3.11 Problem. For a e gl, t > 0, c > 0 we have P°
(3.31)
[
max Ws > a + c; min Ws < a - ci = o(h) t<s
t<s
and for a > 0, r > 0: P°[W+(r) > a; t
(3.32)
TV (r) < t + h; fl, < t + h] = o(h)
P°[14/, > a; fl, < t + h] = o(h)
(3.33) as h 1 O.
PROOF OF PROPOSITION 3.9. For arbitrary but fixed a > 0, b > 0, t > 0, r e (0, t)
we define the function co
F(a, b; t, r) A
(3.34)
f(a, 13; t, r) da d13 6
a
which admits, by virtue of (3.28), (3.32), the interpretation (3.35)
F(a, b; t, r) 1
= lim -P°[W+(r) > a; L+(t) > b; t _<_ F471(r) < t + h] hl.o h 1
= lim -P°[W+(r) > a; L+(t) > b; t ._<_ 1"471(r) < t + h; 13,
t + h].
1,13 h
For every h > 0 we have 1"+(s) = 1"+(t) + s - t,
L(s) = L(t);
V s e [t, t + h]
on the event 114/, > 0, fl, > t + h }. Therefore, with 0 < c < a and
A A {L(t) > b; T - h <1"+(t)
(3.36)
r; 13, > t + hl,
we obtain (3.37)
P°[W.,(r)> a+ e; L +(r) >b; t 51T1(r) a; A]
< P° [ max Ws > a + c; 241-P°[ min t<s
Ws > a;
Al = o(h),
6.4. Elastic Brownian Motion
425
by virtue of (3.31). Dividing by h in (3.37) and then letting h 10, we obtain in conjunction with (3.35), (3.32): (3.38)
1
F(a + e, b; t, t) < lim - PqW, > a; L(t) > b; t -h < r+(t) < a T-1.7i h
Similarly, (3.38)
P°[Wt> a; A] - PG1W+(z)> a- e; L+(z)>b; tl-Z1(T) t+h]
< P° [ max Ws > a; /1]- P°[ min Ws > a -E; Al = o(h), t<s
r<s<14-h
and we obtain from (3.33): 1
(3.39)
F(a - E, b; t, T) _>_ lim -P°[1,11, > a; L(t) > b; T - h < F+(t)
t].
h.Lo h
Letting E ,I, 0 in both (3.38), (3.39) we conclude that 1
F(a, b; t, T) = lim -P°[W, > a; L(t) > b; i - h < F,(1) _.<_ r], hlo h from which (3.30) for a > 0 follows in a straightforward manner.
0
3.12 Remark. From (3.30) one can derive easily the bivariate density (3.40)
P° [24(0) e db; r+ (0 e dr] =
bte-'21"`-') 4irr 3/2 (t - -r)3/2
db di;
b > 0, 0 < r < t
as well as the arc-sine law of Proposition 4.4.11.
The reader should not fail to notice that for a < 0, the trivariate density of (3.30) is the same as that for (W M 0,) in Proposition 2.8.15, for M, = maxo.(s , Ws, et = sup {s .5 t; Ws = AI. This "coincidence" can be explained by an appropriate decomposition of the Brownian path {Ws; 0 < s < t }; cf. Karatzas & Shreve (1987).
6.4. Elastic Brownian Motion This section develops the concept of elastic Brownian motion as a tool for computing distributions involving Brownian local time at one or several points. This device allows us to study local time parametrized by the spatial variable, and it is shown that with this parametrization, local time is related to a Bessel process (Theorem 4.7). We use this fact to prove the DvoretzkyErdi5s-Kakutani Theorem 2.9.13.
6. P. Levy's Theory of Brownian Local Time
426
We employ throughout this section the notation of Section 3.6. In particular, W = {W, F; 0 5_ t < co}, (C/, F), {Px}x RI will be a one-dimensional Brownian family with local time (4.1)
1
L,(a) = lim - meas {0 40 4a
t; 1W- al 5 a}; 05t
s
On a separate probability space (ST, , P'), let R1, R be independent, , y,,, respecexponential random variables with (positive) parameters yi , tively, i.e.,
R e dr) =
P' (R1 e dri,
dr,. i=i
We consider n distinct points al,
an on the real line and define a new process 1P, which is the old Brownian motion W "killed" when local time at any of these points ai exceeds the corresponding level R,. More precisely, with Tr(at) = inf { t > 0; L,(a,) > r}; (4.2)
r > 0, and
0; L,(a,) > R, for some 1 < i 5.
inf { t
= min T 1
we define the new process
if A
{Wt;
0
t < C,
A;
t
C,
and call it elastic Brownian motion with lifetime C. Here, A is a "cemetery" state isolated from R. We may regard l' as a process on C/ g S2 x = F 0 .Y7', fix
= Px x P'.
A. The Feynman-Kac Formulas for Elastic Brownian Motion Our intent is to study the counterpart (4.3)
f(1P-t) exp { -at -J t k(Ifs)ds} dt
u(x) = Ex J
f f(W,) exp 1 -- - ft k(IP) ds} dt dP' dPx.
= n
n'
o
of the function z in (4.4.14). The constant a is positive, and the functions f and k are piecewise continuous on R (Definition 4.4.8), mapping l u {A} into R and [0, co), respectively, and with (4.4)
f(A) = k(A) = 0.
Hereafter, we will specify properties of functions restricted to R; condition
6.4. Elastic Brownian Motion
427
(4.4) will always be an unstated assumption. In order to obtain an analytical characterization of the function u, we put it into the more convenient form U(X) = n
min r, (a,)
r .fw 0
0
f(W)e- -
k(K)ds
JL,(a,,)
1,,(a 1)
dt dri ...drdPx 1=1
rc.
co
f0
k(K)ds
0
dri ...drdt dPx,
fl
1=1
whence u(x) = EX f°D f(W)exp
(4.5)
{-
- ft k(Ws)ds - E yiL,(a,)}dt. 1=1
4.1 Theorem. Let f:
[0, co) be piecewise continuous functions, and let D = Df u Dk be the union of their discontinuity sets. Assume that, for some a > 0, (4.6)
01 and k: fly
EX
e' I f(W,)I dt < co;
V X E kR,
and that there exists a function Ii: R R which is continuous on R, C1 on R\{ai,...,a,,}, C2 on DIVD u {a1,... , an}), and satisfies 1
(a +
(4.7)
= -2a" + f on OlVD u {a
(a,+) -
(4.8)
,a}),
-) = ylii(a,); I < i < n.
If f and a are bounded, then a is equal to the function u of (4.3), (4.5); if f and a are nonnegative, then a > u. PROOF. An application of the generalized ItO rule (3.6.53) to the process X, A ti(W,)exp
{-at -jt k(Ws)ds -
y,L,(a,)};
0
t < co,
1 =1
yields nAs,,
f(W,) exp
EX
{-
- f k(Ws)ds -
o
i=1
y,L,(a,)} dt = a(x) - Ex X. A s,,,
where (4.9)
S,, = inf ft
0; IV
If f and u are bounded, we may let n op and use the bounded convergence theorem to obtain u = u. If f and a are nonnegative, then Ex X A sn > 0 and we obtain a > u from the monotone convergence theorem.
428
6. P. Levy's Theory of Brownian Local Time
4.2 Remark. Theorem 4.1 is weaker than its counterpart Theorem 4.4.9 because the former assumes the existence of a solution to (4.7), (4.8) rather than asserting that u is a solution. The stronger version of Theorem 4.1, which asserts that u defined by (4.5) satisfies all the conditions attributed to u, is also true, but the proof of this result is fairly lengthy. In our applications, the function u with the required regularity will be explicitly exhibited. These comments also apply to Theorem 4.3.
A variation of Theorem 4.1 can be obtained by stopping the Brownian motion W when it exits from the interval [b, c], where -co < b < c < oo are With the convention Tx, = co, define fixed constants not in {al, , (4.10) v(x) = EX [1 {Tb
{-
Tb
ds
Tb 0
i=1
y,LTb(ai)}1;
x
b
c.
4.3 Theorem. Let k: [b, c] [0, co) be piecewise continuous, a > 0, and assume which is continuous on [b, c], Cl on that there exists a function 17: [b, c] (b, c)\{ai, , a}, C2 on (b, c)\(LI, u {al, , a}), and satisfies
(a +
(4.11)
1
k)i5 =
on (b, c)\(D, u la ,
-26
,
a}),
+) - nai-) = yib(ai); 1 5 i 5 n,
(4.12) (4.13)
17(b) = 1,
(4.14)
i5(c) = 0,
(except that (4.13) should be omitted if c = oo). If 15 is bounded, then function v of (4.10); if i3 is nonnegative, then
15 is
the
15 > v.
PROOF. With Y,
i5(
exp -at -
k(14/s)ds - > yiLt(ai)}; o
0
t < co,
i=i
the generalized Ito rule (3.6.53) yields i5(x) = Ex X.Tn AT, A /I A Sn; b < x < c, where S is given by (4.9). If 1i-is bounded, we may let n co to conclude that i5(x) = 1 {Tb Ex 1{ Tb<
YTb = v(x).
The following exercises illustrate the usefulness of Theorem 4.3 in computations of distributions.
