General Theory of Markov Processes
This is Volume 133 in PURE AND APPLIED MATHEMATICS
H. Bass, A. Borel, J. Moser, and S.-T.Yau, editors Paul A. Smith and Samuel Eilenberg, founding editors A list of titles in this series appears at the end of this volume.
General Theory of Markov Processes Michael Sharp Department of Mathematics University of California at San Diego La Jolla, California
ACADEMIC PRESS, INC. Harcourt Brace Jovanovich, Publishers
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Copyright 0 1988 by Academic Press, Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher.
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Library of Congress Cataloging-in-Publication Data Sharpe, Michael, Date General theory of Markov processes I Michael Sharpe. p. cm. - (Pure and applied mathematics : 133) Bibliography: p. Includes indexes. ISBN 0-12-639060-6 1. Markov processes. I. Title. 11. Series: Pure and applied mathematics (Academic Press) : 133. QA3.P8 vol. 133 [QA274.7] 519.2 -dc19 88-18088 CIP Printed in the United States of America 88 89 90 91 9 8 7 6 5 4 3 2 1
To my wife Sheila and my son Colin
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Preface This work is intended to serve as a reference to the theory of right processes, a very general class of right continuous strong Markov processes. The use of the term general theory is meant to suggest both the absence of hypotheses of special type other than those for right processes, and the coordination of the methods with those of the general theory of processes, as exposed in the first two volumes of Probabilith et Potentiel by Dellacherie and Meyer. We do provide in the appendix a fairly extensive discussion and summary of the general techniques needed in the text, with hopes that it may lead the reader to a fuller appreciation of the Dellacherie-Meyer volumes. The original definition of right process (processus droit) was set down twenty years ago by Meyer as an abstraction of certain properties possessed by standard Markov processes, which had been up to that time the largest class of strong Markov processes that could be shown to have an intimate connection with abstract potential theory. The hypotheses of Meyer were weakened in the subsequent lecture notes of Getoor (1975). Right processes in the sense of Meyer or Getoor do form a class large enough to encompass most right continuous Markov processes of practical interest such as Brownian motion, diffusions, LQvy processes (processes with stationary independent increments), Feller processes and so on, constructed from reasonable transition semigroups. However, the form of hypotheses discussed by Meyer and Getoor contains a serious flaw, in that their hypotheses are not invariant under the classical transformations of Markov processes such as killing, time-change, mappings of the state space, and Doob's h-transforms. Motivated by the wish to have a setting which is preserved by essentially all these transformations, we propose hypotheses for right processes weaker than those of either Meyer or Getoor, but which
. . I
Vlll
Preface
remain strong enough to guarantee a rich theory of sample path behavior and close links with potential theory. The point of view of the book is chiefly to study the probabilistic structure of a given right process as expressed through such objects as its homogeneous functionals, its additive and multiplicative functionals, its associated stochastic calculus, and to consider the transformations of right processes that yield other right processes. It has been a constant goal to avoid imposing secondary hypotheses which would limit the domain of applicability. There is only one section concerning the construction of a right process from a nice (Ray) semigroup, and while adequate for constructing some classical examples, it is not of great generality. There is no discussion of construction of Markov processes by solving Stroock-Varadhan type martingale problems. The recent book of Ethier-Kurtz (1986) has much on these matters. Explicit examples of right processes are discussed principally in the exercises. The connections between right processes and abstract potential theory are discussed though not always in full detail. For example, though there is a discussion of the Hunt-Shih identification of hitting operators and reduite of an excessive function on a set, we do not present a complete proof. The reader interested in questions of more direct potential theoretic type is referred to volumes I11 and IV of Dellacherie-Meyer. The sections on multiplicative functionals and homogeneous random measures, the latter a generalization of additive functionals, bring up to date the older books of Meyer and Blumenthal-Getoor. Especially in the sections on LCvy systems and exit systems, there is a penalty to be paid for the breadth of the hypotheses, requiring us to construct kernels on spaces larger than the state space so that the statements of the results will look a bit unusual to experts familiar with their forms under restrictive measurability conditions. However, the applications of these constructions do not appear to be affected in any essential way by this complication. It is a pleasure to thank those individuals whose comments on earlier versions have eliminated many inaccuracies, inconsistencies and irrelevancies. Marti Bachman, Ron Getoor, Joe Glover, Bernard Maisonneuve, Joanna Mitro, Wenchuan Mo, Art Pittenger, Phil Protter, Tom Salisbury and Michel Weil provided me with valuable feedback for which I am very grateful. Thanks are also due to Neola Crimmins, whose expert entry of part of the first draft simplified the task of assembling the final document in Tj$ format. La Jolla, 1988.
Contents Chapter I: Fundamental Hypotheses 1. Markov Property, Transition Functions and Entrance Laws 2. The First Regularity Hypothesis 3. The Natural Filtration 4. Excessive Functions and the Resolvent 5. The Optional and Predictable a-Algebras 6. The Strong Markov Property 7. The Second Fundamental Hypothesis 8. Right Processes 9. Existence Theorem for Ray Resolvents 10. Hitting Times and the Fine Topology
1 1 7 10 14 19 25 30 38 42 51
Chapter 11: Transformations 11. The Lifetime Formalisms 12. Killing at a Terminal Time 13. Mappings of the State Space 14. Concatenation of Processes 15. Cartesian Products 16. Space-time Processes 17, Completion of a Resolvent 18. Right Process in the Ray Topology 19. Realizations of Right Processes 20. Resume of Notation and Hypotheses
61 62 65 75 77 84 86 89 94 97 104
Chapter 111: Homogeneity 21. Measurability and the Big Shift 22. Construction of Projections 23. Relations between the a-Algebras
106 106 110 115
Contents
X
24. Homogeneous Processes and Perfection 25. Co-optional and Coterminal Times 26. Measurability on the Future
123 138 141
Chapter IV: Random Measures 27. Random Measures and Increasing Processes 28. Integrability Conditions 29. Shifts of Random Measures 30. Kernels Associated with Random Measures 31. Dual Projections 32. Integral Measurability 33. Characterization by Potentials 34. Representation of Potentials 35. Homogeneous Random Measures 36. Potential Functions and Operators 37. Left Potential Functions 38. Generating Homogeneous Random Measures
145 146 147 151 152 157 163 166 169 172 175 181 184
Chapter V: Ray-Knight Methods 39. The Ray Space 40. The Entrance Space 41. Meager Sets and Predictable Functions 42. Left Limits and Predictable Projections 43. The Predictable Projection Kernel 44. Topological Characterizations of Projections 45. Accessibility 46. Left Limits in the Original Topology 47. Quasi-Left-Continuity 48. Natural Processes 49. Balayage of Functions
189 190 195 201 205 207 210 213 218 219 222 231
Chapter VI: Stochastic Calculus 50. Local Martingales over a Right Process 51. Decomposition Theorems 52. Semimartingales and Stochastic Integrals 53. Finite Lifetime Considerations
233 233 239 249 254
Chapter VII: Multiplicative F‘unctionals 54. Multiplicative Functionals and Terminal Times 55. Exact Perfection of a Weak MF 56. Exactly Subordinate Semigroups 57. Decreasing MF’s 58. m-Additivity 59. Left MF’s and Exceptional Sets
259 259 263 267 272 275 278
Markov Processes 60. Measurability of a MF 61. Subprocess Generated by a Decreasing MF 62. Subprocess Generated by a Supermartingale MF
xi
284 286 290
Chapter VIII: Additive Functionals 63. Classification of Additive Functionals 64. Fine Support 65. Time Change by the Inverse of an AF 66. Absolute Continuity 67. Balayage of an AF 68. Local Times 69. Additive Functionals of Subprocesses 70. Relative Predictable Projections 71. Exponential Formulas 72. Two Motivating Examples 73. LCvy Systems 74. Excursions from a Homogeneous Set 75. Characteristic Measures of an AF
302 303 304 305 309 315 324 326 329 334 339 342 350 358
Appendices AO. Monotone Class Theorems A l . Universal Completion and Trace A2. Radon Spaces A3. Kernels A4. Lebesgue-Stieltjes Integrals A5. Sketch of the General Theory of Processes A6. Relative Martingales and Projections
364 364 367 369 375 379 385 394
References
404
Notation Index
41 1
Subject Index
413
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1
Fundamental Hypotheses
1. Markov Property, Transition Functions and Entrance Laws
A stochastic process indexed by a subset of the real line has the Markov property if, roughly speaking, the past and future are conditionally independent given the present, for every possible value of the present. See (1.1) below for the precise specification. In this definition, the state space is required to have only measurable structure-no algebra or topology is involved. Nevertheless, because of applications to special examples and our focus on path regularizations which would otherwise take a different form, we shall work exclusively with topological state spaces. The minimal hypothesis on every state space E shall be that E is a Radon topological space. See SAl. This is not a burdensome restriction. Every Polish (:=complete, separable, metrizable) space and every locally compact Hausdorff space with countable base (LCCB) is Radonian. The notation B ( E ) stands for the Borel a-algebra on E , but we shall use the simpler notation E in its place unless clarity dictates otherwise. The notation E" will, following the pattern described in the Appendix, denote the a-algebra of universally measurable subsets of E . Other a-algebras intermediate to E and E" will be introduced later. We shall always denote a generic such a-algebra by E' with the superscript usually being one of O , T , ~ ,referring to the a-algebras on E generated by the Borel, Ray and excessive functions respectively. Thus €O is just another name for E. See $10. In later sections, we shall make a distinction between Eo and E , identifying E with E' instead of Eo. This will require minor reinterpretation of some of the constructs in this chapter, but to do otherwise would lead to serious notational complications later.
Markov Processes
2
The reader is now assumed to be familiar with the terminology established in the Appendix, especially in AO-A3. In particular, given a ualgebra M on a space M , b M (resp., p M ) stands for the class of bounded (resp., positive) M-measurable functions on M. (Positive always refers to values in [O,oo], rather than the positive reals). Let (R, 9, P) be a probability space, I be an index set contained in the ~ a stochastic process indexed by I, real line R, and let X = ( X t ) $ , =be with values in E. That is, ( X t ) t E Iis a collection of measurable maps of (R, 9 ) into (E,E ) . In order to emphasize the dependence here on E , we call X an €-stochastic process. Similar definitions will apply when E is replaced with a larger u-algebra E'. It is, of course, a more demanding condition for X to be an €'-stochastic process, as it is required in this case that for every t E I, {Xt E F } := { w E R : X t ( w ) E F} be in B for every set F in E m rather than for every F in E . Corresponding to a fixed a-algebra E' on E and a fixed €'-stochastic process X on R, the natural a-algebra 3& ( or, more simply, F:) is defined as u { f ( X , ) : r E I , r 5 t , f E E m } . Asimilar definition specifies the ualgebra 3:t of the future from t. Thus, for example, (resp., F )denotes the u-alge%ra generated by the maps f(X,) with T 5 t and f in Eo(:= E ) (resp., f in E"). The process X has the Em-Markovproperty if the a-algebras F&, 3:t are conditionally independent given Xt,for every t E I. That is, For t I, A f 3& - and B E 3&, (1.1)
P{A fl B I X t } = P{A I Xt} P { B I X t } .
The need for the prefix E' is only temporary, as we shall see after the discussion of augmentation procedures in 56. Under the condition (l.l),one may compute, using the well known properties of conditional expectations,
P{ A n B
} = P{ P{A n B 1 Xt} } = p{ P{A I X t } P{B I X t } } = P{ P{B I Xt}; A } .
As A E F: is arbitrary, it follows that (1.1) implies
for every B E 3;t,t E I. That is, prediction of future behavior of X based on the entire past is only as valuable as the predictor based on the present value Xt alone. Conversely, the condition (1.2) implies (1.1) by similar manipulations, and consequently (1.2) is also referred to as the Em-Markov
I: Fundamental Hypotheses
3
property of X. In many respects, (1.2) is more convenient to manipulate and generalize. In the first place, it is reasonable and useful to replace the filtration (F:)with a more general filtration (Gt) to which ( X t ) is &'adapted. This leads us to say that (Xt) is €'-Markovian with respect to (Gt) if X is €'-adapted to ( G t ) , and if, for all t E I and all B E F;t, -
In applications, (1.3) has a more convenient form (1.4)
P{H I G t } = P{H I X t ) ,
H€ P5t.
Formula (1.4) is an immediate consequence of (1.3), starting with the case H = l g , B E F&, and making use of the Monotone Class Theorem (AO.l). The definition above is too crude to be useful except when I is a discrete subset of R. We bring more precision to bear by introduction of the notion of a transition function (P,,t) for X. (1.5) DEFINITION.A family of Markov kernels on ( E , € * )indexed by pairs s,t E I with s 5 t is a transition function on ( E ,E') if, for all T I s 5 t in I and all x E E , B E E' p r , t ( x ,B ) =
J,
p r , s ( x ,d y ) ~ , , t ( yB). ,
In accordance with the discussion of kernels in A3, P,,t(x, dy) is a kernel on (E,E') provided that, for all x E E , P,,t(x,dy)is a positive measure on (E,E'), and for every B E E', x -, P,,t(x, B ) is E' measurable. In addition, P,,t(x,dy)is a Markov kernel if P,,t(z, E ) = 1 for all z E E. The equation in (1.5) is called the Chapman-Kolmogorov equation. Define the action of the Markov kernel P,,t on bE' (resp., pE') by
P,,tf
:=
J P,,t(z,4)fb),
f
E
PE'
u bE',
so that P,,t f E bE' (resp., pE'.) See sA3. We say that a transition function (P8,t)on (E,€') is the transition function for a process (Xt)tE~ with values in E , and satisfying the Markov property (1.4) relative to (Gt) in case (1.6)
P { f (Xt) I G,} = P,,tf (XS), s I t E 1,f E bE'.
(1.7) THEOREM. Let ( X t ) t Ebe ~ €'-adapted to (&), and suppose that is a transition function on (E,E') such that (1.6) holds for every s 5 t E I and every f E bE'. Then X has the Markovproperty (1.4).
PROOF:The class IH of random variables in bF,t for which (1.4) holds is clearly an MVS (see AO) because of monotonicity properties of conditional
4
Markov Processes
expectations. By hypothesis, 'H contains every H of the form f ( X t ) with f E bE'. AS bF>, is generated by the multiplicative class V = UnVn, where V , is the calection of products FlF2 . . S F , with Fj = f j ( X t j ) ,t 5 tl 5 t 2 5 5 t n , fj E bE', it suffices by the Monotone Class Theorem to verify that V C 'H. Proceed by induction on n to get U, c 'H for all n 2 1. By our first remarks above, V1 C 3-1. Suppose, inductively, that V , C 'H and let G = F 1 . . . F,+1 E Vn+l . Compute P { G I Gt} by first conditioning relative to Gt,, so that
However, the random variable being conditioned in the last term is clearly in V,, and thus, by inductive hypothesis, Gt may be replaced by X t . The same calculation with Gt replaced throughout by Xt completes the inductive step by proving P { G I G t } = P { G I X t } , which finishes the proof. We shall be interested primarily in the case I = R+ := [O,m[ though the cases 10, m[, ] - m, m[ and 30,1[ also arise frequently in practice. A family (Pt)t>0of Markov kernels on ( E ,E ' ) is called a Markov transition semigroup or simply a transition semigroup in case
A transition function (P,,t) indexed by s 5 t E R+ is called temporally homogeneous if there is a transition semigroup ( P t )with P,,t = Pt-, for all s 5 t. Starting with a transition semigroup (Pt),P,,t := Pt-, defines a temporally homogeneous transition function. A Markov process X satisfying (1.6) with a homogeneous transition function (P,) has the characteristic property
This is the simple Markov property of X relative to (Pi). The Markov processes considered here will be temporally homogeneous for the most part. See however exercise (1.15), which deals with the socalled space-time process connected with a general Markov process. Suppose now that (Xt)t>o has the Markov property (1.8) relative to (Q, 9,G t , P ) , with transition-semigroup (Pt). The distribution po of Xo is called the initial law of X , and the distribution pt of Xt then satisfies pt = p0Pt for all t 2 0. That is, for f E bE',
I: Fundamental Hypotheses
5
If the index set for X were instead 10, m[, there would be no initial law PO definable as above. However, the pt would obviously satisfy the identities
Pt+e = Pt ps,
(1.9)
t , s > 0.
A family (pt)t>o of positive measures on ( E ,E’) satisfying (1.9) is called an entrance law for the semigroup (Pt).It is called finite in case p t ( E ) < m V t > 0 , bounded if sup,pt(E) < m, probabiZity if p t ( E ) = 1 for all t. If there is a measure PO such that pt = popt for all t > 0, then PO is said to close the entrance law (pt)t>o. A probability entrance law (,ut)t>o need not have a closing element PO. For example, let E be the open right half line R++and let P t ( x ,dy) := ~,+t(dy) -unit mass a location x t. Then, for t > 0, pt(dy) := c t ( d y ) defines a probability entrance law for (Pt)without a closing element. See Chapter V for the compactification theory needed to permit the representation of a closing element for an arbitrary probability entrance law. A (temporally homogeneous) Markov process (Xt)t>o satifying (1.8) and having initial law PO necessarily satisfies the more general identities
+
(1.10) P { f l ( X t l ) f 2 ( X t , ). * * . f n ( X t , ) } = p o ( P t , ( f l 4 , - t , (fz-.-(ft,
-
*Pt,-tn-lfn)*-.)))9
-
for 0 5 tl 5 t z 5 -. 5 t,, f l , . . ,fn E bE’. This is a simple consequence of (1.8) via an induction argument. The last formula is perhaps more intuitive in its differential version, which states that under the same conditions as above, (1.11) P(X0 E dx0,Xt1 E
,xinE d x n }
d21,...
= Po(dxo)Pt,( 2 0 , dx1) * * . Pt,-tn4 (xn-1, d 4 -
In this form, the Markov property corresponds to the Huygens principle in wave propagation-in order to compute P { X t E dx}, one may imagine interposing a barrier at time tl < t and, knowing the position X t , , perform calculations supposing that the process starts afresh at Xi,. The integral version (1.10) asserts that the total probability that X t E dx is obtained by adding the above probabilities over all possible positions Xi,, weighted by the probabilities of reaching the points Xi, in the first place. (1.12) EXERCISE. Formulate the appropriate versions of (1.10) and (1.11) in the case where X is homogeneous Markov with time parameter set 10, m[.
The next pair of exercises is designed to give the reader a little practice with arguments involving completions. This kind of ‘‘sandwiching”will be used repeatedly in later sections. Exercise (1.14) will show that there is no need to maintain any distinction between different €‘-Markov properties, provided the filtration is sufficiently rich.
6
Markov Processes
(1.13) EXERCISE.Let ( P t ) preserve each of the a-algebras E', E l , with E C E' C E' C E". Let ( X t ) t > be ~ defined on (R, Q,Bt, P) with X satisfying (1.8) for all t , s 1 0, f E b&'r Assume that X is €'-adapted to (Qt). Prove that (1.8) holds then for f E bE'. (Hint: choose f i 5 f 5 f 2 with f 1 , f2 E E' and f 2 - f 1 null for the measure g -t P g ( X t + s ) = P P,g(Xt) ( g E bE'). Remember that a conditional expectation is an equivalence class of random variables.) (1.14) EXERCISE.Let E' C E' be a-algebras preserved by (Pt),and assume that X satisfies (1.8) for every f E bE'. Prove that for all f E bE',
(Hint: by (1.13),one may reduce to the case f E bE'. Show usingmonotone classes that, for every H E b3;, there exist H1 5 H 5 H2 with H I ,HZ E b3: and H2 - H I null for the measure G + P G (G E b3:).)
(1.15) EXERCISE.Let ( X t ) t > ~be Markov with transition function (Pa,t). Suppose also that (P8,t)satisfies the measurability condition (s,t , x ) + Ps,t(z,B)l{,st) is in B(R) @ B(R) 8 E'
V B E E'
Show that, with E , denoting unit mass at u E E ,
defines a Markov transition semigroup on (R x E , B(R) 8 E') and the space-time process X t := ( t , X t ) has the Markov property relative to (fz,G, Gt, P), with transition semigroup (p,).
(1.16) EXERCISE.Verify, using the Kolmogorov existence theorem, that if (Pt) is a Markov transition semigroup on the Radon space E , and if(pt)t>o is an arbitrary probability entrance law for (Pt),then there exists a unique probability measure P on the product space R = Elo@[ with product ualgebra Q so that the coordinate maps Xt form a Markov process with transition function (P,) and entrance law ( p t ) . Formulate and check the temporally inhomogeneous version of this result. (1.17) EXERCISE.Let X t be a process with (not necessarily stationary) independent increments in Rd. Show that X is Markovian and satisfies (1.6) for some transition function (Pa,,).
I: Fundamental Hypotheses
7
2. The First Regularity Hypothesis
A stochastic process (Xt)tEf defined on (Q, 0, P) and having values in a topological space E is right continuous in case every sample path t + X t ( w ) is a right continuous map of I into E. The following hypothesis is essentially the first of Meyer's hypotheses droites, which is to say in rough English translation, the regularity hypotheses for right processes. It is formulated as a condition on the transition semigroup rather than on the stochastic process. (2.1) DEFINITION (HD1). A Markov semigroup (P,) on a Radon space E is said to satisfy HD1 if, given an arbitrary probability law p on E, there exists a 0-algebra E' with E c E' C E" and Pt(bE') c bE', and an E-valued right continuous €'-process (Xt)t>O on some filtered probability space ( f l , G , G t , P ) so that X = (R,G,Gt,P,Xt) is (temporally homogeneous) Markov with transition semigroup (Pt) and initial law p. It is implicit in (2.1) that Xt is €'-adapted to ( G t ) and that (1.8) is verified. Obviously, under the conditions of (2.1), one may replace (Gt) by (F:)without affecting anything. Notice that exercise (1.14) shows that if X satisfies all the conditions described in (2.1), and if ( X t ) is €"-adapted to ( G t ) then X also satisfies (2.1) with E' replaced everywhere with E". That is, (2.1) does not really depend on the particular E'. The Markov property in the form (1.4) is rather awkward to manage, and in order to facilitate compututions, we shall make use of Dynkin's setup for Markov processes, which brings in a family of measures governing a Markov process-one for each initial value x E E-rather than one fixed measure P. Loosely speaking, we shall work with a fixed collection of random variables Xt defined on some probability space, and a collection P" of measures specified in such a way that P"(X0 = x}, and, under every P", Xt is Markov with semigroup (Pt). The P" may be thought of as the conditional distributions for P, given X O = x. (There is much of interest to be said in connection with Markov processes run under one distinguished measure P, especially in case P is not necessarily a finite measure and the time-parameter set is the entire real line. We shall not go into such matters.) (2.2) DEFINITION. Let E be a Radon space, (Pt) a Markov semigroup on (E,E') preserving E'. The collection X = (Q,G,Gt,Xt,Ot,P")is a right continuous simple E'-Markov process with state space E and transition semigroup (Pt) in case X satisfies conditions (2.3-5) below: (2.3)
(R, G, G t ) is a filtered measurable space, and Xt is an E-valued right continuous process €'-adapted to ( G t ) ;
8
Markov Processes
(2.4)
(@t)t>ois a collection of shift operators for X, viz, maps of R into itself satisfying, identically for t , s 2 0,
(2.5)
For every x E E , P"(X0 = x} = 1, and the process (Xt)t>ohas the Markov property (1.8) with transition semigroup (Pt) relative to (0,G, Gt, P").
The condition P"(X0 = z} = 1 in (2.5) is not always built into the definition of a simple Markov process. Markov processes enjoying this property are usually called normal. All Markov processes here will be assumed normal, unless explicit mention is made to the contrary. Note that (2.3) imposes the requirement that Gt 3 F; for every t 2 0. (2.6) LEMMA.Given a collection (R,G,Gt,Xt,Ot,Pz) as above, then for every H f bF', x -, P"H is €'-measurable.
PROOF:The formula (1.10) and the fact that the initial law for P" is c z , the unit mass at z, shows that P " { f ~ ( X t , ) . . . f ~ ( X t n is )in } €' whenever f l , . . . ,fn € b&', 0 tl t z 5 - . t, . An application of the MCT completes the proof. Let (R, 8,G t , Xt, 8t,P") satisfy (2.2). Given an arbitrary probability law p on E , define P" on (R,F') by P"(H) := p(dx) P"(H) , H E b P . It is a routine exercise to verify that (Xi) continues to have the Markov property relative to (R, G, Gt, P"),with transition function (Pt)and initial law p. One says that the collection X = (R, 9, Gt, X t , &, P") satisfying (2.2) is a realization of the semigroup (Pt). The idea here is that (Pt)may be the prime object of study, and all information about (Pt) is embodied in X, which may then be studied by the methods of stochastic processes rather than those of functional analysis. Under HD1, there is a realization of (P,) which is in some respects canonical.
< <
- <
(2.7) THEOREM. Let (Pt) be a Markov transition semigroup on E satisfying HD1. Then (Pt)has a right continuous realization (st,6, &, X t , &, P").
PROOF:Let x E E . According to (2.1), there is a process (K)t>o on a filtered probability space (W,%, Hi,P) and a a-algebra €' on E preserved and satisfies by (Pt) such that Y is right continuous, €'-adapted to (Xt), (1.1)and (1.8) with initial law ez, so that P(Y0 = z} = 1. Let R denote the space of all right continuous maps of Rf into E. Let X t ( w ) := w ( t ) denote the coordinate variables on R and let 6' := c ( f ( X t ) : t 2 0,f E bE'},
I: Fundamental Hypotheses
9
Q: := a{f(Xs) : 0 5 s 5 t , f E b&'}. The map @ : W Cl defined by @(w):= w , where W ( S ) := Ys(w)for all s 2 0, is characterized by the formulas Y, X,o@,s 2 0. It follows trivially that @ E Z/Q'. Let P" be the image of P under the map @ so that P"F = P { F o @ } for every F E bQ'. This means that for 0 5 tl 5 t z 5 . . . 5 tn and f1, . . . ,fn E b&', ---$
The result now follows from (1.14), taking 9 := Gu. (2.8) PROPOSITION. Let (0,Q,Qt, Xt, Bt, P") be a right continuous simple &'-Markov process as defined in (2.2). For every F E bF* and all t 2 0, F o e t E b3', and P ' L { F O B t I Qt} = PX'{F}. where f(x) := P"F. This notation, (More precisely, PXt{F}means f(Xt), which looks rather confusing at first, will be used consistently.) Since Qt 3 3:,Qt may be replaced by 3: in the above conditional expectation.
PROOF:The map x -+ P"F is in &' by (2.6). Thus, PXt{F}E bFF C Qt. For F of the form F = fl(Xtl)..-fn(Xt,) with fl, ...,fn E b&' and 0 5 tl 5 t 2 5 ... 5 tn , the sought identity follows at once from the proof of (1.7), and the general case is completed by an appeal to the MCT. The inexperienced reader is cautioned that HD1 (2.1) is a very substantial hypothesis whose verification seems possible only in very special situations, such as under strong analytic conditions on (Pt). See 59 for one such result. Another approach to HD1 is to start with a process known to satisfy HD1 and deform it in some probabilistic manner, verifying that the new process thus obtained continues to satisfy HD1. Examples of this type will be given in the course of the next few chapters, particularly in Chapter 11. The examples will also illustrate why it is desirable to avoid hypotheses that would mandate that (Pt)preserve Bore1 functions. A third avenue to HD1 is to take a Markov process not necessarily satisfying HD1 and regularize it in some manner so that the new semigroup satisfies HD1. Part of the theory of Ray-Knight completions deals with this matter. It should be mentioned that the definitions given in this section differ slightly from those in the earlier work of Blumenthal-Getoor [BG68]. The requirements imposed on (Qt) in this section involve only that the conditional expectations of the H E b F relative to Qk be the same as the conditional expectations relative to FT. If one wishes to condition Q-measurable random variables, further measurability conditions must be imposed on ( Q t ) . We take up this issue in $6.
10
Markov Processes
(2.9) EXERCISE. Show that (2.4) implies Ot : 52 + R E F:+8/FzV s 2 0. (2.10) EXERCISE.Let X be a right continuous simple E’-Markov process with transition semigroup ( P t ) . Show that there exists a function h : E + [0, m] with h E E’ such that for every x E E ,
P”{XT= XOV T 5 t } = exp[-h(z)t]. (Hint: for fixed x, call the probability in question $ ( t ) , and prove that 5 t } depends only on rational times T . Show that it belongs to 3:.)The point x is called instantaneous if h(z) = 00, a holding point if 0 c h(x) < 00, and a trap if h ( z ) = 0. Thus (2.10) states, in case x is a holding point, that the P” distribution of the time spent until X first leaves x is exponential with parameter h(x).
$ ( t + s ) = & ( t ) $ ( s ) . The event {XT= Xo V r
3. The Natural Filtration
We shall suppose throughout this section that, in the sense laid down in $2, (a,B, Bt, Xt, O t , P”)is a right continuous E’-Markov process with semigroup ( P t )on a Radon space E , satisfying the condition HD1 of (2.1). Fix a metric d compatible with the topology of E , and let & ( E ) denote the space of d-uniformly continuous real functions on E . The space C d ( E ) is separable in the uniform norm llfll := sup{lf(x)l : x E E } , thanks to the separability of the &completion of E . Because & ( E ) is a multiplicative (fl,. .. ,fn E Cd(E))is a class generating bE, the class fl(Xt,) fn(Xt,), multiplicative class generating b3”. Putting this together with the related case in which t , 5 t , we get (3.1) PROPOSITION. The 0-algebras
and 3$’ are separable.
Unless (Pt) preserves the Bore1 u-algebra E , this is practically the only virtue of the u-algebra 3”. (3.2) PROPOSITION. For every f E bC(E),the map t continuous on R+.
-+
P t f ( x ) is right
PROOF:Right continuity of t + Xt implies that for f E bC(E),t -, f(Xt) is as. right continuous and bounded. By the Lebesgue dominated convergence theorem, t + Ptf(x) = P”f(Xt) is right continuous. We turn now to the fundamental augmentation procedure which will obliterate much of the distinction between the different u-algebras F and relieve us of much of the burden of carrying around special u-algebras E’.
I: Fundamental Hypotheses
11
(3.3) NOTATION.Given an initial law p, let 3 p denote the completion of 3" relative to P p , and let Np denote the o-ideal of Pp-null sets in 3 p . Define then: (i) 3 := n ( 3 p : p an initial law on E}; (ii) N := n{Np : p an initial law on E}; (iii) 3; := 3; vW'; (iv) 3t:= n{3[ : p an initial law on E}; (v) two random variables G, H E 3 are as. equal if { G # H } E N .
Sets in N are called null. We emphasize that sets in N are null for every Pp. The definition of the filtration (3;) is motivated by the methods of the general theory of processes which require completeness of the o-algebra 3 p relative t o P p whenever martingale methods are applied. Each measure Pp has a unique extension to 3 p . (3.4) PROPOSITION. For every H E b 3 , the function h(x) := P"H is in b€", and for every initial law p, PI(H)= p(dx) P"H.
PROOF:Let H E b 3 . Given a probability law p on E,we may choose H I , E b F such that H I 5 H 5 Hz and Pp(H2 - H I ) = 0. By (2.6), the functions P"H1, P"H2 are in bE", and H2
J p(dx) (P"H2 - P"H1) =
PP(H2 - Hl)
= 0.
Consequently, by sandwiching, x -+ P"H belongs to b P so, p being arbitrary, it belongs to b€". The same sandwiching also proves the second assertion. Similar completion arguments will occur frequently .throughout the development, and we shall usually omit the details in favor of the phrase "by sandwiching". One such example is provided by the following extension of the simple Markov property. (3.5) THEOREM. Let p be an initial law on E , H E b 3 and t 2 0. Then Hoet E b 3 and
In particular, it is also the case that (3-7)
Pp{HoetI 3t} =P ~ ~ H ) .
PROOF:It is clear that H o d t E b p whenever H E b p . Let v = pPt. Sandwiching H relative to Pv and using the simple Markov property (2.8),
Markov Processes
12
one obtains H o o t E b3f'. Since p is arbitrary, Hoot E b 3 . By definition of the conditional expectation as an equivalence class relative to Pp, sandwiching gives (3.6). Let h ( x ) := P"H so that h E bE" by (3.4). Then h(Xt) E b 3 t so 3f may be replaced by 3tin (3.6) to get (3.7). Our original definition of 3 and 3 t is not the one most common in the literature, but the next result will reconcile all differences.
(3.8) PROPOSITION. For every probability p on E , (i) 3f'is the Pp-completion of 3"; (ii) for every t 2 0, 3; = VN P .
e
PROOF:For this proof only, let (M,P)denote the P-completion of a oalgebra M . Obviously (9, P p ) c 3f', and V Np c 3 ; for every p. We shall show that 3% c (3",Pf'), which will imply that 3f'c ( 9 , P f ' ) . Consider A := b(3",Pf'), an algebra of bounded random variables on R which contains I n and which is closed under bounded pointwise convergence. Given f € bE" and t > 0, we may choose g,h € bE such that g 5 f 5 h and ( p P t ) ( h- g) = 0. Then g(Xt) 5 f(Xt) 5 h(Xt) and Pp{h(Xt) - g(Xt)} = 0. Consequently, f(Xt) E A, and so A contains every product fi(Xt,) -.-fn(Xt,) with 0 5 tl 5 t2 5 -.. 5 t , and f i , . . . ,fn E b&". Invoke now the MCT to get A 2 b3". This proves (i), c V N p . From this, (ii) follows. and a similar argument gives The following less trivial result gives us a way to work with the filtration (3t). It will be of much use in some later constructions.
e
e e
e
(3.9) THEOREM. For every t 2 0, 3 t = V qVN = V N . That is, the 0-algebra 3 t is generated by random variables of the form
f (X0)fi(Xtl) f n (Xt, + H, where 0 < tl < t2 < < t,, fo E bE", f l , . . . , fn E bE, H E h/. (H E N means { H # 0) E N with H otherwise arbitrary.) More generally, if *
F E b3t and if x + P"(F . G) is in E' for every G E bE, then there exists F' E 3 : with F - F' E N . PROOF:Clearly V e V NC V N C 3 t . We shall prove the opposite inclusion. Fix a positive H E b3t. For each x E E , we define a measure by Q"(G) := P"(G.H), G E b e . Then Q" on the separable o-algebra Q" << P" on and x + P"G, x -+ QxG are both €"-measurable. By such that Doob's lemma (A3.2) there exists a function h E b(E" 8
e,
e
e)
P"(G H ) = P"(h(z, . ) G) VG E b e .
e.
Set K ( w ) := h ( X o ( w ) , w ) ) E J$' V Since P"{X,-, = x} = 1, we have P" (G . (H - K ) ) = 0 Vz E E , VG E b e . Given an arbitrary G' E b3t
I: Fundamental Hypotheses
13
and x E E, (3.8ii) affirms the existence of a G E b e such that P"(G'-GI = 0, and consequently PZ{G'(K - H ) } = 0 Vx E E , VG' E b3t. From this it follows that P"(H- K ( = 0 Vx E E, hence that { H # K } E N . To get the last assertion, simply replace E'" by E' in the proof above. The preceding proof is a model for a number of results of similar type in the sequel. (3.10) COROLLARY. Let t, s 2 0. Then 3 k + s is generated by products of the form F GOBt, with F E bFt, G E b3,.
-
It is worthwhile also to point out the following special case of (3.9) in which t = 0. This case, which becomes useful if it is known that Fo+ = Fo, is a form of the Blumenthal Zero-One Law. (3.11) COROLLARY. For every G E that, as., G = g(X0).
30
there exists a function g E E" such
(3.12) EXERCISE. Define the process of uniform motion to the right on the state space R as follows. Let R := R, Xt(w) := w t , P" := cz (the unit mass at x) and & ( w ) := w t. The transition semigroup for this process is P t ( z , . ) := cz+t( .). Show that N consists of the empty set 8 alone, and that for all t 2 0, = B(R), and 3 t = = B"(R), the o-algebra of universally measurable subsets of R.
+
+
e
(3.13) EXERCISE. Let ( M , M ) be a measurable space, and let f : R+ x M + R have the properties (i) for every x E M , t -+ f(t,x) is right continuous; (ii) for every t 2 0, x -+ f ( t , x ) is in M. Prove that f E a(R+)8 M - (Hint: f = limn C k l[k/,,(k+l)/n[(t)f((k+ l)/ni ~ 1 . 1 (3.14) EXERCISE. Let (Pt) be a Borel transition semigroup on E. Use (3.2) and (3.13) to show that for every f E pE, ( s , x ) P,f(x) E B+ 8 E . -+
(3.15) EXERCISE.For all t 2 0 and all A E Bt E 3tt,s/3, for all s,t 10.
N , B;'(A)
E N . Moreover,
Markov Processes
14 4. Excessive Functions and the Resolvent
We continue to assume that (a,4, Gt, X t , O t , P") is a right continuous simple €'-Markov process with transition semigroup ( P , ) satisfying HD1 (2.1). The resolvent (Ua),,>O - is the family of kernels on ( E ,P ) defined by rw
(4.1)
U a f ( x ):= P"
e-at f ( X t )d t ,
(Y
2 0, f E pE"
10
In order to justify this as a definition we must settle some measurability problems which arise from the augmentation procedure of the last section.
(4.2) NOTATION. 3*denotes the universal completion of 9 and 3: is the universal completion of It is important to note here that 3*c 3,as a set in 3*must be in the for every finite measure Q, and not just measures Q-completion of (a,9) of the form Pp. It is easy to see that f ( X t ) E 3*i f f E E",t 2 0 , and so, by a routine monotone class argument, 3" c 3*. The following result is given in a more general form than needed here because of the needs of later sections.
(4.3) PROPOSITION. (i) Let H E b3. Then for every initial law p and every finite measure X on (R+,a+), (t,w ) H(Otw) is in the X x Pp-completion of B+ @ 9. For a.a. w , t --$ H(Otw) is A-measurable on R+ and --$
roo
roo
In particular, i f f E E", t + f ( X t ( w ) )is universally measurable on R+ for every w E R. (ii) Let H E b3*. Then for every finite measure Q on (a,F*)and every finite measure A on (R+,D+),( t , w ) ---* H(&w) is in the X x Q-completion of B+ @ 9. For a.a. w , t -+ H(Otw) is Ameasurable on R+ and
If H E 3*, t -+ H(t+(w))is in Z?"(R+)for all w E Q. PROOF:If H E b p has the form H = f~(Xt,)..-fn(Xt,) with 0 5 t l 5 5 t, and fl,. ,f, E &( E ) (the d-uniformly continuous functions on E ) , then ( t ,w ) ---* H ( & ( w ) )is in B+ 8 9because it is right continuous in
I: Fundamental Hypotheses
15
t for every w , and every t-section is clearly in p . By the MCT, the same is true for every H E b p . Given an initial law p, let v be the measure defined on ( E ,E ) by
~ ( f:= )
A(&) P p f ( X t ) ,
f E bE.
(Note that for f E bE, t -, P p f ( X t ) is Bore1 measurable on R+,thanks to an argument similar to that above.) Choose H1,H2 E b p so that H I 5 H 5 Hz and P"(H2 - H I ) = 0. Then 00
PP
~ ( d t( ) H ~ H1l0et= =
r1"
~ ( d tP) ~ { ( H - H1)oet) ~ A(&) P'Lh(Xt),
where h ( z ) := P5(H2 - H I ) E bE". We are given that J v(dz)h ( z )= 0 so there exists k E bpE with 0 5 h 5 k and J v ( d z ) k ( z )= 0. By definition of Y , A(&) P p k ( X i ) = 0, from which the first assertion follows, since Pp X(dt) (Hz - H1)oOt = 0. The last identity in (i) is obtained using the form of Fubini's theorem valid for completed spaces. Assertion (ii) is proved by entirely analogous arguments. (4.4) EXERCISE. For f E pE", ( t , z ) -+ Ptf(z) is measurable relative to the X x p-completion of f3+ €3 E" for every initial law p and every finite measure A on R+. In our discussion of the resolvent, we shall make use of (4.3) with X a finite measure on R+ equivalent to Lebesgue measure, and H := f(Xo),f E bE". To begin with, given an arbitrary z E E , (Y > 0 and f E bE", (4.3) g'ives us (4.5) PZ
e - " t f ( X t ) dt =
I"
e-at P Z ~ ( X dt~=)
e-at
~ t f ( zdt. )
Jr
In view of (4.3ii), the integral e - Q t f ( X t ( w ) dt ) is defined for every w E R, and by (4.3i) the integral is in 3 p for all p, hence in 3. It follows that U Qf , as defined in (4.1), is in E", and so by (4.5),
It follows that for every a > 0, U " ( z , A ) = U Q I A ( x )defines a bounded kernel on (E,E"). The identity (4.5) holds for every f E pE", as one sees
Markov Processes
16
from the monotone convergence theorem, approximating f monotonically from below with bounded positive functions. The case a = 0 in (4.6) requires a restriction on f, generally speaking, to the case f E pE". It is customary to write the operator U o simply as U . Thus, for f E pE", (4.7)
For each fixed x E E , the measure U ( x , - ) is s-finite, as defined in AO. It is now a simple matter to derive the following important identity. (4.8) EXERCISE (RESOLVENT EQUATION). For 0 5 a
5 6 and f E p€",
U"f = U P f + ( P - a ) U " U P f . (Finiteness conditions are not needed.) (4.9) EXERCISE.The resolvent (U")">O of a semigroup ( P t ) satisfying
HD1 completely determines (Pt). (Hint: examine the action on an f C d ( E ) so that t + P t f ( x ) is right continuous for each stronger result, see (36.211.)
2.
E
For a much
In many ways, the resolvent plays a more important role in the study of a Markov process than does the transition semigroup itself, because the resolvent seems to be more closely connected with the properties of the trajectories. This will be illustrated at length in Chapter V. In addition, the resolvent gives the basic potential-theoretic interpretation of a Markov process, and for this reason, U" is called the a-potential operator for the process, with U called simply the potential operator. One may interpret U ( x , B )as the expected total time spent in the set B C E by a particle starting at x. One interpretation of U " ( x , B ) is essentially the same, but in the presence of inflation at the rate a , which makes a time instant at t worth only e-ut times the same instant at time 0. See also (4.22). (4.10) EXERCISE. The Gaussian semigroup (Pt) on Rd is defined by
P t ( z ,d y ) := ( 2 ~ t ) - ~ /exp ' [-ly - x1'/2t] d y where dy denotes Lebesgue measure on Rd. (This is the transition semigroup of so-called standard Brownian motion.) For a > 0, U" f is the convolution off with u", where
I: Fundamental Hypotheses
17
K p being modified Bessel functions of the third kind pMOT54, p.1461. The case d = 1 reduces to
which may be checked by taking the Fourier transform in the variable z . Successive differentiations relative to z allow one to compute u " ( z ) for odd d by elementary means. In particular, for d = 3,
u=(z)=
exp (+Id%) 244
In case d 2 3, the limit as a 1 0 gives the convolution kernel u,which is just the classical Newtonian potential kernel density, aside from an inessential multiplicative constant:
(4.11) DEFINITION.Let a 2 0 and f E pE". Then f is a-super-meanvalued in case e-"tPtf 5 f for all t 2 0, and f is a-excessive in case e-"tPt f t f as t 1 0. (That is, f is a-excessive in case f is a-super-meanvalued and e-atPt f + f as t + 0.) (4.12) EXERCISE.I f f is a-super-mean-valued, then t + e-atPtf is a decreasing function on R+. Define the a-excessive regularization o f f to be the function f := limtllo e-atPt f . Prove that f is indeed a-excessive, and that P t f ( z )= Pt f (z) for all t 2 0, z E E . Show that f is the largest a-excessive function dominated by f .
A function f on E is called excessive if f is 0-excessive, and similarly for super-mean-valued. (4.13) NOTATION. S", S denote the classes of a-excessive, excessive functions respectively.
The resolvent (V") being just the Laplace transform of the semigroup (Pt), as operators on bE", it is sometimes useful to have a specific inversion formula to refer to. A convenient one for our applications is the following, which follows easily from [Fe71, p230, (6.4)]. (4.14) INVERSION FORMULA FOR LAPLACE TRANSFORMS. Let g : R+ + e-atg(t) dt denote R be bounded and right continuous, and Jet ( ~ ( a:= ) the Laplace transform of g. Then, for every t > 0,
18
Markov Processes
(4.15) EXERCISE. (i) (ii) (iii) (iv) (v) (vi)
(vii) (viii)
(ix) (Consult
The classes S",S are closed under monotone increasing limits; if a 2 0 and f E pE", then U" f E S"; S" is a cone containing the positive constants; i f f and g are a-super-mean-valued, then so is f A g; for f E p&",f E S" if and only if @Up+" f t f as P t 00; for f E S" with a > 0, let h, := n (f A n - nun+"( f A n)) . Then U"h, 7 f as n --f 00. That is, every f E S" is an increasing limit of a sequence of potentials of bounded functions; S* c SO i f a I p; for f E S", t + e-atPtf(x) is decreasing and right continuous for every x; for f E S" and t 3 0, Ptf E S". PG88, II] for details if you get stuck.)
(4.16) EXERCISE. A function f is super-mean-valued for the sem'group governing uniform translation to the right (3.12) if and only if it is positive and decreasing. Such an f is excessive if, in addition, it is right continuous. The excessiveregularization of a super-mean-valued f is its right continuous regularization. (4.17) EXERCISE. Iff : Rd + [0,GO] is a-excessive for the Gaussian semigroup of (4.10), then f is lower semicontinuous. (4.18) EXERCISE. I f f E & ( E ) then a u a f ( x ) -+ boundedly as a + GO.
f(x) pointwise and
(4.19) EXERCISE. Write down the semigroup (P,) for the process described as follows. The state space E C R2 is the union of the two halves of the co-ordinate axes, E := { (x,y) : y 2 0 and x = 0 or x 2 0 and y = 0). Starting away from (0,O) E E , the process moves away from the origin in uniform motion at unit speed. Starting at (O,O), choose an axis, each with probability .5, and then continue the uniform motion along that axis. Describe the excessive functions for this process, and show that minimum of two excessive functions need not be excessive. (4.20) EXERCISE. Let f be a-super-mean-valued, and let p be an initial law on E with f dp < 00. Then e-at f (Xi)is a supermartingale over (0,Ff,Pp). In case f E S", its Pp-expectation is right continuous in t. (4.21) EXERCISE. Let X , p be finite measures on E such that, for some a > 0, XU" = pU" as measures on E. Using the resolvent equation, prove that XUP = p u p for all p > 0. Conclude from (4.18) that X = p. (This is the principle of masses in potential theory, a stronger form of which will be discussed in (10.40).)
I: Fundamental Hypotheses
19
(4.22) EXERCISE.Let B E &” and a 2 0. Let S denote an exponentially distributed time (i.e. positive random variable) with P”S = 1 / a for all x E El S being independent of X. Then P(z, B) is the Px-expectation of the time spent by X in B until time S. (4.23) EXERCISE.A function f E p&” is called a-supermedian in case
@Ua+pf 5 f for all ,f3 > 0. I f f is a-super-mean-valued, then f is asupermedian. Iff is a-supermedian, then f := limp,, @Ua+Of 5 f exists and is a-excessive. Show that Upf = Upf for all @ > 0. The function f is called the a-excessive regularization o f f . Show that this definition off is consistent with that of (4.12), in case f is a-super-mean-valued. (4.24) EXERCISE.Show that f is supermedian for uniform motion to the right on R (3.12) if and only if there exists a decreasing function g 5 f with f = g a.e. (Lebesgue). (4.25) EXERCISE.Prove: F C 7*, F: C 3: for all t , and universal completion of F .
