Stochastic Integration and Differential Equations

Stochastic Mechanics Random Media signal Processing and Image Synthesis Mathematical Economics and Finance Stochastic Optimization

Applications of Mathematics Stochastic Modelling and Applied Probability

21

Stochastic Control Stochastic Models in Life Sciences Edited b y

Advisory Board

B. Rozovskii M. Yor D. Dawson D. Geman G. Grimmett I. Karatzas F. Kelly Y. Le Jan B. Bksendal E. Pardoux G. Papanicolaou

Springer Berlin Heidelberg New York Hong Kong London Milan Paris Tokyo

Applications of Mathematics FlemingIRishel, Deterministic and Stochastic Optimal Control (1975) Marchuk, Methods of Numerical Mathematics ig75,2nd. ed. 1982) 3 Balakrishnan, Applied Functional Analysis (1976,znd. ed. 1981) 4 Borovkov, Stochastic Processes in Queueing Theory (1976) 5 LiptserlShiryaev, Statistics of Random Processes 1:General Theory (1977.2nd. ed. 2001) 6 LiptserlShiryaev, Statistics of Random Processes 11: Applications (1978,znd. ed. 2001) 7 Vorob'ev, Game Theory: Lectures for Economists and Systems Scientists (1977) 8 Shiryaev, Optimal Stopping Rules (1978) g IbragimovlRozanov, Gaussian Random Processes (1978) lo Wonham, Linear Multivariable Control: A Geometric Approach (1979,znd. ed. 1985) 11 Hida, Brownian Motion (1980) 12 Hestenes, Conjugate Direction Methods in Optimization (1980) 13 Kallianpur, Stochastic Filtering Theory (1980) 14 Krylov, Controlled Diffusion Processes (1980) 15 Prabhu, Stochastic Storage Processes: Queues, Insurance Risk, and Dams (1980) 16 IbragimovlHas'minskii, Statistical Estimation: Asymptotic Theory (1981) 17 Cesari, Optimization: Theory and Applications (1982) 18 Elliott, Stochastic Calculus and Applications (1982) lg MarchuWShaidourov,DifferenceMethods and Their Extrapolations (1983) 20 Hijab, Stabilization of Control Systems (1986) 21 Protter, Stochastic Integration and Differential Equations (1990,znd. ed. 2003) 22 Benveni~telMCtivierIPriouret, Adaptive Algorithms and StochasticApproximations (1990) 23 KloedenlPlaten, Numerical Solution of Stochastic Differential Equations (1992, corr. 3rd printing 1999) 24 KushnerlDupuis, Numerical Methods for Stochastic Control Problems in Continuous Time (1992) 25 FlemingISoner, Controlled Markov Processes and Viscosity Solutions (1993) 26 BaccellilBrCmaud, Elements of Queueing Theory (1994,znd ed. 2003) 27 Winkler, Image Analysis, Random Fields and Dynamic Monte Carlo Methods (igg5,2nd. ed. 2003) 28 Kalpazidou, Cycle Representations of Markov Processes (1995) 29 ElliotffAggounlMoore, Hidden MarkovModels: Estimation and Control (1995) 30 Hernandez-LermalLasserre,Discrete-Time Markov Control Processes (1995) 31 DevroyelGyorfdLugosi, A Probabilistic Theory of Pattern Recognition (1996) 32 MaitralSudderth, Discrete Gambling and Stochastic Games (1996) 33 EmbrechtslKliippelberglMikosch,Modelling Extremal Events for Insurance and Finance (1997, corr. 4th printing 2003) 34 Duflo, Random Iterative Models (1997) 35 KushnerlYin, Stochastic Approximation Algorithms and Applications (1997) 36 Musiela/Rutkowski,Martingale Methods in Financial Modelling (1997) 37 Yin, continuous-~ime ~ a r k o chains v and Applications (1998) 38 DembolZeitouni, Large Deviations Techniques and Applications (1998) 39 Karatzas, Methods of Mathematical Finance (1998) 40 Fayolle/Iasnogorodski/Malyshev,Random Walks in the Quarter-Plane (1999) 41 AvenlJensen, Stochastic Models in Reliability (1999) 42 Hernandez-LermalLasserre,Further Topics on Discrete-Tie Markov Control Processes (1999) 43 YonglZhou, Stochastic Controls. Hamiltonian Systems and HJB Equations (1999) 44 Serfozo, Introduction to Stochastic Networks (1999) 45 Steele, Stochastic Calculus and Financial Applications (2001) 46 ChenlYao, Fundamentals of Queuing Networks: Performance, Asymptotics, and Optimization (2001) 47 Kushner, Heavy Traffic Analysis of Controlled Queueing and Communications Networks (2001) 48 Fernholz, Stochastic Portfolio Theory (2002) 49 KabanovlPergamenshchikov,Two-Scale Stochastic Systems (2003) 50 Han, Information-Spectrum Methods in Information Theory (2003) (continued after index) I

2

Philip E. Protter

Stochastic Integration and Differential Equations Second Edition

Springer

Author Philip E. Protter Cornell University School of Operations Res. and Industrial Engineering Rhodes Hall 14853 Ithaca, NY USA e-mail: [email protected] Managing Editors B. Rozovskii Center for Applied Mathematical Sciences University of Southern California 1042 West 36th Place, Denney Research Building 308 Los Angeles, CA 90089, USA

M. Yor Universite de Paris VI Laboratoire de ProbabilitCs et Modeles Aldatoires 175, rue du Chevaleret 75013 Paris, France

Mathematics Subject Classification (2000): PRIMARY: 60H05,60H10,60H20 SECONDARY: 60G07,60G17,60G44,60G51 Cover pattern by courtesy of Rick Durrett (Cornell University, Ithaca) Cataloging-in-PublicationData applied for A catalog record for this book is available from the Library of Congress.

Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de

ISSN 0172-4568 ISBN 3-540-00313-4 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science + Business Media GmbH O Springer-Verlag Berlin Heidelberg 2004

Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: Erich Kirchner. Heidelberg Typescmng by the author using a Springer TEX macro package Printed on acid-free paper 4113142~8-543210

To Diane and Rachel

Preface to the Second Edition

It has been thirteen years since the first edition was published, with its subtitle "a new approach." While the book has had some success, there are still almost no other books that use the same approach. (See however the recent book by K. Bichteler [15].) There are nevertheless of course other extant books, many of them quite good, although the majority still are devoted primarily t o the case of continuous sample paths, and others treat stochastic integration as one of many topics. Examples of alternative texts which have appeared since the first edition of this book are: [32], [44], [87], [110], [186], [180], [208], [216], and [226]. While the subject has not changed much, there have been new developments, and subjects we thought unimportant in 1990 and did not include, we now think important enough either to include or t o expand in this book. The most obvious changes in this edition are that we have added exercises a t the end of each chapter, and we have also added Chap. VI which introduces the expansion of filtrations. However we have also completely rewritten Chap. 111. In the first edition we followed an elementary approach which was P. A. Meyer's original approach before the methods of DolBans-Dade. In order to remain friends with Freddy Delbaen, and also because we now agree with him, we have instead used the modern approach of predictability rather than naturality. However we benefited from the new proof of the Doob-Meyer Theorem due to R. Bass, which ultimately uses only Doob's quadratic martingale inequality, and in passing reveals the role played by totally inaccessible stopping times. The treatment of Girsanov's theorem now includes the case where the two probability measures are not necessarily equivalent, and we include the Kazamaki-Novikov theorems. We have also added a section on compensators, with examples. In Chap. IV we have expanded our treatment of martingale representation to include the Jacod-Yor Theorem, and this has allowed us to use the Emery-AzBma martingales as a class of examples of martingales with the martingale representation property. Also, largely because of the Delbaen-Schachermayer theory of the fundamental theorems of mathematical finance, we have included the topic of sigma martingales. In Chap. V

VIII

Preface to the Second Edition

we added a section which includes some useful results about the solutions of stochastic differential equations, inspired by the review of the first edition by E. Pardoux [191]. We have also made small changes throughout the book; for instance we have included specific examples of L6vy processes and their corresponding L6vy measures, in Sect. 4 of Chap. I. The exercises are gathered at the end of the chapters, in no particular order. Some of the (presumed) harder problems we have designated with a star (*), and occasionally we have used two stars (**). While of course many of the problems are of our own creation, a significant number are theorems or lemmas taken from research papers, or taken from other books. We do not attempt to ascribe credit, other than listing the sources in the bibliography, primarily because they have been gathered over the past decade and often we don't remember from where they came. We have tried systematically to refrain from relegating a needed lemma as an exercise; thus in that sense the exercises are independent from the text, and (we hope) serve primarily to illustrate the concepts and possible applications of the theorems. Last, we have the pleasant task of thanking the numerous people who helped with this book, either by suggesting improvements, finding typos and mistakes, alerting me to references, or by reading chapters and making comments. We wish to thank patient students both at Purdue University and Cornell University who have been subjected to preliminary versions over the years, and the following individuals: C. Benei, R. Cont, F. Diener, M. Diener, R. Durrett, T. Fujiwara, K. Giesecke, L. Goldberg, R. Haboush, J. Jacod, H. Kraft, K. Lee, J . Ma, J. Mitro, J. Rodriguez, K. Schiirger, D. Sezer, J. A. Trujillo Ferreras, R. Williams, M. Yor, and Yong Zeng. Th. Jeulin, K. Shimbo, and Yan Zeng gave extraordinary help, and my editor C. Byrne gives advice and has patience that is impressive. Over the last decade I have learned much from many discussions with Darrell Duffie, Jean Jacod, Tom Kurtz, and Denis Talay, and this no doubt is reflected in this new edition. Finally, I wish to give a special thanks to M. Kozdron who hastened the appearance of this book through his superb help with B W ,as well as his own advice on all aspects of the book. Ithaca, NY August 2003

Philip Protter

Preface to the First Edition

The idea of this book began with an invitation to give a course at the Third Chilean Winter School in Probability and Statistics, at Santiago de Chile, in July, 1984. Faced with the problem of teaching stochastic integration in only a few weeks, I realized that the work of C. Dellacherie [42] provided an outline for just such a pedagogic approach. I developed this into a series of lectures (Protter [201]), using the work of K. Bichteler [14], E. Lenglart [145] and P. Protter [202], as well as that of Dellacherie. I then taught from these lecture notes, expanding and improving them, in courses at Purdue University, the University of Wisconsin at Madison, and the University of Rouen in France. I take this opportunity to thank these institutions and Professor Rolando Rebolledo for my initial invitation to Chile. This book assumes the reader has some knowledge of the theory of stochastic processes, including elementary martingale theory. While we have recalled the few necessary martingale theorems in Chap. I, we have not provided proofs, as there are already many excellent treatments of martingale theory readily available (e.g., Breiman [23], Dellacherie-Meyer [45, 461, or EthierKurtz [71]).There are several other texts on stochastic integration, all of which adopt to some extent the usual approach and thus require the general theory. The books of Elliott [63], Kopp [130], MQtivier [158], Rogers-Williams [210] and to a much lesser extent Letta [148] are examples. The books of McKean [153], Chung-Williams [32], and Karatzas-Shreve [121] avoid the general theory by limiting their scope to Brownian motion (McKean) and to continuous semimartingales. Our hope is that this book will allow a rapid introduction to some of the deepest theorems of the subject, without first having to be burdened with the beautiful but highly technical "general theory of processes." Many people have aided in the writing of this book, either through discussions or by reading one of the versions of the manuscript. I would like to thank J. Azema, M. Barlow, A. Bose, M. Brown, C. Constantini, C. Dellacherie, D. Duffie, M. Emery, N. Falkner, E. Goggin, D. Gottlieb, A. Gut, S. He, J. Jacod, T. Kurtz, J. de Sam Lazaro, R. Leandre, E. Lenglart, G. Letta,

X

Preface to the First Edition

S. Levantal, P. A. Meyer, E. Pardoux, H. Rubin, T. Sellke, R. Stockbridge, C. Stricker, P. Sundar, and M. Yor. I would especially like to thank J. San Martin for his careful reading of the manuscript in several of its versions. Svante Janson read the entire manuscript in several versions, giving me support, encouragement, and wonderful suggestions, all of which improved the book. He also found, and helped to correct, several errors. I am extremely grateful to him, especially for his enthusiasm and generosity. The National Science Foundation provided partial support throughout the writing of this book. I wish to thank Judy Snider for her cheerful and excellent typing of several versions of this book.

Philip Protter

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Basic Definitions and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Poisson Process and Brownian Motion . . . . . . . . . . . . . . . . . 4 LBvy Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Why the Usual Hypotheses? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Local Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stieltjes Integration and Change of Variables . . . . . . . . . . . . . . . . 7 8 Na'ive Stochastic Integration Is Impossible . . . . . . . . . . . . . . . . . . Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises for Chapter I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 12 19 34 37 39 43 44 45

Semimartingales and Stochastic Integrals . . . . . . . . . . . . . . . . . . 1 Introduction to Semimartingales . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Stability Properties of Semimartingales . . . . . . . . . . . . . . . . . . . . . 3 Elementary Examples of Semimartingales . . . . . . . . . . . . . . . . . . . 4 Stochastic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Properties of Stochastic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 6 The Quadratic Variation of a Semimartingale . . . . . . . . . . . . . . . 7 It6's Formula (Change of Variables) . . . . . . . . . . . . . . . . . . . . . . . . 8 Applications of It6's Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises for Chapter I1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 51 52 54 56 60 66 78 84 92 94

I

I1

7

I11 Semimartingales and Decomposable Processes . . . . . . . . . . . . . 101 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 2 The Classification of Stopping Times . . . . . . . . . . . . . . . . . . . . . . . 103 3 The Doob-Meyer Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4 Quasimartingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

XI1

Contents 5 Compensators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6 The Fundamental Theorem of Local Martingales . . . . . . . . . . . . 124 7 Classical Semimartingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 8 Girsanov's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 9 The Bichteler-Dellacherie Theorem . . . . . . . . . . . . . . . . . . . . . . . . 143 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Exercises for Chapter I11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

IV General Stochastic Integration and Local Times . . . . . . . . . . . 153 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 2 Stochastic Integration for Predictable Integrands . . . . . . . . . . . . 153 3 Martingale Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 4 Martingale Duality and the Jacod-Yor Theorem on Martingale Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 5 Examples of Martingale Representation . . . . . . . . . . . . . . . . . . . . 200 6 Stochastic Integration Depending on a Parameter . . . . . . . . . . . . 205 7 LocalTimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 8 AzBma's Martingale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 9 Sigma Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 Exercises for Chapter IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 V

Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 243 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 2 The BPNorms for Semimartingales . . . . . . . . . . . . . . . . . . . . . . . . 244 3 ~ x i s t e n c eand Uniqueness of Solutions . . . . . . . . . . . . . . . . . . . . . 249 4 Stability of Stochastic Differential Equations . . . . . . . . . . . . . . . . 257 5 Fisk-Stratonovich Integrals and Differential Equations . . . . . . . . 270 6 The Markov Nature of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 7 Flows of Stochastic Differential Equations: Continuity and Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 8 Flows as Diffeomorphisms: The Continuous Case . . . . . . . . . . . . 310 9 General Stochastic Exponentials and Linear Equations . . . . . . . 321 10 Flows as Diffeomorphisms: The General Case . . . . . . . . . . . . . . . 328 11 Eclectic Useful Results on Stochastic Differential Equations . . . 338 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 Exercises for Chapter V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

VI Expansion of Filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 2 Initial Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 3 Progressive Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 4 TimeReversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 Exercises for Chapter VI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384

Contents

XI11

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

Symbol Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .407

Introduction

In this book we present a new approach to the theory of modern stochastic integration. The novelty is that we define a semimartingale as a stochastic process which is a "good integrator" on an elementary class of processes, rather than as a process that can be written as the sum of a local martingale and an adapted process with paths of finite variation on compacts: This approach has the advantage over the customary approach of not requiring a close analysis of the structure of martingales as a prerequisite. This is a significant advantage because such an analysis of martingales itself requires a highly technical body of knowledge known as "the general theory of processes." Our approach has a further advantage of giving traditionally difficult and non-intuitive theorems (such as Stricker's Theorem) transparently simple proofs. We have tried to capitalize on the natural advantage of our approach by systematically choosing the simplest, least technical proofs and presentations. As an example we have used K. M. Rao's proofs of the Doob-Meyer decomposition theorems in Chap. 111, rather than the more abstract but less intuitive DolBans-Dade measure approach. In Chap. I we present preliminaries, including the Poisson process, Brownian motion, and LBvy processes. Naturally our treatment presents those properties of these processes that are germane to stochastic integration. In Chap. I1 we define a semimartingale as a good integrator and establish many of its properties and give examples. By restricting the class of integrands to adapted processes having left continuous paths with right limits, we are able to give an intuitive Riemann-type definition of the stochastic integral as the limit of sums. This is sufficient to prove many theorems (and treat many applications) including a change of variables formula ("ItG's formula"). Chapter I11 is devoted to developing a minimal amount of "general theory" in order to prove the Bichteler-Dellacherie Theorem, which shows that our "good integrator" definition of a semimartingale is equivalent to the usual one as a process X having a decomposition X = M A, into the sum of a local martingale M and an adapted process A having paths of finite variation on compacts. Nevertheless most of the theorems covered en route (Doob-

+

2

Introduction

Meyer, Meyer-Girsanov) are themselves key results in the theory. The core of the whole treatment is the Doob-Meyer decomposition theorem. We have followed the relatively recent proof due to R. Bass, which is especially simple for the case where the martingale jumps only at totally inaccessible stopping times, and in all cases uses no mathematical tool deeper than Doob's quadratic martingale inequality. This allows us to avoid the detailed treatment of natural processes which was ubiquitous in the first edition, although we still use natural processes from time to time, as they do simplify some proofs. Using the results of Chap. I11 we extend the stochastic integral by continuity to predictable integrands in Chap. IV, thus making the stochastic integral a Lebesguctype integral. We use predictable integrands to develop a theory of martingale representation. The theory we develop is an L2 theory, but we also prove that the dual of the martingale space 3-1' is BMO and then prove the Jacod-Yor Theorem on martingale representation, which in turn allows us to present a class of examples having both jumps and martingale representation. We also use predictable integrands to give a presentation of semimartingale local times. Chapter V serves as an introduction to the enormous subject of stochastic differential equations. We present theorems on the existence and uniqueness of solutions as well as stability results. Fisk-Stratonovich equations are presented, as well as the Markov nature of the solutions when the differentials have Markov-type properties. The last part of the chapter is an introduction to the theory of flows, followed by moment estimates on the solutions, and other minor but useful results. Throughout Chap. V we have tried to achieve a balance between maximum generality and the simplicity of the proofs. Chapter VI provides an introduction to the theory of the expansion of filtrations (known as "grossissements de filtrations" in the French literature). We present first a theory of initial expansions, which includes Jacod's Theorem. Jacod's Theorem gives a sufficient condition for semimartingales to remain semimartingales in the expanded filtration. We next present the more difficult theory of progressive expansion, which involves expanding filtrations to turn a random time into a stopping time, and then analyzing what happens to the semimartingales of the first filtration when considered in the expanded filtration. Last, we give an application of these ideas to time reversal.

Preliminaries

1 Basic Definitions and Notation We assume as given a complete probability space ( 0 , F,P). In addition we are given a filtration (Ft)o
(i) Focontains all the P-null sets of 6 (ii) Ft = 3;1, all t, 0 5 t < co;that is, the filtration IF is right continuous.

nu,,

We always assume that the usual hypotheses hold. Definition. A random variable T : R -+ [0,co] is a stopping time if the event {T 5 t) E Ft7 every t, 0 5 t 5 co.

One important consequence of the right continuity of the filtration is the following theorem. Theorem 1. The event {T stopping time.

< t)

E Ft, 0

< t < co, if

and only if T is a

< u), any E > 0, we have {T Proof. Since {T I t) = n,+,,,,,{T t) E Fu = Ftlso T is a stopping time. For the converse, {T < t) Ut,E,O{T t - E), and {T 5 t - E) E FtTt--,) hence also in F t .

nu,,

<

< =

A stochastic process X on ( 0 , F , P ) is a collection of R-valued or Rdvalued random variables (Xt)o
4

I Preliminaries

If X and Y are modifications there exists a null set, Nt, such that if w $ Nt, then Xt(w) = &(w). The null set Nt depends on t. Since the interval [O, co) is uncountable the set N = UoSt<, Nt could have any probability between 0 and 1, and it could even be non-measurable. If X and Y are indistinguishable, however, then there exists one null set N such that if w 4 N, then Xt(w) = & (w), for all t. In other words, the functions t H Xt(w) and t H & (w) are the same for all w $ N , where P ( N ) = 0. The set N is in F t , all t , since 6 contains all the P-null sets of F . The functions t H Xt(w) mapping [0,co) into R are called the sample paths of the stochastic process X . Definition. A stochastic process X is said to be chdlhg if it a.s. has sample paths which are right continuous, with left limits. Similarly, a stochastic process X is said to be chglhd if it a.s. has sample paths which are left continuous, with right limits. (The nonsensical words chdlhg and chghd are acronyms from the French for continu h droite, limites a gauche and continu h gauche, limites a droite, respectively.) Theorem 2. Let X and Y be two stochastic processes, with X a modification of Y. If X and Y have right continuous paths a.s., then X and Y are indistinguishable.

Proof. Let A be the null set where the paths of X are not right continuous, and let B be the analogous set for Y. Let Nt = {w : Xt(w) f &(w)), and let N = UtEqNt, where Q denotes the rationals in [0,co). Then P ( N ) = 0. Let M = A U B U N , and P ( M ) = 0. We have Xt(w) = &(w) for all t E Q, w $ M. If t is not rational, let t, decrease to t through Q. For w $ M I Xtn(w) = limn,, &, (w) = Xtn(w) = Kn(w), each n , and Xt(w) = limn,, &(w). Since P ( M ) = 0, X and Y are indistinguishable. Corollary. Let X and Y be two stochastic processes which are c&dl&g.If X is a modification of Y, then X and Y are indistinguishable.

CBdlBg processes provide natural examples of stopping times. Definition. Let X be a stochastic process and let A be a Bore1 set in R. Define T(w) = inf{t > 0 : Xt E A).

Then T is called a hitting time of A for X . Theorem 3. Let X be an adapted chdlhg stochastic process, and let A be an open set. Then the hitting time of A is a stopping time.

Proof. By Theorem 1 it suffices to show that {T < t ) E Ft,0 < t

since A is open and X has right continuous paths. Since {X, E A)

F,, the result follows.

< co. But

= X;'(A)

E

1 Basic Definitions and Notation

5

Theorem 4. Let X be an adapted cadlhg stochastic process, and let A be a closed set. Then the random variable T(w) = infit > 0 : Xt(w) E A or Xt- (w) E A)

is a stopping time. Proof. By Xt-(w) we mean lims,t,s
It is a very deep result that the hitting time of a Bore1 set is a stopping time. We do not have need of this result. The next theorem collects elementary facts about stopping times; we leave the proof to the reader. Theorem 5. Let S, T be stopping times. Then the following are stopping times:

(2) S A T = min(S, T); (22) S V T = max(S, T); (iiz) S + T; (iv) a s , where a > 1. The a-algebra Ft can be thought of as representing all (theoretically) observable events up to and including time t . We would like to have an analogous notion of events that are observable before a random time. Definition. Let T be a stopping time. The stopping t i m e a-algebra FT is defined to be all, t {AE F : A n { T S t ) E F ~

> 0).

The previous definition is not especially intuitive. However it does well represent "knowledge" up to time T, as the next theorem illustrates. Theorem 6. Let T be a finite stopping time. Then FT is the smallest a algebra containing all cadlag processes sampled at T . That is,

FT= ~ { X TX; all adapted chdlhg processes). Proof. Let = a{XT; X all adapted c&dl&gprocesses}. Let A E FT. Then Xt = ~ - ~ is a c&dl&g l process, ~ and ~ XT = >In. Hence ~ A E 6,~and FTc 6 . 1 ~ is the indicator function of A : l ~ ( w = )

1,

0,

w E A, w q! A.

I Preliminaries

6

Next let X be an adapted c&dl&gprocess. We need to show XT is FT measurable. Consider X(s, w) as a function from [O, co) x R into R. Construct cp : {T 5 t ) -+ [0,co) x R by cp(w) = (T(w),w). Then since X is adapted and c&dl&g,we have XT = Xocp is a measurable mapping from ({T 5 t), Ftn{T 5 t)) into (R, a ) , where I3 are the Bore1 sets of R.Therefore

is in Ft7and this implies XT E FT. Therefore

6 c FT.

We leave it to the reader to check that if S 5 T as., then Fs c FT,and the less obvious (and less important) fact that Fs n FT= f i ~ ~ . If X and Y are c&dl&g,then Xt = Yt a s . each t implies that X and Y are indistinguishable, as we have already noted. Since fixed times are stopping . each finite stopping time T , then X and times, obviously if XT = YT a . ~for Y are indistinguishable. If X is c&dl&g,let AX denote the process AXt = Xt - Xt-. Then AX is not ciidliig, though it is adapted and for a.a. w, t H AXt = 0 except for at most countably many t. We record here a useful result.

Theorem 7. Let X be adapted and cadlag. If A X T ~ { ~ < , = ) 0 a.s. for each stopping time T, then A X is indistinguishable from the zero process. Proof. It suffices to prove the result on [0,to] for 0 < to < co. The set {t : [AXt[> 0) is countable a.s. since X is chdliig. Moreover { t : IAXtl > O )

1

=

U { t : lAXtl > -) n n= 1

and the set {t : lAXtl > l l n ) must be finite for each n, since to < co. Using Theorem 4 we define stopping times for each n inductively as follows:

Then

~

"

> 1

~

~~ 2

"

'

a.s. on {T~)"' < co). Moreover,

where the right side of the equality is a countable union. The result follows.

Corollary. Let X and Y be adapted and c&dl&g.If for each stopping time T, A X T ~ { ~ < ,= ) AYT1{T
2 Martingales

7

A much more general version of Theorem 7 is true, but it is a very deep result which uses Meyer's "section theorems," and we will not have need of it. See, for example, Dellacherie [41] or Dellacherie-Meyer [45]. A fundamental theorem of measure theory that we will need from time to time is known as the Modtone Class Theorem. Actually there are several such theorems, but the one given here is sufficient for our needs. Definition. A monotone vector space 'H on a space R is defined to be the collection of bounded, real-valued functions f on R satisfying the three conditions: (i) 'H is a vector space over R; (ii) la E 'H (i.e., constant functions are in 'H); and fi 5 . . f, . - ., and limn,, (iii) if (fn)n21 C 'H, and 0 f l and f is bounded, then f E 'H.

< <

<

<

fn =

f,

Definition. A collection M of real functions defined on a space R is said to be multiplicative if f , g E M implies that f g E M . For a collection of real-valued functions M defined on R, we let a{M) denote the space of functions defined on R which are measurable with respect to the a-algebra on R generated by {fP1(A); A E B(R), f E M ) . Theorem 8 (Monotone Class Theorem). Let M be a multiplicative class of bounded real-valued functions defined on a space R, and let A = a{M). If 'H is a monotone vector space containing M , then 'H contains all bounded, A measurable functions. Theorem 8 is proved in Dellacherie-Meyer [45, page 141 with the additional hypothesis that 'H is closed under uniform convergence. This extra hypothesis is unnecessary, however, since every monotone vector space is closed under uniform convergence. (See Sharpe [215, page 3651.)

2 Mart ingales In this section we give, mostly without proofs, only the essential results from the theory of continuous time martingales. The reader can consult any of a large number of texts to find excellent proofs; for example DellacherieMeyer [46], or Ethier-Kurtz [71]. Also, recall that we will always assume as given a filtered, complete probability space (R, 3,IF, P), where the filtration IF = ( F t )jti, ~ is assumed t o be right continuous. Definition. A real-valued, adapted process X = (Xt)o
<

> X,).

8

I Preliminaries

Note that martingales are only defined on [0,co);that is, for finite t and not t = co.It is often possible to extend the definition to t = co.

Definition. A martingale X is said to be closed by a random variable Y if < co and Xt = E{YI.Ft}, 0 5 t < co.

E{IYI)

A random variable Y closing a martingale is not necessarily unique. We give a sufficient condition for a martingale to be closed (as well as a construction for closing it) in Theorem 12.

Theorem 9. Let X be a supermartingale. The function t ++ E{Xt) is right continuous if and only if there exists a modification Y of X which is chdldg. Such a modification is unique. By uniqueness we mean up to indistinguishability. Our standing assumption that the "usual hypotheses" are satisfied is used implicitly in the statement of Theorem 9. Also, note that the process Y is, of course, also a supermartingale. Theorem 9 is proved using Doob's upcrossing inequalities. If X is a martingale then t H E{Xt) is constant, and hence it has a right continuous modification.

Corollary. If X = (Xt)o
Theorem 10 (Martingale Convergence Theorem). Let X be a right continuous supermartingale, supolt<m E{IXtl) < co. Then the random variable Y = limt,, Xt a.s. exists, and E{IYI) < co. Moreover if X is a martingale closed by a random variable 2 , then Y also closes X and Y = E{ZI Volt<m3tTt).2 A condition known as uniform integrability is sufficient for a martingale to be closed.

Definition. A family of random variables (Ua)aEAis uniformly integrable if lim sup

n-cc

S

,

IU,ldP

= 0.

IIUc212n)

Theorem 11. Let ( U a ) , € ~ be a subset of L 1 . The following are equivalent: (i) (Ua)aEAis uniformly integrable. (ii) SUPaEAE{IUaI) < co, and for every E > 0 there exists 6 > 0 such that A E 3, P(A) I 6, imply E{IU,lnl) < E . Ft

denotes the smallest a-algebra generated by (Ft), all t, 0 5 t

< co.

2 Martingales

9

(iii) There exists a positive, increasing, convex function G(x) defined o n = +co and sup, E{G o 1U,I) < m. [0, co) such that lim,,, The assumption that G is convex is not needed for the implications (iii) (ii) and (iii) + (2).

+

Theorem 12. Let X be a right continuous martingale which is uniformly integrable. Then Y = limt,, Xt a.s. exists, E{IYI) < co, and Y closes X as a martingale. Theorem 13. Let X be a (right continuous) martingale. Then (Xt)t20 is Xt exists a.s., E{IYI) < co, uniformly integrable if and only if Y = limt,, and (Xt)o
Theorem 14 (Backwards Convergence Theorem). Let (Xn)n10 be a 0 martingale. Then limn,-, Xn = E{Xol nn=-, 3 n ) a.s. and i n L1.

A less probabilistic interpretation of martingales uses Hilbert space theory. Let Y E L 2 ( R , 3 ,P ) . Since Ft 3 , the spaces L2(R,Ft,P ) form a family of Hilbert subspaces of L2(R,3 , P ) . Let "tY denote the Hilbert space projection of Y onto L2(R,Ft,P). Theorem 15. Let Y E L 2 ( R , 3 ,P ) . The process Xt = "tY is a uniformly integrable martingale.

Proof. It suffices to show E{YIFt} = "tY. The random variable E{YIFt} is the unique Ft measurable r.v. such that JA Y d P = JA E{YIFt}dP, for JA(Y -"t Y)dP. But any event A E Ft. We have JA Y d P = JA "tYdP JA(Y -"tY)dP = J l A ( Y-"tY)dP. Since l AE L2(R,Ft, P ) , and (Y -"tY) is in the orthocomplement of L2(R,Ft,P ) , we have J ~ A ( Y "")dP = 0, and thus by uniqueness E{YIFt} = "tY. Since ll"tYllLz 5 llYllLz, by part (iii) of Theorem 11 we have that X is uniformly integrable (take G(x) = x2).

+

The next theorem is one of the most useful martingale theorems for our purposes.

Theorem 16 (Doob's Optional Sampling Theorem). Let X be a right continuous martingale, which is closed by a random variable X,. Let S and T be two stopping times such that S 5 T a.s. Then Xs and XT are integrable and Xs = E{XTIFS} a.s. Theorem 16 has a similar version for supermartingales.

I Preliminaries

10

Theorem 17. Let X be a right continuous supermartingale (resp. martingale), and let S and T be two bounded stopping times such that S < T a.s. Then Xs and XT are integrable and

If T is a stopping time, then so is t A T = min(t, T), for each t 2 0. Definition. Let X be a stochastic process and let T be a random time. is said to be the process stopped at T if XT = X t A ~ .

xT

Note that if X is adapted and ciidl&gand if T is a stopping time, then

is also adapted. A martingale stopped at a stopping time is still a martingale, as the next theorem shows. Theorem 18. Let X be a uniformly integrable right continuous martingale, and let T be a stopping time. T h e n = (XtAT)O jtjm is also a uniformly

xT

integrable right continuous martingale. Proof. XT is clearly right continuous. By Theorem 16

However for H E Ft we have H1 {T>t) - E 3 ~Thus, .

Therefore

since XT1{T
xTis a uniformly integrable Ft mar-

Observe that the difficulty in Theorem 18 is to show that X T is a martingale for the filtration (3t)o
xT

Corollary. Let Y be an integrable random variable and let S , T be stopping times. Then

2 Martingales

Proof. Let

& = E{YIFt).

11

Then yT is a uniformly integrable martingale and

Interchanging the roles of T and S yields

Finally, E { Y ( ~ ~ A=TYSA~. ) The next inequality is elementary, but indispensable.

Theorem 19 (Jensen's Inequality). Let cp : R -, IW be convex, and let X and cp(X) be integrable random variables. For any a-algebra Q ,

Corollary 1. Let X be a martingale, and let cp be convex such that cp(Xt) is integrable, 0 5 t < co. Then cp(X) is a submartingale. In particular, if M is a martingale, then lMl is a submartingale. Corollary 2. Let X be a submartingale and let cp be convex, non-decreasing, and such that cp(Xt)olt<, is integrable. Then cp(X) is also a submartingale. We end our review of martingale theory with Doob's inequalities; the most important is when p = 2.

Theorem 20. Let X be a positive submartingale. For all p jugate to p (i.e., + = I), we have

:+

> 1, with q con-

We let X* denote sup, IXsI. Note that if M is a martingale with M, then lMl is a positive submartingale, and taking p = 2 we have

E

L2,

This last inequality is called Doob's maximal quadratic inequality. An elementary but useful result concerning martingales is the following.

Theorem 21. Let X = (Xt)olt5, be an adapted process with chdlag paths. Suppose E{(XTI) < co and E{XT) = 0 for any stopping time T , finite or not. Then X is a uniformly integrable martingale.

Proof. Let 0 5 s < t < co, and let A E F S . Let u, co,

if w E A, ifw$A.

12

I Preliminaries

Then un are stopping times for all u 2 s. Moreover

since E{X,,) = 0 by hypothesis, for u 2 s. Thus for A E 3, and s < t, E{Xtln) = E{X,ln) = -E{Xmln\A), which implies E{XtlFs) = X,, and X is a martingale, 0 t co.

< <

Definition. A martingale X with Xo = 0 and E{X:) < co for each t > 0 is called a square integrable martingale. If E{X&) < co as well, then X is called an L2 martingale. Clearly, any L2 martingale is also a square integrable martingale. See also Sect. 3 of Chap. IV.

3 The Poisson Process and Brownian Motion The Poisson process and Brownian motion are the two fundamental examples in the theory of continuous time stochastic processes. The Poisson process is the simpler of the two, and we begin with it. We recall that we assume given a filtered probability space ( R , 3 , IF, P ) satisfying the usual hypotheses. Let (Tn)n20be a strictly increasing sequence of positive random variables. We always take To = 0 a.s. Recall that the indicator function l { t-> ~ nis}defined

Definition. The process N

=

(Nt)olt<m defined by

with values in W~{co)where W = {0,1,2,. . . ) is called the counting process associated to the sequence (Tn),> 1. If we set T = sup, T, , then

as well as [Tn,Tn+l) = { N = n), and [T,w)

= {N = w).

3 The Poisson Process and Brownian Motion

13

The random variable T is the explosion time of N . If T = co as., then N is a counting process without explosions. For T = co, note that for 0 s < t < co we have

<

The increment Nt - N, counts the number of random times Tn that occur between the fixed times s and t . As we have defined a counting process it is not necessarily adapted to the filtration IF. Indeed, we have the following. Theorem 22. A counting process N is adapted if and only if the associated random variables (Tn)nL1are stopping times. - are stopping times (with To = 0 a.s.), then the event Proof. If the (Tn)n>o

for each n. Thus Nt E Ft and N is adapted. If N is adapted, then {Tn {Nt 2 n ) E Ft, each t , and therefore Tn is a stopping time.


Note that a counting process without explosions has right continuous paths with left limits; hence a counting process without explosions is cadlhg. Definition. An adapted counting process N is a Poisson process if

<

(i) for any s, t , 0 s < t < co, Nt - N3 is independent of 3,; (ii) for any s, t , u , v , 0 s < t < co, 0 u < v < co, t - s = v - u , then the distribution of Nt - N, is the same as that of N, - Nu.

<

<

Properties (i) and (ii) are known respectively as increments independent of the past, and stationary increments. Theorem 23. Let N be a Poisson process. Then

n = 0 , 1 , 2 , . . . , for some X 2 0 . That is, Nt has the Poisson distribution with parameter At. Moreover, N is continuous i n probabilit$ and does not have explosions. Proof. The proof of Theorem 23 is standard and is often given in more elementary courses (cf., e.g., Cinlar [33, page 711). We sketch it here. Step 1. For all t 2 0 , P(Nt = 0 ) = e-At, for some constant X 2 0. N is continuous in probability means that for t > 0, limu,t Nu = Nt where the limit is taken in probability.

3 The Poisson Process and Brownian Motion

= P(Nt = 0)

+ oP(Nt = 1) +

15

a n P ( N t = n), n=2

and $(a) $(a)

= cpl(0), the = =

derivative of cp at 0. Therefore

cp(t) - 1 = lim lim t t+O

P(Nt = 0) - 1

t-0

-A

+

a P ( N t = 1) t

1

+ Aa.

Therefore cp(t) = e-At+Aat,hence

Equating coefficients of the two infinite series yields

Definition. The parameter A associated to a Poisson process by Theorem 23 is called the intensity, or arrival rate, of the process. Corollary. A Poisson process N with intensity A satisfies E{Nt) = At, Variance(Nt) = Var(Nt) = At. The proof is trivial and we omit it. There are other, equivalent definitions of the Poisson process. For example, a counting process N without explosion can be seen to be a Poisson process iffor all s , t , 0 5 s < t < co,E{Nt) < co and

Theorem 24. Let N be a Poisson process with intensity A. Then Nt -At and (Nt - At)2 - At are martingales. Proof. Since At is non-random, the process Nt - At has mean zero and independent increments. Therefore

for 0

< s < t < co.The analogous statement holds for (Nt

-

At)2 - At.

16

I Preliminaries

Definition. Let H be a stochastic process. The natural filtration of H, denoted IF0 = (fl)olt<m, is defined by fl = a{Hs; s 5 t). That is, fl is the smallest filtration that makes H adapted. Note that natural filtrations are not assumed to contain all the P-null sets of 3.

Theorem 25. Let N be a counting process. The natural filtration of N is right continuous. Proof. Let E = [0,co] and 8 be the Bore1 sets of E , and let I' be the path space given by I'=( ES, @ B,).

n

~~(0,co) ~~(0,co)

Define the maps .rrt : R

4

I' by

Thus the range of .rrt is contained in the set of functions constant after t. The a-algebra fl is also generated by the single function space-valued random variable .rrt. Let A be an event in $. Then there exists a set An E BIE lo,m) BS such that A = { . r r t + ~ E A,). Next set Wn = {.rrt = .rrt+;). For each w, there exists an n such that s ++ N,(w) is constant on [t,t therefore R = UnLl Wn, where Wn is an increasing sequence of events. Therefore

nn, e+

+ i];

= lim(Wn n {.rrt+& E An)) n

= lim(Wn n

n {xt E An))

= lim{.rr, E n

which implies A E equal.

fl. We conclude

A,), -

e+:C E ,

which implies they are

We next turn our attention to the Brownian motion process. Recall that we are assuming as given a filtered probability space (R, 3,IF, P ) that satisfies the usual hypotheses.

Definition. An adapted process B = (Bt)olt
3 The Poisson Process and Brownian Motion

17

The Brownian motion starts at x if P(Bo = x) = 1. The existence of Brownian motion is proved using a path-space construction, together with Kolmogorov's Extension Theorem. It is simple to check that a Brownian motion is a martingale as long as E{IBol) < cm. Therefore by Theorem 9 there exists a version which has right continuous paths, a s . Actually, more is true. Theorem 26. Let B be a Brownian motion. Then there exists a modification of B which has continuous paths a.s.

Theorem 26 is often proved in textbooks on probability theory (e.g., Breiman [23]). It can also be proved as an elementary consequence of Kolmogorov's Lemma (Theorem 72 of Chap. IV). W e will always assume that we are using the version of Brownian motion with continuous paths. We will also assume, unless stated otherwise, that C is the identity matrix. We then say that a Brownian motion B with continuous paths, with C = I the identity matrix, and with Bo = x for some x E Rn, is a standard Brownian motion. Note that for an Rn standard Brownian motion B, writing Bt = (Btl, . . . , BF), 0 t < cm, then each Bi is an R1 Brownian motion with continuous paths, and the B"S are independent. We have already observed that a Brownian motion B with E{JBoJ)< cm is a martingale. Another important elementary observation is the following.

<

Theorem 27. Let B = (Bt)olt<, be a one dimensional standard Brownian motion with Bo = 0. Then Mt = Bt2 - t is a martingale.

Proof. E{Mt ) = E{B:

-

t ) = 0. Also

and E{BtBsIFs)

= BsE{BtlFs) = B:,

since B is a martingale with B,, Bt E L2. Therefore

due to the independence of the increments from the past. Theorem 28. Let .ir, be a sequence ofpartitions of [a,a+t]. Suppose r.i, C .ir, i f m > n (that is, the sequence is a refining sequence). Suppose moremesh(.ir,) = 0. Let .ir,B = CtZErr, (Bti+l - Bti)2. Then over that lim,,, lirn,,, .ir, B = t a.s., for a standard Brownian motion B .

18

I Preliminaries

Proof. We first show convergence in mean square. We have

where Yi are independent random variables with zero means. Therefore

i + lhas the distribution of Z2, where Next observe that (Bti+l- ~ ~ , ) ~ / ( t-ti) Z is Gaussian with mean 0 and variance 1. Therefore

which tends to 0 as n tends to cm.This establishes L2 convergence (and hence convergence in probability as well). To obtain the a s . convergence we use the Backwards Martingale Convergence Theorem (Theorem 14). Define

for n = -1, -2, -3,. . . . Then it is straightforward (though notationally messy) to show that

Therefore Nn is a martingale relative to Gn = a{Nk, k 5 n), n = -1, -2, . . . . By Theorem 14 we deduce limn,-, Nn = limn,, r n B exists as., and since r n B converges to t in L2, we must have limn,, r n B = t a s . as well. Comments. As noted in the proofs, the proof is simple (and half as long) if we conclude only L2 convergence (and hence convergence in probability), instead of a s . convergence. Also, we can avoid the use of the Backwards Martingale Convergence Theorem (Theorem 14) in the second half of the proof if we add the hypothesis that Enmesh(rn) < co. The result then follows, after having proved the L2 convergence, by using the Borel-Cantelli Lemma and Chebsyshev's inequality. Furthermore to conclude only L2 convergence we do not need the hypothesis that the sequence of partitions be refining.

Theorem 28 can be used to prove that the paths of Brownian motion are of unbounded variation on compacts. It is this fact that is central to the difficulties in defining an integral with respect to Brownian motion (and martingales in general).

4 L6vy Processes

19

Theorem 29. For almost all w, the sample paths t H Bt(w) of a standard Brownian motion B are of unbounded variation on any interval.

Proof. Let A = [a,b] be an interval. The variation of paths of B is defined to

where P are all finite partitions of [a,b]. Suppose P(VA < co) > 0. Let .ir, be a sequence of refining partitions of [a,b] with limn mesh(.ir,) = 0. Then by Theorem 28 on {VA < m}, b

-

a = lim

n+cc

< -

(Bt,+, - B ~ , ) ~ t, E r n

lim sup IBt,+, - Bt,lVA t,Ern

= 0,

since suptiErnIBt,+, - Bti ( tends to 0 a s . as mesh(.ir,) tends to 0 by the a.s. uniform continuity of the paths on A. Since b-a < 0 is absurd, by Theorem 27 we conclude VA = m a s . Since the null set can depend on the interval [a,b], we only consider intervals with rational endpoints a, b with a < b. Such a collection is countable, and since any interval (a, b) = U;==,[a,, b,] with a,, b, rational, we can omit the dependence of the null set on the interval. We conclude this section by observing that not only are the increments of standard Brownian motion independent, they are also stationary. Thus Brownian motion is a Lkvy process (as is the Poisson process), and the theorems of Sect. 4 apply to. it. In particular, by Theorem 31 of Sect. 4, we can conclude that the completed natural filtration of standard Brownian motion is right continuous.

4 L6vy Processes The Lkvy processes, which include the Poisson process and Brownian motion as special cases, were the first class of stochastic processes to be studied in the modern spirit (by the French mathematician Paul Lkvy). They still provide prototypic examples for Markov processes as well as for semimartingales. Most of the results of this section hold for Rn-valued processes; for notational simplicity, however, we will consider only R-valued processes.4 Once again we recall that we are assuming given a filtered probability space (a,F , IF, P ) satisfying the usual hypotheses. Rn denotes n-dimensional Euclidean space. R+ real numbers.

= [O, m) denotes the

non-negative

20

I Preliminaries

Definition. An adapted process X = (Xt)t20 with Xo = 0 a s . is a LQvy process if

(i) X has increments independent of the past; that is, Xt - X, is independent ofF,,O<s
=

(Xt)tlo with Xo = 0 a s . is an intrinsic LQvy

(i) X has independent increments; that is, Xt - X, is independent of X, -Xu if (u, v) n (s, t) = 0; and (ii) X has stationary increments; that is, Xt - X, has the same distribution a s X , - X u i f t - s = v - u > 0 ; and (iii) Xt is continuous in probability. Of course, an intrinsic L6vy process is a L6vy process for its minimal (completed) filtration. If we take the Fourier transform of each Xt we get a function f (t, u) = ft (u) given by ft(u) = ~ { e ' ~ ~ ~ ) , where fo(u) = 1, and ft+,(u) = ft(u)fs(u), and ft(u) # 0 for every (t, u). Using the (right) continuity in probability we conclude ft(u) = exp{-t$(u)), for some continuous function $(u) with $(O) = 0. (Bochner's Theorem can be used to show the converse. If $ is continuous, $(0) = 0, and if for all t 2 0, ft (u) = e c t + ( ~ satisfies ) ft (ui - uj) 2 0, for all finite (ul, . . . ,21,; a1, . . . ,a,), then there exists a Ltvy process corresponding to f .) In particular it follows that if X is a L6vy process then for each t > 0, Xt has an infinitely divisible distribution. Inversely it can be shown that for each infinitely divisible distribution p there exists a L6vy process X such that p is the distribution of X I .

xi,?

Theorem 30. Let X be a Le'vy process. There exists a unique modification Y of X which is ccidlcig and which is also a Le'vy process. tuxt h.

Proof. Let M; = For each fixed u in Q,the rationals in R, the process (M;)olt
4 L6vy Processes

21

t H M ~ ( wand ) t H eiuxt(w),with t E Q+, are the restrictions to Q+ of cbdlAg functions. Let E Q+, A = {(w,u)E fl x R : eiuXt(W),t is not the restriction of a cadlag function).

One can check that A is a measurable set. F'urthermore, we have seen that

J lA(w,u)P(dW) = 0, each u E W. By Fubini's Theorem

hence we conclude that for a.a. w the function t H eiUxdw),t E Q+ is the restriction of a chdlAg function for almost all u E W. We can now conclude that the function t H Xt(w), t E Q+, is the restriction of a cbdlhg function for every such w, with the help of the lemma that follows the proof of this theorem. Xs(w) for all w in the projection onto fl of Next set K(w) = limsGq+,s~t {fl x W) \ A and Yt = 0 on A, all t. Since Ft contains all the P-null sets of F and (Ft)olt<m is right continuous, Yt E F t . Since X is continuous in probability, P{Yt # Xt) = 0, hence Y is a modification of X . It is clear that Y is a Lkvy process as well. The next lemma was used in the proof of Theorem 30. Although it is a pure analysis lemma, we give a proof using probability theory. Lemma. Let x, be a sequence of real numbers such that eiuxn converges as n tends to cm for almost all u E W. Then x, converges to a finite limit. Proof. We will verify the following Cauchy criterion: x, converges if for any increasing sequences nk and mk, then limk,, x,, - xmk - 0. Let U be a random variable which has the uniform distribution on [0, 11. For any real t , by hypothesis a s . eituxnk and eituxmk converge to the same limit. Therefore,

so that the characteristic functions converge,

for all t E W. Consequently (x,, - xmk)U converges to zero in probability, whence limk,, x,, - xmk= 0, as claimed. We will henceforth always assume that we are using the (unique) cadlag version of any given Lkvy process. Lkvy processes provide us with examples of filtrations that satisfy the "usual hypotheses," as the next theorem shows.

22

I Preliminaries

e

Theorem 31. Let X be a Lkvy process and let Gt = v N , where (e)olt
nu,,

Gu. Note that since the proof. We must show Gt+ = Gt, where Gt+ = filtration G is increasing, it suffices to show that Gt = On,, - Gt+*. Thus, we can take countable limits and it follows that if s l , . . . , sn 5 t, then for (~l,...,~n)

For vl, . . . , vn > t and (ul, . . . ,u,), we give the proof for n = 2 for notational convenience. Therefore let z > v > t, and suppose given ul and 212. We have

using that M,Yz

=

is a martingale. Combining terms the above becomes

and the same martingale argument yields

3 IGt} for all ( s l , . . . , s,) and It follows that E{eiCUjX"jlGt+) = E{eiCUjXs all ( ~ 1 ,. . , un), whence E{ZlGt+) = E{ZlGt} for every bounded Z 6 Vols
e.

The next theorem shows that a Lkvy process "renews itself" at stopping times.

4 L6vy Processes

23

. Let X be a Lkvy process and let T be a stopping time. On the set {T < m} the process Y = ( Y t ) ~ < t < defined ~ by Yt = X T + ~- XT is a Lkvy process adapted to 7it = FT+t, Y is independent of FT and Y has the same distribution as X .

Theorem 32.

Proof. First assume T is bounded. Let A E FTand let ( u l , . . . , u,; to,. . . ,t,) be given with uj in a countable dense set (for example the rationals Q)and t j E EX+, t j increasing with j . Recall that M,U' = is a martingale, where ft (uj) = E{e'ujXt). Then

$$$

by applying the Optional Sampling Theorem (Theorem 16) n times. Note that this shows the independence of Yt = X T + ~- XT from FTas well as showing that Y has independent and stationary increments and that the distribution of Y is the same as that of X. If T is not bounded, we let Tn = min(T, n) = T A n. The formula is valid for A, = An{T 5 n ) when A E FT, since then A, E FTAn.Taking limits and using the Dominated Convergence Theorem we see that our formula holds for unbounded T as well, for events A = A n {T < m), A E FT. This gives the result. Since a standard Brownian motion is a L6vy process, Theorem 32 gives us a fortiori the strong Markov property for Brownian motion. This allows us to establish a pretty result for Brownian motion, known as the reflection principle. Let B = (Bt)tlo denote a standard Brownian motion, Bo = 0 as., and let St = supolslt B,, the maximum process of Brownian motion. Since U ~ Bu , where ~ Q ~ denotes ~ the~ rationals; ~ hence , ~ B is continuous, St = S St is an adapted process with non-decreasing paths. Theorem 33 (Reflection Principle for Brownian Motion). Let B =

(Bt)t20 be standard Brownian motion (Bo = 0 a.s.) and St = supolslt B,, the Brownian mm.mum process. For y 2 0, z > 0,

Proof. Let T = inf{t > 0 : Bt = z). Then T is a stopping time by Theorem 4, and P ( T < m ) = 1. We next define a new process X by

~

~

24

1 Preliminaries

The process X is the Brownian motion B up to time T, and after time T it is the Brownian motion B "reflected" about the constant level z. Since B and - B have the same distribution, it follows from Theorem 32 that X is also a standard Brownian motion. Next we define R =inf{t > 0 : Xt = z ) . Then clearly

since (R, X ) and ( T ,B ) have the same distribution. However we also have that R = T identically, whence

by the construction of X . Therefore

The left side of (*) equals

where the last equality is a consequence of the containment { S t 2 z ) > { B t > z + y ). Also the right side of (*) equals P ( S t 2 z; Bt < z - y). Combining these yields P(St 2 z ; B t < z - y ) = P ( B t > z + y ) , which is what was to be proved. We also have a reflection principle for Lkvy processes. See Exercises 30 and 31.

Corollary. Let B = (Bt)t>o - be standard Brownian motion ( B o = 0 as.) and St = S U ~B s .~For~0, ~ ~

Proof. Take y

=0

in Theorem 33. Then

Adding P ( B t > z ) to both sides and noting that { B t > z ) z ) yields the result since P ( B t = z ) = 0.

= {Bt

> z ) n {St 2

4 L6vy Processes

25

A L6vy process is c&dl&g,and hence the only type of discontinuities it can have is jump discontinuities. Letting Xt- = limSTtX,, the left limit at t , we define AX, = xt - xt-, the jump at t. If sup, IAXtI 5 C < cm as., where C is a non-random constant, then we say that X has bounded jumps. Our next result states that a L6vy process with bounded jumps has finite moments of all orders. This fact was used in Sect. 3 (Step 2 of the proof of Theorem 23) to show that E{Nl) < cm for a Poisson process N. Theorem 34. Let X be a LLvy process with bounded jumps. Then E{IXt

In)

<

cm for all n = 1,2,3,.. . . Proof. Let C be a (non-random) bound for the jumps of X . Define the stopping times

Since the paths are right continuous, the stopping times (Tn),?l form a strictly increasing sequence. Moreover [AXT[5 C by hypothesis for any stopping time T . Therefore sup, IX,T" I 5 2nC by recursion. Theorem 32 imand also that the distribution plies that Tn - Tn-1 is independent of FT"-~ of Tn - Tn-1 is the same as that of TI. The above implies that

for somea, 0 < a < 1. But also

which implies that Xt has an exponential moment and hence moments of all orders. We next turn our attention to an analysis of the jumps of a LLvy process. Let A be a Bore1 set in IR bounded away from 0 (that is, 0 @ where is the closure of A). For a L6vy process X we define the random variables

z,

T,"+'

= inf {t

> Tz : AXt E A).

26

I Preliminaries

z,

Since X has chdlhg paths and 0 @ the reader can readily check that { T i 2 t ) E Ft+ = Ft and therefore each TT is a stopping time. Moreover 0 @ ;I and c&dl&gpaths further imply T; > 0 a s . and that lim,,, = cm a s . We define 00

and observe that N" is a counting process without an explosion. It is straightforward to check that for 0 < s < t < co,

and therefore N,/' - N: is independent of FS; that is, N" has independent increments. Note further that N,/' - N b is the number of jumps that Zu = XS+U- X , has in A, 0 5 u < t - S . By the stationarity of the distributions of X , we conclude N,/' - N,/' has the same distribution as NL,. Therefore N" is a counting process with stationary and independent increments. W e conclude that N" is a Poisson process. Let v ( A ) = E { N f ) be the parameter of the Poisson process N" ( v ( A )< cm by the proof of Theorem 34).

Theorem 35. The set function A H N,/'(w) defines a a-finite measure on R \ (0) for each fixed ( t ,w). The set function v ( A ) = E{N:) also defines a a-finite measure on R \ (0). Proof. The set function A ++ N,/'(w) is simply a counting measure: p ( A ) = {number of s < t : AX,(w) E A). It is then clear that v is also a measure. Definition. The measure v defined by

is called the LQvy measure of the L6vy process X . We wish to investigate further the role the Lkvy measure plays in governing the jumps of X . To this end we establish a preliminary result. We let Nt(w,d x ) denote the random measure of Theorem 35. Since Nt(w,dx) is a counting measure, the next result is obvious.

Theorem 36. Let A be a Borel set of R, 0 @ 2, f Borel and finite on A. Then

f (x)Nt(w,dx) =

C

f (AXs)ln(AXs).

O<s
Just as we showed that N,/' has independent and stationary increments, we have the following consequence.

4 Lbvy Processes

Corollary. Let A be a Borel set of R with 0 finite on A. Then

# 3,and

27

let f be Borel and

is a Lkvy process.

x),

For a given set A (as always, 0 # we defined the associated jump process to be J," = AX,ln(AX,). O<&t

C

By Theorem 36 and its corollary we conclude that J :

Hence,

= LXNt(.,dX).

Je is a L6vy process itself, it is defined, and J," < oa as., each t 2 0.

Theorem 37. Given A, 0 $11, the process Xt - J : is a Lkvy process. Proof. It is clear that we need only check the independence and stationarity of the increments. But Xt

-

J," - (X, - )J:

= Xt - X, -

C

AXulA(AXu)

s
for some const ant a > 0. The advantage of doing this is that Yathen has jumps bounded by a, and hence has finite moments of all orders (Theorem 34). We can choose any a > 0; we arbitrarily choose a = 1. Note that

The next theorem gives an interpretation of the L6vy measure as the expected rate at which the jumps of the Livy process fall in a given set.

Theorem 38. Let A be Bowl with 0 $ and let f l~ E L2(dv). Then

2.Let v

be the Lkvy measure of X,

28

I Preliminaries

and also

Proof. First let f = Cj ajln,, a simple function. Then

since N,"j is a Poisson process with parameter v(Aj). The first equality follows easily. For the second equality, let M j = N,"~-&(Ai). The M j are L P martingales, all p 2 1, by the proof of Theorem 34. Moreover, E{Mj) = 0. Suppose Ai, Aj are disjoint. We have

for any partition 0 = to < t l < . . . < tn = t. Using the martingale property we have E{M;M~) = E { C ( M & + ~- M ; ~ ) ( M ~ ~-+MA)). , k

Using the inequality lab( 5 a2

+ b2, we have +

However Ck(Mjk+l- Mjk)' 5 (N:')~ v(Ai)'t2; therefore the sums are dominated by an integrable random variable. Since M j and Mz have paths of finite variation on [O,t]it is easy to deduce that if we take a sequence ( T ~ of partitions where the mesh tends to 0 we have

Using Lebesgue's Dominated Convergence Theorem we conclude

the expectation above is 0 because Ai and Aj are disjoint, implying that M" and M j jump at different times. The second equality is now easy to verify for simple functions. For general f , let fn be a sequence of simple functions such that f n l n converges to f in L2(dv),and the result follows.

)

~

~

~

4 LBvy Processes

29

Remark. The first statement in Theorem 38 remains true if f l AE L 1 ( d v ) . See Exercise 28. Corollary. Let f : R Then

+

R be bounded and vanish in a neighborhood of 0.

O<.
Proof. We need only combine Theorem 38 with Theorem 36. Theorem 39. Let A l , A2 be two disjoint Bore1 sets with 0 $ Then the two processes

xl, 0 $

22.

are independent Lbvy processes. Proof. By Theorem 36 and its corollary we have that J 1 and J 2 are L6vy processes. To show they are independent, we begin by forming for u,v in R,

Then Cu and Du are both martingales, with E{CF) = E{DY) = 0. As in the proof of Theorem 38, let x,: 0 = to < t l < . - . < tn = t be a sequence of partitions of [0,t ] with limn mesh(xn) = 0. Then

Since Cu and Du have paths of finite variation on compacts, it follows by letting mesh(xn) tend to 0, that

The expectation above equals zero because Cu and Du jump at different times, due to the void intersection of Al and A2. We conclude that E{C," DY) = 0, and thus

30

I Preliminaries

which in turn implies, because of the independence and stationarity of the increments that

This is enough to give independence. The preceding results combine to yield the following useful theorem, which is one of the fundamental results about L6vy processes.

+

Theorem 40. Let X be a Livy process. Then Xt = Yt Zt, where Y , Z are Livy processes, Y is a martingale with bounded jumps, Yt E L P for all p 1 and Z has paths of finite variation on compacts.

>

Proof. Let Jt = Co<s
+

While Theorem 40 is the most important result about L6vy processes from the standpoint of stochastic integration, the next two theorems provide a better understanding of Livy processes themselves. Theorem 41. Let X be a Livy process with jumps bounded by a. That is, sup, [ A X , [ 5 a a.s. Let Zt = Xt - E { X t ) . Then Z is a martingale and Zt = 2," Z,d where Z c is a martingale with continuous paths, Zd is a martingale,

+

and Zc and Zd are independent Lkvy processes. Proof. Z has mean zero and independent increments so it is a martingale, as well as a L4vy process. For a given set A we define

4 LBvy Processes

{A

31

a).

For this proof we take a = 1. Let Ak = < 1x1 5 Then ~ " are pairwise independent Lkvy processes and martingales (Theorem 39). Set M n = C;=l ~ " k Then . the martingales Z - M n and M n are independent by an argument similar to the one in the proof of Theorem 39. Moreover Var(Zt) = Var(Zt - M?) Var(MF) where Var(X) denotes the variance of a random variable X . Therefore Var(MF) I Var(Zt) < m for all n. We deduce that Mp is Cauchy in L2 and hence converges in L2 as n tends to m to a martingale Z,d, and Z - M n also converges to a martingale Zc. Using Doob's maximal quadratic inequality (Theorem 20), we can find a subsequence converging as., uniformly in t on compacts, which permits the conclusion that Zc has continuous paths. The independence of Zd and Zc follows from the independence of M n and Z - M n , for every n.

+

Note that a consequence of the convergence of Mp to Z,d in L2 in the proof of Theorem 41 is that the integral ~ - l , o ~ u ~ ox2v(dx) ,ll is finite. Note that this improves a bit on the conclusion in Theorem 38. We recall that for a set A, 0 $ 2, the process N," = JA Nt(-,dx) is a Poisson process with parameter v(A), and thus N," - tv(A) is a martingale. Definition. Let N be a Poisson process with parameter A. Then Nt - At is called a compensated Poisson process. Theorem 41 can be interpreted as saying that a L6vy process with bounded jumps decomposes into the sum of a continuous martingale Lkvy process and a martingale which is a mixture of compensated Poisson processes. It is not hard to show that ~ { e ~ =" e-tu2u2/2, ~ ~ ) which implies that Zc must be a Brownian motion. The full decomposition theorem then follows easily. We state it here without proof (consult Bertoin [12], Bretagnolle [25],Feller [72], or Jacod-Shiryaev [I101 for a proof). Theorem 42 (LQvy Decomposition Theorem). Let X be a Lkvy process. Then X has a decomposition Xt = Bt

+ + ~E{XI-

iI

xNI (-,dx)) ~1211

+

i,

xNt (., dx)

421)

where B is a Brownian motion; for any set A, 0 $ 2 ,N : = JA Nt(., dx) is a : if A and are Poisson process independent of B; N," is independent of N disjoint; N," has parameter v(A); and v(dx) is a measure on R\(0) such that Jmin(l,x2)v(dx) < m.

r

Finally we observe that Theorem 42 gives us a formula (known as the LQvy-Khintchine formula) for the Fourier transform of a L6vy process.

k

I Preliminaries

32

Theorem 43. Let X be a Le'vy process with Le'vy measure v. Then

where a2 +(u) = -u2 2

-

iau

+

iI

+

(1- ehX)v(dx) ~1211

(1- eiux

+ iux)v(dx).

Moreover given v, a2,a, the corresponding Le'vy process is unique in distribution. The Lkvy-Khintchine formula has another, more common expression. (For a complete discussion of the various formulae of Livy and Khintchine for the case of infinitely divisible distributions, see Feller [72, pages 558-5651 .) Indeed, in the n-dimensional case we can express the L6vy-Khintchine formulas as follows (letting (-, -) denote the standard inner product for Rn).

Theorem 44. Let X be a Le'vy process in Rn, with Xo = 0. Then there exists a convolution semigroup of probability measures on Rn such that L(Xt) = pt. The characteristic function of pt is e-t*(u), where is given in terms of a positive definite n x n matrix C and a LEVY measure v on Rn such that

+

(i) v(A) < KJ if A is Bore1 on Rn and bounded away from 0; (ii) I X ~ ~ V< ( ~oaXand ) v{O) = 0.

hxI
Also there exists a bounded, continuous centering j-unction 7 : Rn --+ Rn such that lim,+o = 0, and a constant a t Rn such that

A nice proof of the above theorem can be found in Bertoin [12, pages 13-15]. Several observations concerning this theorem are in order. First, one still has (in the n-dimensional case) the key property that for f : Rn --+ Rn, bounded, vanishing in a neighborhood of 0, that

Second, in the case n = 1, one often takes ~ ( x=) &, possibly at the cost of changing the constant a . In the next section (Sect. 6) we discuss semigroups of Markov processes. The semigroup Pt for the L6vy process X has the form Ptf (x) = f (x y)pt(dy), and one can show in fact that the "infinitesimal generator" of the semigroup Pt is, for f E bC2(Rn),

+

4 LBvy Processes

33

A f (x) = lim Ptf(x) - f ( x ) t

t-0

J

+ f (x + Y) f (x) - (Vf (x),v(y))v(dy). -

Note further that i f f is constant in a neighborhood of x, then lim

t-0

Ptf

(x) - f(x) t

=

J

(x + y) - f(x)v(dy).

We next consider some examples of L6vy processes. For the examples we take n = 1.

Example. If a L6vy process X has continuous paths, then v is identically 0, and X must be Brownian motion with drift. That is, Xt = aBt + a t , for constants a, a, are the only L6vy processes with continuous paths. Example. Assume that v is a finite measure. That is, v(Rn) < m , and that a2 = 0. Then X is a compound Poisson process with jump arrival intensity X = v(Rn). That is, let Nt = Czl lit?Ti) be a Poisson process with arrival intensity A. Let Zt = Czl Uil{t?Ti) where the sequence (Uj)j21 are i.i.d., independent of the sequence ( T j ) j ? l ,and L(U1) = I v . Then one easily verifies ~ { e = ~ee-t*(u), ~ ~where ~ +(u) } = J(1 - eiux)v(dx). Example. Let v(dx) = x r = l ak&@, (dx), where &@, (dx) denotes the positive piak < m and a2= point mass measure at pk E R of size one. Assume 0. Let N k be a sequence of independent Poisson processes with parameters a k . Then Xi = C r = l pk(N: - a k t ) has v as its L6vy measure. Example. Let X be a L6vy process which is a Gamma process. That is, L(Xt) = r ( a , p), with ,6 2 0 and a = at. More precisely, the density for Xt is ft(x) = &xta-le-@xl{x,o). In this case a2 = 0, and taking a = p = 1 gives $t(u) = ~ { e = ~(1 -~i ~ ~) - ~ (See, ~ . ) for example, page 43 of JacodProtter [log] or p. 73 of Bertoin [12].) Then $t(u) = e-t'n(l-iu) = et*(u). But 00 1 $'(u) = -2 = idl(u) = i ~ { e " ~ l = ) i ~"YT-~~X. (1 - iu) 0

1

which implies, by the L6vy-Khintchine formula, that v(dx) = ~ e - x l { x > o ~ d x .

Example. Our last example is that of stable processes. We begin by recalling what a stable distribution is. Extending the notion of the Central Limit Theorem, consider the normalized sums

34

I Preliminaries

We say that a random variable X has a stable law if it is the limit in distribution as n --+ KJ of sums of the above form. One can then prove (see, for example, Breiman [23, page 2001) that either X has a normal distribution, or there exists a number a, 0 < a < 2, called the index of the stable law, and also constants ml 2 0, m2 > 0, and 6 such that E { e i u x ) = eG(u)and

+(u)= iu6

+

mil

00

iux (ehX - 1 - -)-dx 1 +x2

0

+ rn2 J _ O O ( e i ~1~ -)-1i+u xx 2 -

-

1 XI+"

1 dx. /x11+"

One says that a stable distribution is symmetric if ml = m2. By analogy we define a symmetric stable process (in the case n = 1 ) to be a L6vy process where the Ldvy measure has the form v ( d x ) = -dx, with 0 < a < 2, and the constant a 2 in the L6vy-Khintchine forI&la is 0, which, of course, implies that no Brownian component is present in the process. Note that such stable processes have nice scaling properties. If a is the index, then the law of (Xt)t>0 is the same as the law of Consult Bertoin [12] for many other properties of stable processes.

5 Why the Usual Hypotheses? We defined the usual hypotheses at the very beginning of this book, and we saw with Ldvy processes that the natural filtration of a Levy process, once completed, satisfies a fortiori right continuity, and thus the usual hypotheses hold (see Theorem 31). This phenomenon is true much more generally. Indeed it is true for all "reasonable" strong Markov processes, as well as for all counting processes (cf., Theorem 25). In this section we will show that it is true,for a very large class of Markov processes. We first begin with some definitions.

Definition 1. Let X be an Rd-valued process, with X adapted to a filtration IF = (Ft)ostloo. X isMarkov with respect to IF i f F t and a { X , ; s 2 t ) are conditionally independent, given Xt . Definition 2. The process X in Definition 1 is Markov with respect to IF if for any random variable Y E ba{X,; s 2 t ) , E { Y IFt) = E { Y I X t ) , as., all t > 0. The following theorem has a straightforward proof.

Theorem 45. Definitions 1 and 2 are equivalent. Moreover X is Markov with respect to IF if and only if E { f ( X u ) I F t }= E { f ( X u ) I X t ) a.s. for all u 2 t 2 0.

5 Why the Usual Hypotheses?

35

Let X be a c8.dl8.g process and let X be Markov with respect to IF. Since X is Markov, we can associate with it a Markov transition function on (IRd,B), where B denotes the Bore1 sets of Rd. This is a family (Ps,t)s,tEw+of Markov kernels on (Rd,B) such that

for each triple ( s ,t , u) with s < t < u. We then have the fundamental relationship E{f (Xt)lFs) = Ps,t(Xs, f ) a-sNote that, by using indicator functions such as l A= f , we have

Also note that the above implies the equality

which, of course, is weaker. That is, the converse implication is not true in general. The most important special case is Markov processes which are time homogeneous. In this case we have a transition semigroup Pt given by

for s 5 t .

Definition. Let Co denote continuous functions vanishing at infinity. A Markov process is a Feller process if its transition semigroup (Pt)tlo has the following two properties. (a) For each f E Co and each t > 0, Pt f E Co. (b) For each f E Co and each x E Rd, Ptf(x) --t f ( x ) as t

--t

0.

Examples of Feller processes include Brownian motion and the Poisson process. More generally, one can show that diffusions5 and all Lkvy processes are Feller processes.

Theorem 46. Let X be a cddldg Feller process for a filtration IF. Then it is also Markov with respect to the filtration IF+ = (Ft+)oltloo Proof. Let t > 0 be fixed, and assume f E Co. Let = f (Xt ). For 0 5 s 5 t , let = E{f (Xt)lF,) = E{f (Xt)lXs) = Pt-sf (Xs). Then Y is a martingale for 0 I s I t , and it is c8.dl8.g, because Pt f is continuous, and X is chdlhg. Therefore, Y is a martingale with respect to IF+ too, which means that E{ f (Xt)lF,+) = Y,, from which we deduce that E{f (Xt)JFs+)= E{f (Xt)lXs), and we are done. This depends, of course, on how one defines a diffusion; the definition is not yet standardized in the literature.

36

I Preliminaries

We now define Pp as a probability measure such that X is Markov with respect to P p and also L(Xo) = p under P", where L(Xo) denotes the distribution, or "law," of X o Let IF" = (F[)OstIm be the filtration where Ff de= a{X, : s 5 t), and 9= a{X, : s 2 0). notes the P" completion of Also, let F~= n n ( l l p l l =F.:l )

e,e

Theorem 47. If X is Markov with respect to P", or alternatively Markov with respect to P" for all p with Ilpll = 1, then Pp and P are, respectively, right continuous. Proof. We prove the result for IF", from which the result for IF follows. Let 7-l denote all h E b F p such that

Then 7-l 3 { H = f (X,), f E bB), which is obvious for s 5 t , and for s > t it is just Theorem 46. Then 7-l contains all H of the form H = ny=l f,(XSi), for any f E bB. Using the Monotone Class Theorem, 7-l 3 b p . By completion, 7-l 3 b P . Therefore if h E F[+, we have E{hlF[) = E{hlF[+) = h, Ppas. The property that the natural filtration of a Feller process is right continuous when completed is not restricted to Feller processes. Indeed, it can also be shown to be true for more general classes of processes, such as Hunt processes6. However it is simply two elementary properties which yield the right continuity of the filtration in a general setting.

Definition. Let X be a c&dl&gMarkov process for a filtration IF, which we take to be completed and therefore can assume to be right continuous, in view of Theorem 47. Let T be a stopping time. We say that X verifies the strong Markov property at T if

We say that X is strong Markov if the above holds for every stopping time

T. Suppose now that the semigroup (Pt)is Borel. That is, if f E B, then Ptf E B as well. Suppose that for every probability measure p on (Rd,B), there exists a probability space (R,G, Q) and a chdlag Markov process (Yt)tlo defined on this space with L(Xo) = p. Suppose further that every such ciidlhg Markov process Y is strong Markov. Then the filtrations IFp = (F[)Ost<m and IF = (Ft)o
6 Local Martingales

37

6 Local Martingales Recall that we assume as given a filtered probability space (0,F,IF, P) satisfying the usual hypotheses. For a process X and a stopping time T we further recall that xTdenotes the stopped process

Definition. An adapted, chdlhg process X is a local martingale if there Tn = ca a s . exists a sequence of increasing stopping times, Tn, with limn,, such that XtATnl{Tn>Ol is a uniformly integrable martingale for each n. Such a sequence (T,) of stopping times is called a fundamental sequence. Example. Clearly any chdlhg martingale is a local martingale (take Tn = n). Example. We give an example of a local martingale which is not a martingale. Let (Bt)05t<, be a Brownian motion in R3 with Bo = x, where x # 0. Let u(y) = IIyII-', a superharmonic function on R3.As a consequence of It6's formula (cf., Theorems 32 and 41 of Chap. 11) one can show that Xt = u(Bt) is a positive supermartingale (indeed, it is even a uniformly integrable supermartingale). Next let Tn = inf{t > 0 : 11 Bt 11 5 l l n ) . Outside of the ball of radius l l n centered at the origin the function u is harmonic. Again by It6's formula one can show that this implies u(BtATn)is a martingale. Since u(BtATn)is bounded by n it is uniformly integrable. On the other hand, if Bo = x # 0 then limt,, E{u(Bt)) = 0 as is easily seen by a calculation, while E{u(Bo)) = 11~11-~. Since the expectations of a martingale are constant, u(Bt) is not a martingale. (For more on this example, see Sect. 6 of Chap. 11, following Corollary 4 of Theorem 27, as well as Exercise 20 in Chap. 11.) The reason we multiply X t A ~ ,by l{Tn>Ol is t o relax the integrability condition on Xo. This is useful, for example, in the consideration of stochastic integral equations with a non-integrable initial condition. Definition. A stopping time T reduces a process M if M~ is a uniformly integrable martingale. Theorem 48. Let M , N be local martingales and let S and T be stopping times. (a) (b) (c) (d) (e)

If T reduces M and S 5 T a.s., then S reduces M . The sum M + N is also a local martingale. If S, T both reduce M , then S V T also reduces M . The processes M ~ ~, ~ 1 are{ local ~ martingales. , ~ ~ Let X be a cadldg process and let Tn be a sequence of stopping times increasing to ca a.s. such that ~ ~ ~is a local l martingale { ~ for ~ each , n. Then X is a local martingale.

Proof. (a) follows from the Optional Sampling Theorem (Theorem 16) and Theorem 13. For (b), if (Sn),>l, (Tn)n21 are fundamental sequences for M

~

~

38

I Preliminaries

+

and N , respectively, then SnA Tn is a fundamental sequence for M N . For (c), let Xt = Mt - Mo. Then xSVT = xS xT- XSAT is a uniformly integrable martingale. But

+

+

which is in L1, therefore xSVT M O{SVT>Ol ~ =~ ~ is a uniformly~ integrable martingale. The proof of (d) consists of the observation that if (Tn)n21 is a fundamental sequence for M , then it is also one for M T and ~ ~ 1 {by~the , Optional ~ ) Sampling Theorem (Theorem 16). For (e), we have XTn 1{Tn,ol = M n is a local martingale for each n. For fixed n we know there exists a fundamental sequence Unlk increasing to ca a s . as k tends to ca. For each n, choose k = k ( n ) such that P(U"~~(")< Tn n ) < 2-". Then limn u ~ > = ~ ca ( ~as.,) and Unyk(") A Tn reduces X for each n. We take Rm = m a x ( ~ l > ~A( TI,. l ) . . ,u ~ ~ A T,), ~ and ( ~each) Rm reduces X by (c), the Rm are increasing, and lim Rm = ca a s . Therefore X is a local martingale. I7

~

1

~

Corollary. Local martingales form a vector space. We will often need to know that a reduced local martingale, M T , is in LP and not simply uniformly integrable. Definition. Let X be a stochastic process. A property .rr is said to hold locally if there exists a sequence of stopping times (Tn),>l increasing to ca a s . such that ~ ~ ~has property l { .rr, ~ each n~ 1. , ~ ~

>

We see from Theorem 48(e) that a process which is locally a local martingale is also a local martingale. Other examples of local properties that arise frequently are locally bounded and locally square integrable. Theorems 49 and 50 are consequences of Theorem 48(e). Theorem 49. Let X be a process which is locally a square integrable martingale. T h e n X is a local martingale. The next theorem shows that the traditional "uniform integrability" assumption in the definition of local martingale is not really necessary. Theorem 50. Let M be adapted, cbdlag and let (Tn)n>l be a sequence of ~i s a martingale ~ for each l stopping times increasing to ca a.s. If ~ n, then M is a local martingale. It is often of interest to determine when a local martingale is actually a martingale. A simple condition involves the maximal function. Recall that X,* =supslt IXsI and X* =sup, IXsI. Theorem 51. Let X be a local martingale such that E{X,*)

< ca for every

t 2 0. T h e n X i s a martingale. If E{X*) < ca, then X is a uniformly integrable martingale.

~

~

~

,

7 Stieltjes Integration and Change of Variables

39

Proof. Let (Tn)n21 be a fundamental sequence of stopping times for X. If s 5 t, then E{Xt~TnIFs)= XsA*,. The Dominated Convergence Theorem yields E{XtlFs) = X,. If E{X*) < oo,since each IXtI 5 X*, it follows that 17 (Xt)tLois uniformly integrable. Note that in particular a bounded local martingale is a uniformly integrable martingale. Other sufficient conditions for a local martingale to be a martingale are given in Corollaries 3 and 4 to Theorem 27 in Chap. 11, and Kazamaki's and Novikov's criteria (Theorems 40 and 41 of Chap. 111) establish the result for the important special cases of continuous local martingales.

7 Stieltjes Integration and Change of Variables Stochastic integration with respect to semimartingales can be thought of as an extension of path-by-path Stieltjes integration. We present here the essential elementary ideas of Stieltjes integration appropriate to our interests. We assume the reader is familiar with the Lebesgue theory of measure and integration on R+.

Definition. Let A = (At)t>o be a ckdlhg process. A is an increasing process if the paths of A : t 5At(w) are non-decreasing for almost all w. A is called a finite variation process (FV) if almost all of the paths of A are of finite variation on each compact interval of R+. Let A be an increasing process. Fix an w such that t H At(w) is right continuous and non-decreasing. This function induces a measure pA(w,ds) on R+. If f is a bounded, Bore1 function on R+, then f ( s ) p ~ ( w , d s )is well-defined for each t > 0. We denote this integral by f(s)dA,(w). If Fs = F ( s , w) is bounded and jointly measurable, we can define, w-by-w, the integral I(t,w) = F(s,w)dAs(w). I is right continuous in t and jointly measurable. Proceeding analogously for A an F V process (except that the induced measure pA(w,ds) can have negative measure; that is, it is a signed measure), we can define a jointly measurable integral

Jot

Jot

for F bounded and jointly measurable. Let A be an F V process. We define

Then lAlt < co as., and it is an increasing process.

40

I Preliminaries

Definition. For A an F V process, the t o t a l variation process, IAl = ((Alt)t20,is the increasing process defined in (*) above. Notation. Let A be an F V process and let F be a jointly measurable process such that F ( s , w)dA,(w) exists and is finite for all t > 0, a s . We let

and we write F . A t o denote the process F . A = ( F . At)t>0. - We will also : F, ldA, for ( F . The next result is an absolute continuity result write J for Stieltjes integrals.

I

T h e o r e m 52. Let A, C be adapted, strictly increasing processes such that C - A is also an increasing process. Then there exists a jointly measurable, adapted process H (defined on (0, co)) such that 0 5 H 5 1 and

or equivalently

Proof. If p and v are two Borel measures on R+ with p << v, then we can set =

{p,

if v((s1 tl) > 0, otherwise.

Defining h and k by h(t) = liminfstt a ( s , t ) , k(t) = limsuptl, a ( s , t), then h and k are both Borel measurable, and moreover they are each versions of the Radon-Nikodym derivative. That is,

To complete the proof it suffices t o show that we can follow the above procedure in a (t,w) measurable fashion. With the convention = 0, it suffices t o define H ( t , w) = liminf ( 4 4 w) - a r t 1 4). ~ t i , ~ E (C(t, a + W)- C(rt,w)) ' such an H is clearly adapted since both A and C are.

4

Corollary. Let A be an F V process. There exists a jointly measurable, adapted process H , -1 5 H 1, such that

<

IAI=H.A

and

A=H.IAI

l A l t = j HsdAs

and

At=loHsdA,l.

or equivalently

0

7 Stieltjes Integration and Change of Variables

41

Proof. w e define A$ = i(IAlt+At)and A, = $(IAlt-At). Then A+ and Aare both increasing processes, and IAl - A+ and IAl - A- are also increasing processes. By Theorem 52 there exist processes H + and H - such that A t = J ; H:dAsI, A; = fH ; d A s I . It then follows that At = A: -A; = H;)ldA,I. Let Ht H: - H c and suppose H + and H - are defined as in the proof of Theorem 52. Except for a P-null set, for a given w it is clear that IHs(w)1 = 1 d A s ( w ) almost all s. Considering H . A, we have

-

G(H;

This completes the proof.

17

When the integrand process H has continuous paths, the Stieltjes integral J : H s d A s is also known as the Riemann-Stieltjes integral (for fixed w ) . In this case we can define the integral as the limit of approximating sums. Such a result is proved in elementary textbooks on real analysis (e.g., ProtterMorrey [195, pages 316, 3171).

Theorem 53. Let A be a n F V process and let H be a jointly measurable process such that a.s. s H H ( s , w ) is continuous. Let xn be a sequence of finite random partitions of [0,t ] with limn,, mesh(xn) = 0. Then for T k I Sk I Tk+l,

We next prove a change of variables formula when the F V process is continuous. ItB's formula (Theorem 32 of Chap. 11) is a generalization of this result.

Theorem 54 (Change of Variables). Let A be an F V process with continuous paths, and let f be such that its derivative f' exists and is continuous. T h e n ( f (At))t2ois a n F V process and

Proof. For fixed w , the function s ++f 1 ( A S ( w ) )is continuous on [ O , t ] and f 1 ( A S ) d A sexists. Fix t and let xn hence bounded. Therefore the integral mesh(xn) = 0. Then be a sequence of partitions of [0,t ] with limn,,

Sot

42

I Preliminaries

by the Mean Value Theorem, for some Sk,tk 5 Sk 5 tk+l. The result now n follows by taking limits and Theorem 53. Comment. We will see in Chap. I1 that the sums

converge in probability to rt

for a continuous function f and an F V process A. This leads to the more general change of variables formula, valid for any F V process A, and f E C1, namely

The next corollary explains why Theorem 54 is known as a change of variables formula. The proof is an immediate application of Theorem 54. Corollary. Let g be continuous, and let h(t) = J : g(u)du. Let A be an F V process with continuous paths. Then

We conclude this section with an example. Example. Let N be a Poisson process with parameter A. Then Mt = Nt At, the compensated Poisson process, is a martingale, as well as an F V process. For a bounded (say), jointly measurable process H, we have

where (Tn)n>l are the arrival times of the Poisson process N. Now suppose the process H is bounded, adapted, and has continuous sample paths. For 0 5 s < t < oo,we then have

8 Nai've Stochastic Integration Is Impossible

43

The interchange of limits can be justified by the Dominated Convergence Theorem. We conclude that the integral process I is a martingale. This fact, that the stochastic Stieltjes integral of an adapted, bounded, continuous process with respect to a martingale is again a martingale, is true in much greater generality. We shall treat this systematically in Chap. 11.

8 NaYve Stochastic Integration Is Impossible In Sect. 7 we saw that for an F V process A, and a continuous integrand H s d A s as the limit of sums (Theprocess H , we could express the integral orem 53). The Brownian motion process, B, however, has paths of infinite variation on compacts. In this section we demonstrate, with the aid of the Banach-Steinhaus Theorem7, some of the difficulties that are inherent in trying to extend the notion of Stieltjes integration to processes that have paths of unbounded variation, such as Brownian motion. For the reader's convenience we recall the Banach-Steinhaus Theorem.

Sot

Theorem 55. Let X be a Banach space and let Y be a normed linear space. Let {T,} be a family of bounded linear operators from X into Y . If for each x E X the set {Tax) is bounded, then the set {I(TaI()is bounded. Let us put aside stochastic processes for the moment. Let x(t) be a right continuous function on [ O , l ] , and let .rr, be a refining sequence of dyadic mesh(xn) = 0. We ask the question rational partitions of [ O , l ] with limn,, "What conditions on x are needed so that the sums

converge to a finite limit as n + ca for all continuous functions h?" From Theorem 53 of Sect. 7 we know that x of finite variation is sufficient. However, it is also necessary.

Theorem 56. If the sums Sn of (*) converge to a limit for every continuous function h then x is of finite variation. The Banach-Steinhaus Theorem is also known as the Principle of Uniform Boundedness.

44

I Preliminaries

Proof. Let X be the Banach space of continuous functions equipped with the supremum norm. Let Y be R, equipped with absolute value as the norm. For h E X , let

For each fixed n it is simple to construct an h in X such that h(tk) = sign{x(tk+l) - x(tk)), and llhll = 1. For such an h we have

Therefore

each n, and sup, ]ITn11 2 total variation of x. On the other hand for each T,(h) exists and therefore sup, IITn(h)ll < co. The h E X we have limn,, Banach-Steinhaus Theorem then implies that sup, llTn 11 < co, hence the total variation of x is finite. Returning to stochastic processes, we might hope to circumvent the limitations imposed by Theorem 56 by appealing to convergence in probability. That is, if X(s, w) is right continuous (or even continuous), can we have the sums

converging to a limit in probability for every continuous process H? Unfortunately the answer is that X must still have paths of finite variation, a s . The reason is that one can make the procedure used in the proof of Theorem 56 measurable in w, and hence a subsequence of the sums in (**) can be made to converge a s . to +co on the set where X is not of finite variation. If this set has positive probability, the sums cannot converge in probability. The preceding discussion makes it appear impossible to develop a coherent notion of a stochastic integral HsdXs when X is a process with paths of infinite variation on compacts; for example a Brownian motion. Nevertheless this is precisely what we will do in Chap. 11.

Sot

Bibliographic Notes The basic definitions and notation presented here have become fundamental to the modern study of stochastic processes, and they can be found many places, such as Dellacherie-Meyer [45], Doob [55], and Jacod-Shiryaev [110]. Theorem 3 is true in much greater generality. For example the hitting time of a Bore1 set is a stopping time. This result is very difficult, and proofs can be found in Dellacherie [41] or [42].

Exercises for Chapter I

45

The ritsumit of martingale theory consists of standard theorems. The reader does not need results from martingale theory beyond what is presented here. Those proofs not given can be found in many places, for example Breiman [23], Dellacherie-Meyer [46], or Ethier-Kurtz [71]. The Poisson process and Brownian motion are the two most important stochastic processes for the theory of stochastic integration. Our treatment of the Poisson process follows Cinlar [33]. Theorem 25 is in Bremaud [24], and is due to J. de Sam Lazaro. The facts about Brownian motion needed for the theory of stochastic integration are the only ones presented here. A good source for more detailed information on Brownian motion is Revuz-Yor [208], or Hida [88]. Lkvy processes (processes with stationary and independent increments) are a crucial source of examples for the theory of semimartingales and stochastic integrals. Indeed in large part the theory is abstracted from the properties of these processes. There do not seem to be many presentations of L6vy processes concerned with their properties which are relevant to stochastic integration beyond that of Jacod-Shiryaev [110].However, one can consult the books of Bertoin [12], Rogers-Williams [210], and Revuz-Yor [208]. Our approach is inspired by Bretagnolle [25]. Local martingales were first proposed by K. It6 and S. Watanabe [I021 in order to generalize the Doob-Meyer decomposition. Standard Stieltjes integration applied to finite variation stochastic processes was not well known before the fundamental work of Meyer [171]. Finally, the idea of using the Banach-Steinhaus Theorem to show that na'ive stochastic integration is impossible is due to Meyer [178].