4.4 Problem. For any positive numbers a, /3, y, b, justify the formula
6.4. Elastic Brownian Motion (4.15)
429
Ex exp( -ca'b - 13f +(Tb) - YLTA) Y + .i2c4 .N./2(a + 13)
Y + N/2c4
.,./2(a + /3)
sinh(x.,/2(a + /3)) + cosh(x.,./2(a + /3)) ;
0 < x < b,
;
x < 0.
sinh(b,/2(a + II)) + cosh(b.,/2(a + 11) ex
2a
Y + ,.x sinh(b,./2(a + /3)) + cosh(b2(a + /9)) .,./2(a + /3)
With x = 0, we obtain in the limit as al 0, iq i 0:
E° exp(-yL,b(0)) =
(4.16)
y > 0.
;
In other words, LTb(0) under P° is an exponential random variable with expectation E° LTb(0) = b. On the other hand, as a 10, y 10, (4.15) becomes
Exe-or.(To =
(4.17)
cosh(x.i2/3);
0 < x < b,
cosh(b.,/249)
and we recover (4.4.23) by setting x = 0. Can you also derive (4.4.23) from (2.8.29) and Theorem 3.1 without any computation at all?
4.5 Exercise. Let R, be the first time that the Brownian path W falls b > 0 units below its maximum-to-date M, A- max,,,<, Ws; i.e.,
R, = inf {t _- 0; Mt -W, = b}. Show that (4.18)
E° exp( - aRb -yMRb)=---
.1S ../2a cosh(bN/2a) + y sinh(bN/2a)
holds for every a > 0, y > 0. Deduce the formula (4.19)
E° exp(-yMRb) =
1
yb;
y > 0,
which shows that MRb is an exponential random variable under P° with expectation E° M Rb = b. (Hint: Recall Theorem 3.6.17.)
The following exercise provides a derivation based on Theorem 4.1 of the joint density of Brownian motion, its local time at the origin, and its occupation time of [0, co). This density was already obtained from D. Williams's first formula in Proposition 3.9.
6. P. Levy's Theory of Brownian Local Time
430
4.6 Exercise.
(i) Use Theorem 4.1 to justify the Laplace transform formula E°
(4.20)
lia'm)(W)e -2' -flf.(t)-yL,(0)dt o
2e-'12('+9)
=
.,./2(oc + /3) [y + ,./2a + .12(a + /3)] for positive numbers a, /3, y, and a. (ii) Use the uniqueness of Laplace transforms, the Laplace transform formula in Remark 4.4.10, the formula
f.0 o
b
e-At
.,/2/rt3
exp 1
b2
b > 0, A > 0,
1 dt = exp{ - b,./2.1.};
and (4.20) to show that for a > 0, b > 0, 0 < T < t: P°[W, > a; WO) e db; F+(t)e di
(4.21)
b 7t,t112(t
(a +2-rb)2}
W/2 exP {
2(tb-2 T)
db d- r.
(iii) Use (4.21) to prove (3.30).
B. The Ray-Knight Description of Local Time We now formulate the Ray-Knight description of local time, evaluated at time t = Tb, and viewed as a process in the spatial parameter.
4.7 Theorem (Ray (1963), Knight (1963)). Let IR .A; 0 < t < ool, (S, °.), {Pr}r>o be a Bessel family of dimension 2, and let b > 0 be a given number. Then 111?; 0 < t < b} under P° has the same law as {L,.(b - t); 0 < t < b} under P°.
In order to prove Theorem 4.7, it is necessary to characterize the finitedimensional distributions of {R,2; 0 < t < b }. Toward that end, we define recursively (4.22)
fo = 1,
and for n > 1, (4.23)
f(ti, Yi; tz; Yz;
; 4,, yn) = fn-i(tz - t1, Yz; t3 - tl, Y3;
; to - tl, Yn)
n
+ t 1 E Ykfn-k(tk+1 - tk, Yk+1; tk+2 - tk, Yk+2; k=1
; to - tk, yn).
6.4. Elastic Brownian Motion
431
Although we will not need this fact, the reader may wish to verify that
E E vivito), - t1)
; tn, yn) = 1 + E Ytti
fn(ti, Yi; t2, Yz;
1
1 <<sn
-
+ E E E yiyiykti(t; 1 < i<J
+ Y1 Y2
(tn
t1)
+
4-1).
We will, however, need the relation (4.24)
fn+i(t yi;
:
:t
t n, , vn, n1, , vn+1)
= [1 + Yn+1(tn+1 - tn)]fn ( t1, Yl;
; tn-1, 'A-1; tn, Yn +
.
Yn+1
1 + Yfl-F1 (4+1 - tn)
valid for n > 1, which is easily proved from (4.22), (4.23) by induction. , yn, < tn < co and positive numbers yl, 4.8 Lemma. For 0 < t1 < t2 < the Laplace transform of the finite-dimensional distributions of the two-
dimensional Bessel process is given by 1
[exp
(4.25)
n
1
yiR4}] = fn(t1, Yl;
L f=E1
; tn, Yn).
PROOF. The straightforward computation 1
Ex e-vivt212 =
\/ 1 + yt
exp
yx 2
2(1 +
x
gives (4.26)
fir e- YR1/2 =
1
1 + yt
exp
y > 0,
Yr2
2(1 + yt)
t > 0, r > 0,
which proves (4.25) for n = 1. Assume that (4.25) holds for some value of n, < tn < tn+i < co and positive numbers v,i, and choose 0 t1 < Yn, Yn+l From the strong Markov property, (4.26), and (4.24), we obtain E° [exp
1 n+1
2 1E =1
yi/q
}1
'
exp { --i En yin2} C 0 [exp(-- yn+/ 12,2.+,) R,. = rni
=
2
2 1=1
f[0,co),,
X P° [R,1 e dri,... , Rt. e dr.]
=
exp f[0,00).
-! i yiri2 Ern exp 21-1
x
--21 Y.-Filq.1--t.
[R, edri,..., Rt e dr.]
432
6. P. Levy's Theory of Brownian Local Time "
1
exp
y+i(t+1 -
21
yi;
1 + Yn+1tn+1 (
[R, E dr
x
=
Yn+i r,i/2
LEI y,r,2
,
tn)
, Rt. E dr n]
; t., Y.; t.+1, Y.+1).
4.9 Lemma. With 0 =
< b and positive numbers 61, ..., (5, we
<
S1<
have the Laplace transform formula (4.27)
[exp
-
1
OiLT(0}] =
-
(5.; b -
(5.-1;
..; b -
61).
PROOF. Theorem 4.3 implies that the function V(X) -4 Ex
[expf- E SiLTA)}1;
<x
i=i
b
can be found by seeking a bounded, continuous function i3 on (-co, b) which is linear in each of the intervals (-cc, 0), (0, n b) and which satisfies
-
1 < i < n,
= 01,) = 1.
Thus, /3 must be of the form ti(x) = Ci
where 4 = -co,
Lix
(4.30)
0 < i < n,
i4-1],
= b, and we must have Lo = 0,
(4.28) (4.29)
on
= Ci + 1/.1;
Ci-i +
Li -
1 < i < n, 1 < i < n,
= (51(C1_1 +
C + bL = 1.
(4.31)
These 2n + 2 equations can be solved as follows. Let 131_1 = (C1_1 + 1L1_1)/
Co; 1
Bi = B1_1 +
- 0-Li 1
i
Co
from (4.29), as well as (4.33)
L, = L1_1 + oiCoBi_i;
1 < i < n,
from (4.30). This last identity, along with (4.28), gives
n,
6.4. Elastic Brownian Motion
433
Li = co E afpfri; 1 < i < n, J=1
which, substituted in (4.32), yields (4.34)
-
B, = 131_1 +
E 6/3;_i;
1
n.
J=1
The recursion (4.34) determines B1, B2, ... , B, and (4.31) together with the
definition of B gives Co = (1/B). The constants L1, L2, ... L are now determined by (4.33), and C1,
can be found from (4.29).
Having thus solved the equations (4.28)-(4.31), we may conclude from Theorem 4.3 that Co =
1
Bn
= v(0) = v(0) = E° [exp
- i=i oiLTigi)}1.
But comparison of the recursion (4.34) with (4.23) shows that 13; =
JilSitl - bi, Si, Sitl -
6i-1;
-
; Sitl - S1, 61);
1 < 1 < n.
El
PROOF OF THEOREM 4.7. We simply make the identifications t, = b in Lemmas 4.8 and 4.9. Yi =
Armed with the description of local time in Theorem 4.7, one can provide a very simple proof for the Dvoretzky- Erdos- Kakutani Theorem 2.9.13 concerning the absence of points of increase on the Brownian path. PROOF OF THEOREM 2.9.13. We first show that (4.35)
P° [co e
W (w) has a point of strict increase] = 0.
The event in (4.35) is equal to Ur,pe(2 Ar where (1r
and thus it suffices to prove, for given rationals 0 < r < p, that P°(A,,p) = 0. Considering, if necessary, the Brownian motion { Wr-ht - Wr, t-Fri-t; 0 S t < oo }, we may assume that r = 0.