F* is the
5. The Optional and Predictable a-Algebras Because of the fact that we shall be considering a single sample space R with a number of different filtrations and a family of probability measures rather than just one fixed one, it is essential to describe before going much further the modifications needed in the usual definitions (A5) of the optional and predictable a-algebras. We assume given a right continuous realization (Cl,G, G t , Xt,&,P”)of a Markov semigroup (P,) satisfying the first fundamental hypothesis HD1 (2.1). With an initial law p on E , associate the a-ideal NP(G) of PP-null sets in the Pp-completion Gfi of G, and let N(G) := n,Np(G). We use here and below the notation n, for the intersection over all initial laws p on E . If (&) were the natural filtration ( F t )defined in 53, P ( G ) would be the same as NP, as defined in (3.4), but in general it is only the case that N,(G) 3 NP. A real valued process 2-that is, a real valued function Z(t,w ) on R+ x Cl-is called Pp(G)-evanescent in case { w : 3 t 2 0 with Z t ( w ) # 0) E JV’(9).Saying that 2 is Pp-evanescent means that 2 is P p (7)-evanescent. Let P ( G ) denote the class of P”(G)-evanescent processes, Z(G) := nP(G), P := ZP (F) , and Z := Z(3).Processes in Z are called simply evanescent. A subset of R+ x R is called evanescent if its indicator is an evanescent process. Similar obvious definitions apply in other cases. Two processes 2, W are called indistinguishable if ( 2 # W } is evanescent. Similarly, 2, W are Pp(G)-indistinguishable in case {Z # W } E ”’(G).
20
Markov Processes
(5.1) NOTATION. A function on R+ with values in E is rcll provided it is right continuous on [0, oo[and has left limits in E everywhere on 10, oo[, and similarly for lcrl. Then: (i) D E is the space of all rcll maps of R+ into E ; (ii) C E is the space of all lcrl maps of R+ into E . Take now an arbitrary filtration (Gt) of (QG). We don’t assume at this point that (Gt) satisfies the usual hypotheses of the general theory of processes. Define Do(&)(resp., C(Gt)) to be the class of all bounded real processes Zt(w) which are adapted to (Gt) and have all sample paths t --t Zt(w) in the class DR (resp., CR). The fundamental a-algebras of processes are then defined as follows: (5.2) DEFINITION. (i) a{D(Gt)} is the a-algebra of strictly optional processes over (&); in particular, 0’denotes the class of strictly optional processes over (F:) ; (ii) o{C(Gt)}is the a-algebra of strictly predictable processes over (Gt); in particular, P. denotes the class of strictly predictable processes over (F;); (iii) S(Gt):= a{D(Gt)}VZ(G) is the a-algebra of optional processes over ( G t ) ; (iv) W ( G t ) := a{D(Gt)}V P ( G ) is the a-algebra of Pp-optional processes over (Gt); (v) P(Gt) := a{L(Gt)}V Z(G)is the a-algebra of predictable processes over (Gt); (vi) P”(Gt):= o{C(Gt)} V P ( 8 ) is the a-algebra of Pp-predictable processes over ( G t ) ; (vii) n,O’(Gt) is the a-algebra of processes which are nearly optional over (Gt); (viii) n,P,(G,”) is the a-algebra of processes which are nearly predictable over ( G t ) . Following the usual pattern of notation, the filtration is to be inferred to be the natural filtration (Ft)if it is not mentioned explicitly. Thus, for example, 0 is by default O(Ft), and it is called the a-algebra of optional processes. Similar terminology applies to P, P,, and so on. Trivially, every process in 0 is nearly optional, and every process in P is nearly predict able.
(5.3)DEFINITION.An extended real function f on E is an optional function [for X over (Gt)]provided f(Xt) is optional [over (Gt)]. Similar notation applies to the terms nearly optional function [over (Gt)], nearly
I: Fundamental Hypotheses
21
optional [over ( G t ) ] , predictable function [over (Gt)], and so on. The function f is nearly Borel relative to X if, for every initial law p , there exist Borel functions 9,h on E such that g 5 f 5 h and Pp{3t 2 0 with g ( X t ) # h ( X t ) } = 0. For every f E bC(E), f ( X t ) is right continuous and adapted to ( G t ) . It follows from (A5.5iv) that, since (Fr+) satisfies the usual hypotheses, f is an optional function over (Fr+). By a trivial monotone class argument, every Borel function on E is a nearly optional function. The fact that this is not also true for every f E &? is a source of considerable complication in the general theory of Markov processes. Obviously then, every nearly Borel function f is nearly optional for X , but the converse does not seem to be true. Recall that a random variable T defined on a filtered measurable space (R, 9, Gt) is a stopping time, or equivalently, an optional time, relative to ( B t ) in case {T 5 t } E 81 Vt 2 0. See the discussion of optional times in A5 for a more detailed description. In the sequel, we always suppose B, := V&. Given an optional time T over (Gt), the dyadic approximants to T are the optional times T, defined by (5.4)
T n ( w ) :=
l ~ / 2 if~ ( I c - 1)/2" 5 T ( w ) 00 ifT(w)=m.
< k/2n,
Clearly Tn(w) 1 T ( w ) as n --* 00 for every w E R, and Tn(w) > T ( w ) if T ( w ) < 00. Given an optional time T over (&), the c-algebra GT of the past at T is defined by (5.5)
bBT := {G E bGm : G l { T i t ) E bGt V t 2 0).
Given also a set A E G,, define
TA(w):= 00
if w E A, ifw$A.
The following fact is simple to verify. (5.6) LEMMA. Let T be an optional time for ( G t ) . Then A E GT ifand only if TA is an optional time over (Gt). A real process 2 is progressive over ( G t ) if, for every t 2 0, the map + Z,(w)of (0, t] x R into R is measurable relative to B[O,t] C3 G t . Then [DM75, IV-641: (3,w )
22
Markov Processes
(5.7) LEMMA. If T is an optional time over (Bt) and Z is progressive over ( G t ) , then Z d { T < m o ) E G T . The condition that T be an optional time is clearly equivalent to the condition that the indicator of the stochastic interval IT, 00[l := { ( t , w ) € R+ x Q : T f w ) 5 t } be adapted to (&). If H E &, then H ~ ~ T , , is I [in O(&).See also (6.18). The following result is completely trivial in case (0t) satisfies the usual hypotheses of the general theory of processes.
(5.8) LEMMA.Let (&) be an arbitrary filtration of (Q, 0). Suppose that the real process Z is adapted to (Gt) and has almost all sample paths in DR.Then there exists a process Z', strictly optional over (Bt+), such that Z - Z' E Z(B).Consequently, Z is optional over (Bt+).
PROOF:There is no loss in generality if we assume that Z is uniformly bounded, replacing Z if necessary by arctan 2. Let &+ denote the positive rationals. Given an arbitrary bounded real function $ on R+, let
$"(t) :=
sup
e-a(t-8)
$(s),
$ ( t ) := limsup $(s).
s
aTTt,s€Q+
It is an elementary matter to see that $J*is a left continuous, bounded = $ ( t ) for every t > 0. If the function on R+,and that lim,,,$"(t) function $ were in DR,then $ ( t ) = $ ( t - ) for t > 0, and for every E > 0, the set { t > 0 : I$(t) -$(t)I 2 E} would have to be a discrete subset of R+. It follows from the first remarks above that the process 2 E o(Lc(6t))C a{D(Bt+). Given E > 0, let T1:=inf(t>o:Vk3T,s€
& , O < r < S < T + k
1
c},
1 Tz:=inf{t>Tl : V ~ ~ T , S € Q , ~ < T < S < T< +t , I- z a ( w ) - & . ( w ) l > ~ } , k and so on. It is clear that each T, is an optional time for (Bit+). Let 0 2 0 := {w : t + & ( w ) is in DR}and A E ( w ):= { t > 0 : IZt(w)-Zt(w)l 2 E}. For w E $20,the set A'(w) is discrete in R+. Let Rf,(w) := infh"(w), Rq(w) := infAB(w)n]RE(w),co[,and so on. Suppose T l ( w ) < t < 00. For each k , choose T k , S k E 8 with 0 < T k < S k < T k -t l / k < t and IZ,, ( w ) - Z,, ( w ) I > E. Passing to a subsequence if necessary, we may assume that T k converges to T 5 t monotonically. If T k were decreasing, then s k 1 T also, leading to the absurdity IZ,(w) - Zr(w)I 2 E , by right T . By similar reasoning, continuity. Therefore, we may assume that T k If sk cannot increase strictly to T , and we may therefore assume S k 1 T . It follows that IZ,(w) - Z,(w)I 2 6. That is, Tl(w) E A'(w). A similar
I: Fundamental Hypotheses
23
argument applies to show T,(w) E A‘(w) for every n. It is also clear that, for every 0 < Q < c, Tl(w)2 R:(w),and, more generally, T,(w) 2 Rz(w) for all n. Therefore T,(w) 4 00 provided w E Ro. Let I’ := Un>l - [IT, 1 E O{D(Gt+)} and T , := lim, T,. Let
z,’(w)
:= limsup Zt(w) = lim sup e - a f T - t ) ~ t ( w ) . rllt,r€Q a’oo r>t,rEQ
By arguments similar to those above for limits from the left, Z$ is progressive over (Gt+), and for w E 00,Z,‘(w) = Zt(w)for every t 2 0. For every optional time T over (Gt+), Z $ 1 [ ~ 1is in u{D(St+)}.Define
zt~(w) := z0(~)1[01(t) + C z$n(w)lr(t,u)+ Zt(w)lno,T,u\r(t,w). n
The first and last terms on the right side of this expression are clearly in a{D(Gt+)}, and the second term is also by the preceding argument applied to each T,. This proves that 2: E 0{D(&+)}. To complete the proof, observe that IZi(w)- Zt(w)I 5 E for every w E R,-, and every t 2 0, and take 2‘ := limsupk Z 1 / k . The appearance of the 0-algebra ( G t + ) in (5.8) is not entirely natural or welcome. It is possible to modify the proof to get a version of (5.8) that works without taking an infinitesimal peek into the future, but at the cost of replacing each Qt by its universal completion $. In this modification, we make use of the “min” of a set in R+,as distinct from its “inf”. To be specific, if A c R+,then minA :=
(5.9)
inf A if inf A E A, 00 otherwise.
Observe that (5.10)
minA 5 t
(inf A
< t , inf A E A)
or (inf A 2 t , t E A).
(5.11) LEMMA.If T := minA with A progressive over (&), then T is an optional time for the filtration (G;).
PROOF:Let D := inf A. As {D < t } is the projection on R of A n 10, t [ E B([0, t [ )@ Gt, measurability of projections (A5.2) shows that {D < t} is analytic over Gt, hence in its universal completion 8;. Thus D is an optional time for ( S t ; ) . According to (5.7), IA(D)l{D
{T 5 t } = {D < t , ln(D) = 1)
U
{D 2 t , t E A}.
By the remarks in the preceding sentence, the first term on the right is in
G;, as is second term, noting that {D 2 t } = {D < t}“E G;.
Markov Processes
24
( 5 . 1 2 ) LEMMA.Let (Gt) be an arbitrary filtration of (R, (2'). Suppose that the real process 2 has almost all sample paths in DR,and suppose that for every optional time T over (G:), Z T ~ { T < E~ (2.;') (The latter condition is satisfied if, in particular, 2 is progressive over ((2';)). Then there exists a process Z', strictly optional over (G;), such that 2 - 2' E 2(G*).
PROOF:The proof is essentially the same as that of ( 5 . 8 ) , but a little shorter because of the additional assumption on 2. Let Tl(w) := min{t T ~ ( W ):= min{t
> O : IZt(w) - Zt(w)l > E } , > T ~ ( w:) I ~ t ( w-) Z t ( w ) l > E } ,
(Gt),
and so on. Then every Tn is an optional time for by ( 5 . 1 1 ) . Because A' := { t : I Z t ( w ) - Zt(w)l > E } is a s . discrete in R+,the Tn exhaust A' as., so that A'Ar E 2((2'*), where I? := U, [ITn]is strictly optional over ((2';). Let T, := limn T,,and define
+
z;(w> := z o ( ~ ) l [ o ] ( t )C z T n ( w ) l r ( t , w )+ %(w)lno,T,u\r(t,w). n
The first and last terms on the right side of this expression are clearly in cr(D(G;)}, and the second term is also by hypothesis. This proves that 2; E a { D ( B ~ ) }To . complete the proof, observe that IZ;(w) - Zt(w)l 5 E for every w E Ro and every t 2 0, and take 2' := limsupk 21/k. The following lemma points the way to applications of (5.12). ( 5 . 1 3 ) LEMMA.Let 3;denote the universal completion o f c , T an optand cp E b(B([O,t])@&')*.Then cp(T,X T ) ~ { T < €~ 3 ) ;. ional time for (3;) That is Zt := cp(t,X t ) satisfies the secondary condition of (5.12).
PROOF:By definition of F;,we must prove that c p ( T , X ~ ) l p E q 3; for every t 2 0. Given an arbitrary probability Q on (R,F;), use the fact that TtTst) E 3; to choose a positive random variable R E f i such that Q ( T { T < ~#) R { R i t } ) = 0. For every E Gi(E), n
g ( X R ) l { R < t } = d x O ) l { R = O } -k l@x g(Xkt/n)l{(k-l)t/n
e.
k=l
which is obviously in The same is then true by the MCT for all g E b&. X R ) l { R < t ) E b e for $ E b(B([O,t]) @&). Define Again by the MCT, $(R, a measure on ([o, t] X E , B([o,>])C3 8) by A(@) := Q'$'(R,X R ) l { R < t ) , $ E b(B([O,t ] )@ Eo. Choose $1 I cp I $2 with A(& - $1) = 0, $1,$2 E b(17([0,tl) @ 8). Then Q($2(RIX R ) ~ { R <-~$i(K ) X R ) ~ { R <= ~ }0,) and therefore p(T,X T ) l { T < t ) differs from $l(R, X R ) ~ { R < ~only ) on the set {T{T
I: Fundamental Hypotheses
25
6. The Strong Markov Property
Suppose now that (a,G, Gt, X t , Ot, P") is a right continuous Em-Markov process with transition semigroup (Pt) satisfying HD1 (2.1). We begin by augmenting the given filtration ( G t ) in a manner similar to the augmentation of (F:)in 53. Recall from the last section that NP(9)denotes the o-ideal of PP-null sets in the PP-completion G P of 9. (6.1) DEFINITION.(i) Gf := Gt V N P ( G ) ; (ii) L? := nPGP,gt := n&f. Because X is assumed to be €'-adapted to (&), one has 3 ' C G and consequently, NP c ""(8). It is therefore clear that, for every t 2 0, 3; c 8.: Hence, every T which is optional over (Fr+) is also optional over (@+). Observe too that n&+ = Gt+, for if A E npGf+, then A E n,Gf+, = Gt+, for every s > 0: that is, A E G t + . This will be needed in the proofs of (6.4) and (6.9) below. These observations show that further application of the augmentation procedure to ( B t ) is fruitless in the sense C t . Similarly, letting Gt+ := ns>tGs,we have that (Gt)" = Of and (Gf)+ = ( G t + ) p for every t. We shall say that ( G t ) is augmented provided Gt = ct for every t.
(ct)-=
(6.2) PROPOSITION. For every initial law p, t , s 2 0, and F E b 3 , PP
{FoOt I G f } = PXt((F),
and the same formula holds with
6: replaced by G t .
PROOF:This comes from the simple Markov property relative to ( G t ) via an elementary sandwiching argument in the manner of the proof of (3.5). Because of (6.2), whenever (a,9,G t , X t , O t , P") is a right continuous simple Markov process, the filtration (Gt) may be replaced by its augmented version (&). There is, therefore, little harm in building in the additional regularity hypothesis that the underlying filtration (Gt) be augmented already. The technical advantages of this hypothesis are substantial. We first check the manner in which augmentation affects optional times. (6.3) PROPOSITION. Let p be an initial law on E and let T be an optional time for (Gf+). One may then choose an optional time To for the filtration (&+) with P p (T # To) = 0.
PROOF:Let T, denote the dyadic approximants to T constructed above. Clearly, {T, = k/2"} E In view of (6.1), one may choose sets Ak E Gk/2n so that Pp ({T, = k/P)AAk) = 0. Define T,"(w) := inf{k : w E Ak}/2,. Then T," is optional over (Gt) since {T,"5 k/2"} = U{Aj : j 5 k} E G k 1 2 % . It is obvious from the construction that P"{T, # T,"} = 0. Then T o := inf, T," is an optional time for (&+) with PP{T # T o } = 0.
26
Markov Processes
(6.4) PROPOSITION. Let X be €'-Markov relative to ( G t ) , f E E" and let p be an initial law. Then:
(i) if T is optional over (Gr+),then ~ ( X T ) ~ { TE
PROOF:For every f E C d ( E ) , f(Xt) is progressive over (@+) and so, by (5.7),f ( X ~ ) l { ~E
In view of (6.4), f(Xt)OflT1{T<m}= f(Xt+T)l{T<m}is G-measurable } Therefore (6.6) makes sense. One and P t f ( X ~ ) l { ~
PROOF:If (6.8) holds for every f E C d ( E ) , the MCT implies that it holds for every f E bE, and by sandwiching f relative to the measure v(g) := Pp{g(Xt+T)l{T<,}} (g E bE), one sees that (6.8) holds for every f E bE". Now fix T and let A E &+. By (5.6), TA is optional over ( G t + ) , and applying (6.8) to TA yields (6.6). Exactly as in the case of the simple Markov property, there is an inductive extension of (6.6).
I: Fundamental Hypotheses
27
Let (Q8, Bt, X t , Ot, P ” ) be a right continuous real( 6 . 9 ) PROPOSITION. ization of (Pt) having the SMP relative to (&+), and suppose that (gt) is augmented as in (6.1). Then, for every H E b 3 , every initial law p and every optional time T over (&+), HOOTl{T<,) E 0 and
PI’{HoOT1{T<m}I &‘+I = P X ( T ) ( H ) l { T < c o } * PROOF:Let f E b€ and t 2 0. Then H := f(Xt) has the property H o O T ~ { T < , ) E B by (6.4) applied to the optional time t T. It follows then from an application of the MCT that the same is true for every H E b p . To prove the asserted equality for H E b F , it suffices, by the Monotone Class Theorem, to prove it in case H = f l ( X t , ) - . . f n ( X t , ) , where f l , .. .,f,, E b P and 0 5 tl 5 t2 5 . 5 t, . The argument in this instance is exactly parallel to that used in the similar extension of the simple Markov property (1.7),using (6.6) inductively. So far, we have not used anything about augmentations. Given H E b 3 and an initial law p, choose H I , Hz E b P so that H1 5 H 5 HZ and HZ - HI E Nu, where v(g) := P”{g(XT)l{T,,)} ( g E bEU). Using the case just proven, we find
+
p p { p p { ( H 2- Hl)O@T1{T<m)I &+)I = p I ’ { p X ( T ) ( H 2 Hl)l{T<m)} = P”(H2 - Hi) = 0. This obviously yields the desired identity, and shows H o O T ~ { T <E~6). It is important to note that the variables H in (6.9)run through 3 and not 8. See the following definition in this regard. ( 6 . 1 0 ) DEFINITION.Let (O,G, Bt, X t , Ot, P ” ) have the MP relative to the filtration (&+), and assume that (Bt+) is augmented as in (6.1). Call ( G t + ) a Markov filtration for X in case: (i) (Bt) is obtained by augmentation of a filtration (@) such that Gk is separable and @+s = $‘V e,’(@) for all t, s 2 0; (ii) the map z -, P ” ( G ) is in E” for every G E 8,; (iii) for every G E bBm and every t 2 0, P’”{GoOt 1 &+} = P X t ( G ) . The filtration is a strong Markov filtration for X if, in addition, condition (i) holds whenever t is replaced by an optional time T for the filtration (Gt+) (i.e., G$+s = G$ V O$’(G;)), and (iii) is replaced by
Pp{GO~T1{T
Markov Processes
28
(6.11) EXERCISE.Let ( G t ) be a strong Markov filtration for X. Fiddle with the proof of (6.9) to prove that if H E 8, and if T is an optional time for (&+), then HOOT1{T
+
(6.12) EXERCISE.Let (0, 4 , G t , Xi,Ot, P”)have the SMP with (Gt) augmented, and let T be optional over (Gf+). For every G : 0 x R+ x R + R with G E b(G;+ @ B(R+) 8 F),one has
on {T < 00). (Hint: start with G of the form G(w,t,w’) := F(w)$(t)H(w‘) where F E bQg+,$ E bB+ and H E bF.) The form of the SMP given in the last exercise looks puzzling at first because of the obscure notation. Remember that the random variable being conditioned is a function on Sl with the variable in Sl, and that the right side is obtainable by first evaluating G(w, T ( w ) ,w’) P”(dw’),then substituting XT(w)for x. The form of SMP in (6.12)is very powerful. It can be shown that, in particular, every random variable in b F may be .)) for some G E b ( q + @ B ( R + ) @ expressed in the form G (. , T ( T ) ,so that (6.12) gives an expression for conditioning in almost complete generality. As an illustration of the power of (6.12),fix f E bE” and s > 0. The random variable f(Xs)l{~
a ) , & ( . ) (
G(w,t l 4 := f(Xs(W))l[e,m[(t) + f(Xs-t(W’))l[O,s[(t)
since, for all t E [0, 00(, G(w,t , Btw) = f(Xs(w)). It is obvious that G satisfies the conditions of (6.12),and one may therefore conclude that
PC”{f(Xs)l{T
Zt := Ps-tf(Xt)l[o,s[(t) + f(Xdl[s,oo[(t) is a martingale relative to (R, O f , P p ) with final value 2, := f(X,). This is a direct consequence of the simple Markov property. If (R, 4,Gt, Xt,Ot, P“) has the SMP, then (6.12) shows that 2 has the optional sampling property relative to (0,Gf,Pp). That is, P”{Z,l{T<m) I 4;+} = ZT1{T
I: Fundamental Hypotheses
29
Then there exists (6.13) EXERCISE.Let T be an optional time over (Ft”,). an optional time To over such that P”{T # To}= 0. (Hint: use the argument of (6.3).)
(e+)
(6.14) EXERCISE.Let T be an optional time over (3t+). Then there exists an optional time T’” over (F;+)such that PP{T # T”} = 0 for every p. (Hint: use dyadic approximations and (3.91.) (6.15) EXERCISE.Prove that Gf+ = Gt+ V N ” ( Q ) . (Recall (6.1).) (6.16) EXERCISE.Let X be right continuous, strong Markov with semigroup (Pt),and let x be a holdingpoint (2.10) for X . Let T := inf{t : X t # X O } . Prove that T is optional over and P z { X ~= x,T < oo} = 0. That is, X cannot move continuously away from the holding point x.
(e)
(6.17) EXERCISE.Let E = R+,and let X stay at 0 for an exponentially distributed time and then move uniformly to the right at unit speed. Verify that X is simple Markov but not strong Markov. (6.18) EXERCISE.Let T be an optional time for the augmented filtration
( G t ) . Prove that bQT = { H E bg,
: 3 2 E Q(G) with
H~{T<,) = ZT~{T<,)}.
(6.19) EXERCISE.Let T be an optional time for the augmented filtration ( Q t ) . The o-algebra QT- is defined to be the o-algebra generated by sets of the form A n {T < t } as A runs through Gt and t runs through R+.Check that b&-- = { H E bQ, : 3 Y E P ( g ) with Hip<,) = YT~{T<,)). (See also (23.8) and (3f.14).) (6.20) EXERCISE.Show that if T is optional over ( F t ) ,then FT = n,F;. PRINCIPLE). Let (n,g, &, xt, f?t, P ” ) be (6.21) EXERCISE(REFLECTION a right continuous realization of a semigroup (Pt)on R, with X having the SMP relative to (Gt).Suppose that, for every x E El the measure Pt(x, . ) is invariant under the reflection map g!~,(y) := 2x - y of R about x . Let W denote the space o f all maps w of R+ into R, and for any r 2 0, let 9, : W -, W denote the map reflecting the path after time t about its position at time t , specified by
Q r 4 t ) := 4
W { t < T ]
+ ( 2 w ( t )- w(r))l{t>r).
Fix T, an optional time over ( G t ) . Prove using (6.12) that the distribution of ~ T ( Xunder ) P” is the same as that of X , for every z E R. (6.22) EXERCISE. Let T be an optional time over (&+) taking on only countably many distinct values. Prove that the strong Markov property (6.8) holds in this case. (Hint: check the proof of (7.5) in the next section.)
30
Markov Processes
(6.23) EXERCISE.Let (Q, 4, Gt, X t , &,P") be Markovian with semigroup ( P t ) relative to the augmented ( G t + ) , and suppose (Gt) satisfies (6.104.
Suppose also that: (i) 5 P"(G) is in E" for every G E bGk; (ii) for every p, G E bG& and t, Pp{GoBt I @+} = PX'(G). Prove then that ( G t + ) is a Markovian filtration (6.10) for X. Prove, in addition, the corresponding strong Markov version of this result, involving only optional times for the filtration (@+). (Hint: use (6.3)and recall the corresponding results in $3 for the filtration -+
(e+).)
(6.24) EXERCISE.Let (Q, F,Ft, Xt,&, P") be a right continuous, strong Markov realization of ( P t ) . Let H be a separable a-algebra on Q such that Hoot E ? fori every H f 'H, and with 31 independent of 9 relative to Pp for every p. (For example, 'H could be defined from a product space construction.) Let 4: := e V H , and define ( Q t ) by the usual augmentation procedure. Prove that (&+) is a strong Markov filtration for X . Note that Go+ does not have the 0-1 law, in general.
7. The Second Fundamental Hypothesis We assume throughout this section that X = (Q,G,Gt,Xt,et,Pz) is a right continuous simple Markov process with transition semigroup (Pt) on a Radon space E . We take up now the second fundamental hypothesis, which will make the strong Markov property available in a form powerful enough to overcome any poor measurability properties of (Pt). (7.1) HYPOTHESIS (HD2). For everya -+ f (Xt) is a s . right continuous.
> 0 and every f E S", the process
t
The above hypothesis depends on ( G t ) , or at least on the class N(E)of null sets over G. The larger 4 is, the easier it is to satisfy (7.1). For f E S", t -, e - Q t f ( X t ) is a P"-supermartingale over (9t) whenever f(z)< 00. In case f(z)< 00, (7.1) implies that f ( X t ) is a s . (i.e. up to a set in N(G)) rcll (right continuous with left limits) in [0, m]. Because a right continuous supermartingale a s . never achieves the value 00, it follows that, under (7.1), t -+ f(Xt) is rcll, up to a set in N(G).Recalling that 2>01[, denotes the class of maps of R+ into [0,00]which are rcll, (7.1) may be restated as
This condition is at its most forceful in the case ( G t ) = (Ft). The following modification of (7.2) will be encountered later as a key technical hypothesis. In this condition, Nt (resp., N f ) denotes the class of
I: Fundamental Hypotheses
31
sets which are null (resp., Pp-null) in 3p-that is, A E N t if and only if, with Pp(A0) = 0 for every p (resp., for the given p ) , there exists A0 E and h c Ao. (Since f l C 3; c 3;, A is null in 3; if and only if A is null in 3F.)
(7.3)
v t > 0, v a > 0, v f E s",
{s
+
f ( X , ) l [ o , t [ ( s4) 2)[0,1}
E Nt.
It is not really a substantial advance on (7.2), as it will be shown later (18.3) that if (Q, 9,Gt, X t , Bt, P") satisfies (7.2),then a 8-null set may be excised from so that (7.3) is then automatically satisfied. Contrary to what one might suspect initially, HD2 is not a hypothesis on the semigroup (Pt) alone. See the discussion at the end of $19. The following theorem contains a number of important basic facts about processes satisfying HD1 and HD2, as well as given a number of alternate ways to check for HD2. In practice, it is usually difficult to verify HD2 directly, but the conditions (ii)-(vii) are in many cases easy to verify.
( 7 . 4 ) THEOREM. Let X = ( R , 9 , G t , X t , B t , P 5 be ) a right continuous realization of a Markov semigroup ( P t ) satisfying HDl (2.1), the filtration (&) being augmented as in (6.1). The following conditions on X are then equivalent: (i) X satisfies HD2 relative to ( G t ) ; (ii) {t -+ U a f ( X t ( w ) )is not right continuous} E N ( Q )V a > 0, v f E Cd(E); (iii) X satisfies the strong Markov property (6.5) relative to ( G t + ) , and for every a > 0 and every f E C d ( E ) , the function U af is nearly optional relative to ( X ,Gt+); (iv) X satisfies the strong Markov property relative to (Gt+), and for every t 2 0 and f E C d ( E ) , the function Pt f is nearly optional Gt+); relative to (X, (v) V f E C d ( E ) , V t 2 0 , {s + P t f ( X , ) is not right continuous} E
Jw);
(vi) V f E C d ( E ) , V t 2 0, Ptf is nearly optional relative to ( X , G t + ) , and for every initial law p and every optional time T over (G,",),
(vii) V f E C d ( E ) , V t > 0, { s -+ Pt-, f ( X , ) l[o,t[(s) is not right continuous} E N ( 9 ) ; ( snot ) right continu(viii) V f E bE", V t 2 0, { s + Pt-, f ( X , ) l ~ o , ~ [is ous } E N ( 9 ) . Under any of these conditions, the filtrations (8:) have the property: (ix) Pp{H 1 Or+} = P p { H 1 G,"} for every H E b 3 .
32
Markov Processes
PROOF:Obviously (i) implies (ii). The plan of the proof is to show that
(ii)+
(%)a (iv)H (v), (iv)j (vi) =j(vii)+-(viii)+-(ii)3 (i)
The final assertion will then be proved using (viii). Throughout this proof, the term “as.” means “up to a set in N(G)”. (ii)=k(iii): It suffices to prove, thanks to (6.7), that, for every f E pCd(E), t 2 0, all initial laws p, and every optional time T over (G,”,),
p’”{f(xt)oeT 1{T
Let {T,} be the dyadic approximants to T constructed in (5.4), and
Taking expectations, interchanging the order of summation and integration with an appeal to Fubini’s theorem, and then evaluating the right side by means of the Markov property, we find
Letting n -+ m, using the hypothesis that U ” f ( X t ) is 8.5. right continuous, and noting obvious boundedness properties, we obtain (7.5). (iii)+(iv): If f E Cd(E), t + Ptf(z) is right continuous for every fixed z E E. The inversion formula (4.14) then shows that Ptf is measurable relative to the a-algebra generated by functions of the form P g , g E Cd(E),
1: Fundamental Hypotheses
33
each of which is a nearly optional function over (&+) under condition (iii). Consequently, Pt f is also nearly optional over (Gt+). (iv)+(v): By hypothesis, Ptf is nearly optional over (Gt+). In order to prove right continuity of s -, P t f ( X , ) , we call on (A5.9). Fix p and let {T,} be a uniformly bounded sequence of optional times decreasing to T . Then, by the Markov property relative to (G,”,), ppPtf(&‘,,) = ppf
Since t+T,o&
1 t+Tof#t and s
--$
(Xt+T,oOt)
*
f(X,) is right continuous and bounded,
l i m p p P t f ( x T , ) = Ppf ( X ~ + T ~=BPp ~ )p t f ( X ~ ) . n
(v)+(iv): As noted in the proof of (ii)+(iii), the SMP follows from
p’
1
{ f ( x t ) o e T~ { T < c o } = p’ { P t f ( X T ) l{T
for p an arbitrary initial law, f E C d ( E ) , t > 0, and T an optional time over (Gf+). Let (27,) denote the dyadic approximants (5.4) to T . By the same manipulations used to establish (7.5) (see also (6.22)), the identity above holds with T replaced by T,. By right continuity and boundedness of s * P t f ( X , ) and s -+ f(Xt+,), one may let n -+ 00 to obtain the desired identity for T. (iv)+(vi): Let f E Cd(E). The map ( t , w ‘ ) + f ( X t ( w ’ ) ) is in B+ @ 9, and if R E pGg+, then (w,w’)
-+
( R ( w ) , w ’ ) E G;+ €3P / B + @ 9.
By composition, cp(w, w ’ ) := f ( X R ( w l ( ~ ’ ) )E G;+ @ F’.Fix t >_ 0 and let T be optional for (Gf+). Let R = (t - T)lp
v(w,etw) 1{T
l{T
and we get (vi) by applying the form (6.12) of the SMP, so that
p’”{ f ( x t )1 { T < t }
I Gg+} (“1
=
1dw,
w’> l { T ( w ) < t ) PXT@) (dw’)
= P R ( w ) f ( X t ( W ) ) 1{T(w)
(vi)+(vii): Let f E C d ( E ) . For each fixed t 2 0, ( s , w ) + P t f ( X , ( w ) ) is is right in L ? p ( G ) , by hypothesis. For each fixed ( s , w ) , t -, Ptf(X,(w)) continuous. It follows that
34
Markov Processes
Since ( t ,(3,w ) ) -+ (t - s)+ is right continuous in t for every fixed (s,w ) and in Op(G) for every fixed t ,
Composition of these maps shows that
Consequently, for a fixed t 2 0, the map ( 9 , ~ ) p~-8f(xS(w))l[o,t~(8) is in S p ( G ) . Apply now (A5.13ii) to this situation with the filtration (#+). The process above is a martingale, and (vi) asserts that it has the optional sampling property. We may therefore conclude that the martingale is as. right continuous. (vii)+(viii): Given f E C,(E), and t 2 0 and an initial law p, the simple Markov property implies that the process -+
is a martingale over (52,Gf,Pp). According to (vii), s -+ M,f is as. right continuous, so for every s < t
That is, ( M i ) is a bounded, right continuous martingale over (52, Gf+, P p ) . A monotone class argument resting on (A5.13ii) proves then that the same is true for every f E bE. Now let f E bE” and let p be an initial law. Set u := pPt and choose g,h E bE with g 5 f 5 h and u(h - g) = 0. In particular, Pp{h(Xt) # g(Xt)} = 0. Then Mf and M,” are right continuous martingales over (52, Bt+, Pp) and because their final values are P”-a.s. equal, M i and M,” are Pp-indistinguishable. Therefore, M,f is Pp-as. right continuous. (viii)=s(ix): According to (viii), for every f E M u ,
is 8.5. right continuous, and it is a martingale over (fi, Gf, Pp), because of the simple Markov property relative to (G8). It follows that for 0 5 s 5 t ,
I: Fundamental Hypotheses
35
That is, X is simple Markov relative to (0, G,",, P"). Given H E b 3 of the form H := f o ( X o ) f ~ ( X t ,.). .f n ( X t , ) with f o , f i , .. . ,f, E b€" and 0 = t o < tl < . * . < t n , the simple Markov property above implies that
PP { H I Gf+} = n
k-1
n
k=l
j=O
j=O
where h, := f , , and for k = n - 1 , . . ., 1 , hk := f k Ptk+,-tkhk+l. By a monotone class argument, for every H E b F , P"{H I Gf+} may be chosen to be Gf-measurable. By trivial sandwiching, the same is true for every H E b3. (viii)+(ii): For g := U Qf with (Y > 0 and f E C d ( E ) , consider the map
~ ' ( t( 8, , ~ ) ) := p t - a f ( X a ( u ) )l [ o , t [ ( s ) . For fixed t , s + M f ( t ,( s , u ) ) is a s . right continuous, and for fixed (s,w), it is right continuous in t . Therefore, M f is in B+ 8 O P ( G ) for every 1.1. By the MCT, the same is true for every f E bpE. Fix f E bpE, and write M in place of Mf.For a > 0, (3,u)+ e-Qt M ( t , (s,w ) ) d t is in OP(G). But
lm I" e - Q t M ( t ,(s,w)) d t =
= e-us
That is, (7.6)
Ju"
e-"tPt--s f (X,(u)) l p , t [ ( s )d t
1"
e-arPTf ( X a ( u ) )dr.
e-"tM(t, (5, w))d t = e-aa u a f ( x , ( w ) ) .
We have shown therefore that g := U Qf is a nearly optional function over (Gt+). Fix a n optional time T for (Gt+). Since s --+ M ( t , ( s , u ) ) is right continuous and bounded, optional sampling gives us
M ( t ,T ) = P " { f ( X t ) I GF+} 1{T
PP{ee-aTU" f (XT) 1 { ~ < " } }= PI"
/
0
M
e-Qt
M ( t , T )dt
36
Markov Processes
It is evident from this identity that, if {T,} is a decreasing sequence of increases optional times with limit T , then P p {e-aT= U af (XT,) l{T, 0. Given an initial law p, set v = pUa and choose f1, f 2 E bpE with f l I f 5 f 2 and v ( f 2 - f1) = 0. The processes e-atU"fi(Xt) and e-atUaf2(Xt) are a.s. right continuous, and for every t 2 0,
Ppe-at
U'l(f2
- fl)(Xt) = e-otp [PtUa(fi- fl)]
I p U"(f2 - fl) = 0. This shows that e-at Uaf l ( X t ) and e-"t U a f i ( X t ) are indistinguishable relative to P p , and consequently e-"t U a f ( X t )is Pp-a.s. right continuous. Finally, each f E S" is an increasing limit of functions of the form U'lh, h E b p P , so another application of (A5.9) completes the proof. (7.7) COROLLARY. I f X satisfies HD2 relative to its natural filtration (Ft), then ( F f ) and (Ft)are right continuous.
PROOF:Apply (7.4ix) with H E bFt+. (7.8) COROLLARY. Let X = (R, B, Gt, X t , Bt, P") satisfy one of the equivalent conditions of (7.4), and suppose, in addition, that {t -t U a f ( X t ) is not right continuous} E N ( F ) for every a > 0 and every f E Cd(E). Then (R, 7 ,Ft,X t , Bt, P") satisfies the conditions of (7.4).In particular, all the exceptional sets in the conditions (7.4) are all in N ( F ) rather than N(G).
PROOF:It is clear from the form of (6.6) that if X has the SMP relative Under the conditions to (Gt+), then it also has the SMP relative to (Ft+). above, it follows that (R, F,FtlX t , Bt, P")satisfies (7.44. The result is then an immediate consequence of (7.4). In the following statement, saying that a process Y is optional on [ O , t [ means that ( s , w ) t Ys(w) (3 < t ) is in the o-algebra on [O,t[@R gentogether with processes Z with erated by rcll processes adapted to lZ,l vanishing off a set in N t . Similarly, Y is nearly optional on [0, t [if, for every p, there exists 2 optional on [0, t [so that lY,- Z,l vanishes off a set in N f .
(c),
(7.9) COROLLARY. Let X = ( R , F , F t , X t , B t , P "be ) a right continuous &*-Markovprocess satisfying HDI (2.1) with semigroup (Pt). The foltowing conditions on X are then equivalent: (i) X satisfies (7.3);
I: Fundamental Hypotheses
37
(ii) V a > 0, V f E C d ( E ) , {s -+ U " f ( X , ( w ) ) (s < t ) is not right continuous} E Nt; (iii) X satisfies the strong Markov property (6.5) relative to (F,+), and for every t 1 0 and f E C d ( E ) , s U a f ( X s ) (s < t ) is nearly optional on [0, t[; (iv) X satisfies the strong Markov property relative to (F8+),and for every t 2 0 and f E C d ( E ) , the function Ptf is nearly optional on [O,t[; (v) V f E C d ( E ) , V t 2 0, {s -, P t f ( X , ) (s < t ) is not right continuous} E Nt; (vi) V f E C d ( E ) , V t 1 0, Pt f is nearly optional on [0, t[,and for every initial law p and every optional time T over (Ff+), --$
p' { f ( X t ) l { T < t )1 FG+}= Pt--Tf(XT) l{T 0, {s P t - 8 f ( X , ) l [ o , t [ ( ~ ) is not right continuous} E Nt; (viii) Vf E bE", V t 2 0, {s + P t - , f ( X , ) l [ o , t [ ( sis ) not right continuous } E N t . --$
The proof is almost exactly parallel to that of (7.4),and we shall not repeat it.
(7.10) THEOREM. Let (a, 9,Gt, X i , &, P") be a right continuous simple E-Markov process, and suppose that X satisfies HD2 (7.1) relative to (Ft). Then X is a right process relative to the augmented filtration ( c t + ) .
PROOF:We showed in (6.7) that the SMP holds relative to the augmented filtration if it is the case that, for every choice of p , f E C d ( E ) , t > 0, and every optional time T over (@+), P'{f(Xt+T) l{T
(7.11) EXERCISE. Show directly that the example in (4.19) does not satisfy
HD2.
38
Markov Processes
8. Right Processes
(8.1) DEFINITION. A system X = (Q,G, G t , X t , Bt, P")is a right process in the Radon space E with transition semigroup (Pt)provided: (i) X is a right continuous realization (2.2) of (Pt); (ii) X satisfies HD2 (7.1) relative to (Gt); (iii) (Gt) is augmented (6.1) and right continuous. If there is some right process (Q, G, Qt, X t , O t , P") with transition semigroup ( P t ) ,then (Pt)is a right semigroup. We impose no Borel hypothesis on the semigroup (Pt). The right process X is a refined right process in case it also satisfies (7.3). It should be emphasized that part of the condition (i) requires that X be normal. That is, for every 2 E E , P"(X0 = z} = 1. According to (7.10), if (Q,G, G t , X t , Bt, P") is a normal, right continuous Markov process with semigroup (Pi),having the strong Markov property relative to its natural filtration ( F t ) ,if (Pi) maps Borel functions to Borel functions, and if ( G t ) satisfies (8.liii), then (R,Q, Gt, X t , Bt, P") is a right process. As we mentioned earlier, it will be shown in (18.3) that every right process (Q, 8,Gt, Xt,Bt, P") may be modified by deleting a null set in G so that it becomes a refined right process. Refined versions are necessary in some later discussions of transformations preserving right processes-for example, Doob's h-transforms are most easily treated in this framework. A right process possesses each and every property listed under (7.4). In particular, for every initial law p , the filtration (3:)satisfies the usual hypotheses of the general theory of processes relative to the probability space ( Q , F p , P " ) . Suppose (0,G,Gt, Xt,Bt, P") is a right process. It is natural to ask then if (Q, F,Ft, X t , Bt, P") is necessarily a right process. This does not seem to be the case by any obvious argument, but it is trivially true of a refined right process, by (7.9). It is worthwhile to recall attention to some features of right processes which are hidden in the definition (8.1). First of all, (8.1) is somewhat different from the definition given by Getoor [Ge75a],where it is required that (Pt) be realizable on the space W of all right continuous maps of R+ into E . It turns out that the difference between the two definitions is minor in that (a) for right processes whose semigroups map Borel functions to Borel functions, the definitions are equivalent; (b) even in the case of a non-Bore1 semigroup, a right process may always be realized on the space of maps of R+ igto E which amreright continuous for both the original topology and a new topology-the Ray topology-on E . In particular, all results proved under Getoor's hypotheses hold also for right processes
I: Fundamental Hypotheses
39
in the sense of (8.1), at least when interpreted with respect to the Ray topology on E . The necessity of changing the topology on E has to do with the fact that the defining property (8.1) for a right process is not in general invariant under a change of realization. Two of the results in 519 will be concerned with such questions. It will be shown that if one right continuous realization of a Markov semigroup has the strong Markov property, then so does every other right continuous realization. It will be shown also that, under some additional mild hypotheses on X, the right process property is shared by all right continuous realizations of (Pt). The first example of a right continuous, normal, strong Markov process which is not a right process seems to be due to Salisbury [Sa87].The example does not, of course, have a Bore1 semigroup. See the discussion in 519. It would be very interesting to have a satisfying description of the class of all right continuous, normal, strong Markov processes which have some realization as a right process. There are a number of important simplifications in the properties of excessive functions deriving from HD2. Let (Pi) be a right semigroup on E . Then: (8.2) PROPOSITION. (i) for every positive, continuous, concave, increasing function cp on R+, the composition cpo f E So for all f E So; (ii) the minimum of two a-excessive functions is also a-excessive.
PROOF:Let X be a realization of (Pt)as a right process. If f E S", HD2 asserts that f ( X t ) is a.s. right continuous. For every cp satisfying the conditions of (i), cpof(Xt) -+ c p o f ( X 0 ) as t 1 0. Moreover, one may write cp = inf, cpnr where each cpn is an affine function of the form vn(t) = ant + b,, with a,, bn 2 0. Thus, for any sub-probability p on E and any g E PE", p(cpog) L inf, p(pnOg) Iinfn V n ( p ( g ) )= M g ) ) . Applying this to p = e-ot Pt(z, one sees that a ) ,
e-at Pt(cp0.f) L ~ ( e - ~ ~ p t f ) and because cp is increasing, the latter term is bounded by cpof. Therefore cpof is a-super-mean-valued. In particular, t + e-at Pt(cpof)(z)is decreasing, and consequently Fatou's lemma yields lim e-Qt ~t(cpof)(z) = limedat p2c p o f ( x t ) 2
tllo
(pof(z).
tll0
That is, cpof E So.For point (ii), let f,g E Sa.It is clear that h := f A g is a-super-mean-valued, and since h ( X t ) -+ h ( X 0 ) a.s. as t 1 0, another application of Fatou's Lemma gives
h(z)<_ lime-at P"[/L(X~)] = tl0
tl0
~th(xC) 5 h(x).
40
Markov Processes
This shows that h is indeed in S". (8.3) REMARK. The proof of (8.2) does not use the full power of HD2. It requires only the fact that for every f E S", there is a sequence t, 11 0 with f(Xtn) + f(&) a s . as n + 00. The following result displays a much weaker hypothesis under which the same conclusion obtains. Let X = (R,F,Ft,Xt,Bt,PZ)be a process having (8.4) PROPOSITION. the simple Markov property with transition semigroup (Pt) relative to (3t). is P"-trivial for every x E E . Then, for every finite valued Assume that 30+ f E S", f ( X , , ) + f(X0) as. as n -, 00, whenever t, 1 0. Consequently, under these hypotheses, the assertions (i) and (ii) of (8.2) hold for every finite valued a-excessive function.
PROOF:Fix t, 4 0 and f E S" with finite values. The process t + e-'lt f(X,) is a supermartingale over (R, F,3 t , P") for every x E E. Therefore, Y, := e-"tn f(Xt,) is a P"-supermartingale over the index set {t,}, which filters to the left. In addition, sup, P" Y, 5 P" YO= f(x) < 00, so we may apply (A5.17ii) to conclude that for every x E E , the family {Y,} is uniformly integrable relative to P", and limn Y, exists P"-as. as n + 00. As the limit is Fo+-measurable, it is necessarily constant P"-a.s. By uniform integrability and the fact that Y, + f(x) Pz-a.s. as n + 00, this proves that f(Xt,) + f(x) P"-a.s., as claimed. The close connection between the excessive functions for (Pt) and the probabilistic properties of X makes the following definition useful. (8.5) DEFINITION.The a-algebra E" on E is the least a-algebra containing U,~OS". The corresponding a-algebra 3: on s2 is that generated by {f(X,) : 0 5 r 5 t , f E bE"}, and F" := Vt203f= a{f(X,) : T 2 0 , f E bE"}. (8.6) THEOREM.Let X be a right process with semigroup ( P t ) . Then: (i) bEe is the least MVS containing U,,obS*; (ii) Pt maps bE" into itself;: (iii) every f E E" is an optional function; (iv) if H E bFe, then the function x + P"H is in bE"; (v) E c Ee c E".
PROOF:Since S" is closed under finite i d m a , and constants are in Sa, every function f E S" is an increasing limit of the functions f A n E bSa. The class U,>obS" is closed under finite infima, thanks to (8.2) and (4.15). It follows then from the lattice form of the MCT that the least MVS containing u,,obS" is b1-1, where 1-1 is the a-algebra generated by U,,obS". However, every function in S is also in S" for every a > 0, so 1-1 is precisely Ee. In proving (ii), it suffices, by (i), to remark that f E S" implies
I: Fundamental Hypotheses
41
Pt f E S". Every f E bS" is optional because t --+ e-at f ( X t ) is a P"supermartingale which is a.s. right continuous, and therefore has left limits a.s. everywhere on R++.We then invoke (i) to complete the proof of (iii). w i t h o s t 1 st2 GivenHE b&eoftheformfl(Xt,).-.fn(Xt,) and all fj E b&", x P"H is in b P , by repeated application of (ii). The proof of (iv) is accomplished then by an appeal to the MCT. For (v), only the inclusion & c &" requires argument. It suffices to prove that f E & ( E ) implies f E E e . We observed in (4.18) that, for such an f , aUaf + f pointwise on E . Writing f = f+ - f-, it follows that U"f E bS" - bS", and therefore f E bEe.