Exercises for Chapter I For all of these exercises, we assume as given a filtered probability space (0,F , IF, P ) satisfying the usual hypotheses.

Exercise 1. Let S, T be stopping times, S 5 T a s . Show Fs C FT. Exercise 2. Give an example where S, T are stopping times, S 5 T, but T - S is not a stopping time. Exercise 3. Let (Tn),?~ be a sequence of stopping times. Show that sup, T,, inf, T,, lim sup,,, T,, and lim inf,,, T, are all stopping times. Exercise 4. Suppose (Tn)n21 is a sequence of stopping times decreasing to a random time T. Show that T is a stopping time, and moreover that

FT =

n, FT,.

Exercise 5. Let p > 1 and let Mn be a sequence of continuous martingales (that is, martingales whose paths are continuous, a s . ) with ME E LP, each n. Suppose M z -+ X in LP and let Mt = E{XIFt}. (a) Show Mt E LP, all t

2 0.

46

Exercises for Chapter I

(b) Show M is a continuous martingale. Exercise 6. Let N = (Nt),>0 - be Poisson with intensity A. Suppose Xt is an

2(tii-;)1 . X t

integer. Show that E{INt - Xtl) =

-At

Exercise 7. Let N = (Nt),>0 be Poisson with intensity A. Show that N is continuous in L~ (and hence in probability), but of course N does not have continuous sample paths. Exercise 8. Let Bt = (B:, B:, Bt3) be three dimensional Brownian motion (that is, the Bi are i.i.d. one dimensional Brownian motions, 1 i 5 3). Let ra = sup, {ll Bt 11 a ) . The time ra is known as a last exit time. Show that if a > 0, then ra is not a stopping time for the natural filtration of B.

<

<

Exercise 9. Let (N&o be a sequence of i.i.d. Poisson processes with parameter X = l. Let M,i = +(N,i - t ) ,and let M = ( C E O M,i),>o. =, (a) Show that M is well-defined (i.e., that the series converges) in the L2 sense. *(b) Show that for any t > 0, C,,,- AM, = co as., where AM, = M, - M,is the jump of M at time s.

be

Exercise 10. Let (N:),L~ and two independent sequences of i.i.d. Poisson processes of parameters X = 7. Let M,i = + ( ~ ,-i L:). (a) Show that M is a martingale and that M changes only by jumps. *(b) Show that C,,, \AM,[ = cm as., and t > 0. Exercise 11. Show that a compound Poisson process is a Lkvy process. Exercise 12. Let Z be a compound Poisson process, with E{IUlI) Show that Zt - E{Ul)Xt is a martingale.

< co.

Exercise 13. Let Z = (Zt)t20 be a Lkvy process which is a martingale, but Z has no Brownian component (that is, the constant g2 = 0 in the L6vyKhintchine formula), and a finite Lkvy measure v. Let X = v(R). Show that Z is a compensated compound Poisson process with arrival intensity X and i.i.d. jumps with distribution p = i v . *Exercise 14. Let 0 < Tl < T2 < . . . < T, < . - . be any increasing sequence of finite-valued stopping times, increasing to co. Let Nt = Czl l{tLT,}, and let (Uz)i21be independent, and also the (Ui)i21 are independent of the process N . Show that if supiE{IUiI) < co, all i, and E{Ui) = 0, all i, then Zt = Czl Uzl{t2Ti}is a martingale. (Note: We are not assuming that N is a Poisson process here. Furthermore, the random variables (Ui)i21, while independent, are not necessarily identically distributed.) Exercise 15. Let Z be a Livy process which is a martingale, but with no Brownian component and with a Livy measure of the form

Exercises for Chapter I

where efl, (dx) denotes point mass at pk E Show that Z is of the form

]W of

size one, with

47

@ak< oo.

where ( N , ~-) ~is > oan independent sequence of Poisson processes with param- Verify that z is an L2 martingale. eters (ak)k>1.

Exercise 16. Let B = (Bt)o
> 0.

(a) Show that on [0,t] there are only a finite number of jumps larger than e. That is, if Nt = C,,,- l{lAxs12E), where AX, = X, - X,-, then P(Nt < oo) = 1, each t > 0. (b) Conclude that X can have only a countable number of jumps on [0,t].

Exercise 18. Let Z be a LQvyprocess, and let e > 0. Let AZ, = Z, - 2,- , and set J,E = CS
=0

that

Exercise 21. (This exercise shows that 2 is the best constant in Doob's quadratic martingale inequality.) Let (R, 3,P ) be Lebesgue measure on ( 0 , l ) with the Bore1 sets. Define the filtration

and 3 t = 3 for t > 1. Intuitively, with this filtration we know all about what is going on during (0, t], but nothing about what is happening during (t, 1). (a) Show that for Y E L1, the martingale Mt

= E{YIFt}

is given by

48

Exercises for Chapter I

(b) Let Y(w) = (1 - w ) - ~for some a , 0 < also if Mt = E{Y13t}, then

(c) Show sup, Mt =

Note that lim,,l/2(l

CY

< 112. Show that Y

E L2 and

AY, w-by-w, and deduce

-

a)-' = 2.

Exercise 22. (This exercise gives an example of a martingale with each path bounded but which nevertheless is not locally bounded.) Let (R, 3,IF, P ) be as in Exercise 21. Let Y(w) = 1 , and Mt = E{YIFt}. J;; (a) Show that

(b) Let T be a stopping time, T not identically 0. Show that T(w) 2 w on some w-interval (0, e ) with E > 0. $ds = co, and deduce that M cannot be (c) Use (b) to show E{M;) 2 locally an L2 martingale. (d) Conclude that M has bounded paths but that is unbounded for every stopping time T not identically 0.

*Exercise 23. Find an example of a local martingale M and finite stopping times S , T with S I T a.s. such that E { 1 M ~ l l 3 ~<) co, but Ms # E { M T ( 3 . ) a.s. (Hint: Begin by showing that the equality would imply that every positive local martingale is a martingale.) Exercise 24. (This exercise helps to clarify the difference between stopping time a-algebras of the two forms GT- and GT.) Let Z be a Lkvy process, and let Gt = a{Z,; s 5 t) V N , where N are all the null sets in the given complete probability space ( R , 3 , P). For a stopping time T define GT- = a{A n (t < T); A E Gt }; at time 0, set Go- = GO. For a process X , let and take Xo- = 0. xT-= Xt l t t < ~4-) XT(a) (b) (c) (d)

Show GT- = a{T, ZT-} V N . Show GT = a{T, ZT) V N. Show GT = a{GT-, ZT) if T is a.s. finite. Conclude that for a bounded time T if ZT = ZT- a.s., then GT

= GT-.

(Note that if G = (Gtlt>o is the completed natural filtration of a Brownian motion, then GT = GT- for all bounded stopping times.)

Exercises for Chapter I

49

Exercise 25. Let Z be a L6vy process (or any process which is continuous in probability). Show that the probability Z jumps at a given fixed time t is zero. Exercise 26. Use Exercises 24 and 25 to show that with the assumptions of Exercise 24, for any fixed time t, Gt- = Gt, giving a continuity property of the filtration for fixed times (since by Theorem 31 we always have Gt+ = Gt), but of course not for stopping times. Exercise 27. Let S , T be two stopping times with S 5 T. Show 3s- c FT-. Moreover if (Tn)n21is any increasing sequence of stopping times with limn,,

Tn = T, show that

Exercise 28. Prove the first equality of Theorem 38 when f l n E L1(dv). **Exercise 29. Show that if Z is a L6vy process and a local martingale, then Z is a martingale. That is, all Lkvy process local martingales are actually martingales. Exercise 30 (reflection principle for LQvyprocesses). Let Z be a symmetric LQvy process with Zo = 0. That is, Z and -2 have the same distribution. Let St = supslt Zs. Show that P(St 2 z; Zt

< z - y) 5 P(Zt > z + y).

Exercise 31. Let Z be a symmetric Lkvy process with Zo = 0 as in Exercise 30. Show that P(St 2 z) 1 2P(Zt 2 2).

Semimartingales and Stochastic Integrals

1 Introduction to Semimartingales The purpose of the theory of stochastic integration is to give a reasonable meaning to the idea of a differential to as wide a class of stochastic processes as possible. We saw in Sect. 8 of Chap. I that using Stieltjes integration on a path-by-path basis excludes such fundamental processes as Brownian motion, and martingales in general. Markov processes also in general have paths of unbounded variation and are similarly excluded. Therefore we must find an approach more general than Stieltjes integration. We will define stochastic integrals first as a limit of sums. A priori this seems hopeless, since even by restricting our integrands to continuous processes we saw as a consequence of Theorem 56 of Chap. I that the differential must be of finite variation on compacts. However an analysis of the proof of Theorem 56 offers some hope. In order to construct a function h such that h(tk) = sign(x(tk+l) - x(tk)), we need to be able to "see" the trajectory of x on (tk, tk+l]. The idea of K. It6 was to restrict the integrands to those that could not see into the future increments, namely adapted processes. The foregoing considerations lead us to define the stochastic processes that will serve as differentials as those that are "good integrators" on an appropriate class of adapted processes. We will, as discussed in Chap. I, assume that we are given a filtered, complete probability space (R, 3,IF, P) satisfying the usual hypotheses. Deflnition. A process H is said to be simple predictable if H has a represent at ion n

<

where 0 = TI . - . < Tn+l < oo is a finite sequence of stopping times, Hi E 3 ~with ; lHil < oo as., 0 < i n. The collection of simple predictable processes is denoted S.

<

52

I1 Semimartingales and Stochastic Integrals

Note that we can. take TI = To = 0 in the above definition, so there is no "gap" between To and TI. We can topologize S by uniform convergence in (t,w), and we denote S endowed with this topology by S,. We also write Lo for the space of finite-valued random variables topologized by convergence i n probability. Let X be a stochastic process. An operator, Ix, induced by X should have two fundamental properties to earn the name "integral." The operator Ix should be linear, and it should satisfy some version of the Bounded Convergence Theorem. A particularly weak form of the Bounded Convergence Theorem is that the uniform convergence of processes H n to H implies only the convergence in probability of I x ( H n ) to I x ( H ) . Inspired by the above considerations, for a given process X we define a linear mapping Ix : S + Lo by letting

where H E S has the representation

Since this definition is a path-by-path definition for the step functions H(w), it does not depend on the choice of the representation of H in S .

Definition. A process X is a total semimartingale if X is cAdlAg, adapted, and Ix : S , + LO is continuous. Recall that for a process X and a stopping time T, the notation X T denotes the process ( X ~ A T ) ~ > O -

Definition. A process X is called a semimartingale if, for each t E [0,co), X t is a total semimartingale. With our definition of a semimartingale, the second fundamental property we want (bounded convergence) holds. We postpone consideration of examples of semimartingales to Sect. 3.

2 Stability Properties of Semimartingales We state a sequence of theorems giving some of the stability results which are particularly simple.

Theorem 1. The set of (total) semimartingales is a vector space.

Proof. This is immediate from the definition.

2 Stability Properties of Semimartingales

53

Theorem 2. If Q is a probability and absolutely continuous with respect t o P , then every (total) P semimartingale X is a (total) Q semimartingale.

Proof. Convergence in P-probability implies convergence in Q-probability. Thus the theorem follows from the definition of X. Theorem 3. Let ( P k ) k 2 1 be a sequence of probabilities such that X is a Pk semimartingale for each k. Let R = CF=O=I XkPk, where XI, 2 0, each k, and CF=O=, XI, = 1. Then X is a semimartingale under R as well.

Proof. Suppose H n E S converges uniformly to H E S. Since X is a Pk semimartingale for all Pk,Ix(Hn) converges to I x ( H ) in probability for every Pk. This then implies Ix(Hn) converges to Ix(H) under R. Theorem 4 (Stricker's Theorem). Let X be a semimartingale for the filtration IF. Let G be a subfiltration of P, such that X is adapted t o the G filtration. Then X is a G semimartingale.

Proof. For a filtration W, let S(W) denote the simple predictable processes for the filtration W = ('FIt)t20.In this case we have S(G) is contained in S(P). The theorem is then an immediate consequence of the definition. Theorem 4 shows that we can always shrink a filtration and preserve the property of being a semimartingale (as long as the process X is still adapted), since we are shrinking as well the possible integrands; this, in effect, makes it "easier" for the process X to be a semimartingale. Expanding the filtration, therefore, should b e a n d is-a much more delicate issue. Expansion of filtrations is considered in much greater detail in Chap. VI. We present here an elementary but useful result. Recall that we are given a filtered space (R, 3 , P I P ) satisfying the usual hypotheses. Theorem 5 (Jacod's Countable Expansion). Let A be a wllection of events i n 3 such that i f A,, Ag E A then A, n Ap = 0, (a # p). Let 'Fit be the filtration generated by Ft and A. Then every (P, P) semimartingale is a n (W, P ) semimartingale also.

Proof. Let A, E A. If P(A,) = 0, then A, and A; are in To by hypothesis. We assume, therefore, that P(A,) > 0. Note that there can be at most a countable number of A, E A such that P(A,) > 0. If A = Un>l A, is the union of all A, E A with P(A,) > 0, we can also add AC to A without loss of generality. Thus we can assume that A is a countable partition of R with P(A,) > 0 for every A, E A. Define a new probability Q, by Q, (.) = P(.IA,), for A, fixed. Then Q, << PIand X is a (IF, Q,) semimartingale by Theorem 2. If we enlarge the filtration IF by all the 3 measurable events that have Q,probability 0 or 1, we get a larger filtration Jn = (,7p)t20, and X is a (Jn,Q,) ,,; semimartingale. Since &,(A,) = 0 or 1 for m # n, we have Ft c 'Fit c 7 for t 2 0, and for all n. By Stricker's Theorem (Theorem 4) we conclude that X is an (W, Q,) semimartingale. Finally, we have dP = P(A,)dQ,, -

En,,

54

I1 Semimaxtingales and Stochastic Integrals

where X is an (W, Q,) semimartingale for each n. Therefore by Theorem 3 we conclude X is an (W, P ) semimartingale. •

Corollary. Let A be a finite collection of events in 3,and let W = ('Flt)t>o be the filtration generated by Ft and A. Then every (IF,P) semimartingale is an (W, P) semimartingale also.

Proof. Since A is finite, one can always find a (finite) partition 17 of R such that 'Fit = Ft Tt IT. The corollary then follows by Theorem 5. Note that if B = (Bt)t20 is a Brownian motion for a filtration ( 3 t ) t 2 ~ , by Theorem 5 we are able to add, in a certain manner, an infinite number of "future" events to the filtration and B will no longer be a martingale, but it will stay a semimartingale. This has interesting implications in finance theory (the theory of continuous trading). See for example Duffie-Huang 1601. The corollary of the next theorem states that being a semimartingale is a "local" property; that is, a local semimartingale is a semimartingale. We get a stronger result by stopping at Tn- rather than at Tn in the next theorem. A process X is stopped at T- if = Xtl{Olt
xo-

xT-

+

Theorem 6. Let X be a cadlrig, adapted process. Let (T,) be a sequence of positive r.v. increasing t o ca a.s., and let (Xn) be a sequence of semimartingales such that for each n, xTn-= (Xn)Tn-. Then X is a semimartingale.

Proof. We wish to show X t is a total semimartingale, each t > 0. Define Rn = Tnl{TnIt) + m1{Tn>t) Then P{lIxt(H)I 2 c ) 5 P{lI(xn)t(H)1 2 c )

+ P(Rn < co).

But P ( R n < co) = P(Tn I t ) ,and since Tn increases to co a.s., P(Tn 5 t ) --t 0 as n --t co. Thus if H k tends to 0 in S,, given e > 0, we choose n so that P(Rn < co) < e/2, and then choose k so large that P{lI(xn)t(Hk)l 2 c ) < e/2. Thus, for k large enough, P { l I x t ( ~ k ) )2 J c ) < e.

Corollary. Let X be a process. If there exists a sequence (T,) of stopping ~ , ~ ) ) times increasing to co a.s., such that xTn (or ~ ~ n l ~ is~a semimartingale, each n , then X is also a semimartingale.

3 Elementary Examples of Semimartingales The elementary properties of semimartingales established in Sect. 2 will allow us to see that many common processes are semimartingales. For example, the Poisson process, Brownian motion, and more generally all L6vy processes are semimartingales.

3 Elementary Examples of Semimartingales

55

Theorem 7. Each adapted process with ccidlcig paths of finite variation o n compacts (of finite total variation) is a semimartingale (a total semimartingale).

Jr

(dXsI Proof. It suffices to observe that IIx(H)( 5 11 Hll, J ' ldXs I, where denotes the Lebesgue-Stieltjes total variation and ((HI(,= sup(,,,) (H(t,w ) ( .

Theorem 8. Each L2 martingale with cadldg paths is a semimartingale. Proof. Let X be an L~ martingale with Xo = 0, and let H E S. Using Doob's Optional Sampling Theorem and the L2 orthogonality of the increments of L~ martingales, it suffices to observe that

Corollary 1. Each cAdlAg, locally square integrable local martingale is a semimartingale. Proof. Apply Theorem 8 together with the corollary to Theorem 6.

Corollary 2. A local martingale with continuous paths is a semimartingale. Proof. Apply Corollary 1 together with Theorem 51 in Chap. I.

Corollary 3. The Wiener process (that is, Brownian motion) is a semimartingale. Proof. The Wiener process Bt is a martingale with continuous paths if Bo is integrable. It is always a continuous local martingale.

Deflnition. We will say an adapted process X with cadlag paths is decomposable if it can be decomposed Xt = Xo Mt At, where Mo = A. = 0, M is a locally square integrable martingale, and A is c&dl&g,adapted, with paths of finite variation on compacts.

+ +

Theorem 9. A decomposable process is a semimartingale.

+

+

Proof. Let Xt = Xo Mt At be a decomposition of X . Then M is a semimartingale by Corollary 1 of Theorem 8, and A is a semimartingale by Theorem 7. Since semimartingales form a vector space (Theorem 1) we have the result.

Corollary. A L6vy process is a semimartingale.

56

I1 Semimartingales and Stochastic Integrals

Proof. By Theorem 40 of Chap. I we know that a Lkvy process is decomposable. Theorem 9 then gives the result.

Since L6vy processes are prototypic strong Markov processes, one may well wonder if all Rn-valued strong Markov processes are semimartingales. Simple examples, such as Xt = B:'~, where B is standard Brownian motion, show this is not the case (while this example is simple, the proof that X is not a semimartingale is not elementary1). However if one is willing to "regularize" the Markov process by a transformation of the space (in the case of this example using the "scale function" S(x) = x3), "most reasonable" strong Markov processes are semimartingales. Indeed, Dynkin's formula, which states that if f is in the domain of the infinitesimal generator G of the strong Markov process Z, then the process

is well-defined and is a local martingale, hints strongly that if the domain of G is rich enough, the process Z is a semimartingale. In this regard see Sect. 7 of Cinlar, Jacod, Protter, and Sharpe [34].

4 Stochastic Integrals In Sect. 1 we defined semimartingales as adapted, cAd1A.g processes that acted as "good integrators" on the simple predictable processes. We now wish to enlarge the space of processes we can consider as integrands. In Chap. IV we will consider a large class of processes, namely those that are 'Lpredictably measurable" and have appropriate finiteness properties. Here, however, by keeping our space of integrands small-yet large enough to be interestingwe can keep the theory free of technical problems, as well as intuitive. A particularly nice class of processes for our purposes is the class of adapted processes with left continuous paths that have right limits (the French acronym would be cAg1A.d). Definition. We let D denote the space of adapted processes with c&dl&g paths, lL denote the space of adapted processes with c&gl&dpaths (left continuous with right limits) and blL denote processes in lL with bounded paths. We have previously considered S,, the space of simple, predictable processes endowed with the topology of uniform convergence; and Lo, the space of finite-valued random variables topologized by convergence in probability. We need to consider a third type of convergence.

' See Theorem 71 of Chap. IV which proves a similax assertion for Xt 0 < cr

< 1/2.

=

IBtla,

4 Stochastic Integrals

57

Definition. A sequence of processes (Hn)n>l converges to a process H uniformly on compacts in probability (abbreviated ucp) if, for each t > 0, supojsit ( H r - Hs( converges to 0 in probability. IHsl. Then if Yn E D we have Yn + Y in ucp We write H,* = if (Yn - Y): converges t o 0 in probability for each t > 0. We write IDucp, ,,LI S,, to denote the respective spaces endowed with the ucp topology. We observe that D, is a metrizable space; indeed, a compatible metric is given, for X , Y E D, by

The above metric space D,,, is complete. For a semimartingale X and a process H E S , we have defined the appropriate notion Ix ( H ) of a stochastic integral. The next result is key to extending this definition.

Theorem 10. The space S is dense in IL under the ucp topology.

> n } . Then Rn is a stopping time and Proof. Let Y E IL. Let Rn = inf{t : Yn = YRnl{Rn>O)are in blL and converge to Y in ucp. Thus bIL is dense in IL. Without loss we now assume Y E bIL. Define Z by Zt = limu,t Y,. Then u>t Z E D. For E > 0, define T,E = 0 T:+, = i n f i t : t > T,E and IZt - ZT21 > E } . Since Z is c&dl&g, the T: are stopping times increasing to NI a s . as n increases. Let ZE = EnZT;l[T;,TE+l), for each E > 0. Then ZE are bounded and converge uniformly to Z as E tends to 0. Let UE = Yol{o) EnZ ~ ; l ( ~ ; , ~ ; + , p Then U EE bIL and the preceding implies UE converges uniformly on compacts to Yol{o)+ Z- = Y. Finally, define

+

and this can be made arbitrarily close to Y E bIL by taking and n large enough.

E

small enough !El

We defined a semimartingale X as a process that induced a continuous operator Ix from S, into Lo. This Ix maps processes into random variables (note that XTi is a random variable). We next define an operator (the stochastic integral operator), induced by X , that will map processes into processes (note that xT'is a process).

58

I1 Semimartingales and Stochastic Integrals

Definition. For H E S and X a chdlhg process, define the (linear) mapping Jx:S+Dby

for H in S with the representation

Hi E Fri and 0 = To I T i I . . . I T,+l < co stopping times. Definition. For H E S and X an adapted cadlhg process, we call J x ( H ) the stochastic integral of H with respect to X .

We use interchangeably three notations for the stochastic integral:

Observe that J x ( H ) , = I x t ( H ) . Indeed, I x plays the role of a definite inteH,dX,. gral. For H E S, I x ( H ) = Theorem 11. Let X be a semimartingale. Then the mapping J x D,,, is continuous.

:

S,,

+

Proof. Since we are only dealing with convergence on compact sets, without loss of generality we take X to be a total semimartingale. First suppose H k in S tends to 0 uniformly and is uniformly bounded. We will show J x ( H k ) tends to 0 ucp. Let 6 > 0 be given and define stopping times T~ by

Then ~ ~ 1 [E S~ and , ~tends k to ~ 0 uniformly as k tends to co.Thus for every t,

which tends to 0 by the definition of total semimartingale. is continuous. We now use this We have just shown that J x : S, + ID, is continuous. Suppose H~ goes to 0 ucp. Let to show Jx : S,,, --+ ID, 6 > 0, E > 0, t > 0. We have seen that there exists q such that IIHllu_< q implies P(Jx(H)F > 6 ) < 5. Let Rk = inf{s : I H ~ I > q), and set H~ = ~ ~ l ~ ~ ,Then ~ ~ fik ~E S1 and ~ 1lfikll,, ~ ~ I, q~ by) left. continuity. Since R~ 2 t implies ( H k .X ) : = ( H k . X ) : , we have

4 Stochastic Integrals

p ( ( H k .X); > 6) L: p ( ( H k .X); > 6)

59

+ P ( R ~< t)

P((Hk)f > q) = 0.

if k is large enough, since limk,,

We have seen that when X is a semimartingale, the integration operator Jx is continuous on Sucp,and also that S,, is dense in IL,. Hence we are able to extend the linear integration operator Jx from S to IL by continuity, since D, is a complete metric space.

Definition. Let X be a semimartingale. The continuous linear mapping Jx : IL, -+ D, obtained as the extension of Jx : S -+ D is called the stochastic integral. The preceding definition is rich enough for us to give an immediate example of a surprising stochastic integral. First recall that if a process (At)t>o has continuous paths of finite variation with A. = 0, then the ~ i e m a n n - ~ c e l t j e s A,dA, yields the formula (see Theorem 54 of Chap. I) integral of

&

Let us now consider a standard Brownian motion B = (Bt)t>owith Bo = 0. The process B does not have paths of finite variation on compacts, but it is a semimartingale. Let (n,) be a refining sequence of partitions of [0,co)with Btkl ~ t k , t kThen + l ~Bn . E IL for each limn,, mesh(nn) = 0. Let BT = CtkEan n. Moreover, B n converges to B in ucp. Fix t 2 0 and assume that t is a partition point of each n,. Then

and JB(B)t = lim J B ( B " ) ~= lim n-00

n

C

Btk (Btk+l - Btk) tkE.rr,,tk
where we have used telescoping sums. The last term on the right, however, converges a.s. to t, by Theorem 28 of Chap. I. We therefore conclude that

60

I1 Semimartingales and Stochastic Integrals

a formula distinctly different from the Riemann-Stieltjes formula (*). The change of variables formula (Theorem 32) presented in Sect. 7, will give a deeper understanding of the difference in formulas (*) and (**).

5 Properties of Stochastic Integrals Throughout this paragraph X will denote a semimartingale and H will denote an element of IL. Recall that the stochastic integral defined in Sect. 4 will be denoted by the three notations Jx (H) = H . X = J H,dX,. Evaluating these processes at t , we have

To exclude 0 in the integral we write

2

H,dX, when the limit The integral SowHSdX, is defined to be limt,, H,dX, = HoXo H,dX,. For a process Y t ID, exists. Note that we recall that AYt = Yt - &-, the jump at t. Also since Yo- = 0 we have AYo = Yo. Recall further that for a process Z and stopping time T , we let ZT denote the stopped process (Z?) = We will establish in this section several elementary properties of the stochastic integral. These properties will help us to understand the integral and to examine examples at the end of the section. Recall that two processes Y and Z are indistinguishable if P{w : t H Xt(w) and t H &(w) are the same functions) = 1.

Sot

+ g+

Theorem 12. Let T be a stopping time. Then ( H . X)T = H~[,,T]- X H . (xT).

=

Theorem 13. The jump process A(H.X), is indistinguishablefrom H, (AX,). Proofs. Both properties are clear when H t S, and they follow when H E IL by passing to the limit. Indeed, we take convergence in ucp for each t , then a s . convergence by taking a subsequence, and then we choose the rationals (which of course is countable and dense, so the union of null sets is still a null set), and then use the path regularity to get indistinguishability. Let Q denote another probability law, and let HQ . X denote the stochastic integral of H with respect to X computed under the law Q.

Theorem 14. Let Q << P. Then HQ . X is Q indistinguishable from H p . X .

5 Properties of Stochastic Integrals

61

Proof. Note that by Theorem 2, X is a Q semimartingale. The theorem is clear if H E S, and it follows for H E IL by passage to the limit in the ucp topology, since convergence in P-probability implies convergence in Q-probability.

Theorem 15. Let Pk be a sequence of probabilities such that X is a Pk semimartingale for each k . Let R = Cr=O=, X k P k where X k 2 0, each k , and CF"=,k= 1. Then HR . X = Hpk . X , Pk-a.s., for all k such that Xk > 0. Proof. If XI, > 0 then P k << R, and the result follows by Theorem 14. Note that by Theorem 3 we know that X is an R semimartingale.

Corollary. Let P and Q be any probabilities and suppose X is a semimartingale relative to both P and Q. Then there exists a process H e X which is a version of both Hp . X and HQ . X . Proof. Let R

=

(P+Q)/2. Then H R . X is such a process by Theorem 15.

Theorem 16. Let (6 = (Gt)t>o be another filtration such that H is i n both L((6) and IL(P), and such thatX is also a (6 semimartingale. Then H G .X = HF . X . Proof. IL((6) denotes left continuous processes adapted to the filtration (6. As in the proof of Theorem 10, we can construct a sequence of processes H n converging to H where the construction of the H n depends only on H. Thus H n E S((6) n S(P) and converges to H in ucp. Since the result is clear for S, the full result follows by passing to the limit.

Remark. While Theorem 16 is a simple result in this context, it is far from simple if the integrands are predictably measurable processes, rather than processes in IL. See the comment following Theorem 33 of Chap. IV. The next two theorems are especially interesting because they show-at least for integrands in IL-that the stochastic integral agrees with the pathby-path Lebesgue-Stieltjes integral, whenever it is possible to do so.

Theorem 17. If the semimartingale X has paths of finite variation o n compacts, then H . X is indistinguishable from the Lebesgue-Stieltjes integral, computed path-by-path. Proof. The result is evident for H E S. Let H n E S converge to H in ucp. Then there exists a subsequence n,+ such that limn,,,(Hnk - H): = 0 as., and the result follows by interchanging limits, justified by the uniform a.s. convergence.

Theorem 18. Let X , 5? be two semzmartzngales, and let H , 1? E L. Let A = {w : H. (w) = H . (w) and X . (w) = x . ( w ) ) , and-let-B = {w : t H Xt(w) is of finite variation o n compacts). Then H . X = H . X o n A, and H . X is equal to a path-by-path Lebesgue-Stieltjes integral o n B.

62

I1 Semimartingales and Stochastic Integrals

Proof. Without loss of generality we assume P ( A ) > 0. Define a new probability law Q by Q ( A ) = P(A1A). Then under Q we have that H and H as well as X-and - X are indistinguishable. Thus HQ . X = HQ . X,and hence H . X = H . X P - a s . on A by Theorem 14, since Q << P. As for the second assertion, if B = 0 the result is merely Theorem 17. Define R by R ( A ) = P ( A l B ) , assuming without loss that P ( B ) > 0. Then R << P and B = 0 , R-a.s. Hence H E . X equals the Lebesgue-Stieljes integral R-as. by Theorem 17, and the result follows by Theorem 14. The preceding theorem and following corollary are known as the local behavior of the integral.

Corollary. With the notation of Theorem 18, let S , T be two stopping times with S < T . Define C D

= =

{W : Ht(w) = H t ( w ) ;Xt(w) = X t ( w ) ;S ( W )< t 5 T ( w ) ) { w : t H Xt(w) is of finite variation on S ( w ) < t < T ( w ) ) .

T ~ ~ ~ H . X ~ - H . X ~ = H . X ~ - H . X ~ ~ ~ C ~ ~ ~ H . X T - H . X ~ ~ ~ ~ ~ ~ ~ a path-by-path Lebesgue-Stieltjes integral on D. = Xt - XtAs. Then H . Y = H . X - H . xS,and Y does not Proof. Let be viewed change the set [O, S ] ,which is evident, or which-alternatively-can as an easy consequence of Theorem 18. One now applies Theorem 18 to YT to obtain the result.

Theorem 19 (Associativity). The stochastic integral process Y = H . X is itself a semimartingale, and for G E IL we have

Proof. Suppose we know Y = H - X is a semimartingale. Then G . Y = J y ( G ) . If G , H are in S, then it is clear that J y ( G ) = J x ( G H ) . The associativity then extends to IL by continuity. It remains to show that Y = H . X is a semimartingale. Let ( H n ) be in S converging in ucp to H . Then Hn . X converges t o H . X in ucp. Thus there exists a subsequence ( n k )such that Hnk - X converges a.s. to H . X . Let G E S and let Y n k = Hnk . X , Y = H . X . The Y n kare semimartingales converging pointwise to the process Y . For G E S, J y ( G ) is defined for any process Y ; so we have J y ( G ) = G . Y = lim G . Y n k= lim G . ( H n k . X ) n k

-'a

nk-',

= lim

( G H n k ). X

n k -00

which equals limn,,, Jx (GHnk) = Jx ( G H ) , since X is a semimartingale. Therefore J y ( G ) = JX ( G H ) for G E S.

5 Properties of Stochastic Integrals

63

Let Gn converge to G in S,. Then G n H converges to G H in IL,,, and since X is a semimartingale, limn,, Jy(Gn) = limn+, J x ( G n H ) = J x ( G H ) = Jy(G). This implies Yt is a total semimartingale, and so Y = H - X is a semimartingale. Theorem 19 shows that the property of being a semimartingale is preserved by stochastic integration . Also by Theorem 17 if the semimartingale X is an F V process, then the stochastic integral agrees with the Lebesgue-Stieltjes integral, and by the theory of Lebesgue-Stieltjes integration we are able to conclude the stochastic integral is an FV process also. That is, the property of being a n F V process is preserved by stochastic integration for integrands i n IL.2 One may well ask if other properties are preserved by stochastic integration; in particular, are the stochastic integrals of martingales and local martingales still martingales and local martingales? Local martingales are indeed preserved by stochastic integration, but we are not yet able easily to prove it. Instead we show that locally square integrable local martingales are preserved by stochastic integration for integrands i n IL.

Theorem 20. Let X be a locally square integrable local martingale, and let H E IL. T h e n the stochastic integral H . X is also a locally square integrable local martingale. Proof. We have seen that a locally square integrable local martingale is a semimartingale (Corollary 1 of Theorem B ) , so we can formulate H .X . Without loss of generality, assume Xo = 0. Also, if T~ increases to co a.s. and (H. x ) ~is a~ locally square integrable local martingale for each k, it is simple to check that H . X itself is one. Thus without loss we assume X is a square integrable martingale. By stopping H, we may further assume H is bounded, by C. Let H n E S be such that H n converges to H in ucp. We can-then modify H n , call it H ~ such , that fin is bounded by e, fin E S , and H n converges uniformly to H in probability on [0,t ] .Since fin E bS, one can check that fin . X is a martingale. Moreover

and hence ( H n . X)t are uniformly bounded in L~ and thus uniformly integrable. Passing to the limit then shows both that H . X is a martingale and that it is square integrable. See Exercise 43 in Chap. IV which shows this is not true in general.

64

11 Semimartingales and Stochastic Integrals

In Theorem 29 of Chap. I11 we show the more general result that if M is a local martingale and H E IL, then H . M is again a local martingale. A classical result from the theory of Lebesgue measure and integration (on R) is that a bounded, measurable function f mapping an interval [a,b] to R is Riemann integrable if and only if the set of discontinuities of f has Lebesgue measure zero (e.g., Kingman and Taylor [127, page 1291). Therefore we cannot hope to express the stochastic integral as a limit of sums unless the integrands have reasonably smooth sample paths. The spaces D and IL consist of processes which jump at most countably often. As we will see in Theorem 21, this is smooth enough.

Definition. Let a denote a finite sequence of finite stopping times:

The sequence a is called a random partition. A sequence of random partitions a,, . . . TLn a, : T," 5 T;"

<

<

is said to tend to the identity if (i) limn supkT; = m as.; and (ii) llanll = supk IT;+1 - TLI converges to 0 a.s. Let Y be a process and let a be a random partition. We define the process Y sampled at a to be

It is easy to check that

for any semimartingale X , any process Y in S, D, or IL.

Theorem 21. Let X be a semimartingale, and let Y be a process i n D or i n IL. Let (a,) be a sequence of random partitions tending to the identity. Then the processes + Yu-dXs = YTF( x F 1 - X Y ) tend to the stochastic integral S (Y-) . X i n ucp.

Ji

zi

Proof. (The notation Y- means the process whose value at s is given by (Y-), = lim,,,, ,<, Y,; also, (Y-)O = 0, by convention.) We prove the theorem for the case where Y is c&dl&g,the other case being analogous. Also without loss of generality we can take Xo = 0. Y c&dl&gimplies Y- E IL.Let yk E S such that yk converges to Y- (ucp). We have

5 Properties of Stochastic Integrals

65

where !Y denotes the c&dl&gversion of Yk. The first term on the right side and since Y- -Yk + 0, equals Jx (Y- -Yk), and since Jx is continuous in,L,I we have J(Y- - y k ) , d x , + 0 (ucp).The same reasoning applies to the third term, for fixed n, as k + co. Indeed, the convergence to 0 of ( ( Y + ~ )-o Yon) ~ as k -+ co is uniform in n. It remains to consider the middle term on the right side above. Since the Yk are simple predictable we can write the stochastic integrals in closed form, and since X is right continuous the integrals (for fixed (k, w ) ) J(Yk - (~!)"n),dx, tend to 0 as n + co. Thus one merely chooses k so large that the first and third terms are small, and then for fixed k, the middle term can be made small for large enough n. We consider here another example. We have already seen at the end of Sect. 4 that if B is a standard Wiener process, then

showing that the semimartingale calculus does not formally generalize the Riemann-Stieltjes calculus. We have seen as well that the stochastic integral agrees with the LebesgueStieltjes integral when possible (Theorems 17 and 18) and that the stochastic integral also preserves the martingale property (Theorem 20; at least for locally square integrable local martingales). The following example shows that the restriction of integrands to IL is not as innocent as it may seem if we want to have both of these properties. Example. Let Mt = Nt - At, a compensated Poisson process (and hence a martingale with Mt E L P for all t 2 0 and all p 2 1).Let (Ti)i21 be the jump )(t).Then H E D. The Lebesgue-Stieltjes integral times of M. Let Ht = l[O,Tl is

66

I1 Semimartingales and Stochastic Integrals

This process is not a martingale. We conclude that the space of integrands cannot be expanded even to ID,in general, and preserve the structure of the theory already e~tablished.~

6 The Quadratic Variation of a Semimartingale The quadratic variation process of a semimartingale, also known as the bracket process, is a simple object that nevertheless plays a fundamental role.

Definition. Let X , Y be semimartingales. The quadratic variation pro- is defined by cess of X , denoted [X, XI = ([X,X]t)t>o, [X,X ] = x2- 2

I

X-dX

(recall that Xo- = 0). The quadratic covariation of X , Y, also called the bracket process of X , Y, is defined by [X,Y] = X Y -

S

X-dY -

J

Y-dX.

It is clear that the operation (X, Y) + [X, Y] is bilinear and symmetric. We therefore have a polarization identity 1 [X,Y] = I ( [ X + Y , X + Y ] - [ X , X ] - [Y,Y]).

The next theorem gives some elementary properties of [X, XI. ( X is assumed to be a given semimartingale throughout this section).

Theorem 22. The quadratic variation process of X is a cadlag, increasing, adapted process. Moreover i t satisfies the following.

( i ) [X, XIo = X$ and A[X, X] = AX)^. (ii) If a, is a sequence of random partitions tending to the identity, then

x; + C ( x T G 1 xTq2+ [X, X ] -

with convergence i n ucp, where a, is the sequence 0 = Ton < TT < . . . 5 T: < . . . 5 TL and where T," are stopping times. (iii) If T is any stopping time, then [ x T ,X ] = [X, XT] = [XT,xT]= [X, XIT. Ruth Williams has commented to us that this example would be more convincing if M were itself a semimartingale, H bounded and in Dl and H,dM, were not a semimartingale. Such a construction is carried out in [I].

Ji

6 The Quadratic Variation of a Semimartingale

67

Proof. X is chdliig, adapted, and so also is J X - d X by its definition; thus [X, XI is chdlhg, adapted as well. Recall the property of the stochastic integral: A(X- . X ) = X- A X . Then

x,J2 = x,2 2X,X,- + x,2_ = x,2 x,2_+ 2X,-(X,- X,)

(AX)? = (X,

-

-

-

=

A ( x ~ ) , - 2Xs- (AX,),

from which part (i) follows. For part (ii), by replacing X with X = X - Xo, we may assume Xo = 0. Let R, = supi Ty. Then R, < cm a s . and limn R, = cm a s . , and thus by telescoping series

xi

converges u c p to X 2 . Moreover, the series XT; ( ~ ~ T i -x1T ? ) converges in u c p to S X-dX by Theorem 21, since X is c&dlbg.Since b2 - a2 - 2a(b - a) = (b - a ) 2 , and since XT? (xT""+1 - x T ? ) = xT? (xT?+1 - x T ? ) , we can combine the two series convergences above to obtain the result. Finally, note that if s < t , then the approximating sums in part (ii) include more terms (all nonnegative), so it is clear that [X, X ] is non-decreasing. (Note that, a priori, one only has [X,X], [X,XIt as., with the null set depending on s and t ; it is the property that [ X , X ] has chdlhg paths that allows one to eliminate the dependence of the null set on s and t.) Part (iii) is a simple consequence of part (ii) .

<

An immediate consequence of Theorem 22 is the observation that if B is a Brownian motion, then [B,BIt = t , since in Theorem 28 of Chap. I we showed the a.s. convergence of sums of the form in part (ii) of Theorem 22 when the partitions are refining. Another consequence of Theorem 22 is that if X is a semimartingale with continuous paths of finite variation, then [X,X] is the constant process equal to X i . To see this one need only observe that

< - sup lxT,"+l - xTzn IV, i

where V is the total variation. Therefore the sums tend to 0 as llrn/l-t 0. Theorem 22 has several more consequences which we state as corollaries.

Corollary 1. The bracket process [X, Y] of two semimartingales has paths of finite variation on compacts, and it is also a semimartingale.

68

I1 Semimartingales and Stochastic Integrals

Proof. By the polarization identity [X,Y] is the difference of two increasing processes, hence its paths are of finite variation. Moreover, the paths are clearly c&dl&g,and the process is adapted. Hence by Theorem 7 it is a semimartingale. Corollary 2 (Integration by Parts). Let X , Y be two semimartingales. Then X Y is a semimartingale and

Proof. The formula follows trivially from the definition of [X, Y]. That X Y is a semimartingale follows from the formula, Theorem 19, and Corollary 1 above. In the integration by parts formula above, we have (X-)o Hence evaluating at 0 yields

xoyo = (X-)oYo

=

(Y-)o

=

0.

+ (Y-)ox0 + [X,Yl0.

Since [X, YIo = AXoAYo = XoYo, the formula is valid. Without the convention that (X-)o = 0, we could have written the formula

Corollary 3. All semimartingales on a given filtered probability space form an algebra.

Proof. Since semimartingales form a vector space, Corollary 2 shows they form an algebra. A theorem analogous to Theorem 22 holds for [X, Y] as well as [X,XI. It can be proved analogously to Theorem 22, or more simply by polarization. We omit the proof.

Theorem 23. Let X and Y be two semimartingales. Then the bracket process [ X ,Y] satisfies the following.

(i) [X,YIo= XoYo and A[X,Y] = AXAY. (ii) If rn is a sequence of random partitions tending to the identity, then [X, Y] = XoYo + lim C ( x T ? + 1 - XT?)(YT$I - yT? n+cc

1 1

where convergence is in u p , and where rn i s the sequence 0 = Ton I Tr 5 - .. I Tr I . . . 5 TPn, with Tp stopping times. (iii) If T is any stopping time, then [XT,Y] = [X, YT] = [XT,YT] = [X, YIT.

6 The Quadratic Variation of a Semimartingale

69

We next record a real analysis theorem from the Lebesgue-Stieltjes theory of integration. It can be proved via the Monotone Class Theorem.

P,

y be functions mapping [O, cm) to R with a ( 0 ) = P(0) = y(0) = 0. Suppose a , P, y are all right continuous, a is of finite variation, and ,B and y are each increasing. Suppose further that for all s, t with s 5 t , we have T h e o r e m 24. Let a,

Then for any measurable functions f , g we have

In particular, the measure da is absolutely continuous with respect to both d p and dy. Note that Ida1 denotes the total variation measure corresponding to the measure da, the Lebesgue-Stieltjes signed measure induced by a. We use this theorem t o prove an important inequality concerning the quadratic variation and bracket processes. T h e o r e m 25 ( K u n i t a - W a t a n a b e Inequality). Let X and Y be two semi-

martingales, and let H and K be two measurable processes. Then one has a.s.

Proof. By Theorem 24 we only need to show that there exists a null set N , such that for w $ N , and ( s ,t ) with s t , we have

<

<

+

Let N be the null set such that if w $ N , then 0 Jst d [ X rY, X every r, s , t ; s 5 t , with r, s, t all rational numbers. Then

+ rY],, for

The right side being positive for all rational r , it must be positive for all real r by continuity. Thus the discriminant of this quadratic equation in r must be non-negative, which gives us exactly the inequality (*). Since we have, then, the inequality for all rational ( s ,t ) , it must hold for all real ( s ,t ) , by the right continuity of the paths of the processes.

70

I1 Semimartingales and Stochastic Integrals

Corollary. Let X and Y be two semimartingales, and let H and K be two measurable processes. Then

Proof. Apply Holder's inequality to the Kunita-Watanabe inequality of Theorem 25. Since Theorem 25 and its corollary are path-by-path Lebesgue-Stieltjes results, we do not have to assume that the integrand processes H and K be adapted. Since the process [X, X ] is non-decreasing with right continuous paths, and since A[X, XIt = (AXt)2 for all t 2 0 (with the convention that Xo- = O), we can decompose [X, X ] path-by-path into its continuous part and its pure jump part.

Definition. For a semimartingale X , the process [X, XIC denotes the pathby-path continuous part of [X, XI. We can then write

Observe that [X,X]g = 0. Analogously, [X, YIC denotes the path-by-path continuous part of [X, Y].

Comment. In Chap. IV, Sect. 7, we briefly discuss the continuous local martingale part of a semimartingale X satisfying an auxiliary hypothesis known as Hypothesis A. This (unique) continuous local martingale part of X is denoted X C .It can be shown that a unique continuous local martingale part, X C , exists for every semimartingale x . ~(If X is an F V process, then XCr 0, as we shall see in Chap. 111.) It is always true that [Xc,X c ] = [X,XIc. Although we will have no need of this result in this book, this notation is often used in the literature. We remark also that the conditional quadratic variation of a semimartingale X, denoted ( X , X ) , is defined in Chap. 111, Sect. 4. It is also true that (Xc,X c ) = [XC,Xc] = [X,XIc, and these notations are used interchangeably in the literature. If X i s already continuous and Xo = 0, then (X, X) = [X, XI = [X, XIC. See, for example, Dellacherie-Meyer [46, page 3551. See also in this regard Exercise 6 of Chap. IV which proves the result for L' martingales.

6 The Quadratic Variation of a Semimartingale

71

Definition. A semimartingale X will be called quadratic pure jump if [X,XIC= 0.

+

The If X is quadratic pure jump, then [X,XIt = X: xO<s
+

-

(using the notation of Theorem 42 of Chap. I), then Y is a quadratic pure jump semimartingale. Last note that the trivial continuous process Xt = t is quadratic pure jump since [X,X],C = [X,X]t 0. The next theorem gives a simple criterion for a process X to be a quadratic pure jump semimartingale.

Theorem 26. I f X is adapted, c&dl&g,with paths offinite variation on compacts, then X is a quadratic pure jump semimartingale. Proof. Without loss of generality we assume Xo = 0. We have already seen that such an X is a semimartingale (Theorem 7), and that the stochastic integral with respect to X is nothing more than a path-by-path Lebesgue-Stieltjes integral (Theorem 17). The integration by parts formula for Lebesgue-Stieltjes J XdX, comdifferentials applied to X times itself yields x2= J X-dX puted path-by-path. The semimartingale integration by parts formula (Corol[X,XI. lary 2 of Theorem 22), on the other hand, yields x2= 2 JX-dX Moreover

+

+

and x-dXS

+

AXsdXs =

Jlo

t

Xs-dXs

+x

(

~

~

s

)

~

s
Equating the two formulas we deduce [X,XIt theorem.

=

x s 5 t ( ~ ~ s )yielding 2, the

Note in particular that if X is adapted with continuous paths of finite variation, then [X,X]t = x:, a11 t 0.

>

Theorem 27. Let X be a local martingale with continuous paths that are not everywhere constant. Then [X,X] is not the constant process x:, and X 2 - [X,XI is a continuous local martingale. Moreover if [X,XIt = 0 for all t then Xt = 0 for all t.

72

I1 Semimartingales and Stochastic Integrals

Proof. Note that a continuous local martingale is a semimartingale (Corollary 2 of Theorem 8). We have X2 - [X, X ] = 2 X-dX, and by the martingale preservation property (Theorem 20) we have that 2 J X-dX is a local martingale. Moreover A2 J X - d X = 2(X-)(AX), and since X is continuous, A X = 0, and thus 2 J X - d X is a continuous local martingale, hence locally square integrable. Thus x2- [X,X ] is a locally square integrable local martingale. By stopping, we can suppose X is a square integrable martingale. Assume further Xo = 0. Next assume that [X, X ] actually were constant. Then [X,XIt = [X, XIo = X i = 0, for a11 t. Since x2- [X, XI is a local martingale, we conclude x2is a non-negative local martingale, with X: = 0. Thus X: = 0, all t. This is a contradiction. If Xo is not identically 0, we set xt = Xt - Xo and the result follows. The following corollary is of fundamental importance in the theory of martingales.

Corollary 1. Let X be a continuous local martingale, and S 5 T 5 GO be stopping times. If X has paths of finite variation on the stochastic interval (S,T ) , then X is constant on [S,TI. Moreover if [X,X ] is constant on [S,T]n [0, GO),then X is constant there too. Proof. M = xT- xS is also a continuous local martingale, and M has finite variation on compacts. Moreover [M, M ] = [XT - X S , X T - XS] = [X, XIT - [X, XIS, and therefore [M,M ] is constant everywhere, hence by Theorem 27, M must be constant everywhere; thus X is constant on [S,TI. Observe that if t > 0 is arbitrary and we take S = 0 and T = t in Corollary 1, then we can conclude that a continuous local martingale with paths of finite variatiofis on compacts is a.s. constant. While non-trivial continuous local martingales must therefore always have paths of infinite variation on compacts, they are not the only such local martingales. For example, the LQvyprocess martingale Zd of Theorem 41 of Chap. I will have paths of infinite variation if the LQvymeasure v has infinite mass in a neighborhood of the origin. Another example is AzQma's martingale, which is presented in detail in Sect. 8 of Chap. IV.

Corollary 2. Let X and Y be two locally square integrable local martingales. Then [X,Y] is the unique adapted c&dl&gprocess A with paths of finite variation on compacts satisfying the two properties: (i) X Y - A is a local martingale; and (ii) A A = A X A Y , A o =XoYo.

Proof. Integration by parts yields XY

=

I

X-dY

+

I

Y-dX

+ [X, Y],

6 The Quadratic Variation of a Semimartingale

73

but the martingale preservation property tells us that both stochastic integrals are local martingales. Thus XY - [X, Y] is a local martingale. Property (ii) is simply an application of Theorem 23. Thus it remains to show uniqueness. Suppose A, B both satisfy properties (i) and (ii). Then A - B = (XY B ) - (XY - A), the difference of two local martingales which is again a local martingale. Moreover,

Thus A - B is a continuous local martingale, A. - Bo = 0, and it has paths of finite variation on compacts. Corollary 1 yields At - Bt - A0 Bo = 0 and we have uniqueness.

+

Corollary 2 can be useful in determining the process [X,Y]. For example if X and Y are locally square integrable martingales without common jumps such that XY is a martingale, then [X,Y] = XoYo One can also easily verify (as a consequence of Theorem 23 and Corollaries 1 and 2 above, for example) that if X is a continuous square integrable martingale and Y is a square integrable martingale with paths of finite variation on compacts, then [X,Y] = XoYo,and hence X Y is a martingale. (An example would be X a Brownian motion and Y a compensated Poisson process.) Corollary 2 is true as well for X , Y local martingales, however we need Theorem 29 of Chap. I11 to prove the general result. In Chap. I11 we show that any local martingale is a semimartingale (see the corollary of Theorem 26 of Chap. 111), and therefore if M is a local martingale its quadratic variation [M,MIt always exists and is finite a s . for every t 2 0. We use this fact in the next corollary.

Corollary 3. Let M be a local martingale. Then M is a martingale with E{M:) < cm, all t > 0, if and only if E{[M, MIt) < cm, all t > 0. If E{[M, M]t)

< cm, then E{M:)

= E{[M, Mlt).

Proof. First assume that M is a martingale with E{M:) < cm for all t > 0. Then M is clearly a locally square integrable martingale. Let Nt = M: t [M,MI = 2 Ms- dMS, which is a locally square integrable local martingale by Theorem 20. Let (Tn)n>l be stopping times increasing to cm a s . such that N?" is a square integrable martingale. Then E{NFn) = E{No) = 0, all t > 0. Therefore E{M:AT~) = E{[M7M ] ~ A T ~ ) .

So

Doob's maximal quadratic inequality gives E{(M:)~) Therefore by the Dominated Convergence Theorem

E{M:)

= =

lim E{M&~,)

n-00

lim E { [ h f , M ] t ~ ~ n )

n-+00

= E{[M, Mlt),

5 ~E{M;) < cm.

74

I1 Semimartingales and Stochastic Integrals

where the last result is by the Monotone Convergence Theorem. In particular we have that E{[M, MI,) < GO. For the converse, we now assume E{[M, MIt) < GO, all t 2 0. Define stopping times by Tn = inf{t > 0 : (Mt(> n) An.

+

+

Then Tn increase to GO a s . Furthermore (MT")* I n IAMTnl 5 n [M,M];l2, which is in L2. By Theorem 51 of Chap. I, MT" is a uniformly integrable martingale for each n. Also we have that

for all t 2 0. Therefore MT" satisfies the hypotheses of the first half of this MT"],). Using Doob's inequality we theorem, and E{(M,T")~) = E{[M~", have

The Monotone Convergence Theorem next gives E{(M,*)~)= lim E{(M;~,,)~) n-+w

Therefore, again by Theorem 51 of Chap. I, we conclude that M is a martingale. The preceding gives E{M:) < GO. For emphasis we state as another corollary a special case of Corollary 3.

Corollary 4. If M is a local martingale and E{[M, MI,) < GO, then M is a square integrable martingale (that is sup, E{M$) = E{M&)< GO). Moreover E{M:) = E{[M, MIt) for all t, 0 5 t 5 GO. Example. Before continuing we consider again an example of a local martingale that exhibits many of the surprising pathologies of local martingales. Let B be a standard Brownian motion in R3 with Bo = (1,1,1). Let Mt = 11 Bt ( I - ' , where l l ~ l lis standard Euclidean norm in R3.(We previously considered this example in Sect. 6 of Chap. I.) As noted in Chap. I, the process M is a continuous local martingale; hence it is a locally square integrable local martingale. Moreover E{M:) < GO for all t. However instead of t H E{M:) being an increasing function as it would if M were a martingale, limt,, E{M:) = 0. Moreover E{[M, MIt) 2 E{[M, MIo) = 1 since [M,MIt is increasing. Therefore we cannot have E{M:) = E{[M, MIt) for all t. Indeed, by Corollary 3 and the preceding we see that we must have E{[M, MIt) = GO for all t > 0. In conclusion, M = IIBII-' is a continuous local martingale with E{M:) < GO for all t which is both not a true martingale and for which E{M:) < GO while E{[M, MI,) = GO for all t > 0. (Also refer to Exercise 20 at the end of this chapter.)

6 The Quadratic Variation of a Semimartingale

75

Corollary 5. Let X be a continuous local martingale. Then X and [X, X ] have the same intervals of constancy a s . Proof. Let r be a positive rational, and define

T, = inf{t 2 r : Xt f X,). Then M = xTp - X' is a local martingale which is constant. Hence [M,M ] = [X, XIT, - [X,XIr is also constant. Since this is true for any rational r a s . , any interval of constancy of X is also one of [X,XI. Since X is continuous, by stopping we can assume without loss of generality that X is a bounded martingale (and hence square integrable). For every positive, rational r we define

Then E{(XS,.

-

X r ) 2 ) = E{x~,.) - E{x:)

by Doob7sOptional Sampling Theorem. Moreover

by Corollary 3. Therefore E{(Xs, - x , ) ~ ) = 0, and XSp = X, a s . Moreover this implies X, = Xsq a s . on {S, = ST)for each pair of rationals (r, q), and therefore we deduce that any interval of constancy of [ X , X ] is also one of X. Note that the continuity of the local martingale X is essential in Corollary 5. Indeed, let N, be a Poisson process, and let Mt = Nt - t. Then M is a martingale and [M, MIt = Nt; clearly M has no intervals of constancy while N is constant except for jumps.

Theorem 28. Let X be a quadratic pure jump semimartingale. Then for any semimartingale Y we have

' Proof. The Kunita-Watanabe inequality (Theorem 25) tells us d[X, Y], 1s as. absolutely continuous with respect to d[X, XI (path-by-path). Thus [X,XIC = 0 implies [X,YIC= 0, and hence [X, Y] is the sum of its jumps, and the result follows by Theorem 23.

Theorem 29. Let X and Y be two semimartingales, and let H, K E

and, in particular,

L.Then

76

I1 Semimartingales and Stochastic Integrals

Proof. First assume (without loss of generality) that Xo = Yo = 0. It suffices to establish the following result

and then apply it again, by the symmetry of the form [.,.I, and by the associativity of the stochastic integral (Theorem 19). First suppose H is the indicator of a stochastic interval. That is, H = l[O,T], where T is a stopping time. Establishing (*) is equivalent in this case to showing [ x T , Y] = [X,Y ] ~ a, result that is an obvious consequence of Theorem 23, which approximates [X,Y] by sums. Next suppose H = Ul(s,Tl, where S, T are stopping times, S 5 T a s . , and U E 3s. Then H,dX, = U(XT - XS), and by Theorem 23

The result now follows for H E S by linearity. Finally, suppose H E IL and let Hn be a sequence in S converging in ucp to H . Let Zn = H n . X , Z = H . X . We know Z n , Z are all semimartingales. We have J H r d [ X , Y], = [Zn,Y], since Hn E S, and using integration by parts

By the definition of the stochastic integral, we know Zn + Z in ucp, and since Hn + H (ucp), letting n + co we have

=

YZ-

I

Y-dZ-

I

Z-dY

again by integration by parts. Since limn,, J Hrd[X, Y]s = J Hsd[X, Y],, we have [Z,Y] = [H . X , Y] = J Hsd[X, Y], , and the roof is complete.

Example. Let Bt be a standard Wiener process with Bo = 0, (i.e., Brownian motion). B: - t is a continuous martingale by Theorem 27 of Chap. I. Let H E IL be such that E{J: H:ds) < co,each t > 0. By Theorem 28 of Chap. I

6 The Quadratic Variation of a Semimartingale

77

we have [B, BIt = t, hence [H . B , H . BIt = Jot H:ds. By the martingale preservation property, J H,dB, is also a continuous local martingale, with ( H . B)O = 0. By Corollary 3 t o Theorem 27

It was this last equality,

that was crucial in K. It6's original treatment of a stochastic integral. Theorem 30. Let H be a chdldg, adapted process, and let X, Y be two semimartingales. Let a, be a sequence of random partitions tending to the identity. Then H, (xT;i - xT? ) (yT?+i- yT?)

converges in ucp to J H,-d[X,Y], ...I Tan < ...
(Ho- = 0). Here a, = (0

< To" < Tp <

Proof. By the definition of quadratic variation, [X, Y] = X Y -X- .Y -Y- . X , t where X- . Y denotes the process (So X,-dYs)t20. By the associativity of the stochastic integral (Theorem 19)

H-.[X,Y]=H-.(XY)-H-.(X-.Y)-H-.(Y-.X) = H- . (XY) - (H-X-) . Y - (H-Y-) . X = H- . (XY) - ( H X ) - . Y - (HY)- . X. By Theorem 21 the above is the limit of

78

I1 Semimartingales and Stochastic Integrals

7 It6's Formula (Change of Variables) Let A be a process with continuous paths of finite variation on compacts. If H E L we know by Theorem 17 that the stochastic integral H . A agrees a.s. with the path-by-path Lebesgue-Stieltjes integral H , ~ A , . In Sect. 7 of Chap. I (Theorem 54) we proved the change of variables formula for f E C1, namely

lo rt

f ( A t )- f

(-40)

=

f1(AS)dAS.

We also saw at the end of Sect. 4 of this chapter that for a standard Wiener process B with Bo = 0,

Taking f ( x ) = x2/2, the above formula is equivalent t o t

f ( B t )- f ( B o )=

f 1 ( B S ) d B s+

1

1

f"(Bs)dsl

which does not agree with the Lebesgue-Stieltjes change of variables formula (*). In this section we will state and prove a change of variables formula valid for all semimartingales. We first mention, however, that the change of variables formula for continuous Stieltjes integrals given in Theorem 54 of Chap. I has an extension to right continuous processes of finite variation on compacts. We state this result as a theorem but we do not prove it here because it is merely a special case of Theorem 32, thanks to Theorem 26. Theorem 31 (Change of Variables). Let V be an F V process with right continuous paths, and let f be such that f' exists and is continuous. Then ( f (K)) is an F V process and

,,,

Jl+

Recall that the notation = J(o,tl denotes the integral over the half open interval (0, t ] . We wish to establish a formula analogous t o the above, but for the stochastic integral; that is, when the process is a semimartingale. The formula is different in this case, as we can see by comparing equation (*) with equation (**); we must add an extra term! Theorem 32 (Itci's Formula). Let X be a semimartingale and let f be a C2 real function. Then f ( X ) is again a semimartingale, and the following

formula holds:

7 ItG1sFormula (Change of Variables)

79

Proof. Note that the jump part of the stochastic integral f1'(XS-)d[X, XIs is given by C,,, f'1(Xs-)(AXs)2, and this is a convergent series. By adding and subtracting 112 of this series, we can rewrite It6's formula in the equivalent form

which is perhaps less obviously a generalization of the "classical" case, but notationally simpler t o prove. The proof rests, of course, on Taylor's Theorem which says

where IR(x, y)l 5 r(ly - xl)(y - x ) ~ such , that r : R+ ---,R+ is an increasing function with limUlor(u) = 0, which is valid for f E C2 defined on a compact set. We now prove separately the continuous case and the general case. Proof for the continuous case. We first restrict our attention to a continuous semimartingale X , since the proof is less complicated but nevertheless gives the basic idea. Without loss of generality we can take Xo = 0. Define stopping times R, = i n f { t : lXtl 2 m). Then the stopped process XRm is bounded by m, and if Itb's formula is valid for XRm for each m, it is valid for X as well. Therefore we assume that X takes its values in a compact set. We fix a t > 0, and let a, be a refining sequence of random partitions5 of [O,t] tending t o the identity [a, = (0 = Ton 5 TF 5 - . . 5 Trn = t)]. Then

Note that it would suffice for this proof to restrict attention to deterministic partitions.

80

11 Semimartingales and Stochastic Integrals

Jot

The first sum converges in probability to the stochastic integral fl(X,-)dX, by Theorem 21; the second sum converges to f "(X,)d[X, XI, in probabil-

4 Jot

ity by Theorem 30. It remains to consider the third sum But this sum is majorized, in absolute value, by

xiR

and since Ci(XT?+, - XTn)2 converges in probability to [X,XIt (Theorem 22), the last term will tend to 0 if limn,, supi r((XTP+, - XTp)) = 0. However s ++ XS(w) is a continuous function on [0,t], each fixed w, and - TFl = 0 by hypothhence uniformly continuous. Since limn,, supi esis, we have the result. Thus, in the continuous case, f (Xt) - f (Xo) = fl'(Xs-)d[X, X,], for each t, a.s. The continuity of the fl(Xs-)dXs paths then permits us to remove the dependence of the null set on t, giving the complete result in the continuous case. Proof for the general case. X is now given as a right continuous semimartingale. Once again we have a representation as in (* * *), but we need a closer 2 [X,XIt < co a.s., hence analysis. For any t > 0 we have C o < s < t ( A ~ , ) 5 C o < s < t ( A ~ s )is2 convergent. Given E-> 0 and t > 0, let A = A ( E , ~ be ) a set of jump times of X that has a.s. a finite number of times s, and let B = B(E,t) be such that C,,B(AX,)2 5 s2,where A and B are disjoint and A U B exhaust the jump times of X on (0, t]. We write

+ Jot

where

denotes

Ci ~ { A ~ ( T ~ , T ~ Then ,]#B).

and by Taylor's formula

7 ItG's Formula (Change of Variables)

81

As in the continuous case, the first two sums on the right side above converge respectively t o $+f 1 ( X S - ) d X s and J + ; f f . ( X S - ) d [ X , X ] , .The third sum converges t o

1

Assume temporarily that IXsI 5 Ic, some constant Ic, all s 5 t. Then f" is uniformly continuous, and using the right continuity of X we have

where r ( ~ + is ) l i m s ~ r(b). p ~ Next ~ ~ let t o 0 , and

E

tend t o 0. Then r ( & + ) [ XXIt , tends

tends t o the series in (* * *), provided this series is absolutely convergent. Let Vk = infit > 0 : lXtl 2 Ic), with X o = 0. By first establishing (***) for X l l o , v k ) which , is a semimartingale since it is the product of two semimartingales (Corollary 2 of Theorem 22), it suffices t o consider semimartingales taking their values in intervals of the form [-Ic, Ic]. For f restricted t o [-Ic, Ic] we have I f ( y ) - f ( x ) - ( y - x ) f l ( x ) l 5 C ( ~ - X )Then ~ .

and C o < s 5 t I f " ( X s - ) l ( A X s ) 2 F K C o < s l t ( A X s ) 2 F K [ X , X I t < m a.s. Thus the series is absolutely convergent and this completes the proof.

Corollary (Itii's Formula). Let X be a continuous semimartingale and let f be a C2 real function. Then f ( X ) is again a semimartingale and the following formula holds:

Theorem 32 has a multi-dimensional analog. We omit the proof.

Theorem 33. Let X = ( X I , . . . , X n ) be an n-tuple of semimartingales, and let f : Rn -+I% have continuous second order partial derivatives. Then f ( X ) is a semimartingale and the following formula holds:

82

11 Semimartingales and Stochastic Integrals

The stochastic integral calculus, as revealed by Theorems 32 and 33, is different from the classical Lebesgue-Stieltjes calculus. By restricting the class of integrands to semimartingales made left continuous (instead of IL), one can define a stochastic integral that obeys the traditional rules of the LebesgueStieltjes calculus.

Definition. Let X , Y be semimartingales. Define the Fisk-Stratonovich integral of Y with respect to X , denoted Ys- o dX,, by

The Fisk-Stratonovich integral is often referred to as simply the Stratonovich integral. The notation "0" is called Itci's circle. Note that we have defined the Fisk-Stratonovich integral in terms of the semimartingale integral. With some work one can slightly enlarge the domain of the definition and we do so in Sect. 5 of Chap. V. In particular, Theorem 34 below is proved with the weaker hypothesis that f E C2 (Theorem 20 of Chap. V). We will write the F-S integral as an abbreviation for the Fisk-Stratonovich integral.

Theorem 34. Let X be a semimartingale and let f be

c3.T h e n

Proof. Note that f ' is C2, SO that f l ( X ) is a semimartingale by Theorem 32 and in the domain of the F-S integral. By Theorem 32 and the definition, it suffices to establish [f'(x), XIC= f1'(XS-)d[X, XI:. However

i

i sot

Thus

The first term on the right side above is J : f1'(XS-)d[X, XI: by Theorem 29; the second term can easily be seen, as a consequence of Theorem 22 and the

7 ItG's Formula (Change of Variables)

fact that [X, XI has paths of finite variation, to be That is, zero, and the theorem is proved.

Co,,It f (3) (x,-)

83

(AX,)3.

Note that if X is a semimartingale with continuous paths, then Theorem 34 reduces to the classical Riemann-Stieltjes formula f (Xt) - f (Xo) = fl(Xs) o dX,. This is, of course, the main attraction of the Fisk-Stratonovich integral. Corollary (Integration by Parts). Let X , Y be semimartingales, with at least one of X and Y continuous. Then

Proof. The standard integration by parts formula is

+

However [X,YIt = [X, Y],C XoYo if one of X or Y is continuous. Thus adding [X,Y]: to each integral on the right side yields the result. One can extend the stochastic calculus to complex-valued semimartingales. Theorem 35. Let X , Y be continuous semimartingales, let Zt and let f be analytic. Then

=

Xt

+ iYt,

Proof. Using Itb7sformula for the real and imaginary parts of f yields

Since f is analytic by the Cauchy-Riemann equations

9

2 = f' and

= ay

= = f " and axay = if" and if'. Differentiating again gives = ay -f". The result now follows by collecting terms and observing that dZ, = dX, idY, and [Z, Z] = [X + iY, X iY] = [X, X ] 2i[X,Y] - [Y, Y].

+

+

+

Theorem 35 also has a version for Zt = Xt +iY,, with both X and Y c&dl&g semimartingales. The proof is almost the same as the proof of Theorem 35.

84

I1 Semimartingales and Stochastic Integrals

Theorem 36. Let X , Y be cadlag semimartingales, let Zt let f be analytic. Then

=

Xt

+ iYt,and

Observe that for a complex-valued semimartingale the process [Z, Z] is i n general a complex-valued process. For many applications, it is more appropriate t o use the non-negative increasing process

to play the role of the quadratic variation.

8 Applications of It6's Formula As an application of the change of variables formula, we investigate a simple, yet important and non-trivial, stochastic differential equation. We treat it, of course, in integral form.

Theorem 37. Let X be a semimartingale, Xo = 0. Then there exists a (unique) semimartingale Z that satisfies the equation Zt = 1 J~~ 2,-dX,. Z is given by

+

where the infinite product converges. Proof. We will not prove the uniqueness here since it is a trivial consequence of the general theory t o be established in Chap. V. (For example, see Theorem 7 of Chap. V.) Note that the formula for Zt is equivalent t o the formula Zt = exp {xt-

XI:

I,,,

n ( 1 + AX,) exp{-AX,}.

Since Xt - ;[x, XI; is a semimartingale, and ex is C2, we need only show AX,) exp{-AX,) is c&dlAg, adapted, and of finite variation. that n s l t ( l It will then be a semimartingale too, and thus Z will be a semimartingale. The product is clearly cAdlAg, adapted; it thus suffices t o show the product converges and is of finite variation. Since X has cAdlAg paths, there are only a finite number of s such that lAX,l 112 on each compact interval (fixed w ) . Thus it suffices to show

+

>

8 Applications of It& Formula

85

converges and is of finite variation. Let Us = AXslt lAx,lll/2). Then we have +Us) -Us), which is an absolutely convergent series a.s., log V, = C,,t{log(l since C o , , ~ t ( U s ) 2 5 [X,XIt < oo a s . , because I log(1 X) - XI 5 x2 when 1x1 < 112. Thus log(&) is a process with paths of finite variation, and hence so also is exp{log V,) = &. To show that Z is a solution, we set K t = Xt - i [ X , XI:, and let f (x, y) = yex. Then Zt = f (Kt, St),where St = n0,,,,(l - AX,) exp{-AX,). By the change of variables formula we have

+

+

since [K, SIC= [S,SIC= 0. Note that S is a pure jump process. Hence eKs-dS, = ~ o , , 5 t e K s A S s . Also 2, = 2,-(l + AX,), and 2,-AK, = 2,-AX,, so the last sum on the right side of equation (*) becomes

Jl+

Thus equation (*) simplifies due t o cancellation to Zt we have the result.

-

1 = Jot2,-dX,,

and

Definition. For a semimartingale X , Xo = 0, the stochastic exponential of X , written E(X), is the (unique) semimartingale Z that is a solution of Zt = 1 z,-~x,.

+ Jl

The stochastic exponential is also known as the DolBans-Dade exponential. Theorem 37 gives a general formula for &(X).This formula simplifies considerably when X is continuous. Indeed, let X be a continuous semimartingale with Xo = 0. Then

An important special case is when the semimartingale X is a multiple X of a standard Brownian motion B = (Bt)t>o. Since XB has no jumps we have

86

I1 Semimartingales and Stochastic Integrals

Moreover, since E(XB)t = 1

+ X ~ ~ ( X B ) , - ~weB , see that

E(XB)t =

e"t-gt is a continuous martingale. The process E(XB) is sometimes referred to as geometric Brownian motion. Note that the previous theorem gives us &(X) in closed form. We also have the following pretty result. Theorem 38. Let X and Y be two semimartingales with Xo = Yo = 0. Then E(X)E(Y) = E ( X Y [X,Y]).

+ +

Proof. Let Ut = E(X)t and 6 = E(Y)t. Then the integration by parts formula gives that Ut& - 1 = J& Us-dVs J,'+Vs-dUs + [U, V],. Since U and V are exponentials, this is equivalent to

+

+

: Ws-d(X Letting Wt = Ut&, we deduce that Wt = 1 J that W = E(X Y + [X, Y]), which was t o be shown.

+

+ Y + [X,Y]), so

Corollary. Let X be a continuous semimartingale, Xo = 0. Then E(x)-I = E(-X [X,XI).

+

Proof. By Theorem 38, E(X)E(-X

+ [X,XI) = E(X + (-X + [X,XI) + [X, -XI),

since [-X, [X, XI] = 0. However E(0) = 1, and we are done. In Sect. 9 of Chap. V we consider general linear equations. In particular, we obtain an explicit formula for the solution of the equation

where Z is a continuous semimartingale. We also consider more general inverses of stochastic exponentials. See, for example, Theorem 63 of Chap. V. Another application of the change of variables theorem (and indeed of the stochastic exponential) is a proof of LQvy's characterization of Brownian motion in terms of its quadratic variation. Theorem 39 (LQvy'sTheorem). A stochastic process X = (Xt)t20is a standard Brownian motion if and only if it is a continuous local martingale with [X,XIt = t .

8 Applications of It6's Formula

87

Proof. We have already observed that a Brownian motion B is a continuous local martingale and that [B, BIt = t (see the remark following Theorem 22). Thus it remains to show sufficiency. Fix u E R and set F ( x , t ) = exp{iux

+

g t ) . Let Zt = F ( X t , t ) = exp{iuXt

+ ($) t). Since F E C2 we can apply

It6's formula (Theorem 33) to obtain

which is the exponential equation. Since X is a continuous local martingale, we now have that Z is also one (complex-valued, of course) by the martingale preservation property. Moreover stopping Z at a fixed time to, Zto, we have that Zto is bounded and hence a martingale. It then follows for 0 5 s < t that E{exp{iu(Xt - Xs)}IFs}

= exp

{

uz"

-- (t - s)}

Since this holds for any u E R we conclude that Xt - X, is independent of Fs and that it is normally distributed with mean zero and variance (t - s). Therefore X is a Brownian motion. Observe that if M and N are two continuous martingales such that M N is a martingale, then [M,N] = 0 by Corollary 2 of Theorem 27. Therefore if Bt = (B:, . . . ,BF) is an n-dimensional standard Brownian motion, B~B: is a martingale for i # j , and we have that

Theorem 39 then has a multi-dimensional version, which has an equally simple proof.

Theorem 40 (L6vy's Theorem: Multi-dimensional Version). Let X = (X1,.. . ,X n ) be continuous local martingales such that

Then X is a standard n-dimensional Brownian motion. As another application of It6's formula, we exhibit the relationship between harmonic and subharmonic functions and martingales.

Theorem 41. Let X = (X1,. . . , X n ) be an n-dimensional continuous local martingale with values in an open subset D of Rn. Suppose that [Xi,X j ] = 0 if i # j , and [ X i , X i ] = A, 1 5 i 5 n. Let u : D --+ R be harmonic (resp. subharmonic). Then u(X) is a local martingale (resp. submartingale).

88

I1 Semimartingales and Stochastic Integrals

Proof. By ItG's formula (Theorem 33) we have

where the "dot" denotes Euclidean inner product, V denotes the gradient, and A denotes the Laplacian. If u is harmonic (subharmonic), then Au = O(Au 2 0) and the result follows. If B is a standard n-dimensional Brownian motion, then B satisfies the hypotheses of Theorem 41 with the process At = t. That u(Bt ) is a submartingale (resp. supermartingale) when u is subharmonic (resp. superharmonic) is the motivation for the terminology submartingale and supermartingale. The relationship between stochastic calculus and potential theory suggested by Theorem 41 has proven fruitful (see, for example, Doob [56]). Lbvy's characterization of Brownian motion (Theorem 39) allows us t o prove a useful change of time result.

Theorem 42. Let M = (Mt)t>0 - be a continuous local martingale with Mo = 0 and such that limt,,[M, MIt = oo a.s. Let

T,

= inf{t

> 0 : [M,MIt > s).

Define Gs = FT8and B, = MTs. Then (B,, Gs)s20 is a standard Brownian motion. Moreover ([M,M ] t ) t 2 ~are stopping times for (G,),20 and

That is, M can be represented as a time change of a Brownian motion. Proof. The (Ts),20 are stopping times by Theorem 3 of Chap. I. Each T, is finite a.s. by the hypothesis that limt,,[M, MIt = m a.s. Therefore the a-fields Gs = FTaare well-defined. The filtration (G,),>o - need not be right continuous, but one can take 'H, = Gs+ = FTs+ t o obtain one. Note further - are stopping times for that {[M, MIt 5 s) = {T, 2 t), hence ([M,MIt)t>o the filtration G' = (G's)s20. By Corollary 3 of Theorem 27 we have E{M;~) = E{[M, MITs) = s < m , since [M, MITs = s identically because [M, M ] is continuous. Thus the time changed process is square integrable. Moreover

by the Optional Sampling Theorem. Also

8 Applications of It6's Formula

89

Therefore Bz - s is a martingale, whence [B, B], = s provided B has continuous paths, by Corollary 2 t o Theorem 27. We want t o show that B, = MTs has continuous paths. However by Corollary 5 of Theorem 27 almost surely all intervals of constancy of [M, M] are also intervals of constancy of M . It follows easily that B is continuous. It remains t o show that Mt = B[M,M]~. Since B, = MT,, We have that B[M,M],= MqM,,,, , a.s. Since (T,),>O is the right continuous inverse of [M, MI, we have that T [ M , ~2 ] , t, with equality holding if and only if t is a point of right increase of [M, MI. (If (Ts),>0 were continuous, then we would always have > t implies that t [M,MIt is constant that T[M,M1, = t.) However thus by Corollary 5 of Theorem 27 we conclude on the interval (t, , , a.s., and M is constant on ( t , q M , ~ ] ~Therefore ). B[M,M],= M T ~ ~ ,=~Mt we are done. Another application of the change of variables formula is the determination of the distribution of Lkvy's stochastic area process. Let Bt = (Xt, &) be an R2-valued Brownian motion with (Xo,Yo) = (0,O). Then during the times s t o s ds the chord from the origin t o B sweeps out a triangular region of area RsdNs7where R, =

+

Jm

and ys dN, = --dX,

Rs

xs + -dY,. Rs

Therefore the integral At = J: R,d% = J:(-Y,~x,+x,~Y,) is equal t o twice the area swept out from time 0 until time t. Paul L6vy found the characteristic function of At and therefore determined its distribution. Theorem 43 is known as LQvy'sstochastic area formula.

Theorem 43. Let Bt = (Xt, &) be an R2-valued Brownian motion, Bo (0, O), u E R. Let At = J:X,dY, - $ Y,dX,. Then

Proof. Let a ( t ) , P(t) be C' functions, and set

Then

+ &dY, + dt) + Pf(t)dt - a(t)Xt)dXt + (iuXt - a ( t ) x ) d &

- a(t){XtdXt

= (-iu%

=

90

I1 Semimartingales and Stochastic Integrals

Next observe that, from the above calculation,

Using the change of variables formula and the preceding calculations

Therefore eVt is a local martingale provided

Next we fix to > 0 and solve the above ordinary differential equations with a(to) = P(to) = 0. The solution is a ( t ) = u tanh(u(to - t)) p(t) = - log cosh(u(t0 - t)), where tanh and cosh are hyperbolic tangent and hyperbolic cosine, respectively. Note that for 0 5 t I to,

Thus eVt, 0 5 t 5 to is bounded and is therefore a true martingale, not just a local martingale, by Theorem 51 of Chap. I. Therefore,

However, Go = iuAto since a(t0) = P(t0) = 0. AS A0 = Xo = Yo = 0, it follows that Vo = -log cosh(ut0). We conclude that ~ { e ~ " *} ~=oexp{- log cosh(ut0)) 1 cosh(ut0) ' and the proof is complete. There are of course other proofs of LQvy's stochastic area formula (e.g., Yor [246],or LQvy [150]). As a corollary to Theorem 43 we obtain the density for the distribution of At.

8 Applications of It8's Formula

91

Corollary. Let Bt = (Xt, K ) be an IR2-valued Brownian motion, Bo = (0, O), and set At = Jt X s d x - YsdXs. Then the density function for the distribution of At is

Jbt

Proof. By Theorem 43, the Fourier transform (or characteristic function) of At is ~ { e ' ~ "=~ } 1 Thus we need only t o calculate & J G d u . The

w.

integrand is of the form f (2) zo =

E,we have

=

Q(z)

=

+.

-irz

Since cosh(rt) has a pole at

Next we integrate along the closed curve C, traversed counter clockwise, and given by

as shown in Fig. 1 below.

Fig. 1. The closed curve of integration C, = C:

Therefore

+ C: + C: + c:.

Jc f (z)dz = 27ri Res(f, 20). Along c,"the integral is -r

-

e-i(u+y)x cosh(ut)

/-, r

du=e?

e-iux

cosh(ut)

du.

The integrands on C: and C: are dominated by 2e-,, and therefore

92

I1 Semimartingales and Stochastic Integrals

Finally we can conclude

The stochastic area process A shares some of the properties of Brownian motion, as is seen by recalling that At = RsdN,, where N is a Brownian motion by Livy's Theorem (Theorem 39), and N and R are independent (this must be proven, of course). For example A satisfies a reflection principle. If one changes the sign of the increments of A after a stopping time, the process obtained thereby has the same distribution as that of A. One can use this - - As, then St has the same fact t o show, for example, that if St = supo,,,, distribution as IAtl, for t > 0.

Jbt

Bibliographic Notes The definition of semimartingale and the treatment of stochastic integration as a Riemann-type limit of sums is in essence new. It has its origins in the fundamental theorem of Bichteler [13, 141, and Dellacherie [42]. The pedagogic approach used here was first suggested by Meyer [176], and it was then outlined by Dellacherie [42]. Dellacherie's outline was further expanded by Lenglart [I451 and Protter [201, 2021. These ideas were also present in the work of M. Mitivier and J. Pellaumail, and to a lesser extent Kussmaul, although this was not appreciated a t the time by us. See for example [160], [162], and [138]. A similar idea was developed by Letta [148]. A recent treatment is in Bichteler [15]. We will not attempt to give a comprehensive history of stochastic integration here, but rather just a sketch. The important early work was that of Wiener [228, 2291, and then, of course, It6 [93, 94, 95, 97, 981. However, it has recently been discovered, by B. Bru and M. Yor, that the young French mathematician W. ~ o e b l i nhad ~ essentially established It6's formula in early Wolfgang Doeblin (Doblin) was born in 1915 in Berlin, and after acquiring French citizenship in 1936 was known as Vincent Dobbin. He was killed in June 1940.

Bibliographic Notes

93

1940, but was killed later that year in action in the Second World War, and thus his results were lost until recently [26, 271. Doob stressed the martingale nature of the It6 integral in his book [55] and proposed a general martingale integral. Doob's proposed development depended on a decomposition theorem (the Doob-Meyer decomposition, Theorem 8 of Chap. 111) which did not yet exist. Meyer proved this decomposition theorem in [163, 1641, and commented that a theory of stochastic integration was now possible. This was begun by Courrkge [36], and extended by Kunita and Watanabe [134], who revealed an elegant structure of square integrable martingales and established a general change of variables formula. Meyer [166, 167, 168, 1691 extended Kunita and Watanabe's work, realizing that the restriction of integrands t o predictable processes is essential. He also extended the integrals t o local martingales, which had been introduced earlier by It6 and Watanabe [102]. Up to this point, stochastic integration was tied indirectly t o Markov processes, by the assumption that the underlying filtration of a-algebras be "quasi-left continuous." This hypothesis was removed by DolBans-Dade and Meyer [53], thereby making stochastic integration a purely martingale theory. It was also in this article that semimartingales were first proposed in the form we refer t o as classical semimartingales in Chap. 111. A different theory of stochastic integration was developed independently by McShane [154, 1551, which was close in spirit t o the approach given here. However it was technically complicated and not very general. It was shown in Protter [200] (building on the work of Pop-Stojanovic [194]) that the theory of McShane could for practical purposes be viewed as a special case of the semimartingale theory. The subject of stochastic integration essentially lay dormant for six years until Meyer [171] published a seminal "course" on stochastic integration. It was here that the importance of semimartingales was made clear, but it was not until the late 1970's that the theorem of Bichteler [13, 141, and Dellacherie [42] gave an a posteriori justification of semimartingales. The seemingly ad hoe definition of a semimartingale as a process having a decomposition into the sum of a local martingale and an F V process was shown t o be the most general reasonable stochastic differential possible. (See also Kussmaul [I381 in this regard, and the bibliographic notes in Chap. 111.) Most of the results of this chapter can be found in Meyer [171],though they are proven for classical semimartingales and hence of necessity the proofs are much more complicated. Theorem 4 (Stricker's Theorem) is (of course) due t o Stricker [218]; see also Meyer [173]. Theorem 5 is due to Meyer [175]. There are many other methods of expanding a filtration and still preserving the semimartingale property. For further details, see Chap. VI on expansion of filtrations. Theorem 14 is originally due t o Lenglart [142]. Theorem 16 is known t o be true only in the case of integrands in IL.The local behavior of the integral (Theorems 17 and 18) is due t o Meyer (1711 (see also McShane (1541).The a.s.

94

Exercises for Chapter I1

Kunita-Watanabe inequality, Theorem 25, is due t o Meyer [171],while the expected version (the corollary t o Theorem 25) is due t o Kunita-Watanabe [134]. That continuous martingales have paths of infinite variation or are constant a.s. was first published by Fisk [73] (Corollary 1 of Theorem 27). The proof given here of Corollary 4 of Theorem 27 (that a continuous local martingale X and its quadratic variation [X,X ] have the same intervals of constancy) is due t o Maisonneuve [151]. The proof of It6's formula (Theorem 32) is by now classic; however we benefited from Follmer's presentation of it [75]. A popular alternative proof which basically bootstraps up form the formula for integration by parts, can be found (for example) on pages 57-58 of J. Jacod and A. N. Shiryaev [110]. The Fisk-Stratonovich integral was developed independently by Fisk [73] and Stratonovich [217], and it was extended t o general semimartingales by Meyer [171]. Theorem 35 is inspired by the work of Getoor and Sharpe [215]. The stochastic exponential of Theorem 37 is due t o Dolkans-Dade [49]. It has become extraordinarily important. See, for example, Jacod-Shiryaev [110]. The pretty formula of Theorem 38 is due to Yor [237]. Exponentials have of course a long history in analysis. For an insightful discussion of exponentials see Gill-Johansen [82]. That every continuous local martingale is the time change of a Brownian motion is originally due t o Dubins-Schwarz [59] and Dambis [37]. The proof of LQvy's stochastic area formula (Theorem 43) is new and is due t o Janson. See Janson-Wichura [113] for related results. The original result is in Lkvy [150], and another proof can be found in Yor [246].

Exercises for Chapter I1 Exercise 1. Let B be standard Brownian motion and let f be a function mapping R --+ R which is continuous except for one point where there is a jump discontinuity. Show that Xt = f (Bt) cannot be a semimartingale. Exercise 2. Let Q << P (Q and P are both probability measures and Q is absolutely continuous with respect t o P). Show that if Xn --+ X in P probability, then Xn --t X in Q-probability. Exercise 3. Let (X, Y) be standard two dimensional Brownian motion. (That is, X and Y are independent one dimensional standard Brownian motions.) Let 0 < a < 1. Set Bt = axt +

d z ~ .

Show that B is a standard one dimensional Brownian motion. Compute [X,B] and [Y,B].

Exercise 4. Let f : R+ -, R+ be non-decreasing and continuous. Show that there exists a continuous martingale M such that [M, MIt = f (t). Exercise 5. Let B be standard Brownian motion and let H E IL be such that lHtl = 1, all t. Let Mt = H,dB,. Show that M is also a Brownian motion.

Jbt

Exercises for Chapter I1

95

Exercise 6. Give an example of semimartingales Xn, X, Y such that Xp = Xt a.s., all t , but limn,,[Xn, YIt # [X,Y]t a.s., for some limn,, t. Exercise 7. Let Y, Z be semimartingales, and let H n , H E IL. Suppose Hn = H in ucp. Let X n = J H;dY, and X = J H,dY,. Show that limn,, limn,,[Xn, ZIt = [X,ZIt in ucp, all t. Exercise 8. Suppose f and all fn are c 2 , and that fn and f; converge uniformly on compacts t o f and f', respectively. Let X , Y be continuous semimartingales and show that limn,, [f n ( X ), Y] = [f ( X ),Y]. Exercise 9. Let M be a continuous local martingale and let A be a continuous, adapted process with paths of finite variation on compact time sets, with A. = 0. Let X = M + A . Show that [X,X]= [M, MI a.s. Exercise 10. Let X be a semimartingale and let Q be another probability law, with Q << P. Let [X,XIP denote the quadratic variation of X considered as a P semimartingale. Show that [X,x]Q = [X, XIP, Q - a . ~ . Exercise 11. Let B be standard Brownian motion, and let T = infit Bt = 1 or Bt = -2).