Because of the continuity of Brownian local time, Problem 3.6.13(H), Theorem 4.7, and Proposition 3.3.22, we may choose an event n. S./ with P°(S2 *) = 1 such that for every w e (4.36) (4.37) (4.38)
the mapping (t, a)1-* L,(a, co) is continuous, for every b e Q,
f
151\ {b}(Ws(co)) dLs(b, co) = 0,
and
for every beQn (0, co), LT,,,,,)(a, co) > 0; V 0 < a < b.
6. P. Levy's Theory of Brownian Local Time
434
Suppose that w e SP n A0,,, for some p > 0, let 0 be as in the definition of A 0 and choose a (positive) rational number b e (14/e(w), Wy(w)). Let {6,,} ;,°=1 be a
nondecreasing sequence of nonnegative rational numbers with lim, 6,, = Wo(w). From (4.38) we have (4.39)
0 < LTb(c,)(We(w), w) = [LT00(14/0(0), w) - LTb(c,)(b, w)]
+ [LTb(c,)(b, w) - Le(b, co)]
+ [Le(b, w) - LTb(c4(b, w)];
V n > 1,
thanks to (4.37). From the nature of 0, the second difference on the right-hand side of (4.39) vanishes for all n > 1, and we have lim, Tb.(w) = 0. We can now let n co in (4.39), and use the joint continuity property (4.36),
to arrive at the contradiction 0 < LTb(c,)(Wo(w), w) = 0. This shows that P°(A0,) = 0, and leads ultimately to (4.35). Consider now the process 11(w) Wt(w) + t; 0 < t < co, w e O. From Corollary 3.5.2, there exists a probability measure P on (S2, P (A) =
E°[ lA exp { -WT. -
Ti];
1
w) with
VA e .FT
holding for every finite T > 0, and such that iP is standard Brownian motion under P. For every w e 0, the points of increase of W(w) become points of strict increase of 1,r(w), and (4.35) shows that P[w e 0; 14 (w) has a point of increase] = 0.
But the two measures P° and P are equivalent on .FT, and consequently P° [w e 0; W(w) has a point of increase on [0, T]] = 0 for every fixed T e (0, co). The assertion of the theorem follows easily.
n
4.10 Exercise. Verify the Cameron & Martin (1945) formula (
E° exp -13
b
j. o
W2 dt}
=
1
;
13 > 0, b > 0
N/cosh(b N/213)
for the Laplace transform of the integral Jo 14/,2 dt. (Hint: Use (4.17).)
C. The Second Formula of D. Williams The first formula of D. Williams (relation (3.22)) tells us how to compute the distribution of the total time elapsed F+-1(r), given that W+ is being observed and that r units of time have elapsed on the clock corresponding to W. The relevant information to be gleaned from observing 14/, is its local time 1.+(r). The second formula of D. Williams (relation (4.43)) assumes the same observa-
6.4. Elastic Brownian Motion
435
tions, but then asks for the distribution of the local time 4;4,0) of W at a point b < 0. Of course, if b = 0, the local time in question is just L +(r), which is known from the observations. If b < 0, then L,-, i(T)(b) is not known, but as in Williams's first formula, its distribution depends on the observation of W, only through the local time L+(r) in the manner given by (4.43).
4.11 Problem. Under P°, the process {LTb(0); 0 s b < a)} has stationary, independent increments, and 1
E° exp{ -otLTb(0)} =
(4.40)
a > 0, be R.
aim;
4.12 Lemma . Consider the right-continuous inverse local time at the origin
0 s s < oo.
p, A inf ft >_ 0; Lt(0) > 4;
(4.41)
For fixed b 0 0, the process N, A Lp.(b); P°, and Eoe-aNs
(4.42)
= exp
0 ._ s < co is a subordinator under as
a, s > 0.
1 + alb'
PROOF. Let Lt = L,(0), and recall from Problem 3.6.18 that lim, L, = a), P° a.s. The process N is obviously nondecreasing and right-continuous with N, = 0, a.s. P°. (Recall from Problem 3.6.13 (iii) that po = 0 a.s. P°.) We have
the composition property p, = p3 + p, o O; 0 < s < t, which, coupled with the additive functional property of local time, gives P ° -a.s.:
0 < s < t.
isi, - Als = Lpt_.08..(b) 0 Op.;
Note that W, = 0 a.s. According to the strong Markov property as expressed in Theorem 2.6.16, the random variable Lp,_.00..(b) o Op. is independent of Fi,. and has the same distribution as L ts (b) = N,,. This completes the proof that
N is a subordinator. As for (4.42), we have from Problem 3.4.5 (iv) for a > 0, /3 > 0: q --'A E°
ro° exp{ -As - ocLp.(b)} ds = E°
f
exp{ - 134 - aL,(b)} dL,
o
E0
T1,
f
e-pc., dLt
o
+ E° [e-PLTb .i'm exp { -13L, 0 OTb - aL,(b) o 0Tb} d(L, 0 OT,). o
The first expression is equal to [1 - E° e-PLTb]/ 11, and by the strong Markov property, the second is equal to ee-flurb times
436
6. P. Levy's Theory of Brownian Local Time
Eb J
exp {
134 - aL,(b)} dL,
03
= Eb f
exp{
/IL, - aL,(b)} dL,
= Eb[e-21-70(b)
f exp{ - fL, o °To
To
aLt(b) OTo} d(Lt o 0To)
03
= Eqe-"-To(b)]- E° f
oo
exp { -13L, - aL,(b)} dL,
= q Eb[e'LT0(b)] = q- E°[e'LTb]. Therefore, 1
q = -[1 - E°e-PLTb] + q E° [e-I3LTb] E°[e-2LTb], and (4.40) allows us to solve for i
CO
e-fisE°(e-"i's)ds =
(13
+
q
1 + 001
Inversion of this transform leads to (4.42).
4.13 Remark. Recall the two independent, reflected Brownian motions WI_ of Theorem 3.1, along with the notation of that section. If b < 0 in Lemma 4.12, then the subordinator N is a function of W_, and hence is independent of W.+. To see this, recall the local time at 0 for W_: L_(t) A Li(t)(0), and let Lb_(T)
L,w(b)
be the local time of W_ at b. Both these processes can be constructed from W_ (see Remark 3.2 for L_), and so both are independent of The same is true for
FAO = inf {T > 0: LAO > and hence also for Lb-(F-(P.0) = Lrit(rApo)(b).
But Wp. = 0 a.s., and so fi(ps + a) > 11(0 for every e > 0 (Problem 2.7.18 applied to the Brownian motion W o Op.). It follows from Problem 3.4.5 (iii) that TT' (1"_(ps)) = p and so Lb_.(f_(ps))=-- N,. A comparison of (4.42) with (2.14) shows that for the subordinator N, we have m = 0 and Levy measure u(d() =
4.14 Proposition (D. Williams (1969)). In the notation of Section 3, and for fixed numbers a > 0, t > 0, b < 0, we have a.s. P°:
437
6.5. Transition Probabilities of Brownian Motion with Two-Valued Drift
(4.43)
E° [exp { - al,,,I(t)(b)} I W+(u); 0 < u < co] = exp
aL+(-r)
1 + alblj
PROOF. This is obvious for b = 0, so we consider the case b < 0. We have from Problems 3.4.5 (iii) and 3.6.13 (iii) that with p as in (4.41), P4.7.100) = sup {s >
'(i); Ls(0) = L
= inf {s _.- IT'(x); Ws = 0},
i (0(0)}
a.s. P°.
Because W. > 0 for r_ +' (0 _< u _< p,r.v.(0) = pw,), the local time 4(6) must be constant on this interval. Therefore, with N the subordinator of Lemma 4.12, Nwo = L +(,)(b) = 1-;,(0(b), a.s. P°. Remark 4.13, together with relation (4.42), gives
E° [exp { -akso} I W+ (u); 0< u< co] = E° [exp{ - aNt} ]It=1.-,.(r) (xL +(t)
= exp { 1
}
+ albl '
a.s. P°.
4.15 Exercise.
(i) Show that for # > 0, a > 0, b < 0, we have E0 [exp { - fil.. T(b)} I W+(u); 0
u < co }]
fiL T.(0) 1 1
1 + flibi Y
a.s. P°.
(ii) Use (i) to prove that for a > 0, /3 > 0, a > 0, b < 0, Eb[expl -13LT.(b) - aLT.(0)}]
1
f2(a, a; a + Ibl, 11)'
where f2 is given by (4.22), (4.23). (This argument can be extended to provide an alternate proof of Lemma 4.9; see McKean (1975)).
6.5. An Application: Transition Probabilities of Brownian Motion with Two-Valued Drift Let us consider two real constants Coo < Os, and denote by V the collection of Borel-measurable functions b(t, x): [0, co) x 01 [On, Or]. For every b E W and
x ell, we know from Corollary 5.3.11 and Remark 5.3.7 that the stochastic integral equation (5.1)
X, = x +
1 t b(s, Xs) ds + 14(t;
0 s t < co
o
has a weak solution (X, W), (CI,
,Px), {,i} which is unique in the sense of
6. P. Levy's Theory of Brownian Local Time
438
probability law, with finite-dimensional distributions given by (5.3.11) for 0 ti < t2 < < t = t < co, I- e ROI"): (5.2)
Px[(X,,,.