-
<...st,
Note that all assertions of (8.6), with the sole exception of (iii), remain valid under the weaker hypotheses that X is a right continuous simple Markov process with Fo+ trivial under P" for every x E E. This is a simple consequence of (8.4). The result in the next exercise will be important in later work, as it gives a simple multiplicative class generating bEe. (8.7) EXERCISE.Let X be right continuous, simple Markov, and assume that Fo+is P"-trivial for every x E E . Let A := U,,o[bS" - bS"]. Show that A is an algebra of bounded functions on E . (Hint: if h E bS" and 0 5 h 5 1 then 1- (1- h)' E bS".) (8.8) EXERCISE.Let E be a Radon space and let ( 4 t )-t > ~ be a family of mappings of E into E satisfying:
(i) $t E &"I&"V t 2 0; (ii) Vx E E , t + 4t(z)is a right continuous map of R+ into E ; (iii) Vt,s 2 0, V x E E , 4t+&) = cbt(4dx)); (iv) $o(z)= x V x E E . A family satisfying (i)-(iv) will be called a right continuous flow in E . Prove that if one defines Pt(x, to be unit mass at ~ $ ~ ( xthen ) , (Pt)is a Markov semigroup satisfying HD1 and HD2. (Use (7.4vii).) The associated right process is completely deterministic. We shall re-examine this example in (47.8). 0
)
(8.9) EXERCISE.Let (a,8, Gt, Xt,O t , P") be a right process, with (Gt) a strong Markov filtration. Suppose that for every G E b80, a.s., Go00 = G. Prove that P"(A) = 0 or 1 for all A E GO. (Hint: let T ( w ) = 0 if w E A , = 00 if w $ A , and use the strong Markov property to evaluate P"{~A1 o{ ~~ T< ~ ) ) - ) (8.10) EXERCISE.For every f E S", the process
(Ft+). (Hint: use (5.8).)
fox is optional over
42
Markov Processes
(8.11) EXERCISE. Let g E bS and suppose Ptg + 0 as t and let gn := (g - Pt,g)/t,. Prove that gn E bpEe and
+ 00.
Fix t,
110
d n
Ug, = t i 1 /o (Hint: first evaluate
Ptgdt g
as n -+ 00.
s," Pgg, ds for finite t . )
(8.12) EXERCISE. Prove that sup{lf(Xt)l : a 5 t 5 b} is F-measurable for every f E Eel a < b. (Hint: for f E S", the sup differs from the sup over a countable time set only on a null set. Now use the MCT, making use of (8.6) and (8.7).) 9. Existence Theorem for Ray Resolvents
This section contains the fundamental existence theorem for right processes. Suppose given a compact metrizable space E and a family of kernels (Ua),>O on (E,E) satisfying the conditions
aU"(z,E) = 1 for every a > 0; U" maps C(E)into itself, for every a > 0; U" - U P = ( p - ~ ) u " ufor @all a,p > 0.
(9.1) (9.2) (9.3)
Condition (9.3) is, of course, just the resolvent equation of (4.8). We may consider (U") as a family of bounded positive operators on the Banach space C(E) with the uniform norm llfll := sup{]f (.)I : 2 E E } . The condition (9.3) is equivalent to the condition that the operators (Ua)form a resolvent on C(E),in the functional analytic sense. Condition (9.1) asserts that the resolvent is Markovian. In the case U" = &O0 e-atPt dt, it corresponds to the condition Pt(z,E) s 1. As in 54, f E pEu is called a-supermedian in case PU"+Pf 5 f for all p > 0. The class of all asupermedian functions for (U")will be denoted by
s".
(9.4) DEFINITION. A Ray resolvent on a compact metrizable space E is a Markovian resolvent (U") on C(E) satisfying (9.5)
Ua>0
C(E)fl S" separates the points of E .
For any a > 0, f , g E C (E )fl Sa obviously implies f A g E C(E)fl S". Since 3" is a cone of positive functions on E , := c(E)nd"-c(E)n+ is a vector subspace of C(E).Actually, 'Hais closed under lattice operations, for if f , g , h , j E C(E)flS", then
(f -9)
A
( h-
A
= [(f + A - (9 +A1 A [ ( h+ 9 ) - ( g + j ) l = [(f + A A ( h +s)l- (9 + j )
I: Fundamental Hypotheses
43
belongs to 'Ha.Observe though that the vector space 'H" is the same for implies that 'K" C 'Kp. For the reverse all a > 0, for if 0 < LY < @, S" C inclusion, take f E C ( E )[email protected] f = [f (P - a ) U " f ] - (P - a ) U a f displays f as the difference of two functions, the second of which is obviously in C(E)n S a . In order to prove that g := f (@- a)U"f E C ( E )n S",we f 5 f, must show that XUX+"g 5 g for all X > 0. But, we are given XUA+@ and therefore, by the resolvent equation,
+
+
Another application of the resolvent equation yields W+=g
5f
+ (P - ,)UX+"f + (p - a ) ( U " f - .X+,f)
= g.
Thus, under (9.5), 'H" = Uo>o'Ko, and 'K" is a vector lattice separating the points of E , so, by the lattice form of the Stone-Weierstrass theorem, 'Ha is uniformly dense in C(E). It follows in particular that for every a > 0, C(E)n Sa separates the points of E . To summarize: (9.6) PROPOSITION. Let (U") be a Ray resolvent on E . Then: (i) for every a > 0, C ( E )n Sa separates the points of E ; is independent of a; (ii) 'Ha:= C ( E )n - C(E)n (iii) 'Ha is uniformly dense in C ( E ).
s"
If a resolvent (U") on a compact metrizable space E satisfies (9.1)-(9.3), and if a U " f ( z ) f(z) as a + 00 for every f E C ( E ) ,then since $ ( E ) separates E , so does U,>oC(E) r l S a . ---f
(9.7) EXERCISE.Let E = { a , b, c}. Define kernels U" by:
uyc, - ) :=
-),
+
U"(b, * ) := ( a + 1 ) - l € b ( . ) + [a(a l ) ] - l € c ( * ), U"(a, ) := p ( a 1 ) - % b ( . ) [1+ a(1- p)][a(a
-
+
+
+
1)]-lCc(
a ) ,
where 0 < p < 1. Check that (U") is a Ray resolvent on E which does not satisfy the condition a U a f ( z )-, f(z) as a -+ 00 for every f E C(E). We come now to the first major result [Ra59]on Ray resolvents. (9.8) THEOREM (RAY). Let (U") be a Ray resolvent on E . Then there exists a unique Markov semigroup (P,) on ( E ,E ) such that:
(i) for every f E C(E),t + P , f ( z ) is right continuous for all z E E ; e-"t P,f(z) d t for all z E E. (ii) for every f E C ( E ) ,U a f ( z )=
Markov Processes
44
We shall not offer a proof. The reader may consult Meyer [Me66a, X191 and Getoor [Ge75a, (3.6)] for two different approaches. It is worthwhile to point out that the semigroup (P,) obtained in (9.8) does not in general map C ( E ) into itself, nor is it the case that POis the identity operator on C(E). However, it is easy to see that if a U " f ( x ) + f(z)as a + 00 for every f E C ( E ) ,then PO is the identity operator. (9.9) EXERCISE.Show that for the resolvent of (9.7), the corresponding semigroup is given by
In particular, Po(a,- ) #
fa(
.).
(9.10) DEFINITION.A semigroup (Pt) of kernels on a compact space E is a Ray semigroup if it satisfies the conditions of (9.8) with (Ua)a Ray resolvent. The set B of branch points for (Pt)is
B
:= {Z €
E : Po(z, - ) # tz(
a ) } ,
and the set D of non-branch points is D := E - B. Some elementary analytic facts about Ray resolvents and their branch points are described below. These will be useful here and in Chapter V. (9.11) PROPOSITION. Let (Pt) be a Ray semigroup on E and let B be its associated set of branch points. Then: (i) if {g,} is uniformly dense in C ( E ) n s l ,then B = Un{Pogn < g,}; (ii) B is an F, set in E , hence B E E ; (iii) for every t 2 0 and every x E E , Pt(x,.) is carried by the set D of non- branch points for (Pt).
PROOF:It follows from (9.8) that for every f E C(E)and p lima+m aU"'+Pf(z). Therefore, if g E C(E) n
sl,
POg(zc>= Iim
a-+m
aUa+lg(x)
sl,
> 0, Pof(x) =
5 g(z).
s1
s1
If {g,} is uniformly dense in C (E )n then since C(E)n - C(E)n is uniformly dense in C ( E ) , it follows that D = n,{POg, = gn}. This proves (i). Writing
I: Fundamental Hypotheses
45
we see that B is a countable union of closed sets, since both gn and UIC+'gn are in C(E). Finally, B = Un{Pogn < gn} implies
p t ( x ,B ) I
C ~t(xc,{ P o g n < gn))n
However, Pogn I gn and Ptgn(x) - Pt(Pogn)(x)= 0 imply then that Pt(x,B ) = 0 for all z E E . (9.12) LEMMA.I f f E C(E)n S a , then e-atPtf
5 f for all t 2 0.
PROOF:For /3 > 0 and x E E ,
Lw
Be-Pt
[f(x)- e - a t P t f ( x ) ]d t = f(x) - /3Ua+Pf(z).
With x fixed, let g : 10, oo[+ [0, oo[be defined by g(P) := P-'f(z) - U*+Pf(.).
Then g is the Laplace transform of the bounded right continuous function t -+ f(x)-e-atPtf(z). On the other hand, successive differentiations in p, making use of the resolvent equation, lead to
= - (UP)'f(z),
and, more generally,
It follows that dn
(-l)n-g(B)
wn
= n! ( p + l ' f ( x ) - (Ua+P)n+' f(5)).
It is clear by induction on n that the right side is positive for all n and all /3 > 0. We have shown therefore that g ( p ) is a completely monotone function of ,B, so the Hausdorff-Bernstein-Widder Theorem shows that g is the Laplace transform of some positive measure. It follows from uniqueness of the Laplace transform that f(x) - e-atPtf(z) is positive for almost all t > 0, and then right continuity in t completes the proof. Fix now a Ray resolvent and let (Pt) denote the associated Ray semigroup generating ( P ) As . in (9.10), let D denote the non-branch points for (Pt). We showed in (9.11) that Pt(x,. ) is always carried by D. The restriction of the semigroup ( P t )to D is obviously a Markov semigroup on the Lusin space D.
Markov Processes
46
(9.13) THEOREM. The restriction of the semigroup (Pt)to D is a right semigroup which may be realized on the space of right continuous maps of R+ into D having a left limit in E.
PROOF:Let W := ER+,the space of all maps w : R+ -+ E , and let (Y,) denote the co-ordinate process on W . Let p , be the usual (unaugmented) a-algebras on W generated by Bore1 functions of the process Y. Kolmogorov’s theorem permits us to construct, for any initial law p, a measure Pp on (W,F”) such that, for 0 5 tl I t2 5 5 t , and ZO, . ..,x, E E , (9.14) P”{& E d ~ o ,. .. ,yt, E dx,} = PPo(dZo)Pt, (20,dz1)* * * Ptn-tn4(2n-1, d%). Note that the first term on the right side of (9.14) is pP0, not p. ‘This of (yt) constructed below. For every is critical for the regularization (X,) f E C(E) n Sa,(9.12) gives e-atPtf 5 f for all t 2 0, so e-a(t+s)Pt+afI e-asP,f for all t , s 3 0. Using (9.14), we conclude that t -, e-atf(yt) is a Pp-supermartingale over Let A c R+ be countable and dense in R+,and let R A denote the set of all real valued functions on A which are regulated-that is, f E R A in case limrllt,rEAf ( r ) exists for all t 1 0, and limrfft,TEAf(r) exists for all t > 0. Set WO = {w E W : 2 0 1 ~ E R A } .Standard supermartingale regularity results quoted in $A5 show that Pp{Wo} = 1. We noted in (9.6) that C ( E ) - C ( E )n S1 is uniformly dense in C(E), and as C(E) is separable, we conclude that W I:= {w : t + f(yt(w))(t E A ) E R A for every f E C(E)} has full P” measure for every initial law p. Restrict all yt to W1, replacing P p , 30 and fi by their respective traces on W1. The formula (9.14) remains valid since W1 has full Pp measure for every p. For w E Wl and t 3 0, define
e.
ns’
(9.15)
&(w) := lim
sllt,sEA
Y,(w).
We shall demonstrate the following two points. (9.16) P@{Xt = yt} = 1 for all t 2 0 and all p. (9.17) Pp{Xt E D Vt 2 0) = 1 for every p. Suppose that (9.16) and (9.17) have been proved. Then for every p carried by D, we may restrict Xtto WD:= {w E WI : Xt(w) E D Vt 2 0 } , getting a right continuous realization of ( P t )restricted to D. That is, the restriction of (Pt)to D satisfies HD1. It is then an immediate consequence of (7.4ii) that the restricted X also satisfies HD2. We turn back now to proving (9.16). Observe that, if f E C(E) n 6 and g E C(E),
I: Fundamental Hypotheses
47
However, because P 4 P,f(z)is right continuous, this latter limit is equal to pPt[gPof].Since f E dl, Pof 5 f . But, for alla: E E , P t ( x , f - P o f ) = P t f ( x )- P t f ( z ) = 0, so that Pof = f a s . relative to pPt. The above calculation shows then that (9.18)
P P { g ( Y t ) f ( X t )=) @ t [ g f I = P c " { S ( y t > f ( y t ) } .
Since C ( E )n 8" - C (E )n Sa is dense in C ( E ) , (9.18) holds therefore for all f , g E C(E).It follows from a routine monotone class argument that, for every h E b(E 8 E )
Pp h ( X , , & ) = pph(X, Y,). Setting h(z,y) = l{,#g), one obtains (9.16). For (9.17), we shall use the characterization of D given in (9.11). Fix a dense sequence { g n } in C ( E ) n d' so that D = n , { g , = Pog,}. We shall prove that for all n 2 1, g n ( X t ) and Pog,(Xt) are Pp-indistinguishable, and this will prove (9.17). Fix n and let g denote gn. Set hk := k[g - kUk+'g], ( k 3 l ) , so that U'hk = kU"'g t Pog as k 00. Since U'hk E C ( E ) n d', t e - W 1 h k ( X t ) is a right continuous Pp-supermartingale for every p. This implies, taking into account (A5.16), that t -, e-tPog(Xt) is a right continuous Pf'-supermartingale for all p. As we observed above, Pog 5 g and pPt(g - Pog) = 0 for every p, and consequently Pog(Xt) = g ( X t ) Pp-a.s. for each fixed t 2 0. By right continuity in t , t 4 g ( X t ) and t + Pog(Xt) are therefore Pp-indistinguishable for all p, proving (9.17). (9.19) REMARK. The proof of (9.13) shows that the restriction of (Pt) to D may be realized on the space of all paths in D which are right continuous with left limits in E , though not necessarily in D. For example, let E := [0,1], and let P t ( z , be unit mass at y = (x t ) mod 1. Then one may check easily that (Pt) is a Ray semigroup on E , and that the set D of nonbranch points is [0,1[. The paths of X, which may be thought of as uniform motion to the right modulo 1, do have left limits in [0,I ] but not in [0,1[. In Chapter V, we shall see that every right process may be re-topologized in such a way that, after adjoining a polar set (10.9) to its state space, it may be identified with a process constructed as in (9.13) In the special case D = E , where there are no branch points, the right process X on E possesses an important additional property called quasileft-continuity, defined in (9.21) below. Before discussing this, we note the analytic significance of the condition D = E. --$
--$
a )
+
(9.20) PROPOSITION. Let (U")be a Ray resolvent on E , and suppose that no point of E is a branch point. Then U" ( C ( E ) )is uniformly dense in C(E)for every a > 0 . The converse is also true-see (9.33).
48
Markov Processes
PROOF:Let (Pt) be the associated Ray semigroup. Then D = E implies that Po is the identity operator on C(E),and so aU”f + f pointwise as a m for every f E C(E). The subspace J := U a (C(E))of C(E)does not depend on a. This is an easy consequence of the resolvent equation. The remark above shows that that every f E C(E) is a bounded pointwise limit of functions in J . Therefore, for every signed measure v on E with v ( f )= 0 for all f E J’, one has v(f) = 0 for all f E C(E),and hence u = 0. If the uniform closure of J’ were not equal to C(E),we could use the HahnBanach theorem and the Riesz representation theorem to construct a signed measure u on E vanishing on 3. .--)
(9.21) THEOREM.Let (Ua)be a Ray resolvent on E having no branch points. Then the associated right process X is quasi-left-continuousin the sense that, for every increasing sequence {T,} of optional times over (Ft)with limit T , X(Tn) --t X ( T ) a s . on {T < m}.
PROOF:There is no loss of generality in assuming that T is uniformly bounded, for we may otherwise replace each optional time by its minimum with an arbitrary constant time. Let Y := limn X ( T n ) . This limit exists because Xt has limits in E for all t > 0. In order to prove that Y = XT a.s., it is enough to show that for every p and every h E b(E 8 E), (9.22)
Pph(Y, XT) = P’”h(Y,Y).
For, under (9.22), simply take h(z,y) = l{,#y). In order to prove (9.22), we start by showing that for g, k E C(E), Ppg(Y)k(xT) = P’”g(Y)k(Y). In view of (9.20), it suffices to take k := U”fwith f E C(E).In this case, we may use the SMP and bounded convergence to get
= lim Ppg ( X (T,)) k (X(Tn) ) n
= P”g(Y)k(Y).
I: Fundamental Hypotheses
49
In Chapter V, we shall examine in more detail the meaning of quasileft-continuity for a right process. In order to apply the existence theorem (9.13) to the case of a resolvent on a space E which is locally compact, Hausdorff, with countable base (LCCB), we require an artifice which is a special case of one we shall describe systematically in 511. Suppose that E is not compact, and let Co(E) denote the space of real continuous functions on E that vanish at infinity. Suppose given also a Markov resolvent ( U a ) on ( E ,E ) satisfying (9.23) (9.24)
U" (Co(E))C Co(E)VCU> 0; aU" f + f pointwise as cr + 00 for all f E Co(E).
Adjoin a point A as the point at infinity and let EA denote the compactified (metrizable) space E U { A } . Define a resolvent (0") on EA extending (U") in the following manner: (9.25i) Oa(A, = @-lea(.); (ii) for 2 E E , f i " ( z , ( A } )= 0 and l?"(z, . ) ( E = U a ( x , A trivial calculation shows that (Oa) is a Ray resolvent on E A , having no branch points. The right process X on EA having resolvent (Ga) is therefore quasi-left-continuous, by (9.21). In addition, the point A is a trap for X , since p t ( A , - ) = EA( .) for all t 2 0. Let { K , } be an increasing sequence of compacts in E with union E . Then E A \ K , is an open set in E A , so if we let T, := inf{t > 0 : Xt E E A \ K,}, a )
a ) .
then elementary arguments show that T, is optional for X. By quasi-leftcontinuity, T := lim,T, has the property XT = A a s . on {T < 00). However, if p is carried by E ,
P"{Xt = A } =
J
p(&)
P~(z, {A}) = 0
so P"{T < 00) = 0. If we now let ( X t ) denote ( X t ) restricted to { w : X ~ ( WE)E V t 2 0}, then the argument above proves the following result.
(9.26) THEOREM. Let ( U a ) be a Markov resolvent on the non-compact LCCB space E satisfying (9.23) and (9.24). Then there exists a right process X with state space E , having resolvent (U"), such that: (i) X is quasi-left-continuous; (ii) for all t > 0, the set { X , ( w ) : 0 5 s 5 t } as. has compact closure in E ; (iii) as., the left limit X t - := limsTtt X, exists in E for all t > 0.
Markov Processes
50
(9.27) EXERCISE.Let (Pt) be a Markov semigroup on an LCCB space El and suppose Pt maps Co(E) into itself for every t 2 0. Assume also that Pt f -, f pointwise as t 10 for every f E Co(E). Prove that the associated resolvent ( U a ) satisfies (9.23) and (9.24), and that Pt f + f uniformly as t 1 0 for every f E Co(E). (Hint: show that U := U"Co(E) does not depend on a, and that Pt f -, f uniformly for every f E U. Argue that the same is true for f in the uniform closure of U. Use the Hahn-Banach theorem to prove that U is uniformly dense in Co(E). Semigroups of the type described in this exercise are called Feller semigroups.)
Let {pt}t?o be a vaguely continuous convolution semigroup of probability measures on Rd. That is, pt * ps = pt.+s for all t , s 2 0, and f d p t + f (0) for every f E C,(Rd). Define then, for f E bB"(Rd), (9.28)
It is easy to check that ( P t ) is a Feller semigroup on Rd. The processes arising from these particular semigroups are known as Levy processes, or processes with stationary independent increments. Brownian motion, also known as the Wiener process, is the most important special case, the semigroup being given by the Gauss semigroup of (4.10). (9.29) EXERCISE.Let X be a standard Brownian motion on R1,constructed according to the methods of this section with the Gauss semigroup of (4.10), so that pt(dx) := ( 2 ~ t ) - ~ / ~ e - "dx. ' / ~Prove ~ that t + X t is a.s. contin uo us by computing
for fixed t
> 0 and y > 0.
(9.30) EXERCISE.The Poisson process with parameter X on the integers is defined to be the Le'vy process with convolution semigroup
Compute its associated a-potential kernel. (9.31) EXERCISE. Brownian motion on R with drift c is the LCvy process with semigroup 1 pt(dx) := -exp [-I.
rn
- d I 2 / 2 t ] dx.
I: Fundamental Hypotheses
51
Using manipulations with Fourier transforms as in (4.10), show that the corresponding a-potential kernel density is given by u"(2) :=
1 Jwexp [-cy - &=IYl].
(9.32) EXERCISE. Let (U") be a subMarkov resolvent on the LCCB space E (z.e., ffU"1E 5 l ~ )satisfying , (9.23) and (9.24). Adjoin A to E rn the point at infinity. Define 0"on the compactified space EA by setting: (i) O"(A, . ) := CX-'CA( .); (ii) for z E E , O"(z,{A}) := 1 - a U " ( z , E ) and U"(z,* ) I E := U"(2, Prove that (0") is a Ray resolvent on EA without branch points. Prove that A is a trap for the corresponding process. a ) .
(9.33) EXERCISE. Prove that the converse of (9.20) is valid. Namely, if U"(C(E)) is uniformly dense in C ( E ) for some a > 0, then D = E . 10. Hitting Times and the Fine Topology
Assume that ( R , 3 , Ft,X t , Bt , P") is a right process in E with semigroup (Pt). Given B E E , the random set { ( t ,w ) : Xt(u)E B } is in B+ €3 p . Its projection on R is, by the Projection Theorem (A5.2), measurable relative to 3*,the universal completion of p . (Recall that this is the completion of 9 relative to all finite measures on E , not just those of the form Pp.) Similarly, the projection on SZ of { ( t ,u): 0 5 t 5 s,Xt(u) E B } belongs to 3:,the universal completion of Given an arbitrary subset B of E , the debut DB of B and the hitting time TB of B are defined respectively by
c.
D B ( w ):= inf{t 2 O : X t ( w ) E B } ; TB(w) := inf{t > 0 : X t ( w ) E €3);
(10.1) (10.2)
where the infimum of the empty set is defined to be 00. The distinction between D B and TB is extremely important. The hitting times have a more intimate connection with the potential theory of a Markov process than do debuts, and the reasons can usually be traced back to the fact that
+
x,
t TBo& = inf{s > t : E B } 1TB as t 110, but (10.3) (10.4) t DBoBt = inf{s 2 t : X, E B} 1TB (not D B )as t 11 0.
+
Markov Processes
52
See 512 for further discussion of the r61e of (10.3). In view of the initial remarks, if B E E , then since { w : DB(LJ)E B } is the projection on R of { ( t , w ) : 0 5 t 5 s , X t ( w ) E B}, one has {DB < s} E 3:. A similar argument applies to TB,so both DB and TB are optional times over (F;+). Because 3;c Ft c Ft+,DB and TB are also optional over (Ft).The fact that the hitting times of Borel sets are optional over the augmented natural filtration is one of the principal reasons for introducing those augmentations. Hunt's capacity-theoretic proof that hitting times are optional times under such circumstances wits one of the principal forerunners to the development of the so-called general theory of processes. Largely because the sort of operations dealt with in Markov process theory, like the one above, lead one quickly outside the domain of Borel sets, it is essential to consider hitting times for more than just Borel sets, and the class of nearly optional sets introduced in $5 is convenient for this purpose. Recall that B C E is nearly optional for X in case 1B(Xt) is an optional process over ( F f )for every initial lav p . By the theorem on measurability of debuts (A5.1), the debut of the random set { ( t , w ) : X t ( w ) E B } is an optional time over ( F f ) . As p is arbitrary, D B is optional over (Ft). A similar argument shows that the hitting time TB of a nearly optional set is likewise optional over (&), using instead the nearly optional set { ( t , w ) : t > O,Xt(w) E B } . (10.5) DEFINITION. Let B be a nearly optional set in E. A point x E E is regular for B in cme P"{TB = 0) = 1. The set of regular points for B is denoted by B'. If x 4 B', then B is thin at x. According to the 0-1 law (3.11), P"{TB = 0 ) = 0 or 1. The case P"{TB = 0) = 0 means that the process starting at x will almost surely = 0) = 1 means that avoid B during some initial time interval, and P z { T ~ the process starting at x will almost surely visit B infinitely often during every initial time interval. (10.6) PROPOSITION. If B is nearly optional, then B ' E Ee and TB = so that ( B U B') ' = B'. In addition, X (TB)E B U B' a s . on
TBUB~,
{TB
< 00).
PROOF: Let &(x) := P"exp (-TB). Then e-tPt&(x) = P" exp{-(t + TBo&)} T &(x) as t 11 0, because of (10.3). Therefore, c$&is 1-excessive. ' E E e . The right continuity Observe that B' = {x : &(x) = l}, so that B o f t + &(Xt) implies that X ( T B V E) B ' a s . on { T B < ~ m}. In order to simplify the typography, let T denote TB, and let S denote TBP.On { S < T } , the terminal time property of T gives T = S + Toes, and
I: Fundamental Hypotheses
53
consequently
P5{S < T } = Ps(S < T , Toes
> 0)
= P S { P X ( S ) {> T 0); S
< T},
making use of the strong Markov property. But, since X(S) E B" a.s. on ( S < m}, one finds P z { S < T } = 0. As z is arbitrary, this proves that T 5 S a.s., and therefore, a.s., TBUBT = T A S. Observe finally that on { X T 4 B T , T < co}, P X ( T ) { T> 0) = 1, so, using the strong Markov . Toes > 0 property again, Toes > 0 a.s. on { X T 4 BT,T < m} E 3 ~But means that X T + @ ~ B for all sufficiently small t > 0, a.s., and thus X T E B a s . on {XT 4 B T , T < 00). (10.7) DEFINITION. A subset G of E is finely open (for X) provided, for every x E G, there exists a nearly optional set B c E such that BUB' 3 G" andx @ B U B'. In other words, G is finely open provided the process starting at any point of G a.s. remains in G during some initial time interval (0, E[, where E > 0 may depend on w. Another way to state (10.7) is to say that G is finely open if and only if G" is the intersection of a family (not in general countable) of sets of the form ( B U B")",with B nearly optional. Sets of the form ( B U B")' with B nearly optional form a base for a topology on E called the fine topology. It is clear that the sets in E which are open for the fine topology are precisely the finely open sets defined in (10.7). It is also clear that the open sets in E are also finely open. If B C E is nearly optional, its fine closure is just B U B", and B is finely closed if and only if B" C B. Note that (10.6) implies that the set B" is itself finely closed. The functions on E which are continuous relative to the fine topology are called finely continuous. (10.8) PROPOSITION. Iff : E 4 [--00, co]is in E" and if t + f (Xt) is a.s. right continuous at t = 0, then f E E" and f is finely continuous.
PROOF:One may assume f bounded, replacing f by arctan f if necessary. By right continuity of f (Xt) at t = 0,
1
00
aUaf(x) = P S
ae-atf(Xt) dt
-+
f(x) as a
+ m.
Therefore, f E E". For any open interval I, right continuity of t + f (Xt) at t = 0 shows that B := {f E I } is a finely open set, so that f is finely continuous. We shall prove in (49.9) that the fine topology is in fact the coarsest topology on E making all functions in U a > 0 S a continuous. We make no use of this result in this work.
54
Markov Processes
In general, the fine topology does not satisfy the first axiom of countability (see exercise (10.29)), but the property just mentioned permits E with its fine topology to be embedded in a cube, so that the fine topology is completely regular. Some concepts related to fine continuity are discussed in the following definition. They are important in Markov process theory because they are the appropriate and natural notions of exceptional sets for the process and its related potential theory. (10.9) DEFINITION.Let B C E be nearly optional, and let &(z) :=
P"exp{-aTB}. The set B is: (i) polar if #% = 0 for some (and hence all) a > 0;
(ii) thin if 4% < 1 everywhere for some (and hence all) a > 0; (iii) totally thin if B is thin, and if there exist a > 0 and ,L3 < 1 such that c$%(X(TB))5 ,L3 a.s.; (iv) null or of potential zero if U ~ =B0. An arbitrary subset D c E is polar (resp., thin, null) if it is a subset of a nearly optional set which is polar (resp., thin, null), and D is semipolar if it is a countable union of thin sets. Obviously, polar sets are thin, and thin sets are semipolar and finely closed. If B is polar, then sup,,! l B ( X 1 ) vanishes a.s., and this is about as negligible as a set in E can be, other than empty. In $19, we shall describe an example of a non-polar set in E", all of whose Bore1 subsets are polar. The following exercises contain some of the standard facts concerning exceptional sets and the fine topology. (10.10) EXERCISE.I f f E Sa and B := {f = GO}, then B is polar if and only if B is null. (Hint: if f(z) < 00, e - a t f ( X t ) is a P"-supermartingale.) (10.11) EXERCISE. If f , g 6 p€" and {f # g } is null, then Uaf= Uag.
(10.12) EXERCISE. If B is non-empty and finely open, then B is not null. (Be careful-B is not assumed to be in €".)
(10.13) EXERCISE. I f f , g are finely continuous and i f f = g except on a null set, then f = g everywhere. (10.14) EXERCISE.Let B be nearly optional and totally thin. Set T1 := TB,and for n 2 1, define Tn+l := T" +T109p. Show that, for some a > 0 and 13 < 1, P"exp{-aTn+'} 5 PP"exp{-aTn} for all n 2 0. Deduce that T" 7 00 a.s. as n + 00, and, in consequence, { t > 0 : X t ( w ) E B } = {T1(w),T2(u),. . .} is a.s. a discrete subset of R+.This proves that B is, in fact, optional. Show, more generally, that if B is nearly optional and semipolar, and if B' c B is in E", then B' is optional.
I: Fundamental Hypotheses
55
(10.15) EXERCISE.Let B be nearly optional for X. Fix a > 0 and < 1, and D := B f l (4% 5 p}. Prove that D is totally thin. (Hint: D is thin is because, if x E D', 5 is regular for B , hence c$%(x)= 1. Since @j(Xt) a s . right continuous, x is not regular for {& 5 p}. Observe then that, for 2 E D,&(x) 5 $%(x>.) (10.16) EXERCISE.Let B be nearly optional for X. Prove that B \ B' is semipolar. (Hint: write B \ B' = B n {q5% < 1)as u,Bn {@ 5 1- l/n}.) In particular, prove that every semipolar set B is contained in a countable union of totally thin sets, and conclude that {t : XtE B} is a s . countable.
(10.17)EXERCISE.Let B be nearly optional and finely closed for X. Show E B} is right closed in R+. That is, for a.a. that for a.a. w , {t : Xt(w) w , given any sequence t, J. t with Xt, E BVn, Xt(w) E B also. (Hint: let Ro be the set of w with 4h(Xt(w)) right continuous on R+ and such that, for every n 2 1, the set {t : Xt(w)E B,&(Xt(w))5 1 - l f n } is discrete E B for all n 2 l,lim$~(Xt,(w)) in R+. For w E Ro, ift, 1t and Xtn(w) exists. If the limit were less than 1, t, = t would hold for all sufficiently large n.) (10.18) EXERCISE.Theright topology on R+ is defined to be the topology whose open sets are countable unions of intervals of the form [a,b [ , 0 5 a 5 b 5 00. Show that a set in R+ is right closed if and only if if is closed in the right topology, and that a real function f on R+ is right continuous if and only if, for all rational a,{f 2 a} and { f 5 a} are right closed in R+. Use this to prove that i f f is finely continuous and nearly optional, then t -+ f (Xt)is a.s. right continuous. (10.19) EXERCISE.I f f is a-supermedian and f is its a-excessive regularization (4.12), then {f # f} is null. (10.20) EXERCISE.Use (10.10) and (4.10) to show that every singleton is polar for Brownian motion in Rd ( d >_ 2). (10.21) EXERCISE.Show that for the process of uniform motion to the right on R (3.131, the fine topology is just the right topology on R, every polar set is empty, the totally thin sets are the discrete subsets of R, the thin sets are those without accumulation points from the right, the semipolar sets are the countable sets in R, and the null sets are those of Lebesgue measure zero. (10.22) EXERCISE.A process X with state space E a subinterval of R is called a (conservative) regular, linear diffusion if t + Xt(w)is continuous for all w and if, for every interior point x of E , x is regular for both ]x,GO[ nE and ] - 00, x[nE. (The term conservative is used here to indicate that Xt remains in E for all t, and no killing ($11) is permitted.)
56
Markov Processes
Brownian motion in R (4.10)is the prime example of such a process. Show that, for such X, the fine topology and the usual topology of R have the same restriction to the interior of E . (Hint: if G C E and if x is an interior point of G , then show that, for some E > 0, ]x - E , X E[CG . )
+
(10.23) EXERCISE. A subset B C E is called finely perfect if B is nearly optional and B' = B. Show that a nearly optional set B is finely perfect if and only if B is finely closed and, a.s., DB = TB. Show also that for a process as in (10.221,every closed set is finely perfect. (10.24) EXERCISE.Using (10.8) and (10.18), prove that i f f E bE" is finely continuous, then Pt f is finely continuous for every t 2 0. (10.25) EXERCISE. Meyer's hypothesis (L), or the hypothesis of absolute continuity, is a simplifying hypothesis on X to the effect that there exists a finite measure on E such that for every a > 0 and every x E E , U a ( x , - ) << <( The measure E is called a reference measure for X . Processes obeying this hypothesis have a theory which is considerably simpler than the general case. Prove that if X has a reference measure E, and if (Pt) maps E into itself, then: (i) Q a > 0, Qf E pE", there exists g E pE such that Uaf = U"g; (ii) every f E S" is Bore1 measurable. In particular, E" = E ; (iii) two finely continuous functions which agree E-a.e. are necessarily identical. a ) .
(10.26) EXERCISE. Suppose that for every non-empty finely open set G E E", P x { T < ~ m} > 0 for all x E E. (This condition is satisfied if X has the following recurrence property: for all B E E", U ~ isBeither identically zero or identically infinite. See (10.38).) Show that the Section Theorem applied to the random set {t : X t E G } implies that U"(x, G ) > 0 for every a > 0 and all x E E . Show that for every x E E , := U ' ( q is a reference measure for X . (Hint: if U'(y, . ) were not absolutely continuous with respect to U'(x, -), one could choose B E E" with U'(x, B ) = 0 and U'(y, B ) > 0. Prove that, in this case, U1(x,G) = 0, where G := { z : U'(z, B ) > 0). A much finer result is proved in /WWSl], to the effect that X has a reference measure if and only if there is no uncountable disjoint collection of non-empty finely open sets in E.) < ( a )
0
)
(10.27) EXERCISE (THECOMPLETE MAXIMUM PRINCIPLE). Show that every kernel of the form U" corresponding to a right process in E satisfies the following principle: for all a > 0 and all f , g E pEe, if a U" f (x)2 U"g(z) for all x E E such that g(x) > 0, then a U"f 2 U a g everywhere (in E. (Hint: by fine continuity, a Uaf 2 Uag on the fine closure of G := { g > 0). Set T = TG,write a Uaf (x) 2 a P" e-at f (Xt) dt
+ +
+
+
+ S F
I: Fundamental Hypotheses
57
and use the strong Markov property with (10.61.) Some other potential theoretic properties of the resolvent will be give in later chapters. See in particular (36.19), (36.20) and (36.21). It will be shown later (66.7) that for every f E E", there exists g E E" such that {f # g} is null. Therefore, E" may be replaced by E" in the statement above. (10.28) EXERCISE.Let X be the right process governed by the right continuous deterministic flow (cpt) as in (8.8). Prove that X satisfies hypothesis (L) (10.25) if and only if there are only countably many distinct orbits of (cpt). For example, if E = R2 and if cpt(x, y) = (x t , y), then X may be described as uniform motion to the right on the plane. Show that X has at least one non-Bore1 excessive function. Show that the finely continuous functions on E are those whose restrictions to all horizontal lines are right continuous.
+
(10.29) EXERCISE.Suppose every point is polar for X. Show that every countable set is finely closed, and conclude that a sequence (X,) converges to x in the fine topology if and only if X, = x for all sufficiently large n. That is, the fine topology does not satisfy the first axiom of countability in this case.
(10.30) EXERCISE(APPROXIMATION OF HITTING TIMES).Given a fixed c E , prove that there exists an increasing sequence K, of compacts in B such that, a s . P", TK, 1 TB. (Hint: take t k 11 0 and note that t k DAoo(tk) 1 TA for every set A C E. Using (A5.30), choose compacts K , in B so that D~,,d?(tk)1 DBoe(tk) P"-a.s. for every k. Prove then that, Pp-as., TK, 1 TB. See $49 for a more difficult approximation by hitting times of open sets.) p and a nearly Borel B
+
(10.31) EXERCISE.Let Ici denote a class of finely closed Borel sets in E having the property that, for every set B E E , there exists an increasing sequence K , C B with K , E Ki and T K , 1 TB a s . as n + 00. (The compact sets form such a class, by (10.30).) Let f,g , h E pE", and suppose (i) PKf 5 f for every K E Ici; (ii) f + Ug 2 Uh on { h > 0). Prove that f U g 2 U h everywhere. (Hint: suppose first that h is Borel. For each K C { h > 0) with K E Ki,prove that f +Ug 2 PK(f + U S ) 2 PKUh. For a fixed x, approximate T{h>o)by TK a s . P". Finally, having completed the proof for Borel h, fix x E E and approximate a general h E pE" from below by h' E pE with U h ( x )= U h ' ( x) . )
+
(10.32) EXERCISE.Under the same conditions as in (10.31), prove that
f is supermedian in the sense of (4.23). (Hint: since f A n also satisfies (10.31i), we may assume f bounded. Fix p > 0, and set h := a(f aUa+Pf)+, g := a(f - aUa+Pf)-. Show that f + UPg 2 UPh on { h > 0},
58
Markov Processes
by computing U p ( h - g ) ( x ) for h ( z )> 0. Now apply (10.31) to Up. Finally, let p 1 0 . The result of the last two exercises is due to Dynkin /Dy65].)
(10.33) EXERCISE. Under the same conditions as in (10.31),suppose that, for every instantaneous point x E E , there is a sequence K, C E \ {x} with K, E Ki, TK, 1 0 a.s. P",and PK,f(z)7 f(x) as n --+ 00. Prove that f is excessive. (Hint: let f(x) := limt,o Pt f (x). If x is a holding point, it is almost obvious that f(x) = f(z). For x an instantaneous t < TK,} 1 point, K, as above, and t > 0, write Ptf(z) L P"{f(Xt); P"{PK,f(Xt);t_< TK,}= P"{f(X(TK,));t < T K , } . Let t -t 0 and then n --t 00 to get f 2 f . This argument is due to Pat Fitzsimmons. The result is important because it shows that the class of excessive functions is completely determined by the class of hitting operators PK,K E Xi.) (10.34) EXERCISE. Let X denote the Brownian motion on R with drift c, as discussed in (9.31). For x E R, let T, := inf{t > 0 : Xt = x}. Fix a < b, and let T := Ta A Tb. Compute formulas for P"{e-UT; T = Tb} and P5{e-"T;T = Ta},by the following method. (i) Prove that exp{X(Xt- ct) - X2t/2}is a martingale over ( 3 t ) . (ii) Show that P"T < 00 for a < x < b. For c = 0, this may be verified using the martingale property of X z - t, while for c # 0, the martingale property of Xt - ct suffices. (iii) Use the martingale in (i) to obtain
P"exp{X(XT - cT) - A2T/2}= exp{Xx}. (iv) Expand the last displayed expression to get a linear equation involving Ps{e--uT; XT = b } and P"{e-"T; XT = a } , with u = X2/2-Xc. Makinga linear substitution for X leavingu unchanged, one finds a pair of linear equations whose solutions are given by eA(z-a)+2ca - e-A(s-a)+2cs
P"{e-"T; xT = b}
=
eA(b-a)+Zca
;
- e-A(b-a)+Zcb
e A ( b - ~ ) +2 ~e-A(b-~)+2cb ~
PZ{eduT; XT = u } = eA(b-a)+Zca - e-A(b-a)+Zcb
'
Deduce from these expressions formulas for Pz{X~ = b } , P5{T I XT = b } and P"{T 1 XT = u } . (10.35) EXERCISE. Given B c E , L B ( w ):= sup{t : Xt(w) E B } (sup0 := 0) is the last exit time from B. Show by arguments similar to those at the beginning of the section that LB is 3*-measurable if B E E , and using (8.12) and (A5.2), LB is 3-measurable if B E E". The excessive function
I: Fundamental Hypotheses
59
P”{TB < m} is equal also to P”{LB caJJed transient in case LB < m almost surely.
~ B ( I C:= )
> 0). A set B
E E e is
(10.36) EXERCISE (TRANSIENCE). Prove that the following conditions on X are equivalent: (i) 3h E bpE” with U h bounded and U h > 0 on E ; (ii) 3h E bpE“, finely continuous and strictly positive, U h bounded; (iii) 3(hn)c bpE” with Uhn bounded and Uhn t 00; (iv) 3Bn t E , Bn E E“, with U l B , bounded for each n; ( v ) 3Bn 1 E with Bn a transient set in E for each n. In (i)-(iv), the condition is unchanged if “bounded” is replaced by (‘finite”. (Processes satisfying one of these equivalent conditions are called transient. See [ADRSS], [ADR69] or [Get301 for a full discussion of the results in this and the subsequent exercise. In particular, transience is equivalent to the condition that U is a proper kernel (A3). The following hints will help to establish the equivalences. (i)+(ii): Take h as in (i), let g := U’h and use the resolvent equation to see that g satisfies (ii). (i)+(iii): Let hn := nh. (iii)+(i): Let h := h , / ( 2 , b n ) where b, := max(1, llhnII, IlUhnII), and show U h > 0. (iv)+(iii): Let h, := nlg,. (ii)+(iv): Let Bn := { h > l/n}. The “finite” cases are almost the same except that we get (iii finite)+(i bounded), which completes the proof of equivalence of (i)-(iv) and their “finite” versions, from the following steps. Take h, satisfying (iiifinite), and replace it by ( h l V . . .Vh,)An to get an increasing sequence satisfying the same conditions. Let A n , k := {Uhn I k } and gn,k := lAn,khn* Then u g n , k 5 p” f(xt> dt 5 P A , , k u f ( z ) 5 k, T denoting the hitting time of An,k. Then A n , k t E a8 k -+ m and limn limk u g , , k = limn Uh, = 00. Take h := x n , k ( 2 n + k b n , k ) - 1 g n , k (bn,k := maX(1, ((gn,kl(,I(Ugn,kl() to satisfy (i). (i)+(v): Let Bn := { U h > l / n } , and note that PtUh(z) 2 P ” { t TgnoOt < co} so that letting t + 00, P ” { t Tgno8tQt} = 0, hence B, is transient. (v)+(i): Let +B,(z) := P“{Lg > 0) SO that +gn E S, Pt~$g, 1 0 as t m and +B,(z) T 1 as n + 00. Let g n , k := n(+gk - P ~ / , C $ Buse ~ ) ,(8.11) to get u g n , k 14~~ as n 00, and construct a bounded h with U h > 0 as in the preceding segment.)
SF
+
+
-+
-+
(10.37) EXERCISE.For B E E e l let $B(z) := P“{TB < m} = P”{LB > 0) E S, $g := limttooPt4g. Then ?,bg(z) = P”{Xs E B for arbitrarily large s} = P“{Lg = m}, Pt+g = $g and 4g - $g E S . (10.38) EXERCISE.Suppose ulg = oa identically whenever B E 8“ is non-void and finely open. Prove that for any B E E“, +g(x) = 1 for some x E E implies + B = 1identically. (Hint: apply (8.11) to $B-$B, observing that the gn are finely continuous, to get a sequence gn with Ug, 1 4 - $,
Markov Processes
60
and apply the hypothesis to B, := {gn > 0}, concluding that 4g = $ B . Now 4g(z) = 1 implies PZ+g(Xt) = 1, hence P"{Xt E {+g < 1)) = 0, hence Ul{+B
(i) for each B E E", U ~ isBeither identically 0 or identically infinite; (ii) if B E E" is non-void and finely open, then Ulg = DO; (iii) if B E E e is non-polar, then Ps{Tg < 00) = 1; (iv) if B E E" is non-void and finely open, then P"{TB < 0 0 ) = 1; (v) every excessive function is constant; (vi) if B E Ee is non-polar, then LB = DO almost surely. (Processes satisfying one of these equivalent conditions are called recurrent. Hints: (i)+ (ii), (iii)+ (iv) and (v)+ (iv) are obvious. (ii)+ @): A p ply (10.38). (iv)+(v): Iff E S is non-constant, choose 0 < a < b < DO and z E E so that f (z) < a and B := { f > b } is non-void. Then 4g = 1, violating a > f (z) 2 PBf (z) 2 bP"(TB < DO) = b+B(z) = b. (iv)+(i): Suppose B E E" and b := supu1B > 0. Take 0 < a < b and A := {UlB > a } . Show U l B ( z ) >_ p"s," 1B(Xs)ds + 1B(xs) ds 2 p" s," 1B(Xs) ds a so that, letting t -+ DO, U1g 2 U l g a, hence UlB = 00. (ii)+(iii): Let B E E" be non-polar so that 4~ = c for some constant c > 0 (by (iv)). Then for t > 0, c = P"(Tg < 00) = P"(Ts 5 t ) PZ(t< Tg,Tg< 00) = P"(TB t ) cP"(t < TB).Let t -+ 00 to get c(1 - c) = 0, hence c = 1. (v)=+(vi): Let B E Ee be non-polar. Then +B - $g = c for some constant c. But P ~ ( I $B$g) + 0 implies c = 0. (vi)+(i): Let B E E" and suppose Ul~(z) < 00. Then PtU1g(z) + 0 as t t 00. If G := {Ulg > a } # 0, Lg = DO a.s. implies P"(t Tgo&) = 1 for all t, hence P t U 1 g ( z ) 2 a.)
.fE~~~o,
+
+
+
+
+
(10.40) EXERCISE.Let a 2 0, and let p, u be measures on E such that pUa and uUa are a-finite. IfpU" = UP, then p = u. (This generalization of (4.21) comes horn [GG83]. Show that the resolvent equation implies that for /3 > a , pU@ and pUaU@are both a-finite, and hence pU@= uU@ for all p 2 a. Fix y > 0, y 2 a , and choose g E b€, g > 0, with pU7g = uUrg < 00. Let h := Urg E bE", h > 0 , so that hdp and hdu are finite measures. For f E & ( E ) with 0 5 f 5 1, f h is bounded and finely continuous so / 3 V + P ( f h ) -+ f h as /3 + 00. Show pUY+@(f h ) 5 /3Ur+Ph 5 h, and conclude that p ( f h ) = u(f h ) . )
11
Transformations
The existence theorems in $9 give a reasonably large class of right processes to get started with. Those examples exhibit none of the seemingly pathological behavior permitted by the definition of a right process-their semigroups map Borel functions to Borel functions and their state spaces are Lusinian. In addition, the processes are not just right continuous, but also have left limits, at least in a compactification of the state space. The first few sections of this chapter show why it is not desirable to build in these regularity hypotheses as part of the definition of a right process, and additional confirmation will re-appear later as we discuss other transformations in subsequent chapters. In $17 and $18,we shall show that, at the cost of a change of topology on E , the most general right process differs in an inessential way from a right process generated by a Ray resolvent as discussed in $9. Finally, in $19, we describe some features of right processes that are preserved under a change of realization, and in $20, we set down the final form of the hypotheses and notation under which we shall work during the remaining chapters.