>0:

(a) Show that T is a stopping time. (b) Let M = B t n and ~ let N = -M. Show that M and N are continuous martingales and that [M, MIt = [N,NIt = t A T . Show that nevertheless M and N have different distributions.

Exercise 12. Let X be a semimartinagle such that EO<,[AXS[< co a.s., each t > 0. Let f E c2 and show that ~ o < s l tlAf(X,)( < co a.s., each t > 0 as well. Exercise 13. Let X be a semimartingale and suppose Hn and H are in IL. Further suppose that Hn -, H in ucp. Show that HFdX, --+ J H,dX, in ucp.

c(l+

*Exercise 14. Let X be a semimartingale, and let A be a non-negative, continuous, increasing process with A, = co a.s. Suppose limt,, A , ) - ' d ~ , exists and is finite a s . Show that limt,, = 0 a.s.

2

Exercise 15. Let M be a continuous local martingale and assume [M,MI, cm a.s. Show that limt,, = 0 a.s.

&

=

Exercise 16. Let B be a standard Brownian motion and let H be adapted and continuous. Show that for fixed t, lim h-+O

1

Bt+h - Bt

with convergence in probability.

/

t

HsdBs=Ht,

96

Exercises for Chapter I1

Exercise 17. Let Y be a continuous, adapted process with Yo = 0 and with Y constant on [I, GO). Let

Show that X n is a semimartingale, each n, but that limn,, not be a semimartingale.

Xn

=

Y need

Exercise 18. (Continuation of Exercise 17.) Let B be a standard Brownian t motion and let Xn = n B , d ~ l { , > ~ )Solve . n

and show that limn,, X; = Bt a.s., each t, but that limn,, Z solves the equation dZt = ZtdBt.

Zn

# Z, where

Exercise 19. (Related t o Exercises 17 and 18.) Let A: = sin(nt), 0 5 t 5 r / 2 . Show that An is a semimartingale for each n, and that 1dA:l = 1, each n (this means the total variation process, path-by-path, of A), but that

k

lim

sup

o
= 0.

*Exercise 20. (This exercise refers to an example briefly considered in Sect. 6 of Chap. I on local martingles, and again in Sect. 6 of Chap. 11.) Let B be a standard Brownian motion in R3, and let u : IW3 \ (0) 4 R be given by u(y) = Ilyll-'. Assume Bo = x a.s., where x # 0, show the following. (a) (b) (c) ( 4

Mt = u(Bt) is a local martingale. lirnt*, Ex(u(Bt)) = 0. u(Bt) E L2, each t 0. SUP,,, E " ( u ( B ~ ) ~ <)00.

>

Conclude that u(B) is not a martingale.

Exercise 21. Let A, C be two non-decreasing c&dl&gprocesses with A0 = Co = 0. Assume that both A, = limt,, At and C, = limt,, Ct are finite.

+

= J ~ ( A , - A,)dCs Jr(C, (a) Show that A,C, (b) Deduce from part (a) the general formula

- C,-)dA,.

Note that the above formulae are not symmetric. See also Theorem 3 in Chap. VI.

Exercises for Chapter 11

Exercise 22. Let A be non-decreasing and continuous, and assume A, limt,, At is finite, A. = 0. Show that for integer p > 0,

97 =

*Exercise 23 (expansion of filtrations). Let B be a standard Brownian motion on a filtered probability space (0,3 , IF, P),and define a new filtration G by G: = Ft v o(B1) and Gt = fin,,, G.: (a) Show that E{Bt - B, JG,) = (b) Show that

(B1 - B,) for 0 5 s 5 t 5 1.

where /3 is a G-Brownian motion. (Note in particular that the process (Bt - Jot sf::* ds)o
6'1~;::

~ d< s oo

as.

and conclude that B is a G semimartingale. This is a special case of Theorem 3 in Chap. VI.

Exercise 24. With the notation of Exercise 23, let Xt

= Bt - tB1,

0 5 t 5 1.

(a) Show that

(b) Deduce from (a) that X satisfies the stochastic differential equation

( X is called a Brownian bridge from 0 to 0.)

*Exercise 25. Let (Bt, Ft)t>o be a standard Brownian motion. Let a, b and T = infit > 0 : Bt E (-a;b)~). (a) Show that

is a local martingale. (b) Show that

1 a+b XT = e x p { - 0 2 ~ )cos(-0). 2 2

>0

98

Exercises for Chapter 11

(c) For 0 5 0 <

5 ,show that XT is a positive supermartingale, and deduce a+b 1 a-b COS(-S)E{~X~{~S~T))5 cos(-0). 2 2

(d) Use (c) to show that X$ martingale. (e) Conclude that

=

supslT lXsl E L1, and conclude that X is a

Exercise 26. Let ( B t , 3 t ) t > obe standard Brownian motion, and let T = infit > 0 : Bt E (-d,d)'). LG M = B ~ Show . that (a) if d

< 5, then E{exp{i [M, M]T)) < ca;but

(b) if d = 5 , then E{exp{i[M, MIT))

=

ca. (Hint: Use Exercise 25.)

Exercise 27. Let (Bt,3t)tlo be standard Brownian motion, and let Xt = e-at(Xo+a eaSdBS).Show that X is a solution to the stochastic differential equation dXt = -aXtdt adBt.

Sot

+

Exercise 28. Let B be a standard Brownian motion and let E(B) denote the stochastic exponential of B. Show that limt,, E(B)t = 0 a s . Exercise 29. Let X be a semimartingale. Show that

Exercise 30. Let B be a standard Brownian motion. (a) Show that M is a local martingale, where

(b) Calculate [M, MIt, and show that M is a martingale. (c) Calculate E{eBt ). The next eight problems involve a topic known as changes of time. For these problems, let (R, 3 , IF, P) satisfy the usual hypotheses. A change of time R = (Rt),?0 is a family of stopping times such that for every w E R, the function R.(w) is non-decreasing, right continuous, Rt < ca a s . , and Ro = 0. Let Gt = F R t . Change of time is discussed further in Sect. 3 of Chap. IV.

Exercise 31. Show that G = (Gt)tlo satisfies the usual hypotheses. Exercise 32. Show that if M is an IF uniformly integrable martingale and M t := MRt, then M is a G martingale.

Exercises for Chapter I1

Exercise 33. If M is an IF (right continuous) local martingale, show that is a 6 semimartingale.

99

M

*Exercise 34. Construct an example where M is an IF local martingale, but is not a (6 local martingale. (Hint: Let (Xn)nEN be an adapted process. It is a local martingale if and only if IXnldP is a a-finite measure on and E{Xn = XnP1,each n > 1. Find Xn where IXnldP is not a-finite on Fn-2, any n, and let Rn = 2n.)

M

*Exercise 35. Let R be a time change, with s H Rs continuous, strictly increasing, Ro = 0, and Rt < oo, each t 2 0. Show that for a continuous semimartingale X ,

JoRt

H,~x, =

for bounded H

E

IL.

Jo

t

HR,, ~

X

R

~

*Exercise 36. Let R and X be as in Exercise 35. No longer assume that 0, but instead assume that X is a finite variation Rt < co a s . , each t process. Let At = inf{s > 0 : Rs > t).

>

(a) Show that R strictly increasing implies that A is continuous. (b) Show that R continuous implies that A is strictly increasing. (c) Show that for general R, RA, t, and if R is strictly increasing and continuous then RA, = t. (d) Show that for bounded H E !L we have

>

(e) Show that for bounded H

E

IL we have

J,"'

HsdAs

=

Jlo

tAA,

HR,ds.

See in this regard Lebesgue's change of time formula, given in Theorem 45 of Chap. IV.

*Exercise 37. Let R be a change of time and let 6 be the filtration given by = F R t . Let At = inf{s > 0 : R, > t). Show that A = (At)tlo is a change of time for the filtration 6. Show also that if t 4 Rt is continuous a s . , Ro = 0, and R, = co, then RA, = t a s . , t > 0.

Gt

*Exercise 38. Let A, 6, be as in Exercise 37 and suppose that RA, = t a s . , t 2 0. Show that GA, c Ft,each t 0.

>

*Exercise 39. A function is Holder continuous of order cy if If (x) - f (y)l 5 Klx-y la. Show that the paths of a standard Brownian motion are a s . nowhere locally Holder continuous of order cy for any cy > 112. (Hint: Use the fact that = t.) limn,, C,n[O,tl(Bti+,-

Semimartingales and Decomposable Processes

1 Introduction In Chap. I1 we defined a semimartingale as a good integrator and we developed a theory of stochastic integration for integrands in lL, the space of adapted processes with left continuous, right-limited paths. Such a space of integrands suffices to establish a change of variables formula (or "It6's formula"), and it also suffices for many applications, such as the study of stochastic differential equations. Nevertheless the space lL is not general enough for the consideration of such important topics as local times and martingale representation theorems. We need a space of integrands analogous to measurable functions in the theory of Lebesgue integration. Thus defining an integral as a limit of sums-which requires a degree of smoothness on the sample paths-is inadequate. In this chapter we lay the groundwork necessary for an extension of our space of integrands, and the stochastic integral is then extended in Chap. IV. Historically the stochastic integral was first proposed for Brownian motion, then for continuous martingales, then for square integrable martingales, and finally for processes which can be written as the sum of a locally square integrable local martingale and an adapted, c&dl&gprocesses with paths of finite variation on compacts; that is, a decomposable process. Later Dol6iansDade and Meyer [53] showed that the local square integrability hypothesis could be removed, which led to the traditional definition of a semimartingale (what we call a classical semimartingale). More formally, let us recall two definitions from Chaps. I and I1 and then define classical semimartingales. Definition. An adapted, c&dl&gprocess A is a finite variation process ( F V ) if almost surely the paths of A are of finite variation on each compact interval of [O,oo). We write IdA,I or /A(, for the random variable which is the total variation of the paths of A.

Jr

Definition. An adapted, c&dl&gprocess X is decomposable if there exist processes N, A such that

102

I11 Semimartingales and Decomposable Processes

with No = A. = 0, N a locally square integrable local martingale, and A an F V process. Definition. An adapted, cadlag process Y is a classical semimartingale if there exist processes N, B with No = Bo = 0 such that

where N is a local martingale and B is an FV process. Clearly an F V process is decomposable, and both FV processes and decomposable processes are semimartingales (Theorems 7 and 9 of Chap. 11). The goal of this chapter is to show that a process X is a classical semimartingale if and only if it is a semimartingale. To do this we have to develop a small amount of "the general theory of processes." The key result is Theorem 25 which states that any local martingale M can be written

where N is a local martingale with bounded jumps (and hence locally square integrable), and A is an FV process. An immediate consequence is that a classical semimartingale is decomposable and hence a semimartingale by Theorem 9 of Chap. 11. The theorem of Bichteler and Dellacherie (Theorem 43) gives the converse: a semimartingale is decomposable. We summarize the results of this chapter, that are important to our treatment, in Theorems 1 and 2 which follow. Theorem 1. Let X be an adapted, chdldg process. The following are equivalent:

(i) X is a semimartingale; (ii) X is decomposable; (iii) given 0> 0, there exist M , A with Mo = A. = 0, M a local martingale with jumps bounded by 0,A an FV process, such that Xt = Xo+ Mt + A,; (iv) X is a classical semimartingale. Definition. The predictable a-algebra P on R+ x R is the smallest aalgebra making all processes in IL measurable. We also let P (resp. b P ) denote the processes (resp. bounded processes) that are predictably measurable. The next definition is not used in this chapter, except in the Exercises, but it is natural to include it with the definition of the predictable a-algebra. Definition. The optional a-algebra O on R+ x R is the smallest a-algebra making all cadlag, adapted processes measurable. We also let O (resp. bO) denote the processes (resp. bounded processes) that are optional.

2 The Classification of Stopping Times

103

Theorem 2. Let X be a semimartingale. If X has a decomposition Xt =

Xo + Mt + At with M a local martingale and A a predictably measurable F V process, Mo = A.

= 0,

then such a decomposition is unique.

In Theorem 1, clearly (ii) or (iii) each imply (iv), and (iii) implies (ii), and (ii) implies (i). That (iv) implies (iii) is an immediate consequence of the Fundamental Theorem of Local Martingales (Theorem 25). While Theorem 25 (and Theorems 3 and 22) is quite deep, nevertheless the heart of Theorem 1 is the implication (i) implies (ii), essentially the theorem of K. Bichteler and C . Dellacherie, which itself uses the Doob-Meyer decomposition theorem, Rao's Theorem on quasimartingales, and the Girsanov-Meyer Theorem on changes of probability laws. Theorem 2 is essentially Theorem 30. We have tried t o present this succession of deep theorems in the most direct and elementary manner possible. In the first edition we were of the opinion that Meyer's original use of natural processes was simpler than the now universally accepted use of predictability. However, since the first edition, R. Bass has published an elementary proof of the key Doob-Meyer decomposition theorem which makes such an approach truly obsolete. We are pleased t o use Bass' approach here; see [ l l ] .

2 The Classification of Stopping Times We begin by defining three types of stopping times. The important ones are predictable times and totally inaccessible times.

Definition. A stopping time T is predictable if there exists a sequence of - such that Tn is increasing, Tn < T on {T > O), all n, stopping times (Tn)n>l and limn,, Tn = T a s . Such a sequence (T,) is said t o announce T . If X is a continuous, adapted process with Xo = 0, and T = inf{t : IXt 1 2 c), for some c > 0, then T is predictable. Indeed, the sequence Tn = inf{t : lXt 1 2 c - A n is an announcing sequence. Fixed times are also predictable.

i)

Definition. A stopping time T is accessible if there exists a sequence ( T k ) k l l of predictable times such that 00

P ( U { w : T ~ ( w=)T ( w ) < co)) k=l

=P(T

< co).

Such a sequence (Tk)kll is said t o envelop T . Any stopping time that takes on a countable number of values is clearly accessible. The first jump time of a Poisson process is not a n accessible stopping time (indeed, any jump time of a L4vy process is not accessible).

Definition. A stopping time T is totally inaccessible if for every predictable stopping time S,

104

I11 Semimartingales and Decomposable Processes

Let T be a stopping time and A E FT. We define

TA(w) =

T(w), oo,

ifw~A, ifwen.

I t is simple t o check that since A E FT,TA is a stopping time. Note further that T = m i n ( T ~TA=) , = TAA TAc.

A simple but useful concept is that of the graph of a stopping time. Definition. Let T be a stopping time. The graph of the stopping time T is the subset of R+ x R given by {(t,w) : 0 5 t = T(w) < oo); the graph of T is denoted by [TI.

Theorem 3. Let T be a stopping time. There exist disjoint events A, B such that A U B = {T < oo) a.s., TA is accessible and TB i s totally inaccessible, and T = TAA TB a.s. Such a decomposition i s a.s. unique.

Proof.If T is totally inaccessible there is nothing t o show. So without loss of generality we assume it is not. We proceed with an inductive construction: Let R1 = T and take

a1 = sup{P(S

= R1

< 00)

: S i s predictable).

Choose S1 predictable such that P ( S l = R1 < oo) Define Rz = R1{VIZR,).For the inductive step let

>

2 and set Vl

=

S1.

ai = sup{P(S = Ri < oo) : S i s predictable) If ai = 0 we stop the induction. Otherwise, choose Si predictable such that P(Si = Ri < oo) > % and set = Si{sidoes not equal any of v,,l<j 0 and hence P ( W = U) > ai for some i. This contradicts how we chose Si a t step i of the induction. Therefore U is totally inaccessible, and B={U=T
A beautiful application of Theorem 3 is Meyer's Theorem on the jumps of Markov processes, a special case of which we give here without proof. (We use the notation established in Chap. I, Sect. 5 and write IF@ = ( F ~ o l t l , . ) Theorem 4 (Meyer's Theorem). Let X be a (strong) Markov Feller process where the distribution of Xo is given by p, and with its for the probability P@, natural completed filtration IF@. Let T be a stopping time with P @ ( T> 0) = 1. Let A = {W : XT(W)# XT- (w) and T(w) < oo). Then T = TAA TAe, where TA is totally inaccessible and TA=is predictable.

3 The Doob-Meyer Decompositions

105

A consequence of Meyer's Theorem is that the jump times of a Poisson process (or more generally a Lbvy process) are all totally inaccessible. We will need a small refinement of the concept of a stopping time a-algebra. This is particularly important when the stopping time is predictable. Definition. Let T be a stopping time. The a-algebra FT- is the smallest a-algebra containing Fo and all sets of the form A n { t < T), t > 0 and A E Ft. Observe that FT-c FT,and also the stopping time T is FT-measurable. We also have the following elementary but intuitive result. We leave the proof to the reader.

Theorem 5. Let T be a predictable stopping time and let (Tn),_>l be a n . announcing sequence for T . Then FT-= a{Un,- FT,,) = Vn2 FT,,

3 The Doob-Meyer Decompositions We begin with a definition. Let N denote the natural numbers.

Definition. An adapted, c&dl&gprocess X is a potential if it is a nonnegative supermartingale such that limt,, E{Xt) = O. A process (Xn)nEN is also called a potential if it is a nonnegative supermartingale for N and limn,, E{Xn) = 0. Theorem 6 (Doob Decomposition). A potential (Xn)nENhas a decomposition Xn = Mn - An, where An+1 An a.s., A. = 0, An E Fn-l,and Mn = E{A,IFn). Such a decomposition is unique.

>

+

Proof. Let Mo = Xo and A. = 0. Define Ml = Mo (X1 - E{XllFo)), and A1 = Xo - E{XllFo}. Define Mn, An inductively as follows:

<

Note that E{An) = E{Xo) - E{Xn) E{Xo) < oo, as is easily checked by induction. It is then simple to check that Mn and An so defined satisfy the hypotheses. Next suppose Xn = Nn - Bn is another such representation. Then Mn Nn = An - Bn and in particular Ml - Nl = Al - B1E Fo. Thus Ml - Nl = E{Ml - Nl IFo)= Mo - No = Xo - Xo = 0, hence Ml = Nl. Continuing inductively shows Mn = Nn, all n. We wish to extend Theorem 6 to continuous time supermartingales. Note which of course is the unusual measurability condition that An E FnP1 stronger than simply being adapted. The continuous time analog is that the process A be natural or what turns out to be equivalent, predictably measurable.

106

I11 Semimartingales and Decomposable Processes

Throughout this paragraph we assume given a filtered probability space

(R, 3,IF, P ) satisfying the usual hypotheses. Before we state the first decomposition theorem in continuous time, we establish a simple lemma.

Lemma. Let (Yn)n21 be a sequence of random variables converging t o 0 in L ~ Then . sup, E{YnlFt} -+ 0 in L ~ . Proof. Let M p = E{Ynl.Ft) which is of course a martingale for each fixed n . Using Doob's maximal quadratic inequality, E{sup, (Mp)') 5 4 ~ { ( M z ) ' ) = 4E{Y;) + 0.

Definition. We will say a c&dl&gsupermartingale Z with Zo = 0 is of Class D if the collection {ZT : T a finite valued stopping time) is uniformly integrable. The name "Class D" was given by P. A. Meyer in 1963. Presumably he expected it t o come t o be known as "Doob Class" a t some point, but it has stayed Class D for 40 years now, so we see no point in changing it. (There are no Classes A, B, and C.) We now come t o our first version of the Doob-Meyer decomposition theorem. It is this theorem that gives the fundamental basis for the theory of stochastic integration.

Theorem 7 (Doob-Meyer Decomposition: Case of Totally Inaccessible Jumps). Let Z be a cadlag supermartingale with Zo = 0 of Class D, and such that all jumps of Z occur at totally inaccessible stopping times. Then there exists a unique, increasing, continuous, adapted process A with A. = 0 such that Mt = Zt + At is a uniformly integrable martingale. We first give the proof of uniqueness which is easy. For existence, we will first establish three lemmas.

Proof of uniqueness. Let Z = M - A and Z = N - C be two decompositions of Z. Subtraction yields M - N = A - C , which implies that M - N is a continuous martingale with paths of finite variation. We know however by the Corollary of Theorem 27 of Chap. I1 that M - N is then a constant martingale which implies M, - Nt = Mo - No = 0 - 0 = 0 for all t. Thus M = N and A = C.

Lemma 1. Let IF be a discrete time filtration and let C be a non-decreasing process with Co = 0, and Ck E F k P l . Suppose there exists a constant N > 0 suchthat E{C, - C k l F k ) 5 N a s . for all Ic. Then E { C k ) 5 2N2. Proof of Lemma 1. First observe E{C,) = E{E{C, - Co l30)) 5 N . Letting ck = Ck+1 - Ck 0, we obtain by rearranging terms:

>

Thus it follows that

3 The Doob-Meyer Decompositions

Choose and fix a constant v E Z+and let

D, = {k2-,

:0

5 k2-,

107

< v).

Lemma 2. Let T be a totally inaccessible stopping time. For 6 > 0, let R(6) = sup,,,- P ( t < T < t + &IFt).Then R(6) + 0 in probability as 6 + 0.

+

Proof of Lemma 2. Let a > 0 and Sn(6) = inft{t E D, : P ( t < T < t &IFt)> a) A v. First we assume Sn(6) is less than T. Since Sn(6) is countably valued, it is accessible, and since T is totally inaccessible, P(Sn(6) = T ) = 0. Suppose r c {T < t), and also r E Ft. Then

Suppose now P ( T < Sn(6)) > 0. Then for some t E D,, P ( T < t, Sn(6) = t) > 0. Let r = {T < t, Sn(6) = t). Then from the definition of Sn(6) we obtain 1Ft't)lr> alr, (*) E{1(t<~ 0 such that P(R(6) > a) > E, for all 6 tending to 0. Let P > 0 and 6 < P. For n sufficiently large

We also have P ( T = S(6)+6) = 0, since if not ~ ( 6 ) + 6 - A could announce part of T which would contradict that T is totally inaccessible. Thus P({Sn(6) T Sn(6)+6) A ( ~ ( 6 < ) T S(6) +6)) -+ 0 as n -+ co where the symbol A denotes the symmetric difference set operation. Since the symmetric difference tends to 0, if P(Sn(6) < T _< Sn(6) 61Fs,(S) 2 a) 2 E for any 6 > 0, then we must also have

<

<

<

+

since otherwise were it to tend to 0 along with the symmetric difference tending to 0, then we would have (*) + 0, a contradiction. Thus we have P ( s ( 6 ) T < S(6) /?IF9(6)) > a on a set A, with P(A) 2 E. From the definition of S and working on the set A, this implies P(E{l{s
+

<

>

108

I11 Semimartingales and Decomposable Processes

Lemma 3. Let the hypotheses of Theorem 7 hold and also assume IZI 5 N where N > 0 is a constant, and further that the paths of Z are constant after + , - Z, IFt}. Then W(6) --, 0 a stopping time v. Let W(6) = ~ u p , ~ , ~ ,E{Zt in L2 as 6 + 0.

Proof of Lemma 3. Since IZI 5 N we have JW(6))5 2N. Thus it to show W(6) --, 0 in probability. Let E, a > 0 and b = a&. Z? = CsltAZsl{~z,>b},Zt- = Cs5tA z S l { - ~ z ~ > bZt }b, = zt - ( ~ ~ + 6 - Z;l.Ft}l, with W+(6) and W-(6) and Wb(6) = ~ u p ~ ~ , <(E{z,~ analogously. Then

suffices Define t++z~), defined

which by Doob's inequality implies

However since Zb is cBdlBg and lAZbl 5 b, Fatou's Lemma implies 1 b b 2 b2 limsup-E{ sup 12,-ZsI } I - = E . 6-0 a2 T<S
Also, again using that Z is c&dl&g,for each fixed w the number of jumps larger than b is finite. Define inductively TI = inf{t > 0 : AZt > b), T,+l =inf{t > T , : AZt > b). Since IZI 5 N be hypothesis, JAZT< I 5 2N. Choose k such that P(Tk 5 v) < E. Then

By the previous lemma we know that R(6) + 0 in L2, SO by taking 6 small enough, we get the above expression less than 2 ~ The . reasoning for W- is analogous. We achieve W(6) 5 wb(6)+ ~ + ( 6 )W-(6), which gives W(6) + 0 in L2.

+

We return to the proof of Theorem 7.

Proof of existence. First suppose the jumps of Z are bounded by a constant c. Let TN = infit > 0 : IZtl 2 N - c} A N, and Z p = ZtATN.Then IZpI 5

3 The Doob-Meyer Decompositions

109

+

12; 1 c < N, and Z N is constant after TN. Thus we are now reduced to the case considered by the Lemma 2. Fix n and let F E = F+. Define

All the a; 2 0 since Z is a supermartingale, and also a; E q-l. Let A; = a:. Then Lk = Z& + A; is an q discrete time martingale. Define

I;=,

>

Wewant to show that E{supt IBF-BPI2} --+ 0 as n, m + co.Suppose m n. Since Bm and Bn are constant on intervals of the form ( k Y m ,(k + 1)2-m], the sup of the difference occurs at some k2Ym. Fix t and let u = inf{s : s E D, and s 2 t}. Observe that E{AE A r I c } = E{Zt - Z,IFt} is bounded by 2N. Also

which is bounded by 2N. This implies E{Bg

-

B& - ( B y - B,")IFt}

= E{Zt - ZuIFt},

with the right side being nonnegative and bounded by W(2-"), and of course also by E{W(2-n)lFt}. Using Lemmas 1 and 2 we get that E{sup, IBF B,"I2) --+ 0 as n, m t co. Therefore the sequence B: is a Cauchy sequence and we denote its limits by At. Next we show that the process A has continuous paths. We do this by analyzing the jumps of the approximating processes Bn. Note that for t = k2-n, ABr = E{Zk-I - Z A / q - l ) ?K

which are bounded by W(2-"). But then sup, 1 ABFl + 0 in L ~ Thus . there exists a subsequence (nj) such that sup, lAB2 I + 0 a.s., and hence the limit A is continuous. It remains to show that Zt +At is a uniformly integrable martingale. Both are square integrable, so we only need concern ourselves with the martingale property. Let s , t E D,, s < t, and A E Fs. Then E{(Zt At)ln) = E{(Zs A,)lA}. The result follows by taking limits of Zt BF. At the beginning of the proof we made the simplifying assumption that Z had bounded jumps. We now treat the general case. Choose N and let T = infit > 0 : lZtl 2 N ) A N . Then ZT (that is, Z stopped at the time T ) has at most one jump bigger than 2N. By localization and the uniqueness of the decomposition, it suffices to prove the result for this case. Thus without

+

+

+

110

I11 Semimartingales and Decomposable Processes

loss of generality we assume Z has at most one jump greater than or equal to 2N in absolute value, that it occurs at the time T , and that Z is constant after the time T. We let

Since -Z+ and Z- both have decreasing paths, they are both supermartingales. Suppose we can show the theorem is true for -Z+ and Z-. Let -Z+ = M + - A+ and Z- = M - - A- be the two decompositions, with A+ and Aboth continuous. hen 2= z + ( - z + + A + ) - ( 2 - + A F ) = Z + M + - M - is a supermartingale with jumps bounded by 2 N . Let 2 = M - A be the unique decomposition of 2 which we now know exists. Then Z = 2 + M + - M-, and therefore Z = ( M + M + - M - ) - A is the desired (unique) Doob-Meyer decomposition. Thus it remains to show that -Z+ and Z- both have Doob-Meyer decompositions. First observe that lAZTl 5 IZT-I lZTl 5 N lZTl E L1, and hence E{lAZTI) < co. Choose a, E > 0. Next choose R > N so large that E { ~ A Z T ~ l ~ l A z T 1I2 Ea. R l )The cases for when the jump is positive and when the jump is negative being exactly the same except for a minus sign, we only treat the case where the jump is negative. Let Z p = A Z T ~ { ~ ~ ~ ) ~ { - ~ ~ ~ > ~ ) , and Z,d = ZF - 2.: We define Bn- ,BnR,Bnd analogously to the way we defined Bn in equation (*) above. We first show BF- converges uniformly in t in probability. We have:

+

+

The second term on the right side of (**) above is less than 3 ~ :

'The third term on the right side of (**) is shown to be less than 3~ similarly. Since lZRl is bounded by R, the first term on the right side of (**) can be made arbitrarily small by taking m and n large enough, analogous to what we did at the beginning of the proof for Bn. : converges Thus as we did before in this proof, we can conclude that B+ uniformly in t in probability as n + co,and we denote its limit process by A?.

3 The Doob-Meyer Decompositions

111

We prove continuity of A exactly as before. Finally by taking a subsequence such that Bn3+ converges almost surely, we get

nj

E{A&) = E{ lim B Z + ) n j +cc

< ln3i r n i n f ~ { ~ Z +=) '00

by Fatou's Lemma. El-om this it easily follows that Z+ integrable martingale, and the proof is complete.

+ A+

is a uniformly

While Theorem 7 is sufficient for most applications, the restriction to supermartingales having jumps only at totally inaccessible stopping times can be insufficient for some needs. When we move to the general case we no longer have that A is continuous in general (however see Exercises 24 and 26 for supplemental hypotheses that assure that the increasing process A of the Doob-Meyer decomposition is in fact continuous). Without the continuity we lose the uniqueness of the decomposition, since there exist many martingales of finite variation (for example, the compensated Poisson process) that we can add to the martingale term and subtract from the finite variation term of a given decomposition, to obtain a second, new decomposition. Instead of the continuity of A we add the condition that A be predictably measurable.

Theorem 8 (Doob-Meyer Decomposition: General Case). Let Z be a ccidlcig supermartingale with Zo = 0 of Class D. Then there exists a unique, increasing, predictable process A with A. = 0 such that Mt = Zt + At i s a uniformly integrable martingale. Before proving Theorem 8 let us introduce the concept of a natural process. We introduce two definitions and prove the important properties of natural processes in the next three theorems.

Definition. An F V process A with A. = 0 is of integrable variation if the expected total variation is finite: E { J IdA, ~ ~ I ) < co. A shorthand notation for this is E{IAI,) < co. An F V process A is of locally integrable variation if there exists a sequence of stopping times (Tn),?l increasing to co a.s. such ~ = E{IAIT,,} < m , for each n. that E { J IdAal} Definition. Let A be an (adapted) F V process, A. = 0, of integrable variation. Then A is a natural process if

for all bounded martingales M. Here is the key theorem about natural processes. This use of natural processes was Meyer's original insight that allowed him to prove Doob's conjecture, which became the Doob-Meyer decomposition theorem.

Theorem 9. Let A be an F V process, A. = 0 , and E{IAI,) is natural i f and only i f

< co. Then A

112

I11 Semimartingales and Decomposable Processes

for any bounded martingale M . Proof. By integration by parts we have

Then MoAo = 0 and letting Nt = GAS-dM,, we know that N is a local martingale (Theorem 20 of Chap. 11). However using integration by parts we see that E { N Z ) < co,hence N is a true martingale (Theorem 51 of Chap. I). A,- dM, } = E{N,) - E{No) = 0, since N is a martingale, Therefore E so that the equality holds if and only if E { [ M A],) , = 0.

{JF

Theorem 10. Let A be an FV process of integrable variation which is natural. If A is a martingale then A is identically zero.

Proof. Let T be a finite stopping time and let H be any bounded, nonnegative T martingale. Then E{J, Hs-dA,) = 0, as is easily seen by approximating ldAsl t L1 and sums and the Dominated Convergence Theorem, since E { A T ) = 0. Using the naturality of A, E{HTAT) = Hs-dA,) = 0, and letting Ht = E { l ~ ~ , > o ~ then ~ F tshows } that P(AT > 0) = 0. Since E { A T ) = 0, we conclude AT 3 0 as., hence A 0.

-

~ { g

Theorem 11. Let A be an FV process of integrable variation with A. = 0. If A is predictable, then A is natural.

The proof of this theorem is quite intuitive and natural provided we accept a result from Chap. IV. (We do not need this theorem to prove the theorem we are using from Chap. IV.) Proof. Let M be a bounded martingale. First assume A is bounded. Then the stochastic integral JowASdMsexists, and it is a martingale by Theorem 29 in Chap. IV combined with, for example, Corollary 3 of Theorem 27 of Chap. 11. Therefore E { ~ ~ ~ A , ~ =M E{AoMo) ,) = 0, since Ao = 0. However E { J ~ ~ A , - ~=M E{Ao-Mo) ~) = 0 as well, since Ao- = 0. Further,

3 The Doob-Meyer Decompositions

113

since A is a quadratic pure jump semimartingale. Therefore

Since M was an arbitrary bounded martingale, A is natural by definition. Finally we remove the assumption that A is bounded. Let An = n ~ ( A v ( - n ) ) . Then An is bounded and still predictable, hence it is natural. For a bounded martingale M E { [ M ,A],) = lim E { [ M ,An],) = 0, n

by the Dominated Convergence Theorem. Therefore A is natural. The next theorem is now obvious and should be called a Corollary at best. Because of its importance, however, we give it the status of a theorem. Theorem 12. Let M be a local martingale with paths of finite variation on compact time sets. If M is predictably measurable, then M is constant. That is, Mt = Mo for all t , almost surely.

Proof. This theorem is a combination of Theorems 10 and 11. Proof of Theorem 8 (Doob-Meyer: General Case). We begin with the uniqueness. Suppose Z = M - A and Z = N - C are two decompositions. By subtraction we have M - N = A - C is a martingale with paths of finite variation which is predictable. By Theorem 12 we have that M - N is constant and since Mo - No = 0, it is identically zero. This gives the uniqueness. The existence of the decomposition is harder. We begin by defining stopping times Tn,j,where Tn,j is the j-th time lAZtl is in the half-open interval bounded by 2Tn and 2-(n-1), where -co < n < co. We decompose T n j into its accessible and totally inaccessible parts. Since we can cover the accessible part with a sequence of predictable times, we can thus separate each TnTjinto a totally inaccessible time and a sequence of predictable times, with disjoint (stopping time) graphs. Therefore, by renumbering, we can obtain a sequence of stopping times (Si)i21 with disjoint graphs, each one of which is either predictable or totally inaccessible, which exhaust the jumps of Z (in the sense that the jump times of Z are contained in the union of the graphs of the Si) and are such that for each i there exists a constant bi such that bi < lAZs,l 5 2 b i . We define Z o ( t ) = Zt and inductively define Z,+l(t) = Z i ( t ) and A i ( t ) = 0 if Si is totally inaccessible; whereas A i ( t ) = -E{AZsi l F s i - ) l ~ s , 5 t l and then Zi+l(t) = Z i ( t ) A i ( t ) in the case where Si is predictable (note that AZs, E L1). We will show that: each Ai is increasing; Z, is a supermartingale for each i ; and E { E : = ~A j ( m ) }5 C, for each i , where C is a constant not depending on i . We will prove these three properties by induction. We begin by showing that each Ai is increasing. This is trivial if Si is totally inaccessible. Let then

+

114

I11 Semimartingales and Decomposable Processes

Si be a predictable time. Let Sr be an announcing sequence for Si. Since each Zi is a supermartingale, and using the Martingale Convergence Theorem, we have E{AZi(Si)lFs,-) = limn E{AZi(Si)l.Fsl').Now fix n: E{AZi(Si)lFs?}= lim E{Zi(Si)- Z i ( s ~ ) l F s ; ) k

= lim E{E{Zi(Si)- zi(~,")I.Fsk)JFST) Ic

1 0, and thus Ai is increasing. To see that Zi+l is a supermartingale, it will suffice to show that whenever Ul and U2 are stopping times with Ul 5 U2 then E{Zi+l( U l ) ) 2 E{Zi+l(U2)).Again letting S? be an announcing sequence for Si we have

where each of the summands on the right side above are nonnegative. Then let n + co to get E{Zi(Ul))- E{Zi ( U 2 ) )2 0. Finally we want to show that E{c:=, Aj(co))5 C. It is easy to check (see Exercise 1 for example) that the minimum of predictable stopping times is predictable, hence we can order the predictable times so that S1 < S2 < . . . on the set where they are all finite. Let A, be the collection of all j such that j i and such that the S j are all predictable. Then

which is bounded by a constant not depending on i. To complete the proof, we note that because the processes Ai are increasing in t and the expectation of their sum is bounded independently of Ai(t) converges uniformly in t as., as h + co, and i, we have that we call the limit A,(t). Fatou's Lemma gives us that A,(t) is integrable. Each Ai(t) is easily seen to be predictable, and hence A, is the almost sure limit of predictable measurable processes, so it too is predictable. Next set Z , ( t ) = Z, + A, ( t ) = limi Zi ( t ) . Since each Zi is a supermartingale, coupled with the uniform convergence in t of the partial sums of the Ai processes, we get that Z,(t) is a supermartingale. Moreover since each Zi is c&dl&g,again using the uniform convergence of the partial sums we obtain

EL,

3 The Doob-Meyer Decompositions

115

that Z, is also cAdlAg. Since the partial sums are uniformly bounded in expectation, Z,(t) is of Class D. Finally by our construction of the stopping = 0 for all predictable times T. We can then times Si,E{AZ,(T)IFT-) show as before that W ( 6 )+ 0 in probability as 6 -+ 0 for Z,, and we ob. process At = A,(t) + ~ ( t is) then the desired tain Z,(t) = Mt - ~ ( t )The increasing predictable process. The next theorem can also be considered a Doob-Meyer decomposition theorem. It exchanges the uniform integrability for a weakening of the conclusion that M be a martingale to that of M being a local martingale.

Theorem 13 (Doob-Meyer Decomposition: Case Without Class D). Let Z be a cidl&g supermartingale. Then Z has a decomposition Z = Zo + M -A where M is a local martingale and A is an increasing process which is predictable, and Mo = A. = 0. Such a decomposition is unique. Moreover if limt,, E{Zt) > -co, then E{A,) < co.

+

Proof. First consider uniqueness. Let Z = Zo + M - A and Z = Zo N - C be two decompositions. Then M - N = A - C by subtraction. Hence A - C " a uniformly integrable martingale. Then is a local martingale. Let M ~ be

and therefore E { A ~ )5 E{Zo - Zt), using Theorem 17 of Chap. I. Letting n tend to co yields E{At) 5 E{Zo - Zt). Thus A is integrable on [0,to], each to, as is C. Therefore A - C is of locally integrable variation, predictable, and a local martingale. Since A. - Co = 0, by localization and Theorems 10 and 11, A - C = 0. That is, A = C , and hence M = N as well and we have uniqueness. Next we turn to existence. Let Tm = infit : lZtl 2 m) A m. Then Tm increase to co a.s. and since they are bounded stopping times ZTm E L1 each m (Theorem 17 of Chap. I). Moreover the stopped process ZTm is dominated by the integrable random variable max(lZTm1, m). Hence if X = ZTm for fixed m, then 7-l = {XT;T a stopping time) is uniformly integrable. Let us implicitly stop Z at the stopping time Tm and for n > 0 define qn= Zt - E{Znl.Ft}, = 0 when t 2 n. Then Yn is a positive supermartingale with Y," = q:, Mtn - A : Letting N p = E{Znl.Ft}, a of Class D and hence Y," = Y$ martingale, we have on [0,n] that

+

To conclude, therefore, it suffices to show that A? = AT on [0,n], for m 2 n. This is a consequence of the uniqueness already established. The uniqueness also allows us to remove the assumption that Z is stopped at the time Tm. Finally, note that since E{At) 5 E{Zo - Zt), and since A is increasing, we have by the Monotone Convergence Theorem

116

I11 Semimartingales and Decomposable Processes E{A,)

which is finite if limt,,

=

,'t

lim E{At) 5 lim E{Zo - Zt) ,'t

E{Zt) > -co.

4 Quasimartingales Let X be a c&dlhg, adapted process defined on [0, co].'

Definition. A finite tuple of points T = (to, t l , . . . ,tn+1) such that 0 t l < . . . < tn+l = co is a partition of [0, co]. Definition. Suppose that

T

= to

<

is a partition of [0, co] and that Xti E L1, each

ti E T. Define

n

C(X, 7) The variation of X along

= T

C IE{Xti i=O

1.6;)I.

is defined t o be

Var, ( X ) = E{C(X, 7)). The variation of X is defined t o be Var ( X ) = sup Var, ( X ), 7

where the supremum is taken over all such partitions.

Definition. An adapted, c8dlag process X is a quasimartingale on [0, co] if E{IXtl) < co, for each t, and if Var(X) < co. Before stating the next theorem we recall the following notational convention. If X is a random variable, then X + = max(X, 0), X-

=

-

min(X, 0).

Also recall that by convention if X is defined only on [0, co), we set X,

= 0.

Theorem 14. Let X be a process indexed by [0,co). Then X is a quasimartingale if and only if X has a decomposition X = Y - Z where Y and Z are each positive right continuous supermartingales. It is convenient when discussing quasimartingales to include m in the index set, thus making it homeomorphic to [O, t ] for 0 < t < m. If a process X is defined only on [0,m ) we extend it to [O, m] by setting X , = 0.

4 Quasimartingales

117

Proof. For given s 2 0, let C ( s ) denote the set of finite subdivisions of [s, co]. For each T E C ( s ) , set

Y,' = E{C(X,T)'~F,)

and 2: = E{C(X,T)-IF,}

where C(X, T ) + denotes CttE7 E{Xti - Xt,+, IFti)+, and analogously for C(X, 7)-. Also let 4 denote the ordering of set containment. Suppose a,T E C ( s ) with a 4 T. We claim Y t I Y,7 a.s. To see this let a = (to,.. . ,t,). It suffices to consider what happens upon adding a subdivision point t before to, after t,, or between ti and ti+l. The first two situations being clear, let us consider the third. Set

then C

=A

+ E{BIFti), hence

by Jensen's inequality. Therefore

<

Y,7. Since E{Y:) is bounded by Var(X), taking limits and we conclude Yf in L1 along the directed ordered set C ( s ) we define

and we can define 2,analogously. Taking a subdivision with to = s and t,+l co, we see Y,7 - 2; = E{C+ - C-IF,) = X,, and we deduce that Y, - Zs zX,. Moreover if s < t it is easily checked that Y, ',2{fi1FS) ^and zs 2 E{ZtlFs}. Define the right continuous processes K %+, Zt Zt+, with the right limits taken through the rationals. Then Y and Z are positive supermartingales and Y, - 2, = X,. For the converse, suppose X = Y - Z, where Y and Z are each positive supermartingales. Then for a partition T of [0,t]

- -

Thus X is a quasimartingale on [0,t], each t > 0.

118

I11 Semimartingales and Decomposable Processes

Theorem 15 (Rao's Theorem). A quasimartingale X has a unique decomposition X = M A, where M is a local martingale and A is a predictable process with paths of locally integrable variation and A. = 0.

+

Proof. This theorem is a combination of Theorems 13 and 14.

5 Compensators Let A be a process of locally integrable variation, hence a fortiori an F V process. A is then locally a quasimartingale, and hence by Rao's Theorem (Theorem 15)' there exists a unique decomposition

where 2 is a predictable-FV process. In @her words, there exists a unique, predictable F V process A such that A - A is a local martingale.

Definition. Let A be an F V process with A. = 0, with locally integable total variation. The unique F V predictable process A such that A - A is & local martingale is called the compensator of A. The most common examples are when A is an increasing process locally of integrable variation. When A is an increasing process it is of course a submartingale, and thus by the Doob-Meyer Theorem we know that its compensator A is also increasing. We also make the obvious observation that E{At) = E{&) for all t, O < t co.

<

Theorem 16. Let A be an increasing process of integrable variation, and let H t lL be such that HsdAs) < co. Then,

E{G

Proof. Since A - A is a martingale, so also is has constant expectation equal to 0.

J H,d(A,

-

A,), and hence it

In Chap. IV we develop stochastic integration for integrands which are predictable, and Theorem 16 extends with H t lL replaced with H predictable. One of the simplest examples to consider as an illustration is that of the Poisson process N = (Nt)t20 with parameter A. Recall that Nt - At is a martingale. Since the process At = At-is continuous and obviously adapted, it is predictable (natural). Therefore Nt = At, t 2 0. A natural extension of the Poisson process case is that of counting processes without explosions. We begin however with a counting process that has only one jump. Let 77 be the counting process vt = l{t271, where T is a nonnegative random variable. Let F be the minimal filtration making T a stopping time.

5 Compensators

119

Theorem 17. Let P(T = 0) = 0 and P(T > t) > 0, each t > 0. Then the F compensator A of q, where qt = 1{t>71, is given by

where F is the cumulative distribution function of T. If T has a diffuse distribution (that is, if F is continuous), then A is continuous and At = - ln(1 - F ( T A t)). Before proving Theorem 17 we formalize an elementary result as a lemma. Lemma. Let (fl,F,P ) be a complete probability space. In addition, suppose T is a positive F measurable random variable and = a { A~t), where = V N , N are the null sets of F , and Ft = n u > t Then F so constructed is the smallest filtration making T a stopping time. Let Y E L1(F). Then

*

C0

C0

c.

*

*

Proof. By the hypotheses on T the a-algebra is equal to the Bore1 aalgebra on [0,t] together with the indivisible atom (t, 00). Observe that = a { A~u; u 5 t), and the result follows easily. Proof of Theorem 17. Fix to > 0 and let rn be a sequence of partitions of [0,to] with liw,, mesh(rn) = 0. Define A: = C,, E{qti+l - qti IFt,)for 0 5 t 5 to. Then

E{~ti+l lFttt,) = ~ t i +l{rIt,) l

+ E{l{7>t;}~ti+l) P(T > ti) Q7>ti}

by the lemma preceding this proof. The first term on the right above > ~ } = '{ti27}. firthermore, '{7>ti}~ti+l = is ~ t ~ + ~ l { t=~ l{ti+127}1{ti27} '{7>ti} 1{ti+1>7)= l{ti<7
Therefore,

These simple calculations yield

120

I11 Semimartingales and Decomposable Processes

which are Riemann sums converging to XA'(l maining claims of the theorem are obvious.

-

~ ( u - ) ) - ' d F ( u ) . The re-

Corollary 1. If F in Theorem 17 is absolutely continuous, then so also is A. Corollary 2. If At.

T

in Theorem 17 has an exponential distribution, then At

=

T

Proof. Since in this case F is continuous and a distribution function, we can represent it as F ( x ) = 1 - e-d(x). We have that At = - ln(1 - F(TA t)) and equivalently we can write At = $(T At). In the case of this corollary, the function $(x) = x, and the result follows.

Remark. Theorem 17 gives an explicit formula for the compensator of a process qt = l{t27} when one uses the minimal filtration making T a stopping time. A perhaps more interesting question is the following: suppose that one has a non-trivial filtration IF satisfying the usual hypotheses, and a strictly positive random variable L that is not a stopping time. Can one expand the filtration to make a larger filtration G that renders L a G stopping time, and then calculate the G compensator of pt = lit>L)? This has a positive answer with the right hypotheses. To see how to do this, let Xt = l[o,L)(t),which is not an IF adapted process. Let Z denote the optional projection of X onto F. We prove in Chap. VI that Z is a supermartingale. Let Z = M-A be its DoobMeyer decomposition with A predictable. (Note that Mt = E{A,lFt}.) Using the techniques presented in Chap. VI, one can show that if L is "the end of an F predictable set," then the G compensator of pt = l{t>Ll is At, which is an F predictable process! By the end of a predictable set, wemean that there exists an F predictable set A c [0, m] x 0 such that L(w) = sup{t < m : (t, w) E A). In Chap. VI we discuss times that are the ends of optional sets, known as "honest times," and this is of course less restrictive than being the end of a predictable set, since any predictable set is also optional.2 See [115] for this result described here and related ones. Theorem 17 treats the case of a point process with one jump. A special case of interest is counting processes. We have the following.

Theorem 18. Let N be a counting process without explosions, adapted to a filtmtion G satisfying the usual hypotheses. Then the compensator of N , call it A, always exists. Proof. Since N has non-decreasing paths, to ensure the existence of a compensator we need only to show that N is locally of integrable variation. Let T, be the time of the n-th jump of N . Since N has no explosions, the times See Exercise 4.

5 Compensators

121

T, increase to oo a.s. Moreover INtATnI = NtATn< n and thus N is locally of bounded variation, hence certainly locally of integrable variation. be a counting process without explosions and let Let Nt = Ei,, Si = Ti - Ti-l be its interarrival times, where we take To = 0. Let = a { N , ; s 5 t}, and let Ft = V N,the filtration completed by the F-null sets, as usual. (Note that this filtration is already right continuous.) Let us define the cumulative distribution functions for the interarrival times slightly informally as follows (note that we are using the unusual format of P ( S > z) rather than the customary P ( S x)):

c

<

Fl (t) = P(S1 > t) Fi(sl,. . . ,si-1; t) = P(Si > tlSl

=~

1 ,. ..,Si-1 = si-1)

where the s j are in [0,m], j 2 1. (If one of the s j takes the value +GO,then the corresponding cumulative distribution function is concentrated at +m.) Define

We now have the tools to describe the compensator of N for its minimal completed filtration. We omit the proof, since it is simply a more notationally cumbersome version of the proof of Theorem 17.

Theorem 19. Let N be a counting process without explosions and let F be its minimal completed filtration. Then the compensator A of N is given by

on the event {Ti

< t 5 Ti+l).

Corollary. Let N be a counting process without explosions and independent interarrival times. Then the functions cji defined in equation (*) have the simplified form

and the compensator A is given by

Example (hazard rates and censored data). In applications, an intuitive interpretation often used is that of hazard rates. Let T be a positive random

122

I11 Semimartingales and Decomposable Processes

variable with a continuous density for its distribution function F , given by f . The hazard rate X is defined, when it exists, to be 1 X(t) = lim -P(t h-0

h

I T < t + hlT 2 t).

The intuition is that this is the probability the event will happen in the next infinitesimal unit of time, given that it has not yet happened. Another way of viewing this is that P(t 5 T < t + hlT 2 t) = Ah + o(h). By Theorem 17 tAT we have that the compensator of Nt = X(s)ds. Or in is At = language closer to stochastic integration,

So

X(s)ds is a local martingale. Nt - J O In medical trials and other applications, one is faced with situations where one is studying random arrivals, but some of the data cannot be seen (for example when patients disappear unexpectedly from clinical trials). This is known as arrivals with censored data. A simple example is as follows. Let T be a random time with a density and let U be another time, with an arbitrary distribution. T is considered to be the arrival time and U is the censoring and NtU = l{t2X}1{T,U) with time. Let X = T A U, Nt = l{t2x}l{u2T}l .Tt = a{Nu, N ;: u 5 t). Let X be the hazard rate function for T. X is known as the net hazard rate. We define A#, known as the crude hazard mte, by

when the limit exists. We then have that the compensator of N is given by tATAU h#(u)du, or equivalently At = J,' l { X t u ) ~ # ( ~ = ) d ~ X#(u)du = Jo

JIAX

t

l { x 2 u l ~ # ( u ) d uis a local martingale. If we impose the condition that X = A#, which can be intuitively interpreted as P ( s I T < s dslT 2 s) = P ( s 5 T < s dslT 2 s, U 2 s),

+

+

then we have the satisfying result that Nt

-

l

1{xLul X(u)du is a local martingale.

As we saw in Chap. 11, processes of fundamental importance to the theory of stochastic integration are the quadratic variation processes [X, X] = ([X,XI t)t20, where X is a semimartingale.

Definition. Let X be a semimartingale such that its quadratic variation process [X,X ] is locally integrable. Then the conditional quadratic variation of X , denoted (X, X) = ((X, X)t)t20, exists and it is defined to be the compensator of [X, XI. That is (X, X ) = [X,XI.

-

5 Compensators

123

If X is a continuous semimartingale then [X,X] is also continuous and hence already predictable; thus [X,X ] = (X,X ) when X is continuous. In particular for a standard Brownian motion B , [B, BIt = (B, B)t = t , all t 2 0. The conditional quadratic variation is also known in the literature by its notation. It is sometimes called the sharp bracket, the angle bracket, or the oblique bracket. It has properties analogous to that of the quadratic variation processes. For example, if X and Y are two semimartingales such that (X, X ) , (Y, Y), and ( X + Y, X + Y) all exist, then (X,Y) exists and can be defined by polarization (X, Y)

=

1 -((X 2

+ Y, X + Y)

-

(X,X ) - (Y, Y)).

However (X, Y) can be defined independently as the compensator of [X,Y] provided of course that [X, Y] is locally of integrable variation. In other words, there exist stopping times (Tn)n21 increasing to oo a.s. such that E { J ? ~/d[X,Y],I) < m for each n. Also, ( X , X ) is a non-decreasing process by the preceding discussion, since [X,X] is non-decreasing. The conditional quadratic variation is inconvenient since unlike the quadratic variation it doesn't always exist. Moreover while [X,X], [X, Y], and [Y,Y] all remain invariant with a change to an equivalent probability measure, the sharp bmckets i n general change with a change to an equivalent probability measure and may even no longer exist. Although the angle bracket is ubiquitous in the literature it is sometimes unnecessary as one can often use the quadratic variation instead, and indeed whenever possible we use the quadmtic variation rather than the conditional quadratic variation (X,X ) of a semimartingale X i n this book. Nevertheless the process (X,X ) occurs naturally in extensions of Girsanov's theorem for example, and it has become indispensable in many areas of advanced analysis in the theory of stochastic processes. We end this section with several useful observations that we formalize as theorems and a corollary. Note that the second theorem below is a refinement of the first (and see also Exercise 25).

Theorem 20. Let A be an increasing process of locally integrable variation, all of whose jumps occur at totally inaccessible stopping times. Then its compensator A is continuous. Theorem 21. Let A be an increasing process of locally integmble variation, and let_T be a jump time of A which is totally inaccessible. Then its compensator A is continuous at T .

Proof. Both theorems are simple consequences of Theorem 7. Corollary. Let A be an increasing predictable process of locally integrable variation and let T be a stopping time. If P(AT # AT-) > 0 then T is not totally inaccessible.

124

I11 Semimartingales and Decomposable Processes

Proof. Suppose T were totally inaccessible. Let A be the compensator_of A. Then 2 is continuous at T . But since A is already predictable, A = A, and we have a contradiction by Theorem 21. Theorem 22. Let T be a totally inaccessible stopping time. There exists a martingale M with paths of finite variation and with exactly one jump, of size one, occurring at time T (that is, MT # MT- on {T < GO)).

Proof. Define

ut = l{t>T}. Then U is an increasing, bounded process of integrable variation, and we be the compensator of U . A is continuous by Theorem 20, and let A = M = U - A is the required martingale.

5

6 The Fundamental Theorem of Local Martingales We begin with two preliminary results.

Theorem 23 (Le Jan's Theorem). Let T be a stopping time and let H be an integrable random variable such that E{HIFT-) = 0 on {T < GO).Then the right continuous martingale Ht = E{H(Ft}is zero on [0,T ) = { ( t ,w ) : 0 5 t < T(w)).

Proof. Since the martingale (Ht)t20is right continuous it suffices to show that Ht 1{t,Tl = 0 almost surely for all t. Let A E Ft. Then A fl { t < T ) belongs both to FT- and also to Ft. Hence

Since A E Ft is arbitrary, this implies that Htl{t
Theorem 24. Let T be a predictable stopping time and let A be increasing, predictable, and locally of integrable variation. Then AT and AAT are each FT- measurable.

Proof. Let Sn be a sequence of stopping times announcing T . Then AS,, E Fsn c FT- for each n . Since AT- = limn,, As,, we have that AT- E FT-. To show that AT E FT- it suffices to show E { A T H ) = 0 for every bounded random variable H such that E{HIFT-) = 0. Let H be such a random variable and let Ht = E{HIFt},the right continuous martingale with terminal value H. Then Ht(w) = 0 for 0 5 t < T(w) by Le Jan's Theorem (Theorem 23). This implies that the left continuous version of H, Ht-, equals 0 on [0,TI. Then since the hypotheses imply that A is natural, we have roo

6 The Fundamental Theorem of Local Martingales

125

Theorem 25 (Fundamental Theorem of Local Martingales). Let M be a local martingale and let p > 0. Then there exist local martingales N , D such that D is an FV process, the jumps of N are bounded by 2P, and M=N+D. Proof. We first set

Ct =

C

lAMsIl{la~,l~p}~

and we want to show C is locally integrable. By stopping we can assume without loss that M is a uniformly integmble martingale and that Mo = 0. For each w , Ct(w)is a finite sum. Define

R,

= inf{t : Ct 2 n or

1 Mt 1 2 n).

+

+

+

Then ~ A M R1 , I ~ M R1 , IMR_- I I ~ M R1 , n. Moreover CR, I CR,~ A M RI , ~IMR_I 2n, which is in L1. Since R, increases t o co,we have C is locally integrable. By stopping we further assume that E{C,) < oo. Next we define

+

At =

1A M s 1 { ~ ~ ~ . ~ 1 P } 7

O<s
and since C is assumed integrable we deduce that A is a quasimartingale. By Theorems 13 and 14 we know that A decomposes as follows:

where L is a local martingale and B1, B 2 are non-decreasing, nonnegative, predictable processes, such that E { B ~< ) oo, E{B&) < oo. Let

Then /At - ;it

I < C , + B&, + BL 6 L' , and hence the martingales A - 2 and

-

are uniformly integrable. As usual, we write A for the compensator of A. Then

D=A-A. It remains only to show that the_jumps of N are bounded by 2P. To this end, let T be a stopping time. If AAT = 0 a.s., then

This implies lANTl I P, and we are done by Theorem 7 of Chap. I. We therefore assume if A = { @ A T [ > 0 ) , then P(A) > 0. Let R = TA. By Theorem 3 we can decompose R as R = RA A R B , where R A is accessible and R B is totally inaccessible. Then P ( R B < m) = 0 by Theorem 7 , whence

126

I11 Semimartingales and Decomposable Processes

R = RA.Let (Tk)k>l be a sequence of predictable times enveloping R. It will suffice to show AN^, I 5 2PI for each T k . Thus without loss of generality we can take R = T to be predictable. By convention, we set ANT = 0 on {T = oo}. Since we know that axTE FT-by Theorem 24, we have

Since J A ( M- A)TI 5 P, the result follows. If the jumps of a local martingale M with Mo = 0 are bounded by a constant PI then M itself is locally bounded. Let Tn = inf{t : JMtl 2 n). Then IMtATn1 5 n P. Therefore M is a fortiori locally square integrable. We thus have a corollary.

+

Corollary. A local martingale is decomposable. Of course if all local martingales were locally square integrable, they would then be trivially decomposable. The next example shows that there are martingales that are not locally square integrable (a more complex example is published in Dol6ans-Dade [50]).

Example. Let (0,F , P) be complete probability space and let X be a random variable such that X E L1, but X $ L2. Define the filtration

where Ft = V N , with N all the P-null sets of F . Let Mt = E{X1Ft), the right continuous version. Then M is not a locally square integrable martingale. The next example shows another way in which local martingales differ from martingales. A local martingale need not remain a local martingale under a shrinkage of the filtration. They do, however, remain semimartingales and thus they still have an interpretation as a differential.

Example. Let Y be a symmetric random variable with a continuous distribution and such that E{IYI} = oo. Let Xt = Yl{t>ll, and define

- is the completed filtration. Define stopping times T n by where G = (Gtlt>0

Tn=

{

0,

,

if lYl >- n, otherwise.

7 Classical Semimartingales

127

Then Tn reduce X and show that it is a local martingale. However X is not a local martingale relative to its completed minimal filtration. Note that X is still a semimartingale however. The full power of Theorem 25 will become apparent in Sect. 7.

7 Classical Semimartingales We have seen that a decomposable process is a semimartingale (Theorem 9 of Chap. 11). We can now show that a classical semimartingale is indeed a semimartingale as well. Theorem 26. A classical semimartingale is a semimartingale.

Proof. Let X be a classical semimartingale. Then Xt = Mt + At where M is a local martingale and A is an F V process. The process A is a semimartingale by Theorem 7 of Chap. 11, and M is decomposable by the corollary of Theorem 25, hence also a semimartingale (Theorem 9 of Chap. 11). Since semimartingales form a vector space (Theorem 1 of Chap. 11) we conclude X is a semimartingale. Corollary. A chllhg local martingale is a semimartingale.

Proof. A local martingale is a classical semimartingale. Theorem 27. A chdlhg quasimartingale is a semimartingale.

Proof. By Theorem 15 a quasimartingale is a classical semimartingale. Hence it is a semimartingale by Theorem 26. Theorem 28. A chdlhg supermartingale is a semimartingale.

Proof. Since a local semimartingale is a semimartingale (corollary to Theorem 6 of Chap. 11), it suffices to show that for a supermartingale X , the stopped process Xt is a semimartingale. However for a partition r of [0,t ] ,

Therefore Xt is a quasimartingale, hence a semimartingale by Theorem 27. Corollary. A submartingale is a semimartingale.

128

111 Semimartingales and Decomposable Processes

We saw in Chap. I1 that if X is a locally square integrable local martingale and H E IL, then the stochastic integral H . X is also a locally square integrable local martingale (Theorem 20 of Chap. 11). Because of the corollary of Theorem 25 we can now improve this result.

Theorem 29. Let M be a local martingale and let H E IL. Then the stochastic integral H . M is again a local martingale.

Proof. A local martingale is a semimartingale by the corollary of Theorem 25 and Theorem 9 of Chap. 11; thus H . M is defined. By the Fundamental Theorem of Local Martingales (Theorem 25) for P > 0 we can write M = N A where N , A are local martingales, the jumps of N are bounded by P, and A has paths of finite variation on compacts. Since N has bounded jumps, by stopping we can assume N is bounded. Define T by

+

+

Then E { I , " " ~IdAsl} 5 m + P E{IAMTI} < m, and thus by stopping A can be assumed to be of integrable variation. Also by replacing H by H S l ~ s > o ) for an appropriate stopping time S we can assume without loss of generality that H is bounded, since H is left continuous. We also assume without loss that Mo - No = A. = 0. We know H . N is a local martingale by Theorem 20 of Chap. 11, thus we need show only that H . A is a local martingale. Let an be a sequence of random partitions of [0,t] tending to the identity. Then C HTF ( A ~ ~- " tends to ( H . A)t in u p , where on is the sequence 0 = Ton <- TT 5 . - 5 TT 5 . . . . Let ( n k )be a subsequence such that the sums converge uniformly a.s. on [0,t ] .Then

~~q

a

by Lebesgue's Dominated Convergence Theorem. Since the last limit above equals ( H . A),, we conclude that H . A is indeed a local martingale. A note of caution is in order here. Theorem 29 does not extend completely to processes that are not in IL but are only predictably measurable, as we will see in Emery's example of a stochastic integral behaving badly on page 176 in Chap. IV. Let X be a classical semimartingale, and let Xt = X o + Mt At be a decomposition where Mo = A. = 0, M is a local martingale, and A is an F V

+

7 Classical Semimartingales

129

process. Then if the space ( R , 3 , (3t)t2~, P) supports a Poisson process N, we can write

as another decomposition of X.In other words, the decomposition of a classical semimartingale need not be unique. This problem can often be solved by choosing a certain canonical decomposition which is unique.

Definition. Let X be a semimartingale. If X has a decomposition Xt = Xo+ Mt+ At with Mo = A. = 0, M a local martingale, A an F V process, and with A predictable, then X is said to be a special semimartingale. To simplify notation we henceforth assume Xo= 0.

Theorem 30. If X is a special semimartingale, then its decomposition X M A with A predictable is unique.

+

=

+

= N B be another such decomposition. Then M - N = B -A, hence B - A is an F V process which is a local martingale. Moreover, B - A is predictable, and hence constant by Theorem 12. Since Bo- A. = 0, we conclude B = A.

Proof. Let X

Definition. If X is a special semimartingale, then the unique decomposition X = M + A with Mo = Xo and A. = 0 and A predictable is called the canonical decomposition. Theorem 15 shows that any quasimartingale is special. A useful sufficient condition for a semimartingale X to be special is that X be a classical semimartingale, or equivalently decomposable, and also have bounded jumps.

Theorem 31. Let X be a classical semimartingale with bounded jumps. Then X is a special semimartingale.

t = Xo+ Mt+ At be a decomposition of X with Mo= A. = 0, Proof. Let X M a local martingale, and A an F V process. By Theorem 25 we can then also write

Xt

= Xo

+ Nt + Bt

where N is a local martingale with bounded jumps and B is an F V process. Since X and N each have bounded jumps, so also does B. Consequently, it is locally a quasimartingale and therefore decomposes

B is a predictable FV process (Theorem 15). Xt = Xo + {Nt + Lt}+ gt

where L is a local martingale and Therefore

is the canonical decomposition of X and hence X is special.

130

111 Semimartingales and Decomposable Processes

Corollary. Let X be a classical semimartingale with continuous paths. Then X is special and in its canonical decomposition

the local martingale M and the F V process A have continuous paths. Proof. X is continuous hence trivially has bounded jumps, so it is special by Theorem 31. Since X is continuous we must have

for any stopping time T (AAT = 0 by convention on {T = m } ) . Suppose A jumps at a stopping time T. By Theorem 3, T = TA A TB, where TA is accessible and TB is totally inaccessible. By Theorem 21 it follows that P(IAAT,J > 0) = 0. Hence without loss of generality we can assume T is accessible. It then suffices to consider T predictable since countably many predictable times cover the stopping time T. Let Sn be a sequence of stopping times announcing T. Since A is predictable, we know by Theorem 24 that AAT is 3 T - measurable. Therefore AMT is also 3 T - measurable. Stop M so that it is a uniformly integrable martingale. Then

and M , and hence A, are continuous, using Theorem 7 of Chap. I. Theorem 31 can be strengthened, as the next two theorems show. The criteria given in these theorems are quite useful. Theorem 32. Let X be a semimartingale. X is special i f and only i f the process Jt = supsi, [AX,[ is locally integrable. Theorem 33. Let X be a semimartingale. X is special i f and only i f the process X,* = sup,,,- IX, I is locally integrable.

Before proving the theorems, we need a preliminary result, which is interesting in its own right. Theorem 34. Let M be a local martingale and let M t = sup,,,- I M, 1. Then the increasing process M * is locally integrable.

Proof. Without loss of generality assume Mo = 0. Let Tn be a sequence of stopping times increasing to m such that M~~ is a uniformly integrable martingale for each n. Since we can replace Tn with TnAn if necessary, without loss of generality we can assume that the stopping times Tn are each bounded. Set Sn = T,~inf,{lM,l n } . Since MTn is uniformly integrable, and Sn 5 Tn, we have that Msn is integrable. But we also have that M i n I n V IMs, I which is in L 1 . Since Sn increases to m as., the proof is complete.

>

8 Girsanov's Theorem

131

Proof of Theorems 32 and 33. Since X is special the process A of its canonical decomposition X = M A is of locally integrable variation. Writing X,* 5 M; ldAs 1, and since M; is locally integrable by Theorem 34 and since A is of locally integrable variation, we have that X * is locally integrable. Further, note that [AX,1 = IXt - Xt- I 5 2jX; 1, hence Jt 5 X,* and we have that Jt is also locally integrable. For the converse, it will suffice to show that there exists a decomposition X = M A where A is of locally integrable variation, since then we can take the compensator of A and obtain a canonical decomposition. To this end note that IAA, I 5 [AM, 1 [AX, 1 5 2Mt J,, which is locally integrable by the hypotheses of Theorem 32 together with Theorem 34. Since Jt 5 X,* we have that I AA, 1 is locally integrable by the hypotheses of Theorem 33 together with Theorem 34 as well. To complete the proof, let Tn = inf{t > 0 : Jot IdA,I n}, IdA,I = IdA,I ~AAT,,1 2 n sup,<,,, [AA,1, which and note that is in L1 as soon as SUP,
+

+Sot

+

+

1,".

+

I,%-

+

>

+

A class of examples of special semimartingales is L6vy processes with bounded jumps. By Theorem 31 we need only show that a LQvyprocess Z with bounded jumps is a classical semimartingale. This is merely Theorem 40 of Chap. I. For examples of semimartingales which are not special, we need only to construct semimartingales with finite variation terms that are not locally integrable. For example, let X be a compound Poisson process with nonintegrable, nonnegative jumps. Or, more generally, let X be a L6vy process (xlv(dx) = co. with L6vy measure v such that xv(dx) = co and/or

11;

8 Girsanov's Theorem Let X be a semimartingale on a space ( R , 3 , IF, P) satisfying the usual hypotheses. We saw in Chap. I1 that if Q is another probability law on (0, 3 ) , and Q << P, then X is a Q semimartingale as well. If X is a classical semimartingale (or equivalently, if X is decomposable) and has a decomposition X = M A, M a local martingale and A an F V process, then it is often useful to be able to calculate the analogous decomposition X = N + B, if it exists, under Q. This is rather tricky unless we make a simplifying assumption which usually holds in practice: that both Q << P and P << Q.

+

Definition. Two probability laws P, Q on ( R , 3 ) are said to be equivalent if P << Q and Q << P . (Recall that P << Q denotes that P is absolutely continuous with respect to Q.) We write Q P to denote equivalence. N

132

I11 Semimartingales and Decomposable Processes

.

If Q << P, then there exists a random variable Z in L 1 ( d P ) such that and E p { Z } = 1, where E p denotes expectation with respect to the law P. We let

$$ = Z

=

~~{gl3~}

be the right continuous version. Then Z is a uniformly integrable martingale and hence a semimartingale (by the corollary of Theorem 26). Note that if Q -1

%

% (g)

. is equivalent to P, then E L1(dQ) and = We begin with a simple lemma, the easy proof of which we leave as an exercise for the reader. . An adapted, c&dl&gprocess M Lemma. Let Q PI and Zt = EP is a Q local martingale if and only if M Z is a P local martingale. N

Proof. See Exercise 20. Theorem 35 (Girsanov-Meyer Theorem). Let P and Q be equivalent. Let X be a classical semimartingale under P with decomposition X = M + A. Then X i s also a classical semimartingale under Q and has a decomposition X = L C , where

+

is a Q local martingale, and C

=X

-

L is a Q F V process.

Proof. Recall that by Theorem 2 of Chap. I1 it is trivial that X is a Q semimartingale. We need to show it is a classical semimartingale, with the above decomposition being valid. Since M and Z are P local martingales, they are semimartingales (corollary of Theorem 26) and

is a local martingale as well (Theorem 29). Using integration by parts we have

is also a P local martingale. Since Z is a version of Ep{%l3t}, we have $ is a cAdlAg version of E ~ { % I I F ,therefore }; $ is a Q semimartingale. Since P << Q, it is also a P semimartingale. Multiplying equation (*) by Q we have

Note that if we were to multiply the right side of (**) by Z we would obtain a P local martingale. We conclude by the lemma preceding this theorem that

8 Girsanov's Theorem

133

M - ($)[Z, MI is a Q local martingale. We next use integration by parts (under Q):

(5).

Since Let N = J[Z,M]- d orem 29). The above becomes:

5 is a Q local martingale, so also is N (The-

Adding our two Q local martingales yields

which, being the sum of two Q local martingales, is itself one. This establishes the theorem. The Girsanov-Meyer Theorem has another version using the sharp bracket. It transforms a P local martingale into a Q special semimartingale and gives its canonical decomposition. However an extra integrability hypothesis is needed for the theorem to be true. Note that in the predictable version the integrand is replaced by its left continuous version

k.

-&

Theorem 36 (Girsanov-Meyer Theorem: Predictable Version). Let X be a P local martingale with Xo = 0. Let Q be another probability equivalent .f (X, Z) exists for the P probability, then the to P and let Zt = ~ ( g 1 3 t ) I canonical Q decomposition of X is

xt = (Xt

&(x,

-

z),)

+

& d ( ~ , 4,

Proof. Using the same ideas as in the proof of Theorem 35, we want to find a predictable finite variation term A such that Y = Z(X - A) is a P local martingale. Using integration by parts yields

1 t

ZtXt =

2s-dXs

+

+ [Z,X]t = Mt + (X1Z)t = Nt

1

-

t

Xs-dZs

(X, Z)t

+ [Z,X]

+ (X, Z)t

134

I11 Semimartingales and Decomposable Processes

2

where both N = 2,-dX, + X,-dZ, and M = N + [Z,X ] - (X, Z) are P local martingales. Therefore Z X - (X, Z) is a P local martingale. Let A = (X, Z), which of course is a predictable F V process. Moreover we know that A does not jump at totally inaccessible times, and that we can cover its jumps with predictable times. This implies that A is locally bounded. Let Tn = infit > 0 : \At\ > n}, and let AT"- be defined by AT,,-- = J l[o,Tn)(~)dAs. From now on we assume A = AT"-. Then A is still an F V process and is still predictable, IAl 5 n, and the stopping times Tn increase to co a.s. Integration by parts gives us also

If we show that [A, $1 is also a Q local martingale, we will be done. It suffices to show that EQ{[A, $IT} = 0 for every stopping time T . Since A is F V and predictable, by stopping at T - if necessary, we can assume the jumps of A are bounded. And since $ is a Q local martingale, its supremum process is locally integrable (Theorem 34). This is enough to give us that [A, $1 is Q locally of integrable variation. By stopping we assume it is integrable. We then have that A is Q natural, so if T is a stopping time, we replace A with and we still have a natural process, and thus we have EQ{[A, $IT} = 0. Since T was arbitrary, this implies that [A, is a martingale. Of course, the process is implicitly stopped at several stopping times, so what we have actually shown is that [A, $1 is a Q local martingale. We now have that

i]

1 -At

zt

= Q-local martingale

+

which in turn gives us, recalling that A = (X, Z), Xt

1 =

1

(ztxt)

= -Mt

zt

1

= -(Mt

zt

+ (X, Z),)

+ Q-local martingale +

Since z($M) = M = P local martingale, we have that $ M is also a Q local martingale. Thus finally Xt - & d ( ~ ,Z), is a Q local martingale, and the proof is done.

Sot

Let us next consider the interesting case where Q is absolutely continuous with respect to P, but the two probability measures are not equivalent. (That is, we no longer assume that P is also absolutely continuous with respect to = {a), 0) a.s. Q.) We begin with a simple result, and we will assume Theorem 37. Let X be a P local martingale with Xo = 0. Let Q be another probability absolutely continuous with respect to P, and let Zt = ~{%13~},).

8 Girsanov's Theorem

Assume that (X, Z) exists for P. Then At the probability Q, and Xt

-

1,"& d ( ~ ,Z),

=

1; &d(X,

135

Zjs exists a.s. for

is a Q local martingale.

<

Proof. Zo = E{Z} = 1, so if we let R, = inf{t > 0 : Zt lln}, then is bounded on [0,R,]. By R, increase to co, Q-a.s., and the process

&

Theorem 36, x,R" - $ &d(X, z)? is a Q local martingale, each n. Since a local, local martingale is a local martingale, we are done. We now turn to the general case, where we no longer assume the existence of (X, Z), calculated with P . (As before, we take X to be a P local martingale.) We begin by defining a key stopping time: R = inf{t > 0 : Z, = 0, 2,- > 0). Note that Q(R < co) = 0, but it is entirely possible that P ( R < co) > 0. We R } . U is an FV process, and moreover U further define Ut = A X R ~ { ~ ~Then is locally integrable (dP). Let T, increase to co and be such that xTn is a uniformly integrable martingale. Then

Thus U has a compensator 6 , and of course 6is predictable and U - 6is a P local martingale.

Theorem 38 (Lenglart-GirsanovTheorem). Let X be a P local martingale with Xo = 0. Let Q be a probability absolutely continuous with respect to P , and let Zt = E ~ { $ ~ F ~ } ,R = inf{t > 0 : Zt = 0, Zt- > 01, and ~ ~ } . Ut = A X R ~ { ~Then

is a Q local martingale.

< i}.

Proof. Let R, = inf{t > 0 : Zt (Recall that Zo = 1, and also note that it is possible that R, = R.) Then both xRn and ZRn are P local martingales. Also note that A? = &l{Zp.,o} d[xRn,ZRn],, and

yRn= xRn - ARn +

cRnare all P-well-defined. We can define

on [0,R), since d[xRn,ZRn], does not charge (R, m ) , and &I{,:,,

cRn,

=0

at R. Thus we need only to show yRnis a Q local martingale for each fixed n, which is the same as showing that ZRnyRnis a P local martingale. Let us assume all these processes are stopped at R, to simplify notation. We have

136

I11 Semimartingales and Decomposable Processes

Hence, d (ZX) = Z- dX + X- dZ + d [Z,XI = local martingale d[Z,X]

+

d(AZ) = A-dZ+ZdA = local martingale = local martingale d(Z0) = z - d 0 =

+ 0dZ

local martingale

= local martingale

where the last equality uses that we have

ZY

+ ZdA + l{Z>ol d[X, Z] -

+ Z-dU + Z-dU

U-0is a local martingale (dP). Summarizing

=zx-ZA+ZU = local martingale + [Z,X ] local martingale + l { z , o l d [ ~ ,z]) + local martingale + Z-dU -

S

which we want to be a local martingale under dP. This will certainly be the case if d[Z,XI - l{z>old[X,Z] Z-dU = 0. (*)

+

However (*) equals

But AZR = ZR - ZR- = 0 - ZR- = -ZR-, and this implies that equation (*) is indeed zero, and thus the Lenglart-Girsanov Theorem holds. Corollary. Let X be a continuous local martingale under P . Let Q be absolutely continuous with respect to P . Then (X, Z) = [Z,X] = [Z,XICexists, and

which is a Q local martingale. Proof. By the Kunita-Watanabe inequality we have

which shows that it is absolutely continuous with respect to d[X, XI, a.s., whence the result.

8 Girsanov's Theorem

137

We remark that if Z is the solution of a stochastic exponential equation of the form dZ, = 2,-H,dX, (which it often is), then a , = H,.

Example. A problem that arises often in mathematical finance theoy is that one has a semimartingale S = M A defined on a filtered probability space ( R , F , F , P) satisfying the usual hypotheses, and one wants to find an equivalent probability measure Q such that under Q the semimartingale X is a local martingale, or better, a martingale. In essence this amounts to finding a probability measure that "removes the drift." To be concrete, let us suppose S is the solution of a stochastic differential equation3

+

where B is a standard Wiener process (Brownian motion) under P . Let us postulate the existence of a Q and let Z = and Zt = E{ZIFt), which is clearly a cAdlAg martingale. By Girsanov's Theorem

is a Q local martingale. We want to find the martingale Z . In Chap. IV we will study martingale representation and show in particular that every local martingale on a Brownian space is a stochastic integral with respect to Brownian motion. Thus we can write Zt = 1 J ot J,dB, for some predictable then process J . If we assume Z is well behaved enough to define H, = we have Zt = 1 + J ; H,Z,dB,, which gives us a linear stochastic differential t equation to solve for Z. Thus if we let Nt = JoH,dB,, we get that Zt = E(N)t.4 It remains to determine H . We do this by observing from our previous Girsanov calculation that

+

is a Q local martingale. We then choose H, =

2,

-b(s; ST;r 5 S) , which yields h(s, S,)

- Bt + is a local martingale under Q. Letting M -

l

b(szii

is:"ds

denote

this Q local martingale, we get that [M,MIt = [B, BIt = t , and by L6vy's Theorem M is a Q-Brownian motion. Finally, under Q we have that S satisfies the stochastic differential equation

Stochastic differential equations are introduced and studied in some detail in Chap. V. The stochastic exponential & is defined on page 85.

138

I11 Semimartingales and Decomposable Processes

There is one problem with the preceding example: we do not know a priori whether our solution Z is the Radon-Nikodym density for simply a measure Q, or whether Q is an actual bona jide probability measure. This is a constant problem. Put more formally, we wish to address this problem: Let M be a local martingale. When is E(M) a martingale? The only known general conditions that solve this problem are Kazamaki's criterion and Novikov's ~ r i t e r i o n Moreover, .~ these criteria apply only t o local martingales with continuous paths. Novikov's is a little less powerful than Kazamaki's, but it is much easier to check in practice. Since Novikov's criterion follows easily from Kazamaki's, we present both criteria here. Note that if M is a continuous local martingale, then of course E(M) is also a continuous local martingale. Even if, however, it is a uniformly integrable local martingale, it still need not be a martingale; we need a stronger condition. As an example, one can take u(x) = llxll-l and Nt = u(Bt) where B is standard three dimensional Brownian motion. Then N is a uniformly integrable local martingale but not a martingale. Nevertheless, whenever M is a continuous local martingale, then E(M) is a positive supermartingale, as is the case with any nonnegative local martingale. Since E(M)o = 1, and since E(M) is a positive supermartingale, we have E{E(M)t) 5 1, all t. (See the lemma below.) It is easy to see that Z is a true martingale if one also has E{E(M)t) = 1, for all t. We begin with some preliminary results.

Lemma. Let M be a continuous local martingale with Mo = 0. Then E{E(M),} 5 1 for all t 2 0. Proof. Recall that E(M)o = 1. Since M is a local martingale, E(M) is a nonnegative local martingale. Let Tn be a sequence of stopping times reducing E(M). Then E{E(M)tATn) = 1, and using Fatou's Lemma, E{E(M)t)

= E{liminf

n-+m

E ( M ) t A ~ nI ) lim inf E{E(M)tATn) = 1. n--+a

Theorem 39. Let M be a wntinuous local martingale. Then

Proof.

which implies that There also exist partial results when the local martingale is no longer continuous but only has c&dl&gpaths. See Exercise 14 of Chap. V for an example of these results.

8 Girsanov's Theorem

139

and this together with the Cauchy-Schwarz inequality and the fact that 1 gives the result. E{E(M)t)

<

Lemma. Let M be a continuous local martingale. Let 1 < p < co, + 1 = 1. Taking the supremum below over all bounded stopping times, assume &at

Then E(M) is an LQ bounded martingale.

=.

Proof. Let 1 < p < c o a n d r = Jjs+ note that (9 We have

=

1

T h e n s = -and

+ j= 1. Also we

which we use in the last equality of the proof.

We now apply Holder's inequality for a stopping time S:

Recalling that E{E(JqTM)s)

I 1, we have the result.

Theorem 40 (Kazamaki's Criterion). Let M be a continuous local martingale. Suppose sup^ E { ~ ( ; ~ < T co, ) ) where the supremum is taken over all bounded stopping times. Then E(M) is a uniformly integrable martingale.

&

< !. Our hypothesis Proof. Let 0 < a < 1, and p > 1 be such that combined with the preceding lemma imply that E(aM) is an Lq bounded martingale, where = 1, which in turn implies it is a uniformly integrable martingale. However

+

and using Holder's inequality with a c 2 and (1 - a2)-I yields (where the 1 on the left side comes from the uniform integrability):

140

I11 Semimartingales and Decomposable Processes

Now let a increase to 1 and the second term on the right side of the last inequality above converges t o 1 since 2a(l - a) 4 0. Thus 1 E{E(M),), and since we know that it is always true that 1 E{E(M),), we are done.

>

<

As a corollary we get the very useful Novikov's criterion. Because of its importance, we call it a theorem.

Theorem 41 (Novikov's Criterion). Let M be a continuous local martingale, and suppose that

~ { ~ i [ ~< 'm. ~l-)

Then E(M) is a uniformly integrable martingale.

<

E ){ ~ + [ ~ i?,~ and ]T we) need only Proof. By Theorem 39 we have E { ~ + ~ T to apply Kazamaki's criterion (Theorem 40). We remark that it can be shown that 112 is the best possible constant in Novikov's criterion, even though Kazamaki's criterion is slightly stronger. Note that in the case of the example treated earlier, we have [N,NIt = J: H:ds ) where H = b ( ~ ; S ? ' ; r <. ~Clearly whether or not [N, NIt has a finite exponential h(s,Ss) moment will depend on our choice of b and of h. A standard simple solution is to choose b and h so that blh is bounded. Partial results exist that give an analogue of Novikov's criterion in the general (chdlhg) case. See for example Exercise 14 of Chap. V, and the sources [156] and [157].

Example. This example is taken from statistical communicatzon the0y. Suppose we are receiving a signal corrupted by noise, and we wish to determine if there is indeed a signal, or if we are just receiving noise (e.g., we could be searching for signs of intelligent life in our galaxy with a radio telescope). Let x(t) be the received signal, [(t) the noise, and s(t) the actual (transmitted) signal. Then x(t) = s(t) [(t).

+

A frequent assumption is that the noise is "white." A white noise is usually described as a second order wide sense stationary process with a constant spectral density function (that is, E{&) = 0, and V ( r ) = E{XtXt+,) = 6, So, where 6, is the Dirac delta function a t r , and So is the constant spectral =: J ei2""'V(r)dr). Such a process does not exist density; one then has S(v) m in a rigorous mathematical sense. Indeed it can be interpreted as the derivative of the Wiener process, in a generalized function sense. (See Arnold [2, page 531, for example.) This suggests that we consider the integrated version of our signal-noise equation:

8 Girsanov's Theorem

141

where W is a standard Wiener process and where St is thought of as ('J; z(s)ds," the cumulative received signal. The key step in our analysis is the following consequence of the GirsanovMeyer Theorem.

Theorem 42. Let W be a standard Brownian motion on ( R , F , F ,P), and let H E IL be bounded. Let rt

1

~ i d s ) for , some T > 0. Then and define Q by = exp{J; -H,dW, under Q, X is a standard Brownian motion for 0 5 t T.

<

4

~ : d s ) . Then if 4 = E{ZTIF~)we Pro06 Let ZT = exp{J? -H,dW, know by Theorem 37 of Chap. I1 that Z satisfies the equation

By the Girsanov-Meyer Theorem (Theorem 35), we know that

is a Q local martingale. However

since [W,WIt

=t

for Brownian motion. Therefore

hence X is a Q local martingale. Since (Jot H s d ~ ) t r ois a continuous F V process we have that [X,XIt = [W,WIt = t , and by L6vy's Theorem (Theorem 39 of Chap. 11) we conclude that X is a standard Brownian motion.

Corollary. Let W be a standard Brownian motion and H E Then the law of t Xt 0

=

1

Hsds

+ Wt,

< t 5 T < oo,is equivalent to Wiener measure.

IL be bounded.

142

I11 Semimartingales and Decomposable Processes

Proof. Let C[O, T ] be the space of continuous functions on. [O,TI with values in B (such a space is called a path space). If W = (Wt)o
Remark. We have not tried for maximum generality here. For example the hypothesis that H be bounded can be weakened. It is also desirable to weaken the restriction that H G L. Indeed we only needed that hypothesis to be able

Jot

to form the integral H,dW,. This is one example to indicate why we need a space of integrands more general than L. We are now in a position to consider the problem posed earlier: is there a signal corrupted by noise, or is there just noise (that is, does s(t) = 0 a.e., a.s.)? In terms of hypothesis testing, let Ho denote the null hypothesis, H1 the alternative. We have: Ho : XT

=

WT

1

T

HI : XT =

HSds

+ WT.

We then have

by the preceding corollary. This leads to a likelihood ratio test: dpw 5 A, reject Ho, -(w) d ~ x dpw if -(w) > A, fail to reject Ho, d ~ x where the threshold level X is chosen so that the fixed Type I error is achieved. To indicate another use of the Girsanov-Meyer Theorem let us consider stochastic differential equations. Since stochastic differential equations6 are treated systematically in Chap. V we are free here to restrict our attention to a simple but illustrative situation. Let W be a standard Brownian motion on a space ( R , 3 , IF,P) satisfying the usual hypotheses. Let fi(w , s, x) be functions satisfying (i = 1,2): if

Stochastic "differential" equations have meaning only if they are interpreted as stochastic integral equations.

9 The Bichteler-Dellacherie Theorem

143

(i) Ifi(w, s, x) - fi(w, s, y)l I Klx - yl for fixed (w, s); (ii) f i ( ~ , s , x ) ~ 3 , f o r f k e d ( s , x ) ; (iii) fi(w, .,x) is left continuous with right limits for fixed (w, x). By a Picard-type iteration procedure one can show there exists a unique solution (with continuous paths) of

The Girsanov-Meyer Theorem allows us to establish the existence of solutions of analogous equations where the Lipschitz hypothesis on the "drift coefficient" f 2 is removed. Indeed if X is the solution of (* * *), let y be any bounded, measurable function such that y(w, s,X,) E IL. Define

We will see that we can find a solution of

provided we choose a new Brownian motion B appropriately. We define a new probability law Q by

dP

= exp

(iT

y(., X,)dW,

-

:IT

-

~ ( sx,)'ds ,

By Theorem 42 we have that

is a standard Brownian motion under Q. We then have that the solution X of (* * *) also satisfies

which is a solution of a stochastic differential equation driven by a Brownian motion, under the law Q.

9 The Bichteler-Dellacherie Theorem In Sect. 7 we saw that a classical semimartingale is a semimartingale. In this section we will show the converse.