.
. , xtn)
= Ex [1{(}v, ,
e F] }f,
)E ,-) exp
If
t
b(s, Ws)dW, -
1
1 b2(s, Ws) ds}1. 0
o
Here, {147 .97,}, (0,39, {P},,E R is a one-dimensional Brownian family. We shall take the point of view that the drift b(t, x) is an element of control,
available to the decision maker for influencing the path of the Brownian particle by "pushing" it. The reader may wish to bear in mind the special case 00 < 0 < 0,, in which case this "pushing" can be in either the positive or the negative direction (up to the prescribed limit rates 0, and 0,, respectively). The goal is to keep the particle as close to the origin as possible, and the decision maker's efficacy in doing so is measured by the expected discounted quadratic deviation from the origin
J(x; b) = Ex J
e- xi X, dt o
for the resulting diffusion process X. Here, a is a positive constant. The control problem is to choose the drift b. e 0/i for which J(x; b) is minimized over all b eqi:
J(x; b.) = min J(x; b), V x e R.
(5.3)
bell
This simple stochastic control problem was studied by Bend, Shepp & Witsenhausen (1980), who showed that the optimal drift is given by b.(t,x) = u(x); 0 < t < co, x e R and (5.4)
u(x) 4- 101; Bo;
x < 6} x>6 '
1
6
'-4-
1
N/ q + 2a + 0, q + 2aL - Bo
This is a sensible rule, which says that one should "push as hard as possible
to the right, whenever the process Z, solution of the stochastic integral equation (5.5)
Z, = x + f t u(Zs) ds + W,; 0 < t < oo, o
finds itself to the left of the critical point 6, and vice versa." Because there is no explicit cost on the controlling effort, it is reasonable to push with full force up to the allowed limits. If 0, = -0, = 0, the situation is symmetric and 6 = 0. Our intent in the present section is to use the trivariate density (3.30) in order to compute, as explicitly as possible, the transition probabilities (5.6)
fit(x, z) dz = Ex [Z, E dz]
6.5. Transition Probabilities of Brownian Motion with Two-Valued Drift
439
of the process in (5.5), which is a Brownian motion with two-valued, statedependent drift. In this computation, the switching point 6 need not be related to 00 and 0,. We shall only deal with the value 6 = 0; the transition probabilities for other values of 6 can then be obtained easily by translation. The starting point is provided by (5.2), which puts the expression (5.6) in the form (5.7)
z)dz = Ex[l- {w, caz} exP I
f
u(Ws) dllis -
,1
J u2(Ws) ds}1.
L
o
,1
Further progress requires the elimination of the stochastic integral in (5.7). But if we set
u(y)dy =
f(z)
0 z; z < 0 100z; z > 0
we obtain a piecewise linear function, for which the generalized Ito rule of Theorem 3.6.22 gives
f(W) = J(Wo) +
u(His)dW, + (00 - 0,)L(t)
J 0
where L(t) is the local time of W at the origin. On the other hand, with the notation (3.3) we have (Ws) ds = tfi t + (01, - EDF.,(t), 0
and (5.7) becomes (5.8)
/3,(x; z)dz = exp[f(z)
- f(x) - Od
itiO /10 exp
{NO,
- 00) + (eq -
I:" [W, e dz; L(t)e db; 1-+(t)e dr].
It develops then that we have to compute the joint density of (W L(t), 1-+(t)) under F.', for every x e R, and not only for x = 0 as in (3.30). This is accomplished with the help of the strong Markov property and Problem 3.5.8; in the notation of the latter, we recast (3.30) for b > 0, 0 < t < t as
P°[W,eda; L(t) e db;1-+(t)e dr]
J2h(t; b, 0)h(t - r; b - a, 0) da db d-r; 12h(t b, 0)h(r; b + a, 0) da db d-r;
-
and then write, for x > 0 and a < 0:
a<0 a > 0,
440
6. P. Levy's Theory of Brownian Local Time
Px[W,e da; L(t) e db; 1"÷(t)e d-r]
(5.9)
= Px[W, e da; L(t) e db;F+(t)e di; T0
= jo Px[Wie da; L(t)e db; 1-+(t)e d-r1T0 = s] 1' [T0 e ds]
= J i P° [W,_, e da; L(t - s) e db; F +(t - s)e dt - s]- h(s; x, 0) ds 0
= 2h(t; b + x, 0)h(t - t; b - a, 0) da db ch. For x > 0, a > 0 a similar computation gives (5.10)
Px[W,e da; L(t)e db; FAO e dr] = 2h(t - t; b, 0)h(t; b + a + x, 0)dadb dr, and in this case we have also the singular part
Px[Wieda; L(t) = 0, FAO = t] = Px[W,e da; T0 > t] = p_(t; x,a)da
(5.11)
[p(t; x, a) - p(t; x, - a)] da (cf. (2.8.9)). The equations (5.9)-(5.11) characterize the distribution of the triple L(t),1",(t)) under Px. Back in (5.8), they yield after some algebra: f),(x, z) =
(5.12) 2
x > 0, z < 0
c° ft e2be, h(t - r; b - z, -0,)h(t; x + b, -00)d-c db; J0
200
J.te200,+zoo)h(t_ t; b, -01)h(t; x + b + z, -00)dr db
2
0
1:0
1
2nt
[exp
(X -Z
0002} 2t
exp
(x
z2
t
"2
4 ];
x > 0, z > 0.
Now the dependence on 00, 0, has to be invoked explicitly, by writing z; 00,0,) instead of 15,(x, z). The symmetry of Brownian motion gives (5.13)
/5,(x, z; 00, 0, ) = fit( -x, -z; -0,, -00),
and so for x < 0 the transition density is obtained from (5.12) and (5.13). We conclude with a summary of these results.
5.1 Proposition. Let u: R [00, 0,] be given by (5.4) for arbitrary real 5, and let Z be the solution of the stochastic integral equation (5.5). In the notation of (5.6), + 5, z + 5) is given for every z e R, 0 < t < oo by (i) the right-hand side of (5.12) if x Z 0, and
6.5. Transition Probabilities of Brownian Motion with Two-Valued Drift
441
(ii) the right-hand side of (5.12) with (x, z, 00, 01) replaced by (- x, - z, -01, - 00), if x 0.
5.2 Remark. In the special case 01 = -00 = 0 > 0 = (5, the integral term in the second part of (5.12) becomes 2
f
t
so
e20(b-z)h(t - T; b, - 0)h(t; x + b + z, 0)dt db
0
0
ft e_20zh(t - T; b, 0)h(t; x + b + z, 0) dt d0
=2
0
0
\/2nt
=2
e'h(t; x + 2b + z, 0)db
1
(x + z + 002
[exp
2t
0021
(v
96,-26.1 c° exp
20x}
dvi,
x+.
where we have used Prob em 3.5.8 again. A similar computation simplifies also the first integral in (5.12); the result is (5.14)
"Mx, z) = 1
,/2nt
[exp
(x -z - 002}
exp
9e-20z
2t
(v - 0021 dvi 2t
x+ z
x 1
.12nt
(x -z
[exp 120x
002/
exp
2t
z>a
(V -2t9t)2}dvi;
x-z
x > 0, z < O. When 0 = 1 and x = 0, we recover the expression (3.19).
5.3 Exercise. When 01 = -00 = 1, S = 0, show that the function v(t, x) Ex(Z,2); t > 0, x ER is given by (5.15)
1
v(t, x) = +
.N/ t
{03(1
ezixi
- t - 1)exp
-
t
(I xi 2-t t)2
.1}(121(lxi
_)[i 2
(D(Ixl
t)
442
6. P. Levy's Theory of Brownian Local Time
with 41)(z) = (1/.\/2n)Sz e-u212 du, and satisfies the equation
v, = iv - (sgn x) v,
(5.16)
as well as the conditions lim v(t, x) = x2,
lim v(t, x) =
(4.0
t-oco
1
2
5.4 Exercise (Shreve (1981)). With 01 = -0o = 1, 6 = 0, show that the function v of (5.15) satisfies the Hamilton-Jacobi-Bellman equation v, =
(5.17)
-2 vx + min (uvx) 1
on (0, co) x R.
1u15.1
As a consequence, if X is a solution to (5.1) for an arbitrary, Borel-measurable [ - 1, 1] and Z solves (5.5) (under Ex in both cases), then b: [0, co) x (5.18)
0 < t < CO.
Ex Z,2 < Ex X i2;
In particular, Z is the optimally controlled process for the control problem (5.3).
6.6. Solutions to Selected Problems 2.5. If A is a-finite but not finite, then there exists a partition {1-11}711 g_ if of H with 0 < 2(H1) < co for every i > 1. On a probability space (SI, P), we set up independent sequences 10); j e IN,}711 of random variables, such that for every i 1:
(1) a), are independent, (ii) N, A a) is Poisson with EN; = 2(H1), and (iii) P[c ji)E C] = 2(C n H;)12(H;); = 1, 2, ....