62
Markov Processes
11. The Lifetime Formalisms
Suppose given a Radon space E and a semigroup ( P t ) t >of~ positive kernels 5 on ( E ,E"). The semigroup (Pt) is called subMarkovGn provided P t l ~ 1 ~A .subMarkov semigroup (Pt)may be extended to a Markov semigroup (Pt)on a larger space. Simply take an abstract point A not in E and let E A := E U {A} be the Radon space obtained by adjoining A to E as an isolated point. Define pt on (Ea,EX) by
(11.1)
Pt(z,A) :=
{
Pt(x,4,
X E E , A E E Xw i t h A c E , 1 - Pt(x,E), x E E , A = {A} EA(A), x = A.
It is trivial to verify that ( P t ) is a Markov semigroup on E A , with ptf E b & i for every f E bEi. Suppose that (pt)is a right semigroup and that (a,6,Bt, Xt, &, P.) is a right process on EA realizing (pt). It is obvious from (11.1) that A is a trap for the process. That is, FA{Xt= AVt 2 0) = 1. Let 6 := inf{t > 0 : X t = A}. By the SMP, Xt= A for all t 2 6, almost surely. It is reasonable therefore to call (Pt)a right semigroup provided (pt) is a right semigroup in the sense defined in Chapter I. In many respects, the process X constructed above is interesting only while it is in E . Indeed, if (Pt)is the object of interest, adjunction of A is quite artificial. In this situation one may simplify notation and calculations by adopting the following conventions. Think of the point A as a cemetery (coffin, dead point, ...) and call the optional time 6 the lifetime of X. The role played by the augmented semigroup P t is de-emphasized by making the convention that every function f on E is automatically extended to EA by setting f ( A ) := 0. By (ll.l),Ptf then means exactly the same thing as Pt f . Define Xt to be the same as Xt on 0 and let P" := P" for x E E . It is easy to check that the natural a-algebras (Ft) defined by (Xt) and the measures P"(x E E) are in fact identical to the ( F t ) . Moreover, with the convention that sets f(A) := 0 for all f E bE", it is the case that F' is generated by products fi(Xt,) ...fn( Xt,) with 0 5 tl < ... < t,, f1, ...,fn E bE", together with the constant function 151. See (11.14). The process (0,8,&,Xt, Ot, P")is called a right process on E with lifetime and transition semigroup (Pt). Observe that the conventions are in perfect accordance with the strong Markov property, which asserts that for T an optional time, (11.2)
P"{f(Xt)o~T1{T
11: Zlansformations
63
In fact, if one makes the further convention X , := A, the term l{T<w} can be dropped entirely from (11.2). This convention is useful even when C = 00 as., and it will be used henceforward. In the same spirit, it is useful to adjoin to R a dead path [A] with Xt([A]) := A for all t 2 0. Define 0,w := [A] for w E a, and make the convention F ( [ A ] )= 0 for F E F. (11.3) DEFINITION.A family (kt)o
(11.5) (11.6)
k,Okt = k s A t ; W k , + t w ) = k,(&w).
Not all sample spaces R admit killing operators. It is, however, obvious that if R is the space of all right continuous maps of R+ into E A , then R admits killing operators defined in the obvious way. The same is true of the space 0~ of right continuous maps of R+ into E A admitting A as a where <(w):= inf{t : trap. That is, w E RA implies w ( r ) = A V r 2 <(a), X t ( w ) = A } denotes the lifetime of X. The following properties of killing operators are practically obvious: kt E V 8 < t F f / 3 ' for every a-algebra E' c E"; (11.7) for all H E 3', ( t , w ) -+ H ( k t w ) is predictable over (F:); (11.8) C o k t = A t identically; (11.9) (11.10) for every H E Vs
<
Let (kt) be a family of killing operators for X . (11.11) PROPOSITION. (i) For every F E bF;, Fok, = F for all s > t. (ii) If (Zt)t>O is preand if Z vanishes on I]C, 00 [I, then for each s 2 0, dictable relative to (F;), Zt(k,w) = Zt(w)lp,l(t); (iii) if T is an optional time over (F;+)with T 5 C, then T o k , = T A s. PROOF: (i) follows from (11.10) in case F E b e . Given F E bF,* and s > 0, for an arbitrary probability Q on ( R , 7 ' ) , choose F I , F2 E b e with F1 5 F 5 F2 and k,Q(F2 - F l ) = 0. Then Flak, 5 Fok, 5 FzOk,, and as Fjolc, = Fj, ( j = 1 , 2 ) , we obtain QIF - Fok,l = 0. AS Q is arbitrary, this proves (i). In proving (ii), we may assume, by monotone classes, that Z is left continuous. For each t < s, (i) gives Ztok, = Z t . Left continuity implies that Ztok, = Zt for all t 5 s. If t > s, Z t ( k , w ) = 0
Markov Processes
64
since t > C(k,w). This proves the (ii), and (iii) follows by applying (ii) to 2 := 1 ~ 0 , T ~ . The systematic use of killing operators was initiated by AzCma [Az73]. They have become indispensable in a number of situations. It is necessary to exercise more care when working with killing operators than with shift operators, for it may be the case that H E F with H = 0 a s . but, a.s., Hokt # 0. For example, let X be a process with infinite lifetime, and take H = 1 { ~ < ~ Thus } . one cannot pass to the augmented in (11.7), (11.8), (11.10) or (11.11). See however (11.15) a-algebras (3t) and (32.13). Given an arbitrary sample space R for a refined right process X , one may lift X into a larger space fi that does have killing operators. This is the point of the following construction. Let fi := [0,00] x R and let
X t ( r ,w):=
X t ( w ) if t < T , {A ift>r.
<
Note that the lifetime for X satisfies <(r,w) = C(w) A r. On the the product a-algebra Go := B[O,001 @ plet, P” be the image of P” under the map $(w) := ( o o , ~ ) ,which clearly belongs to F’/co.Note that Xta$ = X t . Define
e t ( r , w ):= ( ( T - t)+,etw)(00 - 00 := 0);
&(T,w)
:= (T A t , W ) .
:= a{f(Xs) : f E b&”,s 5 t}. Then b e = For each t 2 0, let {zo$ : F E b e } . ( For F E b e , define F ( T , w ):= F ( w ) so that Fo$ = F . The relation 3 is then evident, and the reverse inclusion uses a simple monotone class argument.) By definition, P5(F)= P ( E o $ ) for every F E b p m . The simple Markov property of X relative to with transition semigroup (Pt) follows at once from this identity. Given f E Sa, let N := { ( r , ~:)t + f ( X t ( r , w ) )is not r.c.}, Nt := {w : s -, f(X,(w)) is not r.c. at some 8 < t } . By the refined right hypothesis, Nt is null in so for each p, there exists Ft E b p c such that l ~ 5, Ft and PPF, = 0. Then Ft(r,w):= Ft(w) is Pp-null, fl = u t , Q + { ( r , w ) : C(r,w) > t,w E N t } , and so
(e)
c,
t€Q+
Hence X is a right process with the same semigroup as X ,and the killing operators for X on fi.
are
(11.14) EXERCISE. Show that i f X is a right process with lifetime C on E , then F” is generated by the multiplicative class composed of In and
II: Transformations
65
products fi(Xt,) ...fn( X i , ) with 0 5 tl < ... < tn and f i ,...,fn E bE”. (Lifetime conventions are being used. Observe first that for any t 2 0, 1{Cit} = 1 - IE(Xt). For .f E E L , f(xt)= ( f l E ) ( X t ) + .f(A)l{c
e. Show that kt E Vg
Let (a,9,Gt, Xt, Ot,P”)be a right process on E with lifetime C and transition semigroup (Pt). (12.1) DEFINITION. An optional time T over (Ft) is a weak terminal time (resp., terminal time) for (Q, 6 ,Gt, X,,Ot, P”)provided (12.2) (resp., (12.3)) is satisfied: (12.2) V t 2 0, t + Toet = T a.s. on {t < T}; (12.3) for every (Gt)-optional time R , R + T O e R = T a s . on { R < T } . Set (12.4)
A := { w : t + T(Otw) = T f w ) for all t < T(w)}.
Then T is an almost perfect terminal time in case A‘ E hf,perfect if Ac is empty. A weak terminal time T is exact provided (12.5)
tn 11 0 implies that, a s , t,
+ To&,
1T.
In case ( G t ) is a strong Markov filtration for X (6.10), it is possible to weaken each of the above to require only that T be optional over (Gt). In this case, we use the term @,)-terminal time. The terminology here is a little different from the classical ones [BG68, 1111 where what we call a weak terminal time (resp., terminal time) is called a terminal time (resp., strong terminal time). We shall prove in $24 that, given a weak exact terminal time T, there exists a perfect terminal time T a.s. equal to T . Of course, a perfect terminal time satisfies (12.3), so this result will imply that a weak exact terminal time is a terminal time. The prime examples of perfect terminal times are the hitting times and debuts of nearly optional subsets of E , which were defined in $10. Recall expecially the discussion preceding (10.5). The calculations (10.3) and (10.4) show that if B is a nearly optional set, its hitting time TB is a perfect exact terminal time but its debut Dg is a perfect terminal time which is not in general exact. The intuitive meaning of the terminal time
66
Markov Processes
condition is that a terminal time represents the first time the path exhibits some particular geometric behavior. To get the flavor of this description, consider the following examples. (In every case, inf0 := 00, as usual.) 0 T ( w ) := inf{t > 0 : X t - ( w ) exists and Xt-(w) E B } , B E E; 0 T ( w ):= inf{t > 0 : X t - ( w ) does not exist in E } ; 0 T ( w ):= inf{t > 0 : f ( X t ) - ( : = lim,ttt f ( X , ) ) E B } , f E S",B E
t3+; 0 0 0
s,"
T ( w ):= inf{t > 0 : f ( X , ) d u > 0 } , f E pE; T ( w ) := inf{t > 0 : ( X t - ( w ) , X t ( w ) )E I?}, I' Borel in E x E ; T ( w ) := inf{t > 0 : u + f(X,(w)) has derivative 0 at u = t } , f E c d (E) ;
T ( w ) := inf{t > 0 : X,(w) E B for uncountably many u < t } , B nearly optional for X. The last example above is called the penetration time of B. The fact IVthat T is an optional time over ( B t ) comes in this case from [DM75, 112)]. In all other cases, optionality comes from measurability of debuts (A5.1). The above examples are all almost perfect exact terminal times. Perfection is attained in the first, second, fourth, fifth and sixth examples. In the other cases, there is a possible null exceptional set on which the process involved may not even be defined. Note that if the function f in the fourth example were only in pEU, then s," f ( X , ) du might fail to be defined on some null set, and the prescription would give only an almost perfect terminal time. We shall encounter other types of terminal times in subsequent chapters. If the underlying process is sufficiently regular-for example, if X has a dual [BG68, VI]-it can be shown that every exact terminal time is a.s. enters some Borel set I? c E x E. See equal to the first time (Xt-,Xt) [Sh73].In particular, if X is Brownian motion in Rd,every exact terminal time is a s . equal to the hitting time of some Borel set. Check out (12.35) for an example in which the terminal time is definitely not a hitting time X). for (X-, The above examples illustrate terminal times for the natural filtration only. Here is an example of a (Bt)-terminal time which is not an (&)-terminal time. Let X = ( R , F , F t , X t , B t , P z )be a right process on E , possibly having a lifetime C. Fix X(dt) := ,t?e-ptdt, a probability on (R++,B(R++)). Let fi := R++ x 52, 6 := B(R++)8 3 and 6t := B(R++) @ 3t. Let := X @ P" and X t ( u , w ) := X t ( w ) . By (11.16), X = ( h , d , & , & , X t , * ) is a right process. Let S ( u , w ) := u. Then S is a (Gt)-terrninal time, and under every Pz,S is independent of 0
*
X.
11: Ransformations
67
Let T be a weak (gt)-terminal time. Then if 0 5 (12.6) PROPOSITION. s 5 t, s T O O , 5 t To& almost surely. If T is exact and t, 1 s, then, a.s., t, Toetn 1 s +TOO,.
+ +
+
PROOF:If 0 5 s 5 t then for all z E E , using the Markov property, P”{s
+
T O O ,
> t +To&}
> t - s +T o O t - , ] ~ O s } - P” PX(’) {T > t - s + T 4 - , } . = P”{[T
However, for every T 2 0, a s . , T+To& 2 T , for if T > T , then r+ToO,. = T a s . , and if T 5 T , then a.s., T TOO,.2 T 2 T. This proves the first assertion. If t , 1 s, t , Toot,, is a s . decreasing and, as., t , Toet,, 2 s TOO,. Using the same calculation as above,
+
+
P”{limt, n
+ To&, > s + T O O , }
+
+
= P”PXB{limt, - s n
=0
+ ToOt,--s > T }
by (12.5).
It will be shown in $55 that, given a weak exact terminal time T, there exists a perfect terminal time T such that T = T a s . , and so that t -+ is right continuous and increasing for 8.8. w. t
+
(12.7) DEFINITION.Let T be a weak terminal time for X . A point is regular for T in case z E reg(T) := {z : P”{T = 0) = 1).
fE
By the Blumenthal 0-1 law (3.11), since {T = 0) E 30, P“{T = 0) is equal to either 0 or 1 for each z E E. If B is a nearly optional set with hitting time TB, then reg(TB) = B“, the set of regular points for B defined in (10.5). By analogy with definitions given for hitting times in (10.9), a weak terminal time T is called thin in case reg(T) is empty, and T is called totally thin provided: (12.8) 3a > 0,p < 1 with Px(T)e-aT < /3 a.s. on {T < m}. Let T be a weak terminal time. Define T n recursively by T1 := T, Tn+’ := Tn + T o O p (n 2 1). It is easy to see from (6.11) that each Tn is an optional time over (Ft). (Note that if we had only assumed terminal times to be optional for (Gt), the T, would not be optional times for (Gt) unless (Gt) were assumed to be a strong Markov filtration in the sense of (6.10).) Given that T satisfies (12.8), it is easy to check (cf (10.14)) that Pxe-aTn 5 for all x E El and consequently Tn m a s . as n -+ 00. In particular, a totally thin terminal time is thin. (12.9) PROPOSITION. Let T be a weak terminal time. Then reg(T) E E“. Let ci > 0 and let 4(z) := P+e-aT, Then 4 is a-super-mean-valued (4.91,
Markov Processes
68
and 4 E S" if and only if T is exact. In this latter case, reg(T) = {z : #(a)= 1) is in Ee.
PROOF:Because of (3.4), reg(") := {z : Pxl{T=O} = 1) is in E". By (12.6), if t 2 0, then t + T o & 2 T as., and so
e - ~ t p t + ( z=) pze-"te--aT08t = pze-a(t+To&) < - pze-aTThus 4 is a-super-mean-valued. Suppose next that T is exact. If t, 11 0, t , + Toot, 1 T as., and this implies that e-"tnPt,4(a) t 4(z), so that $ is a-excessive. Conversely, if 4 is a-excessive, fix a sequence t, 14 0 and let S := liminf (t, + Toet,). By (12.6), the limit exists and, as., S 2 T. Because of bounded convergence, pze-aS
= lim pze-Q(ta+TO@te)- 1'im e-ata&,+(z) = #(z) = Pxe-QT.
It follows that S = T as., so T is exact. (12.10) EXERCISE. Given a weak (&)-terminal time D and t , 11 0, let T := liminf,,,(t, +Do&,). Use part of the proof above to show that: (a) T is a weak (Bt)-terminal time; (b) T is exact; (c) T 2 D a.s.; (d) Pze-OT is the a-excessive regularization of Pze-OD, as defined in (4.10); (e) T = D a.s. on {D > 0). Call T an exact regularization of D. In general, it depends on the sequence {tn), but two exact regularizations of D a s . agree. Part (e) shows that if D is thin, then D is exact. (12.11) DEFINITION. Let R E pG and let Q 2 0. The kernel is defined by
P; f (z) := P"{e-*Rf(XR)),
Ps on (E,E")
f E pE".
As usual, when a = 0 we write just PR for P i . In the notation above, a function f E pEU is a-excessive if and only if PF f t f as t 1 0. If R = TB, the hitting time of a nearly optional set B C El one writes Pg in place of P& . The kernel Pg is called the a-order hitting operator for B and PB is called just the hitting operator for B. For z E El PB(x, - ) gives the distribution of X ( T B )under P" and PBQ(a, .) gives the joint distribution of TB (via a Laplace transform) and ~ ( T B According ). to (10.6), Pg(z, - ) is carried by the fine closure B U ' B of B. (12.12) EXERCISE. For a 2 0. Let R be an F-measurable random time and let T be an optional time over (&). Show that PgP; = P$+R08T, where PF P i means the usual composition of kernels: P$ Pg f := PTQ(Pi$f ) .
We obtain also an important identity when we compose Pi (R an optional time) with the a-potential operator U". This is just another simple application of the SMP on edaR (J," e-qtg(Xt)dt)oOR.
11: Transformations
69
(12.13) EXERCISE.Let R be a (Gi)-optional time and g E pE". Then
LW
PgU"g(x) = P"
e-Otg(Xt) dt.
The identity in (12.13) and (4.15vi) have an important consequence. (12.14) EXERCISE.For every optional time R over (&) and every a 2 0, i f f E S", then P s f 5 f. ( The case a = 0 requires care. See the next proof.) (12.15) PROPOSITION. For T a weak ((&)-terminal time, a 2 0 and f E
Sa,P$ f is a-super-mean-valued, and PTQ f E S" if T is exact. PROOF:To begin with, assume that a > 0 and that f = Uag with g E bpE". By (12.12) and (12.13),
P?P$Uag(x) = pZTo6't U Q g ( z ) = P"
Iw
e-"'g(Xs) ds
t+Tdt
5 P"
Lm
e-"'g(XS) ds
= P,aU"g(x)
since, by (12.6), t+ToOt >_ T almost surely. Therefore, P;U"g is a-supermean-valued. Assume now that T is exact. In view of (12.5) and the second line in the identities above, we conclude that P$Uag is a-excessive. Using now (4.15vi) and (4.15i), we find P$f E Sa for f E Sa,provided a > 0. Suppose now that a = 0. Observe that f E S implies f E Sa for all a > 0 and so P$f E S" for all a > 0. Because Pzf t P T as ~ a + 0, it follows that PT f E S" for all a > 0, which implies PT f E s. (12.16) THEOREM. Let T be a weak (&)-terminal time and a 2 0. Let V" be the kernel defined on ( E ,E") by rT
(12.17) Then, for Q
( 12.18)
V" f (z) := P"
> 0 and f
J,
e-as
E bpE", one has
U"f
f ( X , ) ds,
f E pE"
Dynkin's formula
-V"f = P $ U " f .
Hence, if T is exact, V"f E bSff - bS" for a
> 0, f
E bpE".
PROOF:For a > 0 and f E bpE", (12.18) is just a restatement of (12.13), and the last assertion is a consequence of (12.15).
70
Markov Processes Given a weak (Gt)-terminal time T , define kernels Qt on (IT,&") by Qtf(z) := P"{f(Xt);
(12.19)
t
f E bE".
Then (Qt) forms a subMarkov semigroup on ( E ,E " ) , for if f E beu,
+
Qt+sf(z)= P"{f(Xt+a); t
T}
< T ,s < T O O , } = P"{PXt{(f(Xs); s < T}; t < T } = Px{Qsf(Xt); t < T }
= Px{ f (Xs)oO,; t
= Qt(Qsf)(z>. The resolvent corresponding to (Qt) is the family (V") defined by (12.17). This shows in particular that (V") is a resolvent on (E,&"). In [BG68], a (weak) terminal time T was defined to be exact if the resolvent V" defined by (12.17) was exactly subordinate to U",meaning that for all a > 0 and all f E bpE", U" f - V"f E S". (12.20) EXERCISE. Show that a weak (&)-terminal time T is exact if and only if (V") is exactly subordinate to (U").(Hint: (12.16) gives one and direction. For the other, assume ( V " ) exactly subordinate to (U"), infer that Pze--crT= u"1E - v"1E is a-excessive. Then use [12.10).)
Now fix an almost perfect (&)-terminal time T with T 5 C. We may assume that Q contains a distinguished point [A] satisfying Xt([A]) = A for all t 2 0. See (11.3). If we extend T to vanish at [A], the extended T remains an almost perfect (&&)-terminaltime. By the process (X,T) obtained by killing X at time T, we mean the process X = (fi, 0, Ct, Bt, Pz) constructed as follows:
(iii)
Gt
:;"
is the trace of Et on
(iv) ~ t ( w:= )
{
(v) for z E E , +is
w
E
fi; h, t < T ( w ) ,
otherwise; the trace of Px on 8, and
PA := €[A].
Since Cl \ h is null for all P P , Px is just the restriction of P" to fi. It is an easy consequence of (12.4) that Xt.8, = Xt+a identically on h. The filtration (6t) is in general larger than the natural filtration for X . It is obvious that S E p c is an optional time for (6,) if and only if S is the restriction to h of an optional time s for (Et).
11: Tkansformations
71
It is evident from the construction above that if 0 5 tl . . ,fn E bE", then
<
< tn and
fl,.
(12.22)
P,z{fl(Xt,)...fn(Xtn)} = p ~ t ( f l ( x t , ) . . . f n ( x , , > tn; < T } .
The lifetime conventions of 511 are essential from now on. (12.23) THEOREM. Let F := {x E E : P"{T = 0) = 0}, the set of irregular points for the almost perfect (&)-terminal time T 5 C. (i) If F is nearly optional for X , then for every initial law p, Pp{Xt E F for all t < T } = 1. In this case, the restriction of X to F is a right continuous strong Markov process with state space F and lifetime T whose semigroup and resolvent are respectively the restrictions of ( Q t ) (12.19) and (V")(12.1 7) to F . (ii) If, for all f E C d ( E ) , V a f is nearly optional relative to X , then F is nearly optional, and the restriction of X to F is a right process with lifetime T .
PROOF:Let F be nearly optional for X . Then the process 1 ~ ( X tis) nearly optional. In order to prove the first assertion of (i), it is enough to show f F, R < T } = 0. that if R is any optional time over (Fr)then P ~ { X R (We are using the section theorem here.) By definition of F ,
iFe(xR) = P ~ ( ~ ) ={ o} T = Pp{ToeR = o I F ~ } . Since T = R
+ ToOR as. on { R < T } ,we have P@{XR!$ F, R < T } = Pp{ToflR= 0, R < T } = Pp{T = R, R < T } = 0.
For the other assertions of (i), note that if R is an arbitrary optional time over (Gt) and if t 2 0 and f E bE", then for all 2 E F ,
PZ{f(Xt)o&}
Pz{f(xt)oeR; +
= t R < T ,R < T } = P"{f(Xt)OOR;t < To8R,R < T } .
By the SMP, the last displayed expression is equal to
P ~ { P ~ ( ~ ) { R~ <( Tx }~=) P; " { Q ~ ~ ( X R ) ) . Observing that Q , ~ E \ F ( z )= P " { X t E E \ F , t < T} = 0, the remaining assertions in (i) are immediate. Turning to (ii), we assume that for all f E C d ( E ) , Vlf is nearly optional relative t o X . Then F = { V 1 l > 0) is nearly optional relative to X so the assertions of (i) hold. Since Va(x,. ) is carried by F for all x E F , V * f l ~ is nearly optional for the process X , for all f E C d ( F ) . Thus the restriction of X to F satisfies (7.4ii) and is therefore a right process.
Markov Processes
72
(12.24) COROLLARY. Under the conditions of [12.23), the process X = ( X , T ) restricted to F := {z E E : Pz{T = 0) = 0) is a right process with lifetime T under either (i) T is exact, or (ii) T is the debut of a nearly optional set in E.
PROOF:If T is exact, (12.16) implies that V a f is nearly optional relative to X for all f E bE", so the condition of (12.23ii) is fulfilled. If T is the debut of a nearly optional set B C E , then the resolvent pa for ( X , T B ) satisfies V"f(z) = v'"f(z)l(BUBr)C(XI. The latter function is nearly optional since
B U BT is nearly optional by (10.6).
v"f
is nearly optional and
(12.25) REMARK. We shall prove later (58.11)the stronger result that, just assuming F nearly optional relative to X , it follows that (X, T) restricted to F is a right process. (12.26) EXERCISE. Assume that SZ has killing operators (kt)oCt<,, and Jet T 5 C be an almost perfect (gt)-terrninal time for X. Show that if one For assumes that T is 3*-measurable, then 9 : w --t IEqW)(w)is in 3*/3*. I E E , define Pz to be the image of Pz under 9.Then ( X , ) under Pz is a copy of ( X ,T ) under P5. (12.27) DEFINITION.Let X = ( R , F , 3 t , X t , 0 t , P 5 ) be a right process with state space E . Then F C E is absorbing (resp., quasi-absorbing) for X in case F E E", and for every initial law p carried by F, { X i E F for all t 2 0) has full Pp-measure (resp., full Pp-outer measure). (12.28) EXERCISE. For h E S",{ h < 00) and { h = 0) are absorbing.
In 319, we shall define what it means for one right process to be an extension of another. It turns out that the inverse operation of restriction defined following (12.29) is more naturally connected with quasi-absorbing sets than absorbing sets. (12.29) PROPOSITION. Let F be quasi-absorbing for the right process X on E with semigroup (Pt),and Jet (Qt) be the restriction of (Pt) to F . Then iff : F --t [0, 001 is a-excessive relative to (Q,), there exists an a-excessive function f for (P,) such that f l = ~ f.
PROOF:The condition that F be quasi-absorbing for X obviously implies that Pt(z,F) = 1 for all z E F , so ( Q t ) is a Markov semigroup on F . Define g on E by g(z) := f(z)~ F ( I +00 ) ~ E \ F ( I ) , with, as usual, 0.00 := 0. Then g is a-super-mean-valued for (Pt),for if z E F , e-atPtg(z) = e-'"'Qtf(X)
5 f(z) = gb),
11: Tkansformations
73
while for z E E \ F , e-"tPtg(z) 5 g(z) = 00. Take f to be the a-excessive regularization (4.12) of g relative to (Pt). By the construction of f in (4.12), f(z) = f(z)for all z E F . Suppose now that X = (R, 8 ,Gt, X t , 8t, P") is a right process with state space E , and let F C E be quasi-absorbing for X. By the restriction of X to F we mean the process X = (fi, &, X t , i t , P'")defined as follows. First, d := { w E R : X t ( w ) E F for all t 2 0) and for w E d and t 2 0, Xt(u):= X t ( w ) . Note that &fi c fi, and let 8, be the restriction of Bt to fi. For any initial law p carried by F , P p is the trace of Pp on 8, and and c t are defined as the traces of 6, 8, respectively on fi.
e,
<
(12.30) THEOREM. Under the above conditions, the process X is a right process on F whose semigroup (Qt)is the restriction of ( P t ) to F .
PROOF:We remark first that if F were nearly optional, the result would be an immediate consequence of (12.23). In the general case, we observe first that since !? has Pp-outer measure one whenever p is carried by F , the discussion concerning the trace in (A1.6) applies to prove that, for f E bE and G E c t , there exists G E & ! with G = G n fi and Pp{f(X't+dG) = Pr{f(Xt+,)G). The simple Markov property of X follows. If f is a-excessive for X , let f be a-excessive for X with f l = ~ f, as in (12.29). Then, letting RC denote the space of right continuous maps of R+ into E ,
{WEfi:t~f(xt(w))ERC}=~n{(wER:t-f(xt(w))ERc}. Since Pp{t 7f ( X t ) E Rc) = 1, it follows by the very definition of P p that t + f ( X t ) is Pp-a.s. right continuous. Thus X is a right process. Let F c E with E \ F polar. Then F is nearly optional and absorbing. The process restricted to F is also called the process obtained by deleting the polar set E \ F . This process is then a right process. (12.31) EXERCISE.Use (12.29) to show that if E \ F is polar, then there is a 1-1 correspondence between the a-excessive functions for X and X .
Quasi-absorption leads to an extension of the notion of a polar set. (12.32) DEFINITION. Let X be a right process on E. Then a subset B of E is quasi-polar provided B E E" and, for every initial law p on E , the Pp-outer measure of { w : Xt(w)4 B for all t > 0) is equal to one. (12.33) EXERCISE. Show that if B is quasi-polar for X , then F := E is quasi-absorbing. Show that (12.31) holds in this setting.
\B
74
Markov Processes
Because we insisted that a terminal time be an optional time for the natural filtration (Ft),Theorem (12.23) does not cover all cmes of interest in practice. The following example is especially useful.
(12.34)EXERCISE.Let X = (s1,0,9tlXt,8t, P") be a right process on E , possibly having a lifetime C. Fix p > 0 and let XI0 denote the p-subprocess 9, for X , defined in the following way. Let A := R++ x R, 6 := B(R++)18 and let X(du) := pe-PUdu on (R++,B(R++)).Then for 2 E E take Pz := A 8 P", Xt(u, w ) := X t ( w ) , and set
Let S(u,w):= u. Then, under every Px,S is independent of X, and S has density Pe-0". One may then describe XI0 as X (or X) killed at S. Show that Xp is a right process with lifetime SAC, semigroup Pt = Pf := e-fltPt and resolvent U a := U"+s. Show that a function f on E is a-excessive for XI0 if and only iff is ( a ,8)-excessive for X. Observe that, with the shift operators bt(u,w ) := ( t +u,Btw) on A, one has t+ Soit = S A t identically on fl, but S is not in general an optional time over the natural filtration for X on fl. (Hint: recall (6.24).)
+
(12.35) EXERCISE. Let X be deterministic motion with uniform velocity 1 to the right on the state space E pictured below. Let F denote the open lower left leg of E, and
T ( w ) :=
~up{t> 0 : X , ( W ) E F } ,
Xo
E F,
otherwise.
Show that T is a predictable, exact terminal time, but T is not a hitting time for either X or (X-,X).
Figure (12.36)
11: Ransformations
75
13. Mappings of the State Space Let X = (n,3, Ft,Xt,Bt, P") be a right process with state space E and transition semigroup (Pt). Let (El&)be another Radon space, and let Q : E + E satisfying: (13.1) (13.2)
Q E €"/€" and Q ( E ) = E ;
for all f E b&" and all t 3 0, there exists g E b&" (depending on f and t ) such that P t ( f o Q ) = 9.11; Q(Xt)is a s . right continuous in E. (13.3) (See (13.6) below for another, more easily verifiable, form of (13.2).) We shall show that yt := + ( X t ) is a right process in E . It is an obvious consequence of (13.1) and (13.2) that the function g in (13.2) is uniquely determined. Let Qtf := g. The map f + Qtf is linear and positive, Q t l g = 18,and Qt respects bounded monotone convergence. Therefore, each Qt is a probability kernel on ( k ,EU). Simple manipulations using the identity
Let fl := { w E s2 : show that Qt is a Markov semigroup on (k,€"). t + $(Xt(w))*is right continuous}. By (13.3), P"(fl) = 1 for all 2 E E. For all w E R, t 4 Y,(w) is right continuous on R+. If w E fl and t 2 0, Btw E h also, and YsoBt = Ys+tidentically on h. We may therefore imagine that R is replaced by fl in the original definition of X. Thus if 3" := a{f(yt) : t 1 0 , f E b&"}, then fuc P and a simple calculation using (13.4) proves that if Q(zl) = Q ( z 2 ) = y E E , then PZ1and Pzzagree on F". We denote their common restriction to 9 by P Y . Now let 3 and Tt be the usual completions relative to the family P Y (y E k ) .
(a,$,
(13.5) THEOREM. The process Y = $t, yt, Bt, P Y ) is a right process on the state space E with transition semigroup (Qt). PROOF: It suffices to prove that Y has the simple Markov property, and that for every a > 0 and g E bp€", the function
Vag(y) :=
/
00
e-at Q t g b ) dt
0
has a right continuous composition with yt. The simple Markov property is a trivial consequence of (13.4) applied to calculating
Markov Processes
76
The second property comes from (13.4) again, since Ua(go$) = (Vag)o$ is a s . right continuous along the trajectories of X. (13.6) REMARKS.In practice, the hypothesis (13.2) is awkward to verify because it involves the entire class of universally measurable functions. If one assumes (13.2) only for f E C d ( h ) , then by the MCT, (13.2) holds for all f E b&?. One may then construct Qt as a kernel from (fi,,!?") to (fi,,!?)satisfying (13.4) for all f E bt?. Then, by an elementary completion argument, (13.4) holds for all f E bt". That is, (13.2) is satisfied. (13.7) COROLLARY. Let $ : E + E be injective and satisfy (13.1) and (13.3). Suppose E C $-l(€). Then $(Xt)is a right process on k.
PROOF: We show that (13.2) is satisfied. Since $ is injective, (13.2) is satisfied if we show that the class { f o $ : f E b€"} is exactly b P . This reduces to showing that for A E E", $ ( A ) f ,!?.. Apply Theorem (A2.8)the condition (A2.9) follows from E" C $-'(t?") and the injectivity of $. (13.8) REMARK.By Lusin's theorem (A2.6), E" C $-l(&) if E is a Lusin space and $ is an injective Borel map of E onto E. Let X be Brownian motion on Rd(d 2 1) and $(z)= .1.1 The resulting process @(X)on [0, oo[is called a Bessel diffusion of dimension d. Conditions (13.1-3) are easily verified using (13.6). Other familiar applications to Brownian motion on the line are (i) $(z):= eix, which produces Brownian motion on the circle; (ii) $(z)the periodic extension with period 2 of @(z) := L l[o,l[(z) (2 - z) 1[1,2[(z). The last example gives Brownian motion on [0,1] reflected at 0 and 1.
+
Using (4.10), obtain the following formulas for the a(13.9) EXERCISE. potential kernels of the following processes: (i) X (= Y a Brownian motion on R1) is reflecting Brownian motion on R+,and dy is Lebesgue measure on R+:
1x1,
(ii) X is Brownian motion on the unit circle: i f f is bounded and Borel on the unit circle {eu : 0 5 8 < 27r}, and dt is Lebesgue measure on [0,27r], then
U*f(e")
= / 2 x f(ei(8-t))P ( t ) dt,
1 'cosh ((t - ~ ) / d % ) v"(t) = 6 sinh(rl6) '
77
11: Transformations
(13.10) EXERCISE.Use (13.4) to show that the semigroup ( Q t ) for the reflecting Brownian motion on R+ is given by Qt(Z,
dy) = (2Kt)-1/2[e-(Y--s)2/2t
+ e-(!4+1)2/2t
I dY
where dy is Lebesgue measure on R+. (13.11) EXERCISE.Let Y denote the reffected Brownian motion of (Em), constructed as 1x1, with X = ( R , 3 , F t , X t , B t , P ” a) standard Brownian motion. It is clear that (Ft)is larger than the (augmented) natural filtration for Y . Prove that (Ft)is not a Markov filtration for Y . (Hint: is P”{X,oOt > 0 1 Ft} a function of Y,?)
14. Concatenation of Processes Let X 1 be a right process with lifetime C on E l , and let X 2 be a right process on E2, the spaces El and E2 being imagined disjoint. For j = 1 , 2 , let [P!) denote the semigroup for Xj . We wish to construct a right process X which embodies the following intuitive description: a particle starting in E2 moves according to the law of evolution of X 2 , and a particle starting in El evolves as X 1 until its lifetime C,jumping then into Ez, after which it evolves as X 2 . The resulting process will be called the concatenation of X1and X 2 . The state space E for X will be the topological union of El and E2. That is, E = El U E2, the topology of Ej being identical to the relative topology on Ej inherited from E , and E l , E2 are both open in E. That E is Radonian is obvious. Since the injection mapping i of Ej into E is a homeomorphism of Ej and iEj, Ej = iEj is a universally measurable subset of E , in view of (A2.11). More formally, let X j = (a,, Fj,F:,X i , O i , P s ) for j = 1,2. We may assume, as discussed in (11.3), that 01 contains a distinguished point [ A ’ ] such that X , l ( [ A 1 ] )= A’ for all t 2 0. We then set R := R1 x R2, and define maps Xt : R + E = El U E2 by
Define shift operators dt on R by
Using the fact that (4, = ([ - s)+ identically, one checks the identity
xSoet= Xs+t on a. Let F := F18 F 2 .
Markov Processes
78
The time at which X first enters E2 will be denoted by R, so that
R(w1,w2) = inf{t : Xt(w1,w2) E E2} = < ( w l ) . In order to define measures P" on (52,3) making Xt have the prescribed behavior, we must describe a mechanism to realize the transfer from El into E2. (14.2) DEFINITION.Let Y be a right process and let T be a terminal time for Y. The left germ field 3 T - I for Y at T consists of all 3 T - measurable random variables H satisfying, for all t 2 0, (14.3)
Hoot
= H a.s. on { t < T}.
Recall (6.19) that H f 3 ~ means that there exists a process (Zt), predictable over ( 3 t ) , such that H l { ~ < , } = ZT~{T<,). (See (23.5), (23.8) and (31.14) for further discussion.) For example, if T is finite and Y has a left limit in E at T , then YT- is 3 ~ measurable, and the terminal time property of T shows that YT- is F"[~-]-measurable. In case (Y,) has lifetime 5, then 3 ~ = 3~= 3, for if f E E" and t 2 0, f(Y,) = f(Y,)l{t
3kc-])
(14.4)
Kf(0Zwl) = K f ( w l ) a.s. on { t
< <(wl)},for each t 10.
Measures P" (z E E ) are then defined on ( 0 , F )by the rules: (14.5) for x E E2, P" is carried by {[All} x 522 and it is the image of P; under w 2 + ([A1],w2); for z E E l , P" is defined by the formula (14.6)
P"H :=
J
J I J
Pf(dwl) K(w , d y )
Pg ( O L ~ ) H ( U ~ , WH ~E) b, 3 .
11: Transformations
79
(The symbol K P&g means K(w', dy)P&(ul) g(y) as a function of w'.) From this point on, we write Pif in place of P , ' ( f l ~if~no ) confusion is possible. From (14.7) we see, setting t = 0, that P"f(X0) = f f x ) for x E E l , and obviously the same is true for x E E2. For t 2 0 and f E b&" we let Ptf(x) denote the right side of (14.7). Now let 3 t be the usual filtration for X constructed from := a{f(Xa) : 0 5 s 5 t , f E bE"} and the measures P". The process X := (a,F ,3t, X t , Bt, P") is called the concatenation of X' and X2 with transfer kernel K.
(14.8) THEOREM. The concatenation X ofright processes X' and X2 with transfer kernel K is a right process with semigroup (Pt) given by (14.7).If R := inf{t : Xt E Ez}, then for all x E El and f E b&;, (14.9)
P"{f(XR)1{R
I FR-} = Kf1{R
PROOF:Even though it has not yet been verified that (Pt) is a semigroup, e-atPt dt. Because X is right continuous, it suffices to we set U" := prove that for fixed t 2 0, 2 E E, and f E Cd(E),
{1
00
(14.10) P" (14.11)
e-arf(Xt+r)dr
J
= P"{U"f(Xt). J}
VJ
Eb c ;
t --.* Uaf(Xt) is a.s. right continuous.
The point is that (14.10) is the Laplace transform version of the simple Markov property, and right continuity of u -+ P,f(X,) implies that (14.10) may be inverted. Using (i'.4ii), (14.11) will then prove that X is a right process. By (14.7) we have, letting fl = f i and ~ f~2 = f l ~ ~ ,
U*f(.)
= U?f2(2)1E2(2)
Let g(x) := Pq(e-"CK Ugf2). Then e-"tp,'g(z)= py
[
e--nte-"COf4
= pT[e-OCK vYf2 l { t < ~ } I
by the shift properties (14.4) of the kernel K . It is evident from this expression that g is a-excessive for X1. The fact that U " f ( X , ) is a.s. right
80
Markov Processes
continuous is thereby assured. It remains only to prove (14.10). Observe that
Jd
05
e-"rf(xt+,(wl,
w 2 ) ) dr
= 05
In (14.10) it suffices to take J := f l ( X t , ) . . . fn(Xt,) with 0 t, = t and fl, . . . ,fn E bE" . Observe that ( J l(t
I tl <
<
4 = J l ( 4 l{t
where Jl(wl):= fi(X;' (wl)). . . fn(X:Jwl)) E bTZ"(X1). In addition, Jl{o
G(w1)Ht-C(w')(w2)l{tlC(w')}, with G E b P ( X 1 ) , and for each s 2 0, H, E b P ( X 2 ) . The term
K Jl{R=O) = fi(X,2)*.*fP1(X?")l{R=O} may also be expressed in the above form. To check (14.10), we integrate against J ~ I ~ . . R )and J l l t > ~ separately. } In view of the earlier expression for U"f as a sum of threeterms, it is enough to check that for 51,G and H, as described above, (14.13) ~ " { ~ l ~ i ~ { t <=Rp )~} { u ~ f i ( xJll{tR } } = pz{ ugfi (xt)Gfft -R 1{ t 2 R} }
11: Transformations
81
Note that if 2 E Ez so that R = 0, (14.15) reduces to just the Markov property of X 2 . The equality in (14.13) is trivial because of the Markov property of X1. Fix z E El. We have, using (14.6), P " { 1 z J l l { t < ~ } }=
1
Pi(dul) K(w',dy)
LW
J (wl)e-a(C(w')-t) x SP:(d&l{t
1
e-ar
f Z ( x ?(w2))dr
J ~ ~ ( d wSl ~) ( w ' dy)lIt
e-acoet
{ t
This proves (14.14). Finally, for
2
S
E E l , using (14.6) one finds
J (Srn
P " { I ~ G H ~ - R ~ { ~=~ R pY(dw') }} x spl(d~Z)Ht-CCy1)(U )
0
K(U1, dY) G ( w ' ) l { t ~ ~ ( w l ) }
e-arf2(X?)
dr)
(e;-c(wl)(~q *
Using the Markov property of X2 at the instant t - c ( w l ) , this last expression reduces to
J pi (dw 1
K (w ,dY)G (w ) 1{ t LC (w 1 ) } x
P;(dw2)fLC(d) ( w 2 ) W z (x&(wl) (d)
= P " { u,"fz(Xt)GHt-~ l { t i ~ } } completing the proof of (14.15). For (14.9) note first that bFR- is generated by random variables of the form Gl{t
This equality is a trivial consequence of (14.6). REMARK, Theorem (14.8) clearly extends, without further pain, to the concatenation of any finite number of right processes on disjoint state spaces.
) ~
82
Markov Processes
Extend (14.8) to the concatenation of an infinite se(14.16) EXERCISE. quence { X ” } of right processes on disjoint state spaces. (Hint: all that is needed is the fact that if a right continuous process ( X t ) is given on a space (0,F,Ft, X t , Ot, P“) and if there is an increasing sequence {Tn} of terminal times such that for each n, (X, T,) is a right process, then (X, T) is a right process, where T := limT,.) The construction in (14.17) below lends itself to transfinite recursion. See [Me75], but read with care. (14.17) EXERCISE. In the setting of (14.16),suppose that all the Xn on state spaces E, are copies of one right process X on E. Suppose also that the transfer kernels K , are identical. Show that the natural mapping of UE, onto E satisfies the hypotheses of (13.5). The resulting process in E evolves its X until (, and is then revived by means of the kernel K , evolves again as X , and so on until an infinite number of revivals have occurred, a t which time it dies. (14.18) EXERCISE.A regular step process is a right process X on a Radon space E exhibiting the following behavior. (i) Every point x in E is a holding point for X. That is, by (2.10), there exists X(x) E]O,m] such that
P”{x,= x for all t < s) = e-eX(z)
for all s
1 0.
(ii) There is a kernel Q on ( E ,E ” ) such that if R := inf{t X,} then
> 0 : Xt #
P”{f(x~) 1 FR-}= Qf(x) a.s. on { R < m}. (iii) Let RI := R, and for n 2 2, let Rn := inf{t > &-I : Xt # XR,-~}= R,-I R o h - 1 . Then C 5 limn Given a function X : E -]O,m] with X E E” and given a probability kernel Q on ( E , & ” ) ,a regular step process X may be constructed so as to satisfy (i), (ii) and (iii), by the following argument. Let X‘ denote the process on E such that, starting at x, the process remains at x for an exponentially distributed time with parameter X(x), after which it dies. It can be constructed on fl :=lo, m] x El setting
+
X,‘(S,X)
a.
:=
x A
i f t < s, ift>s,
11: Transformations
83
It is clear that FiC-1 = {f(XC-) : f E E " } . A transfer kernel K is defined by Kf(wl) := JQ(X;-(w'),dg) f ( y ) . Applying (14.17)to a sequence of independent copies of X1, one obtains a regular step process with the properties described, with ( = limn R,. Show from the construction that for each n, R, and X R , are conditionally independent, given that R, < 00. (14.19) EXERCISE. Let X be a regular step process with parameters A, Q, as in (14.18). Show that f E p&" is a-excessive for X if and only if f 2 Pgf,where R := inf {t : Xt # Xo}. (Hint: i f f E So, f 2 Pgf comes from basic facts about supermartingales. I f f 4 Pgf and f is bounded, letting Rn be the time of the nth jump, show that
Then use independence of R and X R on { R < 00) to obtain pUa+p f (2) 5 f (x).Since every null set for X is empty, this implies f E S", by (10.9),) There is a special case of concatenation which is particularly simple. The following discussion makes no use of the preceding construction. Suppose X1 and X 2 are right processes with infinite lifetimes on disjoint Radon spaces El and Ez, respectively. We suppose here that E is a Radon space such that, as sets, E = El U E2, and the topologies of El and E2 are inherited from that of E. We construct now the disjoint union X of X 2 with semigroups (P:), (P,"),respectively. Informally, the processes X1, starting in E j ( j = 1,2), X evolves as X3 with no interaction between the two processes. We construct X on R = 01U 0 2 , 01 and 512 being imagined disjoint, with X t ( w ) := X f ( w ) if w E Rj. For t 2 0 and 2 E El, let Pt(z, .) be the extension of P,'(s, to E which sets Pt(z,E2) := 0. For z E E2 define Pt(z, - ) in a similar manner. It is obvious, using the fact that El, E2 E P , that (Pt) is a Markov semigroup on (E,EU).Given a measure p on (E,E") let p j be the trace of p on Ej and define Pp on ( R , P ) by Pp := P:' Pt*,where Pj"' is considered extended in the obvious way from a measure on (Rj,F&) to a measure on ( R , P ) . It is obvious that under the measure Pp, (Xt) is simple Markov with semigroup (Pt) and initial law p . By construction, t -+ Xt(w) is right continuous in E for all w E R. Finally, a function f E pE" is excessive for (Pt) if and only if f I E , is excessive for ( P l ) ( j = 1,2). It follows that X satisfies the hypothesis (HD2) and so X is a right process with semigroup (Pt). a
)
+
(14.20) PROPOSITION. Let X be a right process on a Radon space E and let E be a compact metrizable space containing E as a subspace. Then
84
Markov Processes
there exists a right process X on E such that X is the restriction of X to the subset E, which is absorbing for X.
PROOF:Let Y denote the constant process on the Radon space I? \ E, and let X be the disjoint union of X and Y on E . The cbnclusion of the theorem is then obvious. This result illustrates, among other things, the advantage of not restricting oneself to working always with a process defined on a path space. Indeed, since E is assumed only universally measurable in E , it would seem in some respects hopeless to recover the set of paths staying in E from the space of paths in E by measurable means. 15. Cartesian Products
Let Xj = (Rj,Fj,F~,X/,O~,P~) ( j = 1 , 2 ) be right processes with state spaces Ej and semigroups Pl. By the Cartesian product X = (X1, Xz)we mean the process on the state space E = El x EZsuch that, given X I E El and 2 2 E Ez,starting at ( X I ,XZ), the coordinate processes are independent X2) copies of X1starting at X I and X2starting at XZ. More formally, (XI, is declared to be a Cartesian product of X1 and X 2 in case each process may be realized as a right process on a single space R and there exist X,") : t 1 0, f E b(€1@€2)} probabilities P z 1 ~ oz n2P ( X 1 , X2):= u{f(Xi, such that : (15.li) for every 22 E Ez, X1 under Pz1~"2 is a copy of X1 under Pzl; for every 21 E El, X2 under Psi+z is a copy of X2under Pz2; (ii) ) El x Ez, P ( X ' > := o{f(Xl): t 2 0,f E (iii) for every ( x ~ , x z E bE1) and P ( X z ) := a(f(X:) : t 2 0, f E bEz} are independent relative to P"1A . For example, we may use SZ = R1 x Rz, defining Px1iz2on F1@J 3 p(X',X2)by Pzl,zz:= P:' 8 P;', the product measure. Set then Xt(w1,w2) := (X,'(w'),X:(w2)), and define shift operators Bt on R by et(W1,W2)
:= (e:wl, e;wz).