144

I11 Semimartingales and Decomposable Processes

Theorem 43 (Bichteler-Dellacherie Theorem). An adapted, ccidlcig pmcess X is a semimartingale if and only if it is a classical semimartingale. That is, X is a semimartingale if and only if it can be written X = M + A, where M is a local martingale and A is an F V process. Proof. The sufficiency is exactly Theorem 26. We therefore establish here only the necessity. Since X is cAdlAg, the process Jt = C o < s 5 tA X s l ~ l a x , l r l )has paths of finite variation on compacts; hence it is an FV process. Let Y = X - J . Then Y has bounded jumps. If we show that X is a classical semimartingale on [0,to], some to, then Y = X - J is one as well; moreover Y is special by Theorem 31. Let Y = Yo 2 2 be the canonical decomposition of Y. Suppose t l > to and X is also shown to be a classical semimartingale on [0,tl]. Then let Y = Yo N + B be the canonical decomposition of Y on 10, tll. We have that, for t I to,

+ +

+

by subtraction. Thus B - 2 is a predictable, FV process which is a local martingale; hence by Theorem 12 we have B = 2.Thus if we take tn increasing to oo and show that X is a classical semimartingale on [0,t,], each n, we have that X is a classical semimartingale on [0, oo). We now choose uo > 0, and by the above it suffices to show X is a classical semimartingale on [0,u O ]Thus it is no loss to assume X is a total semimartingale on [0,uO].We will show that X is a quasimartingale, under an equivalent probability Q. Rm's Theorem (Theorem 15) shows that X is a classical semimartingale under Q, and the Girsanov-Meyer Theorem (Theorem 35) then shows that X is a classical semimartingale under P . Let us take H E S of the special form:

where 0 = To I TI given by

I . . . I Tn-1 < Tn = UO.In this case the mapping

Ix is

The mapping Ix : S, 4 Lo is continuous, where Lo is endowed with the topology of convergence in probability, by the hypothesis that X is a total semimartingale. Let

f3 = {H E S : H has a representation (*) and /HI 5.1). Let ,B = Ix(f3), the image of f3 under I x . It will now suffice to find a probability Q equivalent t o P such that X, E L1(dQ), 0 I t I uo and such that supuEpE Q ( U ) = c < oo. The reason this suffices is that if we take, for a given 0 = to < t l < . . . < t, = UO,the random variables

9 The Bichteler-Dellacherie Theorem

145

Ho = sign(EQ{Xtl - XO~ F o ) ) HI , = sign(EQ{Xt2 - Xt, IFt1 )), . . . , we have that for this H E B,

Since this partition r was arbitrary, we have Var(X) = sup, Var,(X) supUEpEQ(U) = c < co, and so X is a Q quasimartingale.

Lemma 1. lim sup P(IY1 > c) Y Ep

<

= 0.

Proof of Lemma 1. Suppose lim,,, supy Ep P(IY I > c ) > 0. Then there exists a sequence c,tending to co, Yn E 0 , and a > 0 such that P(IYnI > &) 2 a, all n. This is equivalent to

Since Yn E 0 , there exists Hn E B such that I x ( H n ) = Yn. Then Ix($ H n ) = +Ix(Hn) = l Y n E 0 , if cn > 1. But tends to 0 uniformly a.s. which n Cn implies that IX(&H") = lcnY n tends to 0 in probability. This contradicts

k~~

(*>. Lemma 2. There exists a law Q equivalent to P such that Xt E L1(dQ), 0 5 t 5 UO. Proof of Lemma 2. Let Y = supoltluo IXtl. Since X has cadlag paths, Y < oo a.s. Moreover if D is a countable dense subset of [O,uO],then ~ 1. Hence Y is a random variable. Let A, = {m 5 Y < m 1), Y = S U P t E (Xt and set Z = z E = o 2-,lAm. Then Z is bounded, strictly positive, and = A z . Then EQ{IXtI) 5 Y Z E L1(dP). Define Q by setting Ep ( Z ) EQ{Y) = Ep{YZ)/Ep{Z) < oo. Hence EQ{IXtl) < oo, 0 I t I UO.

+

Observe that 0 c L1(dQ) for the law Q constructed in Lemma 2. Lemma 3 below implies that X is an R quasimartingale for R Q P . Hence by Rao's Theorem (Theorem 15) it is a classical R semimartingale, and by the Girsanov-Meyer Theorem (Theorem 35) it is a classical semimartingale for the equivalent law P as well. Thus Lemma 3 below will complete the proof of Theorem 43. We follow Yan [233].

Lemma 3. Let 0 be a convex subset of L1(dQ), 0 E 0 , that is bounded in probability. That is, for any E > 0 there exists a c > 0 such that Q(< > c) 5 E, for any E 0. Then there exists a probability R equivalent to Q, with a bounded density, such that supuEpER(U) < co.

<

Proof of Lemma 3 and end of proof of Theorem 43. To begin, note that the hypotheses imply that 0 c L1(dR). What we must show is that there exists a bounded random variable Z , such that P ( Z > 0) = 1, and such that sUPcEpEQ(ZC) < oo.

146

I11 Semimartingales and Decomposable Processes

Let A E 3 such that &(A) > 0. Then there exists a constant d such that E 0 , by assumption. Using this constant d, let c = 2d, and we have that 0 5 c l A $ 0 , and moreover if B+ denotes all bounded, positive r.v., then clA is not in the L1(dQ) closure of P-B+, denoted 0 - B+. That is, c l A $! 0 - B+. Since the dual of L' is La, and 0 - B+ is convex, by a version of the Hahn-Banach Theorem (see, e.g., Treves [223, page 1901) there exists a bounded random variable Y such that

&(< > d) 5 Q(A)/2, for all

<

Replacing v by aljy,o) and letting a tend to oo shows that Y 2 0 a.s., since otherwise the expectation on the left side above would get arbitrarily large. Next suppose v = 0. Then the inequality above gives

= l {Y E B+ : supcEpEQ{YC) < oo). Since 0 E B+, we know X Now set ? is not empty. Let A = {all sets of the form { Z = 0), Z E X). We wish to show that there exists a Z E X such that Q ( Z = 0) = infAEA&(A). Suppose, then, that Z, is a sequence of elements of X. Let c, = supcEpE{Z,C) and d, = IIZnllLm (Since 0 E 0 , we have c, 2 0). Choose b, such that C b,~, < oo and C b,d, < oo, and set Z = C bnZn. Then clearly Z E X. Moreover, { Z = 0) = nn{Zn = 0). Thus A is stable under countable intersections, and so there exists a Z such that Q ( Z = 0) = infAEA&(A). We now wish to show Z > 0 a.s. Suppose not. That is, suppose Q ( Z = 0) > 0. Let Y satisfy (*) (we have seen that there exists such a Y and that it hence is in X). Further we take for our set A in (*) the set A = { Z = O), for which we are assuming &(A) > 0. Since 0 E 0 and 0 E B+, we have from Lemma 2 that 0 < E{YIA) = E{Y1~Z=o>).

Since each of Y and Z are in X , their sum is in X as well. But then the above implies

This, then, is a contradiction, since Q ( Z = 0) is minimal for Z E X. Therefore we conclude Z > 0 a.s., and since Z E B+, it is bounded as well, and Lemma 3 is proved; thus also, Theorem 43 is proved. We state again, for emphasis, that Theorems 26 and 43 together allow us to conclude that semimartingales (as we have defined them) and classical semimartingales are the same.

Exercises for Chapter I11

147

Bibliographic Notes The material of Chap. I11 comprises a large part of the core of the "general theory of processes" as presented, for example in Dellacherie [41], or alternatively Dellacherie and Meyer [45, 46, 441. We have tried once again to keep the proofs non-technical, but instead of relying on the concept of a natural process as we did in the first edition, we have used the approach of R. Bass [ l l ] , which uses the P. A. Meyer classification of stopping times and Doob's quadratic inequality to prove the Doob-Meyer Theorem (Theorem 13), the key result of the whole theory. The Doob decomposition is from Doob [55], and the Doob-Meyer decomposition (Theorem 13) is originally due to Meyer [163, 1641. The theory of quasimartingales was developed by Fisk [73], Orey [187], K. M. Rao [207], Stricker [218], and MQtivier-Pellaumail [159]. The treatment of compensators is new to this edition. The simple example of the compensator of a process with one jump of size one (Theorem 17)' dates back to 1970 with the now classic paper of Dellacherie [40]. The case of many jumps (Theorem 19) is due to C. S. Chou and P. A. Meyer [30], and can be found in many texts on point processes, such as [24] or [139]. The example of hazard rates comes from Fleming and Harrington [74]. The Fundamental Theorem of Local Martingales is due to J. A. Yan and appears in an article of Meyer [172]; it was also proved independently by DolQans-Dade [51]. Le Jan's Theorem is from [112]. The notion of special semimartingales and canonical decompositions is due to Meyer [171]; see also Yoeurp [234]. The Girsanov-Meyer theorems (Theorems 35 and 36) trace their origin t o the 1954 work of Maruyama [152], followed by Girsanov [83], who considered the Brownian case only. The two versions presented here are due to Meyer [171], and the cases where the two measures are not equivalent (Theorems 37 and 38) are due t o Lenglart [142]. The example from finance theory (starting on page 137) is inspired by [203]. Kazamaki's criterion was published originally in 1977 [124]; see also [125], whereas Novikov's condition dates to 1972 [184]. The Bichteler-Dellacherie Theorem (Theorem 43) is due independently to Bichteler [13, 141 and Dellacherie [42]. It was proved in the late 19701s, but the first time it appeared in print was when J. Jacod included it in his 1979 tome [103]. Many people have made contributions to this theorem, which had at least some of its origins in the works of MQtivier-Pellaumail, Mokobodzki, Nikishin, Letta, and Lenglart. Our treatment was inspired by Meyer [I761 and by Yan [233].

Exercises for Chapter I11 Exercise 1. Show that the maximum and the minimum of a finite number of predictable stopping times is still a predictable stopping time.

148

Exercises for Chapter I11

Exercise 2. Let S be a totally inaccessible stopping time, and let T = S + 1. Show that T is a predictable stopping time. Exercise 3. Let S, T be predictable stopping times. Let A that SAis a predictable stopping time.

=

{S = T). Show

Exercise 4. Let (0,F , P , P) be a filtered probability space satisfying the usual hypotheses. Show that the predictable c-algebra (on R+ x 0 ) is contained in the optional c-algebra. (Hint: Show that a chdlhg, adapted process can be approximated by processes in JL.) Exercise 5. Let S, T be stopping times with S 5 T . Show that (S,T] = {(t, w) : S ( w ) 5 t <_ T(w)) is a predictable set. Show further that [S,T) and (S,T) are optional sets. Last, show that if T is predictable, then (S,T) is a predictable set, and if both S and T are predictable, then [S,T) is a predictable set. Exercise 6. Let T be a predictable stopping time, and let (S,)n21 be a sequence of stopping times announcing T . Show that FT-= Vn Fs,. Exercise 7. A filtration P is called quasi left continuous if for every predictable stopping time T one has FT= FT-.7 Show that if P is a quasi left continuous filtration, then whenever one has a non-decreasing sequence of stopping times (Sn)n21,with lim,,, S, = T , it follows that FT= Vn FS,. (Note: One can have a quasi left continuous filtration such that there exists a stopping time T with FT-# FT.) Exercise 8. Show that the natural completed filtration of a LQvy process is a quasi left continuous filtration. *Exercise 9. Let X be a semimartingale with E{[X, XI,) < oo and suppose X has a decomposition X = Xo + M A, with A predictable. Show that E{[A, A],) 5 E{[X, XI,). (Hint: Recall that if T is a predictable stopping time, then AATl{T<m} = E{AXT~{T<,}IFT-),

+

and also that the set {(t, w) : AAt(w) # 0) is a countable disjoint union of sets of the form {(t,w) : Ti(w) = t), where the stopping times (Ti)i2l are predictable.) Exercise 10. Let X and Y be semimartingales with the processes [X, X] and [Y,Y] locally integrable. Prove the following "sharp bracket" version of the Kunita-Watanabe inequality, for jointly measurable processes H , K:

( Warning: Recall that (X, Y) can exist as the compensator of [X,Y ] even when (X, X ) and (Y, Y) do not exist.) A quasi left continuous filtration is formally defined in Chap. IV on page 189.

Exercises for Chapter I11

149

Exercise 11. Show that if A is a predictable finite variation process, then it is of locally integrable variation. That is, show that there exists a sequence of - such that E{J," IdAsI} < m. stopping times (Tn),>I Exercise 12. Let N be a counting process with its minimal completed filtration, with independent interarrival times. Show that the compensator A of N has absolutely continuous paths if and only if the cumulative distribution functions of the interarrival times are absolutely continuous. Exercise 13. Let T be an exponential random variable and Nt = l{t>T}. Let .Fo = c { T ) and suppose the filtration IF is constant: .Ft = .Fofor all t 2 0. Show that the compensator A of N is At = l{t>Tl. (This illustrates the importance of the filtration when calculating the compensator.) Exercise 14. Let N be a compound Poisson process, Nt = Uil{tLTi) where the times Ti are the arrival times of a standard Poisson process with parameter X and the Ui are i.i.d. and independent of the arrival times. Suppose E{IUil) < ca and E{Ui) = p. Show that the compensator A of N for the natural filtration is given by At = Apt. Exercise 15. Show that if T is exponential with parameter A, its hazard rate is constant and equal to A. Show also that if R is Weibull with parameters ( a , p), then its hazard rate is X(t) = apata-l. Exercise 16. Let T be exponential with parameter X and have joint distribution with U given by P ( T > t, U > s) = exp{-At - p s - Bts) for t 2 0, s 2 0, where A, p, and B are all positive constants and also B Xp. Show that the crude hazard rate of (T, U) is given by ~ # ( t = ) X Bt.

+

<

*Exercise 17. Let M be a martingale on a filtered space where the filtration is quasi left continuous. Show that (M,M) is continuous. (Hint: See the discussion on quasi left continuous filtrations on page 189 of Chap. IV.) Exercise 18. Let X be a semimartingale such that the process Dt = supsst [AXs1 is locally integrable. Show that every decomposition of X , X = Xo + M C, has that the process C is of locally integrable variation.

+

Exercise 19. Suppose that X is locally special, that is, there exists a sequence (Tn)n21 of stopping times increasing to ca a.s. such that xT*= M n An, with An a predictable finite variation process, for each n. Show that X is special. (That is, a semimartingale which is locally special is also special.)

+

Exercise 20. Prove the Lemma on page 132. Let Q

-

P, and Zt = . Show that an adapted, cbdlbg process M is a Q local martingale if and only if M Z is a P local martingale. 'Exercise 21. Let IF c (6 be filtrations satisfying the usual hypotheses, and let X be a (6 quasimartingale. Suppose that & = E{Xtl.Ft}, and that Y can be taken cbdlbg (this can be proved to be true, although it is a little hard to

150

Exercises for Chapter I11

prove). Show that Y is also a quasimartingale for the filtration IF. (Hint: Use Rao's Theorem.)

*Exercise 22. Suppose that A is a predictable finite variation process with 1 E{[A, A]," < cm,and that M is a bounded martingale. Show that [A,M ] is a uniformly integrable martingale. Exercise 23. Let T be a strictly positive random variable, and let .Ft = c{T A s; s 5 t). Show that (.Ft)t>o is the smallest filtration making T a stopping time. Exercise 24. Let Z be a chdlhg supermartingale of Class D with Zo = 0 and suppose for all predictable stopping times T one has E{AZT(.FT-) = 0, a s . Show that if Z = M - A is the unique Doob-Meyer decomposition of Z, then A has continuous paths almost surely. Exercise 25. Let A be an increasing process of integrable variation, and let T be a predictable jump time of A such that E{AATIFT-) = 0. Then its compensator is continuous at T . (This exercise complements Theorem 21.) *Exercise 26. A supermartingale Z is said t o be regular if whenever a E{ZTn) = sequence of stopping times (T,),>l increases t o T, then lim,,, E{ZT). Let Z be a chdlhg supermartingale of Class D with Doob-Meyer decomposition Z = M - A. Show that A is continuous if and only if Z is regular. *Exercise 27 (approximation of the compensator by Laplacians). Let Z be a c&dlhg positive supermartingale of Class D with limt+, E{Zt) = 0. (Such a supermartingale is called a potential.) Let Z = M - A be its DoobMeyer decomposition and assume further that A is continuous. Define

Show that for any stopping time T, limh,o A$ = AT with convergence in L1.

*Exercise 28. Let Z be a c&dl&gpositive supermartingale of Class D with limt,, E{Zt) = 0. Let Z = M -A be its Doob-Meyer decomposition. Let Ah be as given in Exercise 27. Show that for any stopping time T, limh,0 A& = AT, but in this case the convergence is weak for L1; that is, the convergence is in the topology c ( L 1 , Exercise 29 (discrete Laplacian approximations). Let Z be a c&dl&g positive supermartingale of Class D with limt+, E{Zt) = 0. Let Z = M - A be its Doob-Meyer decomposition and assume further that A is continuous. Define X n converges to X in a ( L 1 ,Lm) if X n , X are in L1 and for any a s . bounded random variable Y, E(XnY) -+ E(XY).

Exercises for Chapter I11

Show that lim,,,

A& = A,

151

with convergence in L1

Exercise 30. Use Meyer's Theorem (Theorem 4) to show that if X is a strong (Markov) Feller process for its natural completed filtration Pp, and if X has continuous paths, then the filtration IF" has no totally inaccessible stopping times. (This implies that the natural filtration of Brownian motion does not have any totally inaccessible stopping times.) *Exercise 31. Let ( Q , S , P, P) be the standard Brownian space. Show that the optional c-algebra and the predictable c-algebra coincide. (Hint: Use Meyer's Theorem (Theorem 4) and Exercise 30.) *Exercise 32. Let ( 0 , S , P, P ) be a filtered probability space satisfying the usual hypotheses. Let X be a (not necessarily adapted) chdlhg stochasldt) < oo. Let Rx (Xt ) = tic process such that for X > 0, E{J; e-xt E{J~*e - x s ~ t + s d s l S t } ,the right continuous version. Show that

lxt

is an IF martingale.

'Exercise 33 (Knight's compensator calculation method). Let X be a c&dlhg semimartingale. In the framework of Exercise 32 suppose the limits below exist both pathwise a s . and are in L1, and are of finite variation in finite time intervals:

Show that X is a special semimartingale, and A is the predictable term in its semimartingale decomposition.

General Stochastic Integration and Local Times

1 Introduction We defined a semimartingale as a "good integrator" in Chap. 11, and this led naturally to defining the stochastic integral as a limit of sums. To express an integral as a limit of sums requires some path smoothness of the integrands and we limited our attention to processes in JL, the space of adapted processes with paths that are left continuous and have right limits. The space JL is sufficient to prove It8's formula, the Girsanov-Meyer Theorem, and it also suffices in some applications such as stochastic differential equations. But other uses, such as martingale representation theory or local times, require a larger space of integrands. In this chapter we define stochastic integration for predictable processes. Our extension from Chap. I1 is very roughly analogous to how the Lebesgue integral extends the Riemann integral. We first define stochastic integration for bounded, predictable processes and a subclass of semimartingales known as 'H2. We then extend the definition to arbitrary semimartingales and to locally bounded predictable integrands. We also treat the issue of when a stochastic integral with respect to a martingale or a local martingale is still a local martingale, which is not always the case. In this respect we treat the subject of sigma martingales, which has recently been shown to be important for the theory of mathematical finance.

2 Stochastic Integration for Predictable Integrands In this section, we will weaken the restriction that an integrand H must be in IL. We will show our definition of stochastic integrals can be extended t o a class of predictably measurable integrands. Throughout this section X will denote a semimartingale such that X o = 0. This is a convenience involving no loss of generality. If Y is any semimartin= gale we can set - Yo, and if we have defined stochastic integrals for

154

IV General Stochastic Integration and Local Times

semimartingales that are zero at 0, we can next define

When Yo # 0, recall that we write J:+ HsdY, t o denote integration on (0, t], HsdV, denotes integration on the closed interval [0,t]. and We recall for convenience the definition of the predictable c-algebra, already defined in Chap. 111. Definition. The p r e d i c t a b l e c-algebra P on R+ x S1 is the smallest c algebra making all processes in IL measurable. That is, P = c{H : H E IL). We let bP denote bounded processes that are P measurable.

+

Let X = M A be a decomposition of a semimartingale X , with Xo = Mo = A. = 0. Here M is a local martingale and A is an F V process (such a decomposition exists by the Bichteler-Dellacherie Theorem (Theorem 43 of Chap. 111)). We will first consider special semimartingales. Recall that a semimartingale X is called special if it has a decomposition

where is a local martingale and 2 is a predictable F V process. This decomposition is unique by Theorem 30 in Chap. I11 and it is called the canonical decomposition. Definition. Let X be a special semimartingale with canonical decomposition X = + 3. The 'H2 n o r m of X is defined to be

The s p a c e of semimartingales 'Hz consists of all special semimartingales with finite 'Hz norm. In Chap. V we define an equivalent norm which we denote

11 . llHz. -

T h e o r e m 1. The space of 'H2 semimartingales i s a Banach space.

Proof. The space is clearly a normed linear space and it is easy t o check -that /I . l l X z is a norm (recall that E{T;) = E{[N, N],), and therefore IIXIIX2 = 0 implies that E{E;) = 0 which implies, since E is a martingale, that N = 0). To show completeness we treat the terms X and 2 separately. Consider - - 112 2 first E. Since E{T;} = 11 [N, N], /I ,, , it suffices t o show that the space of L2 martingales is complete. However an L~ martingale M can be identified with M, E L2, and thus the space is complete since L2 is complete. Next suppose (An) is a Cauchy sequence of predictable F V processes in 11 . 112 where llAllp = 11 IdAs 111 L P , p 2 1. TO show (An) converges it suffices

Som

2 Stochastic Integration for Predictable Integrands

155

to show a subsequence converges. Therefore without loss of generality we can assume EnIIAnl12 < m. Then C An converges in 11 . [I1 to a limit A. Moreover

IdAFJ. Therefore C An converges in L' and is dominated in L2 by En to the limit A in 11 . 112 as well, and there is a subsequence converging almost surely. To see that the limit A is predictable, note that since each term in the sequence (An),>'- is predictable, the limit A is the limit of predictably measurable processes and hence also predictable. For convenience we recall here the definition of It,.

Definition. JL (resp. bJL) denotes the space of adapted processes with c8gl&d1 (resp. bounded, c8glhd) paths. We first establish a useful technical lemma.

Lemma. Let A be a predictable F V process, and let H be in IL such that E { r IHsIIdAsI) < m. Then the F V process HsdAs)t20 is also predictable.

(Sot

Proof.We need only to write the integral JotHsdAs as the limit of Riemann sums, each one of which is predictable, and which converge in ucp to Jot HsdAs, showing that it too is predictable. The results that follow will enable us to extend the class of stochastic integrands from blL to b P , with X E 3-1' (and Xo = 0). First we observe that if H E blL and X E z2, then the stochastic integral H . X E 'H2. Also if X = + 2 is the canonical decomposition of X , then H . H .2is the canonical decomposition of H . X by the preceding lemma. Moreover,

x

x+

--

The key idea in extending our integral is to notice that [N,N ] and 2 are -F V processes, and therefore w-by-w the integrals Jot~ ; ( w ) d [ NN],(w) , and I Hs1 Id& 1 make sense for any H E bP and not just H E L.

Sot

Definition. Let X E 'H2 with X = E let H , J E b P . We define dx(H, J) by

+ A its canonical decomposition, and

"c8gl&d1'is the French acronym for left continuous with right limits.

156

IV General Stochastic Integration and Local Times

Theorem 2. For X E 'H2 the space bJL is dense in bP under d x (., .). Proof. We use the Monotone Class Theorem. Define A = {H E bP : for any E > 0, there exists J E blL such that d x ( H l J) < E). Trivially A contains blL. If Hn E A and Hn increases t o H with H bounded, then H E bP, and by the Dominated Convergence Theorem if 6 > 0 then for some N(6), n > N(6) implies dx(H, H n ) < 6. Since each Hn E A, we choose no > N(6) and there exists J E bL such that dx(J, H"0) < 6. Therefore given E > 0, by taking 6 = &/2 we can find J E blL such that dx(J, H ) < E, and therefore H E A. An application of the Monotone Class Theorem yields the result.

Theorem 3. Let X E 'H2 and H" E blL such that Hn is Cauchy under d x . Then Hn . X is Cauchy in 'H2. Proof. Since llHn. X

-

Hm. X 117-12

= dX(Hn,H m )

the theorem is immediate.

Theorem 4. Let X E 'H2 and H E bP. Suppose Hn E blL and JmE blL are two sequences such that lim, d x (Hn, H ) = lim, d x ( Jml H ) = 0. Then H n. X and Jm. X tend to the same limit in 'H2. Proof. Let Y = lim,,, are taken in 'H2. For

E

Hn . X and Z = lim,, Jm. X , where the limits > 0, by taking n and m large enough we have

and the result follows. We are now in a position to define the stochastic integral for H E bP (and X E 3-12).

Definition. Let X be a semimartingale in 'H2 and let H E bP. Let Hn E blL be such that lim,, d x ( H n l H ) = 0. The stochastic integral H . X is the (unique) semimartingale Y E 'H2 such that lim,, Hn . X = Y in 'H2. We write H . X = HsdXs)t20.

(Sot

We have defined our stochastic integral for predictable integrands and semimartingales in 'H2 as limits of our (previously defined) stochastic integrals. In order t o investigate the properties of this more general integral, we need to have approximations converging uniformly. The next theorem and its corollary give us this.

2 Stochastic Integration for Predictable Integrands

157

Theorem 5. Let X be a semimartingale in 'Hz. Then

Proof. For a process H , let H* = sup, IHtl. Let X = N decomposition of X . Then

+ x be the canonical

Doob's maximal quadratic inequality (Theorem 20 of Chap. I) yields

and using (a + b)' 5 2a2

+ 2b2 we have

Corollary. Let (Xn) be a sequence of semimartingales converging to X in , (Xnk - X ) * = 0 'H2. Then there exists a subsequence ( n k ) such that limn, , a.s. Proof. By Theorem 5 we know that ( X n - X ) * = sup, 1x7 - Xtl converges to 0 in L2. Therefore there exists a subsequence converging a.s. We next investigate some of the properties of this generalized stochastic integral. Almost all of the properties established in Chap. I1 (Sect. 5) still hold.2

Theorem 6. Let X , Y E 7-1' and H , K E bP. Then

and H.(X+Y)

=H

.X+H-Y.

Proof. One need only check that it is possible to take a sequence Hn E blL that approximates H in both dx and dy.

Theorem 7. Let T be a stopping time. Then ( H x ) ~= Hl[o,Tl. X = H .(xT). Indeed, it is an open question whether or not Theorem 16 of Chap. I1 extends to integrands in bP. See the discussion at the end of this section.

158

IV General Stochastic Integration and Local Times

Proof. Note that l[O,T] E bIL, so Hl[o,TlE b P . Also, xTis clearly still in x2. Since we know this result is true for H E blL (Theorem 12 of Chap. 11), the result follows by uniform approximation, using the corollary of Theorem 5.

Theorem 8. The jump process (A(H . X),),yo is indistinguishable from (Hs(~x9))s~o. Proof. Recall that for a process J, AJt = Jt - Jt-, the jump of J at time t . (Note that H . X and X are cadlag semimartingales, so Theorem 8 makes sense.) By Theorem 13 of Chap. I1 we know the result is true for H E blL. H ) = 0. By the Let H E b P , and let H n E blL such that limn,,dx(Hn, corollary of Theorem 5, there exists a subsequence (nk) such that lim (Hnk. X - H . X ) * = 0 a.s.

n k+ ,-

This implies that, considered as processes, lim A(Hnk . X ) nk,'

= A ( H .X),

outside of an evanescent set.3 Since each Hnk E blL, we have A(Hnk . X ) = Hnk(AX), outside of another evanescent set. Combining these, we have lim H n k( ( a X ) l { ~ x += ~ ) lim A(Hnk. X ) l { ~ x + o ) nk'00

nk-+OO

= A(H.X

)1{~~Z~)7

and therefore

In particular, the above implies that limn,,, {AX # O), a.s. We next form

HFk(w) exists for all (t, w) in

A = {w : there exists t >0 such that lim HFk(w) # Ht (w) and AXt (w)# 0). nk'm

Suppose P(A) > 0. Then

and if AX, # 0, then lAxsl + lA2isl > 0. The left side of (*) tends to 0 as nk -+ oo, and the right side of (*) does not. Therefore P(A) = 0, and we H n k A X= H A X . conclude A ( H . X ) = limn,,, A set A c R+ x R is evanescent if I n is a process that is indistinguishable from the zero process.

2 Stochastic Integration for Predictable Integrands

159

Corollary. Let X E 'H2, H E b P , and T a finite stopping time. Then

Proof. By Theorem 8, ( H . x ) ~ - = (H . x ) ~- HTAXTltt2T). On the other hand, xT-= - AXTltt2T). Let At = AXTltt2T). By the bilinearity (Theorem 6), H . ( x T - ) = H . ( X T ) - H . A. Since H . ( x T ) = ( H . x ) ~ by Theorem 7, and H . A = H T A X ~ ~-{ ~ >the T )result , follows.

xT

The next three theorems all involve the same simple proofs. The result is known t o be true for processes in blL; let ( H n ) E blL approximate H E bP in dx(., .), and by the corollary of Theorem 5 let nk be a subsequence such that lim (Hnk . X - H . X ) * = 0 a.s.

nk+,

Then use the uniform convergence t o obtain the desired result. We state these theorems, therefore, without proofs.

Theorem 9. Let X E 'Hz have paths of finite variation on compacts, and H E b p . Then H . X agrees with a path-by-path Lebesgue-Stieltjes integral. Theorem 10 (Associativity). Let X 'H2 a n d H . ( K . X ) = ( H K ) . X .

E

'H2 and H , K E b P . Then K . X E

Theorem 11. Let X E 'Hz be a (square integrable) martingale, and H E b P . Then H . X is a square integrable martingale. Theorem 12. Let X , Y E 'Hz and H , K E b P . Then

and in particular

Proof. As in the proof of Theorem 29 of Chap. 11, it suffices to show

> dx(Hn, H ) = 0. Let Tm = inf{t > 0 : Let ( H n ) E blL such that limn,, m). Then (T")are stopping times increasing t o oo a s . and [Y?" I 5 m.4Since it suffices to show the result holds on [0,T m ) , each m, we can assume without loss of generality that Y- is in blL. Moreover, the Dominated Convergence Theorem gives limn,, dx(HnY-, HY-) = 0. By Theorem 29 of Chap. 11, we have

* Recall that Y-

denotes the left continuous version of Y

160

IV General Stochastic Integration and Local Times t

[ H n ~ X , Y I t = ~ H 2 d [ X , Y ] , (t>O), and again by dominated convergence lim [Hn. X , Y] =

n-+m

I'

H,d[X, Y],

( t 2 0).

It remains only to show limn,,[Hn . X, Y] = [ H . X, Y]. Let Zn = H n . X , and let nk be a subsequence such that limnk,,(Znk - Z)* = 0 a.s., where Z = H . X (by the corollary to Theorem 5). Integration by parts yields

where we have used associativity (Theorem 10). We take limits so that lim [ Z n k 7 Y ] = Z Y - Y - . ( H . X ) - Z - . Y nk,'

= ZY - Y- . ( 2 ) - Z= [Z,Y] =

.Y

[H . X, Y].

At this point the reader may wonder how to calculate in practice a canonical decomposition of a semimartingale X in order to verify that X E N2. Fortunately Theorem 13 will show that 7-1' is merely a mathematical convenience.

Lemma. Let A be an F V process with A. = 0 and A E 7-1'. Moreover llAllN2 5 611 SomIdAsl llL2.

Jr

IdA,I E L2. Then

Proof. If we can prove the result for A increasing then the general result will follow by decomposing A = A+ - A-. Therefore we assume without loss of generality that A is increasing. Hence as we noted in Sect. 5 of Chap. 111, the compensator A of A is also increasing and E{&) = E{A,) < m. Let M be a martingale bounded by a constant k. Since A - A is a local martingale, Corollary 2 to Theorem 27 of Chap. I1 shows that

-

is a local martingale. Moreover t

L t 5 k(A,

+ 2,) + 2k

~A(A - X),l 9

I 3k(A,

+ A,)

E L'.

Therefore L is a uniformly integrable martingale (Theorem 51 of Chap. I) and E{L,) = E{Lo) = 0. Hence

2 Stochastic Integration for Predictable Integrands

E{M,(A

-

X),)

= E{[M, A = E{[M, A],)

161

A],) -

E{IM, 21,)

= E{[M, A],),

A

because is natural. By the KunitaWatanabe inequality (the corollary to Theorem 25 of Chap. 11)

where the second inequality uses 2ab 5 a2 + b2. However E{[M, MI,)

= E{M&)

(Corollary 4 of Theorem 27 of Chap. 11) and also [A,A], 5 A& a.s. Therefore

Since M is an arbitrary bounded martingale we are free to choose

Mm = (A - A)ml{,(A-A)m,
and using the Monotone Convergence Theorem we conclude

Consequently

E { A ~ )5 ~E{AL)+ 2E{(A - A ) k ) 5 ~ E { A & ) < oo, and A - 2 is a square integrable martingale, and

-

IlAllW

=

II(A - ~ ) . . I I L Z + IlAmll~zI 311Amll~2

for A increasing.

Remarks. The constant 6 can be improved to I+& 5 4 by not decomposing A into A+ and A-. This lemma can also be proved using the BurkholderGundy inequalities (see Meyer [171, page 3471).

162

IV General Stochastic Integration and Local Times

In Chap. V we use an alternative norm for semimartingales which we denote 11 . (IwP, 1 5 p < oo. The preceding lemma shows that the norms 11 . 1 1 ~ and 2 ii-.- are equivalent. The restrictions of integrands to bP and semimartingales to 'Hz are mathematically convenient but not necessary. A standard method of relaxing such hypothesis is to consider cases where they hold locally. Recall from Sect. 6 of Chap. I that a property 7r is said to hold locally for a process X if there exists a sequence of stopping times (Tn),>o such that 0 = To 5 T1 5 T2 5 . . . 5 Tn 5 . - . and limn,, T n = oo a.s., and such that ~ ~ " l { ~ has n >property ~ ) 7r for each n. Since we are assuming our semimartingales X satisfy Xo = 0, we could as well require only that xTnhas property IT for each n. A related condition is that a property hold prelocally.

Definition. A property 7r is said to hold prelocally for a process X with Xo = 0 if there exists a sequence of stopping times (Tn)n21increasing to oo a s . such that xT"-has property 7r for each n > 1.

+

Recall that XT- = XtltOltThe ~ ) .next theorem shows that the restriction of semimartingales to 'Hz isnot really a restriction a t all.

Theorem 13. Let X be a semimartingale, Xo = 0. T h e n X is prelocally in 'Hz. That is, there exists a non-decreasing sequence of stopping times (Tn), Tn = oo a.s., such that xT"-E 'H2 for each n. limn,,

Proof. Recall that xTn-= Xtl[o,Tn)+ X ~ n - l [ ~ n , , ) . By the BichtelerDellacherie Theorem (Theorem 43 of Chap. 111) we can write X = M + A, where M is a local martingale and A is an F V process. By Theorem 25 of Chap. I11 we can further take M to have bounded jumps. Let P be the bound for the jumps of M . We define

and let Y = x T n - . Then Y has bounded jumps and hence it is a special semimartingale (Theorem 31 of Chap. 111). Moreover

+

where L = M ~ and " C = AT"- - (AMTn)liTn,,). Then [L,L] 5 n P2, SO L is a martingale in 'H2 (Corollary 4 to Theorem 27 of Chap. 11), and also

= L + C E 'H2. hence C E 'H2 by the lemma. Therefore xTn-

2 Stochastic Integration for Predictable Integrands

163

We are now in a position to define the stochastic integral for an arbitrary semimartingale, as well as for predictable processes which need not be bounded. Let X be a semimartingale in .H2. To define a stochastic integral for predictable processes H which are not necessarily bounded (written H E p),we approximate them with Hn E bP.

Definition. Let X E 'Hz with canonical decomposition X = H E P is (.HzlX ) integrable if

+ 2.We say

Theorem 14. Let X be a semimartingale and let H E P be ('Hz,X ) integrable. Let H n = H l t I H l l n ) E bP. Then H n . X is a Cauchy sequence i n

l-t2. Proof. Since Hn E bP, each n , the stochastic integrals H n . X are defined. Hn = H and that (Hnl 5 IHI, each n. Then Note also that limn,,

and the result follows by two applications of the Dominated Convergence Theorem.

Definition. Let X be a semimartingale in .H2, and let H E P be (.HzlX ) integrable. The stochastic integral H . X is defined to be limn,, Hn . X , with convergence in 'H2,where H n = Hi{IHllnl. Note that H . X in the preceding definition exists by Theorem 14. We can "localize" the above theorem by allowing both more general H E P and arbitrary semimartingales with the next definition.

Definition. Let X be a semimartingale and H E P. The stochastic integral H - X is said to exist if there exists a sequence of stopping times Tn increasing to m a.s. such that xTn-E 'H2,each n 2 1, and such that H is ( ' H 2 , x T n - ) integrable for each n. In this case we say H is X integrable, written H E L ( X ) , and we define the stochastic integral by

each n. Note that if m > n then

164

IV General Stochastic Integration and Local Times

where Hk = HIIIHllk),by the corollary of Theorem 8. Hence taking limits we have H . ( x ~ ~ - )= ~H .~( x-T n - ) , and the stochastic integral is well-defined for H E L(X). Moreover let R~ be another sequence of stopping times such E R2 and such that H is ( R 2 , x R e - )integrable, for each !. Again that xReusing the corollary of Theorem 8 combined with taking limits we see that

on [0,R~ A T n ) , each !2 1 and n 2 1. Thus in this sense the definition of the stochastic integral does not depend o n the particular sequence of stopping times. If H E b P (i.e., H is bounded), then H E L(X) for all semimartingales X , since every semimartingale is prelocally in 'Hz by Theorem 13. Definition. A process H is said to be locally bounded if there exists a sequence of stopping times (Sm)mL1increasing to oo a.s. such that for each m 2 1, ( H t ~ s - l ~ ~ - > is bounded. ~ ) ) ~ ~ ~ Note that any process in lL is locally bounded. The next example is sufficiently important that we state it as a theorem. T h e o r e m 15. Let X be a semimartingale and let H E P be locally bounded. T h e n H E L(X). That is, the stochastic integral H . X exists.

Proof. Let (Sm),>1, (Tn)nL1 be two sequences of stopping times, each increasing to oo as., such that ~ ~ ~ is bounded l ~ for each ~ m, ~ and> ~ ) xTn-E 'H2 for each n. Define Rn = min(Sn,Tn). Then H = ~ ~ ~ 1 { ~ n > ~ ) charges only (0, Rn), on (0, Rn) and hence it is bounded there. Since we have that H is ('H2,x R n - ) integrable for each n 2 1. Therefore using the sequence Rn which increases to cc a.s., we are done.

xRn-

We now turn our attention to the properties of this more general integral. Many of the properties are simple extensions of earlier theorems and we omit their proofs. Note that trivially the stochastic integral He X , for H E L(X), is also a semimartingale.

+

T h e o r e m 16. Let X be a semimartingale and let H, J E L(X). T h e n QH ,8J E L(X) and (QH + ,8J ) . X = QH. X + ,8J . X . That is, L(X) i s a linear space.

Proof. Let (Rm) and (Tn) be sequences of stopping times such that H is ('Hz,x R m - ) integrable, each m, and J is ('H2,x T n - ) integrable, each n. Taking Sn = Rn A Tn, it is easy to check that QH+ ,8J is (R2,x S n - ) integrable for each n. T h e o r e m 17. Let X , Y be semimartingales and suppose H E L(X) and H E L(Y). T h e n H E L ( X + Y ) a n d H . ( X + Y ) = H . X + H . Y .

2 Stochastic Integration for Predictable Integrands

165

Theorem 18. Let X be a semimartingale and H E L ( X ) . The jump process ( A ( H - X)s),20 is indistinguishable from (H,(AX,)),>o. Theorem 19. Let T be a stopping time, X a semimartingale, and H E L ( X ) . Then ( H . x ) =~ HipTl . X = H . (xT). Moreover, letting m- equal m, we have moreover

Theorem 20. . Let X be a semimartingale with paths of finite variation on compacts. Let H E L ( X ) be such that the Stieltjes integral IH,IJdX, I exists a.s., each t 2 0. Then the stochastic integral H . X agrees with a path-by-path Stieltjes integral. Theorem 21 (Associativity). Let X be a semimartingale with K E L ( X ) . Then H E L ( K . X ) if and only if H K E L ( X ) , in which case H . ( K . X ) = ( H K ). X . Theorem 22. Let X , Y be semimartingales and let H E L ( X ) , K E L ( Y ) . Then

=I t

[H.X,K.Y]t

HsK,d[X,Y], (t 2 0 ) .

0

Note that in Theorem 22 since H . X and H . Y are semimartingales, the quadratic covariation exists and the content of the theorem is the formula. Indeed, Theorem 22 gives a necessary condition for H to be in L ( X ) , namely H:d[X, XI, exists and is finite for all t 2 0. The next theorem (Thethat orem 23) is a special case of Theorem 25, but we include it because of the simplicity of its proof.

fi

Theorem 23. Let X be a semimartingale, let H E L ( X ) , and suppose Q is another probability with Q << P. If HQ . X exists, it is Q indistinguishable from H p . X . Proof. HQ . X denotes the stochastic integral computed under Q . By Theorem 14 of Chap. 11, we know that HQ . X = H p . X for H E L, and therefore if X E 'Hz for both P and Q , they are equal for H E bP by the corollary (Tn)n>l - be two sequences of stopping times inof Theorem 5. Let (Re)e>l, creasing to m a.s. such that H is ( ' H Z , x R e - )integrable under Q , and H is ('H2,xTn-) integrable under P , each l and n . Let S m = Rm AT^, so that H is ('Hz,xSm-) integrable under both P and Q . Then H . X = limn,, Hn . X on [O,Sm)in both d x ( P ) and d x ( Q ) , where H n = E bP. Since H g . X = H z . X , each n , the limits are also equal.

Much more than Theorem 23 is indeed true, as we will see in Theorem 25, which contains Theorem 23 as a special case. We need several preliminary results.

166

IV General Stochastic Integration and Local Times

Lemma. Let X E 'Hz and X

=

F +?Ibe its canonical decomposition. Then

Proof. First observe that

+

[X,X] = [F,N] 2[F,?I] + --

It suffices to show E{[N,A],) -

= 0,

[?I,a.

since then

-

E{lN, Nl,)

= E{lX,

--

and the result follows since [A,A],

XI,

-

[?I,?Ilm),

2 0. Note that

by the Kunita-Watanabe inequalities. Also E{[M,x],) = 0 for all bounded martingales because A is natural. Since bounded martingales are dense in the space of L~ martingales, there exists a sequence (Mn),>l of bounded martingales such that limn,, E{[Mn - W ,M n -XI,) = 0. Again using the Kunita-Watanabe inequalities we have

and therefore limn,, E{[Mn,x],) -= E{p,A],). each n, it follows that E{[N, A],) = 0.

Since E{[Mn,x],)

= 0,

Note that in the preceding proof we established the useful equality E{[?I,?I]t)for a semimartingale X E 'H2 with E{[X,X]t) = E{[F,F]~) canonical decomposition X = W + 2.

+

Theorem 24. For X a semimartingale in 'Hz,

Proof. By Theorem 5 for IHI 5 1

Since

where X = M + A is the canonical decomposition of X , we have the right inequality. by the For the left inequality we have 11 [M,M]g211~zI 11 [X, lemma preceding this theorem. Moreover if 1 HI I 1, then

~12~11~~

2 Stochastic Integration for Predictable Integrands

167

&;

Next take H = this exists as a predictable process since A is predictable, and therefore A predictable implies that IAl, the total variation process, is also predictable. Consequently we have

We present as a corollary to Theorem 24, the equivalence of the two pseudonorms SUPIHI<~ II(H . X)&IILz and llXllu2. We will not have need of this corollary in this-book,5 but it is a pretty and useful result nevertheless. It is originally due to Yor [241], and it is actually true for all p, 1 5 p < m. (See also Dellacherie-Meyer [46, pages 303-3051 .)

Corollary. For a semimartingale X (with Xo = O),

and in particular S U P I-~ IJ < I (~H .X ) & ~ ~ < Lm Z if and only if llXllu2 < m.

Proof. By Theorem 5 if llXllu2 < m and IHI 5 1 we have

Thus we need to show only the left inequality. By Theorem 24 it will suffice to show that 11 [x,x I ~ ~ I I IL ~SUP Il(H. X)&11~2r lHlll

for a semimartingale X with Xo = 0. To this end fix a t > 0 and let 0 = To < TI < . . . < Tn = t be a random partition of [O,t]. Choose ~ 1 ,. ..,E, non-random and each equal to 1 or -1 and let H = Cy=,E ~ ~ ( ~Then , ~H ~ is, a~ simple ~ I . predictable process and

Let cr = SUPIHILl II(H. X)I$IIL2. We then have However the corollary does give some insight into the relationship between Theorems 12 and 14 in Chap. V.

168

IV General Stochastic Integration and Local Times

If we next average over all sequences E ~ ., ..,E, taking values in the space {+I)", we deduce

Next let u, = {Ty) be a sequence of random partitions of [O,t] tending to the identity. Then Ci(XTr -XTiE1)2 converges in probability to [X,XIt. Let { m k ) be a subsequence so that Ci(XT;k - XTmk)2converges to [XIXIt a.s. "-1 Finally by Fatou's Lemma we have

E{[x, XI,) 5 lirninf E { C ( X ~- ~ ~~ . m k5) a~2 ) rnk-03

1-

1

2

Letting t tend to cm we conclude that

1t remains to show that if S U P I ~ JII(H. < ~ X)&1IL2 < cm, then X E 7f2. We will show the contrapositive. If X <7f2, then S U P -I ~II(H ~ < .~X)&IILZ= cm. Indeed, let T n be stopping times increasing to cm such that xTn-is in 7f2 for each n (cf., Theorem 13). Then

Letting n tend to cm gives the result. Before proving Theorem 25 we need two technical lemmas.

Lemma 1. Let A be a non-negative increasing F V process and let Z be a positive uniformly integrable martingale. Let T be a stopping time such that A = AT- (that is, A, = AT-) and let k be a constant such that Z 5 k on [0,T). Then 5 kE{A,). E{A,Z,) Proof. Since Ao- = Zo- = 0, by integration by parts

2 Stochastic Integration for Predictable Integrands

169

where the second integral in the preceding is a path-by-path Stieltjes integral. Let Rn be stopping times increasing to cm a s . that reduce the local martingale As-dZs)t20. Since dA, charges only [0, T ) we have

(Sot

Jot

ZsdAs 5

Jo

t

kdAs 5 kA,

for every t 2 0. Therefore

The result follows by Fatou's Lemma. Lemma 2. Let X be a semimartingale with Xo = 0, let Q be another probability with Q << P , and let Zt = E p ( S ( 3 t ) . If T is a stopping time such that Zt < k on [O,T) for a constant k, then

Proof. By Theorem 24 we have

5

& sup IHlll

Ep{((H . x ~ - ) * ) ~ ) + ; 2&Ep{[xT-,XT-1:).

where we have used Lemma 1 on both terms to obtain the last inequality above. The result follows by the right inequality of Theorem 24. Note in particular that an important consequence of Lemma 2 is that if Q << P with $$ bounded, then X E 7f2(P) implies that X E 7f2(Q) as well, dQ with the estimate llXllU2(&)< 5 & l l ~ l l ~ z ( ~where ) , k is the bound for D. Note further that this result (without the estimate) is obvious if one uses the equivalent pseudonorm given by the corollary to Theorem 24, since

where again k is the bound for

g.

170

IV General Stochastic Integration and Local Times

Theorem 25. Let X be a semimartingale and H E L ( X ) . If Q << P , then H E L ( X ) under Q as well, and HQ . X = H p . X , Q - a . ~ .

Proof. Let T n be a sequence of stopping times increasing to m a.s. such that integrable under P, each n 2 1. Let Zt = ~ ~ ( g 1 . F the~ ) ~ c&dl&gversion. Define S n = inf{t > 0 : J Z t J> n ) and set Rn = S n A Tn. Then xRnE 7 f 2 ( P )n 7 f 2 ( & ) by Lemma 2, and H is ( 7 f 2 , xRn-) integrable integrable under Q , which will under P . We need to show H is (7f2,xRn-) in turn imply that H E L ( X ) under Q . Let xRn-= N + C be the canonical decomposition under Q . Let H m = H 1 ~ I H 1 5 mThen ).

H is

(xTn-,a2)

integrable and then by monotone convergence we see that H is (7f2,xRn-) under Q . Thus H E L ( X ) under Q , and it follows that HQ . X = H p . X , Q-as. 0 Theorem 25 can be used to extend Theorem 20 in a way analogous to the extension of Theorem 17 by Theorem 18 in Chap. 11. Theorem 26. Let X , be two semimartingales, and let H E L ( X ) , H E Let A = {w : H.(w) = g . ( w ) and X.(w) = W . ( w ) ) , and -let B = {w : t H Xt ( w ) is of finite variation on compacts). Then H .X = He X on A , and H . X is equal to a path-by-path Lebesgue-Stieltjes integral on B.

~(x).

Proof. Without loss of generality assume P ( A ) > 0. Define Q by Q ( A ) = P(A1A). Then Q << P and therefore H E L ( X ) , E L ( W ) under Q as well as processes H and 1? as well as under P by Theorem 25. However under Q the-X and are indistinguishable. Thus HQ .X = H Q . X and hence H'.X = H . X P - a s . on A by Theorem 25, since Q << P . The second assertion has an analogous proof (see the proof of Theorem 18 of Chap. 11). Note that one can use stopping times to localize the result of Theorem 26. The proof of the following corollary is analogous to the proof of the corollary of Theorem 18 of Chap. 11. Corollary. With the notation of Theorem 26, let S , T be two stopping times with S < T . Define

C = {w : H t ( w ) = z t ( w ) ;X t ( w ) = X t ( w ) ;S ( w ) < t 5 T ( w ) ) , D = {w : t H X t ( w ) is of finite variation on S ( w ) < t < T ( w ) ) .

--z.xS

hen H . X T - H . X S =??.WT on^ and H . X T - H . X S equals a path-by-path Lebesgue-Stieltjes integral on D.

2 Stochastic Integration for Predictable Integrands

171

Theorem 27. Let Pk be a sequence of probabilities such that X i s a Pk semimartingale for each k . Let R = CEO=, XkPk where X k 2 0 , each k , and CEO=, X k = 1. Let H E L ( X ) under R. Then H E L ( X ) under Pk and HR . X = Hpk . X , Pk-a.s., for all k such that Xk > 0 .

Proof. If X k > 0 then Pk << R. Moreover since P k ( A ) 5 * R ( A ) , it follows that H E L ( X ) under Pk. The result then follows by Theorem 25. We now turn to the relationship of stochastic integration to martingales and local martingales. In Theorem 11 we saw that if M is a square integrable martingale and H E b P , then H . M is also a square integrable martingale. When M is locally square integrable we have a simple sufficient condition for H to be in L ( M ) . Lemma. Let M be a square integrable martingale and let H E P be such that E{JomH z d [ M ,MI,) < cm. Then H . M is a square integrable martingale.

Proof. If H~ E b P , then H k . M is a square integrable martingale by Theorem 11. Taking H~ = H l i I H l l k ) ,and since H is (7f2,M ) integrable, by Theorem 14 H~ . M converges in 7f2 to H . M which is hence a square integrable martingale. Theorem 28. Let M be a locally square integrable local martingale, and let H E P . The stochastic integral H . M exists (i.e., H E L ( M ) ) and is a locally square integrable local martingale if there exists a sequence of stopping times ( T n ) n y l increasing to cm a.s. such that E{JoT" Hzd[M,MI,) < cm.

Proof. We assume that M is a square integrable martingale stopped at the time T n . The result follows by applying the lemma. Theorem 29. Let M be a local martingale, and let H E P be locally bounded. Then the stochastic integral H . M is a local martingale.

Proof. By stopping we may, as in the proof of Theorem 29 of Chap. 111, assume that H is bounded, M is uniformly integrable, and that M = N + A where N is a bounded martingale and A is of integrable variation. We know that there exists R k increasing to cm a s . such that M ~ E ~7f2,-and since H E b P there exist processes H~ E blL such that 1 1 . M~ ~~ -~H . -~ ~ ~ - tends 1 1 to zero. In particular, H ~ M. ~ tends ~ -to H - M ~ in~ ucp. - Therefore we ~ - to H . M in ucp, with H~ E blL. can take H e such that H e . M ~ tends Finally without loss of generality we assume H e . M converges to H . M in ucp. Since H~ . M = H~ . N H~ . A and H~ . N converges to H . N in ucp, we deduce H ~A .converges to H . A in ucp as well. Let 0 5 s < t and assume Y E bF,. Therefore, since A is of integrable total variation, and since H~ . A is a martingale for H~ E bIL (see Theorem 29 of Chap. 111), we have

+

~

2

172

IV General Stochastic Integration and Local Times E{Y

jts+ H,~A,J

= E{Y

lim H:~A,)

= E{Y

lim

A,

where we have used Lebesgue7s Dominated Convergence Theorem both for the Stieltjes integral w-by-w (taking a subsequence if necessary to have a s . convergence) and for the expectation. We conclude that HsdAs)t20 is a martingale, hence H . M = H - N + H . A is also a martingale under the assumptions made; therefore it is a local martingale.

(Sot

In the proof of Theorem 29 the hypothesis that H E P was locally bounded was used to imply that if M = N + A with N having locally bounded jumps and A an F V local martingale, then the two processes

are locally integrable. Thus one could weaken the hypothesis that H is locally bounded, but it would lead to an awkward statement. The general result, that M a local martingale and H E L(M) implies that H . M is a local martingale, is not true! See Emery's example, which precedes Theorem 34 for a stochastic integral with respect to an IH2 martingale which is not even a local martingale! Emery's counterexample has lead to the development of what are now known as sigma martingales, a class of processes which are not local martingales, but can be thought to be "morally local martingales." This is treated in Section 9 of this chapter. Corollary. Let M be a local martingale, Mo = 0, and let T be a predictable stopping time. Then MT- is a local martingale.

Proof. The notation M ~ -means

where Mop = Mo = 0. Let (Sn)n21 be a sequence of stopping times announcing T. On {T > O), lim l [ O , s n ] = I[O,T) n-oo and since l l o , S n ~ is a left continuous, adapted process it is predictable; hence lI0,T) is predictable. But l[O,T)(~)dMs = M ~ by ~ Theorem 18, and hence it is a local martingale by Theorem 29.

2 Stochastic Integration for Predictable Integrands

173

Note that if M is a local martingale and T is an arbitrary stopping time, it is not true in general that MT- is still a local martingale. Since a continuous local martingale is locally square integrable, the theory is particularly nice.

Theorem 30. Let M be a continuous local martingale and let H E P be such that Hzd[M, MI, < rn a.s., each t 2 0. Then the stochastic integral H . M exists (i.e., H E L(M)) and it is a continuous local martingale. Proof. Since M is continuous, we can take

R~ =inf{t > 0 : lMtl > k). Then IMtARkI 5 k and therefore M is locally bounded, hence locally square integrable. Also M continuous implies [M,M ] is continuous, whence if

I'

~~=inf{t>O: H~~[M,M],>~),

(Sot

is also locally bounded. Then H . M is a locally we see that Hzd[M, square integrable local martingale by Theorem 28. The stochastic integral H . M is continuous because A ( H - M ) = H ( A M ) and A M = 0 by hypothesis. In the classical case where the continuous local martingale M equals B , a standard Brownian motion, Theorem 30 yields that if H t P and Jot Hzds < co a s . , each t 2 0, then the stochastic integral ( H . Bt)t>o = H,dB,)t20 exists, since [B, B]t = t.

(Sot

Corollary. Let X be a continuous semimartingale with (unique) decomposition X = M + A. Let H E P be such that

each t 2 0. Then the stochastic integral ( H . X)t continuous.

=J :

H,dX, exists and it is

Proof. By the corollary of Theorem 31 of Chap. 111, we know that M and A have continuous paths. The integral H . M exists by Theorem 30. Since H . A exists as a Stieltjes integral, it is easy to check that H E L(A), since A is continuous, and the result follows from Theorem 20. In the preceding corollary the semimartingale X is continuous, hence [XIXI = [M, M ] and the hypothesis can be written equivalently as

each t 2 0. We end our treatment of martingales with a special case that yields a particularly simple condition for H to be in L(M).

174

IV General Stochastic Integration and Local Times

Theorem 31. Let M be a local martingale with jumps bounded by a constant p. Let H E P be such that Hzd[M,MI, < cm a.s., t 2 0, and E{H$} < cm for any bounded stopping time T. Then the stochastic integral HsdMs)t>o exists and it is a local martingale.

Sot

(Sot

Sot

Proof. Let Rn = inf{t > 0 : H:~[M, MI, > n), and let Tn = min(Rn, n). Then Tn are bounded stopping times increasing to cm a s . Note that

E{L

T"

H:d[M, MI,}

< n + E{H& < n + P 2 E { ~ g n<} cm,

and the result follows from Theorem 28. The next theorem is, of course, an especially important theorem, the Dominated Convergence Theorem for stochastic integrals. Theorem 32 (Dominated Convergence Theorem). Let X be a semimartingale, and let H m E P be a sequence converging a.s. to a limit H . If there exists a process G E L(X) such that lHml < G, all m, then Hm, H are in L(X) and H m . X converges to H . X in ucp.

Proof. First note that if I JI < G with J E P, then J E L(X). Indeed, let (Tn)n21increase to cm a s . such that G is (7f2,xT"-) integrable for each n. Then clearly

+

and thus J is (7f2,x T " - ) integrable for each n. (Here N 2 is the canonical decomposition of xT" - .) To show convergence in ucp, it suffices to show uniform convergence in probability on intervals of the form [0,to] for to b e d . Let E > 0 be given, and choose n such that P ( T n < to) < E, where xT"-E 7f2 and G is (7f2,xT"-) integrable. Let xT"-= N 2 , the canonical decomposition. Then

+

The second term tends to zero by Lebesgue's Dominated Convergence Theois a square integrable rem. Since IHm - HI < 2G, the integral (Hm - H ) martingale (by the lemma preceding Theorem 28). Therefore using Doob's maximal quadratic inequality, we have

.m

2 Stochastic Integration for Predictable Integrands

175

and again this tends to zero by the Dominated Convergence Theorem. Since convergence in L~ implies convergence in probability, we conclude for 6 > 0,

< lim sup P{sup I ((Hm m+m

-

H ) . x T n - ) t 1 > 6)

tlto

+ p ( T n < to)

and since E is arbitrary, the limit is zero. We use the Dominated Convergence Theorem to prove a seemingly innocuous result. Generalizations, however, are delicate as we indicate following the proof. Theorem 33. Let IF = (.Ft)t>o and G = (Gt)t20 be two filtrations satisfying the usual hypotheses and suppose .Ft c Gt, each t > 0, and that X i s a semimartingale for both IF and G. Let H be locally bounded and predictable for IF. T h e n the stochastic integrals HF . X and HG . X both exist, and they are equaL6 Proof. It is trivial that H is locally bounded and predictable for (Gt)t20 as well. By stopping, we can assume without loss of generality that H is bounded. Let 7f

= {all

bounded, 3 predictable H such that H F . X = H G -X).

Then 7f is clearly a monotone vector space, and 7f contains the multiplicative class blL by Theorem 16 of Chap. 11. Thus using Theorem 32 and the Monotone Class Theorem we are done. It is surprising that the assumption that H be locally bounded is important. Indeed, Jeulin [114, pages 46, 471 has exhibited an example which shows that Theorem 33 is false in general. Theorem 33 is not an exact generalization of Theorem 16 of Chap. 11. Indeed, suppose IF and G are two arbitrary filtrations such that X is a semimartingale for both IF and G, and H is bounded and predictable for both of them. If Zt = .Ftn Gt, then X is still an (Zt)t>osemimartingale by Stricker's Theorem, but it is not true in general that K i s (Zt)t20 predictable. It is an open question as to whether or not HF . X = HG. X in this situation. For a partial result, see Zheng [248]. HF . X and HG . X denote the stochastic integrals computed with the filtrations

(Ft)t?o and (Gt)t20, respectively.

176

IV General Stochastic Integration and Local Times

Example (Emery's example of a stochastic integral behaving badly). The following simple example is due to M. Emery, and it has given rise to the study of sigma martingales, whose need in mathematical finance has become apparent. Let X = (Xt)t20be a stochastic process given by the following description. Let T be an exponential random variable with parameter X = 1, let U be an independent random variable such that P{U = 1) = P{U = -1) = 112, and set X = Ul{t2T).Then X together with its minimal filtration satisfying the usual hypotheses is a martingale in 7f2.That is, X is a stopped compound Poisson process with mean zero and is an L2martingale. Let Ht = 1 ,l{t,o). Therefore H is a deterministic integrand, continuous on (0, cm), and hence predictable. Consequently the path-by-path Lebesgue-Stieltjes integral Zt = HsdXs exists a s . However H . X is not locally in 7 f p for any p 1. (However since it is still a semimartingale7, it is prelocally in 7f2.)Moreover, even though X is an L2 martingale and H is a predictable integrand, the stochastic integral H . X is not a local martingale because E{IZsl) = cm for every stopping time S such that P ( S > 0) > 0.

>

The next theorem is useful, since it allows one to work in the convenient space 7f2 through a change of measure. It is due originally to Bichteler and Dellacherie, and the proof here is due to Lenglart. We remark that the use of the exponent 2 is not important, and that the theorem is true for 7fP for any p 2 1. Theorem 34. Let X be a semimartingale on a filtered complete probability space ( R , 3 ,IF, P ) satisfying the usual hypotheses. Then there exists a probability Q which is equivalent to P such that under Q , X is a semimartingale can be taken to be bounded. i n H2.Moreover,

Before we begin the proof we establish a useful lemma. Lemma. Let ( R , 3 , P) be a complete probability space and let Xn be a sequence of a s . finite valued random variables. There exists another probability Q , equivalent to P and with a bounded density, such that every Xn is in L2(dQ).

Proof. Assume without loss that each of the Xn is positive. For a single ran2-k1Ak.Then dom variable X we take Ak = { k X < k 1) and Y = Ck>l Y is bounded and

<

+

<

2Tn. The For the general case, choose constants a, such that P(X, > a,) Borel-Cantelli Lemma implies that a.s. Xn 5 an for all n sufficiently large. Next choose constants c, such that cnan < cm. Let Yn be the bounded H . X is still a semimartingale since it is of finite variation as. on compact time

sets.

2 Stochastic Integration for Predictable Integrands

177

density chosen for X, individually as done in the first part of the proof, and cnYn so that we have the result. take Y = -

Proof of Theorem 34. Using the lemma we make a first change of measure making all of the random variables [X,XI, integrable. Recall that [X,X] is invariant under a change to an equivalent probability measure. By abusing notation, we will still denote this new measure by P . This implies that if Jt = supslt lAX,l, then J: [X,XIt and thus J is locally in L2. Hence it is also a fortiori locally in L1, and thus X is special. We write its canonical decomposition X = M A, with Xo = Ao. We now make a second change to an equivalent law, this time called Q, again using the lemma, such that IdA,I E L2(dQ) for each n. Then X is special under Q as well, since each [X, XI, remains in L1 for Q. Let X = N C denote its Q decomposition. We apply Girsanov's Theorem to get

<

+

+

s.

where Zt = Ep{ZIFt} with Z = For convenience we write this as Ct = At &. We now need to establish that EQ{(S~l d ~ l ) = ~ }EQ{V1:} < 00. Note that we have EQ{&~) < cc, since & = Mt - At, and At E L2(dQ). It is easy to check that Ep{[M, MI,) I 4Ep{[X, XI,) < cc by an argument similar to the proof of the lemma preceding Theorem 24. Finally L2(dP) C L2(dQ). Since V is predictable, there exists a predictable process H taking values in {-1,l) such that dVsI = HsdVs. Setting S = H - X , we have [S,S] = [X, XI, and the random variables [S,S], are then integrable (dP). S has a decomposition S = L + D where L = H - M and D = H .A, hence under the law Q we have L = H.N+H.V, always using that the stochastic integral is invariant under a change to an equivalent probability measure. We deduce that ~ ~ ~ V , )=~EQ{(J; ) l d ~ , 1 ) ~ coupled ), E Q { (H ~ , ~ v , ) ~ )< cc. Since E Q { ( H with IH. Alt = the result follows.

+

Sot

Corollary (Lenglart's Inclusion Theorem). Let (a,F,IF, P) satisfy the usual hypotheses, and suppose Q is an equivalent probability measure with a bounded density with respect to P . Then 'H2(P) C 'H2(Q). That is, 'H2 with respect to P is contained in the space of 'H2 semimartingales with respect to

QProof. Let X be an 'H2 semimartingale with respect to P . As we saw in the proof of Theorem 34, since Q has a bounded density, we obtain that [X,X] is in L1(dQ), and hence X is special for Q. We then obtain a canonical decomposition of X and show both terms are in 'H2 analogously to how we did it in the proof of Theorem 34.

178

IV General Stochastic Integration and Local Times

3 Martingale Representation In this section we will be concerned with martingales, rather than semimartingales. The question of martingale representation is the following. Given a collection A of martingales (or local martingales), when can all martingales (or all local martingales) be represented as stochastic integrals with respect to processes in A? This question is surprisingly important in applications, and it is particularly interesting (in finance theory, for example) when A consists of just one element. Throughout this section we assume as given an underlying complete, filtered probability space (a,F, (Ft)t20,P) satisfying the usual hypotheses. We begin by considering only L2 martingales. Later we indicate how to extend these results to locally square integrable local martingales.

Definition. The space M2 of L2 martingales is all martingales M such that supt E { M ~ )< cc, and Mo = 0 a.s. Notice that if M E M2, then limt+m E{M:) = E{Mk) < cc, and Mt = E{MmIFt}. Thus each M E M2 can be identified with its terminal value M,. We can endow M2 with a norm

and also with an inner product

for M, N E M2. It is evident that M2 is a Hilbert space and that its dual space is also M2. If E{M:) < cc for each t , we call M a square integrable martingale. If in addition sup, E{M:) = E { M ~ < ) cc, then we call M an L2 martingale. The next definition is a key idea in the theory of martingale representation. It differs slightly from the customary definition because we are assuming all martingales are zero at time t = 0. If we did not have this hypothesis, we would have to add the condition that for any event A E Fo,any martingale M in the subspace F , then M l n E F .

Definition. A closed subspace F of M2 is called a stable subspace if it is stable under stopping (that is, if M E F and if T is a stopping time, then M T E F).8 Theorem 35. Let F be a closed subspace of M2. T h e n the following are equivalent. (a) F is closed under the following operation. For M E F , ( M - M t ) l n E F for A E Ft, any t L 0. (b) F is a stable subspace. Recall that

MT

= M ~ A Tand , MO = 0.

3 Martingale Representation (c)

If M E F and H is bounded, predictable, then (J: H,dM,)t>o

=H

179

.M

E

F. (d) If M E F and H is predictable with E { J ~ H;~[M,MI,) H.MEF.

< co, then

Proof. Property (d) implies (c), and it is simple that (c) implies (b). To get (b) implies (a), let T = t ~where , t, t A = { oo,

ifwE.4, if w A.

e

Then T = tn is a stopping time when A E Ft7and ( M - M t ) l A = M - M ~ ; since F is assumed stable, both M and MT are in F . It remains to show only that (a) implies (d). Note that if H is simple predictable of the special form

<

withAi E Ft",0 5 i 5 n, O = to t l 5 . . . 5 tn+l < co, then H . M E F whenever M E F . Linear combinations of such processes are dense in bL which in turn is dense in bP under dM(.,.) by Theorem 2. But then bP is dense in , < co, as the space of predictable processes 'H such that E{Jom~ z d [ MMI,) is easily seen (cf., Theorem 14). Therefore (a) implies (d) and the theorem is proved. Since the arbitrary intersection of closed, stable subspaces is still closed and stable, we can make the following definition.

Definition. Let A be a subset of M2. The stable subspace generated by A, denoted S(A), is the intersection of all closed, stable subspaces containing

A. As already noted on page 178, we can identify a martingale M E M 2 . another martingale N E M 2 is with its terminal value M, E L ~ Therefore (weakly) orthogonal to M if E{N,M,) = 0. There is however another, stronger notion of orthogonality for martingales in M 2 .

Definition. Two martingales N , M E M 2 are said to be strongly orthogonal if their product L = N M is a (uniformly integrable) martingale. Note that if N , M E M 2 are strongly orthogonal, then N M being a (uniformly integrable) martingale implies that [N,M] is also a local martingale by Corollary 2 of Theorem 27 of Chap. 11. It is a uniformly integrable martingale by the Kunita-Watanabe inequality (Theorem 25 of Chap. 11). Thus M, N E M 2 are strongly orthogonal if and only if [M,N] is a uniformly integrable martingale. If N and M are strongly orthogonal then = E{L,) = E{Lo) = 0, so strong orthogonality implies orE{N,M,) thogonality. The converse is not true however. For example let M E M 2 , and

180

IV General Stochastic Integration and Local Times

let Y E F o , independent of M , with P ( Y Nt = YMt, t 2 0. Then N c M 2 and

=

1) = P ( Y

-1)

=

=

4. Let

so M and N are orthogonal. However M N = Y M is ~ not a martingale (unless M = 0) because E{YM,~~F~} = YE{M,~~F,-,)# 0 = Y M ~ .

Definition. For a subset A of M~ we let A' (resp. A X )denote the set of all elements of M 2 orthogonal (resp. strongly orthogonal) to each element of A. Lemma 1. If A is any subset of M 2 , then A X is (closed and) stable.

Proof. Let M n be a sequence of elements of A X converging to M, and let N E A. Then M n N is a martingale for each n and A X will be shown to be closed if M N is also one, or equivalently that [M,N] is a martingale. However

by the Kunita-Watanabe inequalities. It follows that [Mn,Nlt converges to [M,NIt in L1, and therefore [M,N] is a martingale, and A X is closed. Also A X is stable because M E A X ,N E A implies [MT,N] = [MIN ] is~a martingale and thus M~ is strongly orthogonal to N.

Lemma 2. Let N, M be in M 2 . Then the following are equivalent. (a) M and N are strongly orthogonal. (b) S ( M ) and N are strongly orthogonal. (c) S ( M ) and S ( N ) are strongly orthogonal. (d) S ( M ) and N are weakly orthogonal. (e) S(M) and S ( N ) are weakly orthogonal.

Proof. If M and N are strongly orthogonal, let A = {N) and then M E A X . Since A X is a closed stable subspace by Lemma 1, S ( M ) c {N)X.Therefore (b) holds and hence (a) implies (b). The same argument yields that (b) implies (c). That (c) implies (e) which implies (d) is obvious. It remains to show that (d) implies (a). Suppose N is weakly orthogonal to S(M). It suffices to show that [N,M] is a martingale. By Theorem 21 of Chap. I it suffices to show E{[N, MIT) = 0 for any stopping T. However E{[N,MlT) = E{[N, M ~ ] , ) = 0, since N is orthogonal to M~ which is in S(M). Theorem 36. Let M1,. . . ,M n E M2, and suppose Mi, ~j are strongly orthogonal for i # j . Then s ( M 1 , . . . ,M n ) consists of the set of stochastic

integrals

n

~1

. M I + . . . + ~ nM~. =

CHZ.

~

i=l

i

,

3 Martingale Representation

181

where Hi i s predictable and

xy=l

Proof. Let Z denote the space of processes H~. Mi, where Hi satisfy the hypotheses of the theorem. By Theorem 35 any closed, stable subspace must contain 2. It is simple to check that 2 is stable, so we need to show only that Z is closed. Let

xy=l

Hi . Mi is an isometry from Then the mapping (HI, H ~. ., . ,H n ) -+ @ . ..@ into M 2 . Since it is a Hilbert space isometry its image Z is complete, and therefore closed.

LL~

LL,,

Theorem 37. Let A be a subset o f M 2 which i s stable. T h e n AL i s a stable subspace, and if M E AL then M i s strongly orthogonal to A. That is, AL = A x , a n d ~ ( A=) A L L = A X ' - = A X X .

Proof. We first show that A' = A X .Let M E A and N E A'-. Since N is orthogonal to S ( M ) , by Lemma 2, N and M are strongly orthogonal. Therefore A'- c A X .However clearly A X c AL, whence A X = AL, and thus AL = A X is a stable subspace by Lemma 1. By the above applied to A', we have that (A')L = It remains to show that S(A) = ALL. Since A'-'- = 2, the closure of A in M 2 , it suffices to show that 2 is stable. However it is simple to check that condition (a) of Theorem 35 is satisfied for 2, since it already is satisfied for A, and we conclude 2 is a stable subspace. Corollary 1. Let A be a stable subspace of M 2 . Then each M c M 2 has a unique decomposition M = A + B , with A E A and B E A X .

Proof. A is a closed subspace of M 2 , so each M E M 2 has a unique decomposition into M = A + B with A E A and B E AL. However A' = AX by Theorem 37. Corollary 2. Let M , N E M 2 , and let L be the projection of N onto S(M), the stable subspace generated by M . Then there exists a predictable process H such that L = H . M .

Proof. We know that such an L exists by Corollary 1. Since {M) consists of just one element we can apply Theorem 36 to obtain the result.

182

IV General Stochastic Integration and Local Times

Definition. Let A be finite set of martingales in M2. We say that A has the (predictable) representation p r o p e r t y if Z = M2, where

each Hi predictable such that

Corollary 3. Let A = {M1,.. . ,M n ) c M ~ and , suppose M i , Mi are strongly orthogonal for i # j . Suppose further that if N E M2, N IA in the strong sense implies that N = 0. Then A has the predictable representation property. Proof. By Theorem 36 we have S(A) = Z. The hypotheses imply that s ( A ) I = {0), hence S(A) = M 2 .

Stable subspaces and predictable representation can be considered from an alternative perspective. Up to this point we have assumed as given and fixed an underlying space (a,F, (Ft)t201P), and a set of martingales A in M 2 . We will see that the property that S(A) = M 2 , intimately related to predictable representation (cf., Theorem 36), is actually a property of the probability measure P, considered as one element among the collection of probability measures that make L2 (Ft)t20 martingales of all the elements of A. Our first observation is that since the filtration IF = (Ft)t>ois assumed to be P complete, it is reasonable t o consider only probability measures that are absolutely continuous with respect to P . Definition. Let A c M2. The set of M 2 martingale measures for A, denoted M2(A), is the set of all probability measures Q defined on & such that (i) Q << P, (ii) Q = P on Fo, (iii) if X E A then X E M2(Q), where M ~ ( Q )denotes all (Q, P) L2 martingalesg Lemma. The set M2 is a convex set. Note that while F satisfies the usual hypotheses under P, f i does not contain all of the Q-null sets, and thus F does not completely satisfy the usual hypotheses under Q. This does not create any problems, however.

3 Martingale Representation

Proof. Let Q and R E M 2 ( A ) ,and let S for X E M 2 ( A ) ,

= XQ

+ (1

-

183

A) R , 0 < X < 1. Then

since Q , R E M 2 ( A ) .Also if H E bF,, s < t , then

and S E M 2 ( A ) .

Definition. A measure Q E M 2 ( A ) is an extremal point of M 2 ( A ) if whenever Q = XR ( 1 - X)S with R , S E M 2 ( A ) ,R f S , 0 < X < 1 , then X=Oor 1.

+

Theorem 38. Let A

c M 2 . If S ( A ) = M 2 then P is an extremal point of

M~(A). Proof. Suppose P is not extremal. We will show that S ( A ) # M 2 . Since P is not extremal, there exist Q , R E M 2 ( A ) ,Q # R , such that P = XQ+(l-X)R, 0 < A < 1. Let dQ L, = dP '

and let Lt = E { $ I F ~ } Then . 1 = d p - XL, + (1--A)% _> XL, a . ~ .SO , that L , < a.s. Therefore L is a bounded martingale with Lo = 1 (since Q = P on F o ) , and thus L - Lo is a nonconstant martingale in M 2 ( P ) .However, if X E A and H E bF,, then X is a Q martingale and for s < t ,

and X L is a P martingale. Therefore X ( L - Lo) is a P martingale, and L - Lo E M~ and it is strongly orthogonal to A. By Theorem 37 we cannot have S ( A ) = M 2 .

Theorem 39. Let A

c M 2 . If P is an extremal point of M 2( A ) ,then every bounded P martingale strongly orthogonal to A is null.

Proof. Let L be a bounded nonconstant martingale strongly orthogonal to A. Let c be a bound for ILI, and set L, dQ = (1 - -)dP 2c

L, dR = ( 1 + -)dP.

and

2c

+

We have Q , R E M 2 ( d ) ,and P = +Q + R is a decomposition that shows that P is not extremal which is a contradiction.

184

IV General Stochastic Integration and Local Times

Theorem 40. Let A = {M1,. . . ,Mn) C M 2 , with Mi c o n t i n u o u ~and Mi, M j strongly orthogonal for i # j . Suppose P i s a n extremal point of M 2 ( A ) . Then (a) every stopping t i m e i s accessible;

(b) every bounded martingale i s continuous; ( c ) every uniformly integrable martingale i s continuous; and (d) A has the predictable representation property. Proof. (a) Suppose T is a totally inaccessible stopping time and P ( T < cc) > 0. By Theorem 22 of Chap. 111, there exists a martingale M with AMT =

l{T<m) and M continuous elsewhere. Moreover, the martingale M can be taken of finite variation with Mo = 0. Therefore, since each Mi is continuous, [M,Mi] = 0. Moreover taking (for example) Rn = infit

> 0 : lMtl > n ) ,

I M ~ " I 5 n + 1 and thus M is locally bounded. Indeed, we then have M ~ " are bounded martingales strongly orthogonal to Mi. By Theorem 39 we have M ~ = " 0 for each n. Since limn,, Rn = cc a.s., we conclude M = 0, a contradiction. (b) Let M be a bounded martingale which is not continuous, and assume Mo = 0. Let TE = inf{t > 0 : lAMtl > E ) . Then there exists E > 0 such that for T = TE, P{lAMTI > 0 ) > 0. By part (a) the stopping time T is accessible, hence without loss we may assume that T is predictable. Therefore M ~ is- a bounded martingale by the corollary to Theorem 29, whence N = MT - M ~ = - A M T l t t 2 ~ )is also a bounded martingale. However N is a finite variation bounded martingale, hence IN, Mi] = 0 each i. That is, N is a bounded martingale strongly orthogonal to A. Hence N = M~ - M ~ - 0 by Theorem 39, and we conclude that M is continuous. ( c ) Let M be a uniformly integrable martingale closed by M,. Define Then M n are bounded martingales and therefore continuous by part (b). However 1 P{sup lMr - Mtl > E ) L: - E{IMZ - M,I) t

E

by an inequality of ~ o o b ' ~and , the right side tends to 0 as n tends to oo. Therefore there exists a subsequence (nk) such that limk,, M r k = Mt a . ~ . , uniformly in t. Thus M is continuous. (d) By Corollary 3 of Theorem 37 it suffices to show that if N E AX then N = 0. Suppose N E A X .Then N is continuous by (c). Therefore N is locally bounded, and hence by stopping, N must be 0 by Theorem 39. lo

See, for example, Breiman [23, page 881.

3 Martingale Representation

185

The next theorem allows us to consider subspaces generated by countably infinite collections of martingales.

Theorem 41. Let M E M 2 , Y n E M 2 , n 2 1, and suppose Y z converges to Y, in L2, and that there exists a sequence Hn E L ( M ) such that Gn = $ HFdM,, n 2 1. Then there exists a predictable process H E L ( M ) such that = HSdMS. Proof. If Y z converges to Y, in L ~ then , Y n converges to Y in M 2 . By Theorem 36 we have that S ( M ) = Z ( M ) ,the stochastic integrals with respect to M . Moreover Y n E S ( M ) ,each n. Therefore Y is in the closure of S ( M ) ; but S ( M ) is closed, so Y E S ( M ) = Z ( M ) , and the theorem is proved. Theorem 42. Let A = { M I ,M 2 , . . . ,M n , . . . }, with M i E M 2 , and suppose there exist disjoint predictable sets Ai such that l A i d [ M i M , i ] = d [ M i ,M i ] , i > 1. Let At = C,"=, J: Ini ( s ) d [ M iMi],. , Suppose that (a) E{A,) < co; and (b) for F i c Fa such that for any X i E b F i , we have Xj = E { X i ( F t } = HjdM;, t 2. 0, for some predictable process Hi.

x,"=,

Then M = M~ exists and is in M 2 , and for any Y E b V i p,if = E{YIFt}, we have that Yt = HsdMs, for the martingale M = C z , M i and for some H E L ( M ) .

Sot

. [ N n ,Nn]t = C:=l J: 1~~( s ) d [ M iMi],, , Proof. Let N n = C:=, M ~ Then hence E { ( N $ ) ~ }= E { [ N n ,Nn],} 5 E{A,}, and N n is Cauchy in M 2 with limit equal to M . By hypothesis we have that if xi E b F i then

Therefore if i f j we have that

since Ai n Aj

= 0,

by hypothesis. However using integration by parts we have

186

IV General Stochastic Integration and Local Times

where Hid is defined in the obvious way. By iteration we have predictable Xi. The Monotone Class Theorem representation for all finite products together with Theorem 41 then yields the result.

n,,,

We conclude this section by applying these results to a very important special case, namely n-dimensional Brownian motion. This example shows how these results can be easily extended to locally square integrable local martingales.

Theorem 43. Let X = (XI,. . . ,X n ) be an n-dimensional Brownian motion and let IF = (3t)olt<m denote its completed natural filtmtion. Then every locally square integrable local martingale M for IF has a representation

where Hi is (predictable, and) in L(Xi). Proof. Fix to, 0 < t < to, and assume X is stopped at to. Then letting A = {XI,. . . , X n ) , we have that A c M2. Let M2(A) be all probability measures Q such that Q < P , and under Q all of A c M2(Q).11 Since [Xi,X j ] is the same process under Q as under P we have by Lkvy's Theorem (Theorem 40 of Chap. 11) that X = (X1,. . . ,X n ) is a Q-Brownian motion as well. Therefore, for bounded Bore1 functions f l , . . . ,f,, we have that for tl < t2 .. . < tn 5 to, fl(Xtl) fi(Xti+, -Xti) have the same expectation and we conclude that M ~ ( A ) for Q as they do for P . Thus P = Q on FtO, is the singleton {P). The probability law P is then trivially extremal for M 2 ( A ) , and therefore by Theorem 40 we know that A has the predictable representation property. >O Suppose M is locally square integrable and Mt = MtAt,. Let ( T ~ ) ~ be a sequence of stopping times increasing to cc a.s., To = 0, and such that MTk E M ~each , k 0. Then by the preceding there exist processes such that n

nyz;

>

M~~ = l1

Hip* Xi,

each k

Recall that M ~ ( Q )denotes all L2 Q martingales.

> 0.

3 Martingale Representation

By defining H i =

187

I F l ~ ~ ) ~ l ( ~ kwe- 1have , ~ that k ~ Hi , E L(x~ and ) also

Next let (te)e>o be fixed times increasing to oo with to = 0. For each te we know there exist H~~~such that for a given locally square integrable local martingale N n

By defining Hi = that

zzl

one easily checks that Hi E L ( X i ) and n

Corollary 1. As in Theorem 43, let F be the completed natural filtration of an n-dimensional Brownian motion. Then every local martingale M for F is continuous.

Proof. In the proof of Theorem 43 we saw that the underlying probability law P is extremal for A = { X 1 , .. . ,X n ) . Therefore by Theorem 40(c), every uniformly integrable martingale is continuous. The corollary follows by stopping. Corollary 2. Let X = ( X 1 , .. . ,X n ) be an n-dimensional Brownian motion and let F be its completed natural filtration. Then every local martingale M for F has a representation

where H i are predictable.

Proof. By Corollary 1 any local martingale M is continuous, hence it is locally square integrable. It remains only to apply Theorem 43. Corollary 3. Let X = ( X 1 , . . , X n ) be an n-dimensional Brownian motion and let F be its completed natural filtration. Let Z E Fa be in L 1 . Then there exist Hi predictable in L ( X i ) with J r ( ~ : ) ~
188

IV General Stochastic Integration and Local Times

Proof.Let Zt = E{Zl.Ft}, taking the c&dl&g(and hence continuous) version. By Corollary 2 we have

for some Brownian By Theorem 42 of Chap. I1 we have that Zt = motion B , 0 < t < m . Letting t tend to m shows that Z, = B l i m t + m [ ~ , ~ ] t 7 which shows that [Z, Z], < m a.s. However,

2

21 t

i m ( ~ : ) 2 d s= t-+a Iim

i=l

( ~ : ) ~ d=s t+cc lim [z, zjt = [z,

z], < m

a.s.

i=l

Finally, take t = m , observe that Z, = Z and that .Fo is a.s. trivial. Hence Zo is constant and therefore Zo = E{Z}. Corollary 4. Let X = (X1,. . . , X n ) be an n-dimensional Brownian motion and let F be its completed natural filtration. Let Z E L1(.F,) and Z > 0 a s . Then there exist Ji predictable with Jom(~:)2ds< m a.s. such that

Proof.By Corollary 3 there exist predictable Hi such that if Zt then

t

n

2, = E{Z)

= E{Zl.Ft},

+ C / H:~x:. i=l

0

Therefore

is locally bounded.) (Note that since (Zt)t20 is continuous and never 0, The proof is completed by setting Jt = $H; and taking exponentials of both sides. Corollary 5. Let F be the completed natural filtration of an n-dimensional Brownian motion. If T is a totally inaccessible stopping time, then T = m 8.S.

Proof.This is merely Theorem 40 (a). We end this section with a quite general martingale representation result, which while often true, nevertheless requires a countable number of martingales to have the representation. We then apply it to give a condition for compensators to have absolutely continuous paths. But first we state a definition from measure theory that is not widely known.

3 Martingale Representation

189

Definition. A measurable space (0,F)is said to be separable if there exists a countable family of functions on (or subsets of) 0 which generate the 0algebra F . Theorem 44. Let ( 0 , F , IF, P) be a filtered complete probability space satisfying the usual hypotheses. Assume also that the 0-algebra F = F, and that it is separable. Then there exists a countable sequence of martingales (MI, M 2 , . . .) in L2 such that they are orthogonal, E{[Mi, Mi],) < oo, and such that if N is any L2 martingale, then thereexists a sequence of predictable processes (H1, H Z , .. .) such that Nt = Ci21 H:dMi. That is, the martingales { M ~i, 2 1) have martingale representation.

s

Proof. Since F is separable, there exists a countable basis of L ~ call , it (MO,MI, M 2 , . . .), where M 0 is constant and E{Mi) = 0 for i # 0. One can take such a basis to be orthonormal in the usual way. That is, E{MiMj) = 0 for i # j, and E{(Mi)2) = 1 for all i. If one then multiplies M i by 2-i and defines the martingales M j = E{MiIFt}, we have the construction. The space of stochastic integrals with respect to all of the M i where the sum converges in L2 is easily seen to be a stable subspace. Next let N be any martingale in L2 with No = 0. Then we have

where L is orthogonal to the stable subspace generated by the countable collection of martingales (Mi)i21. This implies that L, is orthogonal to all . these random variables form a basis of the random variables M& = M ~ Since of L2, this implies that L must be constant. Since Lo = 0 we have that L is identically zero. Before we apply Theorem 44 to compensators, we need to recall and formalize two definitions, and prove Lebesgue's change of time formula. The first definition we have already seen in Exercise 7 of Chap. 111, and the second one we saw in Theorem 42 of Chap. I1 as well as Exercises 31 through 36 of that chapter.

Definition. A (right continuous) filtration F on a filtered probability space satisfying the usual hypotheses is called quasi left continuous if for every predictable stopping time T one has FT= FT-. Note that if T is a predictable stopping time, and if Tn is an announcing sequence of stopping times for T, and if X E L1, then lim,,, E{XIFTn) = E{XIFT-). If M is a uniformly integrable martingale, then ~ MT-. E{MTIFT-) = lim E{MT~FT,) = lim M T = n-cc n-+a Therefore E{AMTIFT-) = 0 a.s. and if AMT is FT- measurable, it must be 0 almost surely. So if F is quasi left continuous, no martingales can jump

190

IV General Stochastic Integration and Local Times

at predictable times. Since the accessible part of any stopping time can be covered by a countable sequence of predictable times, we conclude that i f P is quasi left continuous then martingales jump only at totally inaccessible stopping times.

Definition. Let A = (At)t>0 be an adapted, right continuous increasing process, which need not always be finite-valued. The change of time (also known as a time change) associated to A is the process rt = inf{s

> 0 : A , > t).

Some observations are in order. We have that t H rt is non-decreasing and hence rt- exists. Also, since { A t > s ) = U,>o{At > s $ E ) , we have that t H rt is right continuous. It is continuous if A has strictly increasing paths. Moreover A , 2 A,- 2 t , and A(,-)- 5 A(,)- 5 t. (Here we use the convention 70- = 0.) Finally note that (7,- 5 t ) = { s 5 A t ) , which implies that rt- is a stopping time, and since rt = lim,,o T(~+,)- we conclude that rt is also a stopping time.