E7±1 1,(0)); CE if, is a Poisson random measure for every i > 1, and v1, v2, ... are independent. We show that v v1 is a Poisson random measure with intensity A. It is clear that Ev(C) = E Er_1 v;(C n =
As before, MC)
Er=, 2(C n H;) = A(C) for all C E if, and whenever 2(C) < o o , v(C) has the proper distribution (the sum of independent Poisson random variables being Poisson). Suppose 2(C) = co. We set 2 = 2(C n H.), so v(C) is the sum of the independent, Poisson random variables Ivi(C)IT_I, where Evi(C) = A,. There is a number a > 0
such that 1 -e > 2/2; 0 < 2 < a, and so with b A 2(1 - ca): CO
CO
E P[1,;(C) > 1] = E (1 1=1
1=1
-
1
co
- E (A, Ab) =oo 2 i=1
because Er=, 2 = 2(C) = co. By the second half of the Borel-Cantelli lemma (Chung (1974), p. 76), P[v(C) = co] = P[vi(C) 1 for infinitely many i] = 1. This completes the verification that v satisfies condition (i) of Definition 2.3; the verification of (ii) is straightforward.
6.6. Solutions to Selected Problems
443
2.12. If for some ye D[0, co), t > 0, and e > 0 we have n(y; (0, t] x ((, co)) = co, then we can find a sequence of distinct points {tk }T=1 g (0, t] such that I y(th ) y(tk - )1 > (; k > 1. By selecting a subsequence if necessary, we may assume without loss of generality that {tk}ic_, is either strictly increasing or else strictly decreasing to a limit Be [0, t]. But then y(t k)} ,T=, and { y(t, -) }k 1 converge to the same limit (which is y(0 -) in the former case, and y(0) in the latter). In either case we obtain a contradiction. For any interval (t, t + h], the set Ar.,+h(e)
h] such that I y(s) - y(s -)I > e}
{ y E D [0, co); 3 s E (I, .0
.0
=un
k=1 m=1
1
D[O, oo); y(q) - y(r)I > e
U
i
q-r<(11m).reQ
qQorq=i+h
is in .4(D[0, co)), as is
{y D[O, oo); n(y; (0, t] x (/, co)) > ml =
U
o=q0
Let us now fix 0 < t < oo and / > 0 and let
be be the Dynkin system of all sets CE MOO, I] x (e, COO for which n( ; C) is measurable. The a-field MOO, x
((, co)) is generated by the sets of the form (0, -r] x (A, co); 0 < T < t, e < CO, the collection of such sets is closed under finite intersections, and each such set
belongs to gt.e. It follows from Theorem 2.1.3 that n( ; C) is measurable for every CE .4(03, ti x (C, CO)). For CE g((0, co)2), we write C as the disjoint union
/1
c = u U ic n k =1 .i =1
x
- 111}.
2.16. From (2.17) and Definition 2.15 we have EC" = e 0124' 1 , which gives (2.33) and the continuity of 13( ). Because N is not identically zero, kfr(1) is strictly positive. Furthermore, for every y > 0, e-`114")411) = Ee-",a,,, = EC"' N, = EC"A,) = Cr9(Y)4")
and (2.32) follows from (2.33). To see that #(a) = a', set G(a) = fl(ea) and apply Problem 2.2. For 0 < E < 1, comparison of (2.24) and (2.33) yields
ras = ma +
f o..) (1 - e-'1)p(de). (
Corollary 2.14 asserts that the constant m > 0 and a-finite measure p satisfying the equation are unique. If E .-- 1, they are given by m = r, p = 0. If 0 < e < 1,
we set m = 0, p(d/)= red( /f'(1 - t)/I+` and integrate by parts to reduce the integral to a standard Laplace transform: (o..)
(1 -C )p(cli) =
fo.,e) e-2e Cr' de = ea'.
2.24. We have from (2.56) 2
EIED,(e) - 24(0)12 < 2e2 + 2E
sgn( Ws) d Ws) 1=1
ItA
.
444
6. P. Levy's Theory of Brownian Local Time This last expectation is computed as 1+.)4(c)
co
E
E {(t
A (3)) -
coi-
2,
E
A 21_1 )1
i=1
E
1 =1
E 1{J,Dt(."+1
j=0
= E2(ENE) + 1) by virtue of (2.52), (2.53). But now the downcrossing inequality (Theorem 1.3.8 1
(iii)), applied to the submartingale I WI, gives ED,(E) < - EIW,I =
1
It
3.7. The following are obviously valid, modulo P°-negligible events:
{t > 0; L_(t) > L +(t)} = {t
g {t {t
= {t
(t)) > L(ITI (T))}
0;
0; Ff1(t)
(T))
0; L(F1I(t))
L(F;1(r)))
0; L_(t) > L +(t) }.
Therefore, we have a.s. P°: = inf It >_ 0; L_(t) > L +(t)} < inf It
0; 1-11(t)
(r)}
0; L_(t) > L+(t)} = L:1(L+(t)).
inf (t
The third identity in (3.25) follows now from (3.24). For the fourth, it suffices to observe F., (s) = inf {t 0; 1---,.(t)> s }; 0 < s < co, a.s. P°, which is a consequence of Lemma 3.4.5 (v). 3.11. Using the reflection principle in the form (2.6.2) and the upper bound in (2.9.20), we obtain
[ max Ws
a + e; min Ws < a - E] t<s
P° t<s
Px[ max Ws > a +
<
e[W, Edx]
0 <s
-ao OD
a - e]
Px[ min Ws
J
0<s
<
Pa[ min Ws
Pa[max Ws > a + 0<s
= 213°
e[W, E dx]
0<s
[ max Ws > E 0 <s
= 4P° [Wh > e] <
h
e-'2/2h = o(h).
For (3.32), we notice the inequality
P°[W+(t) > a; t
I (r) < t + h; Mt) < t + h]
< P° [ max Ws > a; min W < 0], t<s
t<s
where this term is o(h) as h 10, thanks to (3.31); a similar argument proves (3.33).
6.7. Notes
445
4.11. Let 4 = L,(0) and assume without loss of generality that b > 0. From the additive functional property of local time, its invariance under translation, and the composition property Tb = Ta + Tb 0 OT.; 0 < a < b, we see that P ° -a.s.: LT. - LT. = LT.08,..0 OT. = L Tb on, 0 Ot;
0 < a < b,
where 07(a))(s) 'A- oi(t + s) - (0(t); s > 0. The stationarity and independence of increments follow from the strong Markov property as expressed by Theorem 2.6.16. Formula (4.40) is just a restatement of (4.16).
6.7. Notes Section 6.2: Most of the material here is due to P. Levy (1937, 1939, 1948). The representation (2.14) is a special case of a general decomposition result for processes with stationary, independent increments (Levy processes) into Brownian and Poisson components, obtained by P. Levy (1937); see also Loeve (1978). In our exposition of Theorem 2.7, we follow Ito & McKean (1974) and Williams (1979). Both these books, as well as McKean (1975) and
Chapter VII of Knight (1981), may be consulted for further reading on Brownian local time. Chung & Durrett (1976) and Williams (1977) also deal
with the subject of Theorem 2.23. Taking up a theme of Levy, Ikeda & Watanabe (1981), pp. 123-136, show how to use local time to construct Brownian motion from a Poisson random measure on the space of excursions
away from the origin. This should be read together with Chung's (1976) excellent treatise on excursions. Ito (1961b) applied Poisson random measures to the study of Markov processes. The characterizations of Theorems 2.21 and 2.23 have been generalized to Markov processes by Fristedt &,Taylor (1983);
see also Kingman (1973). Invariance principles are discussed by Perkins (1982a) and Csaki & Revesz (1983); see also Borodin (1981). Perkins (1982b) showed that Brownian local time is a semimartingale in the spatial parameter; McGill (1982) studied its Markov properties. Section 6.3: Theorem 3.1 appears in Ito & McKean (1974), section 2.11, and in Ikeda & Watanabe (1981), pp. 122-124. Proposition 3.9 is taken from
Karatzas & Shreve (1984a); the bivariate density (3.40) was obtained by Perkins (1982b), using different methodology. The use of local time in decomposition of Brownian paths is illustrated by the work of Williams (1974) and Harrison & Shepp (1981). Section 6.4: The strong version of Theorem 4.1 discussed in Remark 4.2, but with n = 1, appears in Ito & McKean (1974), pp. 45-48. For more general results along the lines of Theorems 4.1 and 4.3, the reader should consult Knight (1981), Theorem 7.4.3. Exercise 4.5 comes from Taylor (1975), where applications to finance and process control are discussed; see also Williams (1976), Lehoczky (1977), and Azema & Yor (1979). We follow Ito & McKean (1974) in our approach to the Ray-Knight theorem 4.7 and in Lemma 4.12, and Knight (1981) for the proof of the Dvoretzky-Erthis-Kakutani Theorem 2.9.13.
446
6. P. Levy's Theory of Brownian Local Time
Section 6.5: This material is taken from Karatzas & Shreve (1984a). The control problem of this section was introduced and solved by Bend, Shepp, & Witsenhausen (1980) and has also been solved in the symmetric case of Exercises 5.3, 5.4 by martingale methods (Davis & Clark (1979)), stochastic comparison methods (Ikeda & Watanabe (1977)), and the stochastic maximum principle (Haussmann (1981)). See also Balakrishnan (1980). Stochastic control problems in which the optimal control process is a local time have been studied by a number of authors, including Bather & Chernoff (1967); Benek Shepp & Witsenhausen (1980); Chernoff (1968); Chow, Menaldi & Robin (1985); Harrison (1985); Karatzas (1983); Karatzas & Shreve (1984b, 1985); and Taksar (1985).