A Cartesian product of two right processes is a right (15.2) THEOREM. process. PROOF: Define the kernel Pt on (El x Ez, €1 @ E z ) by
Then for a function f on El x EZof the form f l @ fz (ie., f(z1,zz) := f l ( x l ) f 2 ( ~ ~ )one ) , has Ptf = P,'fl @.P:fz. All we need to check, thanks to (7.4), is the simple Markov property of X relative to (Pt)and the fact that
11: Transformations
85
for t 2 0, s -+ Pt-sf(Xs)l[o,t[(s) is a s . right continuous for all f E C d ( E ) , where d((zl,z2),(yl, yz)) := dl(x1, y1) d z ( z 2 , y2), d j being a metric for Ej. For the simple Markov property it is enough to verify that
+
for all f E Cd(E).If f = f1 @ f2 with fj E Cd,(Ej),this result is clear. Linear combinations of such f are, however, uniformly dense in Cd(E) and the simple Markov property is therefore verified. A similar argument works for the second property. Namely, for f = fl @ f2 with fj E Cd, ( E j ) ,
is obviously P"1JZ-a.s. right continuous for all
( 2 1 , ~ E ~ )E.
(15.3) REMARK.If X' and X 2 have lifetimes C1, C2 respectively, the lifetime of ( X 1 , X 2 )is declared by convention to be C := C1 A C2. This accords with the natural sense of a process being alive only when both components are alive. More importantly, this is the correct convention to take if one wishes to preserve the lifetime formalisms of $11 without essential change. For example, products fl @f2 with fj E b€, will generate b(€l 8 E2), for b(El@€ 2 ) may be identified with the bounded functions on (El x E2) U {A} vanishing at A, with restriction to El x EZ measurable relative to €1 @ € 2 . The process ( X l ,X2) has state space (El x Ez) U {A} rather than ( E l U {A,}) x (E2 U {Az}).
86
Markov Processes
16. Space-time Processes be a right process with state space E and Let X = ( f i,F,Ft,X t,B t,P”) semigroup (Pt). By a space-time process over X ,we mean a Cartesian product in the sense of 915, of X (the space component) with a process R of uniform motion at unit speed (the time component) to either the left or right on an interval I in R. Of course, if R can reach an endpoint of I, R is a process with a finite lifetime and the formalisms of $11 and (15.3) will come into play. Any space-time process over X is, by (15.2), a right process possibly having a distinguished lifetime. It is clear from (15.1) that for every T E I , z E E and F E P ( X ) , (16.3) We shall examine two special space-time situations in detail. To begin with, the forward space-time process ( R ,X ) over X satisfies (16.2)
R is uniform motion to the right on [0,m[.
No lifetime need be specified for R in this case. It is immediate from the definition of ( P t ) in 915 that the semigroup (pt)for ( R , X ) is given by
R ( ( v 4 ,(ds, dy)) := G+t(d.)
Pt(2,dY)-
That is, for f E b(B(R+)@ E ) ” ,
PP9”f(Rt,X= t)
J
f ‘ t ( ~ , d y ) f ( r+ t , ~ ) .
(16.3) EXERCISE. The resolvent Ua for Pt is given by O T ” ( ( T ,z), (ds, dy))
:= e-a(s-r) l[,,,[(S)dS
P8-T(5, dy).
(16.4) EXERCISE. Define g(r,z) := P,-,f(z) lL0,,[(r), where f E p&” and s 2 0 is fixed. Assuming, as will be proved in (18.6), that g E (B+@ E)”, prove that g is excessive for the semigroup ( P t ) . One interesting feature of (16.4) is that it explains one of the important equivalents (7.4viii) to the property HD2. If the space-time process (Rt,Xi) satisfies HD2, then s -+ P t - R , f ( X s ) I[o,~((R,)is PT+-a.s.right continuous for all z E E . It would be a nice proof of this fact, if one were in a situation where it could be proved readily that the space-time process (Rt,Xi)did indeed satisfy HD2.
11: Transformations
87
As the above example shows, it is sometimes useful to realize a right process as a component of its associated space time process. The following observations will be useful in the comparison with the original realization. Let (R,X ) be realized as the right process (Q, F, Ft,(Rt,X t ) , & , P T i Z ) , where Ft is the augmented natural filtration for ( R t , X t ) . Then, for any , is a realization of the semifixed T 2 0 the process ( 0 , 3 , f tXt,Ot,PT*z) group (Pt) as a right process. The filtration (Ft)may be larger than the natural filtration ( 3 t ) on Q induced by X . In order to compare this filtration in a simple way with (Ft), it is convenient to suppose that the subspace flT := { w E 0 : Ro(w) = T } is closed under a family of shifts (0;) satisfying X,(O?w) = Xt+B(w) identically on a,.. (This is trivially the case if ( R ,X ) had been realized by a product construction.) It is then a routine matter to prove that the traces of the objects F, 3t,PT,zon flT are precisely the same as the objects F,Ft,P" constructed on Q,. Thus any result obtained for a space-time process over X involving the (completed) natural filtration is valid when interpreted for the space component X when the time component R has an arbitrary but fixed starting position. We consider now the backward space-time process ( & X ) over X , where (16.5)
R is uniform motion to the 2eft on 10, co[with killing at 0.
Let denote the lifetime of I?, so that t ( w ) = &,(w). Let us suppose that X has a lifetime C. Following the remarks in (15.3), the lifetime of (A, X ) is taken to be A C.
t
(16.6) EXERCISE. Let Pt denote the semigroup for (&X) and Jet U f f be the resolvent. Show that
(16.7) EXERCISE.Let f E bpE' and suppose U f < 00 everywhere. The Fourier transform, Ui" f (2)o f t -+ P t f ( z )lpwr(t) is then well defined. Using the space time process ( R ,X ) , show that for each u E R, U " f ( X t ) is a s . right continuous. (Hint: show that s -+ J g(s t )Pt f ( X , ) dt is a s . right continuous for all g E bB+.)
+
(16.8) EXERCISE.Suppose there exists a a-finite meitSure E on E and a positive function p t ( z , y) on R+ x E x E such that: (i) V t 2 0, Va: E E , 9 -+ pt(z,y) is E" measurable and P t ( z , d y ) = P t ( 2 , Y) t(dy);
88
Markov Processes (ii) V y E E , ( t , z )+ pt(z,y) is (B(R+)63 T-)"-measurable; (iii) identically in t 2 0, s 2 0,z E E , z E E ,
Fix (t,y) E R+ x E . Show that h(r,z) := pt-,.(z, y) l p t [ ( r )is excessive for (Rt,Xi)and deduce that M , := p t - , ( X , , y) l[o,t[(s)is a s . right continuous. Show in addition that (Ms)ols
(16.10) EXERCISE.Show that ifX is uniform motion to the right on R, the associated space-time processes do not satisfy hypothesis (L), even though X does. Verify the claims made in the paragraph preceding (16.11) EXERCISE. (16.5) concerning LIT. (16.12) EXERCISE.A family { ft(z)}troofpositive functions in E" is called an exit law for ( P t )provided Pt f s = fs+t for all 5 , t 2 0. If ( t ,z) -+ f t ( z ) is in (B+@ E ) U , then f is excessive for the space-time process (k, X),and f t - S ( X S )is a.s. rcll on [ O , t [ for all t > 0. (For example, i f f E E", then ft(z) := P t f ( z )is such an exit law.) (16.13) EXERCISE.For 8 f) = ep 8
l?(ep
2 0, set e p ( r ) := e-PT on R+. Show that
va+@f, f E bP.
11: Transformations
89
17. Completion of a Resolvent
Throughout this paragraph, we suppose given a Radon space E and a metric d for the topology of E such that the d-completion of E is compact. In addition to ( E , d ) we suppose given a Markov resolvent (Ua),>O on (El€").That is, the U" are kernels on (E,E2")satisfying
U" = up -t ( p - a )U*UP vp > a; a U" 1E = 1E. Further in the development, we shall need to impose the following additional hypothesis on (U"). (17.1) HYPOTHESIS.Let E' denote the a-algebra on E generated by the family {U"f: a > 0 , f E Cd(E)}. Then E C E'. The most important special case in which (17.1) is automatically satisfied occurs when (U") is the resolvent generated by a Markov semigroup (Pt)t>oon ( E , E " ) satisfying: V f E Cd(E),V x E E , t + P t f ( x )is right continuous on R f , and (17.2)
Pof(x)= f(z). For, under (17.2), aUa f --t f pointwise as a + 00 for all f
E Cd(E). If (U") is a Markov resolvent satisfying (17.1), it follows that the family {U"f: f E Cd(E)}separates the points of E . As in $9, Sa denotes the cone of a-supermedian functions for the resolvent (U"). It is an easy consequence of the resolvent equation that UQfE 3" for all f E pE". It is also easy to see that the cone S" is closed under finite minima and that Sa C Sp if a < p. Recall that Q denotes the rationals, Q+ the positive rationals and Q++ the strictly positive rationals. The &-conegenerated by a family J' of positive bounded functions on E is the set of all positive Q-linear combinations of functions in the class J'. Given a Q-cone J' C bpE", set
It is obvious that U(y)is a Q-cone contained in the &-cone U,>obSa. That A ( Y )is also a Q-cone comes from the trivial identities (a1 A
. . . A a,)
(Aiai)
+ b = (a1 + b) A . . . A
+ (Ajbj) =
+
Ai,j(~i bj),
(a,
+ b),
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valid for all positive reals aa, b j . By the rational Ray cone R generated by the resolvent (U")and a family C' c bpE", we mean the Q-cone defined as follows. Let 3-1 denote the Q-cone generated by C', and let Ro := A(3-1). For n 2 1, let Rn := A(Rn-l+U(Rn-I)), where Rn-l+U(Rn-l) means { f + g : f E Rn-1, g E U(R,-l)}. Then set R := Un>oRn. The following observations concerning the construction are almost immediate. (17.3i) If C' is countable, then 'Fi is countable and, by induction, every R, is a countable Q-cone. (ii) Ro c R1 c . . . C Ua>o bs". (iii) I f f E R, and (Y E Q++, then Uaf E R,,+I. (iv) If f , g E R,, then f A g E R,. (17.4) PROPOSITION. Let C' C bpE", and let 3-1 denote the &-cone generated by C'. Then the rational Ray cone R generated by C' C bE" and (U") is the smallest Q-cone contained in bEU such that: (i) U"(R)c R if (Y E Q++; (ii) f , g E R implies f A g E R; (iii) R contains U(3-1). (17.5) PROPOSITION. Under the conditions of (1 7.4),
(i) R = A(U(3-1)+ WR)); (ii) A(U(R))is uniformly dense in R.
PROOF: Let ' J := A(U(3-1)+ U ( R ) ) . Then J' is a Q-cone stable under finite minima. By construction of R,one sees that J' C R. If R, C J' then U(R,)c U ( R ) C ,7 so R, +U(R,)C 3. Therefore, R, C J' implies that Rn+l= (a,+ U(R,)). Since Ro = U(31)c J', one obtains R c J' by induction, proving (i). If h E 3-1 and a E Q++, the resolvent equation shows that pU"+@U"h = U*h - U"+flh converges uniformly to U"h as p -+ 00. Thus every f E U(3-1)can be approximated uniformly by functions in U(3-1). Therefore U(R)is uniformly dense in U(3-1)+ U(R).From this fact, (ii) follows, using (i). The preceding results make no use of (17.1). However, the following proposition makes essential use of the fact that {U" f : (Y > 0,f E C d ( E ) } separates the points of E. (17.6) PROPOSITION. If the family C' is a countable, uniformly dense subset of pCd(E) and 1.g € C', then the rational Ray cone R constructed from C' and U" is countable, contains the positive rational constant functions, and separates the points of E. PROOF: Since C' is dense in pCd(E) and U(pCd(E))separates E, U(C') separates E. The other assertions are obvious.
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u")
We now construct what is called a Ray-Knight completion (I?, p , of ( E ,d, U") relative to the family C'. From now on, the family C' will be assumed to satisfy: (17.7i) C' C pCd(E) is countable; (ii) 1~E C'; (iii) the linear span of C' is uniformly dense in Cd(E). Under (17.7), R has the properties described in (17.6). Define then a metric p on E by p(x,?/>:=
c
2-nllgnll-11gn(~) - Sn(Y)l.
n> 1
The map 2 -+ (gn(x)) of E into K := n;==,[O,(lg,((] is an injection, and since the product topology on K is generated by the metric p'(u,b) := 2 - n ~ ~ g n ~ ~ -1 ~bnl, u n the injection above is an isometry of ( E , p ) into ( K , p ' ) . It follows that the completion ( E , p ) of ( E , p ) is compact. Each function gn is puniformly continuous. (17.8) PROPOSITION. Let C,(E) be the space of p-uniformly continuous functions on E . Each function f in C,(E) extends to a unique f E C(E). For all a > 0, U"(C,(E)) C C,(E), Ua(Cd(E))c C,(E), and C J E ) is the uniform closure of R - R.
PROOF:The first assertion is obvious by definition of E . As we remarked above, every gn is in C,(E) and so R C C,(E). The set R - R is a vector space over Q which is stable under lattice operations. Let C denote the uniform closure of R - R. Obviously, C C C,(E),and C is a vector space over R that is closed under lattice operations. By the properties of R listed in (17.6), C contains the constants. Let C = {j: f E C}. Then C is a vector space over R that is closed in the uniform topology, is stable under lattice operations and contains the constants. In addition, C contains the functions g,, which separate the points of E . By the lattice form of the Stone-Weierstrass Theorem, C = C(E). It follows that C = C,(E). According to (17.3), Ua(R)c C if a E Q++ and since C is uniformly closed, Ua(C) c C if a > 0, hence U"(C) c C for all a > 0. A similar argument shows that Ua(Cd(E))c C. The topology on E induced by the metric p is called the Ray topology on E . It appears to depend on d and the family C' C pCd(E) originally chosen. We prove later (43.5) that all possible choices lead to the same topology on E , though not in general to the same metric p or the same compactification E . In general, the Ray topology on E is not comparable to the original topology. (17.9) PROPOSITION. If U"(Cd(E))c C d V a > 0, then the Ray topology on E is coarser than the original topology.
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PROOF:By definition of R, C d ( E ) contains R and so C d ( E ) 3 C,(E). Using (A2.1) it follows that every function that is bounded and continuous in the Ray topology is continuous in the original topology. (17.10) PROPOSITION. Let E' be the o-algebra on E generated by the Ray topology. Then E c Er C E". For all a > 0, Ua(bE') C bE'.
PROOF:The hypothesis (17.1) implies E c E'. Since the cone R generates E', it follows that E' C E". For all a > 0, U a ( C p ( E ) )C C,(E) shows U0(bf?) c b&'. We come now to a key technical point. We continue to assume (17.7). The space E with the Ray topology is Radonian. ( 1 7.11) PROPOSITION.
PROOF:The inclusions E C E' C E" imply that the hypothesis of (A2.10) are satisfied. (17.12) PROPOSITION. I f E is a Lusin space and U" maps Bore1 functions to Borel functions, then E is a Lusin space in the Ray topology.
PROOF:It is evident that E = E' so the inclusion map i : ( E , d )+ (E,p) is Borel measurable. By Lusin's Theorem (A2.6), i ( E ) = E is Borel in E. That is, ( E , p ) is homeomorphic to a Borel subset of a compact metric space. We may now construct by continuity a resolvent U" on E that extends U" on E. If f E C ( E ) is the continuous extension of f E C,(E) then by (17.8), U"f E C,(E) and so U"f extends continuously to (U"f)-E C ( f i ) . The map Ua : C(E)+ C(E) defined by (17.13)
UQJ := (U"f)i
J
E C(E);
is clearly a positive linear operator with allU"fll I11f11.
aU" 1E = (aU" 1+
Since
1E,
allU"ll = 1. For each z E E there exists, by the Riesz representation theorem, a positive measure o"(x, - ) on E with total mass a-' such that
U" f(x) =
1
f(y)
P ( Z ,
dy).
It follows then from (17.13) that Ua is determined by a kernel on (2,€), where E is the Borel a-algebra on E . By a simple continuity argument one may show that U" satisfies the resolvent equation (4.6). The sense in which (U") extends ( U a ) is given by the following result.
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(17.14) PROPOSITION. For x E E , U"(z, . ) is carried by E E &" and its restriction to E is equal to U o ( x , -).
PROOF:Since, as we proved in (1'7.11), E E &", we may define a measure v on (I??,&") by setting, for f E b€, v ( f ) := U " ( f l ~ ) ( x ) where , x E E is fixed. We are using here the fact that the trace of &" on E is E": see (A2.2). The identity (17.13) shows that for all f E C,(E), U"$(x) = Uaf(%) = U"(fIE)(Z) = v(f). That is, u"(z, . ) = v(.) as measures on ( E , € " ) , since these measures
agree on c ( E ) . The collection ( E ,p , 0") constructed above is called a Ray-Knight completion of (E,d,U"). It depends not only on E , d and U" but also on the choice of the family C' c bCd(E). (17.15) THEOREM.IfC'satisfies (17.71,then (U") is aRayresolvent on E .
PROOF:In order to show that (U") satisfies the conditions of (9.4), we need show only that if Sa denotes the a-supermedian functions for (U"), then U,>OC(B) fl separates E . This is trivial, for C separates E , every member o f f is continuous, and iff E R, then f is P-supermedian relative to (V") for some > 0. In addition, the following obvious calculation proves that f E 2 is 0-supermedian relative to
s"
(u"):
/!?P+Pf=
(/?u"+")-< $.
(17.16) THEOREM.Let ( P t ) be a right semigroup on E . Then if(pt)t?o is the Ray semigroup (see $9) associated with the Ray resolvent (U*) on E , then for all x E E and all t 2 0, Pt(z,. ) is carried by E and its restriction to E is equal to Pt(x,.). That is, (pi) is an extension of (P,). PROOF: Fix x E E . Since ( P t ) is a right semigroup, S" is closed under finite minima, so the construction of R shows that if f E R then f is a-excessive for some Q > 0. Thus if f E R, t -+ P t f ( x ) = P"f(Xt) is right continuous on R+. On the other hand, if f is the continuous extension of f to E , t -+ ptfl(z)is right continuous on R+ since ( P t ) is a Ray semigroup. The Laplace transforms of the above functions o f t are respectively a -+ U " f ( x ) and Q -+ U" f(z),which agree for all a > 0 by the very construction of U". It follows that Pt f(x) = Pt f (x)for all t 2 0. For fixed z E E and t 2 0, the MCT implies that for every bounded Bore1 function g on 3,ptg(z) = Pt(glE)(Z). By sandwiching, the same equality is seen to be valid for all g E bt?. Since E is universally measurable in E by (17.11), p i l ~ ( z = ) P t l ~ ( x= ) 1, so pt(z, . ) is carried by E . The asserted equality is then evident. (17.17) REMARK.Using (8.4), one sees that (17.16) remains valid under the weaker hypothesis on (Pt) described in (8.3).
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(17.18) EXERCISE.Use (17.16) and (17.10) to show that if (Pt)is a right semigroup on E , then Pt(bET)c bE' for all t 2 0. 18. Right Process in the Ray Topology
Let X = (a,6 ,&, X t , B t , P") be a right process with state space E and transition semigroup (Pt). Construct a &-cone R as in $17 and let {gn} be an enumeration of R. For every n, gn is a-excessive for some a > 0, so by hypothesis HD2, if Ro := {w E R : t + gn(Xt(w)) is rcll for all n 2 l}, then Rg is null in Q. That is, every P" is carried by Ro. Obviously &Ro c Ro. Rename RO as a, and replace 8, Gt and Pp by their respective traces on the new R. It is clear that (R,G,Gt,Xt,Bt,Pz) remains a right continuous realization of (Pt). As we shall now show, this is true also in the Ray topology. Fix the Ray compactification ( E ,p) generated by R. (18.1) THEOREM. When considered as aprocess on E with the Ray topol6 ,Gt, X t , B t , P") is a right process. In addition, the left limit ogy, X = (ao, Xi- taken in the Ray topology o f E exists in E for all t > 0.
PROOF:Since R - R is uniformly dense in C,(E), t + g ( X t ( w ) ) is right continuous with left limits for all w E no and all g E C(E). The a.s. right continuity of a-excessive functions along the trajectories is not affected by the change of topology on E . We have already proved (17.11) that E is a Radon space in the Ray topology, so the proof is complete. We have now shown that, for theoretical purposes, a right process may be studied in its Ray topology, with the following benefits accruing: ( 1 8 3 ) U" maps C,(E) into C,(E) and the semigroup (Pt)preserves the u-algebra E' of Bore1 functions for the Ray topology; (ii) the process has left limits in I?. With the deletion of the fixed set $26 E N(G), we see that for every g E C,(E), t -+ g ( X t ( w ) ) is right continuous for every w E R. It follows that for every f E b€', the map ( s , w ) + f ( X , ( w ) ) of [O,t[xO into R is measurable relative to B([O,t [ )@ 3:. It follows then from (A5.4) that {w : s -+ f o X , ( w ) (s E f O , t [ ) is right continuous} is in the universal completion of q,hence it is in Ft. The following result is an obvious consequence of this observation. (18.3) PROPOSITION. For f E Cd(E),t > 0 and a > 0, the set {s + U" f O X , is not right continuous on [0,t [ } belongs to N t ( F ) .It follows then from (7.9) and (8.1) that (a,E , Gt, X t ,8t, P")is a refined right process, and 3,X t , Ot, P") is a right process. that, in particular, (12,Ft,
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The principal drawback to changing the topology is that, while it does not change right continuity of the process, the left hand limit X t - in the Ray topology may not be the same as the left hand limit in the original topology, even if that limit exists. We use the notation Xf- for the left hand limit of X t at t in the original topology, if it exists. We shall compare the two limits in 553. The following definition extends (5.3) in one direction. (18.4) DEFINITION. A real function f on E is nearly Borel in the strict sense in case, for every p and every t > 0, there exist Borel functions g,h with g 5 f 5 h and (3s E [ O , t [ with g ( X , ) # h ( X , ) } E J V ~ .
It should be emphasized that g and h depend on p and t. The nearly Borel property (5.3) has played an important role in Markov process theory for many years. In the original formulation of the right hypotheses by Meyer [ M e 6 8 ] ,it was required that every a-excessive function be nearly Borel. In some constructions of Markov processes, it does not seem possible to verify that every a-excessive function is nearly Borel. Getoor [Ge75a] dropped this part of the hypothesis and showed that it was not really needed for the development of the theory, at least under his form of the hypotheses, which demanded that the process be realized on path space. Indeed, we shall see below (18.5) that a-excessive functions over a right process are automatically nearly Borel, even in the strict sense, when the original topology is abandoned in favor of the Ray topology. In the next section, we shall see the only really substantial consequence of the hypothesis that every a-excessive function is nearly Borel relative to the original topology. See (19.3). Let X be a right process with semigroup (Pi). Then (18.5) THEOREM. every f E S" is nearly Bore1 relative to the Ray topology. If, in addition, X is a right process relative to the Ray topology, then every f E S" is nearly Bore1 in the strict sense relative to the Ray topology.
PROOF:According to (18.1) we may assume that X is a right process relative to the Ray topology. We showed in (17.10) that U" f E bE' for all f E b&' and a > 0. Given an initial law p and a > 0, let v denote the measure pU"U" on ( E ,E"). Iff E bE", we may choose g , h E bpE with g 5 f 5 h and u(h - g ) = 0 . Then Uag 5 U af 5 U"h and pUa[Uuh- U"g] = F e-"t[U"h(Xt)- U"g(Xt)]dt = 0 . The last 0. In other words, Pp S equality implies that, Pp-almost surely, U a h ( X t )= U"g(Xt)for Lebesguea.a. t 2 0. Then right continuity shows U a h ( X t )= Uag(Xt) for all t 2 0,
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Pp-almost surely. We have shown therefore that every function of the form UQf ( a > 0, f E bE") is nearly €'. If f E S", by (4.12vi) there exists a sequence { f n } C bp€" such that U"fn t f. For each n we may choose gn,hn E bpE' such that gr 5 U" fn 5 h, and P"{gn(Xt)< hn(Xt)for some t 2 0) = 0. Since U"fn increases, it may be assumed that {gn}and {hn} are increasing sequences, replacing successively gn by g1 V . V gn and h, by hl V . * - V h,, if necessary. Then g 5 f 5 h, g and h belong to E' and P p { g ( X i ) < h ( X t ) for some t 2 0) 5 EnP"{g,(Xt)< h n ( X t )for some t 2 0} = 0. Suppose next that X is a right process relative to the Ray topology, and fix t > 0 and p. The essential new feature here is that s + tp(X,) ( s < t ) is measurable relative to ([0,t[xR, B([O,t [ )8 G )for every cp E E'. Follow the same construction as above, noting that the set I' of w such that U"h(X,(w)) # U"g(Xs(w)) for some s E [O,t[ belongs to the Ppcompletion of F f ) by (A5.4). It follows that I' E N t , proving that every function of the form U" f , f E bpE", is nearly (Ray) Bore1 in the strict sense. The rest of the proof in this case proceeds exactly as before. We now compare the Ray completions of X and X = ( R , X ) , the forward space-time process over X described in $16. In the discussion below, (0") denotes the resolvent (16.3) for X , so that if e p ( r ) := e-p' as in (16.13)) one has &(ep 8 f) = eg @ Ua+@f, f E bE". As in $17, let C' C bpCd(E) denote a countable family satisfying (17.7) and set ? := {ep @ f : p E &+, f E C'}. Then c' satisfies (17.7) relative to E := R+ x E, the state space for XIthanks in part to the Stone-Weierstrass theorem. Let R and & denote the respective Ray cones generated by C', (U") and (?)' (0"). (18.6) PROPOSITION. & is the &-cone generated by the class 1s: of functions of the form ep 8 f, /3 E Q++ and f E R.
PROOF:For
fl,
. . ., fn
E bE",
Thus, because R is closed under A, k is also, and as we saw in the discussion following (17.2), this implies the same for the Q-cone generated by k. By (16.13), is stable under fip for every p E Q++, and similarly, 1s: contains f i g ( e p €3 f ) for every f E C'. The result then follows from (17.4) applied to (0"). The preceding result implies that the topology on R+ x E generated by the Ray metric for X is the product of the usual topology on R+ and the topology generated by the Ray metric for X on E .
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(18.7) LEMMA.For every f E bE' and s > 0, the function g ( r , x ) P,-,f(z) l[o,,[(r)is in B+ @ E') and i f f E bE", then g E (a+8 E T ) u .
:=
PROOF:I f f E C,(E), then g ( r ,z) is right continuous in r for every fixed z, and for fixed r it is in ET because (17.18) the semigroup (Pt)preserves E'. The first assertion follows then from (3.13) and a monotone class argument. Now let f E bpE" and let X be a finite measure on (R+x E , B+ @ E'). The formula X(h) := JJ A(&, dz)Ps-Th(z)l[o,,[(r), h E bE, defines a finite measure on ( E , E ) ,so we may choose hl, hz E bE with hl 5 f 5 hz and X(h2 - h l ) = 0. For i = 1 , 2 define g i ( r , z ) := P,-,hi(z) l[o,,[(r). It follows that g1 5 g 5 g2 and X(g2 - 91)= 0. The second assertion follows. 19. Realizations of Right Processes
To study a given right process X = (@3, .&,X t , &, P") or, equivalently, to study a given right semigroup (Pt), one should first remove as many inessential irritants as possible from the setup. According to (18.1), after deleting a null set from the sample space one may assume that the given right process is right continuous in the Ray topology and has left limits in some compactification of E. From now on, we shall usually assume that a given right process has been so modified, unless it is not natural to do so. After this modification has been made, we obtain a measure theoretic simplification due to the fact (17.18) that the Ray-Bore1 a-algebra E' on E is preserved by the semigroup (Pt).We now set (19.1) F = a{f(Xs) : s 2 0,f E E,}; q = a{f(X,) : 0 5 s 5 t , f E E T } . The filtration (G)will replace the former filtration in all applications, The filtration (Z) and we shall use F: from now on as a synonym for enjoys all the properties described in (3.3) for We return now to some of the difficulties mentioned in $8. We we mentioned there that an arbitrary right continuous realization of a right semigroup is not necessarily a right process. We examine here the properties of a process which are preserved under change of realization. Such questions come down ultimately to examining realizations on spaces of paths. We shall set up the technology here so that the related question of extension of a right process may be discussed at the end of the section. Here are the principal results.
(e)
z. (c).
(19.2) THEOREM. Let (Pt) be a Markov semigroup on E satisfying the hypothesis HDI, and let X and XI be right continuous realizations of (Pt). Then: (i) X has the strong Markov property (6.5) relative to (&+) if and only if XI has the strong Markov property relative to (3;+);
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Markov Processes (ii) I f f is nearly Borel relative to X then f is nearly Borel relative to X'.
(19.3) THEOREM. Let (Pt) be a right semigroup (8.1) on E . If the Borel structure on E is so rich that (Pt)has a right continuous realization in which every a-excessive function is nearly Borel, then every right continuous realization of (Pt) is a right process. In particular, (Pt) may be realized as the coordinate process on the space fl of right continuous maps (;. : R+ + E .
In view of (18.5), if the topology on E is the Ray topology generated by a Ray metric p , then (Pi) may be realized on the space of maps of R+ into E which are pright continuous and have left limits in the pcompletion E of E . Theorem (19.3) is an obvious consequence of (19.2), using the characterization of right processes in (7.4). The proof of (19.2) involves a comparison with a path space realization, whose details we now describe. Given a Radon space ( E ,d), let fl denote the space of maps Lj : R+ .+ E which are right continuous. The coordinate process (Xt),>o on fl is defined by X t ( W ) = &(t) for t 2 0. Then 9 := a { f ( X t ) : t > O , f E bE"} and $? := g ( f ( X 8 ) : 0 5 s 5 t, f E bE"}. Suppose given a process (Xt)t>o - on a filtered probability space (R, 0,Gt, P) satisfying: (19.4i) for all w E R, t + X t ( w ) is right continuous in E ; (ii) X has the Markov property relative to ( R , F , F t , P )with semigroup (&) and initial law p.
As in the path space construction in 52, @ : R map defined by (19.5)
R t o @ := X t
-,
denotes the canonical
for all t 2 0.
The following result extends (2.7).
PROPOSITION. With the above data in force, we have: and for all t 2 0, @ E (i) ip E F/+,
(19.6)
e/*;
(ii) if P on (fi,$=") is the image of P under @, then for every fi E b.?%, P(fio@)= P(@, and in particular, under P, ( X t )is simple Markov with semigroup (Pt) and initial law p; (iii) if fi E b$p, then fro@ E bFp and P[&o@] = P[fi]; (iv) if 0 5 t 5 00 and H E b e , then there exists H E b e such that H = H.9; (v) if 9 is an optional time over (3f+), then T o @ is an optional time over (Ff+);
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there exists an optional time (vi) if T is an optional time over (3[*), T over (3:+) such that P{T # To@} = 0; (vii) using the notation of (5.11, the class u { D ) ~ ( e +is) }identical, up to evanescence, to {Zoa : Z E o { ~ R ( e + ) } } .
PROOF:Parts (i) and (ii) were proved in (2.7). It follows from (ii) that if H E b 9 and if Plfil = 0, then P(fiocf,l= 0. With this observation, (iii) is obtained by sandwiching. For (iv), one uses a monotone class argument starting with H := f i ( X t , ) .. .fn(Xt,).Let T be optional over (3f+). Then for all t > 0, lI+o+
fiI2,,with {T,” = lc/2,}
= Let F:(Lj) := 2-,inf{j : Then T: is an optional time over and T:o@ = T:. Let f’:= inf, T,”. Then, a.s., T 0 cf, = inf, T: 0 cf, = inf, T,”= T o ,proving (vi). IV-641, u { D ( e + ) }is generated by the stochastic Finally, from [DM75, Thus by intervals of the form [ T ,5’1 with T , S optional times for mocotone classes, (vii) is an immediate consequence of (v) and (vi). PROOFOF (19.2): Let X be the coordinate process on h. For any initial law p, P” and P’” induce the same measure P” on (h,@’).In order to prove (19.2) it suffices to compare X and X rather than X and X’. To begin with, assume that X is strong Markov relative to (3t+). If is an optional time over (3:+), then for f E C d ( E ) and t > 0, (19.6iii) gives Ak,, E
LJ E Aj,,}.
(e)
(e+).
P”{f@(t
+ T)); T < co} = P ” { f ( X ( t+ T o @ ) ) ; T o @ < m}.
According to (19.6v), T o @ is an optional time over (Ft”,), so the right side is equal to P”{P,f(X(To@)); T o @ < m}. Using (19.6iii) again, we conclude that X is strong Markov. The proof of the converse is similar but uses (19.6vi) instead of (1 9 .6 ~ ) . Turning t o the nearly Borel part of (19.2), we assume first that f is nearly Borel relative to X. Fix p and let g and h be Borel functions with g 5 f 5 h and P”{g(Xt) < h ( X t ) for some t 2 0) = 0. If T := inf{t 1 0 : g ( X t ) < h ( X t ) } then T is the debut of a nearly optional set, and consequently T is an optional time over (3f+). If T := inf{t : g ( X t ) < h ( X t ) } ,then T o @ = T. By (19.6), P”{T < co} = P”{T < m} = 0. Hence f is nearly Borel relative to X . The converse argument is not essentially different.
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Markov Processes
(19.7) THEOREM.Let (52,3,Ft,Xt,0t,Pz) be a right process, and let W denote the collection of all maps w of R+ into EA which are right continuous for both the original topology and the Ray topology, and which admit A as a trap. Let Xt(w) := w(t), and let Pz denote the image of P” under the map @ of 52 into W satisfying X t ( @ ( w ) ) = Xt(w) for all t 2 0, for a.a. w . Then: (i) X is a right process; (ii) f E E” is nearly optional relative to X if and only if it is nearly optional relative to X; (iii) f E E” is nearly Borel relative to X if and only if it is nearly Borel relative to X ; (iv) for every nearly optional set B, the set BTof regular points is the same when computed relative to X as it is relative to X; (v) the fine topology relative to X is the same as that relative to X .
PROOF:The first assertion is an immediate consequence of (18.1), (18.5) and (19.3). Let f be nearly optional relative to X. Given an initial law p, choose a process 2 E u { D ~ ( c + with ) } f ( X ) = 2 up to Pp-evanescence. By (19.6vii), there exists 2 E a { D ~ ( f i + )with } 2 0 @= 2 up to P P evanescence. Then ( f ( X ) - Z ) o @ = f ( X ) - 2 up to Pp-evanescence, hence by (19.6iii), f(X) = 2 up to Pp-evanescence. That is, f is nearly optional relative to 8. The other direction is proved in a similar way, as is (iii). Now let B be a nearly optional set. Let TB,TB denote the hitting times of B by X , X respectively. Clearly ?BO@ = TB,and so by (19.6iii), P”{TB = 0) = P”{?B = 0}, proving (iv). The last assertion follows at once from (iv). We now consider the notion of extension of a Markov process. We set down first the notion of extension of a kernel, which has already been used informally in (17.14) and (17.16). (19.8) DEFINITION.Let K and I? be kernels on Radon spaces ( E ,E ” ) and (E,&”) respectively. Then K is an extension of K provided the following conditions hold: (i) E c E (up to homeomorphism) and E E &”; (ii) for all 2 E E , k(z, - ) is carried by E , and its restriction to E is equal to K ( z , .). Notice that if the topology of E is identical to the relative topology induced by E , then (A2.11) shows that (i) holds automatically. The definition (19.8) leads to obvious definitions of extensions of a subMarkov semigroup ( P t )and a resolvent UQ.In a similar vein, we shall say
11: Ransforma tions
101
that a right continuous simple Markov process X on E is an extension of a right continuous simple Markov process X on E provided the semigroup (pt)for X is an extension of the semigroup (Pt)for X. (19.9) PROPOSITION. Let X and X be right continuous simple Markov processes, X being an extension of X. Let R and d denote the spaces of right continuous maps of R+ into E and E respectively. If p is a probability measure on E and if P p and P p denote the canonical measures induced by x and X respectively on (a,3”) and (d,Fu), then the *@-outer measure ofR is equal to one, and the trace of Pp on (R,P) is equal to P”. REMARK.In comparing the canonical realizations of X and X ,the “outer measure one” feature in (19.9) is an unfortunate fact of life which cannot be removed even if one assumes that the processes are right processes in their Ray topologies. Additional measurability properties of E-for example, E being Lusinian in a Ray topology for X-would permit one to obtain “measure one” if X and X were known to be right processes. PROOF:We shall use X and X to denote the coordinate processes on R and d respectively. As we observed following (19.8), E is universally measurable in E, so an easy argument shows that F is the trace of 3uon R. For any finite subset J of R+,let F? and denote respectively the u-algebras on R and fl determined by
q
:= . { f ( X t ) : t E J,f E
E”};
ji”J
:= . { J ( X t )
:t E
J,f f P } .
The trace of f i on R is precisely Fy. If Pf”,denotes the trace of P p on then for J = (0 < tl < ... < t n } and B l , . . . , B n E iu,one has
e,
,-
Since p is carried by E and equal to
(pt)is an extension of
( P t ) ,the right side is
where 3 j := Bj i l E for j = 1,.. . , n. This observation proves that the trace of P; on (R,Fy)is equal to the trace, Pf”,, of Pf’ on F?. Now let := U J F as ~ J ranges over all finite subsets of R+.Then is a n algebra of
0
102
Markov Processes
sets generating 9.A standard result in measure theory (see, for example, [Ne65,51.51) shows that the Pp-outer measure of R in fl is equal to the idhimum of EnPP(An) over all sequences (A,) c 6 such that R C UA ,., Suppose An E with each Jn a finite subset of R+. Then
en
n
n
n
n
Consequently, R has Pp-outer measure equal to one. The fact that Pp is the trace of P p on R follows immediately from the definition of the trace in (A1.5), and the calculation showing that P’; is the trace of P’”J on ( R , F j ) . The following clever example of a strong Markov process with no realization as a right process is due to Salisbury [Sa87].Among other things, it points out the necessity for caution when using universal measurability as a substitute for Bore1 measurability. Recall that a measure p on E is called diffuse provided p ( { x } ) = 0 for every z E E. Fix E = Rd, d 2 2. A construction in set theory [SS36], based on the continuum hypothesis, shows the existence of an uncountable set N C R with p ( N ) = 0 for every diffuse measure p. Given a countable set C C R, write N’ := {y E N : (y Q)n C # 8}, N” := N \ N’. Note that N‘ is countable, hence N” is uncountable. Then N Q = (N’ Q)U (N” Q), and obviously N“ Q is dense in R and disjoint from C. Replacing N if necessary by N Q, we may therefore assume that N \ C is dense in R for every countable set C c R. Suppose, as we may, that a d-dimensional Brownian motion (Bt)is realized on the space R of all continuous paths in Rd. Let R j := { w E fl : the projection of w on the jthcoordinate is non-constant}. In view of the continuum hypothesis, each Rj has cardinality N 1 , as does N . Denote the first uncountable ordinal by r. We may then enumerate R1 as ( W ? ) ~ < F , and N as ( ~ ~ ) ~Since < r .N is dense in R, there is an index y(0) < r so that wo hits {y7(o)} x Rd-l. Let the first such hit be at the location zo= (yr(o), ZO), zo E Rd-’. Now iterate this procedure. Because N was selected so that N \ C is dense for every countable set C, y(1) := inf{y > $0) : w1 hits {y7} x Rd-’} exists and $1) < r. Let the first hit be at location z1 = ( ~ ~ ( .q), ~ 1 z1 , E Rd-l. Continue this procedure by transfinite
+
+
+
+
+
+
103
11: Transformations
induction to obtain a set 21 = { x o : a < I?} c N x Rd-l c Rd which is hit by every w E 521. Given a diffuse measure u on Rd, use (A3.5) to disintegrate u as 4 d x ) =7 7 bd z ) p ( d z ) , with p the projection of u on the first coordinate, and 77 a finite kernel from R to Rd-'. Write p = p1 pl', where p' is diffuse and pll is purely atomic-i.e., pll lives on some countable set C C R. We may choose a Bore1 set B C R with B 3 N and p I ( B ) = 0. Note that
+
n (C x Rd-')) U ( 2 1 \ (C x Rd-')) c (2, n (C x Rd-l)) U ( ( B\ C ) x Rd-').
21 =
(21
It follows that the u-outer measure of
v(Z1 n (C x Rd-l)) +
21
is bounded above by
L,,
~ ( yRd-') , p(dy) = 0.
This argument proves that the set 2 1 constructed in the preceding paragraph is null for every diffuse measure on Rd. In the same way, construct sets Z j , j = 2 , . . . , d , and let 2 := UjZj. Then 2 is null for every diffuse measure on Rd and every non-constant continuous path w in Rd hits 2. The section theorem fails with a vengeance for the process l ~ ( B ~ ) l ~ ~ > ~ ) , which is, of course, not optional for (Bt).For, given any optional time T for (Bt) and any initial law p, the measure A(f) := PP{f(&); 0 < T < GO} (f E bE) is diffuse, because, by (10.20), ( B t )as. does not hit points in Rd for d 2 2. Define Xt on 52 by
It is easy to see that X is strong Markov on Rd with transition semigroup Q t ( x , * ) : = P t ( z c. ,) ~ z c ( ~ ) + E ~ ( * ) ~ z ( x ) .
Let
ifzE2, if z $ 2. 0 Then f is obviously excessive for ( Q t ) . However, since almost all paths of X starting outside 2 do enter 2, f o x cannot possibly be a s . right continuous. That is, X is strong Markov but not a right process. Suppose X had a second realization XI as a right process. The function f defined above being excessive for XI,it would follow that t f(Xl) would be a s . right continuous. In order for this to happen, it would have to be the case that XI would as. never hit 2, starting outside 2. See [Sa87]for a similar but more difficult construction leading to a right process having another realization as a non-right, strong Markov process.
f ( x ) :=
{
GO
--$
104
Markov Processes
Show directly that the set 2 constructed above is not (19.10) EXERCISE. nearly Borel for ( X t ) . 20. RBsumB of Notation and Hypotheses
This section gathers together the blanket assumptions and terminology which will remain in force for all subsequent chapters. The process X = (a,B, Bt, X,, &,P")being a right process with semigroup (Pt)and resolvent U" means: (20.li) the state space ( E , d ) is a (separable) Radon space. The dcompletion of E is compact and contains E as a is universally measurable subset. A dead state A is adjoined to E as an isolated point to make an extended state space EA; X is a.s. right continuous in E A , Markov with semigroup (P,) (ii) relative to the filtration (Gt), and admitting A as a trap; (iii) the filtration ( B t ) is augmented in the sense described in $6; (iv) for every f E S", the class of a-excessive functions for (P,), the set N := {w : s + f(X,(w))is not right continuous} is null in G. One may then construct a metric p on EA (the Ray metric) such that: (20.2i) U"C,(EA) c C,(EA); (ii) U"Cd(EA) c Cp(EA); (iii) X is as. prcll in the pcompletion EA of EA. The original Borel a-algebra, now denoted E 0 , is abandoned almost completely in favor of the Borel a-algebra € generated by the Ray topology, and Borel means Borel relative to the Ray topology unless otherwise specified. Then: (20.3i) Pt(b€) C b€ for all t; (ii) every f E Sa is nearly Borel; (iii) the representation of X on the space W of all paths w which are both right continuous in E A and Ray-rcll in ?!A , is also a right process. (This is not true of the space of paths which are only right continuous in EA. In this respect, right processes in our sense are not the same as those in the sense of [Ge75].) The original topology on E plays a very minor part in the remaining theory, and some authors, for example [En77], have developed the theory with only measure theoretic structure given on the state space. One may in that case construct a topology on E similar to the Ray topology, but the essential difference in that case is that the process in that topology is only as. equal to X at each fixed time t , rather than indistinguishable from X. Our purpose is more to provide tools which may be applied to a given process which is already known to be right continuous in some topological
11: Transformations
105
space, and by the nature of the applications, such as potential theory, indistinguishability is essential. Let RO denote the set of w for which t -+ X t ( w ) is Ray-rcll in E A and admits A as a trap. Then RO is closed under the shifts Bt, and if R is replaced by RO and the remaining objects d t , 4, Gt, P" are replaced by their traces on Ro, we may then assume X satisfies: (20.4i) X is a right process relative to its augmented natural filtration
1;
(3t
t
X t ( w ) is Ray-rcll in E A for every w ; by (18.3), for every f E Sa,N := { w : s ---t f ( X , ( w ) ) is not right continuous at some s < t } E N t . That is, given p , there exists Nt E f l with N C Nt and PpNt = 0. That is, X is a refined right process. As we noted in $11,R may always be assumed to have killing operators kt satisfying Xtok, = &At and k t o k , = k 8 A t . Concerning the dead state A, it is convenient to assume (20.5i) for all w E a, X t ( w ) = A implies X , ( w ) = A for all s 2 t. Hence the lifetime 5 := inf{t : X t = A } of X satisfies (00, = (( - t)+ identically; (ii) . there exists a dead path [A] E R with X t ( [ A ] )= A for all t 2 0, so that (([A]) = 0. For all remaining sections, we shall assume that the conditions (20.4) and (20.5) are satisfied. Of special importance in many applications is the specialization to Borel right processes, by which we mean right processes X such that: (20.6i) E is Lusinian (in its original topology); (ii) Pt maps bEo to bE0. It is clear that the Ray-Bore1 a-algebra E in this case is identical to the original Borel o-algebra Eo, and by (17.12), E embeds as a Borel set in its Ray-Knight completion I?. Thus Borel right processes avoid some of the nastier measurability issues that can bedevil general right processes. Unfortunately, Borel right processes are not stable under the kinds of transformations discussed in this and later chapters. We shall from now on use the symbol T to denote the class of optional Superscripts will sometimes be introduced to denote the times over (Ft). classes of optional times for other filtrations. See $23. (ii) (iii)
-+
111
Homogeneity
In this chapter, we explore the optional and predictable projection operations and related concepts in the framework of right processes. We also begin the study of homogeneous functionals, a fundamental theme in the study of right processes, and give some important perfection techniques for such functionals. 21. Measurability and the Big Shift
Recall (8.6~)that & c &", where E" denotes the o-algebra generated by the class U,Sa of functions which are a-excessive for some a. Also from (8.6), we know that every f E E" is an optional function. Consequently, every Bore1 function is an optional function for X. The functions in E" do not in general have the property f ( X ) E Oe-the strictly optional processes over (3;) of (5.2). See (5.12) for a positive result along this line, and (24.35) for another version. One of the irritating and inconvenient facts of Markovian life is that functions constructed by taking expectations of 3-measurable random variables will in general prove to be only universally measurable, and such functions are not necessarily optional for X-recall the example at the end of 519. A weaker notion than optionality, sufficing for many applications, will be discussed in 532. The following definition of a measurable process is the analogue of the definitions of optionality and predictability given in §5. (21.1) DEFINITION. A process Zt(w) defined on R+ x R is measurable in case it is measurable relative to M := (B+8 3)V Z,and it is pmeasurable if it is in MP := (B+8 3)V P .
HI: Homogeneity
107
It is clear that P c 0 c M and P’c U p c MP. It does not seem to be the case that n M p = M , nor even that n v c M . See however (23.1) for a positive result in this direction. (21.2) PROPOSITION. If 2 E U p , then there exists a process 2’ E 0 such that Z - 2’ E P.In fact, 2’ may be chosen strictly optional over (*+). In particular, Up = 0 V P.In the predictable case, the analogous results hold, but with the improvement that 2’ may be chosen strictly predictable relative to
(e).
PROOF:As is well known, the c-algebra U p is generated by stochastic intervals of the form [ O , T I , with T optional over (3;). We noted in (6.13) that there exists an optional time To for with Pp{T # T o }= 0. The stochastic intervals 10,T[ and [IO,T o[ differ therefore by a set in P.A simple monotone class argument completes the proof in the optional case. The predictable case is similar, using the generating stochastic intervals I]O,Tl and x {0}, with r E 3:.