Theorem 45 (Lebesgue's Change of Time Formula). Let a be a positive, finite, right continuous, increasing function on [0,m). Let c denote its right continuous inverse (change of time). Let f be a positive Bore1 function on [0,m). If G is any positive, finite, right continuous function o n [0,m ) with G ( 0 - ) = 0, then

and in particular

Here the integrals are taken over the set [O, m ) . Proof. First consider f of the form f ( s ) = l ~ o , , l ( s )for , 0 5 u < m. The left side of the equation then reduces to G ( a ( u ) ) .For the right side note that f (c(s-))l~c(,-),,l = l{c(s-)su) = l { s l a ( u )and l it follows that the right side is also equal to G ( a ( u ) ) .By subtraction and linearity we also have the result for f of the form 1(,,,1.The Monotone Class Theorem gives the result for any f positive with compact support, and approximations then gives the general f ( s ) d a ( s ) = J," f(c(s-))l{,(,-),,}ds, case. If we take G ( s ) = s we get and since c only jumps at most countably often because it is increasing, and f ( c ( ~ - ) ) l { ~ ( , - ) < , ) d s= since ds does not charge countable sets, we have f ( c ( s ) l{c(,),,lds ) and the second statement is proved, whence the theorem.

JF

5,"

3 Martingale Representation

191

Corollary. Let a be a positive, finite, continuous, strictly increasing function on [0,m ) . Let c denote its continuous inverse (change of time). Let f be a positive Bore1 function on [0,m ) . Then

Proof. It suffices to rewrite the left side of the equation as

and observe that c is also continuous and strictly increasing, which implies 1[0,C(t)](~(~)) = 1[0,t]( s ) . Our goal, stated informally in words, is if a filtration is quasi left continuous, then modulo a change of time, all compensators of adapted counting processes with totally inaccessible jumps have paths which are absolutely continuous. This is achieved in Theorem 47. We begin with two definitions.

Definition. Let (0,F , F, P) be a filtered probability space satisfying the usual hypotheses, and let S denote the space of all square integrable martingales with continuous paths a.s. It is easy to see that S is a stable subspace.12 For an arbitrary square integrable martingale M , let M Cdenote the orthogonal projection of M onto S. Then M Cis called the continuous martingale part of M . If we write M = M C ( M - M C )as its orthogonal decomposition, then we have M~ = ( M - MC)where M~ is called the purely discontinuous part of the martingale M .

+

Note that if a local martingale is locally square integrable, we can extend the definition of continuous part and purely discontinuous part trivially, by stopping. Also note that the term "purely discontinuous" is misleading: it is not a description of a path property of a martingale, but rather simply refers to the property of being orthogonal to the stable subspace of continuous martingales. See Exercises 6 and 7 in this regard.

Definition. Let (0,F , F , P ) be a filtered probability space satisfying the usual hypotheses. We call it an absolutely continuous space if for any purely discontinuous locally square integrable martingale M, d(M, M ) t << d t . Theorem 46. Let ( 0 , F,F, P ) be an absolutely continuous space. Then the compensators for all adapted counting processes with totally inaccessible jump times and without explosions, are absolutely continuous.

-

Proof. Let N be a? adapted counting process and let N be its compensa&or, so that X = N - N is a locally square integrable local martingale. Since N is continuous we have [X,XIt = Cs5,(ANs)2 = Nt, since l2

See Exercise 6.

AN,)^

= ANs = 1

192

IV General Stochastic Integration and Local Times

when N jumps at s. Therefore N = ( X , X ) , and since ( X , X ) has absolutely continuous paths by hypothesis, we are done. Remark (Yan Zeng). With a little more work, one can show the following: Let ( 0 , F,F , P) satisfy the usual hypotheses. If it is an absolutely continuous space, then all jump times of square integrable martingales are totally inaccessible. Moreover it is an absolutely continuous space if and-only if for every then dNt << dt a s . totally inaccessible stopping time T, if Nt = l{t>T),

If we take the uninteresting counting process Nt = Ci,, which jumps at constant (and hence predictable) times, then its compensator is simply Nt = Nt, and the martingale X = N - N is identically zero. To avoid these trivialities we assume that the process N is continuous. Note however that fi is not a priori absolutely continuous. This is to explain our hypotheses in the next theorem:

-

-

Theorem 47 (Absolutely Continuous Compensators). Let ( 0 , F , F, P) be a filtered probability space satzsfging the usual hypotheses. Assume that the 0-algebra F = Fa and that it is separable.13 Assume firther that F is quasi left continuous. To prevent trivialities assume also that it is not an absolutely continuous space. Then there exists a change of time rt and a new filtration (6 given by Gt = FTt , such that if N is a (6 adapted counting process with totally inaccessible jump times and without explosions, then dNt << dt.

Proof. Let (Mi)i>l be the countable sequence constructed in Theorem 44. Define the strictly increasing process At = t [M" Milt, and note that E{At) < m for 0 5 t < m. Let C denote the compensator of A. One can check that Ct = t Ci21(Mi, Mi)t. Then C is both strictly increasing and continuous, since F is quasi left continuous. Let

+ xi>,

+

~t = inf{s

> 0 : C, > t) and then Ct = inf{s > 0 : T, > t),

where T is the time change of the theorem statement. Let N be a (6 adapted counting process and let X = N - fi. Assume, by stopping if necessary, that X, E L2. We write ( X , X ) ~to denote the sharp bracket taken using the (6 filtration. Note that Lt = Net is also a counting process, and since T continuous implies that Get C Ft (see Exercise 38 of Chap. 11), we conclude L is adapted to F. We then have [L,LIt = C,,t(AL,)2 = Ltl hence = (L, L):. But also [X,XIt = Nt, and we have [X,Xlt - (x, x): = Nt - (x, x):.

xt

Hence x)Et Nct - (X, l3

= Lt

-

A separable a-algebra is defined on page 189.

x ) E t1

4 Martingale Duality and Jacod-Yor Theorem

193

and by the uniqueness of the compensator of L we have (X, x ) E t = (L, L):. On the other hand, by Theorem 44 we know that Lt - 2, = H:dMj for some predictable processes (HZ),and thus

ci2,C

for some (predictable) nonnegative processes cZ whose existence is assured because d(Mi, Mi)t << dCt a s . The process J, = C i 2 1 ( ~ : ) 2 ~ 2by, Fubini's Theorem. Next, note that since ( L - i , L-X); = (X, x ) E t , we have (X, x): = (L - L - z)!t = J; J,dC,, which by the corollary of Lebesgue's change of t time formula, implies (X, X)Y = J,,ds.

z,

So

4 Martingale Duality and the Jacod-Yor Theorem on

Mart ingale Representat ion We have already defined the space 'H2 for semimartingales; we use the same definition for martingales, which in fact historically came before the definition for semimartingales. We make the definition for all p, p 2 1.

Definition. Let M be a local martingale with cAdlAg paths. We define the 'HP norm of M to be llMllp = E{[M, M]g2})'/p. If lIMllxp < m , we say that M is in 'Hp. Finally we call 'Hp the space of all c&dl&gmartingales with finite 'Hp norm. I t is simple to verify that 11 . ItRp is a norm if we identify martingales that are almost surely equal. Indeed, the only property that is not obvious is the subadditivity. For the case p = 1 for example, we have that [ M N, M N ] ' / ~ 5 [M, Ml1I2 [N,Nl1I2 since by the Kunita-Watanabe [M, Ml1l2[N,Nl1l2, which gives the subadditivity. It is inequality [M,N ] also simple to verify that 'HP is a linear space. We state without proof the important martingale inequalities known as the Burkholder-Davis-Gundy inequalities, and often referred to by the acronym BDG. (For a proof one can consult many different sources, including [I461 and [44].)

+

+

<

+

Theorem 48 (Burkholder-Davis-Gundy Inequalities). Let M be a martingale with cadlag paths and let p 2 1 be fixed. Let M,* = sup,,,- IM,I. Then there exist constants cp and Cp such that for any such M

<

for all t, 0 t 5 m . The constants are universal: they do not depend on the choice of M.

IV General Stochastic Integration and Local Times

194

We have the following simple results concerning 'H1.

Theorem 49. 'H2 c 'H1 and local martingales of integrable variation are a subset of 'H1. Proof. First note that if M is a cAdlAg martingale, then E{ d m ) 5 (E{[M, M ] , ) ) ~ / ~ which gives the first statement. For the second, if M has whence integrable variation then [M,M], = AM,)^ 5 5 IdMsl, and taking expectations, we have the result.

(z,

SF

Theorem 50. 'H2 is dense in 'H1 and bounded martingales are dense in 'H1. Proof. Let M be a local martingale in 'H1. By the Fundamental Theorem of U where Local Martingales (Theorem 25 of Chap. 111) we have M = N N is a local martingale with jumps bounded by 1 and U has paths of locally integrable variation. If (Tn),>1 is a sequence of stopping times increasing to oo a.s., then IIMTnllR1 5 I I M I I ; ~and , also MTn converges to M in 'H1. Choose Tn to be the first time [U,UIt has total variation larger than n, and also [[N,N ] I 5~n ~ 1; that is, T, = inf{t 0 : Sot[u,UIt > n, or [N,NIt > n - 1) with Tn increasing to oo.Then [N, NIT" n and thus NTn is in 'H1 for each n. Therefore we turn out attention to UTn, and since [U, uITn5 n lAUTn1, we need consider only the one jump of U at time Tn : ACTTn. Let E denote ACTTn. is bounded we can compensate Ekl{t>Tn), Letting Ek = (l{IEllk), since and call the resulting martingale Vc. The compensator of (klit>Tn) is 'H2 because & is bounded. We have then that V$ is in 'H2 and converges to Vn as k --+ oo in total variation norm, and hence also in 'H1. This proves that 'H2 is dense in 'H1, and since bounded martingales are dense in 'H2, they too are dense in 'H1.

+

>

<

+

ek

Theorem 51. Let M be a local martingale. Then M is locally in 'H1. Proof. By the Fundamental Theorem of Local Martingales we know that M = N U, where N has jumps bounded by a constant ,B and U is locally of integrable variation. By stopping, we thus assume that N is bounded and U has paths of integrable variation. The result then follows by the previous theorem (Theorem 49).

+

One can further show, by identifying 'Hp with LP through the terminal value of the martingale, that 'Hp is a Banach space (in particular it is complete) for each p 1. One can further show that the dual space of continuous linear = 1, 1 < p < m. This continues nicely functional on 'Hp is 319, where the analogy with LP except for the case p = 1. It can be shown (see [47]) that the dual of 'H1 is not 'Ha. It turns out that a better analogy than LP is that of Hardy spaces in complex analysis, where C. Fefferman showed that the dual of 'H1 can be identified with the space of functions of bounded mean oscillation, known by its acronym as BMO. With this in mind we define the space of B M O martingales.

>

+

4 Martingale Duality and Jacod-Yor Theorem

195

Definition. Let M be a local martingale. M is said t o be in BMO if M is in R2 and if there exists a constant c such that for any stopping time T we have E{(M, - M T - ) ~ ~ F 5 T )c2 as., where Mop = 0 by convention. The smallest such c is defined to be the B M O norm of M , and it is written IIMIIBM~. If the constant c does not exist, or if M is not in 'H2, then we set 11 M ~ ~ B M =O m.

<

Note that in the above definition E{M,$l.Fo) c2 (with the convention that Mo- = 0) and therefore llMllRz I I M I I B ~ Note ~ . in particular that [ ( M ( ( B M=O0 implies that M = 0. Let T be a stopping time and A E FT. Replacing T with TA shows that the above definition is equivalent t o the statement

<

.

for every stopping time T. This in turn gives us an equivalent description of the B M O norm:

where the supremum is taken over all stopping times T. Note that this second characterization gives that (1 . llBIMo is a semi-norm, since it is the supremum of quadratic semi-norms. An elementary property of B M O martingales is that M E B M O if and only if all of the jumps of M are uniformly bounded. Thus trivially, continuous L2 martingales and bounded martingales are in B M O . The next inequality is quite powerful.

Theorem 52 (Fefferman's Inequality). Let M and N be two local martingales. Then there exists a constant c such that

Fefferman's inequality is a special case of the following more general result.

Theorem 53 (Strengthened Fefferman Inequality). There exists a constant c such that for all local martingales M and N , and U an optional process,

Proof. Let Ct = J: u : ~ [ M ,MI, and define H and K by

196

IV General Stochastic Integration and Local Times

Using integration by parts yields

From the definitions of H and K we have

The Kunita-Watanabe inequality implies

and since Id[M, NISI is absolutely continuous with respect to d[M,MI, as a consequence of the Kunita-Watanabe inequality, we have

But

and

But E{E{[N, N],I.F,) we have that

and the result follows.

-

[N, N],-} is bounded by IINII$Mo on (0, m), hence

4 Martingale Duality and Jacod-Yor Theorem

Remark. The constant c in Theorems 52 and 53 can be taken to be can be seen from an analysis of the preceding proof.

197

a,as

Theorem 54. Let N E ?I2. Then N is in B M O if and only if there is a constant c > 0 such that for all M E ?I2,

Moreover IINIIBMo

< &c.

Proof. If N is in B M O , then we can take c = 11 NIIBMO from Fefferman's inequality (Theorem 52) and the remark following the proof of Theorem 53. cllMJIX1;we want Now suppose that M is in ?I2and that IE{[M, N],}I to show that N is in B M O . We do this by first showing that lNol 5 c a.s., and then showing that N has bounded jumps. Let A = {INo/ > c}. Suppose P(A) > 0. Let = *In. Then E{/
<

< <

<

This of course is a contradiction and we conclude INo[ c a.s. We next show IAN1 5 2. Since every stopping time T can be decomposed into its accessible and totally inaccessible parts, and since each accessible time can be covered by a countable collection of predictable times with disjoint graphs, we can assume without loss of generality that T is either totally inaccessible or predictable. We further assume P(T > 0) = 1. Suppose then that P(lANTI > 2c) > 0, and set

Let M be the martingale consisting of < l t t > T ) minus its compensator. Then M is in ?I2 and has at most one jump, which occurs a t T. The jump is given by - E{<(.FT-}, T is predictable, AMT = T is totally inaccessible.

<

Note that we also have IIMIIX1 is less than the expected total variation of M , which in turn is less than 2E{l
= E{AMTANT} = E{:ANT}

1

= E{ lAN~11{lA~~1>2c) P(lANTl > 2c)

which is a contradiction. On the other hand, if T is predictable, then we know that E{ANTI&-) = 0 and thus we are reduced t o the same calculation and

198

IV General Stochastic Integration and Local Times

the same contradiction. We conclude that P(lANTI > 2c) = 0, and thus N has jumps bounded by 2c. Last let T be any stopping time. Let M = N - N~ and 7 = [N, N], [N,NIT. Then M is in 7-12 and [M,MI, = [M,N], = 7.By our hypotheses it now follows that

which in turn implies

This then implies E{[N,N],IFT)

-

[ N , N ] T 5 c2 a.s.7

and since

almost surely, we obtain

E{N& - N;-)

5 6c2

which yields the result. During the proof of Theorem 54 we proved, inter alia, that the jumps of a local martingale in B M O are bounded. We formalize this result as a corollary, which in turn has its own corollary.

Corollary 1. Let N be a local martingale in B M O . Then N has bounded jumps. Corollary 2. Let N be a local martingale in B M O . Then N is locally bounded. Proof. Let c be a bound for the jumps of N which we know exists by the previous corollary. Next let T, = inf{t > 0 : 1 Nt 1 2 n). Then I 5 n c, and N is locally bounded.

IN^^

+

The key result concerning 7-11 and B M O is the Duality Theorem which is Theorem 55 that follows. First let us lay the foundation. For N chosen and fixed in B M O we define the operator LN from 7-11 to R by

for all M in 7-11. Then one can easily check to see that LN is linear, and Fefferman's inequality proves that it is bounded as well, and therefore continuous. If B M O is the Banach space dual of 7-11 then it is also complete, a fact that is apparently not easy to verify directly.

4 Martingale Duality and Jacod-Yor Theorem

199

Theorem 55 (The Dual of ?-ll is BMO). T h e Banach space dual of all (bounded) linear functionals o n ?-ll can be identified with B M O . Moreover if LN i s such a functional then the norms ( ( L N (and ( I(N(IBMO are equivalent.

Proof. Let N be in B M O . By Fefferman's inequality we have

for all M in N 1 . This shows that LN is in the dual of 3-1' and also that llLNll 5 c IINIIBMo. Note further that LN cannot be trivial since L N ( N )= E { [ N ,N],} > 0 unless N is identically 0. Therefore the mapping r ( N ) = LN is an injective linear mapping from B M O into N1*,the dual of 3-1'. Let L be an arbitrary linear functional in the dual of ?-ll.We have

This means that L is also a bounded linear functional on 3.t2. Since 3.t2 is isomorphic as a Hilbert space t o the L2 space of the terminal random variables of the martingales in N 2 ,we have that there must exist a unique martingale N in ?-12 such that for any M in 3-1' we have:

Clearly LN = L on ?-12, and since ?-12 is dense in 3.t1 by Theorem 50, we have that L and LN are the same functional on 3-1'. This shows that B M O equipped with the norm llLNll is isomorphic to N1* and thus, being the dual of a Banach space, it is itself a Banach space and in particular it is complete. Combining equation (*) with Theorem 54 we have that 11LN 11 and 11 N[IBMO are equivalent norms. This completes the proof. While Fefferman's inequality, the space of B M O martingales, and the duality of 3.tm and B M O are all of interest in their own right, we were motivated to present the material in order to prove the important Jacod-Yor Theorem on martingale representation, which we now present, after we recall the version of the Hahn-Banach Theorem we will use.

Theorem 56 (Hahn-Banach Theorem). Let X be a Banach space and let Y be a closed linear subspace. T h e n Y = X if and only i f the only bounded linear functional L which has the property that L ( Y ) = 0 i s the functional which i s identically zero. Theorem 57 (Jacod-Yor Theorem on Martingale Representation). Let A be a subset of ?-12 containing constant martingales. T h e n S(A), the stable subspace of stochastic integrals generated by A, equals ?-12 if and only if the probability measure P i s a n edremal point of M 2 ( A ) ,the space of probability measures making all elements of A square integrable martingales.

200

IV General Stochastic Integration and Local Times

Proof. The necessity has already been proved in Theorem 38. By the HahnBanach Theorem, 3-1l = S ( A ) if and only if L(S(A)) = 0 implies L is identically zero, where L is a bounded linear functional. Let L be a bounded linear functional which is such that L(S(A)) = 0. Then there exists a martingale N in BMO such that L = LN. The local martingale N is locally bounded, so by stopping we can assume it is bounded and that No = 0. (See the two corollaries of Theorem 54.) Let us also assume it is not identically zero, and let c be a bound for N . We can then define two new probability measures Q and R bv

+

Then Q and R are both in M 2 ( A ) , and P = +Q R shows that P not extremal in M 2 ( A ) , a contradiction. Therefore we must have that L identically zero and we have that 3-1l = S ( A ) . As far as N2 is concerned, is a subspace of N1, hence 7-12 c S(A). But by construction of S ( A ) , it contained in 3.t2, and we have martingale representation.

is is it is

5 Examples of Martingale Representation In Sect. 3 we have already seen the most important example of martingale representation, that of Brownian motion. In this section we give a method to generate a family of examples which are local martingales with jumps, and which have the martingale representation property. The limitations are that the family of examples is one dimensional (so that we exclude vectorvalued local martingales such as n-dimensional Brownian motion), and that the descriptions of the jumps are all of the same rather simple kind. The idea is t o construct a class of local martingales 3-1 such that for X E 3.t we have both that X o = 0 and the compensator of [X, XIt is At = t. If X has the martingale representation property, then there must exist a predictable process H such that

j0 HsdXs. rt

[ X ,XIt - t =

The above equation is called Emery's s t r u c t u r e e q u a t i o n , and it is written (in a formal sense) in differential notation as

In order to establish that solutions to Emery's structure equation actually exists we write it in a form resembling a differential equation:

Equation (**) is unusual and different from the stochastic differential equations considered later in Chap. V, since while the unknown is of course the

5 Examples of Martingale Representation

201

local martingale X (and part of the structure equation is t o require that any solution X be a local martingale), no a priori stochastic process is given in the equation. That is, it is lacking the presence of a given stochastic driving term such as, for example, a Brownian motion, a compensated Poisson process, or more generally a LBvy process. Since therefore no probability space is specified, the only reasonable interpretation of equation (**) is that of a weak solution. That is, we want to show there exists a filtered probability space (R, 3,IF, P ) satisfying the usual hypotheses, and a local martingale X , such that X verifies equation (**). It would also be nice to have weak uniqueness which means that if X and Y are solutions of (**) for a given if, possibly defined on different filtered probability spaces, then X and Y have the same distribution as processes. That means that for every A, a Bore1 set on the function space of c&dl&gfunctions mapping R+ to R , we have P(w : t Xt(w) E A) = Q(w : t H Xt(w) E A), where P and Q are the probability measures where X and Y are respectively defined. Inspired by knowledge of stochastic differential equations, it is natural to conjecture that such weak solutions exist and are unique if the coefficient if is Lipschitz c o n t i n u ~ u s . ' This ~ is true for existence and was proven by P. A. Meyer [179]; see alternatively [136]. Since the proof uses weak convergence techniques which are not within the scope of this book, we omit it.

Theorem 58 (Existence of Solutions of the Structure Equation). Let if : R -+R be Lipschitz continuous. Then Emery's structure equation

has a weak solution with both (Xt)t20 and gales.

(Sot if(Xs-)dXs)t>o local martin-

The issue of uniqueness is intriguing. Emery has shown that one has uniqueness when if is linear, but uniqueness for others if's, including the Lipschitz case, is open. The next theorem collects some elementary properties of a solution X .

Theorem 59. Let X be a (weak) solution of (* * *). Then the following hold. (i) E{X;) = E{[X,XIt) = t , and X is a square integrable martingale on compact time sets. (ii) All jumps of X are of the form AXt = if(Xt- ). (iii) X has continuous paths if and only if if is identically 0, in which case X is standard Brownian motion. (iv) If a stopping time T is a jump time of X , then it is totally inaccessible. Proof. We prove the statements in the order given. Since a solution X and the integral term are both required t o be local martingales, we know l4

Lipschitz continuity is defined and discussed in Chap. V

202

IV General Stochastic Integration and Local Times

there exists a sequence (Tn)n>l - of stopping times increasing t o GO such that $(X,-)dX, is in L1. Therefore E{[X, X]tAT,} = E{t A Tn), and applying the Monotone Convergence Theorem to each side of the equality in this equation yields E{[X, XIt) = t which further implies that X is a martingale on [O,t] for each t < GO and that E{x:} = E{[X,X]t} = t. For the second statement, recall that

CAT"

and hence we have A[X, XIt = AX^)^ = $(Xt-)AXt, and dividing both sides by AXt (when it is not zero) gives the result. For the third statement, suppose q$ is identically zero. Then by the second statement X has no jumps and must be continuous. Since X is then a continuous local martingale with [X,XIt = t, it is Brownian motion by LQvy's Theorem. For the converse, if we know that X is continuous, if it is non-trivial it has paths of infinite variation since it is a local martingale. Thus so too does the term $(X,-)dX,. But notice that this term is the right side of equation (* * *) which is of finite variation, and we have a contradiction, so we must have that is zero. For the fourth statement, we implicitly stop X so that it is a uniformly integrable martingale. Next let T be a jump time of X . Then T = TAA TB where A is the accessible part of T and B is the totally inaccessible part of T . Since TAcan be covered by a countable sequence of predictable times with disjoint graphs, we can assume TA is predictable without loss of generality. Thus it suffices to show P ( A ) = 0. However since TA is predictable, and X is a uniformly integrable martingale, we have E{AXTA(.FTA-} = 0. But since AXT, = $ ( X T ~ - ) by part (ii) of this theorem, and $ ( X T ~ - )E FTATA is predictable, which implies that the jump of X at TA is zero, which in turn implies that P ( A ) = 0.

Sot

Let us now consider a special class of structure equations where q$ is assumed to be an afine function. That is, we assume q$ is of the form $(x) = a px. We analyze these special cases when a and P vary. Emery has named the solutions corresponding to affine structure equations the AzQma martingales, since J. AzQma's work on Markov processes and expansion of filtrations led him t o the amazing formula of "the" AzQma martingale given later in Sect. 7. Equation (**) now becomes

+

and when

p = 0, it reduces t o

and when in addition a = 0 we have seen that X is standard Brownian motion. Note that LQvy's Thebrem gives us weak uniqueness in this case, since

5 Examples of Martingale Representation

203

any solution with a = /? = 0 must have the same distribution, namely that of Wiener measure. We have much more, as we see in the next theorem. However it has a long and difficult proof. Rather than present it, we refer the interested reader t o the excellent treatment of M. Emery, in his original paper [69] proving the result.

Theorem 60 (Emery's Uniqueness Theorem). Let X be a local martingale solution of the structure equation

T h e n X i s unique in law. That is, any other solution Y m u s t have the same distribution as does X . Moreover X is a strong Markov process. The uniqueness is especially significant in light of the next theorem. By martingale representation we mean that every square integrable martingale can be represented as a stochastic integral with respect to one fundamental local martingale.

Theorem 61. Consider the equation

o n a filtered probability space ( R ,3 , IF, P ) which satisfies the usual hypotheses. T h e n X has martingale representation for its completed natural filtration if and only i f the law P i s a n extreme point o f t h e convex set of all probabilities o n (a,3,F ) for which X i s a martingale and verifies the equation. Moreover if the equation with filced initial condition Xo has weak uniqueness of solutions, then every solution X of the equation has martingale representation with respect t o the smallest filtration satisfying the usual hypotheses and to which X i s adapted. Proof. By the Jacod-Yor Theorem (Theorem 57) we need t o verify that P is extremal in the set M2 of all probability measures such that X is a square integrable martingale. It is clearly true if P is extremal. Suppose then that P is not extremal, and let Q and R both be in M2,such that P = XQ+ (1-A) R, with 0 < X < 1. Both Q and R are absolutely continuous with respect to P , so under Q and R the terms [X,XI and $(Xs-)dX, are the same a.s. (resp. dQ and d R ) . Therefore X satisfies equation ( 8 ) for both Q and R as well as P . Thus if P is not extremal in M2 then it is also not extremal in the set of probability measures such that X satisfies equation ( 8 ) . To prove the second statement, let denote the canonical path space of cAdlAg paths, with X being the projection process given by Xt(w) = w(t). We have just seen that among the solutions of equation (@I),the ones having martingale representation are those whose law constitutes an extremal probability measure. But if weak uniqueness holds, the collection of all such probabilities consists of only one, and thus extremality is trivial.

204

IV General Stochastic Integration and Local Times

We now examine several special cases as a and /? vary. Let a = /? = 0, and we have seen that X is standard Brownian motion. Because of Theorem 61 we conclude that (one dimensional) Brownian motion has martingale representation, recovering a special case of Theorem 43. Next suppose a = 1 and /? = 0. In this case the equation of Theorem 60 with X o = 0 becomes, in integral form, [X,X], = t

+ (Xt - Xo) = t + Xt.

Therefore Xt = [X, X]t - t and hence X is a finite variation martingale. Moreover AXt = 1, so X only jumps up, with jumps always of size 1. Now let N be a standard Poisson process with arrival intensity X = 1, and let Xt = Nt-t, the compensated Poisson process. Then X satisfies the equation, and by weak uniqueness all such X are compensated Poisson processes with X = 1. We conclude that a compensated standard Poisson process has martingale representation with respect to its natural (completed) filtration. For general a: (and not just a = I ) , but still with p = 0, it is simple to check that

is the unique solution of the equation of Theorem 60 if N is a standard Poisson process. Note that (as is well known) X a converges (weakly) to Brownian motion as a: -+ 0. We now consider the more interesting cases where P # 0. We repeat the equation of Theorem 60 here (in integrated form) for ease of reference:

Observe that X is a solution of the above equation if and only if X solution of d[X,X]t = dt PXt-dXt

+

+ $ is a (88)

+

with initial condition Xo = xo $. Therefore without loss of generality we can assume that a = 0 and we do this from now on. We have two explicit examples for equation (88).When /? = -1, we take

where M is AzBma's martingale15, B is standard Brownian motion, and gt = sup{s 5 t : B, = 0). By Theorem 86 of Sect. 8 of this chapter, we know t that [X,X],C = 0 and [X,XIt = 29,. Integration by parts gives X,-dX, = (t - gt) - gt = t - [X,X]t, since [X, XIt = C s I t ( A X s ) 2 , because of course [X,XI: = 0. This proves the assertion that for /? = -1, Xt = A M t , where J;;

So

l5

AzCma's martingale is treated in detail in Sect. 8 later in this chapter.

6 Stochastic Integration Depending on a Parameter

205

M is AzQma's (original) martingale as presented in Sect. 8, and thus we have martingale representation for Azkma's martingale. Our second (and last) explicit example is for /? = -2. In this case our equation becomes ft

[X,X ] t - t = 2 1 0 Xs-dXs

Sot

and using integration by parts we obtain 2 Xs-dXs = X: - [X,XIt. Equating terms gives X: = t, and since E{Xt) = 0 for all t because xo = 0 we deduce that 1 P ( X t = J t ) = P ( X t = -Jt) =

5

for all t > 0. The jumps of X occur from a change of sign, and they arrive according to a Poisson process with intensity k d t . Such a process can be seen to be a martingale (as in [204]) by constructing a process X with a distribution as above and with filtration IF. Then for 0 < s < t ,

where N is independent of the a-algebra FSand has a Poisson distribution with parameter X = ln($). In this way it is easily seen t o be a martingale. We call it the parabolic martingale. Once again we are able t o conclude: we have martingale representation for the parabolic martingale.

6 Stochastic Integration Depending on a Parameter The results of this section are of a technical nature, but they are needed for our subsequent investigation of semimartingale local times. Nevertheless they have intrinsic interest. For example, Theorems 64 and 65 are types of Fubini Theorems for stochastic integration. A more comprehensive treatment of stochastic integration depending on a parameter can be found in Stricker-Yor [219] and Jacod [103]. Throughout this section (A, A) denotes a measurable space.

Theorem 62. Let Yn(a, t, w) be a sequence of processes that are (i) A €3 B(R+) 8 F measurable, and (ii) for each fixed a the process Yn(a, t,w) is cadlag. Suppose Yn(a, t , .) converges in ucp for each a E A. Then there exists an A 18 B(R+) 18 3 measurable process Y = Y(a, t, w) such that (a) Y (a, t, .) = limn,, Yn(a, t , .) with convergence in ucp; (b) for each a E A, Y is a.s. cadlag. Moreover there exists a subsequence nk(a) depending measurably on a such that limn,(,),,

&nk(a)

= & uniformly in

t on compacts, a.s.

206

IV General Stochastic Integration and Local Times

IY2(a,t,.) - y j ( a , t , .)I. Since yi is c&dl&gin t Proof. Let St,i,j = the function (a,w) H is A 8 3 measurable. By hypothesis we have SE,i,j= 0 in probability. Let no(a) = 1, and define inductively limi,j,,

sE~,~

nk(a) = inf{m

> max(k, n k - ~ ( a ) ):

sup P(SE,i,j> 2

~)~ 22-k}. )

i,j>m

We then define Z k ( a ,t, w) = ~ ~ " ( ~t,)w). ( a , Since each a

H

nk(a) is measurable, so also is Z k . Define

T:,i,j = sup IZi(a,t,w) - Z j ( a , t , w ) J ; tsu

then also (a,w) H T:,i,j(~) is jointly measurable, since Zi have c&dl&gpaths (in t). Moreover by our construction P(Ti,k,k+m> 2-k) 2-k for any m 2 1. The Borel-Cantelli Lemma then implies that limi,j,, Tt,i,j= 0 almost surely, which in turn implies that

<

lim Zi(a, t, .)

exists a s . ,

2-00

with convergence uniform in t. Let Aa be the set where Zi converges uniformly (note that Aa E A 8 3 and P(Aa) = 1, each fixed a ) , and define

Then Y is c&dl>hanks to the uniform convergence, and it is jointly measurable.

Theorem 63. Let X be a semimartingale with Xo = 0 a.s. and let H ( a , t, w) = H,"(w) be A 8 P measurable16 and bounded. Then there is a function Z(a, t , w) in A 8 B(R+) 8 3 such that for each a E A, Z(a, t, w) is a ccidldg, adapted version of the stochastic integral Jot HFdXs. Proof. Let 3-1 = {H E bA@p such that the conclusion of the theorem holds}. E bP and f = f ( a ) E bA, and if H ( a , t , w ) = f ( a ) K ( t , w ) , then

If K = K ( t , w )

Jd'

H ( a , s, .)dXs =

Jd

t

f (a)K(s,

= f (a)

Jd

t

K ( s 7-Id&,

and thus clearly H = f K is in 3-1. Also note that 3-1 is trivially a vector space, and that H of the form H = f K generate b d 8 P. Next let Hn E 3-1 and suppose that Hn converges boundedly to a process H E bA 8 P. By Theorem 32 (for example) we have that Hn . X converges uniformly in t in probability on compacts, for each a. Therefore H E 3-1, and an application of the Monotone Class Theorem yields the result. '"ecall

that P denotes the predictable a-algebra.

6 Stochastic Integration Depending on a Parameter

207

Corollary. Let X be a semimartingale (Xo = 0 a x ) , and let H ( a , t , w) = H,"(w) E A @ P be such that for each a the process Ha E L(X). Then there exists a function Z(a,t,w) = Z t E A @ B(R+) @ 3 such that for each a , Z t H:dX,. is an a s . cidl&gversion of

Sot

> ~ converge Proof. By Theorem 32 the bounded processes z ~= HalIlHa15k).X to H a . X in ucp, each a . But can be chosen cbdlbg and jointly measurable by Theorem 63. The result now follows by Theorem 43. ~~2~

Theorem 64 (Fubini's Theorem). Let X be a semimartingale, Ht = H ( a , t, w) be a bounded A@P measurable function, and let p be a finite measure on A. Let Zf = H:dX, be A @ B(R+) 8 3 measurable such that for = JA Ztp(da) is a ccidlcig each a, Za is a ccidlcig version of Ha . X . Then version of H - X , where Ht = JA H,"p(da). Proof. By pre-stopping we may assume without loss of generality that X E N2, and because the result holds for the finite variation part of the canonical decomposition of X by the ordinary Stieltjes Fubini Theorem, we may further assume that X is a martingale with E{[X, XI,) < co. Next suppose Hf is of the form H ( a , t, w) = K ( t , w) f (a) where K E bP and f is bounded, measurable. Then K E L(X) and J 1 f (a)lp(da) < co. In this case we have Z," = f (a)K . X , and moreover

By linearity the same result holds for the vector space V generated by processes of the form K ( t , w)f (a) with K E bP and f bounded, measurable. By the Monotone Class Theorem it now suffices to show that if Hn E V and limn,, Hn = H , then the result holds for H . Let Z;,, = HE . X , the cbdlbg version. Then by Jensen's and the Cauchy-Schwarz inequalities,

by Doob's quadratic inequality for the martingales ZE and Za, and by Corollary 3 of Theorem 27 of Chap. 11. Continuing, the preceding equals

208

IV General Stochastic Integration and Local Times

and the above tends t o 0 by three applications of the Dominated Convergence Theorem. We conclude from the preceding that

L and therefore

sup IZ;,, - Z;lp(da)

< cc a.s.

t

IZ;lp(da) < cc for all t , a s . Moreover

which tends t o 0. Therefore taking Hn,t = J H;,,p(da) we have Hn . Xt = JA Z:,,p(da) converges in ucp to J ZFp(da). Since Hn . X converges t o H . X by Theorem 32, we conclude H . X = ! Ztp(da). The version of Fubini's Theorem given in Theorem 64 suffices for the applications of it used in this book. Nevertheless it is interesting to determine under what more general conditions a Fubini-type theorem holds.

Theorem 65 (Fubini's Theorem: Second Version). Let X be a semimartingale, let H t = H ( a , t , w ) be A @ P measurable, let p be a finite positive measure on A, and assume

H f d X , be A @ B(B+)@ 3 measurable and Z a cddldg for Letting Z; = = JA Z;p(da) exists and is a cadlag version of H . X , where each a , then Ht =

HF~(da)'

Proof. By pre-stopping we may assume without loss of generality that X E N 2 and that llHa I ( L 2 ( d p ) is ( N 2 ,X ) integrable. Let X = + 7f be the canonical decomposition of X . Then

x

Next observe that

6 Stochastic Integration Depending on a Parameter

209

E{JF

IH:lldxsl) < co and E { J F ( H , " ) % [ ~ , ~ ] ~<} co for and therefore p-almost all a E A. Whence Ha E L ( X ) for p almost all a E A. , the proof of Theorem 64 works as well Next define Hn = H 1 { I H I < n )and here. The hypotheses of Theorem 65 are slightly unnatural, since they are not invariant under the transformation

where cp is any positive function such that Jcp(a)p(da) < co. This can be alleviated by replacing the assumption (JA (Ha)'p(da)) 1/2 t L ( X ) with (J s e ( d a ) ) 1 / 2 t L ( X ) for some positive cp t ~ ' ( d p ) One . can also relax the assumption on p t o be a-finite rather than finite.

Example. The hypotheses of Theorem 65 are a bit strange, however they are in some sense best possible. We give here an example of a parameterized process (Ha)aEA,a positive, finite measure p on A, and a semimartingale X such that (i) (a, t) -+ Ht is A @ P measurable, (ii) Ha E L(X), each a E A, and (iii) J A IHtal~(da)E L ( X ) but such that if Z,O = HfdX, then JA Z;p(da) does not exist a s a Lebesgue integral. Thus a straightforward extension of the classical Fubini Theorem for Lebesgue integration does not hold. Indeed, let A = N = {1,2,3,. . . ) and let p be any finite positive measure on A such that p({a)) > 0 for all a E A. Let X be standard Brownian motion, let to < t l < tz < . . . be an increasing sequence in [ O , l ] , and define

Then 0°1

Ht =

C a= a

-@a - t a - ~ ) - 1 / 2 ~ ~ t a - l < t 5 t a l

1

is in L2(dt), whence Ht = JA IHf lp(da) E L ( X ) , and moreover if t

where the sum converges in L2. However if t

2 1 then

2 1,

210

IV General Stochastic Integration and Local Times

and

because (t, - t a - 1 ) - 1 / 2 ( ~- X t at , - l ) is an i.i.d. sequence and C z l a-' = m. Note that this example can be modified t o show that we also cannot replace the assumption that

( l ( ~ : ) ~ a ( d a )E) L~ ('X~) with the weaker assumption that ( J A ( ~ ~ ) ~ p ( d aE) )Ll(/Xp) for some p

< 2.

7 Local Times In Chap. I1 we established It6's formula (Theorem 32 of Chap. 11) which showed that if f : R -, R is C2 and X is a semimartingale, then f ( X ) is again a semimartingale. That is, semimartingales are preserved under C2 transformations. This property extends slightly: semimartingales are preserved under convex transformations, as Theorem 66 below shows. (Indeed, this is the best one can do in general. If B = (Bt)t>0 is standard Brownian motion aqd Yt = f ( B t ) is a semimartingale, then f must be the difference of convex functions. (See Cinlar-Jacod-Protter-Sharpe 1341.) We establish a related result in Theorem 71, later in this section.) Local times for semimartingales appear in the extension of Itb7s formula from C2 functions to convex functions (Theorem 70).

Theorem 66. Let f : R -+ R be convex and let X be a semimartingale. Then f ( X ) is a semimartingale and one has

where f' is the left derivative of f and A is an adapted, right continuous, increasing process. Moreover AAt = f ( X t ) - f ( X t - ) - f 1 ( X t - ) A X t . Proof. First suppose 1x1 is bounded by n , and in 7 i 2 , and that Xo = 0. Let g be a positive C* function with compact support in ( - m , 0 ] such that 00

S-w g(s)ds = 1. Let fn(t) = n *J-

+

f ( t s)g(ns)ds. Then fn is convex and C2 and moreover f; increases t o f' as n tends to m. By Itb's formula

7 Local Times

21 1

where

The convexity of f implies that An is an increasing process. Letting n tend to co, we obtain

where limn,, A; = At in L2, and where the convergence of the stochastic integral terms is in I f 2 on [0,t]. We now compare the jumps on both sides of the equation (*). Since fl(X,-)dX, = 0 we have that A. = 0. When t > 0, the jump of the left side of (*) is f (Xt) - f (Xt-), while the jump of the right side equals f '(Xt-)AXt AAt. Therefore AAt = f (Xt) - f (Xt-) - f '(Xt-)AX,, and the theorem is established for 1x1bounded by n and in N2. Now let X be an arbitrary semimartingale with Xo = 0. By Theorem 13 we know there exists a sequence of stopping times (Tn)n>l, increasing to co a.s. E N2 for each n. An examination of the proof of Theorem 13 such that xTnshows that there is no loss of generality in further assuming that 1xTn-I 5 n, also. Then let Yn = Xl[o,Tn)and we have

+

which is equivalent to saying

~ - we , can define on [O, Tn). One easily checks that (A"+ l)Tn- = ( A ~ ) ~and A = An on [0, Tn), each n. The above extends without difficulty to functions g : R2 -+ R of the form g(Xt, H) where H is an 3 0 measurable random variable and x c-, g(x, y) is convex for every y. For general X we take xt = Xt - Xo, and then f (Xt) = f (&+ Xo) = g(Xt, Xo), where g(x, y) = f (x + y). This completes the proof.

Notation. For x a real variable let x f , X- be the functions xf max(x, 0) and x- = -min(x,O). For x, y real variables, let x V y = max(x, y) and x A y = min(x, y). Corollary 1. Let X be a semimartingale. Then IX 1, X f , X - are all semimartingales.

212

IV General Stochastic Integration and Local Times

Proof. The functions f (a) = 1x1, g (a) = xf , and h(x) the result then follows by Theorem 66.

= x-

are all convex, so

Corollary 2. Let X , Y be semimartingales. Then X V Y and X A Y are semimartingales.

++

Proof. Since semimartingales form a vector space and x V y = (la - y 1 x y) and x A y = +(x y - lx - yl), the result is an immediate consequence of Corollary 1.

+

We can summarize the surprisingly broad stability properties of semimartingales.

Theorem 67. The space of semimartingales is a vector space, an algebra, a lattice, and is stable under C2, and more generally under convex transformations. Proof. In Chap. I1 we saw that semimartingales form a vector space (Theorem I), an algebra (Corollary 2 of Theorem 22: Integration by Parts), and that they are stable under C2 transformations (Theorem 32: ItG's Formula). That they form a lattice is by Corollary 2 above, and that they are stable under convex transformations is Theorem 66.

Definition. The sign function is defined to be

Note that our definition of sign is not symmetfic. We further define hO(x)=lxl and

(*I

ha(x)=lx-al.

Then sign(x) is the left derivative of ho(x), and sign(x-a) is the left derivative of ha(x). Since ha(x) is convex by Theorem 66 we have for a semimartingale X t

ha(Xt) = IXt - a1 = 1x0 +

6,

sign(X,-

-

a)dXs

+ A:,

(**)

where At is the increasing process of Theorem 66. Using (*) and (**) as defined above we can define the local time of an arbitrary semimartingale.

Definition. Let X be a semimartingale, and let ha and Aa be as defined in (*) and (**) above. The local time at a of X , denoted Lf = La(X)t, is defined to be the process given by

Sit

Notice that by the corollary of Theorem 63 the integral sign(X,- - a)dX, in (**) has a version which is jointly measurable in (a,t, w ) and cadlig in

7 Local Times

213

t. Therefore so does (A:)t2o, and finally so too does the local time L:. We always choose this jointly measurable, cadlag version of the local time, without any special mention. We further observe that the jumps of the process Aa defined in (**) are precisely Cslt{ha(Xs) - ha(Xs-) - ha(Xs-)AX,) (by Theorem 66), and therefore the local time (L:)t>o is continuous in t . Indeed, the local time La is the continuous part of the increasing process Aa. The next theorem is quite simple yet crucial t o proving the properties of La that justify its name.

Theorem 68. Let X be a semimartingale and let La be its local time at a. Then

Proof. Applying Theorem 66 t o the convex functions f (a) g(x) = (x - a ) - we get

=

(x - a)+ and

Next let

and subtracting the formulas we get C: - CL = 0 and hence Dtf = D r . Also D; D; = L,", so that Dtf = Dr = $L:, and the proof is complete.

+

The next theorem, together with the "occupation time density" formula (Corollary 1 of Theorem 70), are the traditional justifications for the terminology "local time."

214

IV General Stochastic Integration and Local Times

Theorem 69. Let X be a semimartingale, and let L; be its locul time at the level a , each a E R. For a.a. w, the measure in t , dL;(w), is curried by the set { S : X,- (w) = X s ( w ) = a ) .

Proof. Since L," has continuous paths, the measure dL:(w) is diffuse, and since { S : X,- ( w ) = a ) and { s : X,- ( w ) = X,(w) = a ) differ by at most a countable set, it will suffice to show that dL;(w) is carried by the set { s : X,- (w) = a ) . Suppose S , T are stopping times and that 0 < S 5 T such that [ S ,T ) c { ( s ,w) : X s - (w) < a ) = { X - < a). Then X 5 a on [S,T ) as well. Hence by the first equation in Theorem 68 we have

However the left side of the above equation equals zero, and all terms on the right side except possibly $ (L? - L;) also equal zero. Therefore $ (L$ - L;) = 0, whence L$ = L;. Next for r E Q, the rationals, define the stopping times S T ( w ) ,r > 0, by

Then define

TT(w) = inf { t > ST(w) : Xt- ( w ) 2 a}. Then [ S T , T T )c { X - < a ) , and moreover the interior of the set { X - < a ) equals UTEQ, ,,,(ST, T,). As we have now seen, dLa does not charge the interior of the set { X - <,a). This set is open on the left, hence it differs from its interior by an at most countable set, and since dLa is diffuse it doesn't charge countable sets. Thus dLa does not charge the set { X - < a ) itself. Analogously one can show dLa does not charge { X - > a). Hence its support is contained in the set { X - = a ) , and we are done. • The next theorem gives a very satisfying generalization of It6's formula (Theorem 32 of Chap. 11). Theorem 70 (Meyer-It6 Formula). Let f be the difference of two convex functions, let f 1 be its left derivative, and let p be the signed measure (when restricted to compacts) which is the second derivative o f f in the generalized function sense. Then the following equation holds:

7 Local Times

where X is a semimadingale and L:

= L:(X)

215

is its local time a t a .

Proof. Notice that equation (*) is trivially true if f is an affine function (i.e., f (a) = a x b). Next let us first assume that p is a signed measure with finite total mass on a compact interval. Define a function g by

+

It is well known that f and g differ by at most an affine function h(x) = ax+b. Therefore without loss of generality we can assume that the function f in (*) is of the form f(x) = J lx - ylp(dy). Then f'(x) = Jsign(x - y)p(dy) and f "(a) = p(dx). Moreover if

i

then

i

1 2 J J , v ~ ( ~= Y)

x

{f(x,) O<s
! = IXt Also, letting H

-

yI

-

1x0

-

-

f(x,-)

-

fl(xs-)~x,}.

(1)

91, one has

St+

sign(Xs- - y)dX,, and let ZY be the B(R) @ Next consider Z: = B(R+) 8 3 measurable version, which we know exists by Theorem 63. By F'ubini's Theorem (Theorem 64) we have

Since L:

= Ht

-

Jt

-

Z t , we have

and combining (4) with (I), (2), and (3), yields the formula (*). It remains to reduce the general case to that of p supported on a compact interval with finite total mass. By linearity it further suffices to treat the case where f is convex. Define

f (n) + f l ( n ) ( x - n), f(-n)+ fl(-n)(x+n),

if x 2 n, if - n I x < n , ifxi-n.

Then fn is also convex and its second (generalized) derivative pn(dx) agrees with p on [-n, n) and is of finite total mass. Let Yn = fn(X), and Tn =

216

IV General Stochastic Integration and Local Times

inf{t > 0 : lXtl > n). Note that Yn = f (X) on [0, T n ) , and that LF, = 0 for all a , la1 2 n, by Theorem 69. Therefore J L:pn(da) = J L;p(da) for t 5 T n , and by the preceding (*) holds for Y = fn(X) = f ( X ) on [0,Tn). Therefore (*) holds on [0,Tn). Since the stopping times (Tn)n>l - increase to cc a s . , we have (*) holds on all of R+ x 52, and the theorem is proved. The next formula gives an interpretation of semimartingale local time as an occupation density relative to the random "clock" d[X, X];.l7

Corollary 1. Let X be a semimartingale with local time (La)aER.Let g be a bounded Borel measurable function. Then a s .

Proof. Let f be convex and C2. Comparing (*) of Theorem 70 with It6's formula (Theorem 32 of Chap. 11) shows that

where p(da) is of course ft1(a)da. Since the above holds for any continuous and positive function f", a monotone class argument shows that it must hold, up to a P-null set, for any bounded, Bore1 measurable function g. We record here an important special case of Corollary 1.

Corollary 2. Let X be a semimartingale with local time (La)aEw Then [X, XI:

=

/

00

Lfda.

-00

Corollary 3 (Meyer-Tanaka Formula). Let X be a semimartingale with continuous paths. Then

Proof. This is merely Theorem 70 with f ( a ) = 1x1, which implies p(da) = 2 ~ ~ ( d apoint ) , mass at 0. The formula also follows trivially from the definition of LO. If Xt = Bt is a standard Brownian motion, then Mt = $'sign(B,)dB, is a ~ d= [[B, ~ B], , = continuous local martingale, and [M,MIt = ~ i ~ n ( ~ , )B], t. Therefore by Lkvy's Theorem (Theorem 39 of Chap. 11) we know that Mt

2

l7

Recall that [X,XICdenotes the path-by-path continuous part oft [X,XI; = 0.

I--+

[X,X]t with

7 Local Times

217

is another Brownian motion. We therefore have, where Bo = x for standard Brownian motion, lBtl = P o l + Pt Lt, (**>

+

So t

where pt = sign(B,)dB, is a Brownian motion, and Lt is the local time of B at zero. Formula (**) is known as Tanaka's Formula. Observe that if f(x) = 1x1, then f " = 26(x), where 6 is the "delta function at 0," which of course is a generalized function, or "distribution." Thus Corollaries 1 and 3 give the intuitive interpretation of local time as L! = 6(Xs)d[X,X],, and LF = J: 6(Xs - a)d[X,X],, for continuous semimartingales. Local times also allow us to give simple examples of functions f that are only slightly more general than convex functions and are such that f (X) need not be a semimartingale when X is one. Let f (x) = Ixla, 0 < a < 1. Then f cannot be expressed as the difference of two convex functions because the slope of f becomes vertical as x decreases to zero. The proof of this result is simpler if we restrict a to 0 < o < 112.

Theorem 71. Let X be a continuous local martingale with Xo = 0, and let 0 < a < 112. Then & = IXt 1" is not a semimartingale unless X is identically zero. Proof. Let us suppose that Y We then have

=

1x1" is a semimartingale. Let P = l/a > 2.

since Y has continuous paths. Using Theorem 69 and the Meyer-Tanaka formula (Corollary 3 of Theorem 70) we have, where Lt = L! is the local time of X at 0,

Sot

The integral 1txs=o}sign(X,)dX, is identically zero. For if Tn is a sequence of stopping times increasing to co a.s. such that XTn is a bounded martingale, then

218

IV General Stochastic Integration and Local Times

Therefore Lt becomes

=

L:

=

Sot l{x,,o}dlXl,.

Since { X ,

= 0)

equals {Y,

= 0),

this

Using the Meyer-Tanaka formula again we conclude that

Since X is a continuous local martingale, so also is the stochastic integral on the right side above (Theorem 30); it is also non-negative, and equal to zero at 0. Such a local martingale must be identically zero. This completes the proof. It is worth noting that if X is a bounded, continuous martingale with X o = 0, then Yt = IXt I", 0 < a < 112 is an example of an asymptotic martingale, or AMART, which is not a semimartingale.18 We next wish to determine when there exists a version of the local time Lf which is jointly continuous in ( a ,t ) ++ L: as., or jointly right continuous in ( a ,t ) ++ L f . We begin with the classical result of Kolmogorov which gives a sufficient condition for joint continuity. There are several versions of Kolmogorov's Lemma. We give here a quite general one because we will use it often in Chap. V. In this section we use only its corollary, which can also be proved directly in a fairly simple way. Before the statement of the theorem we establish some notation. Let A denote the dyadic rational points of the unit cube [O,1In in Rn, and let A, denote all x E A whose coordinates are of the form k Y m , 0 5 k 5 2m.

Theorem 72 (Kolmogorov's Lemma). Let ( E , d ) be a complete metric space, and let Ux be an E-valued random variable for all x dyadic rationals i n Rn. Suppose that for all x , y, we have d(Ux, UY) is a random variable and that there exist strictly positive constants E , C , such that

Then for almost all w the function x continuous function from Rn to E . l8

++

Ux can be extended uniquely to a

For a more elementary example of an asymptotic martingale that is not a semimartingale, see Gut [86, page 71.

7 Local Times

219

Proof. We prove the theorem for the unit cube [0, 11" and leave the extension to Rn to the reader. Two points x and y in A, are neighbors if sup, (xiyil = 2-,. We use Chebyshev's inequality on the inequality hypothesized to get

Let A, = {w : 3 neighbors x, y E A, with d(Ux(w),UY(w)) > 2-",). Since each x E A, 2mn, we have

has at most 3n neighbors, and the cardinality of A, P(A,)

is

5 c~,("~-P)

where the constant c = 3nC. Take a sufficiently small so that a e < P. Then

where 6 = P-ae > 0. The Borel-Cantelli Lemma then implies P(A, infinitely often) = 0. That is, there exists a n mo such that for m > mo and every pair (u, v) of points of A, that are neighbors,

We now use the preceding to show that x I+ Ux is uniformly continuous on A and hence extendable uniquely to a continuous function on [0, lIn. To this end, let x, y E A be such that llx - yll 5 We will show that d(Ux,UY) 5 c2Yak for a constant c, and this will complete the proof. Without loss of generality assume k > mo. Then x = (xl,. . . ,xn) and have dyadic expansions of the y = ( y l , . . . ,yn) in A with llx - yll 5 2-"l form

where a;, b; are each 0 or 1, and u, v are points of Ak which are either equal or neighbors. 7 ~ 2= ul + ak+22-k-2,.. .. We also Next set uo = u, ul = uo ak+12-"', make analogous definitions for vo, vl, v2,. . .. Then ui-1 and ui are equal or neighbors in Ak+', each i, and analogously for vi-1 and vi. Hence

+

220

IV General Stochastic Integration and Local Times

and moreover d(Uu(u),Uu(u)) 5 2Tak. The result now follows by the triangle inequality.

Comment. If the complete metric space ( E ld) in Theorem 72 is separable, then the hypothesis that d(Ux,UY) be measurable is satisfied. Often the metric spaces chosen are one of R, Rd, or the function space C with the sup norm and these are separable. A complete metric space that arises often in Chap. V is the space E = Dn of c&dl&gfunctions mapping [O, co) into Rn, topologized by uniform convergence on compacts. While this is a complete metric space, it is not separable. Indeed, a compatible metric is

However if f,(t) = l[,,,)(t), then d(f,,fp) = 112 for all a, P with 0 5 a < 1, and since there are uncountably many such a, p, the space is not separable. Fortunately, however, the condition that d(Ux,UY) be measurable is nevertheless satisfied in this case, due t o the path regularity of the functions in the function space Dn. (Note that in many other contexts the space Dn is endowed with the Skorohod topology, and with this topology Dn is a complete metric space which is also separable; see for example Ethier-Kurtz 1711 or Jacod-Shiryaev [110].)

p 5

We state as a corollary the form of Kolmogorov's Lemma (also known as Kolmogorov's continuity criterion) that we will use in our study of local times.

Corollary 1 (Kolmogorov's Continuity Criterion). Let (Xa) t t>O,aEWn be a parameterized family of stochastic processes such that t XF is cMl&g as., each a E Rn. Suppose that

for some e , p > 0, C(t) > 0. Then there exists a version X; of XF which is B(R+) @ B(Rn) @ .F measurable and which is c&dl&gin t and uniformly continuous in a on compacts and is such that for all a E Rn, t 2 0,

(The null set {x: # XF) can be chosen independently of t.) In this section we will use the above corollary for parameterized processes Xa which are continuous in t. In this case the process obtained from the corollary of Kolmogorov's Lemma, xa,will be jointly continuous in (a, t) almost surely. In particular, Kolomogorov's Lemma can be used to prove that the paths of standard Brownian motion are continuous.

7 Local Times

221

Corollary 2. Let B be standard Brownian motion. Then there is a version of B with continuous paths, a.s. Proof. Since Bt - B, is Gaussian with mean zero and variance t - s, we know that E{IBt - ~ $ 1 ~5) c(t - s ) ~ (One . can give a cute proof of this moment estimate using the scaling property of Brownian motion.) If we think of time as the parameter and the process as being constant in time, we see that the exponent 4 is strictly positive, and that the exponent on the right, 2, is strictly bigger than the dimension, which is of course 1. Corollary 2 now follows from Corollary 1. Hypothesis A. For the remainder of this section we let X denote a semimartingale with the restriction that Co<, 0. Observe that if (52, F, (Ft)t>o,P) is a probability space where (Ft)tyo is the completed minimal filtration of a Brownian motion B = (Bt)t20, then all semimartingales on this Brownian space verify Hypothesis A. Indeed, by Corollary 1 of Theorem 43 all the local martingales are continuous. Thus if X is a semimartingale, let X = M + A be a decomposition with M a local martingale and A an F V process. Then the jump processes A X and AA are eaual, hence

since A is of finite variation on compacts. Let X be a semimartingale satisfying Hypothesis A, and let

Jt =

C AX,;

that is, J is the process which is the sum of the jumps. Because of our hypothesis J is an FV process and thus also a semimartingale. Moreover & = Xt - Jt is a continuous semimartingale with Yo = 0, and we let

be its (unique) decomposition, where M is a continuous local martingale and A is a continuous F V process with Mo = A. = 0. Such a process M is then uniquely determined and we can write

the continuous local martingale part of X .

Notation. We assume given a semimartingale X satisfying Hypothesis A. If Z is any other semimartingale we write

222

IV General Stochastic Integration and Local Times

This notation should not be confused with that of Kolmogorov's Lemma (Theorem 72). It is always assumed that we take the B(R)@B(R+)@Fmeasurable, cAdlAg version of Za (cf., Theorem 63). Before we can prove our primary regularity property (Theorem 75), we need two preliminary results. The first is a very special case of a family of martingale inequalities known as the Burkholder-Davis-Gundy inequalities.

Theorem 73 (Burkholder's Inequality). Let X be a continuous local mar, T a finite stopping time. Then tingale with Xo = 0, 2 < p < c ~ and

Proof. By stopping, it suffices to consider the case where X and [X,X] are bounded. By ItG1sformula we have

By Doob's inequalities (Theorem 20 of Chap. I) we have (with

a+

= 1)

with the last inequality by Hiilder's inequality. Since E{(x+)P)'-~ < GO, we divide both sides by it to obtain

and raising both sides to the power p/2 gives the result. Actually much more is true. Indeed for any local martingale (continuous or not) it is known that there exist constants cp, Cp such that for a finite stopping time T

for 1 I p < oo. See Sect. 3 of Chap. VII of Dellacherie-Meyer [46] for these and related results. When the local martingales are continuous some results even hold for 0 < p < 1 (see, e.g., Barlow-Jacka-Yor [9, Table 4.1 page 1621).

7 Local Times

223

Theorem 74. Let X be a semimartingale satisfying Hypothesis A. There exists a version of ( x c ) ? such that (a, t, w ) H (xc);(u) is B(R) @ P measurable, and everywhere jointly continuous in (a, t). Proof. Without loss of generality we may assume X - Xo E 'H2. If it is not, we can stop X - Xo at Tn-. The continuous local martingale part of is then just ( x ~ ) ~ "Suppose . -oo < a < b < oo,and let

xTn-

By Corollary 1 of Theorem 70 we have

1

b

at (a, b) = E{(

~ : d u ) ~= } (6 - u)~E{(-

1

a

by the Cauchy-Schwarz inequality. The above implies

at (a, b) 5 (b - a)2 sup E{(L;)~}. QE

By the definition,

and therefore

and the bound is independent of u. Therefore

for a constant r < co, and independent of t. Next using Burkholder's inequality (Theorem 73) we have

The result now follows by applying Kolmogorov's Lemma (the corollary of Theorem 72).

224

IV General Stochastic Integration and Local Times We can now establish our primary result.

Theorem 75. Let X be a semimartingale satis&ing Hypothesis A. Then there exists a B(R) @ P measurable version of ( a ,t , w) I+ L:(w) which is everywhere jointly right continuous in a and continuous in t . Moreover a.s. the limits L:- = limb,,, b
Proof. Since X satisfies Hypothesis A, the process Jt = Co<s
Observe that /St[5 Co<slt lAX,l < co. By Theorem 68 we have

Consider

We have

and

n+

where the convergence is uniform in t on [O, T ], and T > 0. We have analogous results for (j): and also for St, because it is dominated by Co<s5t IAX,l < co. Since we already know that (M): is continuous, the proof is complete. The next three corollaries are simple consequences of Theorem 75. Recall that for a semimartingale X satisfying Hypothesis A, we let Jt = CossitAX,, Y = X - J, and Y = M A be the (unique) decomposition of Y, with A. = 0.

+

Corollary 1. Let X be a semimartingale satisfying Hypothesis A. Let X = M + A + J be a decomposition with M and A continuous and J the jump process of X , and let (L:)tlo be its local time at the level a. Then

7 Local Times

225

An example of a semimartingale satisfying Hypothesis A but having a discontinuous local time is Xt = lBtl where B is standard Brownian motion with Bo = 0. Here La(X) = La(B) + LPa(B) for a > 0, LO(X)= LO(B),and La(X) = 0 for a < 0. J. Walsh [227] has given a more interesting (but more complicated) example. Corollary 2. Let X be a semimartingale satisfying Hypothesis A. Let A be as in Corollary 1. The local time (L:) is continuous in t and is continuous at a = a0 if and only if

6"

1{xS=a,}IdAsI= 0.

Observe that if X = B , a Brownian motion (or any continuous local martingale), then A = 0 and the local time of X can be taken everywhere jointly continuous. Corollary 3. Let X be a semimartingale satisfying Hypothesis A. Then for every (a, t ) we have

and L:-

=

lim O E'

&

St 0

l { a - E 5 x ~ ~ a } d [ X , X la.s. ~,

The lack of symmetry in the above formula stems from the definition of local time, where we defined sign(x) in an asymmetric way:

A symmetrized result follows trivially, namely

Corollary 3 is intuitively very appealing, and justifies thinking of local time as an occupation time density. Also if X = B , a Brownian motion, then Corollary 3 becomes the classical result

Local times have interesting properties when viewed as processes with "time" fixed, and the space variable "a" as the parameter. The Ray-Knight Theorem giving the distribution of ( L ; ~ ~ ) ~ the~ Brownian ~ < ~ , local time sampled at the hitting time of 1, as the square of a two dimensional Bessel diffusion, is one example.

226

IV General Stochastic Integration and Local Times

-

The Meyer-It6 formula (Theorem 70) can be extended in a different direction, which gives rise to the Bouleau-Yor formula, allowing non-convex functions of semimartingales. The key idea is that the function a L& induces a measure on R if L is the local time of a semimartingale X satisfying Hypothesis A and U is a positive random variable.

Theorem 76. Let X be a semimartingale satisfying Hypothesis A, U a positive random variable, La the local times ofX. Then the operation

(x), can be extended uniquely to a vector measure where f (x) = C fil(,i,ai+l~ on B(R) with values in Lo. Proof. Recall that Lo denotes finite-valued random variables. By Theorem 68 we know that

We write St for the last two sums on the right side of the equation above. Note first that (Xu - a)- - (Xo - a)- is Lipschitz continuous in a and hence absolutely continuous in a. Therefore it induces a measure in a. Also SE is c&dl&gand of finite variation in a, and moreover the function a [ ldaS61 I2 Co<slu IAX,l. Finally consider the stochastic integral. Let

-

xg = Ji+1{x,-
Then

Therefore by the Dominated Convergence Theorem for stochastic integrals fi(X:+' - X?) extends to a measure with (Theorem 32) we have that values in Lo and moreover J f (a)daXE = :J f (X,-)dX,, for f bounded, Bore1 measurable on R. Thus daLE can be defined as the sum of three L0valued measures on (R, B(R)), and the theorem is proved.

xi

In the proof of Theorem 76 we proved the following result, which is important enough to state as a corollary.

8 Az6ma1sMartingale

227

Corollary. Let X be a semimartingale satisfying Hypothesis A, U a positive random variable, and X f = Jot ltx3-laldXs. Then daXE can be defined as an Lo-valued measure, and for f E bB(R) we have

Combining Theorem 76 and its corollary yields the Bouleau-Yor formula. We leave its proof to the reader.

Theorem 77 (Bouleau-Yor Formula). Let X be a semimartingale satisfying Hypothesis A, U a positive random variable, f a bounded, Bowl function, : f (u)du. Then and F ( x ) = J

Combining Theorem 77 with the Meyer-It6 formula (Theorem 70) we obtain the following relationship.

Corollary. Let X be a semimartingale satisfying Hypothesis A, and let T be a finite-valued random variable. Let La be the local times of X and let f be a C1 function. Then

Remark. The Meyer-It6 formula (Theorem 70) and the Bouleau-Yor formula can be extended in interesting ways. This was first done in [77],which inspired subsequent work by N. Eisenbaum [62] and others (e.g., [6]). See [80] for a compilation of selected,results in these areas and also new ones.

8 Azhma's Martingale In order to prove the regularity properties of local times in Sect. 7, we needed Hypothesis A: a semimartingale X satisfies Hypothesis A if X is such that Co<slt lAX,l < oo a.s., each t > 0. This hypothesis was needed to prove Theorems 74 through 77 and their corollaries. We present here a counterexample, known as Az6ma1s martingale, that shows that these results do not hold for general semimartingales. In particular, Theorems 81 and 82 show that Az6ma's martingale has a local time that is zero at every level a except a = 0, and therefore there is no regular version in the space variable. Let (R, F,F, (Bt)t>o,P) be a standard Brownian motion, Bo = 0, with - completed (and hence right continuous). We define the filtration F = (Ft't),>0

228

IV General Stochastic Integration and Local Times

Gt

= u{sign(B,); s 5 t), and let (6 = ( G t )-t > ~ denote the filtration ( G t ) t 2 ~ Set completed. (It is then right continuous as a consequence of the strong Markov property of Brownian motion.) Let Mt = E{BtlGt),t > 0. Then for s < t,

and M is a

(6

martingale. We always take the c&dl&gversion of M .

Definition. The process Mt = E{BtlGt} is called Azkma's martingale. Fundamentally related to Azkma's martingale is the process which gives the last exit from zero before time t. We define

To study the process g we need the refEection principle for Brownian motion, which we proved in Chap. I (Theorem 33 and its corollary) and which we recall here for convenience.

Theorem 78 (Reflection Principle for Brownian Motion). Let B be standard Brownian motion, Bo = 0, and c > 0. Then P(sup B, > c) = 2P(Bt > c). sst

An elementary conditioning argument provides a useful corollary.

Corollary. Let B be standard Brownian motion, Bo = 0, and 0 < s < t. Then P ( sup B, > 0, B, < 0) = 2P(Bt > 0, B, < 0). ,<,st Theorem 79 is known as Lkvy's arcsine law for the last exit from zero.

Theorem 79. The process gt is 05slt.

(6

adapted, and P(gt 5 s) = ;arcsinfi,

Proof. For s 5 t we have almost surely

from which it follows that gt is Gt adapted. To calculate the distribution of gt we observe for s < t (using the symmetry of B),

8 AzBma's Martingale

229

P(gt > s) = 2P(gt > s, B, < 0) = 2P( sup B, > 0, B, < 0)

,<us,

= 4P(Bt

> 0, B, < O),

by the corollary of Theorem 78. Since (B,, Bt) are jointly Gaussian, there exist X , Y that are independent N ( 0 , l ) random variables with B,=&X

and

B~=&X+&Y.

Then {B, < 0) = {X < 01, and {B, > 0) = {Y > P({Y >

-&x, x < O)),

-&x)

To calculate

use polar coordinates:

Therefore 4P(Bt > 0, B, < 0) = 1 $ arcsin

8.

$ arcsin fi and

P(gt 5 s) =

Theorem 80. Azkma's martingale M is given by Mt = s i g n ( B t ) 6JG, t 2 0.

Proof. By definition,

However E{I Bt 1 lGt) = E{I Bt llgt; sign(B,), s 5 t). But the process sign(B,) is independent of lBtl for s < gt, and sign(B,) = sign(Bt) for gt < s 5 t. Hence sign(B,) is constant after g,. It follows that

Therefore if we can show E{I Btllgt complete. Given 0 < s < t we define

=

s) =

Note that T is a bounded stopping time. Then

A&,

the proof will be

230

IV General Stochastic Integration and Local Times E{IBtI;gt I s) = 2E{Bt;gt I s and B, > 0) = 2E{Bt; T = t and B, > 0) = 2E{BT; B, > 0) = 2E{B,; B, > O),

since BT = 0 on {T # t), and the last step uses that B is a martingale, and s 5 T. The last term above equals E{IB,I) = Therefore

6~s.

since $ ~ ( I ~ s)t = . r , & i q l

by Theorem 79.

Let IH; = a{M,; s I t), the minimal filtration of Az6ma7smartingale M , - be the completed filtration. and let W = (IHt)t>o

Corollary. The two filtrations IHI and

(6

are equal.

Proof. By definition Mt E Gtl each t 2 0. Hence IHt Theorem 80, sign(Mt) = sign(Bt), so that Gt C IHt.

c Gt. However, by

Next we turn our attention to the local times of Az6ma's martingale. For a semimartingale (Xt)tyo, let (L:(X))t>o - = La(X) denote its local time at the level a.

Theorem 81. The local time at zero of Azkma's martingale is not identically zero. Proof. By the Meyer-It8 formula (Theorem 70) for f (x) = 1x1 we have

= IMtl Since sign(M,-)AM, = AY,, we have that lMtl - L:(M) Set is a martingale. If LO(M)were the zero process, then Y = ]MI would be a non-negative martingale with Yo = 0, hence identically zero. Since Y is not identically zero, the theorem is proved.

Theorem 82. The local times at all levels except zero of Azkma's martingale are 0. That is, L:(M) = 0, t 2 0, i j a # 0.

8 AzBma's Martingale

231

Proof. By Theorem 69 we know that the measure dL:(w) on R+ is carried by the set { s : Ms-(w) = M,(w) = a } , a.s., for each a. However if a # 0, this set is countable. Since s H Lz is a.s. continuous, it cannot charge a countable set. 0 Therefore dL; is the zero measure for a # 0. Since Lg = 0, we have Lf for a # 0.

-

Theorems 81 and 82 together show that Az6ma's martingale provides a counterexample to a general version of Theorem 75, for example. Therefore a hypothesis such as Hypothesis A is necessary, and also we see that Az6ma's martingale does not satisfy Hypothesis A. Since all semimartingales for the Brownian motion filtration F do satisfy Hypothesis A, as noted in Sect. 7, we conclude the following.

Theorem 83. Azkma's martingale M is not a semimartingale for the Brownian filtration F . While the local time at zero, LO(M),is non-trivial by Theorem 81, more information is available. It is actually the same as the Brownian local time, as Theorem 85 below shows. First we need a preliminary result.

Theorem 84. The local time L'(B) of Brownian motion is (6 adapted. Theorem 84 will be proved if one can express L'(B) as the a.s. limit of measurable functionals of the zero set of Brownian motion. Such a result is classical: see, e.g., Kingman [126, page 7301.

Theorem 85. The local times at zero of Brownian motion and Az6ma1s martingale are the same. That is, LO(B)= LO(M).

Proof. As we saw in the proof of Theorem 81, lMtl - L:(M) is a martingale. Since the process LO(M) is non-decreasing, continuous and adapted, it is natural. If A is another natural process such that IMI - A is a martingale, then LO(M)= A by (for example) Theorem 30 of Chap. 111. Thus, it suffices to show that /MI - LO(B)is a martingale for the filtration (6. By Tanaka's formula, lBtl - L:(B) = Nt, where N is an F martingale. Then E{IBtI - L ~ ( B ) I G ~=}E{NtIGt} and by Theorem 84 E{IBtllGt} - L:(B) is a martingale. Since

we have lMt 1

-

L:(B) is a martingale, and the theorem is proved.

232

IV General Stochastic Integration and Local Times

Corollary 1. The process IMI - LO(B)is a

(6

martingale.

Proof. This corollary is proved at the end of the proof of Theorem 85.

Corollary 2. The process l{Bt>0)fid-

-

L:(B) is a

(6

martingale.

= 0, it follows from Theorem 68 Proof. Since l{Ms_>olMs- = 1 that M; - iL:(M) is a martingale. However M: = l{Bt>o)fi,/--gt by Theorem 80, and L:(M) = L:(B) by Theorem 85, and the theorem follows.

The next theorem shows that Az6ma1smartingale is quadratic pure jump.

Theorem 86. For Azkma's martingale M , [M,MIC= 0, and [M,MIt = ;gt. Proof. By Corollary 2 of Theorem 70, [M,MI: =

/

00

Ltda.

-00

By Theorem 82, [M, MIC5 0. We conclude that [M, M]t = ~ o < s
Theorem 87. Let M be Aze'ma's martingale. Then [M,MIt - 2t is a martingale. That is, (M, M ) t = $t. Proof. Recall that Mt2- [M,MIt is a martingale (cf., Theorem 27 of Chap. 11). By Theorems 80 and 86,

Since the above is a martingale, it remains only t o observe that :gt martingale.

-

:t is a

Note that if Xt = s i g n ( B t ) 4 G , then X is a quadratic pure jump martingale with ( X , X ) t = t . Recall that if N is a Poisson process with intensity one, then Nt - t is a martingale. Since [N, NIt = Nt, we have [N, NIt - t is a martingale; thus (N, N ) t = t. Another example of a martingale Y with (Y, Y)t = t is Brownian motion. Since [B,BIt = t is continuous, [B, Blt = (B, B)t = t.

9 Sigma Martingales

233

9 Sigma Martingales We saw with Emery's example (which precedes Theorem 34) a stochastic integral with respect to an 'H2 martingale that was not even a local martingale. Yet "morally" we feel that it should be one. It is therefore interesting to consider the space of stochastic integrals with respect to local martingales. We know these are of course semimartingales, but we want to characterize the class of processes that arise in a more specific yet reasonable way. By analogy, recall that in measure theory a a-finite measure is a measure p on a space (S, B) such that there exists a sequence of measurable sets A, E B Ai = S and p(Ai) < oo, each i. Such a measure can be thought such that of as a special case of a countable sum of probability measures. It is essentially a similar phenomenon that brings us out of the class of local martingales.

Ui

Definition. An Rd-valued semimartingale X is called a sigma martingale if there exists an EXd-valued martingale M and a predictable R+-valued process H E L ( M ) such that X = H . M . In the next theorem, H will always denote a strictly positive predictable process.

Theorem 88. Let X be a semimartingale. The following are equivalent:

(i) X is a sigma martingale; (ii) X = H . M where M is a local martingale; (iii) X = H . M where M is a martingale; (iv) X = H . M where M is a martingale in ' H I . Proof. It is clear that (iv) implies (iii), that (iii) implies (ii), and that (iii) obviously implies (i), so we need to prove only that (i) implies (iv). Without loss assume that Mo = 0. Since X is a sigma martingale it has a representation of the form X = H . M where M is a martingale. We know that we can localize M in IH1 by Theorem 51, so let Tn be a sequence of stopping times tending to oo a.s. such that MTn E IH1 for each n. Set To = 0 and let N n = Let an be a strictly positive sequence of real numbers such ,T,] . a n N n and one can check that N that En an 11 N n ] l X l < oo. Define N = is an IH1 martingale. Set J = l { H = O ) H En a ; l l ( T n - l , ~ nThen ~ . it is simple to check that X = J . N and that J is strictly positive.

+

En

Corollary 1. A local sigma martingale is a sigma martingale.

Proof. Let X be a local sigma martingale, so that there exists a sequence of stopping times tending to oo a.s. such that xTnis a sigma martingale for each n. Since xTnis a sigma martingale, there exists a martingale M n in IH1 such that xTn = Hn . M n , for each n, where Hn is positive predictable. Choose #n positive, predictable so that Il$PHn. MTnll.H1< 2Tn. Set To = 0 and #O = # l l t o ) .Then 4 = #O En>,#nl(Tn-,,~nl is strictly positive and predictable. Moreover # X is an IH1 &artingale, whence X = f # X, and the corollary is established.

+

234

IV Ge~ieralStochastic Integration and Local Times

Corollary 2. A local martingale is a sigma martingale.

Proof. This is simply a consequence of the fact that a local martingale is locally a martingale, and trivially a martingale is a sigma martingale. The next theorem gives a satisfying stability result, showing that sigma martingales are stable with respect to stochastic integration.

Theorem 89, I f X is a local martingale and H E L ( X ) , then the stochastic integral H . X is a sigma martingale. Moreover if X is a sigma martingale (and a fortiori a semimartingale) and H E L ( X ) , then H . X is a sigma martingale.

Proof. Clearly it suffices to prove the second statement, since a local martingale is already a sigma martingale. But the second statement is simple. Since X is a sigma martingale we know an equivalent condition is that there exists a strictly positive predictable process and a local martingale M such that X = + . M . Since H is predictable, the processes H' = H1{H>o) 1 and H 2 = -H1{H
+

+

and since Hi+ are both strictly positive for i = 1, 2 we have by Theorem 88 that H i # . M is a sigma martingale for each i = 1, 2. Sigma martingales form a vector space, and we are done. We also would like to have sufficient conditions for a sigma martingale to be a local martingale.

Theorem 90. A sigma martingale which is also a special semimartingale is a local martingale.

Proof. Any sigma martingale is a semimartingale by definition; here we also assume it is special. Thus it has a canonical decomposition X = M A where M is a local martingale and A is a process of finite variation on compacts which is also predictable. We want to show A = 0. We assume A. = 0. Choose H predictable, H > 0 everywhere, such that H . X is a martingale, . ( H .X ) which we can easily do using Theorem 88. Then ( H A 1) . X = is a local martingale, and hence without loss of generality we can assume H is bounded. Then H . A is a finite variation predictable process, and we have H . A = H . X - H . M which is a local martingale. Thus we conclude that H . A = 0. If we replace H with J H where J is predictable, I JI 5 1 , and is J,dA, = ldA,I, then we conclude that H, JdA,I = 0, and such that since H > 0 everywhere, we conclude that A = 0 establishing the result.

+

(9)

fi

fi

Sot

We now have that any criterion that ensures that the semimartingale X is special, will also imply that the sigma martingale X is a local martingale.

Bibliographic Notes

235

Corollary 1. If a sigma martingale X has continuous paths, then it is a local martingale.

Corollary 2. If X is a sigma martingale and if either X t = sup,,, IX,( or = supsl, 1 AX, 1 is locally integrable, then X is a local martingale. Note that as a corollary to Corollary 2 we have the following.

Corollary 3. If X is a sigma martingale with bounded jumps, then X is a local martingale.

Bibliographic Notes The extension by continuity of stochastic integration from processes in IL to predictable processes is presented here for essentially the first time. However the procedure was indicated earlier in Protter [201, 2021. Other approaches which are closely related are given by Jacod [I031 and Chou-MeyerStricker [31] (see an exposition in Dellacherie-Meyer [46, page 3811). The "IH" in the space IH2 of semimartingales comes from Hardy spaces, and this bears explaining. When the semimartingale X E IH2 is actually a martingale, the space of IHP martingales has a theory analogous to that of Hardy spaces. While this is not the case for semimartingales, the IHP norms for semimartingales are, in a certain sense, a natural generalization of the IHP norms for martingales, and so the name has been preserved. The IHP norm was first introduced by Emery [64],though it was implicit in Protter [197]. Most of the treatment of stochastic integration in Sect. 2 of this chapter is new, though essentially all of the results have been proved elsewhere with different methods. The author benefited greatly throughout by discussions with S. Janson. Theorem 1 (that IH2 is a Banach space) is due to Emery [66]. The equivalence of the two pseudonorms for 'Hz (the corollary of Theorem 24) is originally due to Yor [241], while the fact that IH2(Q) c IH2(P) if is bounded is originally due to Lenglart [144]. Theorem 25 is originally due to Jacod [103, page 2281. The concept of a predictable a-algebra and its importance to stochastic integration is due to Meyer [166]. The local behavior of the stochastic integral was first investigated by McShane for his integral [154], and then independently by Meyer [171],who established Theorem 26 and its corollary for the semimartingale integral. The first Dominated Convergence Theorem for the semimartingale integral appearing in print seems to be in Jacod [103]. The theory of martingale representation dates back to It6's work on multiple stochastic integrals [96],and the theory for M~ presented here is largely due to Kunita-Watanabe [134]. A more powerful theory (for M1) is presented in Dellacherie-Meyer [46]. Absolutely continuous spaces are defined here for the first time. The author benefited from discussions with Yan Zeng on this subject. That the dual of IH1 is BMO for martingales is due to the combined

236

Exercises for Chapter IV

efforts of many researchers, culminating in Meyer's paper [170];we follow the treatment in [87]. The Jacod-Yor Theorem is from [ I l l ] , and Sect. 9 largely follows Emery [69]. The theory of stochastic integration depending on a parameter is due to the fundamental work of Stricker-Yor [219],who used a key idea of DolkansDade [48]. The Fubini Theorems for stochastic integration have their origins in the book of Doob [55] and Kallianpur-Striebel [I201 for the It6 integral, Kailath-Segall-Zakai [I191 for martingale integrals, and Jacod [lo31 for semimartingales. The counterexample to a general Fubini Theorem presented here is due to Janson. The theory of semimartingale local time is of course abstracted from Brownian local time, which is due to Lkvy [149].It was related to stochastic integration by Tanaka (see McKean [153]) for Brownian motion and the It6 integral. The theory presented here for semimartingale local time is due largely to Meyer [171].See also Millar [181].The measure theory needed for the rigorous development of semimartingale local time was developed by Stricker-Yor [219], and the Meyer-It6 formula (Theorem 70) was formally presented in Yor [242], as well as in Jacod [103]. Theorem 71 is due to Yor [244], and has been extended by Ouknine [188]. A more general version (but more complicated) is in Cinlar-Jacod-Protter-Sharpe [34].The proof of Kolmogorov's Lemma given here is from Meyer [177]. The results proved under Hypothesis A are all due to Yor [243] except the Bouleau-Yor formula [22]. See Bouleau [21] for more on this formula. Aziima's martingale is of course due to Azkma [4], though our presentation of it is new and it is due largely to Janson. For many more interesting results concerning Azkma's martingale, see Aziima-Yor [5], Emery [69], and Meyer [179]. Sigma martingales date back to the work of Emery [67],and to Chou [29]. The importance of sigma martingales was clarified through the work in mathematical finance of Delbaen and Schachermayer [38]. See [I101 for a more comprehensive treatment.

Exercises for Chapter IV Exercise 1. Let X be a semimartingale. Show that X is special if and only if the increasing process Ct = [X,XIt is locally integrable; that is, there exists a sequence of stopping times Tn increasing to infinity a.s. such that E {[XIXIT, ) < oo for each time Tn. *Exercise 2. Let X be a special semimartingale with canonical decomposition Xt = Xo + Mt At. Show that the following two inequalities hold:

+

Exercises for Chapter IV

237

Exercise 3. Let ( R , 3 , P) be a complete probability space. Let Xn be a sequence of random variables such that P(IXn( < m ) = 1, all n. Show that there exists another probability Q which is equivalent to P in the sense that the null sets are the same for both measures, with bounded, and every random variable Xn is in L 1 ( d ~ ) . Exercise 4. Let X be a semimartingale. Show that there is a probability measure Q, equivalent to P, such that under Q, X has a decomposition X = M A, where M is a martingale with EQ([M,MIt) < m for each t, 0 5 t < m , and A is a predictable process of finite variation on compacts with E ~ ( G IdA,() < m for each t, 0 5 t < m . (Hint: Use Exercises 1, 2, and 3.)

+

*Exercise 5. Let Xn be a sequence of semimartingales. Show that there exists one probability measure Q, equivalent to P, such that under Q, each X n has a decomposition Xn = M n An, where M n is a martingale with EQ([Mn,MnIt) < m for each t, 0 5 t < m and each n, and An is a predictable process of finite variation on compacts with E ~ ( IdATl) J ~ ~ < m for each t, 0 5 t < m and each n.

+

Exercise 6. Let S denote the space of all square integrable martingales with continuous paths a s . , on a given filtered complete probability space satisfying the usual conditions. Show that S is a stable subspace. (This exercise was referred to in Definition 3.) Exercise 7. Let Z be a L6vy process with E{IZtJ) < m and E{Zt) = 0 for all t > 0. Show that Z is a martingale, and that ZC is either 0 or Brownian motion, where ZCis defined in Exercise 6. Conclude that any martingale L6vy process Z that has no Brownian component is purely discontinuous. Exercise 8. Let N be a standard Poisson process with arrival intensity A. Show that the martingale Nt -At is purely discontinuous. (Note that this gives an example of a purely discontinuous martingale whose sample paths are not purely discontinuous in the sense that they do not change only by jumps.) Exercise 9. Let M be a square integrable martingale. Show that [M,MI: = [MC,MC]t= [MC,MIt, all t 2 0, almost surely. Exercise 10 (example of orthogonal projection). Let T be a stopping time. Show that the collection S of all square integrable martingales stopped at T is a stable subspace. Also show that if M is an arbitrary square integrable martingale, then its orthogonal decomposition with respect to this subspace is M = MT + ( M - MT). Give a description of the orthocomplement of S. Exercise 11 (example of orthogonal projection). Let T be a totally inaccessible stopping time and let M be a square integrable martingale. Let Ut = AMTl{t2T) and let L be the martingale of U minus its compensator. We denote L by I7M. Show that I7M and M - I7M are orthogonal. Let S denote the space of all square integrable martingales which are continuous at the instant T. Show that S is a stable subspace, and that I7M represents the orthogonal projection of M onto the orthocomplement of S.

238

Exercises for Chapter IV

*Exercise 12. Let T be an arbitrary stopping time, T > 0, and let X be a random variable in L1. Show that M = Xl{t2T) is a uniformly integrable martingale if and only if E{X1.FT-) = 0. *Exercise 13. Let T be a stopping time, T > 0, and let M be a square integrable martingale. Show that the decomposition

is an orthogonal decomposition. Show that one of the stable subspaces in question is the space of all square integrable martingales M such that MT is .FT- measurable. (Hint: Use Exercise 12.)

*Exercise 14. Let Z be a Lkvy process on a complete rob ability space (R,.F, P ) and let F be its minimal filtration completed (and satisfying the usual hypotheses). Show that ( R , 3 ,F, P) is an absolutely continuous space.

+

Exercise 15. Consider the structure equation d[X, XIt = dt PXt-dXt . Show that this equation has a scaling property: if X is a martingale solution, then so too is ;xxztfor every X # 0. Note that by the uniqueness in distribution of the solution, the processes (Xt)t20 and (;XXzt)t20 have the same distribution. *Exercise 16. Let X be a martingale solution of the structure equation of Exercise 15. Show that X C= 0, where X Cis defined in Exercise 6.

a*

Exercise 17. Let M be a local martingale in for some p 2 1. Show that M is a uniformly integrable martingale. Show further that if p = 1 then E{M,) 5 where the constant c is universal. That is, c does not depend on M. Exercise 18. Suppose that M is a bounded martingale. Show that

This can improved to replace

with 2.

Exercise 19. Show that there are martingales with finite B M O norm which are not bounded. (Hint: Let M be a bounded martingale and H be a predictable process such that [HI 5 1, and let N = H . M. Then it is easy to see that N is in BMO, and to construct such an N that is not bounded.) Exercise 20. Prove this version of the Fundamental Theorem of Local Martingales. Let M be a uniformly integrable martingale. Show that there exist N and U such that N is in BMO, U is of integrable variation, and M = U + V. If M is a local martingale this holds locally. Exercise 21. Show that an arbitrary local martingale is a semimartingale by using the Burkholder-Davis-Gundy inequalities instead of the Fundamental Theorem of Local Martingales. (See [61].)

Exercises for Chapter IV

239

Exercise 22. Let M be a local martingale. If T is a predictable stopping time, (s) E bP so that = l f O(s)dM, , ~ ) is a local martingale. Give then l[O,T) an example of a local martingale M and a stopping time T such that MT- is not a local martingale.