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Index
A
Abel transform 266 Adapted process 4 Additive functional 202-203
Adelman, 0. 109 Aldous, D. 46 American option 376, 398 Andre, D. 79 Arc-sine law for last exit of Brownian motion from zero 102 for occupation time by Brownian motion of (0,co) 273, 422 for the time Brownian motion obtains its maximum 102 Arnold, L. 236, 395 Arzela-Ascoli theorem 62 Ash, R.B. 1,85 Augmented filtration (see Filtration) Azerna, J. 238, 445 1
Barrier at a boundary point 247-248 Bather, J.A. 446 Bayes' rule 193 Bellman equation (see Hamilton-JacobiBellman equation) Bena, V.E. 200, 438, 446 Bensoussan, A. 398 BernKtein, S. 394 Besse! process 158ff
in the Ray-Knight description of local time 430 minimum of 162 scale and speed measure for 346 stochastic integral equation for 159 strong Markov property for 159 Bhattacharya, R.N. 397 Billingsley, P. 11, 126 Black & Scholes option pricing formula 378-379 Blumenthal, R.M. 127, 238 Blumenthal zero-one law 94 Borel set 1
B
Borel cr-field
Bachelier, L. 126 Backward Ito integral 148 Backward Kolmogorov equation Kolmogorov equations) Balakrishnan, A.V. 446 Barlow, M.T. 396
Borodin, A.N. 445 Brosamler, G.A. 237 Brown, R. 47 Brownian bridge 358-360 d-dimensional 361 maximum of 265
(see
I
460 Brownian family 73 universal filtration for 93 with drift 78 Brownian functional 185 stochastic integral representation of 185-186, 188-189 Brownian local time 126, 399ff as a semimartingale in the spatial parameter 445 as a Bessel process in the spatial parameter 430 defined 203 downcrossing representation of 416 excursion interval representation of 415 existence of 207 joint density with Brownian motion and occupation time of (0,00) 423 Markov properties of 445 Ray-Knight description of 430, 445 Tanaka formula for 205, 215 Brownian martingale 182 Ito integral representation of 182, 184 Brownian motion absorbed 97-100, 265 augmented filtration for 91 boundary crossing probabilities for 262ff complex-valued 253 construction of 48 as a weak limit of random walks 59ff from finite-dimensional distributions 49ff Levy-Ciesielski method 56ff d-dimensional 72 recurrence properties of 161-162 rotational invariance of 158 elastic 425ff equivalence transformations 103-104 excursion intervals of 401 as a Poisson random measure 411 used to represent Brownian local time 415 used to represent maximum-todate 417 existence of 48 filtration for 48
Index
geometric
349
last exit time of 100-103 Markov property for 75 martingale characterization of 156157, 178, 314 maximum (running maximum, maximum-to-date) of 95-96, 102103, 417, 429 negative part of 418-420 occupation time of (0,00) 273, 274, 422
joint density with Brownian motion and local time 423 one-dimensional 47 passage time of (see Brownian passage time)
positive part of 418-420 quadratic variation of 73, 106 reflected 97, 210, 418-420 scale and speed measure for 344 scaling of 104 semimartingale local time for 225 symmetry of 104 time-inversion of 104 time-reversal of 104 transition density for (see Gaussian kernel) with drift 196ff minimum of 197 passage times of 196 scale and speed measure for 346 with two-valued drift 437ff Brownian oscillator 362 Brownian passage time 79, 95, 126 density of 80, 88, 96 for drifted Brownian motion 196 Laplace transform of 96 process of passage times 96, 400ff as a one-sided stable process 411 Levy measure for 411 Brownian sample path law of the iterated logarithm for 112, 127
level sets of 105-106, 201-202, 209 Levy modulus of continuity for 114ff, 127 local Holder continuity of 56, I I l , 113-114, 126-127 local maxima of 107
Index
nondifferentiability of 110, 203 non-monotonicity of 106 no point of increase for (see also Dvoretzky-Erdos-Kakutani theorem) quadratic variation of 106 slow points of 127 unbounded first variation of 106 zero set of 104-105 Burkholder, D.L. 44, 166, 237 Burkholder-Davis-Gundy inequalities 166
C
Cad lag process (see RCLL process) Cameron, R.N. 191, 237 Cameron-Martin formula 434 Cameron-Martin-Girsanov formula (see
Girsanov's theorem) Canonical probability space 71 Capital asset pricing (see Optimal consumption and investment problem) Cauchy problem (see Partial differential equations) Causality 286, 311 Centsov, N.N. 53, 126 Chaleyat-Maurel, M. 237 Change of measure 129 (see also Girsanov's theorem) Change-of-variable formula 128 for functions of continuous semimartingales 149, 153 for convex functions of Brownian motion 214 for convex functions of continuous semimartingales 218-219 Chernoff, H. 446 Chitashvili, R.J. 395 Chow, P.L. 446 Chow, Y.S. 46 Chung, K.L. 4, 6, 11, 45, 46, 125, 127, 236, 445 Ciesielski, Z. 48, 126 Cirel'son, B.S. 304 Cinlar, E. 237 Clark, J.M.C. 446
Class D 24 Class DL 24
461
Cocozza, C. 237 Communication 395 Comparison of solutions to stochastic differential equations (see Stochastic differential equations) Consistent family of finite-dimensional distributions 50, 126 Consumption process 373 Contingent claim 376 fair price for 377 valuation process of 378 Continuous local martingale 36 cross-variation of 36 integration with respect to I45ff moment inequalities for 166 quadratic variation of 36 represented as an ItO integral 170ff represented as time-changed Brownian motion I73ff Continuous semimartingale 149 change-of-variable formula 149, 153 convex function of 218 local time for 218 Tanaka-Meyer formulas for 220 Controllability 355-356 Controllability matrix 356 Convergence in distribution 61 Convex function 212-213 Ito formula for 214 of a continuous semimartingale 218 second derivative measure of 213 Coordinate mapping process 52, 71 Copson, E.T. 277 Countably determined o-field 306 Countably generated cr-field 307 Courrege, P. 236 Covariance matrix 103,355 Cox, J. 398 Cramer-Wold device 61 Cross-variation process as a bilinear form 31-32 for continuous local martingales 36 for continuous square-integrable martingales 35 for d-dimensional Brownian motion 73, 157 for square-integrable martingales 31 Csaki, E. 445 Cylinder set 49
462
Index
D
Dambis, K.E. 174 Daniell, P.J. 50. 126 Davis, B.
127, 166, 253
Davis, M.H.A. 446 Dellacherie, C. 10, 45, 46, 236 Determining class of functions 325 Diffeomorphism theorem 397-398 Diffusion matrix 281, 284 Diffusion process 281-282 boundary behavior 396 degenerate 327, 396 killed 369 on a manifold 396 support of 396 Dini derivates 109 Dirichlet problem 240, 243ff, 364-366 Dispersion matrix 284 Doleans-Dade, C. 236, 396 Donsker, M.D. 126, 280 Donsker's theorem 70
Doob, J.L. 4, 45, 46, 170, 236, 237, 239, 278, 397 Doob-Meyer decomposition of a submartingale 24ff Doob's maximal inequality for submartingales 14 Doss, H. 295, 395 Downcrossings 13-14 representation of Brownian local time 416 Drift vector 281, 284 Dubins, L.E. 174 Dudley, R.M. 188 Duffle, D. 398 Durrett, R. 239, 445 Dvoretzky, A. 108, 127 Dvoretzky-Erdos-Kakutani theorem 108, 433, 445 Dynamical systems 395 Dynkin, E.B. 98, 127, 242. 269, 397 Dynkin system 49
E Economics equilibrium 398 Einstein, A. 126 Elastic Brownian motion 426 El Karoui, N. 237
Elliott, R.J. 10, 46, 236, 237. 396 Elliptic operator 364 Elliptic partial differential equations (see Partial differential equations) Emery. M. 189 Engelbert, H.J. 329, 332, 335. 396 Engelbert-Schmidt zero-one law 216, 396 Equilibrium (see Economics equilibrium) Erdos. P. 108, 433, 445 Ergodic property (see Recurrence: Positive recurrence) Ethier. S.N. 127, 395, 396 European option 376 Excursion intervals of Brownian motion (see Brownian motion) Exit distribution of Brownian motion from a sphere 252 Exit time of Brownian motion from a sphere 253 Expectation vector (see Mean vector) Exponential supermartingale 147, 153. I98ff not a martingale 201 Extension of a probability space 169 Extension of measure (see Kolmogorov extension theorem) Exterior sphere property 365
F Feller. W. 282, 397 Feller property 127 Feller's test for explosions
348-350,
396
Feynman, R.P. 279 Feynman-Kac formula (see also Kac formula) 366 for Brownian motion 267ff for elastic Brownian motion 426-428 Filtering 395 Filtration 3 augmentation of 89, 285 completion of 89 enlargement of 127 generated by a process 3 left-continuous 4, 90
463
Index
right-continuous
"universal"
4, 90
93
usual conditions for 10 Finite-dimensional distributions
2, 50,
64
Fisk, D.L. 46, 236 Fisk-Stratonovich integral 148, 156, 295, 299 Fleming, W.H. 381, 395 Fokker-Planck equation (see Kolmogorov equations) Forward Kolmogorov equation (see Kolmogorov equations) Fourier, J.B. 279 Freedman, D. 125, 127 Freidlin, M. 395,397 Friedman, A. 236, 269, 327, 365, 366, 368, 395, 397 Fristedt, B. 445 Fubini theorem for stochastic integrals 209, 225 Fundamental solution of a partial differential equation 255, 368 as transition probability density 369
Hall, P. 46 Hamilton-Jacobi-Bellman (HJB) equation 384-385, 442 Harmonic function 240 Harrison, J.M. 398, 445, 446 Hartman, P. 279 Haussmann, U.G. 446 Heat equation 254ff, 278-279 backward 257, 268 fundamental solution for 255 nonnegative solutions of 257 Tychonoff uniqueness theorem 255 Widder's uniqueness theorem 261 Heat kernel (see Gaussian kernel) Hedging strategy against a contingent claim 376 Hermite polynomials 167 Heyde, C.C. 46 Hincin, A. Ya. 112, 405 Hitting time 7, 46 Hoover, D.N. 46 Horowitz, J. 237 Huang, C. 398 Hunt, G.A. 127, 278
G
Game theory 395 Gaussian kernel 52 Gaussian process 103, 355, 357 Gauss-Markov process 355ff Geman, D. 237 Generalized Ito rule for convex functions 214 Getoor, R.K. 127, 238 Gihman, 1.1. 236, 394, 395, 397 Gilbarg, D. 366 Girsanov, I.V. 237, 292 Girsanov's theorem 190ff, 302 (see also Change of measure) generalized 352 Green's function 343 Gronwall inequality 287-288 Gundy, R. 166 H
Haar functions 57 Hajek, B. 395 Hale, J. 354
46, 236, 237, 238, 299, 306, 395, 396, 398, 445, 446 Imhof, J.P. 127 Ikeda, N.