(c+)
(21.3) EXERCISE.Using approximation methods parallel to those of the general theory /DM75,IV-641, check that 0 (resp., the trace of P on I]O,oou) is generated by stochastic intervals of the form [IO,T[I (resp., IJ0,TU)for T E T. (21.4) EXERCISE.Use (21.3) and (6.14) to prove that, for every 2 E 0 (resp., P),there exists Z’, strictly optional over (resp., strictly presuch that 2 - 2’ E 2 (resp., (2 - Z’)lgo,oo~E 2). dictable over (3:)), (The point here is that 2’ is chosen independently of the initial law p. We are using the notation established in $5. What is being shown is that U = 0(e+) V Z and P = P ( c )V Z . See (23.20) for a much better result.)
(e+)
(21.5) PROPOSITION. M = (a+c3 3’)V Z , where 3’ := 3; V 9.
PROOF: A monotone class argument based on (3.9), starting with the case Z ( t , u ) := g ( t ) F ( u ) ,g E bB+, F E b 3 , shows that for every Z E B+ 8 3, there exists 2‘ E B+ @ 3’ with 2 - 2’ E Z. In a number of arguments in this and later chapters, we shall make use of the the device of identifying a process ( t , w ) --t &(u) with a random variable defined on fl := R++x R, the sample space for the forward spacetime process X = (I?,X) of (16.5). We shall suppose for simplicity that X does not have a distinguished lifetime (i.e., A is regarded as an ordinary point of E for this argument), so X has lifetime C(T,W) = r . We shall also suppose, for simplicity, that X is specified in its Ray topology: Bore1 functions on E are really Ray-Bore1 functions. It is obvious that @ . ’ := a { f ( X t ): t 2 0, f E B++@F”} is also equal to B + + @ P . Recalling that F*
108
Markov Processes
it follows that P* = (B++@F”)*, denotes the universal completion of 9, and, following the remark after (A1.5), B++ 8 3*C p . Proposition (21.5) may be interpreted in this setup in the following way: given Z E M , one may choose 2 indistinguishable from 2 so that Z l ~ o , ~ ~ is an j. random variable on fi. Unfortunately, being given G E p , one cannot generally interpret G as a measurable process relative to X, for sets adjoined to P by universal completion do not correspond exactly to evanescent sets. However, the situation described below covers the needs of later applications. (21.6) PROPOSITION.Let G E and suppose that for all w , t -+ G ( t , w ) is right continuous (resp., left continuous). Then (t,w ) + G(t,w ) is in
B+ 8 F*. PROOF:Fix a finite measure Q on (n,9). By Fubini’s theorem, {t : + G(t,w ) is &-measurable} has full Lebesgue measure in R++ . One may therefore select a dense subset D of R++ such that for every t E D ,w -, G(t,u ) is Q-measurable. By one sided continuity, w G(t,w ) belongs to the Q-completion of P for every t > 0. Because Q is arbitrary, w + G(t,w ) is F*-measurable for every t > 0, and another application of one-sided continuity shows that ( t ,w ) -+ G(t,w ) belongs to t?+ 63 F*. w
-+
(21.7) EXERCISE.Let H E p , v a finite meaure on R+,and let Q be . (cf. (4.3)) that ( t , w ) -+ H ( t ,w ) is in the a probability on ( f l , P )Prove v 63 &-completion of B+ 8 9. (21.8) EXERCISE.Let (0,F,3 t , Xt, Bt, P“) be a regular step process, as defined in 514, and let T, denote the time of the nth transition, TO:= 0. Prove that F E bFt if and only if F may be expanded a.s. in the form
F=
cf
TI 7 XT, 7 . * .9 T n ,XT, 1{T,
n (XO7
n>O
-
with f n E E” 8 B+ 8 & 8.. 8 B+ 8 E for each n 2 0. (Hint: use monotone classes to show that the set of such F includes every random variable of the form fo(X0)fl(Xt,). . .fn(Xt,), with fo E bE”, f l .. . f n E bE, 0 < tl < . . . t, 5 t . Make use of (3.9).) (21.9) EXERCISE.Under the same conditions as (21.8), prove that every process Y E P is indistinguishable from one of the form
zt =
fn(XO,Zl,.-.tZn,tATn+l)l{T,O
where Z k ( w ) := (Tk(w),X,(W)), and fn E E” 8 (B+8 E ) , 8 f3+ for every n 2 0. (Hint: use monotone classes, bringing in (21.8) to show that for
111: Homogeneity
109
>0
and every F E bF3, there exists Z as above with Z t ( w ) = on 10, m[xflz.) It is convenient to extend the action of the shift operator B t , which acts on random variables, to an operator Ot which acts on processes, by the formula
every s
F(w)l]s,co[(t>
(21.10)
(Sl
w ) := Z(S - t l w l [ t , , [ ( s ) l
Z being an arbitrary real function on R+ x R. Note the following special cases. For Z := l[T,.o[, one finds OtZ = l[t+Toe,,co[, and for Z := f ( X ) , one has O t Z = f ( X ) l ~ t , , ~ . (21.11) PROPOSITION. For every t 2 0, Ot preserves each of the classes holds identically. Z, M , 0 , P. Moreover, the composition OtO, = PROOF: If Z E 2,then {U
:
s
( o ~ z ( s , w )> ( 0)
= { W : S U P ( Z ( S , B ~ W )> ( 0) a>t
c e;'(n),
where A := {w : sups (Z(S,w ) ( > 0). Since A E N , B;'A E N by the simple Markov property. Therefore, Ot(Z)C 2. To see that O t ( M )C M , observe that if Z ( s , w ) = g ( s ) H ( w )with g E f?+, H E bF, then @ t Z ( s , w ) = g(s
- t)l[t,w[(5)H(~t4
which clearly also belongs to M . Then by the MCT and the fact that Ot(Z) c 1,we get @ ( M ) c M . If Z is rcll and adapted to (&), then so clearly is O t Z . This give O t ( 0 ) C 0 ,by a monotone class argument. The predictable case is similar. The semigroup property of Ot follows directly from that of B t . We shall use the shift OT with argument a random time T with values in [0,m]. Bearing in mind the conventions of 511, the meaning of B,w is [A], the path constantly equal to A, at which all random variables vanish by convention. We do not need therefore to complicate formulas with the term {T < m} in formulas involving 0 ~ . (21.12) PROPOSITION. If T is optional for (Ft), then OT preserves the classes Z, M , 0 and P. PROOF: This is entirely analogous to the proof of (21.11). For example, O T ( ~c) 1 by the strong Markov property, and if & ( w ) = g ( t ) H ( w )with g E B+,H E b F , then @TZ(t,fd)= g(t-T)1{T
+SOBT
E T (by (6.11)) and
Markov Processes
110 22. Construction of Projections
We shall describe in this section the details of the constructions of optional and predictable projections in the framework of a fixed right process (0,F,Ft, Xt,Ot, P”). The projections here differ in a couple of important ways from the general projections described in A5, where an optional process 2 is called the optional projection of a measurable process M in case
Though each of the filtrations (3;) satisfies the usual hypotheses of the general theory of processes relative to (0,F ,P”), direct application of the methods of the general theory would lead us to versions of the projections that would depend on the specific p. If, as here, we wish to have versions of the projections not depending on p, we are forced to work over the filtration (Ft), which does not satisfy the usual hypotheses relative to any P p . Another complication is that the filtration ( F t )induced by a right process possesses a great deal of internal structure due to the presence of the shift operators Ot. The general theory does not take such operators into account. We shall therefore develop the theory of projections paying attention to the structure special to this situation. We start with the particular case in which R admits a splicing map. (22.2) DEFINITION. A map ( w , t ,w ‘ ) -+ w/t/w’ of R x R+ x R into R is a splicing map relative to X provided (22.3)
X,-,(w’)
if s < t, if s 1 t .
The space W of all right continuous maps of R+ into E U {A} admits an obvious splicing map. Observe that for F E P , (22.4) (22.5)
F(w/t/Btw) z F(w), F ((w/t/w’)/t/w’’) F(w/t/w”).
c+,
(22.6) LEMMA. Let F E t < s. Then F ( w / s / w ’ ) = F ( w ) . More then Z t ( w / s / w ’ ) = Zt(w) generally, if 2 is strictly optional over for all t < s. In particularl if T is an optional time over then T ( w / s / w ’ ) = T ( w ) for all s > T (w ).
(c+)l
(c+)l
PROOF: The first assertion is practically obvious by monotone classes, starting with the usual F := fl(Xt,) ...fn( Xt,),0 5 tl 5 ... 5 t , < s.
111: Homogeneity
111
For the second, we may assume that all paths of Z are rcll, and consequently, for t < s, zt(w/s/w'>
=
Zk/n(w/s/w')l[(k-l)/n,k/n[(t) k>l,k/n<s
lim n
Zk/n(w)l[(k-l)/n,k/n[(t) k>l,k/n<s
= Zt(w).
The last assertion is an obvious consequence of the second. (22.7) DEFINITION. If R admits a splicing map, the optional projection by kernel II is defined for M E b(B+ 8 P )
(ITM)( t , w ) := / M t ( u / t / w ' ) PX,(W'(dw'). The domain of II can be extended beyond B+ 8 3"under certain conditions. See (22.21) and (24.35). The justification for calling II a kernel is that II is obviously linear, positive, and it respects bounded monotone convergence. An exact specification of its action is as follows.
e.
(22.8) THEOREM. Let Fl := 3; V Assume that R admits a splicing map, and let M E b(B+ @ F").Then: (i) IIM is optional over (Ft), and IIM agrees u p to evanescence with a process strictly optional over (3;); (ii) IIM E b(B+ 8 3") V 2; (iii) V p , IIM is an optional projection o f M relative to ( R , 3 , F t , P P ) ; (iv) for each t 2 0, I I ( O t M ) = Ot(IIM) identically; (v) if M E b(B+ @ F'), then ITM is optional (but not strictly so) over (Ft+); (vi) if M E b(B+ 8 F o ) ,then IIM is strictly optional over (vii) if M E b(B+ 8 F'),then IIM is strictly optional over (Fi).
(e);
PROOF:The class b P is generated by products H := fl(Xt,) ...fn( X t , ) with fl, . . . , fn E bE" and 0 5 t l 5 t z 5 . . . 5 tn . Take M ( t ,w ) := H ( w ) . Then, setting t o := 0, one finds n
Mt(w/t/w') =
l[tk-l,tk[(t)Fk(u)
k=l
where fo := 1~ and, for k = 1 , . . . ,n, k-1
n
j=1
j=k
-k
l[tn,m[(t>Fn+l(W),
112
Markov Processes
and
n n
Fn+l:=
f j ( X t j( ~ 1 ) .
j=1
It follows that (22.9) I I M ( t , w ) =
k=l j=1 j=1 where h, := fn, h,-l := f n - l P t n - t n - l h n,...,hk := fkPtk+l-tkhk+l for k = n - 1 , . . . , l . Since each hk E bE”, P t r - t h k ( X t ) l [ t , - , , , k ~is ( t a) s . right continuous by (7.4). One recognizes then that II M is an as. rcll version of P@{HI F f } , for all p. That is, for M of the above form, IIM is an optional projection of M relative to (0,F,F:, P”) for every p. For f E bE” and t > 0, (s,z) + Pt-, f (x) is ( B ( [ O , t ]€)3 €)*-measurable by (18.7). Thus (16.4) and (5.13)enable the application of (5.12) to prove (i). Observe also that if the f j are all in bE’ (= bE or bEe), then, since this a-algebra is preserved by Pt, I I M belongs to b(B+ @F’)V Z , I I M is adapted to (F:), and is a s . rcll. According to (5.8), this implies that IIM is optional over (F:+), proving (ii) and (v). The class b ( B + @ P )is generated by products N ( t , w ) = l [ , , b [ ( t ) M t ( u )with , 0 5 a < b 5 00 and M of the above form. One then finds N ( t ,w)= l [ a , b [ ( t )M n ( t ,w) and, as in the classical construction of optional projections, but taking account of (6.3),it follows that IIN is an optional projection of N relative to (0,Fp,F:, Pp) for every initial law p. In addition, for fixed s 2 0,
@ s N ( t ,w)= l[a+s,b+s[(~)H(Qsw). Inspection of the formula for II M shows then that II(@,N) = @ , ( I I N ) identically. An application of the MCT then completes the proof of (iii) and (iv). The last two assertions are more delicate. Because we are assuming (20.4) that t -, X t ( w ) is rcll in Ea for every w E 0, ( t , w ) + Xt(w) is in 0’. For every f E C,(E), Pt f (z) is in E as a function of z for every fixed t , and right continuous in t for every fixed 2. It follows from (3.13) that ( t ,x) + Pt f is in B+ @ E . Composing these measurable maps shows that ( s , w ) + P t - , f ( X s ( w ) ) l [ o , t [ ( sE) 0’ for every t > 0, f E bE. With this observation, we avoid the use of (5.8), and continue as in the case above to conclude that for F E bF” and h E bB+, II ( h I8 F ) E O0.The proof of (vi) is now accomplished by an appeal to the MCT. By (vi), for g E bE”, h E bB+ and F E bF”, II ( h @ ( g ( X o ) F ) E ) 0 ,where 0’ := O 0 V ( B + I 8 e ) is the a-algebra of strictly optional processes over (Fi). One more call to the MCT gives (vii).
III: Homogeneity
113
Let 0' denote the a-algebra of strictly optional pro(22.10) COROLLARY. cesses for the filtration (Fl).Then, for every 2 € 0 , there exists Z' E 0' with 2 - 2' E 2.
PROOF:We may assume 2 bounded, as the general case will then follow by a simple limiting procedure. As 2 E bM,there exists M E B+ @F with 2 - M f 2. By (21.5), for every M E b(B+@F)there exists W E b(B+@F') with M - W E Z. Since 2 E 0 , 2-II W E Z. The theorem asserts though that 2' := II W E 0'. REMARKS. The formula (22.7) for the optional projection kernel is called Dawson's formula. The kernel II is a kernel from (R' x R, 0 )to (R+x R,B+ @ P). It would be very agreeable if II were also a kernel from (R+x R, 0 )to (R+x R, M ) enjoying the properties listed in (22.8). This does not seem feasible, one difficulty being that, if M is evanescent, the right side of (22.7) need not even be defined for all ( t , w ) . It is the case , that II extends automatically to the universal completion of ( B + @ P ) but then II M would only belong to the universal completion of 0. This is not good enough for many applications. See $32, though. Item (v) of (22.8) can be strengthened somewhat, but not all the way to a direct analogue of (vi). See (23.6) and (24.35). Suppose now that R is a general sample space, not necessarily admitting splicing operators. The map which associates to Mt(w) := l[.,br(t)H(w) ( H := f~(Xt,) ...fn( Xt,) as in the last proof) the process l[,$q(t)Kt(w), where Kt is defined to be the right side of (22.9), extends to a kernel II from (R+x R, 0) to (R+x R, B+ @ P ) which realizes the optional projection relative to every P p . The proof of this statement is identical to that of (22.8). The only difference is that II lacks in this case the compact expression (22.7). However, II does enjoy all the properties listed in (22.8). For every 2 E b M , there exists "2 E b 0 (resp., (22.11) THEOREM. PZ E bP), unique u p to evanescence, such that for every p, "2 is an optional projection (resp., PZ is a predictable projection) of Z relative to (R, F f , Pp). For every t 2 0, one has, u p to evanescence, (22.12) (22.13)
O ( 0 , Z ) = O,(OZ); p(@tz)lnt,co[
= @t(PZ)lnt,,u.
PROOF:Because of (21.5), for every 2 E b M = b(B+ @ F V Z ) ,there exists 2' E b(B+ @ Fu)such that 2 - Z' E 2. Then II 2' is a version of the optional projection of Z', hence of 2, relative to every Pp. For every t 2 0, OtZ and 0tZ' are indistinguishable because of the Markov property. The fact that @tII 2' = II OtZ' identically then implies (22.12), taking IIZ' for '2. In the predictable case, it is enough by monotone classes to consider
114
Markov Processes
the case 2 E b(B+ @ F ) .If 2, = l[a,w[(t)fl(Xtl)* . . f n ( X t , ) with u 2 0 , 0 5 tl 5 t 2 5 . . . 5 t , and f ~. .,. ,f, E b&”, we have, as in the proof of (22.8), II 2(t,w ) = 1ia,,.[(t)Mt(w), where Mt is a.s. right continuous and a martingale relative to every P p . Since M t -(u) exists for all t > 0 except for w E h E N ,we may define
’z(t,w):= I[a,w[(t)Mt-(w)1AC(w)l]O,oo[(t)4-l[a,oo[(t)l{O}(t)MO(w) to obtain the predictable projection of 2 relative to every Pp. The identity (22.13) is easily checked from this last formula. (22.14) REMARKS. After we study the Ray-Knight compactification procedure in Chapter V, we shall be in a position to describe a predictable projection kernel fi having properties analogous to those of n. See $43. (22.15) COROLLARY. If T E T and 2 E b M , then up to evanescence, (22.16) (22.17)
O(OT2) = oT(o2);
p(@!rz)lnT,oon = @T(PZ)ln!r,oou.
PROOF: We shall prove only (22.16),the proof of (22.17) being quite similar. In view of (6.14), we may assume that T is optional over (F:+). Given p, there exists, by (21.5), a process 2’ E b(B+ 8 9) for which 2 - 2’ E P n I ” , where u(f) := Ppf(XT), f E bE”. Starting with the wherea 2 0, 0 5 tl 5 t 2 5 5 t, case 2; = l[a,w[(t)fl(Xtl)...fn(Xtn), and fl,. .. , f n E C d ( E ) , a simple calculation yields 0 ~ 2E’ b(B+ @F). In order to get (22.16) in this case, it is enough, by right continuity, to prove the identity (@Tz’)1nT,w IJ = 2/11 nT,oo[. Since T is optional over (?+), (22.18)
(22.6) yields
T(w/t/w’) = T ( w )
Vt
> T(w).
Therefore, in computing n ( O ~ Z ’ ) ( t , wwhen ) t (22.18) that for t > T ( w ) ,
rI (OTZ’)(t,w ) = =
J
> T ( w ) , we
find from
(OTZ‘)( t ,w / t / w ’ ) PX,(W)(dw’)
J z’(t - T ( w ) ,e,(,)(w/t/w’))
PX“(”)(dw’)
= @T(rIZ’)(t,w).
The same identity holds, by monotone classes, for all 2’ E b(B+ @I p ) . We obtain (22.16) then by observing that, since 2 - 2‘ E 1”n Z V ,OTZ -
III: Homogeneity
115
0'2' E P by the strong Markov property, O(OTZ)- ~ ( O T Z 'E) 1'"by definition of optional projection, and similarly "2 - II 2 E Zp n Z" implies O'(OZ) - OT(II 2') E Z p , so that ~ O T Z-) OT(OZ) E P. (22.19) REMARK. Optional and predictable projections are linear, positive operators modulo evanescent processes, so the results of (22.11) and (22.15) extend to all positive 2 E M . (22.20) EXERCISE. Find a natural way to extend the kernel II to as large as possible a 0-algebra M n 3 t3+ 8 F" in such a way that II has the properties listed in (22.8). (Keep (22.10) in mind. The following exercise contains one possible candidate.) (22.21) EXERCISE. Let M n denote the smallest u-algebra on R+ x R containing B+ 8 FU,all processes of the form f(X) with f E bEe, and all processes of the type ( t , w ) + l[o,,[(t)Pg-tf(X,(w)) with s > 0 and f E bE". Show that for every M E b M n , IIM is well defined and
(i) (ii) (iii) (iv)
Il M is a version of OM; Ot(Mn) c Mn and IIoO, = O,oII identically on Mn; II ((nM ) N ) = II M . IIN identically for M , N E bMn; TZMn c M n and II (IIM ) = II M identically.
23. Relations between the u-Algebras
We may now use some of the results of §22 to set down some important relations between some of the fundamental u-algebras on R+ x 0. (23.1) THEOREM. Let 2 E M be nearly optional (resp., nearly predictable). Then 2 E 0 (resp., p ) . In fact, in the notation established in $5, M n (fl,O'z) = 0 and M n (n,Pez) = P . In particular, every process adapted to (Ft)which is a.s. right continuous is necessarily in 0.
PROOF:It suffices to suppose 2 E b M . In view of (22.15), O Z is a version of the optional projection of 2 relative to Px.Since 2 E 0'. , Z-OZ E 1'2. Because 2 - O Z E M , it is legitimate to integrate relative to an arbitrary p to prove 2 - "2 E Z and consequently Z E 0. An entirely analogous argument is valid in the predictable case. For a 2 which is a s . right continuous and adapted to (F,), obviously 2 E M , and (A5.5iv) asserts that 2 E 0'" for all p. Hence 2 E 0. (23.2) THEOREM. Let T E T. Then T is a predictable time relative to ( R , F f , P P ) for all p if and only if the graph [ T J of T is in P. If T is predictable, there exists an increasing sequence {T,} of predictable times over (3t) such that T, increases to T a s . and, for all n, T, < T as. on {T > 0). In fact, the {T,} may be chosen so that supI (e-Tn - e-') 5 1/n
116
Markov Processes
for all n. In particular, the announcing sequence {T,} for T may be chosen independently of the initial law p .
PROOF:If T E T, then [TI] E M and so, by (23.1), [TI E n,P@ if and only if [TI] E P. Suppose now that [TI] E P,and let cp(t) := 1 - e-t. Set Mt := (cp(T)- cp(t))+= q ( T ) - cp(T A t ) . Then M E b M , and since O M cp(T A t ) is a the process cp(T A t ) is continuous and adapted to (Ft), right continuous version of Pp{cp(T) IFt} for every p. It follows that "M is a positive, right continuous supermartingale relative to every P p . Let S := inf{t : "Mi = 0). By standard properties of supermartingales, a.s., OMt = 0 for all t 2 S and "Mt- > 0 for all t < S. By optional sampling,
+
P p { " M ~ l ~ ~ < o= o }Pp{Mrlp- 0). As
"Ms,~{s,<~)= P p { M ~ , l { ~ n < o oI F&} ) =P'"{v(T) - c ~ ( s n ) I and cp(T) > cp(S,) Pp-a.s. on { S , > 0 ) E F;,, one gets "Ms, > 0 Pp-a.s. on {S, > 0). Therefore, Pp-a.s., S, 5 S and so T 5 S. This means that, Pp-a.s., T = S. But, since cp(T) - cp(Sn) -+ 0 PJ'-a.s. as n + 00, OMS, + 0 Pp-a.s. as n -+ 00, and so "MT- = 0 Pp-a.s. on {T > 0). Since p is arbitrary, we have proved that T = S and "MT-= 0 a.s. on {T > 0). Define T, := inf{t : "Mt 5 l/n}, so that T, 5 T and T, < T a.s. on {T > 0). The inequality 'MT, 5 1/n on {T, < oo} gives us, using the defining property of optional projections, l/n 2 Pp{"MT,;Tn < m} = P/"{MT,;T, < W } = Pp"((T) - v(Tn)}. That is, T, announces T relative to all P p simultaneously. We shall investigate the characterization of predictable times more fully in $44. The next definition uses the notation D, & established in $5 for the classes of processes which are rcll (resp., lcrl) and adapted to the specified filtration. (23.3) DEFINITION. Let Fi := V C . Define 0-algebras on R+ x R by: (i) R = constant adapted processes := { ~ ( X:Of )E b&"); : f E &}, X" := {f(x) : f E 6"); (ii) X0 := {f(x) (iii) Oo := o ( D ( e ) ) , Po := Oe := o { D ( F f ) } , Pe :=
4J33f)I;
0{&(e)};
(iv) 0' := o{D(Fl)}, 'P' := u{L(F{)); (v) (3* := a{D(F,*)), P* := .{C(F;)); (vi) oS:= p e v Ee.
111: Homogeneity
117
The items in (iii), (iv) and (v) are special cases of the strictly optional and strictly predictable a-algebras defined in (5.2). It is clear from the Blumenthal 0-1 law (3.11) that every optional process with paths a.s. constant in time is necessarily in R VZ. According to (5.8), if 2 is 8.5. rcll and adapted to , then Z is indistinguishable from a process strictly , which is not as strong as saying 2 E O0 V Z. optional over The a-algebra 0" is relatively useless because fox is only 8.5. rcll for f E Sa. It is not in general the case that f o x E 0". That is X" is not in general a subset of 0". The a-algebra 0' of (vi) is an appropriate replacement for 0". See (24.35) for further properties of O S .
(e) (e+)
(23.4) THEOREM.(i) M = R V (B+ 63 9) V 1 = B+ 63 F'; (ii) Po C P' C P, (3' C 0' c 0, Pa C 0", P' C 0' and c X" c 0; (iii) 0 = R vPO v X " V Z = R v 0' V Z = P vXO.
xo
PROOF:(i) is a direct consequence of (21.5), and the assertions in (ii) are obvious, given (5.2). Turning to (iii), i f f E bE and t > 0, then ( s , w ) -t Pt-sf(Xs(w))lro,tr(s) is in Po V Xo by the following argument. The map ( s , w ) -+ ( s , X , ( w ) ) is in Po V Xo/B+ B E , for if cp E bB+ and g E bE, then (3, w ) cP(s)g(X,(w)) is in Po v X". BY (18.71, Pt-sf(.)l[o,t[(s) is in is in B+ 8 E . Thus, by composition, ( s , w ) -, p,-sf(X,(w))lro,tr(s) Po V Xo. The proof of (22.8vi) now shows, after trivial modifications, that H(B+ 63p) C Po VX". Using (i), we get the first pair of equalities in (iii). The last two equalities follow from the first because Po V Xo V 2 C 0' V 2. -+
+
(23.5) COROLLARY. For every T E T
where
X;
e-
:=
{H E
,
9 : 3 Y E Po with H~{T<,,)= YTl{T<m)}
:= a{f(xT)1{T<m} : f E E } .
PROOF:Direct from (23.4), using the definitions of FT-and
and
e-.
(23.6) COROLLARY. The optional projection kernel II is a kernel from
(R+ x $2,P0 V X") to (R+ x R,B+ 8 p ) ,and from (R+ x $2,0') to (R+ x $2, B+ 63 F"). PROOF: The proof of (23.4) also shows the first assertion of (23.6). The second is proved in exactly the same way, replacing the superscript by the superscript everywhere. All that is needed is the fact that for f E bEe, (s,z) -+ Psf(z)is in B+ @ E", hence by composition ( s , w ) -t Pt-,f(X,)lIo,t((s) is in oS:= P" v
x".
118
Markov Processes
(23.7) EXERCISE.Using (23.21, prove that for every predictable time T over (3t), 1-1~34-= FT-. (Actually, nzPTz-= 3 ~ -Just . Jet A E n,3?and consider 1 ~ 1 [ 1 ~ It , ~is, n of . course, trivial that n F , = 3 ~ .The ) result of this exercise is in fact true for T an arbitrary optional time over (Ft), but the proof is considerably more difficult. See (31.14). (23.8) EXERCISE. For T an optional time over ( F f ) ,3; = 3;- V X$. Fussy results like (23.4) are needed because of the conflicting demands imposed by different methods in Markovian theory. On one hand, results based essentially on martingale methods require augmented filtrations, completed a-algebras, evanescent processes and the like, and mandate the definitions of M , 0 and P used here. On the other hand, direct manipulation of the sample paths by killing, shifting and splicing can go awry when we adjoin null sets and evanescent processes, and require the use of the use of the unaugmented a-algebras O0,Po,etc. (The optional projection kernel II is a case in point-its domain is definitely smaller than the general optional projection operator.) We wish to retain the flexibility to use the most appropriate technique, and the profusion of a-algebras is an inevitable consequence. Arguments involving path transformations require that the underlying sample space fl mimic a space of paths. This will involve, among other IVthings, that R is equipped with stopping operators (at)t>o [DM75, 951. If Cl were a genuine space of paths, then atw(s) := w ( s A t ) . In general though, we must define stopping operators axiomatically.
(23.9) DEFINITION. A family (at)t?o of maps of fl into itself is a family of stopping operators provided the following hold, identically on Cl: Xs(atw) = xsAt(w);
at(asw) = atAs(w);
at(esw) = es(at+sW).
The first of these identities shows in particular that at E f l / F " , hence by the usual completion argument that at E 3;/3*.
(23.10) DEFINITION. A sample space R for X is of path type provided: (i) V w E 0,t + X t ( w ) is right continuous in both the original topology and the Ray topology on E A , and has left limits in a Ray compactification of E A ; (ii) Cl admits a lifetime C = inf{t : X t ( w ) = A}, with X , ( w ) = A for all s 2 6, w E SZ; (iii) R admits shift operators 8t and killing operators kt satisfying (1 1.4-6); (iv) R admits splicing maps ( w , t , w ' ) -, w / t / w ' (22.2) satisfying the stronger conditions w/t/$tw' = w,
(w/tJw')/t/w'' = w/t/w";
111: Homogeneity
119
(v) R admits stopping operators at (23.9) with at(k,w) = k,(atw); (vi) for every x E E A , there exists a constant path [x]E R satisfying Xi([.]) = z for all t 2 0. In particular, R contains a dead path [A] with [([A]) = 0; (vii) the shift, killing stopping and splicing operators above satisfy all general identities that the corresponding operators on the canonical space of all right continuous maps of R+ into E A admitting A as a trap would satisfy-for example, at(w/t/wl) = wltl[Xo(41, * * * (23.11) THEOREM. Let R be ofpath type. Then: (i) Po = {Y : & ( w ) = cp(t,k t ( w ) ) for some cp E B+ @ P } ; (ii) B+ B F O = { Z : z t ( w ) = cp(t,k t w , etw) for some cp E B+ 8 Fo 8
PI;
(iii) Oo = { Z : Z t ( w ) = cp(t,a t ( w ) ) for some cp E
B+ 8 9).
PROOF:Though our hypotheses are weaker than those in [DM75, IV-971, their proof carries over word for word to prove (i) and (iii), and to prove To that in (ii), every process Z of the form cp(t, ktw, &w) is in B+ 8 9. check the reverse inclusion, it is enough to show that every process of the form ( t , ~+) f(Xto(w))(independent o f t ) with f continuous in the Ray topology, is of the specified form. To see this, write f(Xt0) = f(Xto)l{O}(t)+ f(xto)1]o,to](t)+ f(XtO)l]tO,m[W The first and third terms in the display may be written as p(t,ktw,Otw’), with cp(t,w,w’) := l{o}(t)f(Xt,(w’)) in the first case, while in the third, cp(t,w , w l ) := l~~,,,[(t)f(Xt,(w)).The second term may be expanded as
--
c r n
lim
IL+W
f(Xt+,-k/Zn (~))l[k/z”,(k+l)/z“(t)r
k=O
a typical term of which may be expressed in the form cp(t,ktw, Btw’), with cp(t,w , w ’ ) := l ~ ( t ) f ( X ~ ~ - ~ (for w ’some ) ) subinterval J of 10, to] and some 0 I a 5 to. REMARK.It is evident from the proof that the corresponding results hold if Po, Oo, F o , etc., are replaced by P e , Oe, F“,etc., defined in (23.4). (23.12) COROLLARY.Up to evanescence, bM is generated by products of the form f(X0) g(t) H o k t KO& with f E bE”, g E bC(R+), H,K E b p .
PROOF: Use (23.1) and (23.11). We make occasional use below of the “min” of a set in R+-see
(5.9).
120
Markov Processes
(23.13) DEFINITION.Let R be ofpath type. The class To (resp., T”, T’, T* consists of those random times T having a representation T = minA, (resp., (F;), (3;) := for some A C R+ x $2, progressive relative to (30” v (3:)). It is clear that To c T’c T” c T*. We emphasize that the progressive processes in (23.13) are not augmented by 2. Recall that F: is the universal completion of Eland it is the same as the universal completion of 3;or 3;.For example, setting A := [ T J ,we see that every (*)-optional time is in To. However, it is not in general true that optional times for are in T*.
(e)
el,
(e+)
(23.14) LEMMA.T E T* if and only if T is an optional time for the filtration (3;).
PROOF:For T E T*,it follows from (5.11) that T is an optional time for (3:). For the converse, it suffices to remark that T = min [ T I , the latter set being progressive relative to (3:). Recall that O*denotes the class of processes strictly optional over (3:). Given T E T*, define (23.15)
3; := { F E F* : 3 2 E O* with 2 ~ 1 { ~ = } .
The following result contains most of the properties needed in algebraic manipulations of F and T*.
(23.16) PROPOSITION. Let s1 be ofpath type. Then: (i) for every right continuous path cp : R+ -, E A , C := { w : X,(atw) = p ( s ) V s 5 t } is an atom of (a,*) and every F E 3; is constant on that atom; (ii) for F E 3*and t 2 0, Fout E 3:; (iii) for T E p3*, T E T*if and only if (23.17)
U ~ W = QW’,
T(w)5 t
I T(u’) = T ( w ) ,
and therefore, for T E T*,U T ( ~ ) W= aT(w)w’ (iv) for T E T*, F E 3*is in 3; if and only if
(23.18)
&
T ( w ) = T(w’);
* F ( w ) = F(w’);
U T ( ~ ) W= U T ( ~ ) W ’
(v) for T E T*, F E 3*j FoaT E F;, and F E 3; + F = FOaT, hence b3; = {FoaT : F E b3*}; the universal completion of pT:= { F E (vi) for T E T*,3; = 9: 3 2 E Oo with Z T ~ { T <=~ F} l { ~ < , i } ; (vii) T E T*, F E b 3 * 3 ( w ‘ , w ) -,F(w’/T(w’)/w) is in (3;8 3*)*; (viii) for every T E p 3 * and G E 3*, G O & E 3*;
e,
(ix) for S , R E T*,S
+ Roes E T’.
111: Homogeneity
121
PROOF:(i): It is clear that C is an atom in (Q,E). Let Q be the measure putting unit mass on the atom C. Extend Q automatically to a on (0, probability on (0,3:). If A E 3: and A n C # 0,then by the definition of completion relative to Q , A n C = C since every proper subset of C in is empty. To complete the proof of (i), take A := { F = c } for an arbitrary constant c. (ii): Let F E F and let Q be a probability on (Q, 3*). Choose F1, FZ E 3” with F1 5 F 5 F2 and atQ(F2 - F I ) = 0. Then Float 5 Foat 5 F20at and Q(F2oat - Float) = 0. Clearly, Fjoat E f i ,and since Q is arbitrary, this shows that Foat E 3;. (iii): For T E T*,(23.17) is an immediate consequence of (ii). For the and (23.17) shows that converse of (iii), note that F := 1{T
e)
3OTc
e.
Markov Processes
122
and so by the usual completion argument, eT E 3*/F, as is in 3*/p, claimed. (ix): By (viii), S+RoBs E p3*. We prove T := S+RoBs E T* by checking (23.17). Let atw = at,’ and R(w) 5 t. Set s := S(w) 5 t , so that (iii) implies S(w’) = S ( w ) = s. Because atw = atw’ +-at-,(B,w) = at-s(Bsw’) and R(B,w) 5 t - s, a second application of (iii) yields R(8,w’) = R(B,w), hence T ( w )= T(w’). The type of decomposition of an optional time described in (23.19) below was first discussed by Courrkge and Priouret [CP65a]. See also [GShBlb]. (23.19) PROPOSITION. Let R be of path type and Jet T E T’(resp., To, T”, T*). Set H ( ( t ,w ) , w’) := T ( w / t / w ’ ) . Then: (i) H E (Po8 30)’ (= (P*8 F*)*); (ii) V s > 0,Vw’ E i2, w + [H(s,w’),w) - 3]+ E To (resp., To, T”, T*h (iii) T ( w )= H((s,w),B,w) for all s 2 0, w E R.
PROOF:(i): In all cases, T is an optional time over (3:), so it suffices to prove that for all F E bp3*, ( ( s , w ’ ) , w ) + F ( w ’ / s / w ) E (P’ 8 37,. By monotone classes, ( ( s , ~ ’ ) , t , w ) + F(w’/s/w) is in Po 8 3” for every F E bF0. Let $((s,w’),w) := w’/s/w so that 11, E Po 8 3”/Fo. The standard completion argument (see A3) then proves $ E (Po8 p ) * / F * , as claimed. (ii): We shall give a detailed proof for the filtration (3;) and then indicate the changes that must be made in the other cases. Let T := min A with A progressive over (3;) and let D := inf A. Fix now s > 0 and w’ E R. For F E b3;, w + F(w‘/s/w) is in bqT-,)+, by a monotone class argument starting from F := fo(Xo)fi(Xt,). . . f,(Xt,), tl 5 5 t , 5 T , fo E b€”, fi, . . . ,f, E b€. If Z t ( w ) := h ( t ) F ( w )with h E bB([O,r]) and F E b3:, then ( t , w ) -+ Zt+s(w‘/s/w) = h(t+s)F(w’/s/w) E B([O,T I ) 63 by the argument in the preceding sentence. Thus by monotone classes, ( t , w ) l { t l r ) -+ Z t ( w ) := Z ~ + ~ ( W ’ / S / W ) E B ( [ o , T ] )8 f l for every Z progressive over (3;). (The process 2 is progressive relative to (3:)in the next two cases, by simple modifications of this argument.) Let Z := 1~ and let T denote min(2 = 1). Clearly, T ( w ) = ( T ( w ‘ / s / w ) - s)+, proving (ii) except in the case T E T*, which is even easier. For T E T*, (i) implies that for all (s,w’), w + [ H ( ( s , w ’ ) , w )- s]+ is F-measurable. We verify (23.17) for this random variable. Suppose [ H ( ( s , w ’ ) , w ) - s]+ 5 t and atw = atw”. Then T ( w ‘ / s / w ) 5 t s and at+,(w’/s/w) = at+,(w’/s/w‘’), hence (23.16iii) gives T(w‘/s/w‘‘) = T ( w ’ / s / w ) , provingin turn that [H((s,w’),w)-s]+ = [ H ( ( s , w ’ ) , w ‘ ’ ) - s ] + . This proves (ii) in this case, by the converse of (23.16iii). Item (iii) is of course a triviality. 7
.
.
e,
(c),
+
111: Homogeneity
123
(23.20) THEOREM.Let R be of path type. Then for every T E T, there exists T’ E T’with T = T’ almost surely.
PROOF:Since lUT1 E 0 , and we may use (23.4iii) to find 2’ E 0’ indistinguishable from l I T 1 . Then T’ := min{Z’ = 1) is a.s. equal to T and T‘ E T’by definition. (23.21) REMARK.The last result may be a little surprising because, unlike our previous attempts (6.14) to choose a good version of an optional time, it demands no infinitesimal peek into the future. The result should be thought of as the strongest available form of the Blumenthal 0-1 law (3.11). It will be very useful in the construction in $62. (23.22) EXERCISE.Show that min A is a predictable time for every A E P. (See (A5.5~)and the subsequent remarks. The min can not be replaced by inf .) 24. Homogeneous Processes and Perfection
The notion of homogeneity recurs in many guises in the theory of (temporally homogeneous) Markov processes. (24.1) DEFINITION.A real process 2 on R+ x R is homogeneous on R+ (resp., R++) in case it satisfies (24.2) (resp., (24.3)) for every T E T: (24.2) (24.3)
OTZ and 2 1 ( 1 ~are ,~n indistinguishable; ln~,~ijO and ~Z Zln~,,n are indistinguishable.
When spelled out, (24.2) requires that, for every T E T, (24.4) {w : Zt+T(w)(w)# Zt(OT(,)(w))for some t 2 0, T(w)
< oa) EN.
Condition (24.3) weakens this to require only “for some t > 0”. The easiest case to understand is perfect homogeneity. (24.5) DEFINITION.A process Z is almost perfectly homogeneous on R+ (resp., R++) provided (24.6) (resp., (24.7)) holds: (24.6) (24.7)
2 0, s 2 0) E N {w : Z t ( d y w )# Zt+,(w) for some t > 0,s 2 0) E N . { w : Zt(Oyw)# Zt+,(w) for some t
If the sets in (24.6), (24.7) are empty rather than null, then the term perfect is used in place of almost perfect. Where the definition of homogeneity permits the exceptional set (24.4) to depend on T , almost perfect homogeneity demands the existence of an
124
Markov Processes
exceptional set independent of the particular T . Our use of the term perfect is not completely standard-in the literature, what we have dubbed almost perfect is more commonly called perfect, and there is no standard term for our use of perfect. Perfect homogeneity appears simpler than the other varieties mentioned above. However, constructions such as optional and predictable projections do not respect perfection unless stringent measurability conditions are satisfied. See for example (24.12). (24.8) EXERCISE.Let R be a random time with values in [ O , o o ] . Then 2 := l ~ R , o ois~ perfectly homogeneous on R++if and only if s R(B,w) = R ( w ) V s for all s, w , or equivalently, R O O , ( R- s)+.
+
=
A process 2 is perfectly homogeneous on R+ in case 2, = Z o o & for all t 2 0. For example, if f E bE", then Z t ( w ) := e-(s-t) f ( X , ( w ) ) ds is perfectly homogeneous on R+.This particular Z is not adapted to (3t).
st"
We shall see later that, as this example suggests, homogeneity actually reflects dependence on the future only. Under mild conditions (see (24.27)), 2 E 0 is homogeneous on R+ if and only if Z - f ( X ) E Z for some optional function f. The process f ( X t ) is, of course, perfectly homogeneous on R+.Homogeneity on R++ is a more delicate matter. It is clear from the conditions (24.3) and (24.7) that the value of 2 at t = 0 is irrelevant to homogeneity on R++,and it is sometimes assumed that the process in question is defined only on R++x 0. The simplest interesting example of such a process is, perhaps, Zt := f(Xt)-, with f E Sa.(Because t + f ( X t ) is only a.s. rcll, & ( w ) has meaning except on an evanescent set of ( t ,w). The value of & ( w ) on that exceptional set may be chosen arbitrarily.) This 2 is then almost perfectly homogeneous on R++.Perfect homogeneity could have been achieved by setting instead 2, := limsupSTt,f(X,). In fact, f could be an arbitrary real function on E in this case. Simple examples suffice to convince one that the latter 2 is not necessarily homogeneous on R+.For X a process with finite lifetime C and f := l ~one, has 2 = l n o , c j , which is not homogeneous on R+. The difference between almost perfect and perfect homogeneity is not very great, absent other measurability properties one may wish to preserve. (24.9) PROPOSITION. Let 2 be almost perfectly homogeneous on R+,and let a,-, := { w : Zt(8,w) # Zt+B(w) for some t , s 2 0) E N . If R ( w ) := inf{t : &w E RE}, then R o e t = (R-t)+ identically, and8,w E 08 foralls 2 R ( w ) . The process &(LO) := Zt(w)ll~(w),m[(t) is perfectly homogeneous on R+, and indistinguishable from 2.
PROOF: Since R ( w ) = inf{r 2 0 : Zt(es/3,-w) = Zt+,(&.w)Vt,s
2 0 } , it
111: Homogeneity
125
follows that
u
+ R(Q,w) = + inf{r 2 0 : Zt(8,Q,8,w) 21.
= inf{v
= Z,+,(6,8,w)Vt,
s 2 0)
2 u : Z ~ ( I ~ ~=+Zt+s(6,w) ~ W ) V t , s 2 0)
= R ( w )V
U.
This proves all assertions save those in the last sentence. As it is obvious it remains only to prove that 2 is perfectly that R vanishes except on 00, homogeneous on R + . For t 2 0,
ZO(6tw) = ZO(6tW)l[R(e,w),~[(o) The second term on the right vanishes unless R ( 6 t w ) = 0, which is equivalent to the condition R ( w ) 5 t . However, in the latter case, Btw E 06, and
zo(etw) =z &)
=
z&).
(24.10) EXERCISE. Modify the statement and proof of (24.9) in such a way that homogeneity on R+ is replaced with homogeneity on R++. It will be necessary to modify the definition of Ro slightly.
A subset M c R+x R is called homogeneous (with qualifiers) in case its indicator is homogeneous (with the same qualifiers). For example, for B E E , the occupation time set {X E B } := {(_t,w) : Xt(w)E B } is perfectly homogeneous on R + . More generally, if E is a Ray compactification of E , and if r E E , then the random set { ( t , w ) : t > O,(Xt-,Xt) E r} is perfectly homogeneous on R++, but not in general on R+. (We are assuming, as usual, that X has been modified according to the conventions of 520, so that X t - exists in E for all t > 0. The last two examples are both optional random sets. The following example exhibits other interesting possibilities.
c@
(24.11) EXERCISE. Let X be a standard Brownian motion on R, and let M := { ( t , w ) : X t ( w ) = 0). Let MD be the random set whose wsection is the set o f t > 0 such that (t,w) E M but ( t u,w) 4 M for all sufficiently small u > 0. (MDis called the set of left endpoints of the intervals contiguous to {X = O}.) Show that MDis perfectly homogeneous on R+, and progressively measurable. Show, using the SMP, that MD contains the graph of no optional time, and conclude that MD4 0.
+
Note the following direct consequence of (22.15) and (22.8). (24.12) PROPOSITION. Let Z E b M U p M . Then:
(i) if Z is homogeneous on R+(resp., R + + ) then "Z is homogeneous on R+(resp., R++)and PZ is homogeneous on R++; (ii) if Z E b M n U p M n (as defined in (22.21)) and if Z is (almost) perfectly homogeneous on R+ (resp., R++), then so is KZ.
Markov Processes
126
A result similar to (ii) can be proved for perfect predictable projections. See (43.5). Where optionality and predictability have to do with measurability on the past, the two kinds of homogeneity turn out to correspond to the analogues for measurability on the future. See $26 for a brief discussion. The papers of Az6ma [Az72a], [Az72b], [Az73] give a complete and very useful description of the reversal of time at a finite lifetime, and the natural prescriptions of optional and predictable processes in the reverse time direction, using the notion of homogeneity. Two special classes of homogeneous processes may be singled out as the analogues of optional and predictable processes. In order to retain the flavor of the original F’rench, they will be denoted Ijd,,cjg (droit et gauche). (24.13) DEFINITION. f j d (resp., fig) is the a-algebra ofprocesses defined on R+ x R (resp., R++x R) generated by rcll (resp., lcrl) processes in M which are perfectly homogeneous on R+ (resp., R++), together with 1. (24.14) PROPOSITION. Let Z E M be 8,s. rcll (resp., lcrl) and almost perfectly homogeneous on R+ (resp., R++).Then 2 E Ad (resp., 49).
PROOF: Modify the proof of (24.9) (resp., (24.10)) so that Ro is replaced by {w : Zt(8,w)# Zt(8,w) for some t , s 2 0, or t + Z t ( w ) is not rcll}, which still has the property w E fig + 8,w E 06 for all u 2 0 . In order to see that the requirement of perfect homogeneity in (24.13) is not too stringent, we need to make a detour at this point into some perfection techniques. The prototype of the method is a familiar result from the elementary theory of stationary sequences, in which one is given a shift operator 8 with iterates 8, on a complete probability space (0,4, P), and one defines the invariant a-algebra for 8 as {G E 0 : Go0 = G}. A random variable H is called almost invariant in case, as., H O B = H . It follows that Hoe, = H a s . for every n 2 1, and consequently I? := limsup, Hoe, satisfies = H as. and H o e = identically. The variable H is the the perfection of H. In the continuous parameter case, in which one is given a semigroup (8,) of shift operators on (0,G,P),the simple method above must be modified as follows. If H E bB and if, for every t 2 0, one has a.s. Hoot = H, (the exceptional set depending on t ) ,set
H ( w ) := lim esssup(H(8,w) : s 2 t}. t+m
Suppose, as one may, that it can be proven (cf. (4.3)) that (t,w)+ H ( 8 t w ) is in the 1 ~ P-completion 3 of D+ 8 8. It is then an elementary consequence of Fubini’s theorem that, for a.a. w, H ( w ) = H(8,w) for Lebesgue a.a. s 2 0, so P{H # H } = 0. In particular, H E 8. On the other hand,
H(erw) = t+m lim esssup{H(O,+,w)
:s
2 t}=H(w)
III: Homogeneity
127
for all T 2 0, w E 0. The perfection results below are obtained by simple modifications of the technique described above. As our first example, take X to be a right process with lifetime C. In (14.2), the left germ field 3[ic-1 at C was defined by 3fic-] := {H E 3:H o o t = H a s . on {t < C} for all t 2 0},
the exceptional set {w : H ( 0 , u ) # H(w)} depending, in general, on t . (24.15) PROPOSITION. Let
N E p31<-1, and set
H ( w ) := Iim esssup(H(8,w) : t 5 s < [(w)}. tt tf ( w )
Then fi = H as., and H(Otw) = H ( w ) for all t < <(w), for all w E 0. If H E 3*, then H E 3*.