MT- ll

Exercise 23. Let X be a continuous semimartingale. Show that [IXI,X] = [X, XI. (Hint: Use local times.) 'Exercise 24. Let B be standard Brownian motion and let a > 0. Show that IBtl-"dt < m a.s. if and only if a < 1. (Hint: Use local times.) Exercise 25. Let X be a semimartingale. Show that L!(X) = iL!(X+) where X+ = max(X, 0) and L!(x) denotes the local time of X at time t and level 0. Exercise 26. Let A be adapted, continuous, and of finite variation on compacts, and let B be standard Brownian motion. Let X denote Lebesgue measure on the line. Show that X(s : B, = A,) = 0 a s . Exercise 27. Let X be a continuous semimartingale. Show that L! E .Fix', each t 2 0, where .Fix' denotes the smallest right continuous completed filtration generated by the process (IXt Exercise 28. Let X be a semimartingale. Show that for each fixed (w, t) the section x H L t ( X ) has compact support. Exercise 29 (Skorohod's Lemma). This result will be useful for the next two exercises. Let x be a real valued continuous function on [O, m ) such that x(0) 2 0. Show there exists a pair of functions (y, a) on [O, m) such that x = y + a, y is positive, and a is increasing, continuous, a(0) = 0, and the measure da, induced by a is carried by the set {s : y(s) = 0). Show further that the function a is given by a(t) = ~ u p , ~ , ( - x ( s V) 0). Exercise 30. Let M be a continuous local martingale with Mo = 0. Let 1 Mtl = Nt L!(M). Show that N is a continuous local martingale with ) supso(-N,). No = 0, and that L ~ ( M =

+

*Exercise 31. Let X be a continuous semimartingale, and suppose Y satisfies the equation d& = -sign(&)dXt. Show that = Xt* - Xt, where X,* = supult Xu. (Hint: Use Tanaka's formula and Exercise 29.) Exercise 32. Suppose that X and Y are continuous semimartingales such that L!(Y - X ) = 0. Show that

Exercise 33. Let X and Y be continuous semimartingales. Show that

240

Exercises for Chapter IV

Exercise 34 (Ouknine's Formula). Let X and Y be continuous semimartingales. Show that

*Exercise 35 (Emery-Perkins Theorem). Let B be standard Brownian motion and set Lt = L!(B). Let X = B cL, c E R fixed. Assume that there exists a set D c R+ x R such that (i) D is predictable for F X ; (ii) P(w : (t,w) E D ) = 1 each t > 0; and (iii) P(w : (Tt(w),w) E D ) = 0 each t > 0, where Tt = inf(s > 0 : L, > t).

+

(a) Show that Bt = lD(s)dX,. Also show that Bt t conclude that .FB= .Fx = .FBfcL, a11 c E R. (b) Let

p ,each t > 0, and

Use the fact that s H ( B ~ A TL~s, A ~ and t ) s +-+ (-BT,_, , LTt - LTt-,) have the same joint distribution to show that D satisfies (i), (ii), and (iii).

*Exercise 36. With the notation of Exercise 35, let Y = IBI +cL. Show that .FYc .FIBI,for all c. *Exercise 37. With the notation of Exercise 36, Pitman's Theorem states that Y is a Bessel process of order 3, which in turn implies that for .FY,with c = 1,

for an FY-Brownian motion P. Use this to show that .FY# .FIB[for c for all c # 1.) (Note: One can show that .FY= 3IBI

=

1.

*Exercise 38 (Barlow's Theorem). Let X solve the equation

where B is a standard Brownian motion. The above equation is said to have a unique weak solution if whenever X and Y are two solutions, and if Xt(w) E A), then p x = py. The p x is defined by px(A) = P{w : t above equation is said to have a unique strong solution if whenever both X and Y are two solutions on the probability space on which the Brownian motion B is defined, then P{w : t H Xt(w) = t H &(w)) = 1. Assume that the equation has a unique weak solution with the property that if X , Y are any two solutions defined on the same probability space, then for all t 0, L!(X - Y) = 0. Show that the equation has a unique strong solution.

-

>

Exercises for Chapter IV

241

Exercise 39. Consider the situation of Exercise 38, and suppose that a and b are bounded Borel. Further, suppose that (a(x) - ~ ( y )I) p(lx ~ - YI) for all -&du = + m for every x, y, where p : [0, m ) -+ 10, m ) is increasing with E > 0. Show that if X, Y are two solutions, then L:(X - Y) = 0.

Jt+

*Exercise 40. Let B be a standard Brownian motion and consider the equation Xt =

J(;t

sign(X,)dB,.

Show that the equation has a unique weak solution, but that if X is a solution then so too is -X. (See Exercise 38 for a definition of weak and strong solutions.) Hence, the equation does not have a unique strong solution. Finally, show that if X is a solution, then X is not adapted to the filtration F B . (Hint: Show first that Bt = l{jx,izo~dlXsj.)

Sot

Exercise 41. Let X be a sigma martingale and suppose X is bounded below. Show that X is in fact a local martingale. *Exercise 42. Show that any L6vy process which is a sigma martingale is actually a martingale (and not just a local martingale). Exercise 43. Let (Un)n21 be i.i.d. random variables with P(Ul = 1) = P(Ul = -1) = 112, and let X = - 2-nUnl{t2q,) where ( q n ) ~ >is l an $l{t2qn) and show enumeration of the rationals in (0,l). Let Ht = that X E 7f2 and H E L(X), but also that X is of finite variation and Y = H - X has infinite variation. Thus the space of finite variation processes is not closed under stochastic integration! *Exercise44. Let X n be a sequence of semimartingales on a filtered complete probability space (0,3,IF, P) satisfying the usual hypotheses. Show there exists one probability Q, which is equivalent to P , with a bounded density such that under Q each X n is a semimartingale in 7f2. Exercise 45. Let B be a standard Brownian motion and let Lx be its local time at the level x. Let A = {W : dLz(w) is singular as a measure with respect to ds)

-

where of course ds denotes Lebesgue measure on R+. Show that P(A) = 1. Conclude that almost surely the paths s L t are not absolutely continuous. (Hint: Use Theorem 69.)

Exercise 46. Let B be a standard Brownian motion and let N be a Poisson process with parameter X = 1, with B and N independent as processes. Let L be the local time process for B at level 0, and let X be the vector process Xt = (Bt,NL,). Let IF be the minimal filtration of X , completed in the usual way. (a) Show that X is a strong Markov process.

242

Exercises for Chapter IV

(b) Let T be the first jump time of X. Show that the compensator of the process Ct = l { t-> T )is the process At = LtAT. Note that this gives an example of a compensator of the first jump time of a strong Markov process which has paths that are almost surely not absolutely continuous.

**Exercise47. Let Z be a L6vy process. Show that if Z is a sigma martingale, then Z is a martingale. (Note: This exercise is closely related to Exercise 29 of Chap. I.)

Stochastic Differential Equations

1 Introduction A diffusion can be thought of as a strong Markov process (in Rn) with continuous paths. Before the development of It6's theory of stochastic integration for Brownian motion, the primary method of studying diffusions was to study their transition semigroups. This was equivalent to studying the infinitesimal generators of their semigroups, which are partial differential operators. Thus Feller's investigations of diffusions (for example) were actually investigations of partial differential equations, inspired by diffusions. The primary tool to study diffusions was Kolmogorov's differential equations and Feller's extensions of them. Such approaches did not permit an analysis of the paths of diffusions and their properties. Inspired by LBvy7sinvestigations of sample paths, It6 studied diffusions that could be represented as solutions of stochastic differential equations1 of the form

where B is a Brownian motion in Rn, a is an n x n matrix, and b is an n-vector of appropriately smooth functions to ensure the existence and uniqueness of solutions. This gives immediate intuitive meaning. If (.Ft)t20 is the underlying filtration for the Brownian motion B , then for small E > 0

where a' denotes the transpose of the matrix a. It6's differential "dB" was found to have other interpretations as well. In particular, "dB" can be thought of as "white noise" in statistical communication theory. Thus if tt is white noise at time t , Bt = &ds, an equation

Sot

More properly (but less often) called "stochastic integral equations."

244

V Stochastic Differential Equations

which can be given a rigorous meaning using the theory of generalized functions (cf., e.g., Arnold [2]). Here the Markov nature of the solutions is not as important, and coefficients that are functionals of the paths of the solutions can be considered. Finally it is now possible to consider semimartingale driving terms (or "semimartingale noise"), and to study stochastic differential equations in full generality. Since "dB" and "dt" are semimartingale differentials, they are always included in our results as special cases. While our treatment is very general, it is not always the most general available. We have at times preferred to keep proofs simple and non-technical rather than to achieve maximum generality. The study of stochastic differential equations (SDEs) driven by general semimartingales (rather than just by dB, dt, dN, and combinations thereof, where N is a Poisson process) allows one to see which properties of the solutions are due to certain special properties of Brownian motion, and which are true in general. For example, in Sect. 6 we see that the Markov nature of the solutions is due to the independence of the increments of the differentials. In Sects. 8 and 10 we see precisely how the homeomorphic and diffeomorphic nature of the flow of the solution is a consequence of path continuity. In Sect. 5 we study Fisk-Stratonovich equations which reveal that the "correction" term is due to the continuous part of the quadratic variation of the differentials. In Sect. 11 we illustrate when standard moment estimates on solutions of SDEs driven by Brownian motion and dt can be extended to solutions of SDEs driven by Lkvy processes.

2 The HP - Norms for Semimartingales We defined an 7f2 norm for semimartingales in Chap. IV as follows. If X is-a special semimartingale with Xo = 0 and canonical decomposition X = N A, then rOO

+

We now use an equivalent norm. To avoid confusion, we write H~ - instead of 7f2. Moreover we will define HP, 1 p 5 m. We begin, however, with a different norm on the space D (Te., the space of adapted cadlag processes). For a process H E D we define

<

Occasionally if H is in IL (adapted and chglhd) we write llHllsP - as well, where the meaning is clear.

2 The HP - Norms for Semimartingales

245

If A is a semimartingale with paths of finite variation, a natural definition of a norm would be llAllp = 11 IdAsl l l ~ pwhere , IdA, (w)l denotes the total variation measure on R+ induced by s H As(w). Since semimartingales do not in general have such nice paths, however, such a norm is not appropriate. Throughout this chapter, we will let Z denote a semimartingale with Zo = 0, a s . Let Z be an arbitrary semimartingale (with Zo = 0). By the BichtelerDellacherie Theorem (Theorem 43 of Chap. 111) we know there exists at least one decomposition Z = N + A , with N a local martingale and A an adapted, c&dl&gprocess, with paths of finite variation (also No = A. = 0 a s ) . For 1 p < m we set

Sow

<

Definition. Let Z be a semimartingale. For 1 < p

< m define

where the infimum is taken over all possible decompositions Z = N + A where N is a local martingale, A E ID with paths of finite variation on compacts, and A. = No = 0. The corollary of Theorem 1 below shows that this norm generalizes the

HP norm for local martingales, which has given rise t o a martingale theory analogous t o the theory of Hardy spaces in complex analysis. We do not pursue this topic (cf., e.g., Dellacherie-Meyer [46]).

Theorem 1. Let Z be a semimartingale ( 2 0 = 0). Then II[Z, Z ] ~ ~ I5 J L ~ IlZIlgp, (1 5 P 5 m).

Proof. Let Z

=M

+ A, Mo = A. = 0, be a decomposition of Z. Then [Z, Z ] Z 2 < [M,M ] Z 2 + [A,~ 1 % ~

where the equality above holds because A is a quadratic pure jump semimartingale. Taking LP norms yields 11[Z,~ ] % ~ 1 1 j p~ ( ~M 7A) and the result follows.

<

Corollary. If Z is a local martingale (Zo = 0), then llZllH~ - = 11 [Z, Z ] Z 211~p. -

V Stochastic Differential Equations

246

Proof. Since Z is a local martingale, we have that Z = Z+O is a decomposition of Z. Therefore 1 1 ~ 1 l g p5 jP(Z, 0) = IIP,~ I ~ ~ I I L P . By Theorem 1 we have 11 [Z, Z]Z21

1 5~ 11 ZII~ H ~ hence , we have equality. -

Theorem 2 is analogous to Theorem 5 of Chap. IV. For most of the proofs which follow we need only the case p = 2. Since this case does not need Burkholder's inequalities, we distinguish it from the other cases in the proof. Theorem 2. For 1 < p < m there exists a constant cp such that for any < cpllZllH.. semimartingale Z, Zo = 0, llZllsP Proof. A semimartingale Z is in D,so llZllS~makes sense. Let Z = M be a decomposition with Mo = A. = 0. Then

+

+A

+

using (a b)P I 2p-1 (ap bp). In the case p = 2 we have by Doob's maximal quadratic inequality that

For general p, 1 < p < m , we need Burkholder's inequalities, which state

for a universal constant cp which depends only on p and not on the local 2 we proved this martingale M. .For continuous local martingales and p using It6's formula in Chap. IV (Theorem 73). For general local martingales and for all finite p 2 1 see, for example, Dellacherie-Meyer [46, page 2871. Continuing, letting the constant cp vary from line to line we have

>

and taking p t h roots yields the result. Corollary. On the space of semimartingales, the HP norm is stronger than the - norm, 1 5 p < m.

aP

Theorem 3 (Emery's Inequality). Let Z be a semimartingale, H E IL, and LP + ' =9 f (I < p < m , 1 < q < m ) . Then

2 The HP - Norms for Semimartingales

247

(Sot

Proof. Let H - Z denote HsdZs)t20.Recall that we always assume Zo = 0 as., and let Z = M + A be a decomposition of Z with Mo = A0 = 0 a.s. Then H . M + H . A is a decomposition of H . Z. Hence IIH.

zllH~ 5 j T ( H .M, H . A ) .

Next recall that [ H . M, H . MI Therefore

-

=JH

z d [ ~MI,, , by Theorem 29 of Chap. 11.

where the last inequality above follows from Holder's inequality. The foregoing implies that IIH . Z l l g 5 IIHllspj,(M, A) for any such decomposition Z = M compositions yields the result.

+ A. Taking infimums over all such de-

For a process X E D and a stopping time T , recall that

A property holding locally was defmed in Chap. I, and a property holding prelocally was defined in Chap. IV. Recall that a property T is said to hold locally for a process X if x ~ ~ ~ hasc property ~ ~ > T for ~ each ) n, where Tn is a sequence of stopping times tending to oo a.s. If the process X is zero at zero (i.e., Xo = 0 as.) then the property T is said to hold prelocally if has property T for each n.

xTn-

aP

H P ) if there exist stopping Definition. A process X is locally in - (resp. times (Tn),>' increasing to co a.s. such that XTnl{Tn,o) is in Zp (resp. H P ) for each n, 1-5 p co. If Xo = 0 then X is said to be prelocall~in ZP - (rzP. HP) if xTnis in Zp (resp. lJP) for each n. -

<

While there are many semimartingales which are not locally in p,all semimartingales are prelocally in H P . The proof of Theorem 4 belowosely parallels the proof of Theorem 13of Chap. IV.

Theorem 4. Let Z be a semimartingale (Zo = 0). Then Z is prelocally in

HP, l < P < c o . -

248

V Stochastic Differential Equations

Proof.. By the Fundamental Theorem of Local Martingales (Theorem 25 of Chap. 111) and the Bichteler-Dellacherie Theorem (Theorem 43 of Chap. 111) we know that for given E > 0, Z has a decomposition Z = M+A, Mo = A. = 0 a.s., such that the jumps of the local martingale M are bounded by E . Define inductively

The sequence (Tk)kZl are stopping times increasing to m a.s. Moreover

+

is a decomposition of ZTk- . Also, since [M,M ] T ~= [M,M ] T ~ - ( A M T ~ ) ~ , we conclude

Therefore ZTk- E H m and hence it is in Bp as well, 1 5 p

< m.

am

Definition. Let Z be a semimartingale in - and let a > 0. A finite sequence of stopping times 0 = To TI 5 . . . 5 Tk is said to a-slice Z if Z = ZTkand II(Z - ~ ~ ~ ) ~ ~ +a , ~0 - il l k ~- -1. If such a sequence of stopping times exists, we say 2% a-sliceable, and we write Z E S ( a ) .

< <

< <

Theorem 5. Let Z be a semimartingale with Zo = 0 a.s.

(i) For a > 0, if Z E S ( a ) then for every stopping time T, ZT E S ( a ) and zT-E S(2a). (ii) For every a > 0, there exists a n arbitrarily large stopping time T such that ZT- E S ( a ) .

<

Proof. Since ZT- = M ~ + ( A ~ - - A M T ~ [ Tandsince , ~ ) ) ~ IIzTIlrrllzll~2 l(Zllw-, SO that (i) follo~s. always, one concludes 1IZTNext consider (ii). If semimartingales Z and Y are a-sliceable, let Tf and T ! be two sequences of stopping times respectively a-slicing Z and Y. By reordering the points T f and T: and using (i), we easily conclude that Z + Y is 8a-sliceable. Next let Z = M + A, Mo = A. = 0 as., with the local martingale M having jumps bounded by the constant P = a/24. By the preceding observation it suffices to consider M and A separately.

<

3 Existence and Uniqueness of Solutions

>

249

>

k}. For A, let To = 0, Tk+1 = inf{t TI :k;J IdA,l 2 a / 8 or J,' ldA,I Then ATk- E S(a/8) for each k, and the stopping times (Tk) increase to oo as. = inf{t Rk : [M,M]t - [MIMIRk For M , let Ro = 0, p2 or [MIMIt k). Then MRk- E H m , each k, and moreover

>

>

>

Hence

Thus for each k, MRk- E S((1 follows.

+h

) ~ ) and , since

p

=

a/24, the result

3 Existence and Uniqueness of Solutions In presenting theorems on the existence and uniqueness of solutions of stochastic differential equations, there are many choices to be made. First, we do not present the most general conditions known to be allowed; in exchange we are able to give simpler proofs. Moreover the conditions we do give are extremely general and are adequate for the vast majority of applications. For more general results the interested reader can consult Jacod [103, page 4511. Second, we consider only Lipschitz-type hypotheses and thus obtain strong solutions. There is a vast literature on weak solutions (cf., e.g., Stroock-Varadhan [220]). However, weak solutions are more natural (and simpler) when the differentials are the Wiener process and Lebesgue measure, rather than general semimartingales. A happy consequence of our approach to stochastic differential equations is that it is just as easy to prove theorems for coefficients that depend not only on the state Xt of the solution at time t (the traditional framework), but on the past history of the process X before t as well. We begin by stating a theorem whose main virtue is its simplicity. It is a trivial corollary of Theorem 7 which follotvs it. Recall that a process H is in lL if it has cAglAd paths and is adapted.

Theorem 6. Let Z be a semimartingale with Zo = 0 and let f : R+ x R x R R be such that (i) forjixed x, (t,w) H f ( t , w , x ) is in

lL; and

+

250

V Stochastic Differential Equations

(ii) for each (t,w), If (t, w, 2)- f (t, w, y)I 5 K(w)lx-y) for some finite random variable K . Let Xo be finite and F0measurable. Then the equation

admits a solution. The solution is unique and it is a semimartingale. Of course one could state such a theorem for a finite number of differentials d ~ j 1, 5 j 5 d, and for a finite system of equations. In the theory of (non-random) ordinary differential equations, coefficients are typically Lipschitz continuous, which ensures the existence and the uniqueness of a solution. In stochastic differential equations we are led to consider more general coefficients that arise, for example, in control theory. There are enough different definitions to cause some confusion, so we present all the definitions here in ascending order of generality. Note that we add, for technical reasons, the non-customary condition (ii) below to the definition of Lipschitz which follows.

Definition. A function f : R+ x Rn 4 R is Lipschitz if there exists a (finite) constant k such that (i) If(t,x) - f(t,y)l I klx - yl, each t E R+, and (ii) t H f (t,x) is right continuous with left limits, each x E Rn.

f is said to be autonomous if f (t, x) = f (x), all t 2 0. Definition. A function f : R+ x R x Rn + R is random Lipschitz if f satisfies conditions (i) and (ii) of Theorem 6. Let Dn denote the space of processes X = (X1,. . . ,Xn) where each X i E D (1 2 i 2 n).

Definition. An operator F from Dn into D1 = D is said to be process Lipschitz if for any X, Y in Dn the following two conditions are satisfied:

xT-

(i) for any stopping time T , = YT- implies F(X)T(ii) there exists an adapted process K E L such that

= F ( Y ) ~ - ,and

Definition. An operator F mapping Dn to D1 = D is functional Lipschitz if for any X, Y in Dn the following two conditions are satisfied:

xT-

(i) for any stopping time T , = yT- implies F ( x ) ~ - = F ( Y ) ~ - ,and (ii) there exists an increasing (finite) process K = (Kt)t>o - such that IF(X)t F(Y)tl 5 Kt)lX- Y(lf as., each t _> 0.

3 Existence and Uniqueness of Solutions

251

Note that if g(t,x) is a Lipschitz function, then f (t,x) = g(t-,x) is random Lipschitz. A Lipschitz, or a random Lipschitz, function induces a process Lipschitz operator, and if an operator is process Lipschitz, then it is also functional Lipschitz. An autonomous function with a bounded derivative is Lipschitz by the Mean Value Theorem. If a function f has a continuous but not bounded derivative, f will be locally Lipschitz; such functions are defined and considered in Sect. 7 of this chapter. Let A = (At)t2o be continuous and adapted. Then a linear coefficient such as f (t, w,x) = At(w)x is an example of a process Lipschitz coefficient. A functional Lipschitz operator F will typically be of the form F ( X ) = f (t, w; X,, s 5 t), where f is defined on [0,t] x R x D[O,t] for each t 0; here D[O,t] denotes the space of c&dl&gfunctions defined on [0,t]. Another example is a generalization of the coefficients introduced by It6 and Nisio [loll, namely

>

for a random signed measure p and a bounded Lipschitz function g with constant C(w). In this case, the Lipschitz process for F is given by Kt(w) = C ( ~ ) l l p ( w ) ~where ll, I I p ( ~ ) ~denotes ll the total mass of the measure p(w,du) on [O, t]. Lemmas 1 and 2 which follow are used to prove Theorem 7. We state and prove them in the one dimensional case, their generalizations to n dimensions being simple.

Lemma 1. Let 1 5 p < oo,let J E SP,let F be functional Lipschitz with F(0) = 0, and suppose sup, IKt(w)l k a.s. Let Z be a semimartingale in HM such that llZllHm Then the equation - 5

&.

<

has a solution in SP. - It is unique, and moreover

+

Proof. Define A : SP+ Spby A(X)t = Jt Jot F(X),dZ,. Then by Theorems 2 and 3 the operatoris 1/2 Lipschitz, and the fixed point theorem gives existence and uniqueness. Indeed

252

V Stochastic Differential Equations

Since J J F ( X ) ) I = S ~IIF(X) - F ( O ) I J p we , have IIXIIsp 5 which yields tLe estimate.

<

II JIIsP

-

+ ~IIXIISP, -

sP,

Lemma 2. Let 1 p < co, let J E let F be functional Lipschitz with F(0) = 0 , and suppose sup, I Kt ( w ) ] k < oo a.s. Let Z be a semimartingale such that Z E s(&). Then the equation

<

has a solution i n s p . It is unique, and moreover llXllsp C ( k ,Z ) is a constant depending only on k and Z .

< C ( k ,Z)II JllsP, where -

Proof. Let z = llZllHm and j = IIJIISP. Let 0 = T O Y T I., . ,Te be the slicing times for Z , and consider the equations, indexed by i = 0,1,2, . . .,

-

Equation ( i ) has the trivial solution X 0 since JO- = ZO- - 0 fo r a l l t , SPnorm is 0. Assume that equation ( i ) has a unique solution X i , and its and let xi = JJXillSP.Stopping next at Ti instead of Ti-, let Y i denote the unique solution ofYi = JT+ $ F ( Y ~ ) , - ~ and ~ set , y2 = I J Y i l l S-P . Since Y i = X i {AJTi F(Xi)Ti-AZTi)l[Ti,,)l we conclude that

+

+

llYillsp - 5

llxillsp - + 211Jllsp - + IIF(xi)llsPllzllg-

< xa + 2 j = xa(l

+ kxZz

+ k z ) + 2j;

hence yi

< 2j + x i ( l + k z ) .

We set for U E D, DiU = (U - u ~ ~ ) ~ Since , + I each - . solution X of equation ( i 1) satisfies XTi = Y i on [O, Ti+l),we can change the unknown I - the , equations U = Di J + $ F ( Y i U),-dDiZ,. by U = X - ( Y ~ ) ~ %to+get However since F ( Y i + 0 ) need not be 0 we define G i ( - )= F ( Y i .) - F ( Y i ) , and thus the above equation can be equivalently expressed as

+

+

+

We can now apply Lemma 1 to this equation to find that it has a unique solution in SP, - and its norm uiis majorized by

3 Existence and Uniqueness of Solutions

We conclude equation ( i dominated by (using (*))

253

+ 1) has a unique solution in SP - with norm xi+'

Next we iterate from i = 0 to

e - 1 to conclude that

Finally, since Z = ZTe-, we have seen that the equation X = J+J F(X),-dZ, has a unique solution in g,and moreover X = X' J - JTe-. Therefore

+

llXlIs. 5 X'

+ 2j1and hence C ( k ,Z ) < 2 + 8{(2:y2"2:-1}.

Theorem 7. Given a vector of semimartingales Z = ( Z 1 , . . ,Z d ) , ZO = 0 processes Ji t D, 1 5 i 5 n, and operators Fj which are functional Lipschitz (1 5 i 5 n, 1 5 j 5 d ) , the system of equations

<

(1 5 i n) has a solution in Dn, and it is unique. Moreover if (Ji)iln is a vector of semimartingales, then so is ( X i ) i s n . Proof. The proof for systems is the same as the proof for one equation provided we take F to be matrix-valued and X , J and Z to be vector-valued. Hence we give here the proof for n = d = 1. Thus we will consider the equation

Assume that maxi,j sup, ~ i " ( w 5) k tion t

xt = (

+

~ t

Jo

< m a.s. Also, by considering the equa-

Jo

F ( o ) , - ~ z ~+)

t

~(x)s-dzs,

where G ( X ) = F ( X ) - F(O), it suffices to consider the case where F ( 0 ) = 0. We also need Lemmas 1 and 2 only for p = 2. In this case c2 = 4. Let T be an arbitrarily large stopping time such that J ~ t- S2 - and such that Z T t s(&). Then by Lemma 2 there exists a unique solution in $ of t

x(T),= JT-

+

Jo

F(x(T)),-~zT-.

By the uniqueness in S2 one has, for R > T, that x ( R ) ~ -= x ( T ) ~ -and , therefore we can define a process X on R x [0, m) by X = X ( T ) on [O,T). Thus we have existence.

254

V Stochastic Differential Equations

Suppose next Y is another solution. Let S be arbitrarily large such that ( X - Y)S- is bounded, and let R = min(S,T), which can also be taken arbitrarily large. Then XR- and yR- are both solutions of

we know that xR-= yR- by the uniqueness and since ZR- E S(&), established in Lemma 2. Thus X = Y, and we have uniqueness. We have assumed that maxi,' supt Kjl'(w) k < m a s . By proving . . on [O,to], for to fixed, we can reduce the Lipsexistence and uniqueness chitz processes K;" to the random constants K::(W), which we replace with K(w) = maxi,j K:~(W).Thus without loss of generality we can assume we have a Lipschitz constant K(w) < m a.s. Then we can choose a constant c such that P ( K c ) > 0. Let Rn = { K c n), each n = 1,2,3,.. . . Define a new probability Pn by Pn(A) = P ( A n Rn)/P(Rn), and note that Pn << P. Moreover for n > m we have Pm<< Pn. From now on assume n > m in the rest of this proof. Therefore we know that all P semimartingales and all Pn semimartingales are Pmsemimartingales, and that on R, a stochastic integral calculated under Pmagrees with the the same one calculated under Pn7 by Theorem 14 of Chap. 11. Let Yn be the unique solution with respect to Pn which we know exists by the foregoing. We conclude Yn = Ym on am,a s . (dPm). Define

<

< +

<

Ca

and we have Y = Yn a.s. (dPn) on en,and hence also a.s. (dP) on Rn7each n. Since R = Ur=l(Rn \ a.s. (dP), we have that on R,:

a.s. (dP), for each n. This completes the proof. Theorem 7 can be generalized by weakening the Lipschitz assumption on the coefficients. If the coefficients are Lipschitz on compact sets, for example, in general one has unique solutions existing only up to a stopping time T ; at this time one has lim SUP^,^ lXtl = 00. Such times are called explosion times, and they can be finite or infinite. Coefficients that are Lipschitz on compact sets are called locally Lipschitz. Simple cases are treated in Sect. 7 of this chapter (cf., Theorems 38, 39, and 40), where they arise naturally in the study of flows.

3 Existence and Uniqueness of Solutions

255

We end this section with the remark that we have already met a fundamental stochastic differential equation in Chap. 11, that of the stochastic

exponential equation

There we obtained a formula for its solution (thus a fortiori establishing the existence of a solution), namely

The uniqueness of this solution is a consequence of Theorem 7, or of Theorem 6. A traditional way to show the existence and uniqueness of solutions of ordinary differential equations is the Picard iteration method. One might well wonder if Picard-type iterations converge in the case of stochastic differential equations. As it turns out, the following theorem is quite useful.

Theorem 8. Let the hypotheses of Theorem 7 be satisfied, and in addition let (XO)i= H a be processes in D (1 5 i 5 n). Define inductively

and let X be the solution of

Then Xm converges to X in ucp. Xm = Proof. We give the proof for d = n = 1. It is easy to see that if lirn,,, X prelocally in then lirn,,, X m = X in ucp; in any event this is proved X m = X prelocally in Theorem 12 insect. 4. Thus we will show that lirn,, in - We first assume supt Kt 5 a < m a s . Without loss of generality we can assume Z t S ( a ) , with a = &, and that J t - Let 0 = To 5 TI 5 . . . 5 Tkbe the stopping times that a-slice Z . Then (Xm)T1- and xT1are in - by Lemma 1. Then

s2,

s2.

s2

by Theorems 2 and 3. Therefore

s2.

V Stochastic Differential Equations

256

so that lim,, Since

II(Xrn+'-

XK+'

= 0. We next analyze the jump at =X z t l

TI.

+ F(X,)T~ -AZTl,

we have

where z

=

llZllwm < 00. Therefore -

Next suppose we know that

where y = 11 (X1 - ~ ) ~ lls2. k -Then -

where

Ze+l = ( 2 - ~ ~ e ) ~ e +Therefore l-. by iterating on e, 0 5 e I k ,

Note that the above expression tends to 0 as m tends to m. Therefore X m Xm = tends to X prelocally in - by a (finite) induction, and hence lim,, X in ucp. It remains to remove the assumption that supt Kt I a < m a.s. Fix a t < m; we will show ucp on [O,t]. Since t is arbitrary, this will imply the result. As we did at the end of the proof of Theorem 7, let c > 0 be such that P ( K t 5 c) > 0. Define en = {w : Kt(w) 5 c n), and Pn(A) = P(Als2,). Then Pn << P , and under Pn,limm,,Xm = X in ucp on [O,t]. For E > 0, choose N such that n N implies P(Rk) < E. Then

z2

+

>

hence lirn,,, P ( ( X m - X); > 6) must be zero.

I E.

Since E > 0 was arbitrary, the limit

4 Stability of Stochastic Differential Equations

257

4 Stability of Stochastic Differential Equations Since one is never exactly sure of the accuracy of a proposed model, it is important to know how robust the model is. That is, if one perturbs the model a bit, how large are the resulting changes? Stochastic differential equations are stable with respect t o perturbations of the coefficients, or of the initial conditions. Perturbations of the differentials, however, are a more delicate matter. One must perturb the differentials in the right way to have stability. Not surprisingly, an perturbation is the right kind of perturbation. In this section we will be concerned with equations of the form

Xt=Jt+

I"

F(X),-dZ,,

(*I

where J n , J are in Dl Zn, Z are semimartingales, and F n , F are functional Lipschitz, with Lipschitz processes Kn, K , respectively. We will assume that the Lipschitz processes Kn, K are each uniformly bounded by the same constant, and that the semimartingale differentials Zn, Z are always zero at 0 (that is, Zg = O a s . , n 2 1, and Zo = O a.s.). For simplicity we state and prove the theorems in this section for one equation (rather than for finite systems), with one semimartingale driving term (rather than a finite number), and for p = 2. The generalizations are obvious, and the proofs are exactly the same except for notation. We say a functional Lipschitz operator F is bounded if for all H E Dl there exists a non-random constant c < m such that F ( H ) * < c.

Theorem 9. Let J , Jn t D; Z, Z n be semimartingales; and F , Fn be functional Lipschitz with constants K , Kn, respectively. Assume that

s2

(i) J , Jn are in 2' (resp. Hz) and limn,, Jn= J in (resp. H ~ ) ; (ii) Fn are all bo
z2

Then limn,, X n = X in (resp. in H2), where X n is the solution of (*n) and X is the solution of (*T. Proof. We use the notation He Zt to denote H,dZ,, and He Z to denote the process (H .Zt)t20. We begin by supposing that J , (Jn)n21 are in 8' - and Jn converges to J in 2'. - Then S ( a ) is defined in Sect. 2 on page 248.

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V Stochastic Differential Equations

Let Yn = ( F ( X ) - F n ( X ) ) - . Z

For U

ED

+ F n ( X n ) . ( Z - Zn). Then

define Gn by Gn (U) = Fn( X ) - Fn( X - U).

Then Gn(U) is functional Lipschitz with constant a and Gn(0) = 0. Take U = X - X n , and (**) becomes

By Lemma 2 preceding Theorem 7 we have

Since C(a, Z) is independent of n and limn,, it suffices t o show limn,, llYn llsz= 0. But -

11 J

-

JnIISz - = 0 by hypothesis, -

by Theorem 2 and Emery's inequality (Theorem 3). Since llZllHm hypothesis and since lim ( ( F ( X )- Fn(X)((sz = lim (12- Zn (IHz n-03

n-03

< 03 by

= 0,

again by hypothesis, we are done. E ~ then , Note that if we knew J n , J E - and that Jn converged to J in -

a2

We have seen already that limn,, IIX -Xnllsz- = 0, hence it suffices to show - = 0. Proceeding as in (* * *)we obtain the result. limn,, llYnllHz -

Comment. Condition (ii) in Theorem 9 seems very strong; it is the perturbation of the semimartingale differentials that make it necessary. Indeed, the hypothesis cannot be relaxed in general, as the following example shows. We take f2 = [O, 11, P to be Lebesgue measure on [O,l], and (Ft)t20 equal to 3 , the Lebesgue sets on [O, 11. Let $(t) = min(t, 1), t 2 0. Let fn(w) 2 0 and set Zp(w) = $(t)fn(w), w E [O, 11, and Zt(w) = $(t)f (w).

4 Stability of Stochastic Differential Equations

Let F n ( X ) = F ( X ) = X , and finally let JF = Jt = 1, all t equations ( * n ) and (*) become respectively

259

2 0. Thus the

which are elementary continuous exponential equations and have solutions

We can choose f n such that limn,, E { f ; } = 0 but limn,, E { f ; } # 0 for p > 2. Then the Zn converge to 0 in H2 but X n does not converge to X = 1 (since f = 0) in 9, for any p 2 l.Tndeed, limn,, E { f ; ) # 0 for p > 2 implies limn,, ~ { e ~ f#" 1} for any t > 0. The next result does not require that the coeficients be bounded, because there is only one, fixed, semimartingale differential. Theorem 10, 11, and 13 all have H2 - as well as - versions as in Theorem 9, but we state and prove only the S2 - versions.

s2

Theorem 10. Let J , Jn E D; Z be a semimartingale; F , Fn be jknctional Lipschitz with constants K , K n , respectively; and let X n , X be the unique solutions of equations ( * n ) and (*), respectively. Assume that

- and limn,, Jn = J in S2; (i) J n , J are in s2 (ii) lim F n ( X ) = F ( X ) in s-2 , where X is the solution of (*); and n-03 (iii) max(supn Kn, K ) 5 a < oa a.s. for a non-random constant a, and Z

E

S(&).

s2

Then limn,, X n = X in - where X n is the solution of (*n) and X is the solution of (*). Proof. Let X n and X be the solutions of equations (*n) and (*), respectively. Then

We let Yn = ( F ( X ) - F n ( X ) ) - . Z, and we define a new functional Lipschitz operator Gn by Gn(U) = F n ( X ) - F n ( X - U). Then Gn(0) = 0. If we set U

= X - X n , we

obtain the equation

260

V Stochastic Differential Equations

a m , by Emery's inequality (Theorem 3) we have Yn + 0 in a-2 , Since Z E and hence also in - (Theorem 2). In particular llYn llsz - < oa, and therefore by Lemma 2 in Sect. 3 we have

s2

where C ( a ,Z) is independent of n, and where the right side tends to zero as n + oa. Since U = X - X n , we are done. We now wish to localize the results of Theorems 9 and 10 so that they hold for general semimartingales and exogenous processes J n , J. We first need a definition, which is consistent with our previous definitions of properties holding locally and prelocally (defined in Chap. IV, Sect. 2). Definition. Processes M n are said t o converge locally (resp. prelocally) (resp. HP) and if there exists a i n SP(resp. H P ) to M if M n , M are in II(Mn sequence of stopping times Tkincreasing t r o a a.s. such that limn,, ~ ) ~ ~ l= ~ 0 (resp. ~ ~limn,,, ~II(Mn ~ -l M l ) ~~ ~ -~= I I0)~ for each k 2 1 (resp. - replaced by HP). -

sP

zP

T h e o r e m 11. Let J , Jn E D; Z be a semimartingale (Zo = 0); and F, Fn be jknctional Lipschitz with Lipschitz processes K , Kn , respectively. Let X n , X be solutions respectively of

Assume that

s2;

(i) Jn converge to J prelocally i n (ii) F n ( X ) converges to F ( X ) preGcally in - where X is the solution of (*); and (iii) max(supn Kn, K ) 5 a < oa a.s. (a not random).

s2

s2

Then limn,, Xn = X prelocally i n - where X n is the solution of (*n) and X is the solution of (*). Proof. By stopping at T - for an arbitrarily large stopping time T we can assume without loss of generality that Z E S(&) by Theorem 5, and that

Jn converges to J in S~and F ( X n ) converges to F ( X ) in s-2 , by hypothesis. Next we need only toapply Theorem 10. We can recast Theorem 11 in terms of convergence in ucp (uniform convergence on compacts, in probability), which we introduced in Sect. 4 of Chap. I1 in order to develop the stochastic integral.

4 Stability of Stochastic Differential Equations

261

Corollary. Let Jn,J E D; Z be a semimartingale (Zo = 0); and F , Fn be functional Lipschitz with Lipschitz processes K , K n , respectively. Let X, X n be as in Theorem 11. Assume that (i) Jn converges to J in ucp, (ii) F n ( X ) converges to F ( X ) in ucp, and (iii) max(supn K n , K ) 5 a < oo a.s. (a not random). Then lim X n = X in ucp. n-+w

Proof. Recall that convergence in ucp is metrizable; let d denote a distance compatible with it. If Xn does not converge t o 0 in ucp, we can find a subsequence n' such that inf,~d ( x n ' , 0) > 0. Therefore no sub-subsequence (x"") can converge to 0 in ucp, and hence xn"cannot converge to 0 prelocally in s2as well. Therefore to establish the result we need to show only that for any subsequence n', there exists a further subsequence n" such that x""converges prelocally to 0 in s-2 . This is the content of Theorem 12 which follows, so the proof is complete.

Theorem 12. Let H n , H E D. For Hn to converge to H in ucp it is necessary H ~ =' H , and suficient that there exist a subsequence n' such that lirn,~,, prelocally in s-2 . Proof. We first show the necessity. Without loss of generality, we assume that H = 0. We construct by iteration a decreasing sequence of subsets (Nk) of N = {1,2,3,. . . ), such that lim

n-00

sup IHrl = 0 a.s.

n E N k O<s
By Cantor's diagonalization procedure we can find an infinite subset N' of N such that lim sup IHrl = 0 a.s., n-03 n ~ N 1O<s
each integer k > 0. By replacing N with N' we can assume without loss of generality that Hn tends to 0 uniformly on compacts, almost surely. We next define

Sn = inf Tm. m2n Then Tn and Sn are stopping times and the Sn increase to oo a.s. Indeed, IHr(w)l < 1. for each k there exists N(w) such that for n 2 N(w), supols
>

>

262

V Stochastic Differential Equations

Then

which tends t o 0 a.s. Moreover, when m (Lm)*5

SUP

Ols<S,

2 n,

lHFl 5

sup lHFl 5 1. O<s
Hence by the Dominated Convergence Theorem we have that (Lm)*tends t o 0 in L2, which implies the result. For the sufficiency, suppose that Hn does not converge to H in ucp. Then there exist to > 0, 6 > 0, E > 0, and a subsequence n' such that P{(H~'H ) & > 6) > E for each n'. Then I I ( H ~ H):ol(Lz ' 2 6& for all n'. Let T~ tend t o oa a.s. such that (H"' - H ) ~ ~ tends - to 0 in s-2 , each k. Then there exists K > 0 such that p(Tk < to) <

Lim

nl+w

<

for all k > K. Hence

b f i II(Hn - H)t,ll~z5 lim 1l(Hn' - ~ ) ~ ~ -+l -.l ~ z n'+w = 2

We conclude 6& ucp.

5

y ,a contradiction. Whence Hn converges t o H in

Recall that we have stated and proven our theorems for the simplest case of one equation (rather than finite systems) and one semimartingale driving term (rather than a finite number). The extensions t o systems and several driving terms is simple and essentially only an exercise in notation. We leave this t o the reader. An interesting consequence of the preceding results is that prelocal and prelocal convergence are not topological in the usual sense. If they were, then onewould have that a sequence converged to zero if and only if every subsequence had a sub-subsequence that converged t o zero. To see that this is not the case for for instance, consider the example given in the comment following heo or ern 9. In this situation, the solutions X n converge t o X in ucp.By Theorem 12, this implies the existence of a subsequence n' such that limn,,, x"' = X , prelocally in s 2 . However we saw in the comment that X n does not converge t o X in - It is still a priori possible that X n converges t o X prelocally in z 2 , however. In the framework of the example a stopping time is simply a noi-negative random variable. Thus our counterexample is complete with the following real analysis result (see Protter [199, page 3441 for a proof). There exist non-negative functions f n on [ O , l ] such 1 that limn,, f , ( ~ ) ~ d= x 0 and limsup,,, SA(fn(x))pdx = +oa for all p > 2 and all Lebesgue sets A with strictly positive Lebesgue measure. In conclusion, this counterexample gives a sequence of semimartingales Xn such that every subsequence has a sub-subsequence converging prelocally in SP, but the sequence itself does not converge prelocally in - (1 5 p < oa).

zP

s2

s2.

So

zP,

4 Stability of Stochastic Differential Equations

263

Finally, we observe that such non-topological convergence is not as unusual as one might think at first. Indeed, let X n be random variables which converge to zero in probability but not a.s. Then every subsequence has a subsubsequence which converges to zero a.s., and thus almost sure convergence is also not topological in the usual sense. As an example of Theorem 10, let us consider the equations

where W is a standard Wiener process. If (Jn- J),* converges to 0 in L2, each t > 0, and if Fn,Gn, F , G are all functional Lipschitz with constant K < oa and are such that ( F n ( X ) - F(X))F and (Gn(X) - G(X))t converge to 0 in L2, each t > 0, then ( X n - X); converges to 0 in L2 as well, each t > 0. Note that we require only that F n ( X ) and Gn(X) converge respectively to F ( X ) and G(X) for the one X that is the solution, and not for all processes in D. One can weaken the hypothesis of Theorem 9 and still let the differentials vary, provided the coefficients stay bounded, as the next theorem shows. Theorem 13. Let J n , J E D; Z, Zn be semimartingales (Zz = Zo = 0 a.s.); and F , Fn be jknctional Lipschitz with Lipschitz processes K , K n , respectively. Let X n , X be solutions of (*n)and (*), respectively. Assume that

s2;

(i) Jn converges to J prelocally in (ii) F n ( X ) converges to F ( X ) preGcally in s 2 , and the coeficients F n , F are all bounded by c < oa; (iii) Zn converges to Z prelocally in H ~ and ; (iv) max(supn Kn, K ) 5 a < oa a.s. not random). Then lim X n n-+m

=X

prelocally in s-2 .

Proof. By stopping at T- for an arbitrarily large stopping time T we can assume without loss that Z E s(&) by Theorem 5, and that Jn converges to F ( X ) , and Zn converges to Z in If2, to J in s 2 , F n ( X ) converges in all by hypothesis. We then invoke~heorem9, and the proof is complete.

s2

The assumptions of prelocal convergence are a bit awkward. This type of convergence, however, leads to a topology on the space of semimartingales which is the natural topology for convergence of semimartingale differentials, just as ucp is the natural topology for processes related to stochastic integration. This is exhibited in Theorem 15. Before defining a topology on the space of semimartingales, let us recall that we can define a "distance" on D by setting, for Y, Z E D

264

V Stochastic Differential Equations

r(Y)

2TnE{1 A sup

=

O
n>O

IKl),

and d(Y, Z) = r ( Y - 2 ) . This distance is compatible with uniform convergence on compacts in probability, and it was previously defined in Sect. 4 of Chap. 11. Using stochastic integration we can define, for a semimartingale X , i ( X ) = sup r (H . X ) Iff151

where the supremum is taken over all predictable processes bounded by one. The semimartingale topology is then defined by the distance d ( ~Y), = ? ( X - Y). The semimartingale topology can be shown to make the space of semimartingales a topological vector space which is complete. Furthermore, the following theorem relates the semimartingale topology to convergence in HP. For its proof and a general treatment of the semimartingle topology, see ~ m e [66] r ~ or Protter [201].

Theorem 14. Let 1 5 p < oa, let X n be a sequence of semimartingales, and let X be a semimartingale.

(i) If X n converges to X in the semimartingale topology, then there exists a subsequence which converges prelocally in gP. (zi) If X n converges to X prelocally in HP, - %en it converges to X in the semimartingale topology. In Chap. IV we established the equivalence of the norms llXllHz - and SUJII ~ I\(He < ~ X(IS~ in the corollary to Theorem 24 in Sect. 2 of that chapter. Given this result,Theorem 14 can be seen as a uniform version of Theorem 12. We are now able once again to recast a result in terms of ucp convergence. Theorem 13 has the following corollary.

Corollary. Let J n , J E D; Zn, Z be semimartingales (Zg = Zo = 0); and F n , F be functional Lipschitz with Lipschitz processes K , Kn, respectively. Let Xn, X be solutions of (*n)and (*), respectively. Assume that (i) Jn converges to J in u p , (ii) F n ( X ) converges to F ( X ) in ucp where X is the solution of (*), and moreover all the coefficients Fn are bounded by a random c < oa, (iii) Zn converges to Z in the semimartingale topology, and (iv) max(supn Kn, K ) 5 a < oa a.s. Then lim X n = X in ucp. n+m

Proof. Since Zn converges to Z in the semimartingale topology, by Theorem 14 there exists a subsequence n' such that Zn' converges to Z prelocally in H ~Then . by passing to further subsequences if necessary, by Theorem 12 wemay assume without loss that Jn converges to J and F n ( X ) converges

4 Stability of Stochastic Differential Equations

265

to F ( X ) both prelocally in z 2 , where X is the solution of (*). Therefore, by for this subsequence. We Theorem 13, X n converges to X prelocally in have shown that for the sequence (Xn) there isalways a subsequence that converges prelocally in s-2 . We conclude by Theorem 12 that Xn converges to X in ucp.

a2

The next theorem extends Theorem 9 and the preceding corollary by relaxing the hypotheses on convergence and especially the hypothesis that all the coefficients be bounded. T h e o r e m 15. Let J n , J E D; Zn, Z be semimartingales (22 = Zo = 0); and F n , F be jknctional Lipschitz with Lipschitz process K , the same for all n. Let X n , X be solutions respectively of

Assume that (i) Jn converges to J in ucp; (22) F n ( X ) converges to F ( X ) in ucp, where X is the solution of (*); and (iii) Zn converges to Z in the semimartingale topology. Then X n converges to X in ucp. Proof. First we assume that sup, Kt(w) 5 a < m. We remove this hypothesis at the end of the proof. By Theorem 12, it suffices to show that there exists a subsequence n' such that xn'converges to X prelocally in s2.Then by Theorem 12 we can assume with loss of generality, by passing to asubsequence if necessary, that Jn converges to J and F n ( X ) converges to F ( X ) both prelocally in Moreover by Theorem 14 we can assume without loss, again by passing t o a subsequence if necessary, that Z n converges to Z prelocally in B ~ ,and that Z E S(-&-). Thus all the hypotheses of Theorem 13 are satisfied except one. We do not assume that the coefficients Fn are bounded. However by pre-stopping we can assume without loss that IF(X)I is uniformly bounded by a constant c < m. Let us introduce t r u n c a t i o n o p e r a t o r s Tx defined (for x 2 0) by

s2.

Tx(Y) = min(x, sup(-x, Y)). Then Tx is functional Lipschitz with Lipschitz constant 1, for each x Consider the equations

2

0.

266

V Stochastic Differential Equations

Then, by Theorem 13, Yn converges to X prelocally in s 2 . By passing to yet another subsequence, if necessary, we may assume that F ( x ) tends to F ( X ) and Yn tends to X uniformly on compacts almost surely. Next we define

The stopping times Sk increase a.s. to m. By stopping at sk-, we have for n 2 k that (Yn - X ) * and ( F n ( X ) - F ( X ) ) * are a.s. bounded by 1. (Note that stopping at sk-changes Z to being in S(&) instead of S ( & ) by Theorem 5.) Observe that

whence ( T ~ + ' + ' F ~ ) ( Y ~=) Fn(Yn). We conclude that, for an arbitrarily large stopping time R, with Jn and Zn stopped at R , Yn is a solution of

which is equation (*n). By the uniqueness of solutions we deduce Yn = X n on [0, R). Since Yn converges to X prelocally in - we thus conclude X n converges to X prelocally in s2. It remains only to removethe hypothesis that sup, Kt(w) 5 a < m. Since we are dealing with local convergence, it suffices to consider supso Ks 5 K t , > 0, and for a fixed t. Since Kt < m a.s., let a > 0 be such that P ( K t define am= {w: Kt(w) 5 a + m}.

z2,

< a)

Then amincrease to a.s. and as in the proof of Theorem 7 we define Pm by Pm(A) = P(A1am), and define = 3t/n,, the trace of 3 t on a m . Then Pm << P , so that by Lemma 2 preceding Theorem 25 in Chap. IV, if Zn converges to Z prelocally in H ~ ( P ) , then Zn converges to Z prelocally in H~(P,) as well. Therefore by the first part of this proof, Xn converges to X in ucp under Pm, each m 2 1. Choose E > 0 and m so large that P(R&) < E . Then

+P(aL) 5 lirn Pm((Xn- X),* > 6) + E n-+w

lim P ( ( X n - X),* > 6) 5 lim Pm((Xn - X); > 6) 71-00

71-03

and since E > 0 was arbitrary, we conclude that X n converges to X in ucp on [0,t ] .Finally since t was arbitrary, we conclude X n converges t o X in ucp.

4 Stability of Stochastic Differential Equations

267

Another important topic is how t o approximate solutions by difference solutions. Our preceding convergence results yield two consequences (Theorem 16 and its corollary). The next lemma is a type of Dominated Convergence Theorem for stochastic integrals and it is used in the proof of Theorem 16.

Lemma (Dominated Convergence Theorem). Let p, q, r be given such that 1P. + 41' . = $, where 1 < r < oo. Let Z be a semimartingale in H q , and let Hn E SPsuch that IHn(5 Y E - all n 2 1. Suppose limn,, H E ( w ) = 0, all ( t , Then

w).

zP, F

Proof. Since Z E H q , there exists a decomposition of Z, Z = N

that

+ A, such

1

00

j q ( N 7 A )= II[N. ~

1 +% ~ IdAsill~q< m.

Let C n be the random variable given by

The hypothesis that 1 Hn1 5 Y implies

However

Thus Cn is dominated by a random variable in Lr and hence by the Dominated Convergence Theorem Cn tends t o 0 in LT. We let an denote a sequence of random partitions tending t o the identity.3 Recall that for a process Y and a random partition a = (0 = To TI 5 . . . 2

<

Note that if Y is adapted, c&dl&g(i.e, Y E D), then (Y,"),>o is left continuous with right limits (and adapted, of course). It is convenient to have a version of Yo E D, occasionally, so we define

Random partitions tending to the identity are defined in Chap. 11, preceding Theorem 21.

268

V Stochastic Differential Equations

Theorem 16. Let J E z 2 , let F be process Lipschitz with Lipschitz process K 5 a < m a.s. and F(0) E 5'. - Let Z be a semimartingale in S ( & ) , and let X ( a ) be the solution of

for a random partition a . If a, is sequence of random partitions tending to the identity, then X ( a n ) tends to X in z-2 , where X is the solution of (*) of Theorem 15. Proof. For the random partition a, (n fixed), define an operator Gn on D by

Note that G n ( H ) E D for each H E D and that G n ( H ) - = F(HUnC)rm. into itself, as Then G n is functional Lipschitz with constant K and sends the reader can easily verify. Since F(0) E s 2 , so also are G ~ ( O )E n 2 1, and an argument analogous to the proof ofTheorem 10 (though a bit simpler) shows that it suffices t o show that J; Gn(X),-dZ, converges t o J,' F(X),-dZ, in - where X is the solution of

z2

s',

z2,

Towards this end, fix (t,w) with t > 0, and choose E > 0. Then there exists 6 > 0 such that IX,(w) - Xt-(w)l < E for all u E [t - 26, t). If mesh(a) < 6, then also IXzC(w) - X,(w)l < 2~ for all u E [t - 6, t). This then implies that IF(XuC)(w)- F(X)(w)l < 2 a ~ Therefore . ( F ( X u n + )- F(X)),O" (w) tends t o 0 as mesh(a,) tends t o 0. Since, on (0, m ) , lim

mesh(on)+O

F( X )On

= F ( X )-

,

we conclude that lim mesh(u,)+O

F ( x " ~ +=)F"( ~ X)-,

where convergence is pointwise in (t, w). Thus lim

mesh(on)+O

Gn(X)-

= F(X)-.

However we also know that

which is in 5' and is independent of a n . Therefore using the preceding lemma, the omi in zed Convergence Theorem for stochastic integrals, we obtain the ' of Gn(X),-dZ, t o F(X)s-dZs, and the proof is comconvergence in 3 plete.

Ji

4 Stability of Stochastic Differential Equations

269

Remark. In Theorem 16 and its corollary which follows, we have assumed that F is process Lipschitz, and not functional Lipschitz. Indeed, Theorem 16 is not true in general for functional Lipschitz coefficients. Let Jt = lit>ll, Zt = t A 2, and F ( Y ) = Yll{t211. Then X , the solution of (*) of ~ h e o r e k15 is given by Xt = (t A 2)1{,>1), but if a is any random partition such that i 0 for t 5 1, and therefore F(X(a)"f) = 0, Tk # 1 a.s., then ( ~ ( a ) " f ) = (Here X ( a ) denotes the solution to equation (*a) and X(a)t = Jt = lit>ll. of Theorem 16.) Corollary. Let J t D; F be process Lipschitz; Z be a semimartingale; and let a, be a sequence of random partitions tending t o the identity. Then lim X(a,) = X

n-00

in ucp

where X(a,) is the solution of (*a) and X is the solution of (*), as in Theorem 16. Proof. First assume K 5 a < co, a s . Fix t > 0 and E > 0. By Theorem 5 we can find a stopping time T such that ZT- t s(&), and P ( T < t) < E. Thus without loss of generality we can assume that Z t s(&).

sk= sk= co

By letting

inf{t 2 0 : 1 Jtl > k), we have that skis a stopping time, and limk,, a.s. By now stopping at sk-we have that J is bounded, hence also in and Z t S(&). An analogous argument gives us that F(0) can be assumed - as well; hence Z t S(&). We now can apply bounded (and hence in 2') Theorem 16 to obtain the result. To remove the assumption that K 5 a < co a s . , we need only apply an argument like the one used at the end of the proofs of Theorems 7, 8 and 15.

s',

Theorem 16 and its corollary give us a way to approximate the solution of a general stochastic differential equation with finite differences. Indeed, let X be the solution of

xt = ~t +

1 t

~(~)s-dzs

(*I

where Z is a semimartingale and F is process Lipschitz. For each random partition a, = (0 = Ton 5 T r 5 - . . T;'-), we see that the random variables X(an)T2 verify the relations (writing a for a,, X for X(a,), Tk for Tp)

<

Then the solution of the finite difference equation above converges to the solution of (*), under the appropriate hypotheses. As an example we give a means to approximate the stochastic exponential.

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V Stochastic Differential Equations

Theorem 17. Let Z be a semimartingale and let X = &(Z),the stochastic exponential of Z. That is, X is the solution of

Let a, be a sequence of random partitions tending to the identity. Let

Then lim Xn = X in ucp. n-m

Proof. Let Yn be the solution of

equation (*a) of Theorem 16. By the corollary of Theorem 16 we know that Yn converges to X = &(Z) in ucp. Thus it suffices to show Yn = Xn. Let a, = (0 = Ton 5 TF I - .. I TFn). On ( T r ,T,IL+ we have

Inducting on i down t o 0 we have

for Tp < t TGl. Since ZT;+l - ZT? = 0 for all j > i when Tp we have that Yn = Xn, and the theorem is proved.


5 Fisk-Stratonovich Integrals and Differential Equations In this section we extend the notion of the Fisk-Stratonovich integral given in Chap. 11, Sect. 7, and we develop a theory of stochastic differential equations with Fisk-Stratonovich differentials. We begin with some results on the quadratic variation of stochastic processes.

Definition. Let H, J be adapted, c&dl&gprocesses. The quadratic covariation process of H, J denoted [H,J] = ([H,JIt)t>o, if it exists, is defined to be the adapted, c&dl&gprocess of finite variation-on compacts, such that for any sequence an of random partitions tending to the identity,

5 Fisk-Stratonovich Integrals and Differential Equations

lim Son(H, J ) = lim HoJo+ C(H~."+

n-+m

n-m

=

-

271

(J~;+I - J ~ ? )

i

[H,Jl

with convergence in ucp, where a, is the sequence 0 = Ton 5 Tr 5 . . . 5 Trn. A process H in D is said t o have finite quadratic variation if [H,HIt exists and is finite a.s., each t 2 0. If H, J, and H+ J in D have finite quadratic variation, then the polarization identity holds:

For X a semimartingale, in Chap. I1 we defined the quadratic variation of X using the stochastic integral. However Theorem 22 of Chap. I1 shows every semimartingale is of finite quadratic variation and that the two definitions are consistent.

Notation. For H of finite quadratic variation we let [H,HICdenote the continuous part of the (non-decreasing) paths of [H,HI. Thus,

where A[H, HIt

=

[H,H]t

-

[H,HIt-, the jump at t.

The next definition extends the definition of the Fisk-Stratonovich integral given in Chap. 11, Sect. 7.

Definition. Let H E D, X be a semimartingale, and assume [H,X ] exists. The Fisk-Stratonovich integral of H with respect to X , denoted Hs- o

Sot

dX,, is defined to be

To consider properly general Fisk-Stratonovich differential equations, we need a generalization of It6's formulas (Theorems 32 and 33 of Chap. 11). Since ItB's formula is proved in detail there, we only sketch the proof of this generalization.

Theorem 18 (Generalized ItB's Formula). Let X = (X1,. . . ,X n ) be an n-tuple of semimartingales, and let f : R+ x R x Rn --+ R be such that (i) there exists an adapted F V process A and a function g such that

(s, w) H g(s, W, X) is an adapted, jointly measurable process for each x, t and supxEK(g(s,W, X)1 ldAs 1 < co a.s. for compact sets K.

So

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V Stochastic DifferentialEquations

(ii) the function g of (i) is C2 in x uniformly i n s on compacts. That is,

a.s., where rt : R x R+ --+ R+ is an increasing function with limulo rt ( u ) = 0 a.s., provided x ranges through a compact set (rt depends on the compact set chosen). (iii) the partial derivatives f x i , f x i x J , 1 I i, j 5 n all exist and are continuous, and moreover

Then

<

Proof. W e sketch the proof for n = 1. W e have, letting 0 = to 5 tl 5 . . . tm = t be a partition o f [ O , t ] , and assuming temporarily I k for all s 5 t , k a constant,

1x1

sf:+'

g(u,w , X ~ ~ + ~The ) ~integrand A ~ . is not Consider first the term adapted, however one can interpret this integral as a path-by-path Stieltjes

5 Fisk-Stratonovich Integrals and Differential Equations

273

integral since A is an FV process. Expanding the integrand for fixed (u, w) by the Mean Value Theorem yields

where

xuis in between Xu and Xtk+,

Therefore

and since A is of finite variation and X is right continuous, the second sum tends to zero as the mesh of the partitions tends t o zero. Therefore letting rn denote a sequence of partitions of [0,t] with limn,, mesh(^,) = 0,

Next consider the second term on the right side of (*), namely

Here we proceed analogously t o the proof of Theorem 32 of Chap. 11. Given E > 0, t > 0, let A(&,t) be a set of jumps of X that has a.s. a finite number of times s, and let B = B(E,t) be such that CsEB(AXs)2 E ~ where , A UB exhaust the jumps of X on (0, t]. Then

<

By Taylor's formula, and letting A k X denote Xtk+,- X t k ,

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V Stochastic Differential Equations

By Theorems 21 and 30 of Chap. 11, the first sums on the right above cont verge in ucp to fi(s-,w, X,-)dX, and fz.x(s-, w,x,-)d[X, XI,, respectively. The third sum converges a.s. to

&

By condition (ii) on the function g, we have lim sup n

C

R(tk, w, Xt,, Xt,,.)

5 rt (w, &+)IX,Xlt

tkEnn,B

where rt(w,&+)= limsup rt(w,6). 6 1 ~

Next we let E tend to 0; then rt(w,&+)[X,XIttends to 0 a.s., and finally combining the two series indexed by A we see that

tends to the series

which is easily seen to be an absolutely convergent series (cf., the proof of The+fxx(s-, w, x,-) (AX,)' into orem 32 of Chap. 11). Incorporating - Co<s
$

f r x ( ~ - , ~ , X s - ) d [ X , X ]yields S the term !j f,,(s-, w,X,-)d[X,X]:. We further note that f (s,w,Xs) - f(s-,w,X,) = g(s,w,X,)AA,, and since C o < s < tg(s, W, X,)AA, is absolutely convergent, the theorem is proved. The assumption that IX,( 5 k for s 5 t is removed as it was in Theorem 32 of Chap. 11. 5

5 Fisk-Stratonovich Integrals and Differential Equations

275

Note that a consequence of Theorem 18 is that for f satisfying the hypotheses, we have f (t, ., X t ) is a semimartingale when X = ( X 1 , . . . ,X n ) is an n-tuple of semimartingales. Also, an important special case is when the process A, = s; in this case the hypotheses partly reduce to assuming f is absolutely continuous in t. The next theorem allows an improvement of the Fisk-Stratonovich change of variables formula given in Chap. I1 (Theorem 34).

Theorem 19. Let X = (X1,. . . ,x d ) be a vector of semimartingales and let f : R+ x R x Rd --+ R be such that (i) there exists a n adapted FV process A and a function g such that

where (s, w) ++ g(s, W , X) is an adapted and jointly measurable process for each x. t (ii) for each compact set K, So sup,,, Ig(s,w,x)IldA,I < co a.s. t (iii) fxi exists and is continuous in x and fxi (t, w, x ) = So gx% (s, w, x)dA,. Let

& = f ( t , w , x l ,..., xd).

Then Y is an adapted process of finite quadratic variation, and moreover

Proof. First assume that for fixed (t, w) the function f and all its first partials are bounded functions of x . Then by optional stopping at times T- we can assume without loss of generality that f and all its first partials (in x ) are bounded functions. Let fk be a sequence of functions on R+ x R x Rd which are C2 on Rd such that fk and its first partials converge uniformly on Rd respectively t o f and its corresponding first partials. Moreover we can take all the f k bounded and Lipschitz continuous with Lipschitz constant c, independent of k. For simplicity take d = 1. Let an be a sequence of random partitions tending t o the identity. We write, for a process 2 ,

where an = { 0 = T,n 5 T;2 5

. . . 5 Tn).

Since fr, and their first partials converge t o f and its first partials uniformly, we have fk - fe is Lipschitz with constant &kt,which tends t o 0 as k, C tend t o co.Therefore, letting T denote an for a given n and writing f k ( X ) for f k ( t ~ w , X twe ) ~ have

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V Stochastic Differential Equations

by the Lipschitz properties. Since fk(X) and fe(X) are semimartingales, we know (by Theorem 22 of Chap. 11) that So, (fk(X)) and Sun(fe(X)) converge in ucp respectively to [fk(X),fk(X)] and [fe(X), fe(X)]. By restricting our attention to an interval (s, t], we then have from (*) that

I{ [ f k ( x ) ,f k ( x ) ] t - [fk (X),f k (X)]S)- {[fe(X),fe (X)]t - [fe(x),fe ( x ) l s )1 5 ( d e + 2~ke){[X,Xlt- [X,XIs). Since this is true for all 0

< s < t < oo, we deduce that

Therefore d[fk( X ), f k (X)] is a Cauchy sequence of random measures, converging in total variation norm on (0, oo), a.s. In addition, by Itb's formula (Theorem 18) and Theorem 29 of Chap. I1 we have that

from Theorem 23 of Chap. I1 we also know that

Therefore

and since we have a s . convergence in total variation norm, we can pass to the limit t o obtain

Our identification of the limit removes the dependence on the representation of Y by f , because

and by taking k large enough the first two terms on the right side can be taken small in probability independently of n; one then takes n large enough in the

5 Fisk-Stratonovich Integrals and Differential Equations

277

Sgn (f (X))t = third term to make it small in probability, and we have lim,,, l/t, in probability. Therefore l/t = [Y, Y]t and this completes the proof for f and its first partials bounded functions of X, and d = 1. The proof for general d is exactly analogous. For general f satisfying the hypotheses, let gm : EXd -+ EXd be CW such that gm(x) = x if 1x1 5 m, and gm has compact support. Defining Km = f(t,w,gm(Xt)), let Tm(w) = inf{t 2 0 : x ( w ) # Km(w)). Then Tm increase to oo a s . Also, since the quadratic variation is a path property, [Ym,Ym]t(w) = [Y, Y]t(w), for all t < Tm(w). Therefore Y is of finite quadratic variation, and since by the preceding (for d = 1)

which, on [0,T,),

is equal to

whence the result. Note that the subspace of processes Y that have a representation Y = f (X1, . . . ,X d ) , where X = (X1, . . . ,X d ) is a finite-dimensional semimartingale (d is not fixed) and f E C1, is a vector subspace of the space of processes of finite quadratic variation. As an immediate consequence of Theorem 19 we have the following extension of Theorem 34 of Chap. 11.

Theorem 20. Let X be a semimartingale and let f be C2. Then

Proof. Note that f' is C1, so that f1(X) is in the domain of definition of the F-S integral by Theorem 19. Also by definition we have

By Theorem 19 and polarization we know that

and thus the result follows by It6's formula (Theorem 32, Chap. 11). Theorem 20 has an obvious generalization to the multi-dimensional case. We omit the proof which is an analogous consequence of Theorem 33 of Chap. 11.

278

V Stochastic Differential Equations

Theorem 21. Let X = (X1, . . . ,X n ) be an n-tuple of semimartingales, and let f : Rn + R have second order continuous partial derivatives. Then f (X) is a semimartingale and the following formula holds:

As a corollary of Theorem 21, we have the Stratonovich integration by parts formula.

Corollary 1 (Stratonovich Integration by Parts Theorem). Let X and Y be semimartingales. Then

Proof. Let f (x, y) = xy and apply Theorem 21.

Corollary 2. Let X and Y be semimartingales, with a t least one of X or Y continuous. Then

Recall that since Xo- = 0 by convention for a c&dl&gprocess X , we did t not really need to write So+; the formula also holds for For stochastic differential equations with Fisk-Stratonovich differentials we are limited as t o the coefficients we can consider, because the integrands must be of finite quadratic variation. Nevertheless we can still obtain reasonably general results. We first describe the coefficients.

fi.

Definition. A function f : R+ x R x Rd -+ R is said to be Fisk-Stratonovich acceptable if (i) there exists an adapted F V process A and a function g such that

where (s, w) H g(s, W , X )is an adapted, jointly measurable process for each x; t (ii) for each compact set K, fo s u p x C]g(s, ~ W , x)lldA,J < oo a s . ; (iii) fxi is C1 and rt

5 Fisk-Stratonovich Integrals and Differential Equations

279

(g

.f ) (t, w, x ) (iv) for each fixed (t, w), the functions x H f (t, w, x) and x H are all Lipschitz continuous with Lipschitz constant K(w), K < oo a.s. (1 5 i < d ) . 4 We will often write "F-S acceptable" in place of "Fisk-Stratonovich acceptable."

Theorem 22. Given a vector of semimartingales Z = (Z1,. . . , Z"), semimartingales Ji ( 1 I i < d), and F-S acceptable functions f j ( 1 < i < d, 1 5 j < k), then the system of equations

has a unique semimartingale solution. Moreover the solution X of (*) is also the (unique) solution of

Proof. We note that equation (**) has a unique solution as a trivial consequence of Theorem 7. Since X is a d-dimensional semimartingale, we know that fj(s-, ., Xs-) is in the domain of definition of the F-S integral by Theorem 19. Further, as a consequence of Theorem 19, we have that5

and the equivalence of (*) and (**) follows. Therefore the existence of a unique semimartingale solution of (**) is equivalent to the existence of a unique semimartingale solution of (*), and we are done. Note that if Jmis of finite variation, the terms involving [Jm, ZjIC disappear.

(g.

By f ) ( t ,w , x ) we mean the usual product of functions s ( t ,w , x ) . f ( t ,w , x ) . Since we are taking the continuous parts of the quadratic variations, we need not write f (s-, w , X,-), etc.

280

V Stochastic Differential Equations In Chap. I1 we studied the stochastic exponential of a semimartingale. The

F-S integral allows us to give a version that has a more natural appearance.

Theorem 23. Let Z be a semimartingale, Zo the equation

=

0. The unique solution of

rt

is given by

xt = XO

exp{Zt)

n

(I O<s
+A

Z ~ ) ~ - ~ ~ ~ ,

and it is called the Fisk-Stratonovich exponential, XO&F-S(Z). Proof. By Theorem 22 the equation above is equivalent to

Therefore

where the second equality is by Theorem 38 of Chap. 11. Using the formula for the stochastic exponential established in Theorem 37 of Chap. 11, the result follows. The most interesting case is when the semimartingale is continuous.

Corollary 1. Let Z be a continuous semimartingale, Zo = 0. Then the unique ~ of the exponential equation solution, E F - (Z),

is given by EF-S(Z)t = exp{Zt }.

Corollary 2. Let B be a Brownian motion with Bo = 0. Then the unique solution of

is given by Xt

rt

= Xo exp{Bt}.

5 Fisk-Stratonovich Integrals and Differential Equations

281

The simplicity gained by using the F-S integral can be surprisingly helpful. As a n example let Z = (Z1,. . . , Zn) be an n-dimensional wntinuous semimartingale and let x = (xl, . . . ,xn)' be a column vector in EXn. (Here ' denotes transpose.) Let I be the identity n x n matrix and define

(The matrix a(x) represents projection onto the hyperplane normal to x.) Note that x1a(x) = 0. We want to study the system of F-S differential equations rt

where X

=

(X1,. . . , X n ) ' and Z

=

(Z1,. . . , Zn)'

Theorem 24. The solution X of equation (* * *) above exists, is unique, and it always stays on the sphere of center 0 and radius IIxoll. Proof. Since a is singular a t the origin, we need to modify it slightly. Let g(x) be a Cm function equal to a ( x ) outside of a ball centered a t the origin, No, such that XQ @ No. Let Y be the solution of

Then Y exists and is unique by Theorem 22. Let T = inf{t > 0 : Yt E No}. Since Yo = XQ and Y is continuous, P ( T > 0) = 1. However for s < T , g(Y,) = a(Y,). Therefore it suffices t o show that the function t ++ llYtll is constant, since llYoll = IIxoll. This would imply that Y always stays on the sphere of center 0 and radius IIxo11, and hence P ( T = oo) = 1 and g(Y) = a ( Y ) always. To this end, let f (x) = C:,(X~)~, where x = ( x l , . . . ,xn)'. Then it suffices to show that df (Yt) = 0, ( t L 0). Note that f is C2. We have that f o r t < T ,

Therefore IlYtll

=

Ilxoll for t < T and thus by continuity T

= oo

a.s.

Corollary. Let B = (B', . . . ,Bn)' be n-dimensional Brownian motion, let a(x) = I - f$, and let X be the solution of

Then X is a Brownian motion on the sphere of radius IIxoll.

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V Stochastic Differential Equations

Proof. In Theorem 36 of Sect. 6 we show that the solution X is a diffusion. By Theorem 24 we know that X always stays on the sphere of center 0 and radius IIxoll.It thus remains to show only that X is a rotation invariant diffusion, since this characterizes Brownian motion on a sphere. Let U be an orthogonal matrix. Then U B is again a Brownian motion, and thus it suffices to show

The above equation shows that U X is statistically the same diffusion as is X , and hence X is rotation invariant. The coefficient a satisfies

and therefore

and we are done. The F-S integral can also be used to derive explicit formulas for solutions of stochastic differential equations in terms of solutions of (non-random) ordinary differential equations. As an example, consider the equation

Sot

where Z is a continuous semimartingale, Zo = 0. (Note that g(X,) o d s = g:X,)ds, so we have not included ItG7scircle in this term.) Assume that f is C and that f , g, and f f ' are all Lipschitz continuous. Let u = u(x, z) be the unique solution of

Ji

Then

&% = f1(u(x,z))%, and g ( x , 0 ) = 1, from which we conclude that

Let Y = (Yt)t20 be the solution of

which we assume exists. For example if g ~ ~ ( x ' Z s )is) Lipschitz, this would (x7 2s) suffice.

5 Fisk-Stratonovich Integrals and Differential Equations

283

Theorem 25. With the notation and hypotheses given above, the solution X of (*4) is given by xt = u(K1 Zt). Proof. Using the F-S calculus we have

Since

j:

z s

dYs ds

- = exp{-

f ~ ( u (v )~) d, v ) g ( u ( ~zS)), ,

we deduce

By the uniqueness of the solution, we conclude that X t

= u ( K , Zt).

We consider the special case of a simple Stratonovich equation driven by a (one dimensional) Brownian motion B. As a corollary of Theorem 25 we obtain that the simple Stratonovich equation

+ h(Xo)), where

has a solution Xt = h-'(Bt

The corresponding It6 stochastic differential equation is dXt =

, 1

f (Xt)f '(Xt )dt + f (Xt)dBt.

By explicitly solving the analogous ordinary differential equation (without using any probability theory) and composing it, we can obtain examples of stochastic differential equations with explicit solutions. This can be useful when testing simulations and numerical solution procedures. We give a few examples.

Example. T h e equation 1 dXt = - - a 2 x t d t 2 has solution X t

= sin(aBt

+ aJ=dBt

+ arcsin(X0)).

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V Stochastic Differential Equations

Example. The following equation has only locally Lipschitz coefficients and thus can have explosions: for m # 1,

has solution Xt = (x;-"

a ( m - 1)Bt)&.

-

Example. Our last example can also have explosions:

has the solution Xt = tan(t

+ Bt + arctan(X0)).

The Fisk-Stratonovich integrals also have an interpretation as limits of sums, as Theorems 26 through 29 illustrate. These theorems are then useful in turn for approximating solutions of stochastic differential equations. Theorem 26. Let H be cadlag, adapted, and let X be a semimartingale. Assume [ H , X ] exists. Let an = ( 0 = T," 5 T;" 5 .. . 5 T,?) be a sequence of random partitions tending to the identity. If H and X have no jumps in common (i.e., Co,,It AHSAX, = 0 , all t 2 0), then lim

n-cr,

C -21(HTp + HT?+~)(xT?+1 - xqn1 i

equals the F-S integral & H,-

o dX,

in ucp.

Proof. It follows easily from the definition of [H,XI a t the beginning of this section that AIH,XIt = AHtAXt and limn,, C(HTGl - H~"(X~"+' xT;) = [ H , X ] - HoXo. Thus if H and X have no jumps in common we conclude [ H , X ] = [H,XIC HoXo. Observing that

+

the result follows from Theorems 21 and 22 of Chap. 11. Corollary 1. If either H or X in Theorem 21 is continuous, then

where Hop= 0 , in ucp.

5 Fisk-Stratonovich Integrals and Differential Equations

285

Corollary 2. Let X and Y be continuous semimartingales, and let an = (0 = Ton < Tr < . . . 5 Ten} be a sequence of random partitions tending to the identity. Then lim

n+o3

x, 1

1.,

(YTp + YT;+, ) (xTCl - x T z n

) = J'yS0 o~x,,

with convergence in ucp.

Proof. By Theorem 22 of Chap. 11, any semimartingale Y has finite quadratic variation. Thus Corollary 2 is a special case of Theorem 26 (and of Corollary 1). Theorem 27. Let H be cddldg, adapted, of finite quadratic variation, and suppose C,,,,, lAH,I < co a.s., each t > 0. Let X be a semimartingale, and let an = {T~)oli
n+o3

{x

1

~(HT* HT~+, -

x

AH,)(x~;+~ - xT;)} =

T?<s
i

1'

H ~ odXs, -

0

with convergence i n ucp. Proof. First observe that H~ = Ht -Co<slt AH, defines a continuous process of finite quadratic variation and that [A,XI = [H,XIC HoXo.Next we note that

+

1 -(HT+HT;,2

1 C AH,)=HT;+~(HT;+, TP<s
pHT;-

C

AH,)

T;<s
Therefore by Theorems 21 and 22 of Chap. 11, or directly from the definition as in the proof of Theorem 26, we have

and the result follows.

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V Stochastic Differential Equations

For the general case we have a more complicated result. Let H be a c&dl&g, adapted process of finite quadratic variation. For each E > 0 we define

Then JEis also a c&dl&g,adapted process, and it has paths of finite variation on compacts.

Theorem 28. Let H be cadlag, adapted, of finite quadratic variation, and let X be a semimartingale. Assume [H,XI exists. Let a, = {qn)be a sequence

of random partitions tending t o the identity. Let JEbe as defined above. T h e n lim lim

E - 3 0 n-03

1 {C-(HT? + HT;+, + (Jgp+, - J;~))(x~;+~-xT:)) 2 i

with convergence i n ucp Proof. Let H E = H

- J E .Then the jumps of

Therefore the first limit at t

and letting

E

HEare bounded by E. Moreover,

> 0 is

tend to 0 gives the result. That is, since

and since [HE,HEIt = CO<s
Theorem 29. Let X be a semimartingale and Y a continuous semimartingale, and let f be C1. Let p be a probability measure on [O,1] let a = J Xp(dX), and let a, = {T7)i20 be a sequence of random partitions tending to the identity. Then

5 Fisk-Stratonovich Integrals and Differential Equations

lim

n-cc

cilf + (YT?

287

~(YT;,, - Y T ~ ) ) , L L ( ~ ~ )-( X x T~?;)~

i

with convergence in ucp. In particular zf integral &+ f (Y, ) 0 dX, .

if =

112 then the limit is the F-S

Proof. We begin by observing that

&+

f (Y,-)dX, The first sum on the right side of the above equation tends to in ucp. Using the Fundamental Theorem of Calculus, the second sum on the right above equals

which in turn equals

where

Fun(w)

< sup{lfl(Y~? + X~(YT+ - YT;)) i

- fl(Y~?)J).

Since f 1and Y are continuous, Fun (w) tends to 0 on compact time intervals. Also, on [0,t ] , lim sup n-cc

C (YT;,

-

Y T ~ ) ( X ~; ~xT"l

< [Y,~ 1 : ' ~[x,x]:'~,

i

and the result follows by Theorem 30 of Chap. 11. Corollary. Let X be a semimartingale, Y be a continuous semimartingale, and f E C1. Let on = {T,la)05i5kn be a sequence of random partitions tending to the identity. Then

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V Stochastic Differential Equations

with convergence in ucp.

Proof. Let ,u(dX) = E ( ~ ~ ~ ) ( point ~ X ) mass , at 112. Then a! = So A,u(dX) = 112, and we need only apply Theorem 29. 1

For Brownian motion, the Fisk-Stratonovich integral is sometimes defined as a limit of the form lim

f ( B * ~ + ;)(Bt,+l + ~ - Bt,) = a

I'

f ( B s )0 d B s ;

that is, the sampling times are averaged. Such an approximation does not hold in general even for continuous semimartingales (see Yor [238, page 524]), but it does hold with a supplementary hypothesis on the quadratic covariation, as Theorem 30 reveals.

Theorem 30. Let X be a semimartingale and Y be a continuous semimartingale. Let ,u be a probability measure o n [O,1] and let a! = SX,u(dX). Further suppose that [X,YIt = A d s ; that is, the paths of [X,Y] are absolutely continuous. Let an = { t l ) be a sequence of non-random partitions tending to the identity. Let f be C1. T h e n

n-cc

i

with convergence in ucp. I n particular if a = 112 then the limit i s the F-S integral Jd+ f (Y,) o d X s .

+

Proof. We begin with a real analysis result. We let t? denote ti X(ti+l - t i ) , where the ti are understood to be in an. Suppose a is continuous on [0,t ] . Then

which tends to 0. Therefore

1

n-cc

1%: I' a

)=A

a(s)ds.

(*)

Moreover since continuous functions are dense in L1([O,t ] ,d s ) , the limiting result (*) holds for all a in L' ([o,t ], d s ) .

5 Fisk-Stratonovich Integrals and Differential Equations

289

Next suppose H is a continuous, adapted process, and set

Then taking limits in ucp we have

and using integration by parts, this equals

By Theorem 21 of Chap. 11, this equals

However since [X, YIt lim n

C Ht, (x': i

-

= Jot

Jsds, by the result (*) we conclude

xta) (Y':

- Y'.

) = lim n

x

Ht, { [x, Y]': - [X,Y]"} (r r )

i

We now turn our attention to the statement of the theorem. Using the Mean Value Theorem we have

The first sum tends in ucp to S f (Ys)dX, by Theorem 21 of Chap. 11. The second sum on the right side of (* * *) can be written as

V Stochastic Differential Equations

290

The first sum above converges to

by (**), and the second sum can be written as

where

K;."

=

Cf l ( ~ ,

-

Yt;)l(t:,ti+*](s).

Then limn KXyn- X converges to 0 locally in ucp by the Dominated Convergence Theorem (Theorem 32 of Chap. IV). Finally consider the third sum on the right side of (* * *). Let

Then lim n

C i l p ( d X ) F n ( ~ ) l Y , t xiI/x"+' -

- Xti

/

i

since lim Fn = 0 a.s. and the summations stay bounded in probability. This completes the proof. Since Y is continuous, [Y,XI = [Y,XIC,whence if a! = 112 we obtain the Fisk-Stratonovich integral. Corollary. Let X be a semimartingale, Y a continuous semimartingale, and f be C1. Let [X,Y] be absolutely continuous and let on = { t l ) be a sequence of non-random partitions tending to the identity. Then

f (Ytc+tt+l)(xti+l -xt') =

lim

n-oz

i

with convergence in ucp.

2

6 The Markov Nature of Solutions

Proof. Let ,u(dX) = E ( ~ ~ ~ ) (point ~ X mass ) , at 112. Then apply Theorem 30.

291

X,u(dX) = 112, and

Note that if Y is a continuous semimartingale and B is standard Brownian motion, then [Y,B] is absolutely continuous as a consequence of the KunitaWatanabe inequality. Therefore, i f f is C1 and an are partitions of [0,t], then

with convergence in probability.

6 The Markov Nature of Solutions One of the original motivations for the development of the stochastic integral was to study continuous strong Markov processes (that is, diffusions), as solutions of stochastic differential equations. Let B = (Bt)t>obe a standard Brownian motion in Rn. K. It6 studied systems of differentialequations of the form rt

and under appropriate hypotheses on the coefficients f , g he showed that a unique continuous solution exists and that it is strong Markov. Today we have semimartingale differentials, and it is therefore natural to replace dB and ds with general semimartingales and to study any resulting Markovian nature of the solution. If we insist that the solution itself be Markov then the semimartingale differentials should have independent increments (see Theorem 32); but if we need only to relate the solution to a Markov process, then more general results are available. For convenience we recall here the Markov property of a stochastic process which we have already treated in Chap. I. Assume as given a filtered probability space (R, 3 , (Ft)t>o, - P) satisfying the usual hypo these^.^ Definition. A process Z with values in Rd and adapted to lF = (Ft)tyo is a simple Markov process with respect to lF if for each t 0 the a-fields Ft and a{Zu; u 2 t ) are conditionally independent given Zt.

>

Thus one can think of the Markov property as a weakening of the property independent increments. It is easy to see that the simple Markov property equivalent to the following. For u t and for every f bounded, Bore1 measurable, E{f ( ~ u ) l 3 t )= E{f (Zu)Ia{Zt)). (*) One thinks of this as "the best prediction of the future given the past and the present is the same as the best prediction of the future given the present."

>

See Chap. I, Sect. 1 for a definition of the "usual hypotheses" (page 3).

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V Stochastic Differential Equations

Using the equivalent relation (*), one can define a transition function for a Markov process as follows, for s < t and f bounded, Borel measurable, let Ps,t(Zs,f) = E{f(Zt)IFs). Note that iff (x) = lA(x),the indicator function of a set A, then the preceding equality reduces to P ( z t E AIFs) = Ps,t(zs,1 ~ ) . Identifying 1~ with A, we often write Ps,t(Zs,A) on the right side above. When we speak of a Markov process without specifying the filtration of 0algebras (Ft)t>o,we mean implicitly that = o{Zs; s 5 t), the natural filtration generated by the process. It often happens that the transition function satisfies the relationship

for t 2 s. In this case we say the Markov process is t i m e homogeneous, and the transition functions are a semigroup of operators, known as the transition semigroup (Pt)t>o. - In the time homogeneous case, the Markov property becomes P(zt+s E = Ps(Z~,A)-

A stronger requirement that is often satisfied is that the Markov property hold for stopping times. Definition. A time homogeneous simple Markov process is strong Markov if for any stopping time T with P ( T < co) = 1, s 0,

>

or equivalently E{f (ZT+~)IFT)= P s ( Z ~f ), , for any bounded, Borel measurable function f . The fact that we defined the strong Markov property only for time homogeneous processes is not much of a restriction, since if X is an Eld-valued simple Markov process, then it is easy to see that the process Zt = (Xt,t) is an Eld+'-valued time homogeneous simple Markov process. Examples of strong Markov processes (with respect to their natural filtrations of 0-algebras) are Brownian motion, the Poisson process, and indeed any L6vy process by Theorem 32 of Chap. I. The results of this section will give many more examples as the solutions of stochastic differential equations. Since we have defined strong Markov processes for time homogeneous processes only, it is convenient to take the coefficients of our equations to be autonomous. We could let them be non-autonomous, however, and then with an extra argument we can conclude that if X is the solution then the process = (Xt, t) is strong Markov. We recall a definition from Sect. 3 of this chapter.

6 The Markov Nature of Solutions

Definition. A function f : R+ x Rn exists a finite constant k such that

+

293

R is said to be Lipschitz if there

<

(i) (f(t,x) -f(t,y)l klx-YI, e a c h t ~ R +and , (ii) t H f (t, x) is right continuous with left limits, each x E Rn

f is said to be autonomous if f ( t ,x) = f (x). In order to allow arbitrary initial conditions, we need (in general) a larger probability space than the one on which Z is defined. We therefore define

where 8 denotes the Borel sets of Rn, and E~ denotes the Dirac point mass measure at y. For a point a = (y, w) E fi,we further define XO(a)= y,

nu,,

when

a = (Y,w).

(*I

-0

Finally let Ft = F U .A random variable Z defined on R is considered to be extended automatically to by the rule Z(Z) = Z(w), when Z = (9, w). We begin with a measurability result which is an easy consequence of Sect. 3 of Chap. IV.

Theorem 31. Let Z j be semimartingales (1 5 j 5 d), H z a vector of adapted processes in D for each x E Rn, and suppose (x, t, w) H H,"(w) is 8 €9 B+ €9 3 meas~rable.~ Let Fj be functional Lipscliitz and for each x E Rn, Xx is the unique solution of

There exists a version of X x such that (x, t, w) H X,"(w) is 8 €9 B+ €9 3 measurable, and for each x, XF is a ccidlcig solution of the equation. Proof. Let X0(x, t, w) = H," (w) and define inductively

The integrands above are in L,hence by Theorem 63 in Chap. IV there exists measurable, c&dl&gversions of the stochastic integrals. By Theorem 8 the processes Xn converge ucp to the solution X for each x. Then an application of Theorem 62 of Chap. IV yields the result.