Increasing process 23 integrable 23 natural 23 Increasing random sequence 21 integrable 21 natural 22 Independent increments 48 Indistinguishable processes 2 Inequalities Burkholder-Davis-Gundy 166 Doob's maximal 14 downcrossings 14 Gronwall 287-288 Jensen 13 Kunita-Watanabe 142 Lenglart 30 martingale moment 163ff submartingale 13-14 uperossings 14
464
Index
Infimum of the empty set 7 Infinitesimal generator 281 Integration by parts 155 Intensity measure (see Poisson random measure) Invariance principle 70. 311. 445 Invariant measure 353, 362 Irregular boundary point 245 Ito. K. 80. 127. 130, 149. 236. 278, 279, 283, 287, 293. 394, 395. 397. 405, 416, 445 Ito integral (see also Stochastic integral) 148 backward 148 multiple 167 Ito's rule 128
for C functions of continuous semimartingales 149. 153 for convex functions of Brownian motion
214
for convex functions of continuous semimartingales 218-219
J Jacod. J. 237 Jensen inequality 13 Jeulin, T. 127, 238
K Kac. M. 279 Kac formula (see also Feynman-Kac formula) 267, 271-272 Kahane. J.P. 127 Kakutani, S. 108, 278 Kallianpur, G. 395, 397 Karatzas, I. 279, 398, 425. 445. 446 Karlin. S. 395 Kazamaki, N. 237 Kellogg. O.D. 278 Kelvin transformation 252 Khas'minskii, R.Z. 395. 397 Kingman, J.F.C. 404. 445 Kleptsyna, M.L. 396 Kliemann, W. 395 Knight. F.B. 126, 127. 179. 430. 445 Kolmogorov, A.N. 126. 236. 239. 282
Kolmogorovtentsov continuity theorem for a random field 55 for a stochastic process 53 Kolmogorov equations backward 282, 369, 397 forward 282. 369. 397 Kolmogorov extension theorem 50 Kopp, P.E. 236 Kramers-Smoluchowski approximation 299 Krylov, N.V. 327. 395 Kunita, H. 130. 149. 156, 236, 397, 398
Kunita-Watanabe inequality 142 Kurtz, T.G. 127, 395, 396 Kussmaul, A.U. 236 Kusuoka. S. 398
L
Lamperti. J. 294 Langevin, P. 358. 397 Laplace, P.S. 279 Large deviations 280 Last exit time 100-103 Law of a continuous process 60 Law of an RCLL process 409 Law of large numbers for Brownian motion 104 for Poisson processes 15 Law of the iterated logarithm 112, 127 LCRL process 4 Lebesgue. H. 278 Lebesgue's thorn 249 Le Gall. J.F. 225. 353. 396 Lehoczky. J.P. 398. 445 Lenglart. E. 44 Lenglart inequality 30 Levy P. 46, 48. 97. 126. 127, 202. 211. 236. 237. 253. 400, 405, 415. 416. 445 Levy martingale characterization of Brownian motion 156-157, 178 Levy measure 405, 410 Levy modulus of continuity 114 Levy process 445 Levy zero-one law 46 Lifetime
369
465
Index
Liptser, R.S. 23. 46. 200. 236. 395 Localization 36 Local martingale 36 not a martingale 168, 200-201 Local martingale problem (see also Martingale problem) 314 existence of a solution of 317-318 uniqueness of the solution of 317 Local maximum (see Point of local maximum) Local time for Brownian motion (see Brownian local time) for continuous semimartingales 218. 396
for reflected Brownian motion in stochastic control 446 Loeve, M. 126, 445
219
M Malliavin's stochastic calculus of variations 398 Marchal, B. 237
Markov, A.A. Markov family
127
74
Markov process 74 Markov property 71. 127 equivalent formulations of 75-78 for Brownian motion 74-75 for Poisson processes 79 Markov time (see Stopping time) Martin. W.T. 191, 237 Martingale (see also Continuous local martingale: Local martingale; Square-integrable martingale; Submartingale) II, 46 convergence of 17-19 convex function of 13 exponential (see Supermartingale, exponential) last element of 18. 19 moment inequalities for I63ff. 166 transform 21. 132 uniform integrability of 18. 19 Martingale problem (see also Local martingale problem) 311, 318, 396
existence of a solution of 317-318. 323. 327 strong Markov property for the solution
of
322
time-homogeneous
320
uniqueness of the solution of 327 well-posedness 320, 327 Maruyama. G. 237. 397 Mathematical biology 395 Mathematical economics 395 Maximum principle for partial differential equations 242, 268 Maximum principle for stochastic control 446
McGill, P. 445 McKean. H. 80, 125, 127, 236, 237, 278, 279. 396. 397. 416. 437. 445
Mean-value property 241 Mean vector 103. 355 Measurable process 3 Meleard, S. 397 Menaldi. J.L. 446 Merton. R.C. 381, 398 Mesure du voisinage (see Brownian local time)
Metivier, M. Meyer, P.A.
236
5, 12. 45, 46, 212, 218,
236, 237
Millar, P.W.
236
Minimum of the empty set 13 Modification of a stochastic process Modulus of continuity 62. 114
Mollifier 206 Moment inequalities for martingales
Multiple It6 integral
I63ff 167
N
Nakao, S. 396 natural scale 345 Nelson. E. 47. 282. 299. 395. 397 Neurophysiology 395 Neveu. J. 46
Novikov, A.A.
279
Novikov condition Nualart, D. 398
I98ff
2
466
0 Occupation time 203, 273, 274 Ogura, Y. 395 One-sided stable process 410 Optimal consumption and investment problem 379ff, 398 Optimal control (see Stochastic optimal control) Option pricing 376ff, 398 Optional sampling 19, 20 Optional time 6 Orey, S. 237 Ornstein, L.S. 358 Ornstein-Uhlenbeck process 358, 397
P Paley, R.E.A.C.
110, 236 Papanicolaou, G. 238 Parabolic partial differential equation (see Partial differential equation) Pardoux, E. 395 Parthasarathy, K.R. 85, 306, 322, 409 Partial differential equations Cauchy problem 366ff, 397 fundamental solution for 368-369 degenerate 397
Dirichlet problem 240, 243ff, 364- 366 elliptic 239, 363 parabolic 239, 267, 363 quasi-linear 397 Passage time for Brownian motion (see Brownian passage time) Pathwise uniqueness (see Stochastic differential equation) Pellaumail, J. 236 Perkins, E. 396, 445 Picard-LindelOf iterations 287 Piecewise C' 271 Piecewise continuous 271 Piecewise-linear interpolation of the Brownian path 57, 299 Pitman, J. 46, 237 Pliska, S.R. 398 Poincare, H. 278 Point of increase 107 for a Brownian path 108 Point of local maximum 107 for a Brownian path 107
Index
Poisson equation 253 Poisson family 79 Poisson integral formulas for a half-space 251 for a sphere 252 Poisson process 12 compensated 12, 156 compound 405 filtration for 91 intensity of 12 strong law of large numbers for 15 weak law of large numbers for 15 Poisson random measure 404 intensity measure for 404 on the space of Brownian excursions 445 used to represent a subordinator 410 Pollack, M. 352 Population genetics 395 Port, S. 253, 278 Portfolio process 373 Positive recurrence 353, 371 Potential 18 Predictable process 131 random sequence 21-22 stopping time 46 Principal eigenvalue 279 Priouret, P. 396 Process (see Stochastic process) Progressive measurability 4 of a process stopped at a stopping time 9 Progressively measurable functional 199-200 Prohorov, Yu.V. 126 Prohorov's theorem 62 Projection mapping 65 Protter, P. 237, 396 p-th variation of a process 32 Pyke, R. 126
Q
Quadratic variation for a continuous local martingale 36 for a process 32 for a square-integrable martingale 31
467
Index
R
Shiryaev, A.N.