PROOF:Define A
c R’
x 0 by
:= ( ( 4w> : H(BtW)l{t,((w)}
# H(4{t
Fix a law p on E . The map (tlw) -, H(0,w) is, by (4.3), in the t? C3 P P completion of B+ 8 P . Fubini’s theorem applied to t? 8 Pfi shows that, for PP-a.a. w, the Lebesgue measure of the w-section of A vanishes. It follows that Pp{H # H} = 0. In particular, H E F implies I? n p 3 P = 3. If H E 3*, (4.3ii) shows that (t,w) -+ H(Otw) is in the t? x Q-completion of B+ @ 9 ,and so arguing as above, H is in the Q-completion of p . AS Q is arbitrary, E 3*. Now, C(0,w) = ([(w)- r ) + identically in T , w, so, changing the variable s T to u in the second step,
+
identically in
T , w.
(24.16) PROPOSITION. Let 2 E B+ 8 3*and assume: (i) for all t > 0, s 2 0, and all p , PP{ZtoOa (ii) t + Zt is a s . lcrl. Define 2, for t > 0 by
(24.17)
Z,(w):= ~imesssup{Zt-,(e,w) rttt
:T
# Z,,,}
= 0;
5 s < t}.
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Markov Processes
and Z is almost perfectly homoThen 2, 2 are indistinguishable on R++, geneous on R++.Therefore 2 € fig.
PROOF:Let
6 := R++ x s2,and let O,(T,
w ) :=
{1
-t,Otw)
if t
if t 2
T,
so that ht is the shift operator on R++x R for the backward space-time process (k, X ) on R++x E , as in 516. Here, R is uniform motion to the left killed as it approaches the origin. See the discussion surrounding on R++, (21.6). There is no loss in generality if one assumes that X has infinite lifetime, embedding A as an ordinary point of E if necessary. Let 1denote the lifetime for (k, X),so that ( ( T , w) = T . Condition (24.13) amounts to the assertion that the random variable H ( T , w ) := Z,(w) on f2 belongs to the left germ field at 1. The prescription (24.16) for 2 amounts precisely to the perfection H of H in (24.1), and accordingly, Priz{H # H } = 0 for all T > 0, w E 0. Moreover, satisfies H ( & ( r , w ) ) = H(T,w) for all 0 < t < T and all w E R. That is, Zt(O,w) = Zt+B(w) for all t > 0, s 2 0, w E R. In addition, Pz{Zt# Z t } = 0 for all 2 f E , t > 0. Once we show that t + Zt is 8.s. left continuous, it will follow that Z - 2 E 2,hence that 2 is a.s. lcrl, and in particular, 2 E fig, so that Z E fig also. For left continuity of Z , observe that for each fixed t > 0,
p q z t - - s o e, ~ , - , ~ e , ~o = for all T , s €10, t[ and all p. It follows that, for 8.8. w , T Zt-,(O,w) is essentially constant on 10, t [ ,and its essentially constant value is Z t ( w ) . As t varies over the rationals, we put together a null exceptional set RO outside of which Zt-,(O,w) = Z t ( w ) for a.a. T €10, t [ ,for all rational t > 0. However, a.s. left continuity of t + Z t ( w ) shows then that if w $ 00, the map T + Zt-,(O,w) is essentially constant on 10, t[for all t > 0, hence that &-,(O,w) = Z t ( w ) for a.a. T € ] O , t [ , for all t > 0 and all w $ Ro. Left continuity o f t + & ( w ) for w 4 00is then evident. .--)
(24.18) PROPOSITION. Let Z E U+ 8 3*, and assume: (i) for all t , s 2 0, Z t o O , = Zt+, as.; (ii) t + Zt is a s . rcll. For t > 0, define Z t using (24.171, and set 20 := limsuptLloZt. Then Z - Z E 2,and 2 is almost perfectly homogeneous on R+.Consequently,
z€ 4 d .
PROOF:Since (24.183) implies (24.16i), the proof of (24.16) (with right continuity replacing left continuity in the final stage) shows that (Zt),,0 is
111: Homogeneity
129
almost perfectly homogeneous on R++and 2- Z l ~ o , mE ~2. Obviously, 2 0 = 2 0 a.s. and t -+ 2, is a s . rcll. Provided w 4 {t -+ 2 2 is not rcll} E N , one has, for every s > 0,
Consequently, 2 is almost perfectly homogeneous on R+. REMARK. The condition 2 E B+ 8 3* in (24.16) and (24.18) is not a real restriction, in view of (21.5). The perfection methods above apply to many other functionals of interest. Though the perfection of a weak, exact terminal time is a special case of a more general perfection of multiplicative functionals, we shall give a separate argument at this point because it is simple and very useful. We shall say that a function g on an interval I C R with values in [-00, m] is essentially increasing provided
e 8 C { ( S , t ) E I x I : s 5 t , g ( s ) > g ( t ) } = 0,
(24.19)
C denoting, as usual, Lebesgue measure on R. A routine calculation shows that an essentially increasing function g on R+ has an essential right limit g, defined by
g(t)= limesssup{g(s) : t < s < T } = limessinf{g(s) : t < s < r}. Tilt
Tilt
Then g is a right continuous increasing function, and 3 is an essential regularization of g in the sense that g ( t ) = g ( t ) for (Lebesgue) 8.8. t 2 0. (24.20) LEMMA.Let T be a weak terminal time, as defined in (12.1). Then, for a.a. w, t t T(Otw) is essentially increasing. --.)
+
PROOF:If 0 5 s 5 t , then since t - s
+ Toet-,
2 T a.s., one has a.s.
For every initial law 1-1, this leads to
I“LY
l ~ , < t ~ P ’ ” { s + T o O>s t+ToBt}dsdt = 0 ,
and Fubini’s theorem will give the desired conclusion, once we prove the measurability of the map
130
Markov Processes
+
For this, it is clearly sufficient to show that (s,t ,w ) + t T ( 8 t w ) is in the ds €3 dt c 3Pp(dw)-completion of B+ €3 B+ €3 F,and this is in turn an obvious consequence of (4.3). Fix now a weak terminal time T , and define T by
T ( w ) := lim esssup{s + T(O,w) : o < s < t } .
(24.21)
tll0
Note that, for every T T
> 0,
+ T(8,w) = tIim esssup{r + s + T(8,erw): o < s < t}. ll0
Because of the preceding lemma and the above observations, it follows that for w not in Ro := { w : t -, t T(&) is not essentially increasing} E N ,
+
(24.22) (24.23)
t -+ t + T ( 8 t w ) is increasing and right continuous; t + T(Otw) = t + T(Btw) for (Lebesgue) a.a. t 2 0.
Assume now that the weak terminal time T is exact. Given an initial law p , Fubini's theorem permits one to choose a sequence t, 11 0 such that
Pp{t,
+ Foet, # t, + Toet, for some n} = 0.
Exactness and (24.22) then imply Pp{T
# T } = 0.
(24.24) PROPOSITION. Let T be a weak, exact terminal time. Then the random variable T defined bay(24.21) is a s . equal to T, and T is an almost perfect (exact) terminal time.
PROOF: For w not in the null set RO U {T # T } , t + T(Btw) = T ( w ) for Lebesgue a.a. t < T ( w ) . In view of (24.23), t + T(Btw) = T(w)= T(w)for all t < T ( o ) . (24.25) REMARK. T may be further modified so as to be perfect rather than almost perfect. Let R' denote the set of w for which (24.22) holds, so that (R')" E N . Then R' has the properties (i) w E R' =+ 8,w E R'Vs 2 0; (ii) t9t-w E R'Vn, tn 1 t + 8tw E 0'. Let R ( w ) := inf{t : Btw E a'} so that R ( w ) = 0 a.s. and as in (24.9), Roet = ( R - t)+ identically and 8,w E R' for all s 2 R ( w ) . Let S ( w ) := R ( w ) + T ( f 3 R ( w ) ( W ) so that S = T a.s., and for t < S(w),t < R ( w ) so that
t + s(e,w) = t + ~ ( o ~ +wT(oR(e,w)(etw)) ) = R ( w )v t + W R ( W ) " t 4 = R b )+T(&z(w)4 = S(w). Thus S is a perfect terminal time. See 555 for further c a e s of this type.
III: Homogeneity
131
(24.26) LEMMA.Let Z E f i d . Then Zlno,mo E fig.
PROOF:Suppose Z a.s. rcll and perfectly homogeneous on R+. Take R' := { t -+ 2, is rcll}, and
Clearly Zn is almost perfectly homogeneous on R++, and its paths are a.s. lcrl. Therefore Z" E f i g . As W := limsupn,,Zn E f i g and W Zlno,ma E 1,the result follows. Fix now Z E b M with Z homogeneous on R+.The function f(x) := P"Z0 is in bE", and for every initial law p and every T E T, the SMP gives
If f were known to be nearly optional, one could then conclude that "Z = fox,and that f would in fact be optional. For a number of applications, it is important to have conditions on Z which guarantee that this is so. The following assertion contains two specific conditions of this nature. (24.27) PROPOSITION. Let Z E bM be homogeneous on R+. Under either of the conditions (24.281, (24.29) on Z , there exists an optional function f on E with OZ - f OX E Z. Then: (24.28) Z is measurable relative to the smallest u-algebra on R+ x R containing all bounded real processes that are homogeneous on R+and are a.s. right continuous at 0. In this case, f may be chosen in bE"; (24.29) Z E M n (see (22.21)) and Z is perfectly homogeneous on R+.
PROOF:Under (24.28), we reduce the argument by monotone classes to the case where Z is bounded, 8.5. right continuous at 0, and homogeneous on R+. Let f(x) := P"Z0 E bE". It suffices to prove that f E Eel so that, in particular, f is optional. As P"f(X,) = P"Zt for all t 2 0, one has
Since a U a f E Eel it follows that f E E".
132
Markov Processes
Under (24.29), one knows (23.8) that
n2
E 0. It is also clear that
20( B t (w / t / w I ) ) = 20(w') , and consequently
(We made this calculation assuming the existence of a splicing operator on R. The same result obtains, in fact, for an arbitrary R-just do the calculation on the canonical space.) We may now derive useful representations of some of the a-algebras introduced earlier in this section. Recall from (23.3) that X" denotes the a-algebra { f ( X ) : f E E " } on R+ x R. Given a function f on E such that f ( X t ( w ) has left limits a.s. for t > 0, let f ( X ) - denote the the map t ~ . define the a-algebra X? on process ( t , w ) -+ lim,ttt f ( X s ( w ) ) l ~ t , ~We R++ x R by --+
(24.30) DEFINITION. X5 := { f ( X ) - : f E b(U,Sa)}. Actually, to be careful, f(X)- should be defined everywhere on R++X O by taking instead, say, limsupUtTtf(X,(w)).The actual limit exists, of course, except on an evanescent set. The latter point will come up again later in this section, so we set down the appropriate terminology here.
(24.31) DEFINITION. A set N c R is shift-null provided Ut>,&JtlN is null. The class of shift-null sets is denoted N e . For example, iff E Sa,N := { w : t -+ f(Xt(w)) is not right continuous} is clearly in N e . The relevance of N o to homogeneity is through the observation that F vanishes except on a shift-null set if and only if ( t , w ) -+ F(Btw) is in 2.
(24.32) PROPOSITION. (i) If 2 E bBd, then there exists f E bEe with "2 = fox. Consequently, in the notation of (23.2), C3 n f j d = Xe V 2. (ii) If 2 E bB9, then PZ may be chosen in bX5. It follows that
pn~g=xe_vz.
111: Homogeneity
133
PROOF:Assertion (i) is an obvious recasting of (24.27). For (ii), if 2 E bpsjg is lcrl, then Wt := Zt+ is rcll, and a.s., for all t , s 2 0,
That is, W E bBd. The proof of (24.27) shows that "W may be chosen of the form fox with f E bEe. As W is rcll, "W is a.s. rcll by (A5.9). I claim now that PZ = (OW)-, and once this is proven, (ii) will follow. It suffices to prove now that, for an arbitrary predictable time T with finite values,
(To obtain from this the apparently more general equality
replace T by T A n in the first equality, and then let n + m.) Let {T,} announce T relative to P p . Then ZT = limn W(T,) boundedly, and .E&- = V F g n , so by the Blackwell-Dubins lemma (A5.29),
€""{ZT 1 Fg-}= limPP{W(T,) I 3;n} = lim"W(T,) = "WT-, n as claimed. We turn now to a representation of f j d and f i g in terms of perfect objects with the best possible measurability properties. (24.33) LEMMA.Let F E b3'". Then, for every finite measure X on R+, there exists G E 3"such that G = Foot A(&) except on a shift-null set.
SF
PROOF:Recall (8.7) that U,,o(bSa -bS") is an algebra of functions on E generating E". It follows by monotone classes that it is sufficient to prove the lemma for F a generating element of the form f l ( X , , ) . . - f n ( X t n with ) fl, . . . ,f, E UbS". Let N := { w E 52 : t + fj( X t w ) ) is not right continuous for some j } , so that N E N @ . Then t -+ F(&) is right continuous except when w E N . Let 6 denote the trace of 3"on R \ N . The map ( t , w ) + F ( & w ) on R+ x (R \ N ) is then clearly in B+ 8 6. By Fubini's theorem, the map R \ N 3 w -+ S,"F(Btw) A(&) is 6-measurable. By definition of trace, the existence of G E 3ewith G = F(&w) X(dt) except on N is thereby assured.
S F
134
Markov Processes
(24.34) THEOREM. (i) Let Z E f j d . Then there exists H E T esuch that Zt - H o o t E 1. (ii) f i g is generated by 2 and the multiplicative class of processes of the form t -+ ( F o e t ) - , with F E bFe and t + Foot a.s. rcll.
PROOF:For (i), we may assume, thanks to monotone classes, that Z is almost perfectly homogeneous on R+, that all its trajectories are rcll, and 0 5 Z 5 1. Then P"Z0 E €" by (24.27). By that same result, if G E b p so that t -+ GOOt is in f i d ,then 2 -+ P"(Z0G) is in €". It follows then from Doob's Lemma (A3.2), 9 being separable, that there exists @ E Ee 8 9 such that P" (ZoG)= P" (a(.,
. )G(
a ) )
for every G E b p .
Define F ( w ) := @ ( X O ( Ww) ,) so that F E F eand
P"(ZoG) = P" ( F G ) for every G E b p . The last identity extends automatically to all G E bT. It follows that 20 = F a s . and therefore, as., Zt = Z o o O t = Foet for each fixed t . Set
Clearly W r = 2; a s . for each fixed t. However, both processes are continuous in t , and consequently W" - Z" € 2. According to (24.33), there exists H" E bF" such that {Wg # H a } E Ne. Hence Wr - H"o& E 1, and therefore Z,ol - H"o& E 1.As a -+ 00 through, say, integral values, Z,"(w) -+ Z t ( w ) for all t, w . The proof of (i) is then complete once we set H := limsup, H". For (ii), as we pointed out in the proof of (24.32), fig is generated by processes Zt- with Z E f i d having rcll trajectories. Therefore (ii) follows from (i) and the first observation in the proof of (24.32). Observe that (24.34) implies that if F E b 3 and if t --t Foet is a s . rcll, then there exists G E 3"such that F - G E Ne. That same result also shows that every Z E f i d (resp., fig) has a perfect, not just almost perfect, version. REMARK.It will be proved in §45, with some difficulty, that 0 n fig = X? V Xo V 2. It should also be the case that fig = X? V f i d ,but the proof seems elusive. We are now in a position to describe the correct replacement @ for 0" which overcomes the difficulties with 0" raised in sections 21-23.
111: Homogeneity
135
Let R be closed under the splicing map, and let M e (24.35) THEOREM. be the a-algebra on R+ x R generated by the bounded real processes Z such that (a) 2, E 3"for all t 2 0; (b) there exists Ro E Ne such that t -+ Z t ( w ) is rcll provided w 4 00.Let Ze denote the class of processes W for which {sup, IWt( > 0) E Me. Then: (i) B+ 8 3"c M e c B+ @ 3"V Z?; (ii) for every F E 3", Zt := Foot is Me-measurable; (iii) for every Z E 3 9 , there exists 2' E Me with Z - 2' E Z and 2' almost perfectly homogeneous on R++; (iv) Os c 0" v P ; (v) for Z E Ze, JJ Z ( t ,w) is defined and vanishes for all (t,w ) ; (vi) for every Z E bM", n Z ( t , w ) is defined for all t 2 0 and for all s, t 2 0 , w E R, O t r I Z ( s , w ) = rIO,Z(s,w); (vii) for every 2 E bM", n Z E Os. PROOF: Items (i) and (ii) are proved by evident monotone class arguments, and (iii) follows at once from (24.34). Item (iv) comes from monotone classes and (ii). Observe that if N E r/*, then for all t 2 0, w f R, {w' : w f t f w l E N } E N , for wltfw' E N + W' E 0 ; l N . It follows by inspection of (22.7) and (22.9) that for Z E Ze, I I Z ( t , w ) is defined and vanishes for all t , w. This proves (v). To prove (vi), it is enough by monotone classes to consider Z bounded and rcll for w 4 N E N e , with Zt E 3"for all t . Then II Z is defined because of (v), and the commutation relation is clear by inspection of (22.9). In proving (vii), note that it is The conclusion in this enough by (i) and (v) to suppose Z E b(B+ 8 3"). case comes from (23.6). Theorem (24.34) can be used to prove a simple form of perfection theorem for multiplicative functionals. By a raw weak multiplicative functional (RWMF) is meant a right continuous process m E p M such that if N,,t := { w : rn,+,(w) # m t ( w ) m , ( O t w ) } , then Ns,t E N for all pairs s, t >_ 0. The RWMF m is called almost perfect if U,,t$V,,t E N , perfect if U,,t#,,t = 0. (Multiplicative functionals will be studied in detail in Chapter VII. For two typical examples, consider mt := exp(- j,"f ( X , ) d r ) , f E p€", and mt := 110,Tn with T a terminal time. (24.36) THEOREM. Let m be a RWMF such that as., (i) mt > 0 for all C; (ii) mt = 0 for all t 2 C; (iii) the trajectories of m are rcll. Then there exists an almost perfect RMF m' E B+ 8 3"with m - m' E Z.
t<
PROOF:Modifying m on an evanescent set, we may suppose by (21.5) that m E f?+ @ F'. The projection 520 on R of the set { ( t ,w ) : m.(w)is not rcll at t } is by (A5.2) analytic over Fulhence in 3*.Modifying m to vanish
136
Markov Processes
identically if w E 0 0 , we may assume m is rcll for all w and m E For each s 2 0, let
W:(w)
B+ 8 3*.
:= m t + , ( w ) / m t ( w ) l { t < ~ } , (O/O := 0).
Then t -, W,B satisfies the conditions of (24.18), so W" E Ad. By (24.34), there exists F" E 3"such that N" := {sup, IW," - F"o&l > 0) E N for each s 2 0. Let N := U r E ~ + N ET N . Define
mi(#):= limsup F ( w ) ~llt,T€Q+
The latter expression for m: shows that m' E 8, t L 0,
m:(e,w) =
limsup
B+ @ 3e.Fix w 4 N . For all
FT(e,W)
TLlt,TEQ+
= m,+8(w)/m3(w)1{3
Setting s = 0, we get m:(w) = mt(w)for all t 2 0 if w $ N . Thus, from the last displayed equality, for w $? N , m:+S(w)
= mt+3(w)
= mt(w)(mt+3(w)/rnt(w))1{~
= m:(w)mb(etw).
Thus m' is an almost perfect RMF indistinguishable from m. REMARK.For rn as in (24.36), ZTm' is defined and in 0' by (24.35). If m E 0 , then ZTm' - m E Z,proving that there exists m"(:=nm') E 0' indistinguishable from m. The almost perfect multiplicative property of m' is lost under the projection operator, and m'' is in general only a weak
MF. The following result is the analogue of [DM75, IV-901 with the direction of time inverted. It suggests that for time reversed, measurable processes perfectly homogeneous on R++ are to f i g (resp., A d ) as the progressive processes are to 0 (resp., P). (24.37) PROPOSITION. (i) Let A E M be almost perfectly homogeneous on R++.Then its closure is in 39 and its set A' of right accumulation points is in 4id. (ii) Let 2 E M be almost perfectly homogeneous on R++.Then Ut := limsup,ll,Z3 is in Ad and Wt := limsup,,tZ, is in fig. (Caution: the second limsup is as s -+ t, not as s T t.)
HI: Homogeneity
137
PROOF: (i) For w not in some null exceptional set if and only if s + u E A(w). Define
no, s E A(d,w),
s
> 0,
V , ( w ) := inf{s > t : s E A ( w ) } . Then, since A E M , V, E 3by (A5.2). In addition, t -, V,(w) is increasing and right continuous for every w . Note that for w 4 $20and t 2 0, &(O,w)
= inf{s = inf{s
> t : s E A(B,w)} +u > t +u:s +u E A(w)}- u
= %+,(w)
- u.
Therefore, the process V, - t is a.5. rcll and almost perfectly homogeneous on R+,hence V, - t E f i d . But vt - t 2 0, and V , ( w )- t = 0 if and only if t is a right accumulation point of A(w). Define next yt := &- - t , so that Y is a s . lcrl and almost perfectly homogeneous on R++,hence Y E fig. But yt 2 0, and & ( w ) = 0 if and only if t E A(u). This proves (i). Now take 2 as in (ii). For each u E R, Wt(w)2 a if and only if, for every E > 0, t is in the closure of AE(w) := ( s > 0 : Z,(w) > a - E } . By (i), {W 2 u} E f i g , so W E 49. The other case is similar but even easier. (24.38) EXERCISE.For Z E p M homogeneous on R++, let 'p(r,z) := P"2, for T > 0. Show that 'p is excessive for the backward space-time process (R, X) of $16. Conclude that if Z E b M is homogeneous on R++, then for every T > 0, t -+ Pxt Z,-t is a right continuous martingale on the time interval [0, T [ . (24.39) EXERCISE.Let H E 31c-1.Show that there exists 2 E 4 9 such that, a.s., Hl{c<m}= Zcl{c
Markov Processes
138 25. Co-optional and Coterminal Times
Co-optional times are to homogeneous processes, or more properly, to processes in f i g , as optional times are to optional processes. They were introduced under the name return time by Nagasawa "a641 in connection with time reversal of a Markov process. (25.1) DEFINITION.Let L be an 3-measurable random variable with values in [O,CQ], and let Nt := { w : L(0tw) # ( L ( w ) - t)+}. Then L is a co-optional time provided Nt E JV for every t. The cc-optional time L is perfect if Nt = 8 for all t, almost perfect if Ut>oNt E N . Thus L is a perfect co-optional time provided (25.2)
L(etw)= (L(w) - t)+for all t 2 O,W E 0.
The most familiar co-optional time is C, which is both perfectly cooptional and a perfect, exact terminal time. The same would be true of any random time L which is the hitting time of an absorbing set for X. (25.3) PROPOSITION. Let L E p 7 . Then L is (almost) perfectly cooptional if and only if one of the following equivalent conditions holds: (25.4) (25.5)
10,L l is (almost) perfectly homogeneous on R+; 10, LI] is (almost) perfectly homogeneous on R++.
PROOF:Let L satisfy (25.4). Then, (for w not in some null set,)
This condition is clearly equivalent to the (almost) perfect co-optional time property. The other case is similar, and the converse is equally simple. (25.6) THEOREM. Let L be a co-optional time. Then there exists a perfect co-optional time L with L = L almost surely.
PROOF: We may assume L E 31". Since L is co-optional, Z := lU0,Lg satisfies the weak homogeneity conditions (24.18i,ii). Proposition (24.18) gives us a process 2 indistinguishable from 2 such that 2 E fid.By (24.34), we may suppose Zt = Foot for some F E 3e.Set z ( w ) := sup{t : Z t ( w ) # 0}, (sup8 := 0). Then L = L a.s. so L E 3 and for all w E 52,
for every s 2 0. Consequently, L is perfectly co-optional.
III: Homogeneity
139
Let L be a co-optional time. Then, for every T E T, (25.7) COROLLARY. Lo& = ( L - T ) + a.s. on {T < oo}.
PROOF:Take L as in (25.6). By the SMP, Loor = ( E - T ) + a s . , and since L O O T = ( E - T ) + a s . on {T < oo}, the asserted equality follows. Just as the debut of a random set M C R+ x fl is defined to be the random variable D M ( w ):= inf{t 2 0 : ( t , w ) E M } , (inf0 := oo),the end of M is the random variable
L M ( w ):= sup{t 2 0 : ( t , ~E)M } , (25.8) LEMMA.Let M LM E p 3 (resp., F*).
C
R+ x R, M E
(sup0 := 0).
f l p M p (resp.,
B+ @ 3*).Then
PROOF:The event { L M > t } is the projection on R of nt,m[ fl M , which is in (B(R+)@ 3 p ) V Zp for every initial law p and every t >_ 0. It follows that { L M > t } is P-measurable, by (A5.2). Since p is arbitrary, LM E fl,P = 3 . The other case is similar. (25.9) PROPOSITION. A random variable L E p F is ((almost) perfectly) co-optional if and only if L is the end of some measurable set M which is ((almost) perfectly) homogeneous on R++. In fact, there exists F E p 3 " such that a.s., L = sup{t : F O B t = 0).
PROOF:If L is co-optional, take M = I]O, L J and use (25.3). Conversely, if L is the end of M , then L is 3-measurable by (25.8), and co-optionality of L follows as in the proof of (25.6). The (almost) perfect case is handled similarly. The last assertion comes then from (25.6) and (24.34). Note that (24.37) implies that the set M in (25.9) may without loss of generality be assumed to be closed in R++ and in fig. Thinking of homogeneous sets as optional sets for time run backwards, co-optional times are the time-reversion of optional times. The appropriate analogue of a terminal time is given in the following definition. (25.10) DEFINITION. A random variable L E p F is an ((almost) perfect) coterminal time provided L is the end of an optional random set M which is ((almost) perfectly) homogeneous on R++. This is a little different from the definition set down in [MSW73]. Ours corresponds to what they called an exact coterminal time. See the discussion below for an alternative description of coterminal times. (25.11) EXERCISE. Let M c R++ 8 R, M E 0 , and suppose that M is homogeneous on R++. Then D M is an exact terminal time which is (almost) perfect if M is (almost) perfectly homogeneous on R+.
140
Markov Processes
Let 2 E b p M , say 0 5 2 5 1. For o (25.12) EXERCISE.
> 0, put
Show that V a is right continuous and measurable, and letting (Y + CQ, deduce that limsupSllt Z, is measurable, and hence, applying the result to 1 - Z, liminf,llt Z,is also measurable. By a similar argument, show Z, and lim inf,TTt 2, are also in M . Use this to prove that that limsupSTT, the pathwise closure M of a random set M E M is also measurable (cf. (A5.6ff)). Show also that if M is ((almost) perfectly) homogeneous on R++,then so is M . In case M is also optional, use the result cited above to conclude that M is also optional. Observe that in (25.9) and (25.10), M may be assumed closed-that is, every w-section of M is closed in R++. (25.13) EXERCISE.If L is co-optional and T is optional, then LOOT is co-optional. (Recall Loem := 0 by the conventions of $11.) (25.14) EXERCISE. Let L be co-optional and define c(x) := P”{L > 0 } , c’(z) := P”(0 < L < m}. Then c and c’ are excessive, o l ~ o , L=~ COX, ~ ( l [ o , L [ l { ~ < m=} )C’OX, Ptc(z) = P”{L > t}, Ptc’(x) = P”{t < L < G o } . (25.15) EXERCISE. Let B be nearly optional in E with hitting time TB, and let LB := sup{t : X t E B } . Then L B is a perfect coterminal time and CB(Z) := P”{LB > 0) is equal to P“{TB < m} = PBl(z), in the notation of $12. (The excessive function CB is called the equilibrium potential for the set B. See $49.) (25.16) EXERCISE.Define the invariant a-algebra Inv for X as { F E 3 : as., Foot = F V t 2 0). (This is essentially the same as 31~-1, the left germ at infinity.) If L is co-optional, then there exists A E Inv with { L = CQ} = A almost surely. Conversely, given a set A E Inv, let L(w) := 00 if w E A, = 0 otherwise. Then L is almost perfectly co-optional, and if c(z) := P 5 {L = m}, then c(Xt) is a right continuous version of the martingale P”{A I 3t}. (25.17) EXERCISE.Let X be recurrent (10.39). Use the last exercise to prove that for every A E Inv, P x ( A )is either identically 0 or identically 1. Let L be an almost perfectly co-optional time. Prove that Px{O < L < 00) = 0, and derive the dichotomy L = 00 a.s. or L = 0 almost surely. (Hence for every non-polar set B E E , a.s., Xt E B for arbitrarily large t.) (25.18) EXERCISE.Let R have killing operators, as specified in $11. Suppose L € p 3 satisfies the following conditions identically in (t,w ) : (25.193) L(etw) = ( L ( w )- t)+; (ii) L(ktw) = L(w) if L(w) < t;
III: Homogeneity
141
(iii) L(ktw)5 t A L(w); (iv) t L(ktw) is increasing; (v) Lo kt E Ft for every t 2 0; (vi) (exactness of L ) limt,, L(ktw)= L(w). Define Lt(w) := lim,llt L(k,w) so that t -+ Lt(w) is an optional, right continuous, increasing process. Show that the random set M := { ( t ,w ) : t 2 0 , Lt(w) = t} is optional and perfectly homogeneous on R++,and that L is the end of M . That is, if L satisfies the conditions (25.19), then L is a perfect coterminal time. (Hint:let L' denote the end of M . That L' = L on { L < m} is easy. Because of (iii) and (vi), it suffices to prove that L'okt -+ L' as t + 00.) -+
(25.20) EXERCISE. Let L be an almost perfect coterminal time. Prove that the conditions (25.19) hold except for w in some null set. 26. Measurability on the Future
Homogeneity, as described in the two preceding sections, corresponds to a kind of progressive measurability in the reverse time direction. One aim of this section is to make this assertion precise. Note first the following result, which is a direct consequence of (23.12). (26.1) PROPOSITION. M =P V fid. Informally, since P describes measurability on the strict past, (26.1) states that f j d is rich enough to describe measurability on the future. Given a random time R-that is, R E p3-there are at least two reasonable ways to define the a-algebra of events subsequent to R. The critical issue seems to be whether to include the value of R itself in the future. We choose not to do so. The following definition could be modified to make R measurable on the future by applying it to a space-time process over X . See [GShBla] for the details. Fix R E p 7 . Define two o-algebras of the future from R as follows. - and 32R- of the future from R (26.2) DEFINITION.The a-algebras F>R are specified by: (26.3) b F > R := {H E b 3 : H is constant on {R = GO}, and there exists Z Ebfid with H~{R<,) = Z R ~ ~ R < ~ ) } ; (26.4) b 3 2 ~ -:= { H E b 3 : H is constant on { R = oo}, and there exists Z E bfig with H 1 { O < ~ < m=) Z R ~ { ~ < R < , ) } .
For example, i f f E bE and t 2 0, then H := f(XR+t)l{R<m} belongs and F>R-. It follows from (24.26) that F ~ cRF>R-. The to both F>R latter a-algebra contains information from the infinitesimal past a t R, just
142
Markov Processes
as 3 R contains information from the infinitesimal future at R, at least in case R is an optional time. For a second example, take f E bS" and H := f(XR)-l{R0 - does contain 3:,so that F>o - and 3differ only by Pp-null sets, for any p .
(26.5) EXERCISE.If G E bT*, then G O # R ~ { R <E ~b)3 , and, for any p , it differs from some member of b.75.~ - only on a Pfl-null set. (26.6) EXERCISE.Give an example of a random time R with the property R 4 3>R-. (Hint: the state space need have only one point, in addition to the death state.)
(26.7) EXERCISE.Let R be perfect co-optional time, and Rt := ( R - t)+ fort 2 0. Prove that: (i) (ii)
3>Rt- 3 3>R-; n t > 0 7 > R t - = FzR-..
(26.8) EXERCISE.Fix a perfect co-optional time R over X, let Rt := := 3 2 ~ $ - .
(R - t)+,and define Gt
(i) Apply (26.7) to prove that Gt is a right continuous filtration. (ii) For E fig, let &(w) := Z ( R ( w ) - t ) + ( w ) l { R ( w ) > t )its , reverse from R. Prove that 2 is optional over (Gt).Prove also that if Z E f i d , then 2 is predictable over ( G t ) . (iii) Let T 5 R be a co-optional time. Prove that R O ~isTan optional time for (Gt).
z
The converses of the assertions in (26.8) are true under the further hypothesis that R < 00 identically. This is content of the next proposition. In the following statement, it is important that the strictly optional and predictable a-algebras relative to (Gt) be interpreted without any augmentation by evanescent processes. Thus, for example, a process is strictly optional relative to ( G t ) provided it is in the a-algebra generated by the rcll processes adapted to (Gt).
(26.9) PROPOSITION. Let R be a perfect co-optional time with R ( w ) < 00 for all w E a, and let ((&) be the filtration defined in (26.8). Then: (i) if 2 is strictly optional over (Gt), then 2 E fig; (ii) if 2 is strictly predictable over (&), then 2 E 5 j d ;
III: Homogeneity
143
(iii) R- S is a co-optional time for every optional time S for (Gt) with S 5 R.
PROOF:We prove only (i), as the proof of (ii) is quite similar and (iii) follows at once from (i), making use of (25.9). By monotone classes, it suffices to assume that 2 is rcll and adapted to (&).Then 2 is lcrl. Suppose H E &. By finiteness of R, (26.4) implies that there exists Z t E fig with H l { o < R t l = Zk,1{0<~~1, and therefore there exists a null set s10 outside of which, for all u 2 0, HOf?,l{O
zt "8,
= ZRtoB, (8,)
l{O
forallu < ( R - & O ~ , ) + .But u < (R-Rt+,)+ ifandonlyifu < (t+u)AR, which certainly holds on {RtoB, > 0) = { R > t u } and the last identity shows then outside R', & o B , = &+, for all t , u 2 0. Thus 2 E fig.
+
( 2 6 . 1 0 ) PROPOSITION. Suppose 5 < 00 identically on 0, and set R := C. Under the conditions of (26.9), fig is generated by stochastic intervals of the form 10, L 1, with L a co-optional time, and f i d is generated by those of the form 10, L [ .
PROOF:Let 2 E fig be lcrl. Define LO := 5. Given c > 0, the set M := { ( t , w ) : IZt(w) - Z,,,)(u)[> E } is homogeneous on R++,so its end L1 is a co-optional time. Clearly L1 < LO. Define L, recursively to be the end of { ( t , w ) : ( Z t ( w ) - ZL,-~(,)(W)(> E } . Clearly Lo > L1 > .. , and a.s., L, = 0 for all sufficiently large n, for otherwise, 2 would fail t o be lcrl. Inductively, each L , is a co-optional then a.s., sup,[& time, and if we set W, := Wt( 5 E . It suffices to prove therefore that each process of the form Y := F l n L r , L i t l (L' 5 L" co-optional, F E 3>p) is in the a-algebra generated by the 10, L l with L co-optional. Given integers n, k, the process Y' := l { F E [ ( k - l ) / 2 n , k / 2 n ( ) n n L ' , L " n is in fig. Let L"' := sup{t : &' = 1) (sup 8 := 0), a co-optional time, and L"" := L'AL"'. Then Y' = 1 L ~ I1, I and Y is approximated to within 1/2, by linear combinations of such processes, completing the proof for the fig case. The fidcase is similar. The a-algebra of the past at an arbitrary random time R is defined by
-
ILttii
Markov Processes
144
then 3 <- R = 3 R . This is the Note that if R is an optional time for (Ft), content of (6.18). Convenient definitions of the a-algebra of the strict present at R are (26.12) (26.13)
~F[[R] := {C + Z R ~ ~ R: c< E~ R, ) E b(O flfid)}; bFb1 := {C + f(XR)l{R
qR1
Fbl,
Similar definitions apply to and Fbl, but which we have also called Xk, seems to be of greatest utility. For example, (24.323) gives (26.14)
Fb1 C ~ F [ CRFb1 ] VN.
In like manner,'define the left a-algebra F[R-] at R by
+
so that , by (24.32ii), Fp-1 differs only by null sets from 3E::= a { c ~ ( X R ):-c E R, f E S"}. Define also the mixed a-algebra F[(R-,R] at R by
Intuitively, F[R-] contains only information in the so-called left germ at R, meaning only information infinitesimally prior to time R enters its specification. The a-algebra F [ R - , R ] admits information from both the infinitesimal past and the strict present at R . For example, i f f E S", the random ~< in ~F[R-], ) while { ~ ( X R )#- ~ ( X R R) ,< variable ff := f ( X ~ ) - l t is m} E F[[R-,R]. According to (24.35ii), F[[R-,R]and F[R-]V F [ R ] differ only by null sets. It can be proved that X has Markov properties at random times R other than optional times. See [GShBla] for an extensive collection of examples.
Iv
Random Measures
This chapter is concerned with random measures (RM's) on R+.The operations of dual optional projection and dual predictable projection of random measures, and the means of characterizing and generating random measures by means of their potentials are the central features. Sections 27 and 28 treat in detail the finiteness conditions required of increasing processes and random measures in order that they possess dual projections, a complete discussion of which occupies 530-32. Potentials of random measures and the representation thereof is the topic of 533-34. Homogeneity of random measures and potential functions and potential operators for the homogeneous case are discussed in 535-38. Additive functionals appear as distribution functions of homogeneous random measures. The most important results of this chapter come in $38, where the representation of potential functions of class (D) in terms of potentials of homogeneous random measures is described. Throughout this chapter, it 3t,X t , B t , P")is a right process satisfying the is supposed that X = (R, 3, conditions laid down in 520. Generally, the lifetime does not enter the formulation of the results, and the original dead point must be considered as an ordinary point of E. In Chapter VIII, we shall discuss the extension of these objects to processes with a distinguished lifetime.
Markov Processes
146
27. Random Measures and Increasing Processes By a random measure (RM) on R+, we mean a kernel n = K ( W , d t ) from (a,F)to (R+,a+). That is, for each w E R, K ( W , .) is a positive measure on (R+,f3+),and for every g E pa+, w + n(w,g) := ! n ( w , d t ) g ( t ) is 3-measurable. Given a RM K and a set A c R+ x R with A E M , we say that n is carried by A provided, for a.a. w , n(w, is carried by the w-section A ( w ) := {t 2 0 : (t,w ) E A) of A. We also say in this case that n does not charge A". Random measures n , y are said to be indistinguishable if, for a.a. w, n(w, . ) = y(w, as measures on Rf.This is a more elementary notion than that of indistinguishability of processes, at least for RM's whose masses are a s . finite, for such RM's are determined by their values on some countable generating class of sets or functions. The simplest kind of RM is that generated by an increasing process (At)olt<,, by which we shall mean an adapted process such that t + At(w) is right continuous and increasing with values in [O,oo] for 8.8. w . The exceptions to these standards come when A is called a raw increasing process, in which case A is assumed measurable rather than adapted, and when A is a left increasing process, in which case A is assumed a s . left continuous rather than right continuous. It is not assumed that A0 = 0, nor At. For maximal flexibility, it is convenient to that A , = A,- := limt,, allow At to explode at a finite time. It is necessary to impose the following blanket restriction on the increasing process A. a )
a )
(27.1) DEFINITION. Let S E T. An increasing process (At)o
(i) as., At < 00 for all t €10, S[; (ii) a.s., if A s ( w ) = 00 and 0 < S(w), then A s - ( w ) = does not jump to 00.)
00.
(i.e., A
Given S E T and increasing process A satisfying (27.1), we may associate with A a RM n carried by [O,Sn, setting (27.2)
K(W,
[O, t ] ):= At(w),
t 5 S(w).
It is to be the convention in such formulas that n(w, . ) does not charge ] ) 0 if A s - ( w ) = 00. It is clear the interval ]S(w),001, and that K ( [ S ( W ):= from the usual construction of measures from increasing functions that two RM's K , n' both satisfying (27.2) are indistinguishable on [O,S[. Note that n(w, (0)) = Ao(w). It is easy to see that a necessary and sufficient condition for a RM K carried by NO, Sl to be generated by an increasing process A satisfying the conditions (27.1) is that
IV: Random Measures
147
(27.3)Fora.a. w,either (i) the restriction of K ( W , . ) to [O,S(w)[ is a Radon measure and 44 [S(w)l) = 0, or (ii) n(w, [O, S(w)]) < 00. It is clear that two RM’s n, n’ satisfying (27.3)and (27.2)are necessarily indistinguishable on [O, S ] . This is the principal reason for introducing the condition (27.1).Note that when we deal with increasing processes generating RM’s on a stochastic interval 1O,S], there is no loss of generality in assuming that A stops at S , so that At AtAs. Throughout the remaining sections, given an increasing process A, we set Ao- := 0 and AAt := At - At- for t 2 0 (00 - 00 := 0).
=
28. Integrability Conditions We say that a RM IC is integrable provided P5n(., R+) < 00 for all 3: E E . It is clear that a RM K is integrable if and only if it is generated by an increasing process A satisfying (27.1) with PzA, < 00 for all 3:. The following finiteness condition on the RM n will turn out to be appropriate for the dual projection operations discussed later in this chapter. In most cases, ‘H will be one of f f , M , 0,P.
(28.1) DEFINITION.Let ‘H denote a a-algebra on R+ x 0. A RM n is o-integrable on 3t in case there exists a strictly positive %-measurable process Y such that supz P“(dw) Y , ( w ) K ( W , d t ) < 00.
s
s
We remark that, in case ‘H 3 ff, where ff denotes, as in (23.3),the oalgebra of constant (in time) adapted processes, we may replace the term bounded by finite in the above definition, because, in the latter case, Y,(w) may be replaced by K(w)/(l+f(Xo(w)), where f(x) := P” Y, n(. , d t ) , to give a uniformly bounded integral. One should note also that the smaller ?-I is, the more stringent is the condition in (28.1). For example, n is a-integrable on ff if and only if n is integrable. In case the RM K is generated by an increasing process A, there is another useful finiteness condition to consider.
s
(28.2)DEFINITION.Let A satisfy (27.1) and let p be an initial law. Then A is (0, Pp) (resp., (P,Pp))-locally integrable over 10, S’[ provided there exists an increasing sequence {T,} in T such that: (i) limn Tn 2 S , Pp-a.s.; (ii) P ~ [ A ( T , - ) ~ ~ T , > <~00) ]for all n 2 1 (resp., P’{A(Tn)l{T,,>O)} < 00 for all
3 1).
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The increasing process A is called uniformly 0 (resp., P)-locally integrable over lo, S[ provided the sequence {T,} may be chosen so that: (iii) lim, T, 2 S as.; (iv) supz P”{A(T,-)l{Tn>O)}< 00 for all n (resp., supP”{A(Tn)l{T,>O)} < 00 for all n). In (31.16), we shall prove that A is uniformly 0 (resp., P)-locally integrable over 10, S[ if and only if, for every x E E , A is (0,P”) (resp., (P, P”))-locally integrable over 00, S’J. The proof is not obvious. Let A satisfy (27.1) and let K be the RMgenerated (28.3) PROPOSITION. byA on 10,SJ. Then: (i) if A is adapted to ( F t ) (resp., if A E P) then A is uniformly 0 (resp., P)-locally integrable over [ O , SU; (ii) if A is uniformly 0 (resp.,P)-locally integrable over [ 0, S [ , then the restriction of K to 10,S[ is o-integrable on 0 (resp., P); (iii) if A is adapted to (Ft), if A stops at S, and if there exists a (finite valued!) process Y E p P such that as., AAt 5 yt for all t 2 0, then K is c-integrable on P. (We shall refer to this domination property by saying that A has predictably bounded jumps.) If A A is uniformly bounded, then A is uniformly P-locally integrable over 10, S[.
PROOF:(i) Let S, := inf{t : At 2 n}. Obviously limn S, 1 S a.s. and A ( S , - ) l ~ s , > o )5 n for all n 1 1. If A is adapted to (Ft), then S, E T, so A is uniformly 0-locally integrable over [IO, S f . If A is predictable, then [&, C Q ~= { ( t , w ) : A t ( w ) 2 n} is predictable, so each S, is a predictable time. Let ,S := lim, Sn, so that S, 2 S. According to (23.2ii), we may select an announcing sequence {Sk} for S n in such a way that, for all x,
It may be assumed, increasing the Sk if necessary, that Sf I S,k 5 all k 2 1. Set T, := S:”, so that the sequence {T,} increases and
On the other hand,
... for
IV: Random Measures
149
Tn T S, a.s. as n 2 00. It follows that A is uniformly P-locally integrable over [IO, S[ . (ii) We may plainly suppose that K is carried by [ O , S [ , replacing At if necessary by At A As-. Suppose that A is uniformly 0-integrable on [IO,S[, and let {T,} satisfy the conditions (iii) and (iv) of (28.2). Let T := limn T, 2 S and set SO
00
zt
:=
I[o,T,[(~) (2n(l
+ P X o { A ~ n -Tn ; > O}))-l
f
I[T,m[(t).
n=l
Then 2 is a strictly positive optional process, and
Pz/ 2, dAt 5
PzAT,-l(T,,.o}
(2n(l
+ P x { A ~ , , - ;Tn > O}))-'
5 1.
n
In the predictable case, the same argument applies, but using instead the strictly positive predictable process
C 1no,T,i ( t ) (2n(1+
~t :=
P ~ o { A T , ; Tn
> o}))-l
n
+ Pxo (1 +A;'
I{A,,>o)) loon(t)
+ 1ns,,a(t) + 1~s~n{T,<st/n}-
(It should be noted here that the last two terms are indicators of predictable sets.) (iii) Since A is adapted to ( F t ) ,Sn := inf{t : At 2 n} E T. Let S, := limn Sn 2 S. Observe that because (27.1) rules out the possibility that A jumps to infinity, S, < S, as. on (0 < S, < 00). That is, Sn A n TT S, a s . on {S, > 0}, which implies that S, is predictable. But, by the hypothesis that A stops at S , S, = 00 if S, > S. Now, A(Sn-) ~ { s , > o )I Set ~t :=
C lno,s,, n ( ~ ) P ~ ( ~ + Y , +) I(1- +Y{' '
1{y,>o)) lnon (t)+1usw,,~ (t).
n
Clearly 2 is a strictly positive predictable process, and
+ P"{AAs, 2-n(n + Ys,)-'} + P"{Ao (1 + Yc' l{y,>o})}
I 4.
If Y is uniformly bounded, then 2 is uniformly bounded away from 0, and the second assertion in (iii) is evident. This completes the proof of (28.3). For K a RM and Z E p M , we may define a new RM Z * h: by (28.4)
( 2 * K ) ( . , M ) := K (
- ,Z.M),
The following assertion is obvious, but useful.
M E pM.