' 23 denotes the Borel sets on Rn; B+ the Borel sets on R + .

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V Stochastic Differential Equations

We state and prove the next theorem for one equation. An analogous result (with a perfectly analogous proof) holds for finite systems of equations.

Theorem 32. Let Z = (Z1,. . . , Zd) be a vector of independent LLvy processes, Zo = 0, and let (fj) 1 j < d, 1 < i < n, be Lipschitz functions. Let Xo be as in (*) and let X be the solution of

<

Then X is a Markov process, under each PYand X is strong Markov if the

f j are autonomous. Proof. We treat only the case n = 1. Let T be an P stopping time, T < co a.s. Define GT = o{z$+, - 2;; u 0 , l < j < d). Then GT is independent of .FT under pY, since the Z j are L6vy processes, as a consequence of Theorem 32 of Chap. I. Choose a stopping time T < co a.s. and let it be fixed. For u 0 define inductively

>

>

Also, let X(x, T, u) denote the unique solution of

taking the jointly measurable version (cf., Theorem 31). By Theorem 8 we know that X(x,T,u) is GT measurable. By approximating the stochastic integral as a limit of sums, we see by induction that Yn(x,T,u) is GT measurable as well. Under px we have X(Xo,T, u) = X(x, T, u) as., and Yn(XO1 T, u) = Yn(x,T, u) px-a.s., also. By uniqueness of solutions and using Theorem 31, for all u 0 a.s.

>

There is no problem with sets of probability zero, due to (for example) the continuity of the flows. (See Theorem 37.) Writing Ex to denote expectation on 2 with respect to px,and using the independence of .FTand GT (as well as of FTand c T ) , we have for any bounded, Bore1 function h

6 The Markov Nature of Solutions

295

where j(y) = E{h(X(y, T, u)). The last equality follows from the elementary fact that E{F(H, .)I'H) = f (H), where f (h) = E{F(h, -)), if F is independent ' and H is H ' measurable. This completes the proof, since the fact that of H Ex{h(X(Xo,0, T u ) )IFT)is a function only of X(x, 0, T) implies that

+

It is interesting to note that Theorem 32 remains true with Fisk-Stratonovich differentials. To see this we need a preliminary result.

Theorem 33. Let Z = (Z1,. . . , Zd) be a vector of independent Livy processes, Zo = 0. Then [Zi,ZjIc = 0 if i # j , and [Zi, Zi]: = a t , where a = E{[Zi, Zi]i).

-

Proof. First assume that the jumps of each Zi are bounded. Then the moments of Zi of all orders exist (Theorem 34 of Chap. I), and in particular M: Z: - E{Z;) is an L~ martingale for each i, with E{z:) = ~E{Z;). By independence M i M j is also a martingale and hence [Mi,M j ] = 0 by Corollary 2 of Theorem 27 of Chap. 11. Therefore [Zi,Zj]: = [Mi,Mj]: = 0 as well. [Zi,ZiIt = [Mi,Milt. It is an immediate consequence Next consider A: of approximation by sums (Theorem 22 of Chap. 11) that Ai also has independent increments. Since

-

also clearly has independent increand the process Ji = Co,s,t(AM:)2 ments, we deduce that [Mi,Ma]: has independent increments as well. Thereis a finite variation continuous martingale, fore [Mi,Mi]: - E{[Mi, M~],c) hence constant by Theorem 27 of Chap. 11. Since [Mi,Mi]6 = 0, we deduce that [Zi,Zi]: = [Mi,Mi]: = E{[M~,Mi]:). Since [Mi,MZIt is also a L6vy process we have E{[Mi, Milt) = tE{[Mi, Mill) (by the stationarity of the increments), and also

by the same reasoning. Therefore [Zi,Zi]: = tE{[Zi, Zi]:) by subtraction. If the jumps of Z are not bounded, let J,Z = Co<slt A Z ~ l i , Asz-i , > l lThen . = Zi - Ji and Ji are independent LCvy processes, and Ji is a quadratic ... pure jump semimartingale. It therefore follows that [Zi, Zj] = [Z2,2-71= 0 and that [Zi,ZiIC = [Za,ZaIC,and the theorem is proved.

zi

A

A

.

A

.

.

Theorem 34. Let Z = (Z1, . . . ,Zd) be a vector of independent Le'vy proj I d, 1 I iI n, be non-random F-S acceptcesses, Zo = 0, and let (f;) 1 I able function^.^ Let Xo be as in Theorem 32 and X be the solution of F-S acceptable functions are defined on page 278.

296

V Stochastic Differential Equations

Then X is a Markov process and X i s strong Markov if the coefficients f j are autonomous. Proof. We treat only the case n = 1. By Theorem 33 [Zi,Zj],C = 0 if i # j and [Zi,Zi]: = sit, ai 2 0. Therefore, by Theorem 22 the equation is equivalent to

and since the process & = t is a L6vy process we need only to invoke Theorem 32 to complete the proof. If the differentials are not L6vy processes but only strong Markov processes which are semimartingales, then the solutions of equations such as (**) in Theorem 32 need not be Markov processes. One does, however, have the following result.

Theorem 35. Let Z be a strong Markov processes with values in R, Zo = 0, such that Z is a semimartingale. Let f and g be Lipschitz functions. Let Xo be as in Theorem 32 and X be the solution of

Then the vector process (X, Z) is Markov under p, each y E R, and strong Markov i f f and g are autonomous. Proof. First recall that X is defined on 0 = R x R and Z is automatically extended to as explained at the beginning of this section. The probability -Y P = ey x P is such that F y ( x o = y ) = 1, each y E R. As in the proof of Theorem 32 for a fixed stopping time T (T < co a.s.) and x E R define inductively for u 2 0

and let X(x, T, u) denote the unique solution of

6 The Markov Nature of Solutions

297

Next define GT = 0{ZT+, - ZT; u 2 0) for the same stopping time T . As in the proof of Theorem 32 we see that Yn(x,T,u) is GT measurable for each 1 and that X ( x , T , u) is GT measurable as well. Next let h be Borel n measurable and bounded, and let G E G~ and bounded. Observe that

>

) the uniqueness where j : R2 + R is Borel. Since XT+, = X ( X T , T l ~by of the solution, the theorem now follows from the Monotone Class Theorem (Theorem 8 of Chap. I). Theorem 35 can also be shown t o be true with Stratonovich differentials, but the proof is more complicated since the quadratic variation process is an additive functional of Z , rather than a deterministic process (as is the case when Z is a L6vy process). For this type of result we refer the reader t o Cinlar- Jacod-Protter-Sharpe [34]. Each of Theorems 32, 34 and 35 can be interpreted with X having an arbitrary initial distribution p. Indeed for p a probability measure on (R, B), ~ (A dE B ~@ ) F, . Then the concludefine Fpon by F p ( n ) = ~ R ~ x ( ~ ) for sions of the three theorems are trivially still valid for ppland the distribution of Xo is p. Traditionally the most important Markovian solutions of stochastic differential equations are diffusions. Suppose as given a space (R, F, (Ft)t20, P) satisfying the usual hypotheses.

Definition. An adapted process X with values in Rn is a diffusiongif it has continuous sample paths and if it satisfies the strong Markov property. A restatement of Theorems 32 and 34 yields the following.

Theorem 36. Let B = (B1,. . . ,B ~ be ) a standard Brownian motion on dl 1 i n, gi be autonomous Lipschitz Rd, Bo = 0, and let (fj) 1 j functions. Let X i be as in Theorem 32 and let X be the solution of

< <

< <

ThenX is a diffusion. Ifthe coeficients (fj)l<j
< n, are non-random

The definition of a diffusion is not yet standardized. We give a general definition.

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V Stochastic Differential Equations

then Y is a diffusion. In equation (***) of Theorem 36 the coefficients fj are called the diffusion coefficients and the coefficients gi are called the drift coefficients. Note that a diffusion need not be semimartingale, even though of course the solutions of equations such as (* * *) are semimartingales. Indeed any deterministic continuous function with paths of unbounded variation is a diffusion which is not a semimartingale. Another interesting example is provided by Tanaka's formula. The process

is a diffusion and it is a semimartingale, where B is standard Brownian motion on I[$. Since the paths of L: are singular with respect to Lebesgue measure, and since a semimartingale decomposition is unique for continuous processes (the corollary of Theorem 31 of Chap. 111), we see that IBtJcannot be represented as a solution of (* * *). Another example is that of I B ~which ~ ~is /a time ~ homogeneous diffusion but not a semimartingale (Theorem. 71 of Chap. IV). An intuitive notion of a diffusion is to imagine a pollen grain floating downstream in a river. The grain is subject to two forces: the current of the river (drift), and the aggregate bombardment of the grain by the surrounding water molecules (diffusion). The coefficient f (t, x) then represents the sensitivity of the particle at time t and place x to the diffusion forces. For example, if part of the river water is warmer at certain times and places (due to sunlight or industrial effluents, for example), then f might be larger. Analogously g would be larger when the river was flowing faster due to a steeper incline. We give three simple examples of diffusions.

Example. The stochastic exponential exp{Bt - i t ) is a diffusion where f (t,x) = x and g(t,x) = 0. Example. Consider the simple system

The process X can be used as a model of Brownian motion alternative to Einstein's. It is called the Ornstein-Uhlenbeck Brownian motion, or simply the Ornstein-Uhlenbeck process. Note that here the process X has paths of finite variation and hence the process (T/t)t20is a true velocity process for X. Using integration by parts we can verify that

6 The Markov Nature of Solutions

299

is an explicit solution for V. Indeed

and we are done. Since Vo and the Brownian motion B are independent (by construction), we see that when Vo has a Gaussian distribution then V is a Gaussian process. If a is negative and we take cr2 = -l/(2a), then V is a stationary Gaussian process. Example. Consider next the equation

for B a standard Brownian motion. For each to, 0 < to < 1, there is a solution which is a diffusion on [0,to]. By the uniqueness of solutions if tl < to then the solution for [O,to] agrees with the solution for [O, tl] on the interval [0,tl]. Thus we have a solution on [O,l). If we can show that limt,1 X t = 0, then the solution extends by continuity to [O,11 and we will have constructed a diffusion X on [O,l] with Xo = X1 = 0, known as the Brownian bridge.1° An application of integration by parts shows that the solution of the Brownian bridge equation is given by

. see that Write Xt = f (t)Mt, where f (t) = (1-t) and Mt = ~ o t ( l - s ) - l d ~ sTo lirnt,l Xt = 0, we study Mt, making the change of variables t = u(l u)-'. Then 0 t < 1 corresponds to 0 u < oo, and define

<

<

+

N is clearly a continuous Gu martingale, and moreover [N, N], = u; hence N is a standard Brownian motion by Lkvy's Theorem (Theorem 39 of Chap. 11). It is then easy to show, as a consequence of the Strong Law of Large Numbers, that1' Nt lim - = 0 a s . t-w t The Brownian bridge is also known as tied down Brownian motion, and alternatively as pinned Brownian motion. See, for example, Breiman [23, page 2651.

300

V Stochastic Differential Equations

Let y(t, w) = Nt(w)/t so that limt,o y(t-l, w) = 0 a s . Thus, limt-0 tNllt a.s. We then have, replacing t with (1 - t)

=0

and therefore limu,l(l - u)Mu = 0 a.s., and hence limt-1 Xt = 0 a.s. We now know that X is a continuous diffusion on [O,l], and that Xo = X1 = 0 a.s. Also X is clearly a semimartingale on [0, I), but it is not obvious that X is a semimartingale on [0, 11. One needs to show that the integral g d s < co a.s. To see this calculate E{Xt}, 0 5 t < 1,

By the Cauchy-Schwarz inequality

E{lXtl) 5 E{X,"}'~~ =d

m..

Therefore

whence e d s < co a.s. Therefore the solution X is a semimartingale on [O, 11. Finally we remark that one can construct a similar Brownian bridge from any value a to any value b, on an interval of arbitrary finite length T . Using the interval 10, T ] one has the equation

with solution

7 Flows of SDE: Continuity and Differentiability

301

7 Flows of Stochastic Differential Equations: Continuity and Differentiability Consider a stochastic differential equation of the form

Obviously there is a dependence on the initial condition, and we can write the solution in the form X ( t , w, x), or XF(w). The study of the flow o f a stochastic differential equation is the study of the functions q5 : x -+ X ( t , w,x) which can Rn for ( t ,w) fixed, or as mapping Rn + Dn, be considered as mapping Rn where Dn denotes the space of c&dl&gfunctions from R+ to Rn, equipped with the topology of uniform convergence on compacts. It is important to distinguish between Vn and Dn. The former is a function space, and it is associated in the literature with weak convergence results (see, e.g., Billingsley [17], or Ethier-Kurtz [71], or Kurtz-Protter [136] for recent results in a semimartingale context); the latter is the space of stochastic processes with c&dl&gpaths, and which are adapted t o the underlying filtration. Note that we have already encountered flows in Sect. 6 (cf., Theorem 31), where we proved measurability of the solution with respect to a parameter (which can be taken to be, of course, the initial condition). We will be interested in several properties of the flows: continuity, differentiability, injectivity, and when the flows are diffeomorphisms of Rn. We begin with continuity. We consider a general system of equations of the form -+

ft

where X,3i and H,3i are column vectors in Rn, Z is a column vector of m semimartingales with Zo = 0, and F is an n x m matrix with elements (Fk). For x fixed, for each y we have that = X! -XF is a solution of the equation

xt

where F ( Y ) = F ( X x + Y)

-

F(Xx).

Theorem 37. Let Hz be processes in Dn, and let x H Hz : Rn -4 Dn be prelocally Lipschitz continuous in p,some p > n. Let F be a n n x m matrix of functional Lipschitz operators (%'A),1 i n, 1 a m.12 Then there exists a function X ( t , w, x ) o n R+ x R x Rn such that

< <

< <

(2) for each x the process X;(w) = X ( t , w, x ) is a solution of (*), and (ii) for almost all w, the flow x H X(.,w,x) from Rn into Dn is continuous in the topology of unzform convergence o n compacts. l2

See page 250 for the definition of functional Lipschitz.

302

V Stochastic Differential Equations

Proof. We recall the method of proof used to show the existence and uniqueness of a solution (Theorem 7). By stopping at a fixed time to, we can assume the Lipschitz process is just a random variable K which is finite a.s. Then by conditioning13 we can assume without loss of generality that this Lipschitz constant is non-random, and we call it c < oo. By replacing H t with X t JotF(O),-dZs, and then by replacing F with G given by G(Y)t = F ( Y ) t - F(O)t, we can further assume without loss of generality that F(0) = 0. Then for /3 = Cp(c), by Theorem 5 we can find an arbitrarily large stopping time T such that ZT- E S(/3),and H z is Lipschitz continuous in SPon [0,T). Then by Lemma 2 (preceding Theorem 7) we have that for the solution X of (**)

+

and some (finite) constant Cp(c,2 ) . Choose p > n, and we have

due to the Lipschitz hypothesis on x H H z . By Kolmogorov's Lemma (Theorem 72 of Chap. IV) we have the result on Rn x [0,T). However since T was arbitrarily large, the result holds as well on Rn x 52 x R+. For the remainder of this section we will consider less general equations. Indeed, the following will be our basic hypotheses, which we will supplement as needed.

Hypothesis (HI). Za are given semimartingales with Zg = 0, 1 I a

fk : Rn + R are given functions, 1 I i and f (x) denotes the n x m matrix (f: (x)).

Hypothesis (HZ).

I m.

< n, 1 I a <_ m,

We will study the system of equations

which we also write

(@@I where it is understood that Xt and x are column vectors in Rn, f (X,-) is an n x m matrix, and Z is a column vector of m semimartingales. To study the differentiability of the flow we will need a more general theorem on existence and uniqueness of solutions. Indeed, we wish to replace our customary (global) Lipschitz condition with a local Lipschitz condition. l3

See the proofs of Theorems 7, 8, or 15 for this argument.

7 Flows of SDE: Continuity and Differentiability

303

Definition. A function f : Rn -t R is said to be locally Lipschitz if there exists an increasing sequence of open sets Ak such that Uk Ak = Rn and f is Lipschitz with a constant Kk on each Ak. For example, iff has continuous first partial derivatives, then it is locally Lipschitz, while if its continuous first partials are bounded, then it is Lipschitz. If f , g, are both Lipschitz, then their product f g is locally Lipschitz. These coefficients arise naturally in the study of Fisk-Stratonovich equations (see Sect. 5).

Theorem 38. Let Z be as in ( H I ) and let the functions (f:) in (H2) be locally Lipschitz. Then there exists a function ((x, w) : Rn x R + [0,oo] such that for each x ((x, .) is a stopping time, and there exists a unique solution of

llXtll = oo a.s. on {( < oo). Moreover up to ((x, .) with lim~up~,~(,,.) x H ((x, W) is lower semi-continuous, strictly positive, and the flow of X is continuous on [0,((x, -)).

Remark. Before proving the theorem we comment that for each x fixed the stopping time T(w) = ((x, w) is called an explosion time. Thus Theorem 38 assures the existence and uniqueness of a solution up to an explosion time; and at that time, the solution does indeed explode in the sense that lim sup,,, llXt 11 = +oo on {T < oo). Note however that if the coefficients (f:) in (H2) are (globally) Lipschitz, then a.s. ( = oo for all x (Theorem 7). Proof. Let Ae be open sets increasing to Rn such that there exist (he), a sequence of CW functions with compact support mapping Rn to [0, 11 such that he = 1 on Ae. For each e we let Xe(t, w, x) denote the solution (continuous in x) of the equation

xt = x +

JIO

t

ge(Xs-)dZs

where the matrix ge(x) is defined by he(x)f (x). Define stopping times, for x fixed,

By Theorem 37 the flow is uniformly continuous on compact sets, hence the functions Se are lower semi-continuous. Then on [O,Se) we have that Xe = xe+i by the uniqueness of the solutions, since they both satisfy equation (@@). We wish to show that the relation

holds for all x E Rn simultaneously. Choose an x and let

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V Stochastic Differential Equations

Then P(Ax) .= 1. We set A = U? ,, Ax, where Qn denotes the rationals in Rn. Then P(A) = 1 as well, and w~thoutloss of generality we take A = R. Next for arbitrary y E Rn, let yn -+ y, with yn E Qn. Then

and therefore the relation (L) holds for y as well, by the previously established continuity. Observe that S ~ ( W , XI) S ~ + ~ ( W , and X ) , let ((x, w) = supe Se(w,x), for each x. Then ( is lower semi-continuous because the Se are, and it is strictly positive. Further, we have shown that there exists a unique function X(., w,x) on [O,i(x,w)) which is a solution of (@@), and it is equal to Xe(.,w,x) on [O,Se(w,x)),each e 2 1. Indeed, on [O,Se(w,x)) we have X ( . , Wx) , = Xe(.,w,x), and since Xe(.,w,x) E Ae, we have he(Xe(.,w,x)) = 1, wherefore Xe is a solution of (@@) on [0,Se). By our construction we have that X(Se(w,x), w,x) belongs to A:. Indeed, letting Se(w,x) = s, then X(s-, w, x) = Xe(s-, w, x) = u E &,for some value u. Since u E &we have he(u) = 1, and thus X and Xe have the same jump at s andwe canconclude X(s,w,x) = Xe(s,w,x).But Xe(s,w,x) or Xe(s-,w,x) must be in A: by right continuity and the definition of Se; therefore X(s, w, x) or X(s-, w, x) is in A:. This shows that lim JIXt11 = oo on {( < oo). It is tempting to conclude that if there are no explosions at a finite time for each initial condition x a s . (null set depending on x), then also ((x, -) = oo a.s. (with the same null set for all initial values x). Unfortunately this is not true as the next example shows. Example. Let B be a complex Brownian motion. That is, let B1 and B2 be two independent Brownian motions on R and let Bt = B,' iB:, where i2 = -1. Consider the stochastic differential equation

+

where the initial value z is complex. This equation has a closed form solution given by z Z(t, w, 2) = 1 zBt(w)'

+

Indeed, if we set f(x) = z(l + zx)-l and Zt = f(Bt), then fl(Bt) = -z:, iB:, we have by It6's formula and since Bt = B,'

+

since d[B1,B1],

= d[B2,B2], = ds,

we see that Zt = z - SoZ:dB,. t

7 Flows of SDE: Continuity and Differentiability

305

For z fixed we know that P ( 3 t : Bt = -l/z) = 0, since Brownian motion in R2 a.s. does not hit a specified point. Therefore Z does not have explosions in a finite time for each fixed initial condition Zo = z. On the other hand for any wo and to fixed we have Bt,(wo) = zo for some zo E C,and thus for the initial value z = -l/zo we have an explosion at the chosen finite time to. Thus each trajectory has initial conditions such that it will explode at a finite time, and P { w : 3 z withi(w,z) < c Q ) = ~ . We next turn our attention to the differentiability of the flows. To this end we consider the system of n + n 2 equations (assuming that the coefficients (f:) are at least C1)

D;,

= 6;

+

c cJ m

n

afg --(X,-)D~,_~Z:, a=l j=l o 3 %

< <

n ) where D denotes an n x n matrix-valued process and 6; = 1 if (1 i i = k and 0 otherwise (Kronecker's delta). A convenient convention, sometimes called the Einstein convention, is to leave the summations implicit. Thus the system of equations (D) can be alternatively written as

We will use the Einstein convention when there is no ambiguity. Note that in equations (D) if X' is already known, then the second system is linear in D. Also note that the coefficients for the system (D) are not globally Lipschitz, but if the first partials of the (fg) are locally Lipschitz, then so also are the coefficients of (D).

Theorem 39. Let Z be as in ( H I ) and let the functions (f:) in (H2) have locally Lipschitz first partial derivatives. Then for almost all w there exists a function X ( t , w,x) which is continuously differentiable in the open set {x : ((x,w) > t), where i is the explosion time (cf., Theorem 38). If ( f i ) are globally Lipschitz then ( = CQ. Let Dk(t, w, x) = &X(t, w, x). Then for each x the process (X(., w, x), D(-,w, x)) is identically cadlag, and it is the solution of equations (D) on [0,((x, .)). Proof. We will give the proof in several steps. In Step 1 we will reduce the problem to one where the coefficients are globally Lipschitz. We then resolve the first system (for X ) of (D), and in Step 2 we will show that, given X , there exists a "nice" solution D of the second system of equations, which depends

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V Stochastic Differential Equations

continuously on x. In Step 3 we will show that DL is the partial derivative in xk of X i in the distributional sense.14 Then since it is continuous (in x), we can conclude that it is the true partial derivative.

Step 1. Choose a constant N. Then the open set {x : ((x,w) > N ) is a countable union of closed balls. Therefore it suffices to show that if B is one of these balls, then on the set r = {w : Vx E B, ((x, w) > N ) the function x H X(t, W, x) is continuously differentiable on B. However by Theorem 38 we know that for each w E r, the image of X as x runs through B is compact in Dn with 0 5 t 5 N , hence it is contained in a ball of radius R in Rn, for R sufficiently large. We fix the radius R and we denote by K the ball of radius R of Rn centered at 0. Let

We then condition on A. That is, we replace P by PA,where PA(A) = P(AlA) = Then PA<< P, so Z is still a semimartingale with respect to PA(Theorem 2 of Chap. 11). This allows us to make, without loss of generality, the following simplifying assumption: if x E B then ((x, w) > N , and X(t,w,x) E K , 0 I t I N. Next let h : Rn + R be CW with compact support and such that h(x) = 1 if x E K and replace f with f h . Let Z be implicitly stopped at the (constant stopping) time N. (That is, Z N replaces 2 . ) With these assumptions and letting PAreplace P, we can therefore assume-without loss of generalitythat the coeficients i n the first equation in (D) are globally Lipschitz and

w.

bounded. Step 2 In this step we assume that the simplifying assumptions of Step 1 hold. We may also assume by Theorem 5 that Z E S(P) for a P which will be specified later. If we were to proceed to calculate formally the derivative with respect to xk of Xi, we would get

Therefore our best candidate for the partial derivative with respect to xk is the solution of the system

l4

These derivatives are also known as derivatives in the generalized function sense.

7 Flows of SDE: Continuity and Differentiability

307

and let D be the matrix ( D i ) . Naturally X, = X ( s , w , x ) , and we can make explicit this dependence on x by rewriting the above equation as

where the summations over a and j are implicit (Einstein convention). We now show that D k = (DL,. . . , DE) is continuous in x. Fix x, y E Rn and let V,(W)= Dk(s, W,x) - Dk(s, W,y). Then

where

=

ELl J,' $(X:-)~Z:.

Note t h a t by Step 1 we know t h a t

are in S(c@)for a &(x,Y-) is bounded; therefore since Za t S ( @ ) ,the constant c. If @ is small enough, by Lemma 2 (preceding Theorem 7) we have that for each p 2 there exists a constant Cp(Z) such that

>

If we let However we can also estimate IIHlls. -

then H,i = E;=, we have that

Jot J&dZ:,

and therefore by Emery's inequality (Theorem 3)

IlHllg 5 cpll Jllgpllzlly- , which in turn implies

l"11gp 5 cP(z)ll ~llrnp. We turn our attention t o estimating 11 Jllsp. Consider first the terms -

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V Stochastic Differential Equations

By the simplifying assumptions of Step I , the functions are Lipschitz in K , and Xx takes its values in K . Therefore Theorem 37 applies, and as we saw in its proof (inequality (* * *)) we have that

Next consider the terms DL:. We have seen that these terms are solutions of the system of equations

and therefore they can be written as solutions of the exponential system

with

ye"," = Sos eaf3 (Xc-)dZ,*.

As before, by Lemma 2,

Recalling the definition of J and using the Cauchy-Schwarz inequality gives

which in turn combined with previous estimates yields IlVllgp

I C(P, Z)IIx - yll.

Since V was defined to be K ( w ) = Dk(s, W, 2) - D l c ( ~W , , Y), We have ~ h o w n that (with p > n )

and therefore by Kolmogorov's Lemma (Theorem 72 of Chap. IV) we have the continuity in x of Dk(t, W , x). Step 3. In this step we first show that Dk(tl w,x), the solution of equations (*4) (and hence also the solution of the n 2 equations of the second line of (D)) is the partial derivative of X in the variable xk in the sense of distributions (i.e., generalized functions). Since in Step 2 we established the continuity of D k in x, we can conclude that D k is the true partial derivative.

7 Flows of SDE: Continuity and Differentiability

309

Let us first note that with the continuity established in Step 2, by increasing the compact ball K , we can assume further that Dk(s,w,x) E K also, for s 5 N a n d all x E B. We now make use of Cauchy's method of approximating solutions of differential equations, established for stochastic differential equations in Theorem 16 and its corollary. Note that by our simplifying assumptions, Y = (X, D ) takes its values in a compact set, and therefore the coefficients are (globally) Lipschitz. The process Y is the solution of a stochastic differential equation, which we write in vector and matrix form as

Let a, be a sequence of partitions tending to the identity, and with the notation of Theorem 16 let Y(a) = ( X ( a ) ,D ( a ) ) denote the solution of the equation of the form

a =Y+

Jo

t

f(~~+):dz,.

For each (a,) the equations (D) become difference equations, and thus trivially

The proof of the theorem will be finished if we can find a subsequence r, &(aTq) = Dk, in the sense of such that lirn,,, X(aTq)= X and lirn,,, distributions, considered as functions of x. We now enlarge our space S1 exactly as in Sect. 6 (immediately preceding Theorem 31):

where 8'" denotes the Bore1 sets of RZn.Let X be normalized Lebesgue measure of K . Finally define P=XxP. We can as in the proof of Step 2 assume that Z E S(P) for and then, lim (X(a,), D(a,) = (X, D ) in T',

P small enough

s',

by Theorem 16. Therefore there exists a subsequence r, such that

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V Stochastic Differential Equations

The function M = M(w, x) is in L1(X x P ) , and therefore for P-almost all w the function x H M(w, x) E L1(dX). For w not in the exceptional set, and t fixed it follows that

+

X a.e. Further, it is bounded by the function M(w, .) I((X(t, w, .), D(t, w, .))I( which is integrable by hypothesis. This gives convergence in the distributional sense, and the proof is complete. We state the following corollary to Theorem 39 as a theorem.

Theorem 40. Let Z be as in ( H I ) and let the functions (fk) in (H2) have locally Lipschitz derivatives up to order N , for some N , 0 5 N I m. Then there exists a solution X ( t , w, x) to

which is N times continuously differentiable in the open set {x : <(x, w) > t), where is the explosion time of the solution. If the coeficients (fk) are globally Lipschitz, then C = m.

<

Proof. If N = 0,then Theorem 40 is exactly Theorem 38. If N = 1, then Theorem 40 is Theorem 39. If N > 1, then the coefficients of equations (D) have locally Lipschitz derivatives of order N - 1 at least. Induction yields (X, D ) E c N - l , whence X E CN. Note that the coefficients (f:) in Theorem 40 are locally Lipschitz of order N if, for example, they have N + 1 continuous partial derivatives; that is, if f: E CN+l(Rn), for each i and a , then (fk) are locally Lipschitz of order N .

8 Flows as Diffeomorphisms: The Continuous Case In this section we will study a system of differential equations of the form

where the semimartingales Za are assumed to have continuous paths with Zo = 0. The continuity assumption leads t o pleasing results. In Sect. 10 we consider the general case where the semimartingale differentials can have jumps. The flow of an equation such as (*) is considered to be an Rn-valued function cp : Rn + Rn given by cp(x) = X ( t , w, x), for each (t, w). We first consider the possible injectivity of cp, of which there are two forms.

8 Flows as Diffeomorphisrns: The Continuous Case

311

Definition. The flow cp of equation (*) is said to be weakly injective if for each fixed x , y E Rn, x # y, P{w : 3 t : X ( t , w, x) = X ( t , w , y)) = 0. Definition. The flow cp of equation (*) is said to be strongly injective (or, simply, injective) if for almost all w the function cp : x + X(t, w, x) is injective for all t. For convenience we recall here a definition from Sect. 3 of this chapter. Definition. An operator F from Dn into D is said t o be process Lipschitz if for any X , Y E Dn the following two conditions are satisfied. (i) For any stopping time T, XT- = YT- implies F(X)T- = F( Y )T-. (ii) There exists an adapted process K E lL such that

Actually, process Lipschitz is only slightly more general than random Lipschitz. The norm symbols in the above definition denote Euclidean norm, and not sup norm. Note that if F is process Lipschitz then F is also functional Lipschitz and all the theorems we have proven for functional Lipschitz coefficients hold as well for process Lipschitz coefficients. If f is a function which is Lipschitz (as defined at the beginning of Sect. 3) then f induces a process Lipschitz operator. Finally, observe that by Theorem 37 we know that the flow of equation (*) is continuous from Rn into Rn or from Rn into Dn a s . , where Dn has the topology of uniform convergence on compacts. T h e o r e m 41. Let Za be continuous semimartingales, 1 5 a 5 m, H a vector of adapted ccidlcig processes, and F an n x m matrix of process Lipschitz operators. Then the flow of the solution of

is weakly injective.15 Proof. Let x, y E Rn, x # y. Let X x , XY denote the solutions of the above equation with initial conditions x, y respectively. We let u = x - y and U = X x - XY. We must show P{w : 3t : Ut(w) = 0) = 0. Set V = F ( X x ) - F(XY)-. Then V E IL and IVI I K ( U ( .Further, the processes U and V are related by Ut=u+

I'

V,dZs.

Let T = infit > 0 : Ut = 0); the aim is to show P ( T = m) = 1. Since U is continuous the stopping time T is the limit of a sequence of increasing stopping l5

We are using the Einstein convention on sums.

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V Stochastic Differential Equations

times Sk strictly less than T. Therefore the process l[O,T) = limk-+ml[O,Sk~ is predictable. We use ItG's formula (Theorem 32 of Chap. 11) on [O,T) for the function f ( x ) = log 11x11. Note that

Therefore on [0,T), log IIUt Il - 1% llu11

Since dUi = C , V i > a d ~the a , foregoing equals

All the integrands on the right side are predictable and since llVll 5 KllUll they are moreover bounded by K and K 2 in absolute value. However on {T < cm) the left side of the equation, log IlUtll - log IIuII, tends to - m as t increases to T; the right side is a well-defined non-exploding semimartingale on all of [0,m ) . Therefore P ( T < m ) = 0, and the proof is complete. In the study of strong injectivity the stochastic exponential of a semimartingale (introduced in Theorem 37 of Chap. 11) plays an important role. Recall that if Z is a continuous semimartingale, then XoE(Z) denotes the (unique) solution of the equation

and E(Z),

= exp{& - !j

[z,21,). In particular, P(infs 0) = 1.

Theorem 42. For x E Rn,let Hx be in IDk such that they are locally bounded uniformly in x . Assume further that there exists a sequence of stopping times (Te)e21 increasing to m a.s. such that 11 (H3: - H ? ) ~1ISr I- 5 K11x - y 11, each

8 Flows as Diffeomorphisms: The Continuous Case

C 2 1, for a constant K and for some r > n . Let Z

=

313

( Z 1 , .. . ,Z k ) be k

semimartingales. Then the functions

have versions which are continuous as functions from Rn into 27, with 27 having the topology of uniform convergence on compacts. Proof. By Theorem 5 there exists an arbitrarily large stopping time T such that Z T - E E m . Thus without loss of generality we can assume that Z E E m , and that is bounded by some constant K , uniformly in x. Furtherwe T 5 K11x - yII. Then assume IIHT - H! ]I S-

where we have used Emery's inequality (Theorem 3). The result for HF-dZ, now follows from Kolmogorov's Lemma (Theorem 72 of Chap. IV). For the second result we have

and the result follows.

Theorem 43. Let F be a matrix of process Lipschitz operators and Xx the solution of (*) with initial condition x , for continuous semimartingales Z a , 1 5 a 5 m. Fix x , y E Rn.For r E R there exist for every x , y E Rn with x # y (uniformly) locally bounded predictable processes H a ( x ,y), Ja7i3(x,y), which depend on r , such that

where

1 t

A,(x, Y ) t

=

+

1 t

H , Y ( Xy, ) d ~ , ~ J ~ Y . ~y() X d [, z a Izi3IS.

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V Stochastic Differential Equations

Proof. Fix x, y E Rn and let U = X x - XY, V = F ( X x ) - - F(XY)-. It6's formula applies since U is never zero by weak injectivity (Theorem 36). Using the Einstein convention,

Let (., .) denote Euclidean inner product on Rn. It suffices to take

(where Va is the a-th column of V); and

One checks that these choices work by observing that dU; = Finally the above allows us t o conclude that

xz=,v,~'*~z,*.

and the result follows. Before giving a key corollary t o Theorem 43, we need a lemma. Let fi" be the space of continuous semimartingales X with Xo = 0 such that X has a (unique) decomposition X=N+A where N is a continuous local martingale, A is a continuous process of finite variation, No = A. = 0, and such that [N, N], and ldAsl are in Lm. Further, let us define

Lemma. For every p, a < m , there exists a constant C(p, a ) < m such that if llXllem < a, then II&(X)IIgp I C ( P , ~ ) . Proof. Let X

=N

+ A be the (unique) decomposition of X. Then 1

ll&(x)ll;P - = E{sup exp{p(Xt t

5 E{epX*)

IX. Xlt)})

(recall that X * = sup lXtl) t

8 Flows as Diffeomorphisms: The Continuous Case

315

<

since lAtl a , a s . We therefore need to prove only an exponential maximal inequality for continuous martingales. By Theorem 42 of Chap. 11, since N is a continuous martingale, it is a time change of a Brownian motion. That where B is a Brownian motion defined with a different is, Nt = a', we have filtration. Therefore since [N,N],

<

and hence ~ { e p ~ 5 ' ) E{e~p{pB;~)). Using the reflection principle (Theorem 33 of Chap. I) we have

Note that in the course of the proof of the lemma we obtained C ( p ,a) 2l/p exp{a pa2/2).

+

=

Corollary. Let - m < r < m and p < m , and let (AT(x,y)t)t20 be as given in Theorem 43. Then &(AT(x,y)) is locally in S P , uniformly in x, y. Proof. We need to show that there exists a sequence of stopping times Te increasing to m as., and constants Ce < m , such that 11 &(AT(x, y))Te Ce for all x, y E Rn, x # y. By stopping, we may assume that Z" and [Z", 201 are in 1, and that /Hal and IJalOI b for all (x, y), 1 5 a , ,B m. Therefore C for a constant C , since if X E - and if K is bounded, predictable, then K - X E fioo and I J K .Xlliim 5 IIKlls- IIXllem, as can be proved exactly analogously to Emery's inequalities ( ~ x e o r e m3). The result now follows by the preceding lemma.

<

<

1,

<

<

Comment. A similar result in the right continuous case is proved by a different method in the proof of Theorem 62 in Sect. 10. Theorem 44. Let Z a be continuous semimartingales, 1 5 a 5 m, and F an n x m matrix of process Lipschitz operators. Then the flow of the solution of

is strongly injective on Rn Proof. It suffices to show that for any compact set C C Rn, for each N , there exists an event of probability zero outside of which for every x, y E C with x # Y, inf IIX(s,w,x) - X(s,w, y)II > 0. s
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V Stochastic Differential Equations

Let x, y, s have rational coordinates. By Theorem 43 a s .

The left side of the equation is continuous (Theorem 37). As for the right side, E(A,(x, y)) will be continuous if we can show that the processes HF(x, y) and Jr>fl(x,y), given in Theorem 43, verify the hypotheses of Theorem 42. To this end, let B be any relatively compact subset of Rn x Rn \ {(x,x)) (e.g., B = B1 x B2 where B1, B2 are open balls in Rn with disjoint closures). Then IIx - yllT is bounded on B for any real number r. Without loss we take r = 1 here. Let U(x, y) = X x - XY, V(x, y) = F ( X x ) - - F(XY)-, and let Va(x, y) be the a-th column of V. Then for (x, y) and (XI,y') in B we have

The first term on the right side of (**) above is dominated in absolute value by

where we are assuming (by stopping), that F has a Lipschitz constant K . Since U(x, y) - U(x1,y') = U(x, x') - U(y, y'), the above is less than

By the lemma following Theorem 43, and by Holder's and Minkowski's inequalities we may, for any p < m , find stopping times Te increasing to m a s . such that the last term above is dominated in norm by KeIl (x, y) - (XI,yf)ll for a constant Ke corresponding to G.We get analogous estimates for the second and third terms on the right side of (**) by similar (indeed, slightly simpler) arguments. Therefore Ha satisfies the hypotheses of Theorem 42, for (x, y) E B . The same is true for Ja3P, and therefore Theorem 42 shows that A, and [A,, A,] are continuous in (x, y) on B. (Actually we are using a local version of Theorem 42 with (x, y) E B C RZn instead of all of R2"; this is not a problem since Theorem 42 extends to the case x E W open in Rn, because

zP

8 Flows as Diffeomorphisms: The Continuous Case

317

Kolmogorov's Lemma does-recall that continuity is a local property.) Finally since A, and [A,, A,] are continuous in (x, y) E B we deduce that &(A,(x, y)) is continuous in {(x,y) E : x # y). We have shown that both sides of

can be taken jointly continuous. Therefore except for a set of probability zero the equality holds for all (x, y, s) E Rn x Rn x R+. The result follows because &(A,(x, Y ) ) is ~ defined for all t finite and it is never zero.

Theorem 45. Let Za be continuous semimartingales, 1 5 a 5 m, and let F be an n x m matrix of process Lipschitz operators. Let X be the solution of (*). Then for each N < ca and almost all w lim

inf (IX(s,w,x)ll=ca.

11xll-+m S I N

Proof. By Theorem 43 the equality

is valid for all r E R. For x # 0 let Yx = IIXx - Xoll-l. (Note that Yx is well-defined by Theorem 41.) Then

Define YOo = 0. The mapping x by d(x, y) = #jj#,.Indeed,

w

~ l l x l ( inspires -~ a distance d on Rn \ (0)

By Holder's inequality we have that

and therefore by the corollary to Theorem 43 we can find a sequence of stopping times (Te)e>l - increasing to ca a.s. such that there exist constants Ce with (using (* * *)) II(Yx - ~ ' 9 (Iq, ~5 'd(x, y)Ce. Next set

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V Stochastic Differential Equations

Then 11 (Y" -

11-5,. 5 C[ llx - 11' on Rn, and by Kolmogorov's Lemma (Theorem 72 of chap. IV), there exists a jointly continuous version of (t,x) H @, on Rn. Therefore l i m l l x l l . + oexists ~x and equals 0. Since

~ we have the result. (Yx)-l = l l ~ ~ 1 1 ~ 1 -1 -X"(l, Theorem 46. Let Za be continuous semimartingales, 1 5 a 5 m, and F be an n x m matrix of process Lipschitz operators. Let X be the solution of (*). Let cp : Rn -+ Rn be the flow cp(x) = X ( t ,w,x). Then for almost all w one has that for all t the function cp is surjective and moreover it is a homeomorphism from Rn to Rn.

Proof. As noted preceding Theorem 41, the flow cp is continuous from Rn t o Dn, topologized by uniform convergence on compacts; hence for a.a. w it is continuous from Bn to Rn for all t. The flow cp is injective a.s. for all t by Theorem 44. Next observe that the image of Rn under cp, denoted cp(Rn), is closed. Indeed, let cp(Rn) denote its closure and let y E cp(Rn). Let (xk) denote a sequence such that limk,, cp(xk) = y. By Theorem 45, l i m s u ~ ~ 1)xkJI . + ~< 00, and hence the sequence (xk) has a limit point x E Rn. Continuity implies cp(x) = y, and we conclude that cp(Rn) = cp(Rn); that is, cp(Rn) is closed. Then, as we have seen, the set {xk) is bounded. If xk does not converge to x, there must exist a limit point z # x. But then cp(z) = y = cp(x), and this violates the injectivity, already established. Therefore cp-' is continuous. Since cp is a homeomorphism from Rn to cp(Rn), the subspace cp(Rn) of Rn is homeomorphic to a manifold of dimension n in Rn; therefore by the theorem of the invariance of the domain (see, e.g., Greenberg [84, page 82]), the space cp(Bn) is open in Bn. But cp(Rn) is also closed and non-empty. There is only one such set in Rn that is open and closed and non-empty and it is the entire space Rn. We conclude that cp(Bn) = Rn. Comment. The proof of Theorem 46 can be simplified as follows: extend cp to the Alexandrov compactification R: = Rn U (00) of Rn to p by

Then p is continuous on R: by Theorem 45, and obviously it is still injective. Since R: is compact, p is a homeomorphism of R: onto p(R2). However R: is topologically the sphere Sn,and thus it is not homeomorphic to any proper subset (this is a consequence of the Jordan-Brouwer Separation Theorem (e.g., Greenberg [84, page 791). Hence F(R:) = R:.

8 Flows as Diffeomorphisms: The Continuous Case

319

We next turn our attention to determining when the flow is a diffeomorphism of Bn. Recall that a diffeomorphism of Rn is a bijection (one to one and onto) which is CM and which has an inverse that is also Coo. Clearly the hypotheses on the coefficients need to be the intersection of those of Sect. 7 and process Lipschitz. First we introduce a useful concept, that of right stochastic exponentials, which arises naturally in this context. For given n, let Z be an n x n matrix of given semimartingales. If X is a solution of

where X is an n x n matrix of semimartingales and I is the identity matrix, then X = &(Z), the (matrix-valued) exponential of Z. Since the space of n x n matrices is not commutative, it is also possible to consider right stochastic integrals, denoted rt

( z : H)t

=

] (dZs)Hs, 0

where Z is an n x n matrix of semimartingales and H is an n x n matrix of (integrable) predictable processes. If ' denotes matrix transpose, then ( Z : H)

=

(H' . Z')',

and therefore right stochastic integrals can be defined in terms of stochastic integrals. Elementary results concerning right stochastic integrals are collected in the next theorem. Note that Y- dZ and [Y,Z] denote n x n matrix-valued processes here. Theorem 47. Let Y, Z be given n x n matrices of semimartingales, H an n x n matrix of locally bounded predictable processes. Then,

Sot

+

+

(i) YtZt - -020 = Ys-dZs S,~(~Y,)Z,- [Y, 21,; (ii) [ H . Y, Z] = H . [Y,Z]; and (iii) [Y,Z : H] = [Y, Z] : H. Moreover if F is an n x n matrix of functional Lipschitz operators, then there exists a unique n x n matrix of D-valued processes which is the solution of

Proof. The first three identities are easily proved by calculating the entries of the matrices and using the results of Chap. 11. Similarly the existence and uniqueness result for the stochastic integral equation is a simple consequence of Theorem 7. Theorem 47 allows the definition of the right stochastic exponential.

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V Stochastic Differential Equations

Definition. The right stochastic exponential of an n x n matrix of semimartingales Z , denoted .SR(Z),is the (unique) matrix-valued solution of the equation ft

We illustrate the relation between left and right stochastic exponentials in the continuous case. The general case is considered in Sect. 10 (see Theo( &(Z1)'. 2 ) rem 63). Note that ~ ~ =

Theorem 48. Let Z be an n x n matrix of continuous semimartingales with + Zo = 0. Then & ( Z ) and E ~ ( -+z[ Z ,Z ] ) are inverses; that is, & ( Z ) I ~ ( - Z [ Z ,Z ] )= I .

+

Proof. Let U = & ( Z )and V = ~ ~ ( -[ Z 2 ,Z ] ) .Since UoVo = I , it suffices to show that d(Ut&) = 0, all t > 0. Note that dV = (-dZ + d[Z,Z ] ) V , (dU)V = (UdZ)V, and d [U,V ]= Ud [ Z ,V ]= -Ud [ Z ,Z ]V . Using Theorem 47 and the preceding,

+

d ( U V ) = UdV (dU)V + d[U,V ] = U(-dZ + d[Z,Z ] ) V + UdZV - Ud[Z,Z ] V = 0, and we are done. The next theorem is a special case of Theorem 40 (of Sect. 7 ) ,but we state it here as a separate theorem for ease of reference.

Theorem 49. Let ( Z 1 , .. . , Z m ) be continuous semimartingales with Z i = 0, 1 I i I m , and let 1 I i I n, 1 I a I m, be functions mapping Rn to R, with locally Lipschitz partial derivatives up to order N , 1 5 N 5 oo, and bounded first derivatives. Then there exists a solution X ( t , w, x ) to

(fi),

such that its flow cp : x

-+ X ( x , t , w)

is N times continuously differentiable on

Rn. Moreover the first partial derivatives satisfy the linear equation

where 6; is Kronecker's delta.

9 General Stochastic Exponentials and Linear Equations

321

Observe that since the first partial derivatives are bounded, the coefficients are globally Lipschitz and it is not necessary to introduce an explosion time. Also, the value N = oo is included in the statement. The explicit equation for the partial derivatives comes from Theorem 39. Let D denote the n x n matrix-valued process

The process D is the right stochastic exponential ER(Y), where Y is defined

Combining Theorems 48 and 49 and the above observation we have the important following result.

Theorem 50. With the hypotheses and notation of Theorem 49, the mat* Dt is non-singular for all t > 0 and x E Rn, a.s. Theorem 51. Let (Z1,. . . ,Zm) be continuous semimartingales and let ( f i ) , 1 5 i 5 n, 1 5 a 5 m, be functions mapping Rn to R, with partial derivatives of all orders, and bounded first partials. Then the flow of the solution of

is a diffeomorphism from Rn to Rn. Proof. Let cp denote the flow of X . Since ( f : ) l
9 General Stochastic Exponentials and Linear Equations Let Z be a given continuous semimartingale with Zo = 0 and let &(Z), denote the unique solution of the stochastic exponential equation

Then Xt = &(Z)t = exp{Zt - i [ Z , Zit) (cf., Theorem 37 of Chap. 11). It is of course unusual to have a closed form solution of a stochastic differential

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V Stochastic DifferentialEquations

equation, and it is therefore especially nice to be able to give an explicit solution of the stochastic exponential equation when it also has an exogenous driving term. That is, we want to consider equations of the form

where H E D (chdlhg and adapted), and Z is a continuous semimartingale. A unique solution of (**) exists by Theorem 7. It is written EH(Z).

Theorem 52. Let H be a semimartingale and let Z be a continuous semimartingale with Zo = 0. Then the solution EH(Z) of equation (**) is given

Proof. We use the method of "variation of constants." Assume the solution is of the form Xt = Ct Ut, where Ut = E(Z)t, the normal stochastic exponential. The process C is chdl8g while U is continuous. Using integration by parts,

If X is the solution of (**), then equating the above with (**) yields

or dHt = Utd{Ct

+ [C, Z]t).

Since U is an exponential it is never zero and 1/U is locally bounded. Therefore

Calculating the quadratic covariation of each side with Z and noting that [[C,Z], Z] = 0, we conclude 1

[- . H, Z] = [C,Z]. U

Therefore equation (* * *) becomes 1

-dH = dC U

Sot

1 + -d[H, U

and Ct = U; 'd(H, - [H, Z],). Recall that Ut the theorem is proved.

Z], = E(Z)t

and Xt = Ct Ut , and

9 General Stochastic Exponentials and Linear Equations

323

(Z), it is worthwhile Since &(z), = l/&(Z)t appears in the formula for IH to note that (for Z a continuous semimartingale)

d(&)= and also

dZ - d [Z,Z] E(Z)

*

= I ( - Z + [Z, Z]). &(Z) A more complicated formula for I H ( Z ) exists when Z is not continuous (see Yoeurp-Yor [236]).The next theorem generalizes Theorem 52 to the case where H is not necessarily a semimartingale.

-2-

Theorem 53. Let H be cadlag, adapted (i.e., H E D), and let Z be a continuous semimartingale with Zo = 0. Let Xt = IH(Z)t be the solution of

Then Xt = IH(Z), is given by

Proof. Let Yt

= Xt - Ht.

Then Y satisfies

where K is the semimartingale H- . Z. By Theorem 52,

and since KO= 0 and [K, Z]t

=

H,- d[Z,Z], ,

from which the result follows. Theorem 54 uses the formula of Theorem 52 to give a pretty result on the comparison of solutions of stochastic differential equations.

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V Stochastic Differential Equations

Lemma. Suppose that F is functional Lipschitz such that if Xt(w) = 0, then F(X)t-(w) > 0 for continuous processes X. Let C be a continuous increasing process and let X be the solution of

with xo > 0. Then P ( 3 t

> 0 : Xt 5 0)

= 0.

>

Proof. Let T = infit > 0 : Xt = 0). Since Xo 2 0 and X is continuous, X, 0 for all s < T on {T < 00). The hypotheses then imply that F(X)T- > 0 on {T < oo), which is a contradiction. Comment. In the previous lemma if one allows xo = 0, then it is necessary to add the hypothesis that C be strictly increasing at 0. One then obtains the same conclusion. Theorem 54 (Comparison Theorem). Let (Za)l H(X)tfor any continuous semimartingale X. Finally, let X and Y be the unique solutions of

where xo > yo x, 5 yt} = o .

and F and Z are written in vector notation. Then P ( 3 t > 0 :

Proof. Let

Then Ut = Ct

Next set

+

U,dN,,

and by Theorem 52

9 General Stochastic Exponentials and Linear Equations

325

and define the operator K on continuous processes W by

+

Note that since G and H are process Lipschitz, if Wt = 0 then G(W&(N) Y)t- = G(Y)t-. Therefore K has the property that Wt(w) = 0 implies that K(W)t- > 0. Note further that K(V) = G(U Y) - H(Y) = G(X) - H(Y). Next observe that

+

where Dt = J;+ E(N);'dA,. Then D is a continuous, adapted, increasing process which is strictly increasing at zero. Since V satisfies an equation of the type given in the lemma, we conclude that a.s. Vt > 0 for all t. Since E ( N ) , ~is strictly positive (and finite) for all t 2 0, we conclude Ut > 0 for all t > 0, hence Xt > Yt for all t > 0.

Comment. If x o > yo (i.e., x o = yo is not allowed), then the hypothesis that A is strictly increasing at 0 can be dropped. The theory of flows can be used to generalize the formula of Theorem 52. In particular, the homeomorphism property is used to prove Theorem 55. Consider the system of linear equations given by

where H = (Hi), 1 5 i 5 n is a vector of n semimartingales, X takes values in Bn, and Aj is an n x n matrix of adapted, cadlag processes. The processes Z j , 1 5 j 5 m, are given, continuous semimartingales which are zero at zero. Define the operators Fj on IDn by

where Aj is the n x n matrix specified above. The operators Fj are essentially process Lipschitz. (The Lipschitz processes can be taken t o be IIAill which

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V Stochastic Differential Equations

is cAdlAg, not cAglAd, but this is unimportant since one takes F ( X ) t - in the equation.) Before examining equation (*4), consider the simpler system ( 1 5 i , k 5

n),

where

Letting I denote the n x n identity matrix and writing the preceding in matrix notation yields

where U takes its values in the space of n x n matrices of adapted processes in D.

Theorem 55. Let A j , 1 5 j 5 m, be n x n matrices of cadlag, adapted processes, and let U be the solution of (*5). Let XF be the solution of (*4) where Ht = x , x E Rn. Then XF = Utx and for almost all w, for all t and x the matrix Ut(w) is invertible.

Proof. Note that Ut is an n x n matrix for each ( t , w ) and x E B n , so that Utx is in Rn. If X: = U t x , then since the coefficients are process Lipschitz we can apply Theorem 46 (which says that the flow is a homeomorphism of R n ) to obtain the invertibility of Ut(w). Note that U is also a right stochastic exponential. Indeed, U = I R ( v ) , where & = C?==, A:- dz;, and therefore the invertibility also follows from Theorem 48. Thus we need t o show only that X: = Utx. Since Utx solves (*4) with Ht = x, we have Utx = X,3i a.s. for each x . Note that a.s. the function x H U ( W ) Xis continuous from Rn into the subspace of Dn consisting of continuous functions; in particular ( t ,x ) ++ Ut(w)x is continuous. Also as shown in the proof of Theorem 46, ( x ,t ) ++ X,3i is continuous in x and right continuous in t . Since Utx = X: a.s. for each fixed x and t , the continuity permits the removal of the dependence of the exceptional set on x and t . Let U-l denote the n x n matrix-valued process with continuous trajectories a.s. defined by ( U - ' ) t ( w ) = (Ut(w))-'. Recall equation (*4)

9 General Stochastic Exponentials and Linear Equations

327

where H is a column vector of n semimartingales and 2; = 0. Let [H,Zj] denote the column vector of n components, the ith one of which is [Hi, Zj].

Theorem 56. Let H be a column vector of n semimartingales, Z j (1 5 j 5 m) be continuous semimartingales with 2; = 0, and let Aj, 1 5 j 5 m be n x n matrices of processes in D. Let U be the solution of equation (*5). Then the solution X H of (*4) i s given by

Proof. Write xHas the matrix product UY. Recall that U-' exists by Theo~ a~semimartingale, that we need to find explicitly. rem 48, hence Y = l 7 - l is Using matrix notation throughout, we have

Integration by parts yields (recall that U is continuous)

by replacing X with UY on the right side above. However U satisfies (*5) and therefore

and combining this with the preceding gives

or equivalently dY = u - ' ~ H- U-'~[u,Y].

(*6) Taking the quadratic covariation of the preceding equation with Z, we have

since [UP1d[U,Y], Zj] = 0, 1

<j

5 m. However since U satisfies (*5),

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V Stochastic Differential Equations

since U equals U-. Substitute the above expression for d[U,Y] into (*6) and we obtain m

and since XH = UY, the theorem is proved.

10 Flows as Diffeomorphisms: The General Case In this section we study the same equations as in Sect. 8, namely

except that the semimartingales (Za)lsasm are no longer assumed to be continuous. For simplicity we still assume that Zo = 0. In the general case it is not always true that the flows of solutions are diffeomorphisms of Rn, as the following example shows. Example. Consider the exponential equation in R,

Let Z be a semimartingale, Zo = 0, such that Z has a jump of size -1 at a stopping time T, T > 0 a.s. Then all trajectories, starting at any initial value x, become zero at T and stay there after T, as is trivially seen by the closed form of the solution with initial condition x:

Therefore, injectivity of the flow fails, and the flow cannot be a diffeomorphism of R. We examine both the injectivity of the flow and when it is a diffeomorphism of Rn. Recall the hypotheses of Sect. 7, to which we add one, denoted (H3). Hypothesis ( H I ) . Za are given semimartingales with Zg = 0, 1 < cr Hypothesis (H2). f: : Rn -+ R are given functions, 1 5 i and f (x) denotes the n x m matrix (f:(x)). The system of equations

< m.

< n, 1 < cr 5 m,

10 Flows as Diffeomorphisms: The General Case

329

may also be written

where Xt and x are column vectors in Rn, f (Xs-) is an n x m matrix, and Z is a column vector of m semimartingales.

Hypothesis (H3). f is Cm and has bounded derivatives of all orders. Note that by Theorem 40 of Sect. 7, under (H3) the flow is Cm. The key to studying the injectivity (and diffeomorphism properties) is an analysis of the jumps of the semimartingale driving terms. Choose an E > 0, the actual size of which is yet to be determined. For (Zo)llolm we can find stopping times 0 = To < TI < T2 < . . . tending a.s. to co such that z",j= (zff)Tj- - (zo)Tj-l have an HOO - norm less than E (cf., Theorem 5). Note that by Theorem 1, [ Z o ~ jZff>j]Z2 , < E as well, hence the jumps of each Z a > jare smaller than E. Therefore all of the "large" jumps of Zo>joccur only at the times (Tj), j 2 1. Let X;(X) denote the solution of (*) driven by the semimartingales Zo?j. Outside of the interval (Tj-1, Tj) the solution is

Next define the linkage operators

using vector and matrix notation. We have the following obvious result.

Theorem 57. The solution X of (*) is equal to, for Tn 5 t < Tn+l,

where

xo+ = x 21- = x $ 1 - ( ~ ) 7 2 2 - = X$2-(~1+), X,

-

X I + = H1(x1-) x2+ = H2(x2--)

= X T,n ( ~ ( n - l ) + ) , xn+ = Hn(xn-).

Theorem 58. The flow cp : x -, Xt(x,w) of the solution X of (*) is a diffeomorphism if the collections of functions

are diffeomorphisms.

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V Stochastic Differential Equations

Proof By Theorem 57, the solution (*) can be constructed by composition of functions of the types given in the theorem. Since the composition of diffeomorphisms is a diffeomorphism, the theorem is proved.

-

-

We begin by studying the functions x x$~ (x, w) and x x,"' '(x, w). Note that by our construction and choice of the times T j , we need only to consider the case where Z = Zj has a norm in H" smaller than E. The following classical result, due to ~ a d a m a r d underlies , our analysis.

Theorem 59 (Hadamard's Theorem). Let g : Rn (2) (22)

+ Rn

be C". Suppose

limllzll-oo Ilg(x)II = a, and the Jacobian matrix gt(x) is an isomorphism of Rn for all x.

Then g is a diffeomorphism of Rn. Proof. By the Inverse Function Theorem the function g is a local diffeomorphism, and hence it suffices to show it is a bijection of Rn. To show that g is onto (i.e., a surjection), first note that g(Rn) is open and non-empty. It thus suffices to show that g(Rn) is a closed subset of Rn, since Rn itself is the only nonempty subset of Rn that is open and closed. Let be a sequence of points in Rn such that limi,, g(xi) = y E Rn. We will show that y E g(Rn). Let xi = tivi, where ti > 0 and llvill = 1. By choosing a subsequence if necessary we may assume that vi converges to v E S n , the unit sphere, as i tends to co.Next observe that the sequence (ti)i2l must be bounded by condition (i) in the theorem: for if not, then ti = IIxill tends to co along a subsequence and then llg(xik)11 tends to co by (i), which contradicts that limi,, g(xi) = y. Since - is bounded we may assume lirni-+" ti = to E Rn again by taking a subsequence if necessary. Then limi,, xi = tov, and by the continuity of g we have y = lirni," g(xi) = g(tov). To show g is injective (i.e., one-to-one), we first note that g is a local homeomorphism, and moreover g is finite-to-one. Indeed, if there exists an infinite sequence (x,),?~ such that g(xn) = yo, all n, for some yo, then by condition (i) the sequence must be bounded in norm and therefore have a cluster point. By taking a subsequence if necessary we can assume that xn tends to 2 (the cluster point), where g(xn) = yo, all n. By the continuity of g we have g(2) = yo as well. This then violates the condition that g is a local homeomorphism, and we conclude that g is finite-to-one. Since g is a finite-to-one surjective homeomorphism, it is a covering map.16 However since Rn is simply connected the only covering space of Rn is Rn (the fundamental group of Rn is trivial). Therefore the fibers g-l(x) for x E Rn each consist of one point, and g is injective. and x e The next step is to show that the functions x ct X&~(X,W) X;+'(X, w) of Theorem 58 satisfy the two conditions of Theorem 59 and are l6

For the algebraic topology used here, the reader can consult, for example, Munkries 1183, Chapter 81.

10 Flows as Diffeomorphisms: The General Case

331

thus diffeomorphisms. This is done in Theorems 62 and 64. First we give a result on weak injectivity which is closely related to Theorem 41.

Theorem 60. Let 2" be semimartingales, 1 I cr 5 m with Z$ = 0, and let F be a n n x m matrix of process Lipschitz operators with non-random Lipschitz < E, constant K . Let Hi E D, 1 J i I n (ccidlcig, adapted). ~f C:, llZLYllNm for E > 0 suficiently small, then the flow of the solution of

is weakly injective.17

XY denote the solutions of the above equaProof. Let x,y E Rn,and let Xx, tion with initial conditions x, y, respectively. Let u = x - y, U = Xx- XU, and V = F ( X x ) - - F(XY)-. Then V E IL and ( V (5 K(U-1. Also,

Therefore AU, = C , V:AZF and moreover (using the Einstein convention to leave the summations implicit)


1

- I

4

if E is small enough. Consequently IJU,JJ2 11 u,- 1). Define T = infit > 0 : Ut- = 0). Then Ut- # 0 on [0, T)and the above implies Ut # 0 on [0,T)as well. Using It6's formula for f (x)= log IIxlI,a s in the proof of Theorem 41 we have

For s fixed, let 17

We are using vector and matrix notation, and the Einstein convention on sums. The Einstein convention is used throughout this section.

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V Stochastic Differential Equations

so that the last sum on the right side of equation (**) can be written Co,s,t J,. By Taylor's Theorem

where f (x) = log llxll, and Is denotes the segment with extremities Us and U,-. Since

and since l/AUsll

< ~ I I u ~ - ~ / , we deduce

which in turn implies

Therefore

on [0, T ) .Returning to (**), as t increases to T , the left side tends to -co on {T < c o ) and the right side remains finite. Therefore P ( T < co) = 0, and U and U- never vanish, which proves the theorem.

Theorem 61. Let (Za)11a5, be semimartingales, Zg = 0, F a n n x m matrix of process Lipschitz operators with a non-random Lipschitz constant, and Hi E D, 1 5 i 5 n. If (IZa\(Hm< E for E > 0 suficiently small, then for r E R there exist uniformly Gcally bounded predictable processes Ha(x, y ) and Ka>@(x, y ) , which depend on r , such that

where X x is the solution of

The semimartingale A, is given by

10 Flows as Diffeomorphisms: The General Case

333

where Jt = Co,,5tA,, and where A, is a n adapted process such that IA,I I C~(AZ,~)~. Proof. Fix x, y E Rn, x # y, and let U = X x - Xu, V = F ( X x ) - - F(XY)-. By Theorem 60 U is never zero. As in the proof of Theorem 43, by It6's formula,

Let L, denote the summands of the last sum on the right side of the above equation. If g(x) = Ilxllr, then

where I, is the segment with boundary Us and Us-. However,

However as in the proof of Theorem 60, we which is less than have llAU,ll 2 +llU,-ll, which implies (for all r positive or negative) that llxllr 5 CrllUs- /Ir for all x between Us- and Us. Hence (the constant C changes) IL,I I ~

1 l ~ ~ s 1 1 ~ 1 I ~ s -

I ~IIus-ll~ll~zsll~. Therefore, let A, = IIU,-II-rL,,

and we have

an absolutely convergent series with a bound independent of (x, y). To complete the proof it suffices t o take

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V Stochastic Differential Equations

as in the proof of Theorem 43. Note that l ~ u s - f o )is indistinguishable from the zero process by weak injectivity (Theorem 60). These choices for Ha and Kal@are easily seen to work by observing that U; = C:=, I/,"ladZF, and the preceding allows us to conclude that

and the result follows.

Theorem 62. Let (Za)15as, be semimartingales, Zg = 0, F a n n x m matrix of process Lipschitz operators with a non-random Lipschitz constant, and Hi E ID,1 L i I n. Let X = X(t,w, x) be the solution of

If C:=,

llZallim almost all w

< E for E > 0 sufzciently small, then for each N < co and lim

inf IIX(s, w,x)ll = co.

11x11+00 S I N

Proof. The proof is essentially the same as that of Theorem 45, so we only sketch it. For x # 0 let Yx = ( ( X x- XO((-l, which is well-defined by weak injectivity (Theorem 60). Then

by Theorem 61, where A,(x, y) is as defined in Theorem 61. Set YW = 0. Since llZllHm < E each Za has jumps bounded by E, and the process Jt defined in heo or em 61 also has jumps bounded by C,E~.Therefore we can stop the processes A,(x, y) at an appropriately chosen sequence of stopping times (Te)e>1 increasing to co a.s. such that each A,(x, y) E S(E)for a given E, and for each l , uniformly in (x, y). However if Z is a semimartingale in S(E), t then since £(A,(x, y)) satisfies the equation Ut = 1 Us-dAr(x, Y ) ~by , Lemma 2 of Sect. 3 of this chapter we have

+ So+

10 Flows a s Diffeomorphisms: The General Case

335

<

where C(p, z ) is a constant depending on p and z = 11 A, (x, y) llNm kt, the bound for l , provided of course that E is sufficiently small. We Gnclude that for these Te there exist constants Ce such that

where p Set

> n , and where d is the distance on Rn \ {0) given by d(x, y) =

I ,C:llx Then II(Y"- ~ y ) ~ '-l l & -

M.

- y((pon Rn, and by Kolmogorov's Lemma

(Theorem 72 of Chap. IV) we conclude that limllxll,oY" exists and it is zero. Since (Yx)-' = l l ~ " l l " l l - ~ - X"ll, the result follows. If cp is the flow of the solution of (*), Theorem 62 shows that

and the first condition in Hadamard's Theorem (Theorem 59) is satisfied. Theorem 63 allows us to determine when the second condition in Hadamard's Theorem is also satisfied (see Theorem 64), but it has an independent interest. First, however, some preliminaries are needed. For given n , let Z be an n x n matrix of given semimartingales. Recall that X = I ( Z ) denotes the (matrix-valued) exponential of 2, and that I R ( Z ) denotes the (matrix-valued) right stochastic exponential of Z, which was defined in Sect. 8, following Theorem 47. Recall that in Theorem 48 we showed that if Z is an n x n matrix of continuous semimartingales with Zo = 0, then I ( Z ) I R ( - Z + [Z, Z]) = I , or equivalently I ( - Z + [Z, z ] ) E ~ ( Z )= I . The general case is more delicate.

Theorem 63. Let Z be an n x n matrix of semimartingales with Zo = 0. ' is a wellSuppose that Wt = -Zt + [Z, 21,"+ C,,,,,- (I + ~ 2 , ) -(AZ,)2 defined semimartingale. Then

for all t

> 0.

Proof. Let U = I ( W ) , V = I R ( Z ) , and Jt = Co.,s,t(IThen,

and therefore

+ Az,)-'(AZ,)~.

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V Stochastic Differential Equations

since d[Z,[ Z ,Z ] = 0. By Theorem 47

d(UV) = U-dV

+ (dU)V- + d[U,V ] ;

using the above calculations several terms cancel, yielding

2, Since [ Z ,Z]t = [ Z ,z],C+ C o < s s t ( A ~ s )and [J,Z ] , =

C

AJ,AZ, =

C ( A Z , ) ~ (+I AZ,)-', O<s
O<s
the preceding becomes

Since

+

( ~ 2= , )( ~ ( ~ 2 , ()A~Z , ) ~ ) (+I az,)-l,

the above equation implies d(UV)t = 0, all t therefore Ut& = I , all t 2 0.

2

0. However UoVo = I , and

Corollary. Let Z be a square matrix of semimartingales. If llZllHm - < E for E > 0 sufficiently small, then I R ( Z ) t is invertible for all t 2 0.

Proof. If 11 ZllHm < E then the jumps of Z are bounded by E > 0 and therefore the process W-of Theorem 63 is a well-defined semimartingale. Theorem 64. Let ( Z a ) l S a s , be semimartingales with Zo = 0, and let f be a matrix of coeficients satisfying Hypotheses (Hz) and (H3). Let X be the unique solution of

xt

t

=x

+ f(xs-)dzs.

(*)

The Jacobian matrix DL(t,w, x ) = & x i ( t , w, x ) is invertible for each t 2 0 provided llZllHm - < E for suficiently small E > 0.

10 Flows as Diffeomorphisms: The General Case

337

Proof. By Theorem 39 (in Sect. 7) the Jacobian matrix D satisfies the right stochastic exponential equation

bfi

and the matrix semimartingale differential g ( X s - ) d Z F satisfies the hypotheses of the corollary of Theorem 63, whence the result. Before stating the principal result of this section, we need t o define two subsets of Rm; recall that under Hypotheses (Hl), (H2), and (H3), that Z = (Z")llollm is a given m-tuple of semimartingales and that f (x) = (fA(x)) is an n x m matrix of C" functions. Let 2) = {z E Rm : H ( x ) = x

Z = {z E Rm : H ( x ) = x

+ f (x)z is a diffeomorphism of Rn) + f (x)z is injective in Rn).

Clearly 2) c Z .

Theorem 65. Let Z and f be as given i n Hypotheses (HI), (H2), and (H3), and let X be the solution of

The flow of X is a.s. a diffeomorphism of Rn (resp. trajectories of X fiom different initial points a.s. never meet) for all t if and only if all the jumps of Z belong to 2) (resp. all the jumps of Z belong to Z ) . Proof. Recall the processes

xgj(x, w) and x;+' (x, w) defined in Theorem 57,

+

and the linkage operator H ~ ( x = ) x f (x)AZT,, defined immediately preceding Theorem 57. By hypothesis the linkage operators H j ( x ) are clearly diffeomorphisms of Rn (resp. injective), and by Theorems 62 and 64, Hadamard's conditions are satisfied (Theorem 59), and therefore the functions x ++ (x,w) and x w x;+'(x, W) are diffeomorphisms of Rn if E > 0 is taken small enough in the definition of the stopping times (Tj)j21, which it is always possible t o do. Therefore by Theorem 58 the flow cp : x -+ Xt(x,w) is a s . a diffeomorphism of Rn for each t > 0. The necessity is perhaps the more surprising part of the theorem. First observe that by Hadamard's Theorem (Theorem 59) the set 2) contains a neighborhood of the origin. Indeed, if z is small enough and x is large enough then 11 f (x)zII 5 11x11/2 since f is Lipschitz, which implies that IIH(x)II 11x11/2 and thus condition (i) of Hadamard's Theorem is satisfied. On the other hand H f ( x ) = I f '(x)Z is invertible for all x for llzll small enough because f '(x) is bounded; therefore condition (ii) of Hadamard's Theorem is satisfied. Since f (x) is C* (Hypothesis (H3)), we conclude that 2) contains a neighborhood of the origin.

x:+

>

+

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V Stochastic Differential Equations

To prove necessity, set

rl = {w : 3 s > 0 with AZ,(w) r2= {W : 3 s > 0 with AZ,(w)

E DC),

E ZC).

Suppose P ( r l ) > 0. Since V contains a neighborhood of the origin, there exists an E > 0 such that all the jumps of Z less than E are in 2). We also take E so small that all the functions x ++ X+i (x) are diffeomorphisms as soon as the linkage operators H~ are, all k < i. Since the jumps of Z smaller than E are in D, the jumps of Z that are in DC must take place at the times Ti. Let ~j

= {w : AZTi E D, all

i < j, and AZTj E VC).

Since P ( r 1 ) > 0, there must exist a j such that P(Aj) > 0. Then for w E Aj, H XT~-(X,W) is a diffeomorphism, but Hj(x,w) is not a diffeomorphism. Let wo E Aj and to = Tj (wo). Then x ++ Xto(x, wo) is not a diffeomorphism, and therefore

x

P{w : 3 t such that x

+ Xt(x,w)

is not a diffeomorphism)

> 0,

and we are done. The proof of the necessity of the jumps belonging to Z to have injectivity is analogous.

Corollary. Let Z and f be as given in Hypotheses (Hl), (H2), and (H3), and let X be the solution of

Then different trajectories of X can meet only at the jumps of Z. Proof. We saw in the proof of Theorem 65 that two trajectories can intersect only at the times Tj that slice the semimartingales Z" into pieces of norm less than E. If the Z" do not jump at Tjo for some jo, however, a n d paths of X intersect there, one need only slightly alter the construction of Tjo (cf., the proof of Theorem 5, where the times Tj were constructed), so that Tj, is not included in another sequence that €-slices (Z")llollm, to achieve a contradiction. (Note that if, however, (Za)llalm has a large jump at Tjo, then it cannot be altered.)

a"

11 Eclectic Useful Results on Stochastic Differential

Equations We begin this collection of mostly technical results concerning stochastic differential equations with some useful moment estimates. And we begin the moment estimates with preliminary estimates for stochastic integrals. The first result is trivial, and the second is almost as simple.

11 Eclectic Useful Results on Stochastic Differential Equations

Lemma. For any predictable (matrix-valued) process H and for any p and for 0 5 t 5 1, we have

339

>

1,

Proof. Since 0 < t < 1, we have that d s on [ O , l ] is a probability measure, and since f ( x ) = X P is convex for p > 1, we can use Jensen's inequality. The result then follows from Fubini's Theorem.

Lemma. For any predictable (matrix-valued) process H and multidimensional Brownian motion B, and for any p 2 2, and for 0 5 t 5 1, there exists a constant cp depending only on p such that

Proof. The proof for the case p = 2 is simply Doob's inequality; we give it for the one dimensional case.

where the last equality is by Fubini's Theorem. For the case p Burkholder's inequality (see Theorem 48 of Chap. IV,

>

2, we use

where the last inequality follows from Jensen's inequality (recall that p 2 2 ~ a convex function, and since 0 5 t 5 1 we have used so that f ( x ) = X P / is that d s on [O,l] is a probability measure) and Fubini's Theorem.

Theorem 66. Let Z be a d-dimensional Lkvy process with E{II Zt llLp)< for 0 t 1, H a predictable (matrix-valued) process and p 2 2. T h e n there exists a finite constant Kp such that for 0 5 t 5 1 we have

< <

Proof. We give the proof for the one dimensional case.18 Since the L6vy process Z can be decomposed Zt = bt aBt Mt, where M is a purely discontinuous martingale (that is, M is orthogonal in the L 2 sense to the stable subspace generated by continuous L 2 martingales), it is enough to prove the inequality separately for Zt = bt, and Zt = a B t , and Zt = Mt. The preceding two lemmas prove the first two cases, so we give the proof of only the third.

+

l8

+

An alternative (and simpler) proof using random measures can be found in [105].

340

V Stochastic Differential Equations

In the computations below, the constants Cp and the functions Kp(.) vary from line t o line. Choose the rational number k such that 2k 5 p < 2kf1. Applying the Burkholder-Gundy inequalitieslg for p _> 2 we have

Set CYM:= E{[M, MI1)

=

E{ClAMSl

Ix12uM(dx) < a.

s< 1

Since [M, M ] is also a L6vy process, we have that [M, MIt - f f ~ist also a martingale. Therefore the inequality above becomes

We apply a Burkholder-Gundy inequality again t o the first term on the right side above t o obtain

We continue recursively t o get

Next we use the fact that, for any sequence a such that Ilalllq is finite, llalll2 I Ilalllq for 1 q 2. As 1 5 p2-k < 2 we get

< <

whence l9

The Davis inequality is for the (important) case p

= 1 only.

11 Eclectic Useful Results on Stochastic Differential Equations

E{{C

5

341

E{CI H ~ A M ~ ~ P ) .

s
s
Note that C s < t lAMsIP is an increasing, adapted, c&dl?ig process, and its (non-random) compensator is At = t Jx(PuM(dx),which is finite by hypothesis. Since /HIP is a predictable process,

is a martingale with zero expectation. Therefore we have

It remains t o show that, for any 1

< i 5 k,

Let AM := J Ix12uM(dx),SO that

is a probability measure. Denote 2' by q. One has to show

the preceding inequality is obvious. On the other hand, if

then it is sufficient t o prove that

But the bound on AM and Jensen's inequality give the result.

342

V Stochastic Differential Equations

Theorem 66 can be used t o prove many different results concerning solutions of stochastic differential equations driven by L4vy processes. We give two examples. (These types of results are true more extensively (for example in n dimensions) with similar proofs; see for example [105].) Our first example is a moment estimate. As a convenience, we continue to work on the time interval [O, 11.

Theorem 67. Let a be continuously differentiable with a bounded derivative, JxlPv(dx)< and let Z be a Le'vy process with Le'vy measure v such that a, which i s equivalent to E{IZtIP) < a for p 2 2 and for all t , 0 5 t 5 1. Let Xx denote the unique solution of the stochastic differential equation

hx,,,

Then we have E{sup IXflp) 5 K ( l S

+ IxlP)

where K is of course a positive constant. Before proving this theorem we need t o recall Gronwall's inequality.20

Theorem 68 (Gronwall's Inequality). Let a be a function from R+ to itself, and suppose rs

for 0 5 s 5 t . Then a ( t ) 5 cekt. Moreover if c = 0 then a vanishes identically. The proof is simple and usually taught in elementary differential equations courses.

Proof of Theorem 67. This is a consequence of Theorem 66 and Gronwall's inequality. The next corollary follows easily from Theorem 67.

Theorem 69. Let a and Z be as in Theorem 67. Let g be continuous and for q > 0 define

11 g I , = inf{a > 0 : Ig(x)/ I a ( 1 + 1xIq)). For q = 0 we let 1) g 110 be the L" norm. If Pt denotes the transition semigroup of the Markov process X x of equation (*), then we have for q 2 0

20

Extensions of this classic version of Gronwall's inequality are in the exercises for this chapter. See also [182].

11 Eclectic Useful Results on Stochastic Differential Equations

343

We now turn t o our second example, which consists of moment estimates on the flows of the solution Xx of equation (*). We first establish some notation. For a L6vy process Z with L6vy measure v and with finite p t h moment for p 2, we decompose it as Zt = bt cBt Mt where B is a standard Brownian motion and M is a martingale independent from B and L2 orthogonal t o all continuous martingales. We then define

>

+

+

>

which is finite since by assumption E{IZt lP) < and p 2. Here we consider the multidimensional case: both Z and the solution Xx of equation (*) are assumed to be d-dimensional, where d 1.21 We let x:"" represent a k-th derivative of the flow of Xx.

>

Theorem 70. For k 2 1 assume that a is differentiable k times and that all partial derivatives of a of order greater than or equal to one are bounded. Assume also that Z has moments of order kp with p 2 2. With the notation defined right before this theorem, we have that there exists a constant K ( k , p , a, qp) such that for 0 5 s 5 1,

Proof. We recall from Theorem 39 that Xx together with the unique solution of the system of equations

x ~ ,constitute (~)

as long as k 2 2, and with a slightly simpler formulation if k = 1. (Note that this is the same equation ( D ) as on page 305.) We can write the second equation above in a slightly more abstract way

if k 21

> 2; and if k = 1 the equation is the simpler

One could also allow the dimensions of Z and X x to be different, but we do not address that case here.

344

V Stochastic Differential Equations

We next observe that the components of F ~ , ~ ( x .(.~. ,)x(k-if , l)) are sums of terms of the form

nn

k-if1

a3

x(j)>lr, where

jaj = k,

and where x(j)ll is the 1-th component of x(j) E EXd3, and an "empty" product equals 1. We thus want now t o prove that

for all z = 2,. . . ,N and for some constant K = K ( k , p , a , qkp) (in the remainder of the proof K = K ( k , p , a , q k p ) varies from line t o line). And of course, it is enough to prove that if G is any monomial as in (**), then

We will use Theorem 66 and the fact that V i a is bounded. Note that Theorem 66 implies that

<

p' p, where the constant K(p, qp) depends only on p, qp, and the for 2 I dimensions of H and Z. For this we see that the left side of the previous equation is smaller than

by Holder's inequality, since Cjjaj = k. The recurrence assumption yields that each expectation above is smaller than some constant K(p, k , a, qkp), and we obtain the required inequality. We now turn t o the positivity of solutions of stochastic differential equations. This issue arises often in applications (for example in mathematical finance) when one wants t o model a dynamic phenomenon where for reasons relating t o the application, the solution should always remain positive. We begin with a very simple result, already in widespread use.

11 Eclectic Useful Results on Stochastic Differential Equations

345

Theorem 71. Let Z be a continuous semimartingale, a a continuous function, and suppose there exists a (path-by-path) unique, non-exploding solution to the equation rt

with X o > 0 almost surely. Let T = inf{t > 0 : X t = 0). Then P ( T < a ) = 0. In words, if the coefficient of the equation is of the form x a ( x ) with a ( x ) continuous on [0,a ) , then the solution X stays strictly positive in finite time if it begins with a strictly positive initial condition. We also remark that in the applied literature the equation is often written in the form

Note that if a ( x ) is any function, one can then write a ( x ) = x ( q ) , but we then need the new function to be continuous, well-defined for all nonnegative x , and we also need for a unique solution to exist that has no explosions. This of course amounts to a restriction on the function a . Since x a ( x ) being Lipschitz continuous is a sufficient condition to have a unique and non-exploding solution, we can require in turn for a to be both bounded and Lipschitz, which of course implies that x a ( x ) is itself Lipschitz.

Proof of Theorem 71. Define T, = inf{t > 0 : X t = l l n or X t = Xo V n ) , and note that P ( T n > 0) = 1 because P ( X 0 > 0) = 1 and Z is continuous. Using ItB1s formula up to time Tn we have

and since a is assumed continuous, it is bounded on the compact set [ l l n ,n], say by a constant c. Since the stopping times Tn t T we see that on the event { T < a)the left side of the above equation tends to a while the right side remains finite, a contradiction. Therefore P(T < a ) = 0. Theorem 71 has an analogue when Z is no longer assumed to be continuous.

Theorem 72. Let Z be a semimartingale and let a be a bounded, continuous function such that there exists a unique non-exploding solution of

almost surely with X o > 0 almost surely. If in addition we have IAZsl 5 for all s > 0 for some E. with 0 < E. < 1, then the solution X stays positive a.s. for all t > 0.

346

V Stochastic Differential Equations

Proof. 11 o ,1 refers t o the La norm of a. The proof is similar t o the proof of Theorem 71. Using ItG's formula gives

and let us consider the last term, the summation. We have

< 1 by the Mean Value Theorem, applied w-by-w, which implies

where lHsl the above is

which is a convergent series for each finite time t. The only ways X can be zero or negative is t o cross 0 while continuous, which cannot happen by an argument analogous t o the proof of Theorem 71, or to jump to 0 or across into 0] at time s-. Then we negative values. Assume X has not yet reached (-a, have

~Axs~<xs~~a(xs~)Azs~<xs~(l-~)<Xs~, which implies that X cannot cross into ( - a , O ] by jumping. Hence we have that X must be positive on [0,a ) . We now turn t o something completely different. In Sect. 6, Theorem 32, we showed that an equation of the form

with Z a vector of independent Levy processes, had a (unique) solution which was strong Markov. We now show what amounts t o a converse, which is perhaps surprising. That is, if one wants a solution of a stochastic differential equation t o be time homogeneous Markov, one essentially must have that the driving semimartingale is a Levy process!

Bibliographic Notes

347

Theorem 73. Let ( a , 3 , IF,P ) be a filtered probability space with Z a cadlag semimartingale defined o n the space. Let f be a Bore1 function which is never 0 and is such that for every x E R the equation

has unique (strong) solution, which we denote X x . If each of the processes X x is a time homogeneous Markov process with the same transition semigroup, then Z is a Ldvy process.

For the proof of this theorem, we assume the reader is familiar with the abstract theory of Markov processes, as can be found (for example) in [215]. Proof. Let a' be the function space of cBdlBg functions defined on R+ with Xt(w) = w(t), the projection operator. Let IF' be the canonical filtration and let (8;) be the shift operators. If PL denotes the law of X", then we have that (a',IF', 81, X', PL) is a Markov process in the Dynkin (or Blumenthal-Getoor) sense. Since f is never zero, we can write

We can therefore define on a', relative to each law PL, the stochastic integral

and we have that the law of Z' under PL is the law of the process Z - Zo. However Z' is also an additive functional, and hence the Markov property implies

where we have also used that the law of Z' under PL is the law of the process Z - Z o This in turn implies that Z;+, - 2,' is PL independent of 3;, and hence also independent from 2:. for all r < t , and it also implies that the law of Z;+, - Z; is the same as the law of Z, - Zo. Using the identity of the laws of L with that of the process Z - Zo once again, the result follows. Z' under P

Bibliographic Notes The extension of the HP norm from martingales to semimartingales was implicit in Protter [197]and first formally proposed by Emery [64]. A comprehensive account of this important norm for semimartingales can be found in

348

V Stochastic Differential Equations

Dellacherie-Meyer [46].Emery's inequalities (Theorem 3) were first established in Emery [64], and later extended by Meyer [174]. Existence and uniqueness of solutions of stochastic differential equations driven by general semimartingales was first established by Doleans-Dade [51] and Protter [198], building on the result for continuous semimartingales in Protter [197]. Before this Kazamaki [I231 published a preliminary result, and of course the literature on stochastic differential equations driven by Brownian motion and Lebesgue measure, as well as Poisson processes, was extensive. See, for example, the book of Gihman-Skorohod 1811 in this regard. These results were improved and simplified by Dolkans-Dade-Meyer [54] and Emery [65]; our approach is inspired by Emery [65]. Metiver-Pellaumail [161] have an alternative approach. See also Metivier [158]. Other treatments can be found in Doleans-Dade [52] and Jacod [103]. The approach of L. Schwartz [214] is nicely presented in [44]. The stability theory is due to Protter [199], Emery 1651, and also to Metivier-Pellaumail 11621. The semimartingale topology is due to Emery 1661 and Metivier-Pellaumail [162], while a pedagogic treatment is in DellacherieMeyer [46]. The generalization of Fisk-Stratonovich integrals to semimartingales is due to Meyer [171]. The treatment here of Fisk-Stratonovich differential equations is new. The idea of quadratic variation is due to Wiener [230]. Theorem 18, which is a random It6's formula, appears in this form for the first time. It has an antecedent in Doss-Lenglart [58], and for a very general version (containing some quite interesting consequences), see Sznitman [222]. Theorem 19 generalizes a result of Meyer [171], and Theorem 22 extends a result of Doss-Lenglart [58]. Theorem 24 and its corollary is from It8 [99]. Theorem 25 is inspired by the work of Doss [57] (see also IkedaWatanabe [92] and Sussman [221]). The treatment of approximations of the Fisk-Stratonovich integrals was inspired by Yor [238]. For an interesting application see Rootzen [211]. For more examples of solutions of stochastic differential equations with formulas, see [128]. The results of Sect. 6 are taken from Protter 11961 and Cinlar-JacodProtter-Sharpe [34]. A comprehensive pedagogic treatment when the Markov solutions are diffusions can be found in either Stroock-Varadhan [220] or Rogers-Williams [209, 2101. Work on flows of stochastic differential equations goes back to 1961 and the work of Blagovescenskii-Freidlin [18] who considered the Brownian case. For recent work on flows of stochastic differential equations, see Kunita [131, 1331, Ikeda-Watanabe [92] and the references therein. There are also flows results for the Brownian case in Gihman-Skorohod [81],but they are L2 rather than almost sure results. Much of our treatment is inspired by the work of Meyer [I771 and that of Uppman [224, 2251 for the continuous case, however results are taken from other articles as well. For example, the example following Theorem 38 is due to Leandre [140], while the proof of Theorem 41, the nonconfluence of solutions in the continuous case, is due to Emery [68]; an alter-

Exercises for Chapter V

349

native proof is in Uppman 12251. For the general (right continuous) case, we follow the work of Leandre [141]. A similar result was obtained by FujiwaraKunita [78]. Our presentation on moment estimates follows [205] and [105], although it dates back to [16]. A perhaps simpler treatment using random measures can be found in [105]. See also [106]. The results on when solutions of stochastic differential equations are always positive is well known, at least in the Brownian case, but a reference is not easily found in the literature. The general case may be new. Theorem 73 is from [108].

Exercises for Chapter V Exercise 1. Let X be a solution of the stochastic differential equation

with initial condition Xo # 0. Show that X is defined on [O,T), where T = inf{t > 0 : Bt = l/Xo). Moreover show that X is the unique solution on [0,T ) . Finally show that X explodes at T.

a.