Random field 55, 126 Random process (see Stochastic process) Random shift 83 Random time 5 Random walk 70, 126 Rao, K.M. 46 Ray, D. 127, 430 RCLL process 4 Realization 1 Recurrence 370, 397 of d-dimensional Brownian motion 161-162 of a solution to a stochastic differential equation 345, 353 Reflected Brownian motion (see Brownian motion, reflected) Reflection principle 79-80 Regular boundary point 245, 248-250 Regular conditional probability 84-85, 306-309 Regular submartingale 28 Resolvent operator 272 Revesz, P. 445 Revuz, D. 44 Riesz decomposition of a supermartingale 18 Rishel, R.W. 381, 395 Robbins, H. 43, 197, 261, 279 Robin, M. 446 Rosenblatt, M. 279
Shiryaev-Roberts equation 168 Shreve, S.E. 398, 425, 442, 445, 446 Siegmund, D. 43, 197, 261, 279, 352
a-field Borel 1 generated by a process at a random time 5 of events immediately after an optional time 10 of events immediately after t,.?-121 4 of events prior to a stopping time 8
of events strictly prior to t>0 4 universal
Sample path (see also Brownian sample path) I Sample space
1
Samuelson, P.A. 398 Scale function 339, 343 Schauder functions 57 Schmidt, W. 329, 332, 335, 396 Schwarz, G. 174 Sekiguchi, T. 237 Semigroup 395 Semimartingale (see Continuous semimartingale) Sharpe, M.J. 237 Shepp, L.A. 438, 445, 446 Shift operators 77, 83
73
Simon, B.
280 Simple process 132
Skorohod, A.V.
236, 323, 395, 397 Skorohod equation 210 Skorohod metric 409 Skorohod space of RCLL functions 409 Smoluchowski equation 361, 397 Sojourn time (see Occupation time) Speed measure 343, 352 as an invariant measure 353, 362 Square-integrable martingale cross-variation of 31, 35 ItO integral representation of Brownian 182 metric on the space of 37
orthogonality of 31 quadratic variation of Stability theory 395 State space
S
23, 46, 200, 236, 395
31
1
Stationary increments 48 Stationary process 103 Statistical communication theory 395 Statistical tests of power one 279 Stochastic calculus 128, 148, 150 Stochastic calculus of variations 398 Stochastic control (see Stochastic optimal control) Stochastic differential 145, 150, 154 Stochastic differential equation 281ff, 394
approximation of 295ff, 395 comparison of solutions of 293, 395, 446 explosion time of 329 functional 305
468
Index
Stochastic differential equation (cont.) Gaussian process as a solution of 355, 357 linear 354ff one-dimensional 329ff Feller's test for explosions 348350, 396 invariant distribution for 353 nonexplosion when drift is zero 332 pathwise uniqueness 337, 338, 341, 353, 396 strong existence 338, 341, 396 strong uniqueness 338, 341, 396 uniqueness in law 335, 341 weak existence 334, 341 weak solution in an interval 343 pathwise uniqueness 301, 302, 309310 strong existence 289, 310-311, 396 strong solution 283, 285 strong uniqueness 286, 287, 291, 396 uniqueness in law 301, 302, 304305, 309, 317 weak existence 303, 310, 323, 332 weak solution 129, 300 weak solution related to the martingale problem 317-318 well-posedness 319-320 with respect to a semimartingale 396 Stochastic integral 129ff characterization of 141-142 definition of 139 with respect to a martingale having absolutely continuous quadratic variation 141 Stochastic integral equation (see Stochastic differential equation) of Volterra type 396 Stochastic maximum principle 446 Stochastic optimal control 284, 379, 395, 438, 446 Stochastic partial differential equations 395 Stochastic process adapted to a filtration 4 finite-dimensional distributions of 2 Gaussian 103 1
355ff
Gauss-Markov
LCRL 4 measurable 3 modification of 2
of class D 24 of class DL 24 progressively measurable
4
RCLL 4 sample path of simple
I
132
state space of stationary 103 stopped at a stopping time 9 zero-mean 103 Stochastic systems 395 Stone, C. 253, 278 Stopping time 6 accessible 46 predictable 46 totally inaccessible 46 Stratonovich integral (see Fisk-Stratonovich integral) Stricker, C. 168, 189 Strong existence (see Stochastic differential equation) Strong Markov family 81-82 universal filtration for 93 Strong Markov process 81-82 augmented filtration for 90-92 classification of boundary behavior of 397 Strong Markov property 79, 127 equivalent formulations of 81-84 extended 87 for Brownian motion 86, 127 for Poisson processes 89 for solutions of the martingale problem 322 Strong solution (see Stochastic differential equation) Strong uniqueness (see Stochastic differential equation) Stroock, D.W. 127, 236, 238, 283, 311, 322, 323, 327, 395, 396, 397, 398, 416 Submartingale (see also Martingale) 11 backward 15 convergence of 17-18 1
469
Index
Doob-Meyer decomposition of 24ff inequalities for 13-14 last element of 12-13, 18 maximal inequality for 14 optional sampling of 19-20 path continuity of 14, 16 regular 28 uniform integrability of 18 Subordinator 405 Supermartingale (see also Martingale) 11 exponential 147, 153, 198ff not a martingale 201 last element of 13 Sussmann, H. 395 Synonimity of processes 46
T Taksar, M. 446 Tanaka, H. 237, 301, 397 Tanaka formulas for Brownian motion 205, 215 Tanaka-Meyer formulas for continuous semimartingales 220 Taylor, H. 395, 445 Taylor, S.J. 445 Teicher, H. 46 Tied-down Brownian motion (seeBrownian bridge) Tightness of a family of probability measures 62 Time-change for martingales 174 for one-dimensional stochastic differential equations 330ff for stochastic integrals 176-178 Time-homogeneous martingale problem (see Martingale problem) Toronjadze, T.A. 395 Trajectory Transformation of drift (see Girsanov's theorem) Transition density for absorbed Brownian motion 98 for Brownian motion 52 for diffusion process 369 for reflected Brownian motion 97 1
Trotter, H.F. 207, 237 Trudinger, N.S. 366 Tychonoff uniqueness theorem
255
U
Uhlenbeck, G.E. 358 Uniformly elliptic operator 364 Uniformly positive definite matrix 327 Uniqueness in law (see Stochastic differential equation) Universal filtration 93 Universal cr-field 73 Universally measurable function 73 Uperossings 13-14 Usual conditions (for a filtration) 10 379 Utility function
V
Value function 379 Van Schuppen, J. 237 Varadhan, S.R.S. 127, 236, 238, 279, 283, 311, 322, 323, 327, 396, 397
Variation of a process 32 Ventsel, A.D. 237 Veretennikov, A. Yu. 396 Ville, J. 46
Wald identities 141, 168, 197 Walsh, J.B. 395 Wang, A.T. 212, 237 Wang, M.C. 358 Watanabe, S. 46, 130, 149, 156, 232, 236, 237, 238, 291, 292, 293, 299, 306, 308, 309, 395, 396, 397, 398, 445, 446 Weak convergence of probability measures 60-61 Weak solution (see Stochastic differential equation) Weak uniqueness (see Stochastic differential equation, uniqueness in law) Wentzell, A.D. 39, 82, 117, 127 White noise 395
470 Widder, D.V. 256. 277, 279 Widder's uniqueness theorem 261 Wiener. N. 48. 110. 126, 236. 278 Wiener functional (see Brownian functional) Wiener martingale (see Brownian martingale) Wiener measure d-dimensional 72 one-dimensional 71 Wiener process (see Brownian motion) Williams. D. 127. 445
first formula of 421-422. 423 second formula of 436-437 Williams, R. 236 Wintner, A. 279 Witsenhausen, H.S. 438. 446 Wong. E. 169, 237. 395
Index
Y
Yamada. T. Yan. J . A .
291. 308, 309. 395. 396 189
Yor, M. 44, 46. 126. 127. 168. 237, 238. 445 Yushkevich, A.A. 98. 127, 242
Zakai, M. 169. 395. 398 Zaremba, S. 278 Zaremba's cone condition 250 Zero-one laws Blumenthal 94 Engelbert-Schmidt 216, 396 Levy 46 Zvonkin, A.K. 396 Zygmund, A. 110. 236