Exercise 2. Let ~ ( x =) Note that 0 is Lipschitz continuous everywhere except at x = 0. Let X be a solution of the equation

with initial condition Xo. Find an explicit solution of this equation and determine where it is defined. Exercise 3. Let X be the unique solution of the equation

with Xo given, and 0 and b both Lipschitz continuous. Let the generator of the (time homogeneous) Markov process X be given by

where Ptf(x) equals the transition function22 P,(Xo, f ) when Xo = x almost surely. A is called the infinitesimal generator of X . Use It6's formula to show that i f f has two bounded continuous derivatives (written f E Cz),then the limit exists and A is defined. In this case f is said to be in the domain of the generator A. 22

This notation is defined in Sect. 5 of Chap. I on page 35.

350

Exercises for Chapter V

Exercise 4. (Continuation of Exercise 3.) With the notation and terminology of Exercise 3, show that the infinitesimal generator of X is a linear partial differential operator for functions f E Ci given by

Exercise 5. Let f E Cz, let 0 and b be Lipschitz continuous, let X be the unique solution of the stochastic integral equation

and let A denote its infinitesimal generator. If Xo = x, where x E R, prove Dynkin's expectation formula for Markov processes in this case:

Exercise 6. As a continuation of Exercise 5, show that

for y E R, and that limslo PsAf = Af for (for example) all f in C" with compact support.

Exercise 7. As in Exercise 5, extend Dynkin's expectation formula by showing that it still holds if one replaces in equation (**) the integration upper limit t with a bounded stopping time T, and use this to verify Dynkin's formula in this case. That is, show that iff E Cz and 0 and b are both Lipschitz, and X is a solution of equation (*) of Exercise 5, then f (XT) - f (XO)-

Jd

T

A f (X.)ds is a local martingale.

Exercise 8. Let Z be a L6vy process with L6vy measure v and let X satisfy the equation

where f is Lipschitz continuous. The process X is taken to be d-dimensional, and Z is n-dimensional, so f takes its values in IKd x R". We use (column) vector and matrix notation. The initial value is some given x E IKd. Let Px denote the corresponding law of X starting at x. Show that if g E C" with compact support on IKd, then

Exercises for Chapter V

351

where Vg is a row vector and b is the drift coefficient of Z, + denotes transpose, and A is of course the infinitesimal generator of X. Exercise 9. In the setting of Exercise 8 show that

g(Xt) - g(Xo) -

Jd

t

Ag(X,)ds is a local martingale

for each law P". Show further that it is a local martingale for initial laws of the form Pp, where P p is given by: for A c fl measurable, Pp(A) = J,, Px(A)p(dx) for any probability law p on IRd. Exercise 10. Let Z be an arbitrary semimartingale, and let To = 0 and (T,),?l denote the increasing sequence of stopping times of the jump times of Z when the corresponding jump size is larger than or equal to 1 in magnitude. Define E, = sign(AZTn) with EO = 1. For a semimartingale Y define U"(Y)t to be the solution of the exponential equation

for T, 5 t < T,+l. Show that the (unique) solution of the equation

is given by Xt

=

En,o - Xtnl[~n,~n (t), + ,where ) Xf

= X T ~ U ~ ( E , Zfor ) ~ ,T,

I

t < T,+l. Exercise 11. Show that if M is an L~ martingale with AM, > -1, and (M, M), is bounded, then E(M), > 0 a s . and it is square integrable. (See [147].) Exercise 12. Let M be a martingale which is of integrable variation. Suppose also that the compensator of the process E,,, [AM,/ is bounded. Show that the local martingale E(M) is also of integrabie variation.

+

Exercise 13. Let Z be a semimartingale with decomposition Z = L V where L is a local martingale with AL, # -1 for all finite s, and V has paths of finite variation on compacts. Let

Show that E(Z) = E(L)E(U). (See 11571.)

352

Exercises for Chapter V

*Exercise 14 (MQmin's criterion for exponential martingales). Let M be a local martingale, let Jt = I,,, AMsl{laM,I1+),and let Lt = Jt - Jt and Nt = Mt - L,. (Both L and N are of course also local martingales.) Let

and assume that the compensator of A is bounded. Show that this implies that E(M) is a uniformly integrable martingale. (Note: This is related to Kazamaki's and Novikov's criteria, but without the assumption of path continuity. Also see [156].)

Exercise 15 (Gronwall's inequality). . Let Ct be a c&dl&gincreasing prot cess and suppose 0 5 At < a + SoA,- dC, for t 2 0. Show that At < aeCt, each t 2 0. (Note: The classical Gronwall inequality is usually stated with Ct = t. Also see [85].) The next five exercises outline the Me'tivier-Pellaumailmethod for showing the existence and uniqueness of solutions of stochastic differential equations.

*Exercise 16. Let M be an L~ martingale. Show that for any predictable stopping time T, E { ( E { A M ~ I F ~ - ) ) ~<) E{(M, M)T-). Show also that for any totally inaccessible stopping time T, we have E{(E{AMT~FT-})~) 1 E{(M, M)T). Conclude that for an arbitrary stopping time T we have

Exercise 17 (MQtivier-Pellaumail inequality). Let M be an L~ martingale and let M,*_ = sup,,, IMsl Show that for any stopping time T we have E{(M;-)~) 1 4E{(M, M)T- + [M,M]T- ). (Hint: Use Exercise 16).

Exercise 18. Let M be a locally square integrable local martingale. Show that for any stopping time T we have

Exercise 19 (extended Gronwall's inequality). Let A and C be two increasing processes with A adapted and E{A,) < co, and C such that 0 5 C, < k everywhere for some constant k. Suppose that for every stopping time T and for some positive constant a we have

Then E{A,) 5 cuek. (Hint: Use Lebesgue's change of time lemma, together with Gronwall's inequality of Exercise 15. Also see [110, page 5761.)

Exercises for Chapter V

353

Exercise 20. Use Exercises 16 through 19 as needed to show, via a Picard

iteration method, that a unique solution exists to the stochastic differential equation dXt = f (X,-)dZs where f is Lipschitz continuous and Z is an arbitrary semimartingale. Exercise 21. Show that there are an infinite number of solutions (on the same probability space) to the equation

where B is a standard Brownian motion. (See [121, page 2931.) *Exercise 22. Let 0 satisfy a weakening of the Lipschitz condition of the = co and 'i form lo(y) - o(x)l I K ( ( X- y() for all x, y E R, with J: +du increasing with 'i(0)= 0. (Such a condition is a variant of a standard condition for uniqueness of solutions in the theory of ordinary differential equations; see for example [35, page 601. Let b be Lipschitz continuous. Show that there exists a unique solution of the equation

where B is standard Brownian motion. (See [232].) Exercise 23. Let 0 and b be Lipschitz continuous and let X be the unique solution of equation (*) of Exercise 5. Show that 2xb(x) + C T ( X5) ~c + k(xI2, where c and k are positive constants. Use It6's formula on X; to show the ct)ekt. (See [135].) following moment estimate: E{x;) 5 (E{X;)

+

*Exercise 24 (Euler method of approximation). Let 0 and b be continuous and such that a unique path-by-path solution X of equation (*) of Exercise 5 exists. Let ((k)k>l - be i.i.d. with mean zero and variance T ~Let . Xo be independent of ((k)kll and let Xk be defined inductively by

Define X,(t) = X[,tl where [nt] denotes the integer part of nt. Let B,(t) = 1 [ntl J" z k = l 6 , V, (t) = and finally observe that

9,

By Donsker's Theorem we know that (B,, V,) converges weakly (in distribution) to (TB,V), where B is a standard Brownian motion and V(t) = t. Show that B, is a martingale, and that X, converges weakly to X. (See 11361 and [137].)

354

Exercises for Chapter V

*Exercise 25. Let Z be a L6vy process which is a square integrable martingale. Let f be continuously differentiable with a bounded derivative and let Xx be the unique solution of

Show that Xx is also a square integrable martingale. *Exercise 26. In the framework of Exercise 25, let

Show that

xixbe the solution of

x'"is also a square integrable martingale.

**Exercise27. In the framework of Exercises 25 and 26, for each measurable function g with at most linear growth, set

Assume that f is infinitely differentiable, and that g is twice differentiable, bounded and both of its first two derivatives are bounded. Show that the function (t,x) H Ptg(x) is twice differentiable in x and once differentiable in t, that all the partial derivatives are continuous in ( t ,x), and further that

Expansion of Filtrations

1 Introduction By an expansion of the filtration, we mean that we enlarge the filtration (3t)t20 to get another filtration ('lit)t>o such that the new filtration satisfies the usual hypotheses and Ft C 'lit, each t 0. There are three questions we wish to address: (1) when does a specific, given semimartingale remain a semimartingale in the enlarged filtration; (2) when do all semimartingales remain semimartingales in the enlarged filtration; (3) what is a new decomposition of the semimartingale for the new filtration. The subject of the expansion of filtrations began with a seminal paper of K. It6 in 1976 (published in [loo] in 1978), when he showed that if B is a standard Brownian motion, then one can expand the natural filtration (3t)tlo of B by adding the 0-algebra generated by the random variable B1 to all Ft of the filtration, including of course 3 0 . He showed that B remains a semimartingale for the expanded filtration, he calculated its decomposition explicitly, and he showed that one has the intuitive formula

>

where the integral on the left is computed with the original filtration, and the integral on the right is computed using the expanded filtration. Obviously such a result is of interest only for 0 5 t 5 1. We will establish this formula more generally for L6vy processes in Sect. 2. The second advance for the theory of the expansion of filtrations was the 1978 paper of M. Barlow [7] where he considered the problem that if L is a positive random variable, and one expands the filtration in a minimal way to make L a stopping time, what conditions ensure that semimartingales remain semimartingales for the expanded filtration? This type of question is called progressive expansion and it is the topic of Sect. 3.

356

VI Expansion of Filtrations

2 Initial Expansions Throughout this section we assume given an underlying filtered probability space ( R , 3 , (Ft)tlo, P) which satisfies the usual hypotheses. As in previous chapters, for convenience we denote the filtration (3t)t20 by the symbol IF. The most elementary result on the expansion of filtrations is due to Jacod and was established in Chap. I1 (Theorem 5). We recall it here.

Theorem 1 (Jacod's Countable Expansion). Let A be a wllection of events i n 3 such that if A,, Ag E A then A, n Ag = 4, a # p. Let 'Ht be the filtration generated by Ft and A. Then every ((Ft)t20,P ) semimartingale is an (('Ht)t>o, - P ) semimartingale also. We also record a trivial observation as a second elementary theorem.

Theorem 2. Let X be a semimartingale with decomposition X = M + A and let G be a 0-algebra independent of the local martingale term M . Let W denote the filtration obtained by expanding IF with the one 0-algebra G (that is, 'Ht = Ft V G, each t > 0 and 'lit = 'Ht+). Then X is an W semimartingale with the same decomposition.

Proof. Since the local martingale M remains a local martingale under theorem follows.

W,the

We now turn to L6vy processes and an extension of It6's first theorem. Let Z be a given Ldvy process on our underlying space, and define H = ('Ht)tlo to be the smallest filtration satisfying the usual hypotheses, such that Z1 is 'Ho measurable and Ft c 'Ht for all t 0.

>

Theorem 3 (It6's Theorem for LQvy Processes). The Lkvy process Z is an W semimartingale. If moreover E{IZtl) < co, all t 0, then the process

>

is an W martingale o n [0,co). Proof. We begin by assuming E{z')< co, each t > 0. Without loss of generality we can further assume E{Zt) = 0. Since Z has independent increments, we know Z is an IF martingale. Let 0 5 s < t 5 1 be rationals with s = j / n and t = kin. We set Yi= Z - - Z * . 71

c:=;'

Then z1- 2, = Y, and Zt - Zs = are i.i.d. and integrable. Therefore

n

Y,. The random variables

2 Initial Expansions

-

t-s -(Z1 1-s

357

-2s).

The independence of the increments of Z yields E{Zt - Z,I'H,) = E{Zt Z,IZ1 - 2,); therefore E{Zt - Z,I'H,) = e ( Z 1- 2,) for all rationals, 0 5 s
<

<

t-s - - ( - 2 ) 1-s -

I

<

1 1-u --( 2 1 - Zs)du 1-u1-S

There is a potential problem at t = 1 because of the possibility of an explosion. Indeed this is typical of initial enlargements. However if we can show v d s ) < m this will suffice to rule out explosions. By the stationarity and independence of the increments of Z we have E{IZl - Z,I) 5 E { ( Z l - z , ) ~ ) + a ( l - s)+ for some constant a and for all s, 0 s 1. Therefore E { J ~ v d s ) 5 a J,' e d s < m . Note that if t > 1 then 1 z -2 .Ft = 'Ht, and it follows that M is a martingale. Since Zt = Mt i-, ds we have that Z is a semimartingale. Next suppose only that E{IZtl) < m , t 2 0 instead of Z being in L2 as we assumed earlier. We define

E{G

<

< <

+ So

(Since Z has c&dl&gpaths a.s. for each w there are at most a finite number of jumps bigger than a fixed size on each compact time set; hence each Jj is finite a.s.) By the results of Chap. I on L6vy processes we have that & = Zt - J i J: is a L6vy process with bounded jumps, and hence it is square integrable. Additionally, the processes Y, J1, and J2 are jointly independent. Combining these facts with the preceding proof yields

+

is an W' martingale, where W' is the smallest right continuous filtration obtained by expanding IF with ( Y l ,Jll, J;). Note that W c The same argument, plus the observation that since Ji does not jump at time t = 1 and is

w'.

358

VI Expansion of Filtrations

therefore constant in a (random) neighborhood of 1, which in turn yields that t J'-J" lJ;-J.ti 1-s ds < m a s . , shows that (J;-So e d s ) , -, ~is a local martingale for

Ji

HI'. Moreover it is a martingale for H' as soon as E{$ IJi~:ids) < m . But the function t H E{J;) = ait for all i by the stationarity of the increments. Hence

Since Y, J1,and J2are all independent, we conclude that M is an W' martingale. Since M is adapted to W, by Stricker's Theorem it is also an W martingale, and thus Z is an W semimartingale. Finally we drop all integrability assumptions on Z. We let

Then X is also a Lkvy process, and since X has bounded jumps it is in L p for all p 2 1, and in particular E{X:) < m , each t 2 0. Let W(X1) denote IF expanded by the adjunction of the random variable X I . Let K = W(X1) V W ( J ~ ) ,the filtration generated by W(X1) and H ( J ~ ) .Then W(Z1) c K. But X is a semimartingale on W(Xl), and since J1is independent of X we have by Theorem 2 that X is a (K, P ) semimartingale. Therefore by Stricker's Theorem, X is an (W, P ) semimartingale; and since is a finite variation process adapted to W we have that Z is an (H, P) semimartingale as well.

Jt

The most important example of the above theorem is that of Brownian motion, which was ItG's original formula. In this case let W = W(B1), and we have as a special case the W decomposition of Brownian motion:

Note that the martingale P in the decomposition is continuous and has [P,Plt = t , which by LBvy's theorem gives that P is also a Brownian motion. A simple calculation gives ItG's original formula, for a process H which is IF predictable:

2 Initial Expansions

359

where since the random variable B1 is .HOmeasurable, it can be moved inside the stochastic integral. We can extend this theorem with a simple iteration; we omit the fairly obvious proof.

Corollary. Let Z be a given LBvy process with respect to a filtration IF, and let 0 = to < t l < . . . < t, < oo. Let W denote the smallest filtration satisfying the usual hypotheses containing IF and such that the random variables Zt,, . . . , Ztn are all 'Ho measurable. Then Z is an W semimartingale. If we have a countable sequence 0 = to < t l < . . . < t, < . . . , we let T = sup, t,, with W the corresponding filtration. Then Z is an W semimartingale on [O, 7 ) . We next give a general criterion (Theorem 5) to have a local martingale remain a semimartingale in an expanded filtration. (Note that a finite variation process automatically remains one in the larger filtration, so the whole issue is what happens to the local martingales.) We then combine this theorem with a lemma due t o Jeulin to show how one can expand the Brownian filtration. Before we begin let us recall that a process X is locally integrable if there exist a sequence of stopping times (T,),?l increasing to oo a.s. such >O] 1) < oo for each n. Of course, if Xo = 0 this reduces t o that E{IXTn the condition E{IXT,,1) < oo for each n.

Theorem 4. Let M be a n IF local martingale and suppose M is a semimartingale in a n expanded filtration W. Then M is a special semimartingale in H. Proof. First recall that any local martingale is a special semimartingale. In particular the process M; = supso IM, ( is locally integrable (see Theorem 33 of Chap. 111), and this of coursefemains locally integrable in the expanded filtration H, since stopping times remain stopping times in an expanded filtration. Since M is an H semimartingale by hypothesis, it is special because M,* is locally integrable (see Theorem 34 of Chap. 111).

Theorem 5. Let M be an IF local martingale, and let H be predictable such

Sot

that H:d[M, MI, is locally integrable. Suppose W is a n expansion of IF such that M is an W semimartingale. Then M is a special semimartingale in W and let M = N A denote its canonical decomposition. The stochastic integral process ($H,dM,)t>o - is an W semimartingale if and only if the process HsdAs)t>0 - exists as a path-by-path Lebesgue-Stieltjes integral a.s.

+

(6

Proof. First assume that E{JomH:d[M, MI ,) < oo, which implies that H .M HsdM,)t?o) is a (where H . M denotes the stochastic integral process square integrable martingale, and hence by Theorem 4 it is a special semimartingale in the W filtration. Let M = N A be the canonical W deH . A is the composition of M , and it follows that H . M = H . N canonical W decomposition of H . M . By the lemma following Theorem 23 of Chap. IV we have H;d[N, N],) 5 E { J ~ H:d[M, MI,) < oo and E{JOmH:d[A, A],) 5 E { J ~ H:d[M, ~ MI,) < oo. This allows us t o conclude that H is (.H2, M ) integrable, calculated in the H filtration.

(Sot

+

E{Jr

+

360

VI Expansion of Filtrations

To remove the assumption E { S H ~ z d [ M ,MI,) < m , we only need to recall that H . M is assumed to be locally square integrable, and thus take M stopped at a stopping time T, that makes H . M square integrable, and we are reduced to the previous case.

Theorem 6 (Jeulin's Lemma). Let R be a positive, measurable stochastic process. Suppose for almost all s, R , is independent of 3,; and the law (or distribution) of R , is p which is independent of s , with p({O)) = 0 and J r x p ( d x ) < m . If a is a positive predictable process with J: a,ds < m a.s. for each t , then the two sets below are equal almost surely:

Proof. We first show that { J r R,a,ds < oo) C {JomaSds < oo) a s . Let A be an event with P ( A ) > 0 , and let J = l A ,Jt = E { J I F t ) , the c&ll&gversion of the martingale. Let j = inft Jt. Then j > 0 on { J = 1). We have a.s.

Consider next

by Jensen's inequality, and this is equal to ( E { 1 ~ 1 3-~ p(0, } u])+,where p is the law of Rt. Continuing with (*) we have

where @ ( x )= JoW(x - p(0, u])+du.Note that @ is increasing and continuous on [O, 11, and @ > 0 on (O,1] because p({O)) = 0. Choose A , = {Som R,a,ds n ) for the event A in the foregoing. Then

<

which implies that

SoW @(J,)a,ds < oo a.s. But

a,ds < m a.s. on and therefore J r aSds < oo a s . on { J = 1). That is, A,. Letting n tend to oo through the integers gives {Som Rsa,ds < oo) C

2 Initial Expansions

361

{Som a,ds < m ) a s . We next show the inverse inclusion. Let T, = inf{t > 0 : > n ) . Then Tnis a stopping time, and n 2 E{J,". a,ds). Moreover

Sot a,ds

where a is the expectation of R,, which is finite and constant by hypothesis. R,a,ds < m a.s. Let w be in a,ds < m ) . Then there exists Therefore an n (depending on w ) such that Tn(w) = m. Therefore we have the inclusion {Som R,a,ds < m ) 2 {Soma,ds < m ) a s . and the proof is complete.

{Jr

We are now ready to study the Brownian case. The next theorem gives the main result.

Theorem 7. Let M be a local martingale defined o n the standard space of canonical Brownian motion. Let H be the minimal expanded filtration containing B1 and satisfying the usual hypotheses. Then M is a n H semimartingale i f and only if the integral -$==ld[M, B],( < m a.s In this case t A l B -B ;-,a d [ M ,B ] , is an H local martingale. Mt - So

fi

Proof. By the Martingale Representation Theorem we have that every IF loH,dB,, where H is cal martingale M has a representation Mt = Mo ~ : d s< m as., each t > 0. By Theorem 5 we know predictable and IH,l v d s is finite that M is an H semimartingale if and only if IH I and R , = l ~ , < l ) Then as., 0 5 t 5 1. We take a, = &, ~ .

-+

a,R,ds, which is finite only if & a,ds < m a.s. by 1~~l-d~ = Jeulin's Lemma. Thus it is finite only if %ds < m a.s. But 1

and this completes the proof. As an example, let 112 < a

< 1, and define

362

VI Expansion of Filtrations

&1

&1

Then H is trivially predictable and also Hzds < m. However H, A d s is divergent. Therefore M = H . B is an IF local martingale which is not an H semimartingale, by Theorem 7, where of course H = H(B1). Thus we conclude that not all IF local martingales (and hence a fortiori not all semimartingales) remain semimartingales in the H filtration. We now turn to a general criterion that allows the expansion of filtration such that all semimartingales remain semimartingales in the expanded filtration. It is due to Jacod, and it is Theorem 10. The idea is surprisingly simple: recall that for a c&dl&gadapted process X to be a semimartingale, if H n is a sequence of simple predictable processes tending uniformly in (t,w) to zero, then we must have also that the stochastic integrals H n . X tend to zero in probability. If we expand the filtration by adding a a-algebra generated by a random variable L to the IF filtration at time 0 (that is, o{L) is added to Fo), then we obtain more simple predictable processes, and it is harder for X to stay a semimartingale. We will find a simple condition on the random variable L which ensures that this condition is not violated. This approach is inherently simpler than trying to show there is a new decomposition in the expanded filtration. We assume that L is an (IE, &)-valuedrandom variable, where E is a standard Borel spaceland & are its Borel sets, and we let H(L) denote the smallest filtration satisfying the usual hypotheses and containing both L and the original filtration IF. When there is no possibility of confusion, we will write W in place of H(L). Note that if Y E 7-L: = Ft V a{L), then Y can be written Y (w) = G(w,L(w)) , where (w ,x) H G(w, x) is an FtFt & measurable function. We next recall two standard theorems from elementary probability theory.

Theorem 8. Let X n be a sequence of real-valued random variables. Then X n converges to 0 in probability if and only if limn,, E{min(l, IXnl)) = 0. A proof of Theorem 8 can be found in textbooks on probability (see for example [log]). We write 1A lXnl for min(1, IXnl). Also, given a random variable L, we let Qt(w, dx) denote the regular conditional distribution of L with respect to Ft, each t 2 0. That is, for any A E & fixed, Qt(., A) is a version of E{lfLEA)IFt},and for any fixed w, Qt(w, dx) is a probability on &. A second standard elementary result is the following.

Theorem 9. Let L be a random variable with values in a standard Borel space. Then there exists a regular conditional distribution Qt(w, dx) which is a version of E { ~ ( LlFt). ~ ~ ~ ) For a proof of Theorem 9 the reader can see, for example, Breiman [23, page 791.

(IE,&)is a standard Borel space if there is a set r E B, where B are the Borel subsets of R,and an injective mapping c$ : IE -+ r such that c$ is & measurable and 4-I is I3 measurable. Note that (Rn,Bn)are standard Borel spaces, I 5 n 5 co.

2 Initial Expansions

363

Theorem 10 (Jacod's Criterion). Let L be a random variable with values in a standard Bowl space (E, I ) ,and let Qt(w, dx) denote the regular conditional distribution of L given Ft, each t 2 0. Suppose that for each t there exists a positive a-finite measure qt on (E, &) such that Qt(w, dx) << qt(dx) a.s. Then every F semimartingale X is also an H(L) semimartingale. Proof. Without loss of generality we assume Qt (w, dx) << qt(dx) surely. Then by Doob's Theorem on the disintegration of measures there exists an & @I Ft measurable function qt (x, w) such that Qt (w, dx) = qt(x, w)qt(dx). Moreover = P ( Y E IE) = 1, we have since E{JEQt(., dx)) = E{E{1{YEE)13t})

Hence for almost all x (under qt (dx)), we have qt (x, .) E L1 (dP). Let X be an IF semimartingale, and suppose that X is not an W(L) semimartingale. Then there must exist a u > 0 and an E > 0, and a sequence Hn of simple predictable processes for the W filtration, tending uniformly t o 0 but such that inf, E{1 A IHn . XI) 2 E . Let us suppose that t n 5 u, and

V a{L). Hence J," has the form gi(w, L(w)), where (w, x) H with J," E FtZ gi(w, x) is Fti@I I measurable. Since Hn is tending uniformly t o 0, we can take without loss lHn1 5 l l n , and thus we can also assume that lgi 1 5 11.. We write n-1

Ht"'x(~= )

C gi(w7 x)l(ti,t,+,l(t), i=O

and therefore (x, w) H H;'x(w) and (x, w) H ( H n J . X)t(w) are each & @I 3, measurable, 0 5 t 5 u. Moreover one has clearly H n . X = H n > = . X Combining . the preceding, we have

where we have used Fubini's Theorem t o obtain the last equality. However the function hn(x) = E{(1 A IHn>". X,l)qU(., x)) 5 E{q,(., x)) E L1(drl,), and since hn is non-negative, we have

364

VI Expansion of Filtrations

lim E{1 A IHn . X,I) = lim n-00

n-00

S.

E{(l A IHn . Xu()qu(.,x))qU(dx)

by Lebesgue's Dominated Convergence Theorem. However q,(., x) E L1(dP) for a.a. x (under (dq,)), and ifwe define dR = cq,(., x ) d P to be another probability, then convergence in P-probability implies convergence in R-probability, ER{(l A (Hnyx. X,I)) = 0 as well, which imsince R << P . Therefore limn,, plies 1 0 = lim -E R { ( ~ A IHnyX . Xu[)) n-cc c = E p { ( l A IHnlX.X u I ) q u ( . , ~ ) ) for a.a. x (under (dq,)) such that q,(., x) E L1(dP). Therefore the limit of the integrand in (*) is zero for a.a. x (under (dq,)), and we conclude lim E{(1 A IHn .

n'cc

which is a contradiction. Hence X must be a semimartingale for the filtration ' ,O = Ft v a{L). Let X = M A be a decomposition of X under HO,where H WO. Since W0 need not be right continuous, the local martingale M need not be right continuous. However if we define ~t = Mt if t is rational; and lim,Lt,,EQ M, if t is not rational; then Mt is a right continuous martingale 7-L.: Letting At = Xt - M ~we , have that for the filtration W where 'Ht = Xt = M~+At is an W decomposition of X , and thus X is an H semimartingale.

+

nu,,

A simple but useful refinement of Jacod's Theorem is the following where we are able t o replace the family of measures qt be a single measure q.

Theorem 11. Let L be a random variable with values in a standard Bore1 space (IE,&), and let Qt (w, dx) denote the regular conditional distribution of L given Ft, each t 2'0. Then there exists for each t a positive a-finite measure qt on (IE, &) such that Qt(w, dx) << qt(dx) a.s. if and only if there exists one positive a-finite measure q(dx) such that Qt(w, dx) << q(dx) for all w, each t > 0. In this case q can be taken to be the distribution of L. Proof. It suffices t o show that the existence of qt for each t > 0 implies the existence of q with the right properties; we will show that the distribution measure of L is such an q. As in the proof of Theorem 10 let (x, W) ++ qt (x, w ) be & @I 3 t measurable such that Qt (w,dx) = qt(x, w)qt (dx). Let a t ( x ) = E{qt(x,w)), and define rt (2,w) =

if a t (x) > 0, otherwise.

2 Initial Expansions

365

Note that at(x) = 0 implies qt(x, .) = 0 a.s. Hence, qt(x, w ) = rt(x, w)at (x) a.s.; whence rt (x, w)at(x)qt(dx) is also a version of Qt(w, dx). Let q be the law of L. Then for every positive & measurable function g we have

from which we conclude that at(x)qt(dx) rt(w, x)q(dx), and the theorem is proved.

=

q(dx). Hence, Qt(w, dx) =

We are now able t o re-prove some of the previous theorems, which can be seen as corollaries of Theorem 11.

Corollary 1 (Independence). Let L be independent of the filtration IF. Then every IF semimartingale is also an W(L) semimartingale. Proof. Since L is independent of F t , E{g(L)lFt} = E{g(L)) for any bounded, Bore1 function g. Therefore

, in particular Qt (w ,dx) from which we deduce Qt (w, dx) = ~ ( d x )and a s . , and the result follows from Theorem 11.

<< q(dx)

Corollary 2 (Countably-valued random variables). Let L be a random variable taking on only a countable number of values. Then every IF semimartingale is also an H(L) semimartingale. Proof. Let L take on the values al, 122, 123,. . .. The distribution of L is given P ( L = Q ~ ) E , ~ ( ~where x ) , &,=(dx)denotes the point mass at by q(dx) = CEO=, a i . With the notation of Theorem 11, we have that the regular conditional distribution of L given F t , denoted Qt(w,dx), has density with respect t o q given bv P(L = aj/Ft) l{z=aj}. P ( L = a,)

-

C j

The result now follows from Theorem 11.

366

VI Expansion of Filtrations

Corollary 3 (Jacod's Countable Expansion). Let A = (Al, A2,. . . ) be a sequence of events such that Ai n Aj = 8, i # j , all in F, and such that UE, Ai = R. Let W be the filtration generated by IF and A, and satisfying the usual hypotheses. Then every F semimartingale is an W semimartingale.

xzl

Proof. Define L = 2-'lAi. Then W = W(L) and we need only to apply the preceding corollary. Next we consider several examples.

Example (ItB's example). We first consider the original example of It6, where in the standard Brownian case we expand the natural filtration IF with a{B1). We let W denote W(B1). We have

where qt(dx) is the law of B1 - Bt and where we have used that B1 - Bt is independent of Ft. Note that qt (dx) is a Gaussian distribution with mean 0 and variance (1-t) and thus has a density with respect to Lebesgue measure. Since Lebesgue measure is translation invariant this implies that Qt(w, dx) << dx a.s., each t < 1.However at time 1we have E{g(B1)IF1) = g(B1), which yields Ql(w, dx) = ~ i g ~ ( ~ ) } (which d x ) , is a s . singular with respect to Lebesgue measure. We conclude that any IF semimartingale is also an W(B1) semimartingale, for 0 5 t < 1, but not necessarily including 1. This agrees with Theorem 7 which implies that there exist local martingales in IF which are not semimartingales in W(B1). Our next example shows how Jacod's criterion can be used to show a somewhat general, yet specific result on the expansion of filtrations.

Example (Gaussian expansions). Let IF again be the standard Brownian filtration satisfying the usual hypotheses, with B a standard Brownian motion. g ( ~ ) 2 d< s oo, g a deterministic function. Let Let V = Jomg(s)dBs, where a = inf{t > 0 : g(s)2ds = 0). If h is bounded Borel, then as in the previous example

Jr

where qt is the law of the Gaussian random variable hmg(s)dBs. If a = oo, then qt is non-degenerate for each t, and qt of course has a density with respect to Lebesgue measure. Since Lebesgue measure is translation invariant, we conclude that the regular conditional distribution of Qt (w, dx) of V given 3 t also has a density, because

2 Initial Expansions

367

Hence by Theorem 10 we conclude that every IF semimartingale is an H(V) semimartingale. Example (expansion via the end of a stochastic differential equation). Let B be a standard Brownian motion and let X be the unique solution of the stochastic differential equation

where a and b are Lipschitz. In addition, assume a and b are chosen so that for h Bore1 and bounded, E{h(Xl)I.Ft}

=

J h ( x ) ~ ( l -t,Xt,x)dx

where ~ ( -lt, u , x) is a deterministic f ~ n c t i o n .Thus ~ Qt(w, dx) = ~ ( -l t,Xt(w),x)dx, and Qt(w,dx) is a.s. absolutely continuous with respect to Lebesgue measure if t < 1. Hence if we expand the Brownian filtration IF by initially adding X I , we have by Theorem 10 that every IF semimartingale is an H(X1) semimartingale, for 0 5 t < 1. The mirror of initial expansions is that of filtration shrinkage. This has not been studied to any serious extent. We include one result (Theorem 12 below), which can be thought of as a strengthening of Stricker's Theorem, from Chap. 11. Recall that if X is a semimartingale for a filtration H, then it is also a semimartingale for any subfiltration (6, provided X is adapted t o (6, by Stricker's Theorem. But what if a subfiltration IF is so small that X is not adapted t o it? This is the problem we address. We will deal with the optional projection Z of X onto IF. Definition. Let H = (Ht)t20 be a bounded measurable process. I t can be shown that there exists a unique optional process O H , also bounded, such that for any stopping time T one has

The process

O

H

is called the optional projection of H

We remark that O H t = Ht a.s. for each fixed time t, but the null set depends on t. Therefore were we simply t o write as a process (E{Ht13t})tlo, instead of OH, it would not be uniquely determined almost surely. This is why we use the optional projection. The uniqueness follows from Meyer's section theorems, which are not treated in this book, so we ask the reader t o accept it on faith. Sufficient conditions are known for this to be true. These conditions involve Xo having a nice density, and requirements on the differentiability of the coefficients. See, for example, [I901 and [193].

368

VI Expansion of Filtrations

Definition. Let ' H = (Ht),?0 be a bounded measurable process. It can be shown that there exists a predictable process pH, also bounded, such that for any predictable stopping time T one has

The process pH is called the predictable projection of H It follows that for the optional projection, for each stopping time T we have OHT = E { H T I F T ) a.s. on {T < oo) whereas for the predictable projection we have that

for any predictable stopping time T . For fixed times t > 0 we have then of course OHt = E{HtlFt} a.s., and one may wonder why we don't simply use the "process" (E{Ht13t})t20instead of the more complicated object OH. The reason is that this is defined only almost surely for each t, and we have an uncountable number of null sets; therefore it does not uniquely define a process. (Related t o this observation, note in contrast that in some cases ( E { H t13t})t20 might exist even when OH does not). We begin our treatment with two simple lemmas. Lemma. Let IF c (6, and let X be a (6 martingale but not necessarily adapted to IF. Let Z denote the optional projection of X onto IF. Then Z is an IF martingale.

Proof. Let s < t. Then

and the result follows. The Az4ma martingale, a projection of Brownian motion onto a subfiltration t o which it is not adapted, is an example of the above lemma. The projection of an increasing process, however, is a submartingale but not an increasing process. Lemma. Let IF c G,and let X be a (6 supermartingale but not necessarily adapted t o IF. Let Z denote the optional projection of X onto IF. Then Z is an IF supermartingale.

Proof. Let s < t. Then

and the result follows.

3 Progressive Expansions

369

The limitation of the two preceding lemmas is the need to require integrability of the random variables Xt for each t 2 0. We can weaken this condition by a localization procedure.

Definition. We say that a G semimartingale X starting at 0 is an IF special, G semimartingale if there is a sequence (Tn)n>l of IF stopping times increasing a.s. to oo, and such that the stopped XTn can be written in the form XTn = M n + An where M n is a G martingale with Mg = 0 and where An has integrable variation over each [O,t], each t > 0, and with A. = 0. Theorem 12 (Filtration Shrinkage). Let G be a given filtration and let IF be a subfiltration of 6. Let X be an IF special, G semimartingale. Then the IF optional projection of X , called 2, exists, and it is a special semimartingale for the IF filtration.

<

Proof. Without loss of generality we can assume Tn n for each n 2 1. We set To = 0 and let X n = xTn-XTn-1, and N n = ~~n - ~ ~ n with 1 N o = 0. For each n there are two increasing processes Cn and Dn, each starting at 0, with X n = N n Cn - Dn, and moreover we can choose this decomposition such that the following holds:

+

and where t 5 Tn-l implies Cp = DF = Ntn = 0, and t 2 Tn implies Cp - Cpn = Dp - DFn = N; - NFn = 0. The integrability condition implies that the IF optional projections of Cn, Dn, and N n all exist and have c&dl&g versions. By the previous two lemmas the optional projection of N n is an IF martingale, and those of Cn and Dn are IF submartingales. Therefore letting OXn, ONn, OCn, and ODn denote the respective IF optional projections of Xn, N n , C n , and Dn, we have that OXn =O Nn+OCn -O Dn exists and is a special IF semimartingale. Since TnP1and Tn are IF stopping times, we have that also the IF optional projections ONn, OCn, and ODn and hence OXn are all null over the stochastic interval [O,Tn-l] and constant over (T,, oo). Then En,, OXn is a c&dl&g version of OX = Z and thus Z is a special IF semimartingale.

3 Progressive Expansions We consider the case where we add a random variable gradually to a filtration in order t o create a minimal expanded filtration allowing it to be a stopping time. Note that if the initial filtration IF = (3t)t>o is given by Ft = {(8,R) for all t, then G = (G)t>o given by Gt = u{L s; s 5 t ) is the smallest expansion of IF making ~a stopping time. Note that Gt = u{L A t} as well. Let L be a strictly positive random variable. Let A = {(t,w) : t 5 L(w)).

A

370

VI Expansion of Filtrations

Then L = sup{t : (t, w) E A). In this sense every positive random variable is the end of a random set. Instead however let us begin with a random set A c a+ x R and define L t o be the end of the set A. That is, L(w) = sup{t : (t, w) E A) where we use the (unusual) convention that sup(@)= 0-, where (0-) is an extra isolated point added t o the non-negative reals [0, oo] and which can be thought of as 0- < 0. We also define 30-= 3 0 . The purpose of (0-) is t o distinguish between the events {w : A(w) = 8) and {w : A(w) = {0)), each of which could potentially be added to the expanded filtration. The smallest filtration expanding IF and making the random variable L a stopping time is Go defined by Gf = 3 t V u{L A t); but Go is not necessarily right continuous. Thus the smallest expanded filtration making L a stopping time and satisfying the usual hypotheses is (6 given by Gt = G.: Nevertheless, it turns out that the mathematics is more elegant if we consider expansions slightly more rich than the minimal ones. In order t o distinguish progressive expansions from initial ones, we will change the notation for the expanded filtrations. Beginning with our usual filtered probability space satisfying the usual hypotheses ( R , 3 , IF, P ) m where of course IF denotes the filtration ( 3 ) t > o ,and a random variable L which is the end of a random set, we define the egPanded filtration t o be IFL and is given by

nu,,

This filtration is easily seen to satisfy the usual hypotheses, and also it makes L into a stopping time. Thus (6 c IFL. There are two useful key properties the filtration IFL enjoys. Lemma. If H is a predictable process for IFL then there exists a process J which is predictable for IF such that H = J on [0,L]. Moreover, if T is any stopping time for IFL then there exists an IF stopping time S such that SAL=TALa.s.

r

r

Proof. Let be an event in 3;. Then events of the form (t, oo) x form a generating set for p(IFL), the predictable sets for IFL. Let H, = l j c t , o o l x r > ( ~ ) and then take J t o be J, = l(t,,)xr,(s). The first result follows by an application of the Monotone Class Theorem. For the stopping time T, note that it suffices to take H = l[O,Tl,and let J be the IF predictable process guaranteed by the first half of this lemma, and take S = inf{t : Jt = 0). We next define a measure pL on [0, oo] x R by

for any positive, measurable process J. For such a measure pL there exists an which is null at 0- but which can jump at both 0 increasing process -

3 Progressive Expansions

371

and +oo. We will denote AL = (AtL)t20, the (predictable) compensator of ljtlL)for the filtration IF. Therefore if J is an IF predictable bounded process we have

We now define what will prove t o be a process fundamental t o our analysis. The process Z defined below was first used in this type of analysis by J. Az6ma [3]. Recall that if H is a (bounded, or integrable) IF^ process, then its optional projection O H onto the filtration IF exists. We define

is decreasing, hence by the lemma preceding Theorem 12 Note that we have that Z is an IF supermartingale. We next prove a needed technical result.

Theorem 13. The set { t : 0 I t I oo, Zt- = 0) is contained in the set (L, oo] and is negligible for the measure dAL.

>

Proof. Let T(w) = inf{t 0 : Zt(w) = 0, or Zt-(w) = 0 for t > 0). Then it is a classic result for supermartingales that for almost all w the function t H Zt(w) is null on [T(w),oo]. (This result is often referred t o as "a nonnegative supermartingale sticks at zero.") Thus we can write {Z = 0) as the stochastic interval [T, oo], and on [O,T) we have Z > 0, Z- > 0. We have E { A ~ A $ ) = P ( T < L) = E { Z T ~ { ~ < = ~ 0, } )hence d~~ is carried by [0,TI. Note that since is carried by the graph of L, written [L], we have L 5 T, and hence we have Z > 0, Z- > 0 on [0, L). Next observe that the set {Z- = 0) is predictable, hence 0 = E{l{Z,-,o~dl{L>t)) = E{l{Z,-,~}dA~) and hence {Z- = 0) is negligible for dAL. Note that this further implies that P(ZL- > 0) = 1, and again that {Z- = 0) c (L, oo]. We can now give a description of martingales for the filtration IFL, as long as we restrict our attention t o processes stopped at the time L. What happens after L is more delicate. For an integrable process J we let PJ denote its predictable projection.

Theorem 14. Let Y be a random variable with E{IYI) < oo. A right continuous version of the martingale & = ~(~13:) is given by the formula

Moreover the left continuous version Y- is given by

372

VI Expansion of Filtrations

Proof. Let OL denote the optional a-algebra on R+ x R, corresponding t o the filtration IFL. On [0,L), OL coincides with the trace of O on [0,L). (By O we mean of course the optional a-algebra on R+ x R corresponding to the underlying filtration IF.) Moreover on [L, oo), OL coincides with the trace of the a-algebra B(R+) 633, on [L, oo). The analogous description of PLholds, with [0,L) replaced by (0, L], and with [L,oo) replaced with (L, oo). It is then simple to check that the formulas give the bona jide conditional expectations, and also the right continuity is easily checked on [0,L) and [L, oo) separately. The second statement follows since Z- > 0 on (0, L] by Theorem 13. We now make a simplifying assumption for the rest of this paragraph. This assumption is often satisfied in the cases of interesting examples, and it allows us t o avoid having t o introduce the dual optional projection of the measure EL~{L>o}. Simplifging assumption to hold for the rest of this paragraph. We assume L avoids all IF stopping times. That is, P ( L = T ) = 0 for all IF stopping times T . Definition. The martingale M L given by Mk = A: damental L martingale.

+ Zt is called the fun-

Note that it is trivial t o check that M L is in fact a martingale, since AL is the compensator of 1 - Z. Note also that M& = A&,, since Z, = 0. Last, note that it is easy to check that M L is a square integrable martingale. Theorem 15. Let X be a square integrable martingale for the IF filtration. Then - is a semimartingale for the filtration IFL. Moreover XtALtAL 1 =d(X, ML), is a martingale in the IFL filtration.

&

Since C Proof. Let C be the (non-adapted) increasing process Ct = has only one jump at time L we have E{XL) = E { r ~ , d C ~ ) . - ~ i nXc eis a martingale it jumps only at stopping times, hence it does not jump at L, and using that AL is predictable and hence natural we get

Suppose that H is a predictable process for IFL, and J is a predictable process for IF which vanishes on {Z- = 0) and is such that J = H on (0, L]. We are assured such a process J exists by the lemma preceding Theorem 13. Suppose first that H has the simple form H = hl(,,,) for bounded h E 3;. If j is an Ft random variable equal to h on {t < L), then we can take J = jl(t,,) and we obtain H . X, = h(XL - Xt)llt,L). In this way we can define stochastic integrals for non-adapted simple processes. We have then E{(H . X),) = E { ( J . X)L), and using our previous calculation, since J . X is another square integrable martingale, we get

3 Progressive Expansions

373

Since (X, M L ) is IF predictable, we can replace H by because it has the same predictable projection on the support of d(X, M L ) . This yields

Last if we take the bounded IFL predictable process H t o be a stochastic interval [0,T A L], where T is an IFL stopping time, we obtain E{XTAL TAL 1 =d(X, ML),) = 0, which implies by Theorem 21 of Chap. I that

So

XTAL-

1 SOTAL =d(X,

ML), is a martingale.

We do not need the assumption that X is a square integrable martingale, which we made for convenience. In fact the conclusion of the theorem holds even if X is only assumed t o be a local martingale. We get our main result as a corollary t o Theorem 15.

Corollary. Let X be a semimartingale for the IF filtration. Then (XtAL)t20 is a semimartingale for the filtration IFL.

+

Proof. If X is a semimartingale then it has a decomposition X = M D. The local martingale term M can be decomposed into X = V + N , where V and N are both local martingales, but V has paths of bounded variation on compact time sets, and N has bounded jumps. (This is the Fundamental Theorem of Local Martingales, Theorem 25 of Chap. 111.) Clearly V and D remain finite variation processes in the expanded filtration IFL, and since M has bounded jumps it is locally bounded, hence locally square integrable, and since every IF stopping time remains a stopping time for the IFL filtration, the corollary follows from Theorem 15. We need t o add a restriction on the random variable L in order t o study the evolution of semimartingales in the expanded filtration after the time L.

<

Definition. A random variable L is called honest if for every t oo there exists an Ft measurable random variable Lt such that L = Lt on {L 5 t). Note that in particular if L is honest then it is Fmmeasurable. Also, any stopping time is honest, since then we can take L = L A t which is of course Ft measurable by the stopping time property.

Example. Let X be a bounded c&dl&gadapted processes, and let x* : = supsl, X: and X+* = sup, X.: Then L = inf{s : X z * = X+*), is honest which is Ft because on the set {L < t) one has L = inf{s : X, = X:*), measurable. Theorem 16. L is an honest time zf and only zf there exists an optional set A c [0,oo]x R such that L(w) = sup{t < oo : (t, w) E A).

374

VI Expansion of Filtrations

This is often described verbally by saying "L is honest if it is the end of an optional set."

Proof. The end of an optional set is always an honest random variable. Indeed, on {L 5 t ) , the random variable L coincides with the end of the set An([o,t]x fl), which is Ft measurable. For the converse we suppose L is honest. Let ( L t ) p o be an IF adapted process such that L = Lt on {L 5 t ) . Since we can replace Lt with Lt A t we can assume without loss of generality that Lt 5 t . There is also no loss of generality t o assume that Lt is increasing with t , since we can further replace L, with sup,,, L,. Last, it is also no loss, now that it is increasing, t o take it right conti~uous.We thus have that the process (Lt)~?ois optional for the filtration IF. Last, L is now the end of the optional set { ( t ,w ) : Lt(w) = t ) . When L is honest we can give a simple and elegant description of the filtration F ~ .

Theorem 17. Let L be an honest time. Define

Then G = (Gt)t>o constitutes a filtration satisfying the usual hypotheses. Moreover L is a stopping time. A process U is predictable for (6 if and only if it has a representation of the form

where H and K are IF predictable processes. Proof. Let s < t and take H E G,, of the form ( A n {L > s ) ) U ( B n {L 5 s ) ) with A, B E 3,.We will show that H E Gtl which shows that the collection G is filtering to the right.3 Since L is an honest time, there must exist D E Ft such that {L 5 s ) = D n {L 5 t ) . Therefore

U ( B n D)] E F t . The fact that each Gt is a a-algebra, and also with [ ( A n DC) that G is right continuous, we leave to the reader. Note that {L I t ) E Gt which implies that L is a G stopping time, as we observed at the start of the proof of Theorem 16. For the last part of the theorem, let U = ( U t ) t 2 ~be a cAdl8g process adapted to the G filtration. Let Ht and Kt be 3 t measurable random variables such that

U t = H t on { L > t ) and & = K t on { L I t ) . Next set H, = liminf,p H,; K t = liminf,7t J , for all t > 0. Suppose that s is a rational number. Take t = 0 and we have the equality That is, if

s

< t , then G, C Gt.

3 Progressive Expansions

375

One can extend this result to predictable processes by the Monotone Class Theorem. Theorem 18. Let X be a square integrable martingale for the IF filtration. Then X is a semimartingale for the G filtration if L is an honest time. Moreover X has a (6 decomposition

Before beginning the proof of the theorem, we establish a lemma we will need in the proof. It is a small extension of the local behavior of the stochastic integral established in Chap. IV. Lemma (Local behavior of the stochastic integral at random times). Let X and Y be two semimartingales and let H E L ( X ) and J E L ( Y ) . Let U and V be two positive random variables with U < V . ( U and V are not assumed t o be stopping times.) Define

A

=

{W : Ht ( w ) = Jt ( w ) and X t ( w ) - X u ( w ) = Y , ( w ) - Y u ( w ) , for all t E [ U ( w ) V , (w))).

Let Wt = H . X t and Zt = J . Y,. Then a.s. on A, Wt( w ) - Wu( w ) = Zt ( w ) Zu ( w ) for all t E [ U ( w ) V , (w)).

Proof. We know by the Corollary to Theorem 26 of Chap. IV that the conclusion of the lemma is true when U and V are stopping times. Let (u, v ) be rationals with 0 < u < v , and let

Then also Wt - Wu = Zt - Z,, all t E [ a ,v ) , a.s. on A, times and hence a fortiori stopping times. Next let

since u , v are fixed

{ U ( W )< u < v < V ( W )and , W t ( w ) - W,(w) = Z t ( w ) -Z,(w) w: u,vEQ+ U
1

for all t E [ u ,v )) .

The intersection in the definition of A is countable, so null sets do not accumulate. Finally let u decrease t o U ( w ) and then v increase t o V ( w ) , which gives the result.

376

VI Expansion of Filtrations

Proof of Theorem 18. We first observe that without loss of generality we can assume Xo = 0. Let H be a bounded IF predictable process. We define stochastic integrals at the random time L by

lL

HSdxs = ( H . X)L,

lm H ~ ~ X= "( H . X),

- (H . X ) L

where H . X = ( H . Xt)t20 is the IF stochastic integral process. That is, we use the usual definition of the stochastic integral, sampling it at the random time L. When H is a simple predictable process, these are reasonable definitions. Moreover we know from the lemma that if H and J are both bounded IF preHsdXs = dictable processes and if also H = J on (L, oo],then JsdXs a.s., so the definition is well-defined. Since X is an IF martingale in L2 with Xo = 0, we have E{J~, HsdXs) = 0. Applying the equalities (*) on page 372 to H . X we have

where the second equality uses (**) on page 373. Recalling E { J HsdXs) ~ =0 and combining this with the above gives (where we have changed the name of the process H t o K for clarity slightly later in the proof)

We next replace K with K l j Z _

to obtain

because the predictable projection of &l(L,,)(~) is l{Z,_
+

Up to this point we have not used the hypothesis that L is an honest time. Now we do. Let us take U to be a simple (6 predictable process, and using Theorem 17 we have U = H ~ [ o ,+ L K~(L,,] ]

4 Time Reversal

377

where H and K are IF predictable processes (not necessarily simple predictable, however). If we further take U bounded by 1 and take the supremum over all such simple predictable processes of the elementary (6 stochasUsdXs, we get that the variation of X , Var(X) as defined in tic integrals Chap. 111, is less than or equal the expected total variation of

SF

Thus the process X is a (6 quasimartingale, and hence also a semimartingale. We also note that by the preceding we have that E{XT - aT)= 0 for all (6 stopping times T , and thus it is a martingale for the (6 filtration. The next theorem is really a corollary of Theorem 18. T h e o r e m 19. Let X be a semimartingale for the IF filtration. Then X is a semimartingale for the (6 filtration i f L is an honest time.

Proof. Let X = M + A be a decomposition of X in the IF filtration, where M is a local martingale and A is a finite variation process. Since A remains a finite variation process in the larger (6 filtration, we need only concern ourselves with M. By the Fundamental Theorem of Local Martingales (Theorem 25 on page 125), we know that we can write M = N V where N is a local martingale with bounded jumps, and V is a martingale with paths of finite variation on compacts. Again, V remains a semimartingale in the (6 filtration since it is of finite variation, and further N can be locally stopped with IF stopping times T, tending t o ca a s . to be a square integrable martingale for each T,. Then by Theorem 18 each XTn is also a (6 semimartingale. Since X is locally a (6 semimartingale, it is an actual (6 semimartingale.

+

4 Time Reversal In this section we apply the results on initial expansions of filtration t o some elementary issues regrading time reversal of semimartingales, stochastic integrals, and stochastic differential equations. Definition. Let Y = (K)o
Note that time as a superscript denotes reversal. Let IF denote the forward filtration, and correspondingly let If = denote a backward filtration. As an example, if B is a standard Brownian motion, 3$' = a{Bs;s 5 t ) , and

378

VI Expansion of Filtrations

= 6 V N is the Ft = 3$' V N , where Af are the P-null sets of 31, while analogous backward filtration for B. Our first two theorems are obvious, and

we omit the proofs.

Theorem 20. Let B be a standard Brownian motion on [0, 11. Then the time reversal B of B is also a Brownian motion with respect to its natural filtration. Theorem 21. Let Z be a Lkvy process on [O, 11. Then 5 is also a Lkvy process, with the same law as -2, with respect to its natural filtration. In what follows let us assume that we are given a forward filtration IF = (3t)05t51 and a backward filtration (6 = (Gt)oltll. We further assume both filtrations satisfy the usual hypotheses.

Definition. A c&dl&gprocess Y is called an (IF, (6) reversible semimartingale if Y is an IF semimartingale on [0, 11 and ? is a (6 semimartingale on [0,1). Note the small lack of symmetry: for the time reversed process we do not include the final time 1. This is due to the occurrence of singularities at the terminal time 1, and including it as a requirement would exclude interesting examples. Let rt = (to,. . . ,tk) denote a partition of [0,t] with to = 0, tk = t, 0 ( t 5 1, and let

Definition. Let H, Y be two c&dl&gprocesses. The quadratic covariation of H and Y , denoted [H,Y] , is defined to be lim ST: (H, Y)

n-oo

=

[H, Y]t

when this limit exists in probability as mesh(rt) 4 0, and is such that [H,Y] is c&dl&g,adapted, and of finite variation a.s.

Theorem 22. Let Y be an (F,(6) reversible semimartingale, and let H be a chdlhg process such that Ht E F t , and Ht E G'-~, 0 t 5 1. Suppose the quadratic variation [H,Y] exists. Then the two processes [H,Y] and Xt = J: H,-dY, are (IF, G)reversible semimartingales. Moreover

<

<

= GS, and Proof. Since Ht E G1-,,O 5 t 1, we have that HI-, E thus it is adapted and left continuous, so the stochastic integral J: is well-defined. We choose and fix t, 0 < t < 1. Let T be a partition of [l - t, 11, T = ( ~ 1 ,... , sn), chosen such that AHss = 0 and AYSi = 0 a.s.,

4 Time Reversal

379

i = 2 , . . . , n - 1. Note that one can always choose the partition points {si) in this manner because

since for each w, s H Y,(w) has only countably many jumps, and hence 1 l ~ l A K l > o }= d s0 a.s. But then P(lAYsI > 0)ds = 0, which in turn implies that P(lAY,l > 0) = 0 for almost all s in [0, 11. We now define three new processes:

So

Let rn be a sequence of partitions of [O,1] with limn,, lim CTn= [H,Y] 1-

-

n-00

[H,Y] l-t

mesh(rn) = 0. Then

+ AHl-tAY1-t

-

Since Y is (IF, 6) reversible by hypothesis, we know that CT is 6 adapted. Hence [H,Y] is adapted and moreover since it has paths of finite variation by hypothesis, it is a semimartingale. Since H is c&dl&gwe can approximate the stochastic integral with partial sums, and thus

Since Y,,+,- - Yse_ = -(PIps< - Pl-si+l), we have

1 t

lim n-oo

=

Combining ATn, BTn, and CTn, we get

~ l - ~ d F ~

380

VI Expansion of Filtrations

Since we chose our partitions rnwith the property that AYsi = 0 a.s. for each partition point si, the above simplifies to

which tends to 0 since s l decreases to 1- t , and sn-1 increases to 1. Therefore

o = n-00 lim

-

A~~+ lim B~~+ lim n+00

n-00

cTn = -Xt +

h'

-

H~-,~P"[H,Y I ~ .

This establishes the desired formula, and since we have already seen that [H,Y]t is a semimartingale, we have that X is also a reversible semimartingale as a consequence of the formula. We remark that in the case where P is a (6 semimartingale on [O,1] (and not only on [0, I)), the same proof shows that x is a (6 semimartingale on [0,11.

Example (reversibility for LQvy process integrals). Let Z be a LQvy process on [0, 11, and let fi = ('l?t)olt
we have that Zt = Zl-(l-t) E since Zt = Zt- a.s. Therefore Ht E Using Theorem 22 we conclude that Xt = Jot f(Zs-)dZs is an (IF, lh)reversible semimartingale.

Example (reversibility for Brownian integrals). Let B be a standard Brownian motion on [0, 11. Let L: denote Brownian local time at the level x and let p be a signed measure on R.Let IF be the minimal filtration of B satisfying the usual hypotheses, and let (fit)o
4 Time Reversal

381

We will show U is an (IF, a ) reversible semimartingale. The hypotheses on f imply that its primitive F is the difference of two convex functions, and therefore Mt = F ( B t ) - F(Bo)is an IF semimartingale. Moreover Mt = F(B1-t) - F(B1), and since we already know that BlVtis an fi semimartingale by Theorem 3, by the convexity of F we conclude that M is also an fi semimartingale. Therefore Jot f (B,)dBs is an (IF, fi) reversible semimartingale as soon as $ j, Lpv(da) is one, where 77 is the signed measure 'second derivative' of F . Finally, what we want to show is that At = S , L:p(da) is an (F,Ik) reversible semimartingale, for any signed measure p. This will imply that U is an (IF, fi) reversible semimartingale. Since A is continuous with paths of finite variation on [O,l], all we really need to show is that At E X t , each t. But At = Al-t - A1 = jR(Ly-t L?)p(da), and

where A: is the local time at level x of the standard Brownian motion B. Therefore At = -JR(A;-B1)p(da). Since A: is fi adapted, and since B1 E X O , we conclude A is @d adapted, and therefore U is an (IF,@ reversible semimartingale. We next consider the time reversal of stochastic differential equations, which is perhaps more interesting than the previous two examples. Let B be a standard Brownian motion and let X be the unique solution of the stochastic differential equation

where a and b are Lipschitz, and moreover, cr and b are chosen so that for h Bore1 and bounded,

where ~ ( -1t, u, x) is a deterministic function. As in the example expansion via the end of a stochastic differential equation treated earlier on page 367,

382

VI Expansion of Filtrations

we know (as a consequence of Theorem 10) that if IF is the natural, completed Brownian filtration, we can expand IF with X I to get W : 'lit = FU V a { X 1 ) , and then all IF semimartingales remain W semimartingales on [ O , l ) . We fix ( t ,w) and define qh : R 4 R by qh(x) = X ( t , w , x ) where Xt = X (t,w, x ) is the unique solution of

nu,t

for x E R.Recall from Chap. V that qh is called the flow of the solution of the stochastic differential equation. Again we have seen in Chap. V that under our hypotheses on a and b the flow qh is injective. For 0 < s < t < 1 define the function qhStt to be the flow of the equation

It now follows from the uniqueness of solutions that Xt = qhStt(Xs),and in particular X 1 = qht,l(Xt), and therefore qh<:(x1) = X t , where 4;; is of course the inverse function of qhs,t. Since the solution Xs,t of equation (*) is E = a { B , - B u ; s u , v t ) measurable, we have qht,l E F ' - ~ ,where = a{Bs;O s t ) V N where N are the null sets of 3. Let 'lit = P V a { X 1 ) and we have B is an fi = ('lit)o
nu,,

< < <

<

Theorem 23. Let B be a standard Brownian motion and let X be the unique solution of the stochastic differential equation

for 0 < t 5 1, where a and b are Lipschitz, and moreover, a and b are chosen so that for h Bore1 and bounded,

where ~ ( -1t , u , x ) is a deterministic jkction. Let W be given by 'lit = V a { X l } . Then B is an (IF, reversible semimartingale, and xt = X l p t satisfies the following backward stochastic differential equation

nu,, Fu

a)

for 0 < t 5 1. In particular, X is an

(IF,a) reversible semimartingale.

Bibliographic Notes

383

Proof. We first note that B is an (F,fi) reversible semimartingale as we saw a'(X,)a(X,)ds, in the example on page 367. We have that [a(X),B] = which is clearly of finite variation, since a' is continuous, and thus a l ( X ) has paths bounded by a random constant on [0, 11. In the discussion preceding this and of course c ( X t ) E Ft . theorem we further established that a(Xt) E Therefore by Theorem 22 we have

Sot

-

Observe that [a(X), B], = J:-, a'(X,)a(X,)ds. Use the change of variable v = 1 - s in the preceding integral and also in the term

XI-t

= XI

+

I'

a(X1-,)dBS

+

I'

a'(X1-,)a(Xl-,)ds

1 Sl-, b(Xs)ds t o get

+

I'

b(Xl-,)ds,

and the proof is complete.

Bibliographic Notes The theory of expansion of filtrations began with the work of K. Ito [loo] for initial expansions, and with the works of M. Barlow [7] and M. Yor [239] for progressive expansions. Our treatment has benefited from some private notes P. A. Meyer shared with the author, as well as the pedagogic treatment found in [44]. A comprehensive treatment, including most of the early important results, is in the book of Th. Jeulin [114]. Excellent later summaries of main results, including many not covered here, can be found in the lecture notes volume edited by Th. Jeulin and M. Yor [I171 and also in the little book by M. Yor [247]. Theorem 1 is due to J. Jacod, but we first learned of it through a paper of P. A. Meyer [175], while Theorem 3 (It6's Theorem for Mvy processes) was established in [lo?], with the help of T. Kurtz. Theorems 6 and 7 are of course due to Th. Jeulin (see page 44 of [114]); see also M. Yor [245]. Theorem 10, Jacod's criterion, and Theorem 11 are taken from [104]; Jacod's criterion is the best general result on initial expansions of filtrations. The small result on filtration shrinkage is new and is due to J. Jacod and P. Protter. Two seminal papers on progressive expansions after M. Barlow1s original paper are those of Th. Jeulin and M. Yor [I151 and [116]. The idea of using a slightly larger filtration than the minimal one when one progressively expands, ~) dates to M. Yor [240], and the idea to use the process Zt =O 1 1 ~ >originates with J . Az6ma [3]. The lemma on local behavior of stochastic integrals at random times is due to C. Dellacherie and P. A. Meyer [46], and the idea of the proof of Theorem 18 is ascribed to C. Dellacherie in [44]. The results on time reversal are inspired by the work of J . Jacod and P. Protter [I071 and

384

Exercises for Chapter VI

also E. Pardoux [190]. Related results can be found in J. Picard [193]. More recently time reversal has been used to extend It6's formula for Brownian motion to functions which are more general than just being convex, although we did not present these results in this book. The interested reader can consult for example H. Follmer, P. Protter and A.N. Shiryaev [77] and also H. Follmer and P. Protter [76] for the multidimensional case. The case for diffusions is treated by X. Bardina and M. Jolis [6].

Exercises for Chapter VI Exercise 1. Let ( 0 , F , IF, B , P ) be a standard Brownian motion. Expand IF by the initial addition of a(B1). Let M be the local martingale in the formula

Show that the Brownian motion M = (Mt)olt
Exercise 2. Show that the processes Ji defined in the proof of It6's Theorem for L6vy processes (Theorem 3) are compound Poisson processes. Let N i denote the Poisson process comprised of the arrival times of J i , and let Gi be the natural completed filtration of Ni. Further, show the following three formulae (where we suppress the i superscripts) hold. Nt (a) E{JtIJl,R} = l { ~ , t l } l ~ ; J i . (b) E(JtIJ1,Ei) = tJi. (c) E(JtlJ1) = t J i .

Exercise 3. In Exercise 1 assume Bo = x and condition on the event {B1= y). Show that B is then a Brownian bridge beginning at x and ending at y. (See Exercise 9 below for more results concerning filtration expansions and Brownian bridges. Brownian bridges are also discussed in Exercise 24 of Chap. 11, and on page 299.) Exercise 4. Let ( 0 , F , IF, 2,P ) be a standard Poisson process. Expand IF by the initial addition on a{Z1). Let M be the local martingale in the formula

Show that M = (Mt)0
Exercise 5. Let B be standard Brownian motion. Show there exists an F1 measurable random variable L such that if G is the filtration IF expanded 1. Deduce that B is not a semiinitially with L, then Et = 3 1 for 0 t martingale for the expanded filtration (6. Show further that L can be taken f (s)dBs where f is non-random. to be of the form L =

< <

S,'

Exercises for Chapter VI

385

Exercises 6 through 9 are linked, with the climax being Exercise 9.

Exercise 6. Let B denote standard Brownian motion on its canonical path space on continuous functions with domain R+ and range R, with Bt(w) = w(t). IF is the usual minimal filtration completed (and thus rendered right continuous a fortiori). Define Y,' = P{Ba E UIFt}. Show that

where

Y,' for t

= g(u, t , Bt)

and

< a. Show also that 5'

g(u, t, x) = =

-(u - x ) ~

1

J-j

liBaEU)for t

2(a - t)

1

2 a.

Exercise 7. (Continuation of Exercise 6.) Show that Y,U_= 0 except on the is a positive martingale which is null set {B, = u) and infer that (q")o
where (37

-(u, dx for t

S,x) =

U-x 2(a - t )

1

< a.

Exercise 9. (Continuation of Exercises 6, 7, and 8.) Show that one can define a probability P, on C([O,a],B) such that P,(Ba = u) = 1 with P, absolutely continuous with respect to P on Ft for t < a, and singular with respect to P on F a . (Hint: Sketch of procedure. Let JZa = C([O,a), R), be the space of continuous functions mapping [0, a] into R, and define MF = Y,,/V on Ft for (of course) t < a. Let P, denote the (unique) probability on JZa = C([O,a), R) given on Ft by dPu = MpdP, and confirm that

is a Brownian motion under P,. Next show that A: = / & ~ [ B , M " ] ~ verifies dA: = g ( u , s , ~ t ) U-Bt g(u,t,Bt) a-t' Finally show that

386

Exercises for Chapter VI a-t a

t a

Bt - -B o - - u =

(a - t)

l

&dp3

Sot

under P,, and thus (a - t) (a - s)-'dpS, which is defined only on [O, a), has the same distribution under P, as the Brownian bridge, whence lim(a - t ) l & d p S

t-a

=O

a.s., and deduce the desired result.)

Exercise 10 (expansion by a natural exponential random variable). Let IF be a filtration satisfying the usual hypotheses, and let T be a totally inaccessible stopping time, with P ( T < oo) = 1. Let A = (At)t2o be the - At. Note and let M be the martingale Mt = compensator of that AT = A,. ~ e 6t be the filtration obtained by initial expansion of IF with a{A,). For a bounded, non-random, Bore1 measurable function f let f (As)dMs which becomes in 6, Mf =

where F ( t ) = Jot f (s)ds, since A is continuous. Show that E{f (A,)) = E{F(At)), and since f is arbitrary deduce that the distribution of the random variable AT is exponential of parameter 1.

Exercise 11. Let IF, T, A, and 6 be as in Exercise 10. Show that if X is an IF martingale with XT- = XT a.s., then X is also a 6 martingale. Exercise 12. Show that every stopping time for a filtration satisfying the usual hypotheses is the end of a predictable set, and is thus an honest time. Exercise 13. Let B = (Bt)tZo be a standard three dimensional Brownian motion. Let L = sup{t : llBtll 5 I), the last exit time from the unit ball. Show that L is an honest time. Exercise 14. Let M L be the fundamental L martingale of a progressive expansion of the underlying filtration IF using the non-negative random variable (a) Show that for any IF square integrable martingale X we have E X L = E X ~ M ~ . (b) Show that M L is the only IFL square integrable martingale with this property for all IF square integrable martingales.

Exercise 15. Give an example of a filtration IF and a non-negative random variable L and an IF semimartingale X such that the post L process Yt = Xt - XtAL = XtvL - Xt is not a semimartingale. (Hint: Use the standard Brownian filtrations and let

Exercises for Chapter VI

387

where f is the non-random function of Exercise 5.)

Exercise 16. Let ( 0 ' 3 , F, P ) be a filtered probability space satisfying the usual hypotheses, let L be a non-negative random variable, let G, = 3, V a{L), and let (6 be given by Let X be a IF semimartingale. Give a simple proof (due to M. Yor [239]) that Xtltt
2

Jot

Jot

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l+Sot

References

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Symbol Index

AL 371

Vn

IlAllP

dx(H,J )

245

SowldAsl

101,245

155

& ( X ) 85 ~ ~ ( 320 2 )

d L , d X 180 -

220,301

A

154

EH(Z) 322

A"

118

EF-s(Z) 280

IAI, (IAlt)t>o

40,101

(S. f ) ( t , w , x) (Ft)t>o 355

JAlw 101 E{IAlw)

< co

111

F.A

E{IAIT,)

< co

111

F 3,356

40

blL

56,155

FL 370

b0

102

FM 36

bP

102,154

FT 5

BMO

D

56,244

Dn Ducp

16

195

Di 252

250,301 57

279

F

105

--O

Ft 293,309

IF

377

F-S acceptable

279

404

Symbol Index

F-S integral

Z(M)

39,101

FV

gt

82

58

Jx

228

181,185

j p ( N , A ) 245

('Ft2, X)integrable

163

L(X)

163

H . A

40

L a ( X ) t 212,230

H .X

58,60,156

L;

212

HQ . X

60,165,170

IL 56,101,102,153,155,244

He;.X

61

L(G)

61

HG.X

175

Lucp

57

[ H , Y ] 378

Lo

HP -

( M ,N )

244

J H,dX,

58

J,O" H,dX,

58,60

52,56

M~

178

372

M ~ - 172

~ , t H , o d X , 82,271

M2

J:

M 2 ( A ) 182

H,dX,

60,156

J& H,dX,

60,78

178,186

IIMIIBMo

W ( L ) 362

N

154

W'

357

0

102

'Ft2

154,244,193

OL

'Ft2(P) 169

'FtP

-

Hw

193 314

372

PM 36 P

P

102,154,206 293,309

-

P

( H l ) 302,328

Q

( H 2 ) 302,328

Qt(w,dx)

( H 3 ) 329

Q

Ix

52,144

195

131

4,214

Rn 19

362

Symbol Index

IF+ S

19 51

S(W)

53

I

244

IlXllFP

244

(X,X)

70,122

Su

52,56

( X c ,X c )

SUC,

57

{X- 5 a )

S(a)

248

OXn

369

S(A)

179

xvy

211

xAy

211

T'

265

70

TA 104 [TI

104

uq

57

Var(X)

116

Var,(X) X(u,) X*

116 269

11,244

X+*

373

x,+*373 Xc

70,221

xT

10,178

XT-

54,172

xTn 369 X:

293,301

[X,XI

66

[X,XIc

70,70,216,271

[X,YI 66 [ X c , X c ] 70 AXt

25,60,158

IlXllgp

245

a-slice

248

214

405

Subject Index

a-sliceable 248 B M O see bounded mean oscillation F optional projection 369 F special, G semimartingale 369 F-S see Fisk-Stratonovich F V see finite variation process 3-1' norm 154 3-1' norm 193 \-

,

converge (pre)locally in norm 245

approximation of the compensator by Laplacians 150 arcsine law 228 associated jump process 27 asymptotic martingale (AMART) 218 autonomous 250,293 AzBma's martingale 204 and last exit from zero 228 definition of 228 local time of 230

260

sp

-

(pre)locally in 247 converge (pre)locally in 260 norm 244 ucp see uniform convergence on compacts in probability absolutely continuous 131 Absolutely Continuous Compensators Theorem 192 absolutely continuous space 191 accessible stopping time 103 adapted process decomposable 55 definition of 3 space IL of with c&gl&dpaths 155 with bounded c&gl&dpaths 56 with chdlag paths 56 with c&gl&dpaths 56 angle bracket 123 announcing sequence for stopping time 103

Backwards Convergence Theorem 9 Banach-Steinhaus Theorem 43 Barlow's Theorem 240 Bichteler-Dellacherie Theorem 144 Bouleau-Yor formula 226,227 bounded functional Lipschitz 257 bounded jumps 25 bounded mean oscillation B M O norm 195 B M O space of martingales 195 Duality Theorem 199 bracket process 66 Brownian bridge 97,299,384 Brownian motion absolute value of 217 arcsine law for last exit from zero 228 AzBma's martingale 228 Brownian bridge see Brownian bridge completed natural filtration right continuous 19

408

Subject Index

conditional quadratic variation of 123 continuous modification 17 continuous paths 221 definition of 16 Fisk-Stratonovich exponential of 280 Fisk-Stratonovich integral for 288 geometric 86 Girsanov-Meyer Theorem and 141 It6 integral for 78 LBvy's characterization of 88 Lkvy's Theorem (martingale characterization) 86,87 last exit from zero 228 last exit time 46 local time of 217,225 martingale characterization (LBvy's Theorem) 86,87 martingale representation for 187 martingales as time change of 88, 187 maximum process 23 on the sphere 281 Ornstein-Uhlenbeck 298 quadratic variation of 17,67, 123 reflection principle 23,228 reversibility for integrals 380 semimartingale 55 standard 17 starting a t x 17 stochastic area formula 89 stochastic exponential for see geometric Brownian motion stochastic integral exists 173 strong Markov property for 23 Tanaka's formula 217 tied down or pinned see Brownian bridge time change of 88 unbounded variation 19 white noise 141,243 Burkholder's inequality 222 Burkholder-Davis-Gundy inequalities 193 cadlag 4 c&gl&d 4 canonical decomposition

129,154

censored data 121 centering function 32 change of time 88,190 exercises in Chap. I1 98 Lebesgue's formula 190 change of variables see It6's formula, see Meyer-It6 formula Change of Variables Theorem for continuous FV processes 41 for right continuous FV processes 78 classical semimartingale 102, 127, 144 Class D 106 closed martingale 8 Comparison Theorem 324 compensated Poisson process 3 1,42, 65 compensator 118 Absolutely Continuous Compensators Theorem 192 Knight's compensator calculation method 151 compensator of for F 371 compound Poisson process 33 conditional quadratic variation 70, 122 Brownian motion 123 polarization identity 123 continuous local martingale part of semimartingale 70,221 continuous martingale part 191 converge (pre)locally in - ZP 260 convex functions and Meyer-It6 formula 214 of semimartingale 210 countably-valued random variables corollary 365 counting process 12 explosion time 13 without explosions 13 crude hazard rate 122

sp,

decomposable adapted process 55,101 diffeomorphism of IIBn 319 diffusion 243,291,297 examples 298 rotation invariant 282 diffusion coefficient 298 discrete Laplacian approximations 150

Subject Index Dolkans-Dade exponential see stochastic exponential Dominated Convergence Theorem for stochastic integrals in H' - 267 in ucp 174 Doob class see Class D Doob decomposition 105 Doob's maximal quadratic inequality 11 Doob's Optional Sampling Theorem 9 Doob-Meyer Decomposition Theorem case without Class D 115 general case 111 totally inaccessible jumps 106 drift coefficient 143,298 removal of 137 Duality Theorem 199 Dynkin's expectation formula 350 Dynkin's formula 56,350 Einstein convention 305 Emery's example of stochastic integral behaving badly 176 Emery's inequality 246 Emery's structure equation see structure equation Emery-Perkins Theorem 240 enveloping sequence for stopping time 103 equivalent probability measmes 131 Euler method of approximation 353 evanescent 158 example Emery's example of stochastic integral behaving badly 176 expansion via end of SDE 367 Gaussian expansions 366 hazard rates and censored data 121 It6's example 366 reversibility for Brownian integrals 380 reversibility for L6vy process integrals 380 example of local martingale that is not martingale 37'74 example of process that is not semimartingale 217

409

example of stochastic integral that is not local martingale 176 examples of diffusions 298 Existence of Solutions of Structure Equation Theorem 201 expanded filtration 370 expansion by a natural exponential r.v. 386 expansion via end of SDE 367 explosion time 13,254,303 extended Gronwall's inequality 352 extremal point 183 Fefferman's inequality 195 strengthened 195 Feller process 35 filtration countably-valued random variables corollary 365 ' definition of 3 expansion by a natural exponential r.v. 386 filtration shrinkage 367 Filtration Shrinkage Theorem 369 Gaussian expansions 366 independence corollary 365 It6's example 366 natural 16 progressive expansion 355,369 quasi left continuous 148,189 right continuous 3 filtration shrinkage 367 Filtration Shrinkage Theorem 369 finite quadratic variation 271 finite variation process definition of 39,101 integrable variation 111 Fisk-Stratonovich acceptable 278 approximation as limit of sums 284 integral 82,271 integral for Brownian motion 288 Integration by Parts Theorem 278 It6's circle 82 It6's formula 277 stochastic exponential 280 flow of SDE 301 of solution of SDE 382

410

Subject Index

strongly injective 311 weakly injective 311 Fubini's Theorem for stochastic integrals 207,208 function space 301 functional Lipschitz 250 fundamental L martingale 372 fundamental sequence 37 Fundamental Theorem of Local Martingales 125

for an n-tuple of semimartingales 81 for complex semimartingales 83,84 for continuous semimartingales 8 1 for Fisk-Stratonovich integrals 277 for semimartingales 78 generalized 271 ItaMeyer formula see Meyer-It6 formula It6's example 366 It6's Theorem for LBvy processes 356

Gamma process 33 Gaussian expansions 366 generalized It6's formula 271 generator of stable subspace 179 geometric Brownian motion 86 Girsanov-Meyer Theorem 132 predictable version 133 graph of stopping time 104 Gronwall's inequality 342,352 extended 352

Jacod's Countable Expansion 366 Jacod's Countable Expansion Theorem 53,356 Jacod's criterion 363 Jacod-Yor Theorem on martingale representation 199 Jensen's Inequality 11 Jeulin's Lemma 360

Hadamard's Theorem 330 Hahn-Banach Theorem 199 hazard rates 121 hitting time 4 Hijlder continuous 99 honest random variable 373 Hunt process 36 Hypothesis A 221 increasing process 39 independence corollary 365 independent increments of LBvy process 20 of Poisson process 13 index of stable law 34 indistinguishable 3,60 infinitesimal generator 349 injective flow see strongly injective flow integrable variation process 111 Integration by Parts Theorem for Fisk-Stratonovich integrals 278 for semimartingales 68'83 intrinsic L6vy process 20 It6 integral see stochastic integral It6's circle 82 It6's formula

Kazamaki's criterion 139 Knight's compensator calculation method 151 Kolmogorov's continuity criterion 220 Kolmogorov's Lemma 218 Kronecker's delta 305 Kunita-Watanabe inequality 69 LBvy Decomposition Theorem 31 LBvy measure 26 LBvy process associated jump process 27 bounded jumps 25 chdlag version 25 centering function 32 definition of 20 has finite moments of all orders 25 intrinsic 20 is a semimartingale 55 It6's Theorem for 356 LBvy measure 26 reflection principle 49 reversibility for integrals 380 strong Markov property for 23 symmetric 49 L6vy's arcsine law 228 LBvy's stochastic area formula 89 LBvy's stochastic area process 89 L6vy's Theorem characterizing Brownian motion 86.87

Subject Index LBvy-Khintchine formula 31 last exit from zero 228 Le Jan's Theorem 124 Lebesgue's change of time formula 190 Lenglart's Inclusion Theorem 177 Lenglart-Girsanov Theorem 135 linkage operators 329 Lipschitz bounded functional 257 definition of 250,293 functional 250 locally 251,254,303 process 250,311 random 250 local behavior of stochastic integral 62,165,170 local behavior of the stochastic integral a t random times 375 local convergence in Ep, 260 - Sp local martingale B M O 195 cAdl&gmartingale is 37 compensator of 118 condition to be a martingale 38,73 continuous part of semimartingale 22 1 decomposable 126 definition of 37 example that is not a martingale 37,74 fundamental sequence for 37 Fundamental Theorem 125 intervals of constancy 71,75 not locally square integrable 126 not preserved under shrinkage of filtration 126 pre-stopped 172 preserved by stochastic integration see stochastic integral, preserves reducing stopping time for 37 time change of Brownian motion 88 local property 38,162 local time and Bouleau-Yor formula 227 and change of variables formula see Meyer-It6 formula, see Meyer-Tanaka formula and delta functions 217 continuous in t 213

411

definition of L; 212 discontinuous, example of 225 occupation time density 216,225 of AzBma's martingale 230 of Brownian motion 217 of semimartingale 212 regularity in space 224 support of 214 locally bounded process 164 247 locally in Ep, locally integrable process 359 locally integrable variation process 111 locally Lipschitz 251,254,303 locally special semimartingale 149

sp

MBmin's criterion for exponential martingales 352 MBtivier-Pellaumail inequality 352 MBtivier-Pellaumail method 352 Markov process definition of 34 diffusion 243,29 1,297 Dynkin's expectation formula 350 Dynkin's formula 56 equivalence of definitions 34 infinitesimal generator 349 simple 291 strong 292 time homogeneous 35,292 transition function 35,292 transition semigroup 292 martingale L2 projection 9 'HPnorm 193 B M O 195 asymptotic (AMART) 218 AzBma's 204,228 Backwards Convergence Theorem 9 Burkholder's inequality 222 cMl&g modification 8 closed 8 continuous martingale part 191 definition of 7 Doob's inequalities 11 fundamental L martingale 372 178 in L' Jacod-Yor Theorem 199 local 37

412

Subject Index

Martingale Convergence Theorem 8 martingale representation 203 maximal quadratic inequality 11 measures M (A) 182 not locally square integrable 126 Optional Sampling Theorem 9 orthogonal 179 predictable representation property 182 purely discontinuous part 191 quasimartingale 116 sigma martingale 233 space M' of 178 square integrable 73,74 strongly orthogonal 179 submartingale 7 supermartingale 7 uniformly integrable 8 with exactly one jump 124 Martingale Convergence Theorem 8 martingale representation 203 mathematical finance theory 137 maximal quadratic inequality 11 Meyer's Theorem 104 Meyer-It6 formula 214 Meyer-Tanaka formula 216 modification 3 Monotone Class Theorem 7 monotone vector space 7 closed under uniform convergence 7 multiplicative collection of functions 7 natural filtration 16 natural process 111 net hazard rate 122 Novikov's criterion 140 oblique bracket 123 occupation time density 216,225 optional a-algebra 102,372 optional projection 367,369 Optional Sampling Theorem 9 Ornstein-Uhlenbeck process 298 orthogonal martingales 179 Ouknine's formula 240 partition 116 path space 142 path-by-path continuous part

70

Perkins-Emery Theorem 240 Picard iteration 255 pinned Brownian motion 299 Pitman's Theorem 240 Poisson process arrival rate 15 compensated 31,42,65,118 compound 33 definition of 13 independent increments 13 intensity 15 stationary increments 13 polarization identity 66,123,271 potential 105,150 pre-stopped local martingale 172 predictable a-algebra 102,154 predictable compensator of for F 371 predictable process predictable a-algebra 102 simple 51 predictable projection 368 predictable representation property 182 predictable stopping time 103 predictably measurable process 102, 105 prelocal convergence in BP, 260 - Sp prelocally in Bp,ZP 247 preserved by stochastic integration see stochastic integral, preserves process Lipschitz 250,311 process stopped at T 10,37 progressive expansion 355,369 projections optional projection 367 predictable projection 368 property holds locally 38,162,247 property holds prelocally 162,247 purely discontinuous part of martingale 191 quadratic covariation 66,378 polarization identity 66,271 quadratic covariation process 270 quadratic pure jump 71 quadratic variation conditional 122 continuous part of 70,271

Subject Index covariation 66 finite 271 of a semimartingale 66 polarization identity 66,271 quasi left continuous filtration 148, 189 quasimartingale 116 random Lipschitz 250 random partition definition of 64 tending to the identity 64 Rao's Theorem 118 Ray-Knight Theorem 225 reduced by stopping time 37 reflection principle for Brownian motion 23,228 for LBvy processes 49 for stochastic area process 92 regular conditional distribution 362 regular supermartingale 150 representation property 182 reversibility for Brownian integrals 380 reversibility for LBvy process integrals 380 reversible semimartingale 378 Riemann-Stieltjes integral 41 right continuous filtration 3 right stochastic exponential 319,320 right stochastic integral 319 rotation invariant diffusion 282 sample paths of stochastic process 4 scaling property 238 semimartingale (F, (6) reversible 378 (3-1', X) integrable 163 3-1' norm 154 absolute value of see Meyer-Tanaka formula bracket process 66 canonical decomposition 129 classical 102,127,144 continuous local martingale part 70, 221 convex function of 210 definition of 52 equivalent norm 244

413

example of process that is not 217 Integration by Parts Theorem 68, 83 local martingale as 127 local time of 212 locally special 149 preserved by stochastic integration see stochastic integral, preserves quadratic pure jump 71 quadratic variation of 66 quasimartingale as 127 space 7-L2 of 154 special 129,154 stochastic exponential of 85 submartingale as 127 supermartingale as 127 topology 264 total 52 semimartingale topology 264 separable measurable space 189 sharp bracket 123 sigma martingale 233 preserved by stochastic integration see stochastic integral, preserves sign function 212 signal detection 140 simple Markov process 291 simple predictable process 51 Skorohod topology 220 Skorohod's Lemma 239 smallest filtration making r.v. a stopping time 119,370 special semimartingale 129,154 canonical decomposition 129 square integrable martingale 178 preserved by stochastic integration see stochastic integral, preserves stable law 34 index 34 stable process defintion of 33 scaling properties 34 symmetric 34 stable subspace generated 179 o f M 2 178 stable under stopping 178 standard Bore1 space 362 stationary Gaussian process 299

414

Subject Index

stationary increments of LBvy process 20 of Poisson process 13 statistical communication theory 140 Stieltjes integral see Riemann-Stieltjes integral stochastic area formula 89 stochastic area process definition of 89 density 91 LBvy's formula 89 properties 92 reflection principle 92 stochastic differential equation 255, 32 1 flow of 301 flow of solution 382 weak solution 201,240 weak uniqueness 20 1 stochastic exponential appproximation of 269 as a diffusion 298 definition of 85 Fisk-Stratonovich 280 for Brownian motion see geometric Brownian motion for semimartingale 85 invertibility of 335 right 319,320 uniqueness of 255 stochastic integral X integrable, L ( X ) 163 Associativity Theorem 62,159,165 behavior of jumps 60,158,165 does not preserve FV processes in general 241 Dominated Convergence Theorem in H' 267 ~ o z n a t e dConvergence Theorem in ucp 174 example that is not local martingale 176 for lL 59 for b P 156 for S 58 for P 163 for Brownian motion 78 Fubini's Theorem 207,208 local behavior at random times 375

local behavior of 62,165,170 preserves FV processes, integrands in lL 63 continuous local martingales 173 local martingales 128,171,174 locally square integrable local martingales 63, 171 semimartingales 63 sigma martingales 234 square integrable martingales 159, 171 right 319 said to exist 163 stochastic process adapted 3 decomposable 55,101 definition of 3 finite variation 39,101 increasing 39 indistinguishable 3,60 locally bounded 164 locally integrable 359 modification of 3 natural 111 potential 105 sample paths of 4 stopped 10,37 strong Markov 36 stopped process 10,37 stopping time a-algebra 5,105 accessible 103 announcing sequence 103 change of time 98 definition of 3 enveloping sequence 103 explosion time 13,254,303 fundamental sequence of 37 graph 104 hitting time 4 hitting time of Bore1 set is 5 predictable 103 reducing 37 totally inaccessible 103 Stratonovich integral see FiskStratonovich, integral Stratonovich Integration by Parts Theorem 278 strengthened Fefferman inequality 195

Subject Index Stricker's Theorem 53 strong Markov process 36,292 strong Markov property 36,292 for Brownian motion 23 for L6vy process 23 strongly injective flow 31 1 strongly orthogonal martingales 179 structure equation 200,238 Existence of Solutions Theorem 201 submartingale 7 supermartingale 7 regular 150 symmetric LBvy process 49 symmetric stable process 34 Tanaka's formula 217 tied down Brownian motion 299 time change 190 time change of Brownian motion 88 time homogeneous Markov process 292 time reversal 377

415

total semimartingale 52 total variation process 40 totally inaccessible stopping time 103 transition function for Markov process 292 transition semigroup 292 truncation operators 265 uniform convergence on compacts in probability 57 uniformly integrable martingale 8 usual hypotheses 3,291 variation 116 variation along

T

116

weak solution of SDE 201,240 weak uniqueness of SDE 201 weakly injective flow 31 1 white noise 140,243 Wiener measure 141 Wiener process 137,140

Applications of Mathematics

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Asmussen, Applied Probability and Queues (2nd ed. 2003) Robert, Stochastic Networks and Queues (2003) 53 Glasserman, Monte Carlo Methods in Financial Engineering (2003) 51

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