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(28.5) PROPOSITION. If n is a-integrable on a a-algebra 7-1 C M and if Z E p M is dominated by an 7-1-measurableprocess with finite values, then 2 * n is a-integrable on 7-1. Most RM's will be assumed to be a-integrable on M. Thus, for a.a. w , the measure n(w, .) will be a-finite on (R+, a+). One more trivial but important observation that follows directly from the definition (28.1) is that if n is a-integrable on 3-1 C M, there exists an integrable RM y and a 2 E H with finite values such that n = Z * y. (Take Z := Y-', where Y * n is integrable.) Indeed, if 7-1 3 ff, one may even assume that P"y( . ,R+) 5 1 V x E E . Since an integrable RM is generated by an increasing process ( A t ) satisfying (27.1) with S = 00 a.s., we have therefore a simple way to generate all RM's that are a-integrable on a given a-algebra starting from those with bounded expected total mass. (28.6) DEFINITION. Let H c M be a a-algebra ofprocesses on R+ x 0. A RM n belongs to provided it has a representation of the form K = 2 * y,with 2 E p% finite valued, and y an integrable RM with A t ( w ) := ~ ( w[0, , t ] )an %-measurable process. ~
d (resp., P ) we say that IC is an optional (resp., predictable) random measure. Of course, M is just the class of RM's that are a-integrable on M. In case n E
(28.7) THEOREM. Let
K
E
d (resp., P ) and 2 E p M .
zt n(&)
= Pfi
J,+
Then:
"2, n(dt)
r
PROOF:Because we may represent n as W *y with a finite valued W E p 0 and y integrable and optional, it suffices to assume n integrable. In this case, the equality stated is a standard property of optional projections. See (A5.18). The predictable case is similar, based on (A5.19). Let 2 E p M (resp., p 0 , pP). Show that there exists (28.8) EXERCISE. n E M (resp., 6,P)such that, a.s., K ( . ,{t}) = Zt( for all t 2 0 if and of random times (resp., optional times, only if there exists a sequence {h} predictable times) with disjoint graphs such that { Z > 0) and U I& I] are indistinguishable. Show that if At := &,<slt 2, is finite for all t > 0, a.s., then A is right continuous, increasing and measurable (resp., optional, Z , l { ~ , > ~ l enumerating ~), its predictable). (Hint: consider A7 := Co<s5t jumps Tl,Tz,. . . so that A" = X I ,Z ( T k ) l[Tk,m[.) a )
IV: Random Measures
151
29. Shifts of Random Measures
We define now, for n E &I a,shift operation 6, that is, in certain respects, the dual of the shift operator 0, defined in (21.10). (29.1) DEFINITION. Fort 2 0 and n E M ,6 ) tdenotes ~ the RM
Passing from the case g = 1~ to positive linear combinations and then to their increasing limits, it follows that for all g E pB+, (29.2)
g(s)
6t n(w, d s ) =
One then obtains the following result, which will be important for many calculations involving 6,. (29.3) PROPOSITION. Fort 2 0, n E M and 2 E p M (with finite values),
6,(Z * n) = ( O J ) * (6,n). PROOF:For any g E pB+, by (29.2))
For a n of the form (27.2), where A satisfies (27.1),
That is, 6,n corresponds to the increasing process @A. Since Ot preserves M , 0 and P,it follows from (29.3) and the definition (28.6) that the dual result obtains also.
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(29.4) PROPOSITION. Each of M ,6 , P is preserved under 6,. (29.5) EXERCISE.Let K E M ,(resp., 6 , P ) and let & ( w ) := ~ ( w{ ,t } ) be the process of atoms of K . Then 2 E pM (resp., PO, pP), and for every T E T, the process of atoms of 6~K is 0 ~ 2 . (29.6) EXERCISE.Using (28.8) and (29.5), show that if K E M (resp., 6 , the decomposition of K ( W , . ) into its purely atomic part K ~ ( w ., ) and its diffuse part ~ ' ( w .) , defines RM's K ~ K', E M (resp., 6 , P ) . Show that for every T E T, ( 6 )K ~) =~ ~ T ( K ' ) and ( 66)'~ = ~ T ( K " ) .
9) then
(29.7) EXERCISE. Show that for K , 2 as in (29.3) and for any random time T , &(z * K ) = ( o T z ) * ( ~ T K ) . 30. Kernels Associated with Random Measures
For the purpose of constructing dual projections of RM's in $31, and for the representation of potentials in §34, the key technical device is an association between integrable RM's and kernels from ( E ,E") to R+ x R with a o-algebra W to be specified below. The kernel in question is to be the Markovian analogue of the Dolbans-Dade P-measure associated with an integrable increasing process. See (A5.21ff). (30.1) DEFINITION.Let W be a a-algebra on R+ x R. A finite kernel Kx(dt,dw) from ( E ,E") to (R+8 R, W ) is said to respect X in case, for e v e r y x E E , K"I2I = O f o r e v e r y Z E W n P .
The a-algebras W on R+ x R will usually be between B+ 8F" and M . The next result shows that, between these bounds, the exact nature of W is largely irrelevant. (30.2) LEMMA.If K is a finite kernel from ( E ,€") to (R+ x R, B+ 8 F") respecting X, then K extends in a unique way to a kernel from ( E ,E") to (R+x 0,M ) respecting X.
PROOF:A monotone class argument based on (3.9) shows that, given any initial law p and M E b M , there exists M p E b(B+ 8 9 )such that { M # MI"} E P . For x E E , the fact that K respects X shows that M is contained in the completion of B+ 8 9 relative to K " , so the measure K" extends in a unique way to a measure K x on (R+ x R, M ) , and K xvanishes on sets in M n P . It remains to show that for M E b M , x --+ K " ( M ) is in E". Given an initial law p, choose M p as above. Clearly { M # M p } E 2'. for p-a.a. x E E , so { M E =# M p } E Z'x for p-a.a. x E E. Consequently, for p-a.a. x, K " ( M ) = K"(M'") = KX(MP)
IV: Random Measures
153
and because the latter term is in &%, so is z --+ P ( M ) . Suppose now that K: is an integrable RM. Then the formula
defines a finite kernel from (E,&”)to (R+x R,M) respecting X. The converse will be of great importance in subsequent constructions. (30.4) THEOREM. Let K ( = K“(dt,d w ) ) be a finite kernel fiom (I?,&”) to (R+x R, B+ @ P ) ,and suppose that K respects X. Then there exists a unique integrable R M
K:
generating K via (30.3).
PROOF:Because of (30.2), we may consider K extended to be a kernel from ( E ,&”) to (R+x R, M ) . If K: and y are integrable RM’s determining the same kernel K , take 2 := l p t ] @ G with G E b 3 in (30.3) to obtain P”{G K:“O,tI>) = P”{Gy([O, tl)). Letting G run through b 3 one finds that K:([O, t ] ) = y([O,t , ] )as., and, letting t vary through the positive rationals, it follows that K: and y are indistinguishable. We have now proved uniqueness of 6. For existence, fix t 3 0 and define finite measures QT on the a-algebra F” by
Clearly z -+ Q : ( G ) is in &” for each G E b p . (See (30.5) for a variant.) QF << P” on 30. As F” is Since K “ ( .) does not charge sets in F , separable, Doob’s lemma (A3.2) gives a function 4t E p(&%@ F”) such ) on 9. That is, that QT(dw) = $ ~ ( z , w P”(dw)
for all z E E and G E bF”. Now set B t ( w ) := 4 t ( X o ( w ) , w ) . The map X o ( w ) being in 3$/&”,one obtains Bt E P. (Compare with (30.5).) We now have K5(lpt]@ G) = Px(BtG) for all 2 E E and G E bF0. For H E b 3 , choose G1,Gz E b p with GI 5 H 5 GZ and P”(G2 GI) = 0. As l p t ] @ (Gz - GI) E 2‘=,it follows that K”(lptl @ H ) = P ” ( B t H ) for all H E b 3 . It is clear from this that if 0 5 s 5 t then B, 5 Bt as., and that for any sequence t, 1 t , Bt,, 1 Bt a s . as n + 00. Let & ( w ) := sup{B, : s 5 t , s E Q+}. The discussion above shows that Bt E F for every t 2 0, t -+ B t is increasing and left continuous, and, as., & 5 Bt for every t 2 0. Moreover, a.s., &+ = Bt. Finally, let
w
-+
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Markov Processes
At(w) := B t + ( w ) so that At E P , t --t At(w) is right continuous and increasing, and, a s . for each t, At = Bt. The identity
extends by the MCT to give 2 E B+ €3 3.
K " ( 2 ) = P z 1 2 , dAt,
Taking K to be the RM generated by A via (27.2), the proof is complete. The construction above is susceptible to some variants which are of considerable interest. By obvious modifications at the places noted in the proof, one obtains
(30.5) PROPOSITION. Let E' be a a-algebra on E with E c E' c E", and let, as in 31, 3' := o{f(X,) : s 2 0 , f E &'}. Suppose that K is a kernel from (E,€') to (R+x Q,B+€3 9) respecting X . Then At := n([O,t]), with K as in (30.4), may be chosen in B+ €3 3'. The next result is less obvious, and will have an interesting interpretation in (31.13). Observe first of all that if K E 8 is integrable with corresponding kernel K then for t 2 0 and G E b F , K"(l[o,t]€3 G ) = P"
I,,
GK(~s)
That is, for every t 2 0 and G E b 3 ,
(30.6)
K"(l[O,t]€3 G ) = K"(l[O,t]€3 P"{G
I3t)).
(30.7) PROPOSITION. Let E' and 3' be as in (30.5), and let K be the kernel generated by the integrable RM K . Suppose that for every t 2 0 and €3 G ) is in E', and that K satisfies (30.6) for every G E b e , 2 -+ K"(l[0,~1 every t 2 0 and every G E b p . Then At := K ( [0,t]) is indistinguishable from a process adapted to (3:+).
PROOF:We indicate only the modifications needed in the proof of (30.4). Define the measures Qf on f l instead of on 9 by setting Q f ( G ) := K Z ( l p t l €3 G ) for G E b e . Then QZ:<< P" on Doob's lemma gives q5t E p(Eo €3 f l )with QF(dw) = dt(z,w)P"(dw)on Since w -, X,(w) is in 3;/&', &(w) := q5t(Xo(w),w)E 3: for all t 2 0, and for every 2 E E
e. e.
I?': Random Measures
155
and G E b e , K " ( ~ I @ ~ ,G) ~ I= P"(BtG). A completion argument shows that the same formula remains valid also for G E b 3 t . The property (30.6) is needed now to establish that B, 5 Bt a s . if s 5 t , and that t , 1 t implies that, a.s., Bt, 1 Bt. Then A , constructed as in (30.4), is adapted to (F:+), and At = Bt a s . for every t 2 0. The correspondence between RM's and kernels allows one to obtain a technically important absolute continuity theorem for RM's. (30.8) DEFINITION. Let n and y be RM's in for a.a. w., n(w, . ) << y ( w ,
2. Then n << y provided,
a ) .
(30.9) THEOREM. Let n,y E M (resp., 6,P).If IC Z E p M (resp., p o l pP ) such that n = Z * y.
<< y,then there exists
PROOF:Without loss of generality, it may be assume that n and y are integrable. Let K and G be the corresponding kernels from (El€")to (R+x 0 , M ) . It is easy to see that for every z E E , K " ( . ) << G"( . ) on is separable, Doob's lemma (A3.2) yields (R+x R, B+ 8 9). As B+ @ $ E p(E" @ B+ @ 9) with
K"(dt,dw) = $(x,t,w)G"(dt,dw) on p(B+ @ 3F").Let & ( w ) := $ ( X , ( w ) , t , w ) so that Z E p(B+ @ F). For every M E bp(B+ @ 9) and z E El K " ( M ) = G " ( Z M ) . The uniqueness part of (30.4) shows then that n = 2 * y. If n and y are optional (resp., predictable) Z may be replaced by OZ (resp., P Z ) , thanks to (28.7). The following result shows that the density Z in (30.9) is essentially unique. (30.10) THEOREM. Let n E M . If Z ' , 2' E pM satisfy Z1 * n = Z 2 * n, then IZ' - Z 2 [ * n = 0.
PROOF:Replacing n if necessary by W * n with W E M strictly positive, it may be assumed that n and Z1 * n are integrable. One has then Z ' 1 { ~ ' ~ ~*2n}= 22 l { Z l > Z 2 } * 6
so, for all x E E ,
P"J ( Z t - 2,")l{,t2z:) n(dt) = 0. That is, (21- 2 2 ) l{Zl>ZZ} -
Symmetrically, (22
and so JZ1- Z 2 ) * n = 0.
- 21) l { Z 2 >-Z l }
*
n = 0.
* K. = 0,
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Markov Processes
(30.11) EXERCISE. Show that if K"( . ) and G"( - ) (defined in the proof of (30.9)) satisfy the conditions of (30.7), then Z (in (30.9)) may be chosen to be in t3+ 8 3' (resp., O', P', with the obvious definitions).
It is sometimes natural to consider, instead of a single RM n, a family of RM's indexed by a measurable space (V, V ) . One important example occurs in connection with local times in 568. A more trivial example is the family 6tn indexed by t E R+,with 6 a fixed RM. K,
(30.12) DEFINITION. A family {n, : v E V} of RM's indexed by a measurable space (V, V) is called (V,V)-measurable if ( v , w ) + n,(~, . ) is a kernel from (V x R, V 8 7 )to (R+, a+). (30.13) LEMMA.I4 for each v E V , n, is generated by a finite valued increasing process A" satisfying (27.1), then {n, : v E V} is (V,V)measurable if and only if the map ( t , v , w ) 4 A i ( w ) is B+ @ V 8 3measurable.
PROOF:If {n, : v E V} is (V,V)-measurable, then for fixed t 2 0, ( v , w ) + A:(w) = /c,(w, [0, t]) is V83-measurable. Right continuity in t shows that ( t ,v,w ) -+ A i ( w ) is B+ @ V 8 3-measurable. Conversely, the condition on A" implies that ( v , w ) -+ /c,,(w, [0,t , ] )is V 8 3-measurable for all t 2 0. The MCT shows then that (v, u)+ /c,(w, B ) is V 8 3-measurable for all B E B+. If {n, : v E V} is a (V, V)-measurable family of integrable RM's, it is a routine matter to show that if K, is the kernel generated by n, (30.3) then ( q v ) -+ K:(M) is E" 8 %measurable for all M E p M . That is, K,"( is a kernel from ( E x V, E" 8 V ) to (R+x R, M ) . Using the method of proof of (30.4), the following argument produces a measurable family of RM's, being given such a kernel. (30.14) THEOREM. Let {K,; v E V} be a family of finite kernels from (E,E") to (R+x R,M) respecting X. Suppose that ( z , v ) + K,"(M)is E" C3 V-measurable for all M E b(B+ 8 9). Then there exists a family {A";v E V } of increasing integrable processes such that for all w E V, A" generates K,, and such that (t,v ,w ) 4 AP(w) is B+ 8 V @ 3-measurable. PROOF: We indicate only the changes necessary in the proof of (30.4). These changes are mainly notational. For fixed t 2 0, let Q:>" be the measure defined on (R, 9 )by G E b3". QT7"(G):= KE(lLo,tl 8 G ) , Then Q:'" << P5 on 3". By Doob's lemma, there exists & ( z , v , w ) E E" 63 V 8 9 such that Q:'"(dw) = +t(x, v , w ) P " ( d w ) on 9. Then set B r ( w ) := $t(Xo(w),v,w ) . Regularizing Br as in the proof of (30.4) gives the family AY(w) with the desired properties. 0
)
IV: Random Measures
157
31. Dual Projections To begin with, recall that the classical version (As) of the dual optional (resp., dual predictable) projection for the system (a,3:,P”) states that if the right continuous increasing process A is in MP and if PPA, < 00, then there exists a unique increasing right continuous process Bt adapted (resp., predictable relative t o (3;)) such that for every positive to (3;) process (Zt), optional (resp., predictable) over (F;), roo
(31.1)
P”J,
lo Zt dBt. F m
Zt dAt = P”
Recall also that, given A as above, if for every positive measurable process 2 with optional projection O Z one has
1
00
(31.2)
Zt dAt = P”
“Zt dAt,
then A is optional relative to (3;). An analogous assertion is valid in the predictable case. As in the case of optional and predictable projections constructed in 522, we wish to obtain a construction of dual projections independent of the initial law, and taking into account the shift operation of $29. Before proceeding with this construction, we make a few simple comments on the classical case. To begin with, there is no difficulty in extending to the case where A0 > 0, letting Bo := PP{Ao [ I$}in both the optional and predictable cases. Next, where in the classical theory one extends the projection to locally integrable increasing processes, it is just as simple to extend the construction to RM’s that are o-integrable on 0 or P,respectively. This follows from the discussion in 528 where we showed, for example, that if (c is o-integrable on 0, then (c = W * A where A is an integrable increasing process and W is a positive optional process, so that one may define the dual optional projection of K: as W * B, where B is the dual optional projection of A in the classical sense. With this definition, (31.1) remains valid. Moreover, if 7 E d has the property
then
Markov Processes
158
Taking a Y E 0 strictly positive such that Y * 7 is integrable, this leads to the equality pP
L+
2,yt y( * ,d t ) = P P
L+
Wf zt yt dBt,
zE PO,
from which it follows, taking 2 = 1, that ( Y W )* B is integrable. Then Y,y( ,d t ) = &Wt dBt since (31.1) uniquely determines the dual optional projection of integrable RM's. Because Y > 0, we find that 7(.,d t ) = Wt dBt. That is, (31.1) uniquely determines the dual optional projection for a-integrable RM's. A similar discussion holds in the predictable case. The following result will be useful in proving that dual projections commute with shifts.
-
(31.3) PROPOSITION. Let T E T. Then the restriction of M to [rT,m[I is generated by processes of the form (31.4)
G@TZ+W~(IT,~~~,
with G E bFT, 2 E b M and W E bZ.
PROOF:In view of (3.10), it suffices to prove that for f E Cd(E),g E bE", s > 0 and CY > 0 the process M t ( w ) := g(XO)e-at f ( x s ( w ) )~ [ T ( W ) , W [ ( ~ ) is a limit of sums of products of the form G O T Z described above. Since sums of such products form an algebra, it is enough to work separately with
M,'
:= eWatl[T,m[(t);
M," := f ( ~ ~ [) T , ~ i ( t )M: ;
:=
dxo)l[T,oo[(t).
A typical term of the series may be written in the form G OTZ by setting
G := l{k/2n
C
bFT.
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159
If R is a R M that is o-integrable on 0 (resp., P) there (31.5) THEOREM. exists a unique RM K O in 8 (resp., RP in P ) such that, for everyp, no (resp., R P ) is a dual optional (resp., dual predictable) projection of R relative to the system (R,3:, P p ) . Moreover, for T E T, (31.6)
=(
&(RO)
(31.7)
lnT,,u
* &(.PI
6R )~” , = 1nT,,u
and
* (&KIP.
PROOF:Because of the preceding discussion, it suffices to show the existence of K O and K? in case R is integrable. Once (31.6) and (31.7) are proven for integrable R , one obtains the general case for a RM of the form Z * R with Z E p 0 (resp., p P ) using (29.3) and (21.12). We assume now that R is integrable, and associate with R the kernel K” from (I?,€”) to (R+ x 52, M) given by (30.2). Define “K”(2)and PK”(2) for Z E b M by setting OK”(Z)= K 2 (” Z ) , pK”(Z) = KZ(pZ).
(31.8) (31.9)
Because of the properties of optional and predictable projection discussed in $22, it is clear that OK and PK are kernels from ( E ,E ” ) to (R+ x R, M) , and that for each x E E , O K ” and PK“ do not charge processes in M n2‘. . By (30.4), there exist integrable RM’s and RP generating O K ” and PK”, respectively. For each 2 E E and Z E b M , using (31.8)
P”
J,t
Zt( * ) /GO(
’
, d t ) = P“
and by integration relative to p ( d z ) ,
L+zt
( . ) KO(
(31.10) pp
. ,d t ) = P p
L+
“Z,( . ) K (
a
,dt).
It follows from (31.10) that (31.2) is satisfied and so R” is optional. The fact that RO is indeed the dual optional projection relative to (R, ( F f ) ,Pp)is clear now. A completely analogous discussion holds in the predictable case. All that remains now is to prove (31.6) and (31.7) when IE is integrable. Since &R and & K O are carried by [ T ,m i , to prove (31.6), it suffices, by the MCT and (31.3), to prove that for z E E , G E b3T and Z E b M , (31.11)
P”~T,mlG(*)B~Z(s, . ) (&~O)(.,ds)
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160
However, on each side, Z may be replaced by "Z, and in the right hand may be replaced by & R . Finally, using (29.3) we see that term, (&Ic)" (31.11) is equivalent to the equality
P ~ { G ~*IC")(.,[T,~O[)} ~ ( ~ z = P"{G~T("Z*IC)(.,[T,~~[)}. Writing
* I E " ) ( W , [T,w [ ) = ("2 * & " ) ( O T W , R+)
&("Z
and similarly for the right hand term, and then taking conditional expectations, (31.11) reduces to the evident equality
P X ( T ) ( o Z * ~ o ) ( . ,= RPX(T)(oZ*~)(.,R't). +) The predictable case is similar, except that, by (22.13), one knows only "(@rZ)lnT,oojl = @T(*Z).1jz-,wu , and one obtains only
(31.12) REMARK. Since lnT,,u
*
(&K)
= &(lno,wu
* K),
the shift property (31.7) for the dual predictable projection will take the . not charge UOI]. Of same form as (31.6) if it is assumed that, a.s., r ~ does course, in the classical case, this assumption is built in. The message here is that when one permits n to charge [ O n , one should expect meaningful results only for dual optional projections. (31.13) REMARK. Suppose P"A, < 00 for each 2, with A an increasing process satisfying (27.1). Suppose also that At E Fe (resp., p ) for all t 2 0. It follows that the kernel K " ( - )constructed in (30.2) is a kernel from ( E , E e )(resp., ( E , & ) )to (R+x Q, B+ @ p ) .For 2 E b(B+ @ 9) let k"(2) := K " ( I I ( 2 ) )where II is the optional projection kernel of $22. Since K" respects X,k " ( Z ) = OK"(Z), as defined in (31.8), for all 2 E b ( B + @ p ) . However, in view of (22.8), K" is a kernel from ( E , E e )(resp., ( E ,E ) ) to (R+ x 0,B+ @ 9).Using (30.7), one concludes that since OK" obviously satisfies (30.6), the dual optional projection A" of A may be (resp., A similar result will hold for chosen to be adapted to (F:+) dual predictable projections, once we construct the predictable projection kernel in 543.
(e+)).
IV: Random Measures Let T E T. Then (31.14) PROPOSITION.
161
FT- = nF&-.
PROOF:Clearly 3T- c n3&-. Fix F E b (n3&-) with 0 5 F 5 1. Then, by definition of FF-,F l{T
This implies Pp J lyt - Z,"l n(dt) = 0, hence Pp{lY, - Zgl; T < m} = 0. That is, YT = F, Pp a s . on {T < m}. Since Y E P , F 1 { ~ < differs ~ ) from YT l{T
ck,l
PROOF: If A satisfies (27.1), then we may express A as the sum Ak where, for k 2 1, A! := (At A k) - (At A (k - 1)). Each Ak is a uniformly bounded, increasing, right continuous process, and A: 5 1 for t 2 S . From (31.5), we obtain right continuous increasing processes (Ak)" and (Ak)P which are respectively the dual optional and dual predictable projections
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162
of Ak relative to every Ps. Let A: := Ck(Ak)f and A: := Ck(Ak):. Suppose first that A is (8, Ps)-locally integrable over lo, S [ and let {T,} satisfy the conditions (i) and (ii) of (28.2). Since T, E T,
since (Ak)” is a dual optional projection of Ak. Summing on k,
for all n 2 1. That is, A: < m for all t < limnT,, Ps-almost surely. Let T := inf { t : A: = m}. Then P2{T 2 lim,T,} = 1, and
by monotone convergence. Since T 2 S, Ps-a.s.,
Ps{(Ak)$ - (Ak)$-; T < m} = Ps{Ak
- Ak-;
T < m} = 0.
It follows that P”{A+- < 00, T < m} = 0. Consequently, A” satisfies the conditions (27.1). Since A: is adapted to (Ft), (28.3) shows that A” is uniformly 0-locally integrable on 00, T [ . Choosing now an increasing sequence {R,} with limR, 2 T a.s. and sups P”[A(&-); R, > 01 < 00, so A is U-locally integrable on 0 0, S [ . The predictable case involves only obvious changes in the above argument. (31.17) REMARK. In the course of the proof of (31.16) we actually showed that if A is uniformly 0 (resp., P)-locally integrable on [ O , S [ ,then the dual optional (resp., dual predictable) projection no (resp., np) of the RM n generated by A is also generated by an increasing process A” (resp., AP) satisfying the conditions (27.1). We showed also that inf { t : A: = m} >_ S a.s., so if S = m as., A” and AP are a.s. finite valued. (31.18) EXERCISE. Let theRMn b e d i n i t e o n 0 (resp., P). Set Z t ( . ) := n(. ,{ t } ) ,the process of atoms of n. Show that no( { t } ) = “Zt( .) (resp., .”( * ,{ t } )= pzt(* )I. a ,
IV: Random Measures
163
32. Integral Measurability We describe now another completion procedure which is natural and useful in connection with dual projections. A typical situation indicating the need for such a completion is in dual projections with integrand f ( X , ) , f E bpE”. In general this is not an optional process, but one wishes to compute certain integrals as though it were. For example, one would like to know that r
r
for every K E M , a-integrable on 0 . The following definition permits one to make such computations. (32.1) DEFINITION.A real process 2 on R+ x R is integrally measurable (resp., integrally optional, integrally predictable) if for every initial law p and every RM K with s u p , P Z ~ ( R + )< 00, there exist W 1 , W 2 E M (resp., 0,P)such that PpJr(W,2 - W,’)n(dt) = 0 and W 15 Z 5 W 2 . The class of integrally measurable processes is denoted by M i , and similarly for Ot,P i .
The following is a trivial restatement of (32.1). We state only the measurable case. Recall (30.3). (32.2) PROPOSITION. The process 2 is in M t if and only if Z is in the Completion of M relative to the family of measures f p(da) K”(. ), where p is an initial law and K“( ) is a bounded kernel from ( E ,E ” ) to (R+xR, M ) respecting X .
The following two theorems state the principal uses of integral measurability. (32.3) THEOREM. Let Z E p M t (resp., p o t , pPt) and let n E M (resp., Then there exists y E M (resp., d, P ) such that, for a.a. w, t + &(w) is measurable relative to n(w, d t ) , and y ( w , dt) = & ( w ) ~ ( wd t, ) as measures on R+.The kernel K; associated with y by (30.2) satisfies
d, P ) .
K;(dt,dw) = Z&)K:(dt,dw).
For each t 2 0, OtZ E M t (resp., Ot, P t ) and, letting 2 * n denote y as in (28.4), one has, for all t 2 0, &(Z * K ) = Q t Z * & K . PROOF: This is all rather trivial, but we prove the measurable case to < 00 (see give the flavor of the argument. We may assume sup, P”K(R+) (28.1)) and 2 bounded. In view of (32.2), K5(dt,dw) := Zt(w)KE(dt,dw)
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defines for each 2 E E a finite measure on (R+ x Q, M ) . Given p, choose W’, W 2 E b M with W1 5 Z 5 W 2 and P” l(W,“ - W,’) n(dt). Then for any M E bpM, M t ( w ) Kx(dt,dw) is sandwiched by
s
The functions on the right are in E” and their difference is p-null. It follows that KZ( ) is a bounded kernel from ( E ,E ” ) to (R+ x R, M ) respecting X. By (30.3) there exists an integrable RM y such that I?”( . ) = K;( All except the last sentence in the theorem is now obvious. Given t >_ 0, sandwich Z relative to v = pPt and n. The identity asserted follows then from (29.3).
-
0
(32.4) THEOREM.Let 2 E pOt (resp., pPt). If n E 0 (resp., P), then for every initial law p ,
L+
p’”
M
)
.
is o-integrable on
k+
2, n(dt) = P’”
2, nO(dt).
(resp., the same formula holds with no replaced by np.)
PROOF:We treat just the optional case. Choose W1, W 2 E PO with W1 5 Z 5 W 2 and P’”J(W? R ( d t ) = 0.
w;)
Noting that Pfil(W,”- W t )no(&) = 0, the result follows. (32.5) COROLLARY. Let Z E M t (resp., Ot, F t ) . Let R E p T (resp.,
R E T). Then ZR 1 { ~ < E~ 3 ) (resp., FR,3 ~ - ) . PROOF:Take n ( w , d t ) := ~ ~ ( ~ ) (l{~(,)<,) dt) and use n 3’” = 3 (resp., nFg = 3 R , flFg- = 3 R - (31.14)). Having now shown that as far as integrands relative to a RM are concerned, integral measurability is just as good as measurability, we turn to criteria which ensure that a given process is integrally measurable. (32.6) PROPOSITION. Foreveryf E E”, theprocess f ( X t ) isin Ot. Forevery H E 3*(the universal completion of p ) ,the process (t, w ) 4 H ( & w ) is in M t . If H E b3* and h ( z ):= P”H, then for n E 6, (32.7)
Hoot
n(dt) = P’”
PROOF:If f E E”, then given an initial law p and an integrable RM n with sup,P”n{R+} < 00, define the measure v on ( E , E ) by v(g) :=
IV: Random Measures
165
s
PI" g(Xt) ~ ( d t ) g, E bE. Sandwiching f relative to Y proves the first assertion. If H E 3*, given p and K as above, let Q be the finite measure on (S2,p) determined by
Q(G) := PpJ
GoOt
GEbp.
~(dt),
0
Sandwiching H relative to Q proves the second assertion. Finally, if H E b 3 * and h ( x ) := P"H then h E bE", and, given p and K , sandwiching H by H I 5 H2 in bP' relative to Q and letting h k ( z ) := P5Hk (k = 1,2), we have PpJ H k o 1 3 , K ( & ) = Pp h k o X t ~ ( d t ) . The result is obtained by noting that
s
(32.8) EXERCISE.Call 2 E 0 t an integrally optional projection of M E Mt in case P p Z ~ l p < = ~ PpMT1{T
s
(32.9) EXERCISE.Use (30.9) to show that if 2 E p M t (resp., p o t , p P t ) and if K E M (resp., d, P ) then there exists W E p M (resp., PO, pP) such that 2 * K = W * K , independently of the initial law. (32.10) EXERCISE. Let K E M ,2,W E Mt (finite valued), and suppose * K . Show that (2- W (* IE = 0. (Hint: mimic the proof of
2*K =W (30.1O).)
(32.11) EXERCISE. If 2 is bounded and progressive over (3t), 2 is not * K = 0. necessarily in Ot, but for any K E d one has ( 2(32.12) EXERCISE.Let f E Eu and let Tl,Tz, E T have graphs disjoint , win) 0. In up to evanescence. Then ( t , w ) + f ( X , ( ~ ) ) l { ~ [ l ~ , n } ( t is particular, if { f # 0) is semipolar, then f is an optional function. (For a predictable version of this result, see (41.10).) 7 . '
(32.13) EXERCISE. Let R be of path type, as defined in (23.10). Show that if H E 3*, then (t,w) + H ( k t w ) is in Ptand ( t , w ) +- H ( a t w ) is in Ot .
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Markov Processes
33. Characterization by Potentials
The class of u-integrable RM's may be constructed in a simple way from integrable ones. We restrict our attention for the moment to integrable RM's, studying their characterization by means of their potentials. For future reference, we record and discuss the Markovian versions of some classical definitions.
(33.1) DEFINITION. Let Y E M . Then: (i) Y is of class ( D ) in case the family {IYTI l{T<m): T E T} is P"-uniformly integrable for every x E E; (ii) Y is regular if, for all x, for every bounded T E T, P Z Y exists ~ and T, T T , T, E T implies P Z Y ~ ,+ , PSY,; (iii) Y is a strong supermartingale provided Y E 0 and, for every bounded pair S 5 T E T, P"{Y, A 0) > -00 and Ys 2 P Z { Y ~1 3 s ) vxc; (iv) Y is a potential provided Y is positive, as. right continuous, Yt + 0 a.s. as t + 00, and Y is a supermartingale relative to (a,3iz,P") for every 2 E E ; (v) Y is a left potential in case Y is a positive, strong supermartingale and yt ---i 0 a s . as t + m. The least familiar of these objects are the strong supermartingales, introduced by Mertens [Mer73]. The reader may consult [DM75, VII] for a complete discussion of their properties, one of the deepest of which is (33.2) if Y is a strong supermartingale, then t -+ % ( w ) has right and left limits everywhere on R+,for a.a. w . In general, strong supermartingales are neither right continuous nor left continuous. It is of course true, by optional sampling, that every right continuous supermartingale is a strong supermartingale. The class (D) condition is often verified in Markov process theory by means of the following alternative conditions.
(33.3) PROPOSITION. Let Y be a positive, strong supermartingale. Then the following conditions on Y are equivalent: and yt -+ 0 a.s. as t + 00; (33.4) Y is of class (0) for every increasing sequence {T,} C T with limT, = 00 as., (33.5) P" YT, l{Tn<m) + 0 as n + 00 for all x E E ; 00 (33.6) if & := inf {t : yt 2 n } , then P" YR, 1 { ~ , < ~+) 0 as n for all x E E . -+
The proof of (33.3) is classical [DM75, VI-251, using the fact that, for .-* 0 as., Fatou's lemma gives S 5 T E T (Ym := 0). (33.7) PZ{Y-p I3.9} 5 Ys,
Y positive and satisfying yt
IV: Random Measures
167
Let n be a RM. (33.8) DEFINITION. (i) The potential of n is the optional projection of the decreasing process t + n( ,It, m[ ). (ii) The left potential of n is the optional projection of the process t +,[t,4).
-
--$
The following result is a standard exercise using the properties of optVI-31. ional projections-compare with [DM75, The potential of an integrable RM is a potential of (33.9) THEOREM. class (0)over X. The left potential of an integrable RM is a regular left potential of class (0)over X. Because indicators of stochastic intervals of the form IT, mu, with T E
T, form a multiplicative class generating the restriction of P to 10, m i , we have the following important result. (33.10) THEOREM. Two integrable RM’s on R+ not charging [On have the same dual predictable projection if and only if they generate the same potential. In particular, a predictable, integrable RM not charging [On is completely determined by its potential.
PROOF:If two integrable RM’s not charging [I01 generate the same potential, the kernels K” given by (30.2) agree on P by the above remark. Thus, the kernels PKz of (31.9) also agree. Since stochastic intervals [IT,m[ generate 0, (33.10) has the analogue: (33.11) THEOREM. Two integrable RM’s, possibly charging [On, have the same dual optional projection if and only if they generate the same left potential. In particular, an optional, integrable RM is completely determined by its left potential.
Because of (33.10) and (33.11), it is useful to observe the effect of the shift operators on the corresponding potentials. (33.12) PROPOSITION. Let K. be an integrable RM, and let T E T. (if If Y is the potential generated by n, then the potential Y generated by 6~K. satisfies lnT,,n = OT Y . (ii) If 2 is the left potential generated by n, and if 2 is the left potential generated by 6~K then 2 lnT,,n = OT 2.
PROOF:Let W s ( . ):= n(.,]s,m[) for s 2 T ( w ) ,
W&)
and W,(w):= (&n)(.,]s,m[). Then
= K ( B ~ W , ] S - T ,OO[) =
(o~w)~(w).
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168
In other words, l[T,,[I ii' = OT W , and taking optional projections gives (i), applying (22.16). The fact that Q T K does not charge [[On is obvious from the definition of &. To get (ii), use essentially the same argument, but with W s ( ). := K ( [s,m[ ). a ,
(33.13) PROPOSITION. Let K ~ , Kbe~ integrable RM's and let T E T. If Y', Y2 denote the corresponding potentials and Z1, Z 2 denote the corresponding left potentials, then: (i) if n' and K~ are predictable, and if Y1 lnT,,f = Y2 l[T,,i then 1lT,co[
* K'
= llT,ca[
(ii) ifn' and ri2 are optional and if 2'
*
K
lnT,,n
2
;
= Z2 lnT,,n
then
PROOF:By an elementary argument, the potential of liT,mn * d is
Under the conditions of (i), since
*
l ~ T , o o ~nJ
does not charge [On for
j = 1,2, we derive their equality from (33.10). The proof of (ii) is exactly
analogous, using instead (33.11). The classical definition of the potential Y generated by an increasing process A satisfying (27.1) with A , = A,-, P"A, < 00 for all 2,A0 = 0 and At E Ft for all t 2 0, is that Yt := Mt - At, where Mt is a right continuous version of the martingale P " { A , I Ft}.This is in accordance with (33.8), because of the following argument. The process Mt may be chosen independently of x-that is,M is the optional projection of Zt(w) := A,(w). If A is predictable, we have Pyt = PMt - At and since PMt = Mt-, PYt = Mi- - At = yt- - (At - A t - ) . Setting AAt := At - At-, this leads to AAt = Yt- - Pyt. From this we obtain the following important fact. (33.14) THEOREM. Let Y be the potential of an integrable predictable RM n not charging [On. Then the atomicpart of n is given by K ( . ,{ t } )= yt- - P y t . The RM n is diffuse (that is, K ( W , . ) is a diffuse measure on R+ for a.a. w ) if and only if Y is regular. (33.15) EXERCISE.Let A be a left continuous increasing process adapted to (Ft)with A0 = 0 (but, possibly, A@+> 0). Suppose P"A, < 00 for all x E E and let K. be the RM generated by A. Then the left potential Z of K may be written Zt = Mt - A t , where M is a right continuous version,
IV: Random Measures
169
independent of z, of P"{A, I Ft}. Show that the atomic part of K is given by K ( ., {t}) = 2, - Zt+ and that K is diffuse if and only if 2 is right continuous. Show that ( Z t + ) is a potential of class (0). (33.16) EXERCISE. Let X be a Poisson process with rate 1 on E := { 0 , 1 , 2 , .. . }, and let TO,T I , .. . be the hitting times of 0 , 1 , . . . respectively. Let K denote the RM putting mass 1 at each of TO,.. . ,T,. Compute the potential of lno,mo * K and the left potential of K . (33.17) EXERCISE.Let Y E 0 . Suppose that for all z, for every bounded T E T, P"[YT A 01 > -GO, and for bounded R 5 S E T, P S Y , 2 P"Ys. Show that Y is a strong supermartingale. (Hint: for A E 3 R , let R' := R In, + S l ~ = and , show that R' E T.) 34. Representation of Potentials
We are now ready to formulate an important means of constructing random measures. Recall first the representation of potentials in the general theory of processes. Let (W,6, G t , P ) be a filtered probability space satisfying the usual hypotheses, and let P , 0 and M denote respectively the predictable, optional and measurable a-algebras for that filtration. Given a potential ( & ) t l o of class (D) relative to (W,8, G t , P ) , one constructs a measure py on (R+ x W,P) such that for every optional time T ,
The definition of py on the algebra of finite unions of stochastic intervals JS,2'1 is obvious from (34.1), but some analysis is needed to show that py extends to a countably additive measure on P . This was the approach taken by Dol6ans-Dade [DD68]. Her proof is presented in [De72], for example. See also A5. A newer proof which extends to the left potential case is given in [DM75, VI-81. One extends py to a measure on (R+ x W,M ) by setting p y ( M ) := p y ( P M ) for M E b M . Then, in the same manner as (30.3), one produces a predictable, right continuous increasing process A with A0 = 0, A, = A,- and P{Am - A t 1 Gt} = yt. That is, the predictable RM tc(dt) := dAt has potential Y. As we saw in 533, it is equivalent to say that A is the unique predictable, right continuous increasing process with A0 = 0 such that Mt = yt At is a martingale. The decomposition
+
(34.2)
Yt = Mt
- At,
where Mt is a right continuous version of P{A, Meyer decomposition of the potential Y.
[ G t } , is called the Doob-
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Markov Processes
The corresponding facts for left potentials are of more recent origin. We mentioned in 533 that the paths of a left potential 2 almost surely have VII] that if right and left limits everywhere on R+. It is shown in [DM75, Z is a regular left potential of class (D) relative to (W,0,Q t , P), then there exists a unique measure vz on (R+ x W,O) such that, for every optional time T, (34.3)
vz(UT,OO[I) =PZT1{T<m}.
As with the measure py corresponding to an ordinary potential, one may produce an optional, right continuous increasing process A with A0 2 0 and P{A, - AT- 1 GT} = ZT for every optional time T, (Ao- := 0). In ) dAt has left potential 2. As was pointed other words, the RM ~ ( d t := out in (33.15), one has a Doob-Meyer type decomposition (34.4)
2, = Mt - At-,
with Mt a right continuous version of P{Am I Bt}. The next theorem gives the Markovian version of the constructions described above. (34.5) THEOREM. (9 Let Y be a potential of class (0)for X. Then there exists a unique predictable, integrable RM, not charging [I0I], with potential Y . (ii) Let Z be a regular left potential of class (0) for X. Then there exists a unique optional, integrable RM, possibly charging [Ion, having left potential 2.
PROOF:Uniqueness comes from (33.10) and (33.11). Let Y be a potential of class (D) for X. For x E E , let K"( .) be the measure on (R+ x R, P'z), not charging [ O ] , such that for every optional time T over
(c"),
(34.6)
K"( ] T , o o [ )=P"{YT;T
< OO},
and K " ( N ) = 0 for N E Z'z. See the above discussion. Now restrict each K" to P. Since stochastic intervals IT, 001, with T E T, generate P up to evanescent sets, (34.6) shows that ( K Z ) x Eis~a finite kernel from (E,E") to (R+ x R , P ) . Without changing notation, extend K" to a kernel from (E,E") to (R+ x R,M) by setting K 5 ( M ):= K"(PM) for M E bM. It is evident that for all x E E , K " ( . ) does not charge P"-evanescent sets in hi. Consequently, (30.4) displays the unique integrable RM K. corresponding to the kernel K". Since K " ( M ) = K"(pM) for all M E b M , K. is predictable (31.2). Taking M := l ~ T , m we~find
= P"{K.( * , IT( * ) , OO[); T
< OO}.
IV: Random Measures
171
Thus, since Y E 0,Y is the optional projection oft -, K ( . , It, co[). Finally, K: does not charge (I01 because K" { JT 0 I]} = 0 for all z. In the case of a regular left potential of class (D), one constructs a kernel K" from ( E , € " ) to (R+ x R , 0 ) , not charging evanescent sets, such that for T E T
K"(JTT,ooU)=P5{Z~;T
(e+)
-
PROOF:We use (30.7). For any G E bF and s 2 0, the processes lpS1@G and l[O,s~ @ P ' { G I FS)have the same optional projection. By construction of the kernels K" in (34.6), the condition (30.6) is satisfied. Observe now then by composition, w -+ Z T ( ~ ) ( W ) that if T is an optional time over is in 9 so K"( JTT,001) = P"ZT is &-measurable. It follows by the MCT that K " ( M ) is €-measurable for every M strictly optional over
(e+),
(e+). (e+).
Recall now (22.8) that the optional projection kernel II has the property It follows that that if M E b(B+ @ F'),then HM is optional over for every M E b(B+ @9), z + K " ( M ) = K"(rIM) is €-measurable. The case is similar. conclusion then follows from (30.7). The (3:+) (34.8) REMARK.After obtaining ($43) a kernel for the predictable projection mapping b(B+ @ p )into processes predictable over ( E ) trivial , modifications of the proof above will show that if Y is a potential of class (D) adapted to (resp., (Fg+)) then the predictable RM K. generated (resp., by Y may be chosen so that A t ( . ) := K ( . , [0, t]) is adapted to (3f+))-
(c+)
(e+)
(34.9) EXERCISE.Using (30.14) and the method of proof of (34.7) show that if {Z" : v E V} is a jointly measurable family of regular left potentials of class (0)induced by a measurable space (V, V ) , then one may choose the optional increasing processes AT generated by the Z" so that ( t ,w , w ) -+ AT(w) is jointly measurable. (The same argument will apply to measurable after the predictable projection kernel families of potentials of class (0) becomes available in $43.)
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35. Homogeneous Random Measures We apply now the preceding results to a special family of RM’s that are the Markovian analogues of measures on the state space. (35.1) DEFINITION.Let K be a RM in M-see (28.1). Then K is homogeneous on R+ (resp., R++)provided, for every T E T, (35.2) (resp., (35.3)) holds: (35.2) 6~K and 1uT,.3i * K are indistinguishable; (35.3) liT,.ou * 6~K and l ~ T , o o* ~K are indistinguishable. A homogeneous random measure (HRM) is a R M which is homogeneous on R++. Obviously, the mass of
K
at [ O n is irrelevant to homogeneity of
K
on
R++,and one usually assumes that such a K does not charge [ O n . Moreover, if K is homogeneous on R+ and if K does not charge [ O n , then K is homogeneous on R++. From (24.1) and (32.3), it follows immediately that (35.4) PROPOSITION. If K is homogeneous on R+ (resp., R++)and if 2 E pMt ($32)is homogeneous on R+ (resp., R++),then Z * K is homogeneous on R+ (resp., R++).
For example, since d t ) := dt (Lebesgue measure on R+)is clearly homogeneous on R+ (and therefore on R++), for any f E pEU, K ( . ,d t ) := .f(Xt) dt is homogeneous on R+ and R++. For RM’s generated by increasing processes, the homogeneity condition may be formulated in a different manner. )
(35.5) DEFINITION.(i) Let At be a right continuous increasing process with finite values such that A0 = 0. Then A is a raw additive functional (RAF) provided At E 3 for all t 2 0, and, for every T E T, (35.6)
a&, Q s 2 0, AT+* = AT
+A s 0 6 ~ .
We say that A is an additive functional (AF), if, in addition, At E 3t for each t 2 0. (ii) Let A be a left continuous increasingprocess with finite values such that A0 = 0, but possibly Ao+ > 0. Then A is a raw left additive functional (RLAF) provided, for each t 2 0, At E 3,and A satisfies (35.6). We say that A is a left additive functional (LAF) if, in addition, At E 3t for each t 2 0. The definition of an AF given in (35.5) corresponds to what some authors refer to as a strong AF. The following result is straightforward.
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173
(35.7) PROPOSITION. Let finite values.
K
be a RM and A an increasing process with
(i) Suppose K ( W , [0,t]) = At(w) for all t 1 0, and A0 = 0 a.s. Then K is homogeneous on R++if and only if A is a RAF. (ii) Suppose K ( W , [ O , t [ ) = At(w), for all t 2 0. Then K is homogeneous on R’ if and only if A is a RLAF. The next result is a direct corollary of (31.5). (35.8) THEOREM. (i) Let K be a RM, homogeneous on R+.If K is a-integrable on 0 , then K O is also homogeneous on R+. (ii) Let IE be a RM, homogeneous on R++.If K is a-integrable on 0 (resp., P) then KO (resp., K P ) is also homogeneous on R++.
We remind the reader that (28.2) and (28.3) give criteria on an increasing process A guaranteeing that the associated RM is a-integrable on 0 or
P. Just as with homogeneous processes, there is a notion of perfect additivity for AF’s. Let (35.9)
N s , t := {w : At+s(u) #
A t ( w ) + AS(Qtu)}.
Then A is called an almost perfect (resp., perfect) RAF provided A is right continuous, increasing, finite valued, measurable and U {N,,t : s 2 0, t 2 0) is null (resp., empty). (Almost) perfect AF’s, RLAF’s and LAF’s are defined in a similar way. We shall say that A is a weak RAF in case N,,t E JVfor every pair s, t. Our weak RAF’s are the same objects that some other workers call RAF’s. There are corresponding definitions of weak AF and weak LAF. The following theorem follows directly from (24.36). Let A be right continuous, increasing, finite valued (35.10) THEOREM. and measurable. If A is a weak RAF, then there exists an almost perfect RAF indistinguishable from A and in D+ @J F e . (35.11) EXERCISE.Let L be an 3-measurable random time. Then
At
:= 1UL,mU (t) 1{0
is an RAF if and only if 1 ! 0 , ~ [ 1is homogeneous on R+.This is the case if and only if L is co-optional (25.3).
(35.12) PROPOSITION. Let A be a RAF. Then the process
AAt l]O,m[(t):= (At - At-)
1 ] 0 , 4 )
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belongs to f i g .
PROOF:We may suppose that A is almost perfect. For h > 0, yt := A(t+h)- - At- is left continuous, and outside some fixed null set in 52, for t , s > 0,
Consequently Y E f i g . Letting h decrease to zero sequentially one obtains AA E 5 s . There is an obvious converse to (35.12).
xO
Let Y E pfig. Then At := Y,is an almost (35.13) PROPOSITION. perfect RAF, provided At < 00 a.s. for all t >_ 0. I f , in add%on, Y E Onfig (resp., P n f i g ) then A is optional (resp., predictable).
PROOF:Additivity of A is obvious, and measurability comes from (28.7). (35.14) EXERCISE. Let T be a thin terminal time whose iterates Tn increase a.s. to 03. Show that Enl u T n ] is in fig. (Hint: T may be assumed almost perfect because of (24.25). A simple calculation yields do Ot lnTn = l U p + i n l{Tn
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175
36. Potential Functions and Operators
We begin with a simple case.
(36.1) DEFINITION.Let n be a R M , homogeneous on R++. Then u,(z) := P”n(R++)is the potential function for K . Note the restriction to R++in (36.1). If n is generated by a RAF A with finite values, we also write u,(z) = u ~ ( z=) P“A,. The function u, may take infinite values, but in any case, u, is excessive on E , for by the simple Markov property and homogeneity of tc,
It is also the case that for every T E T,
Since u, is an optional function, this proves that un(Xt) is the optional projection of the process t -+ tc( It, m[ ), possibly taking infinite values. If u, is finite everywhere, then, in the language of $33, u,(Xt) is the potential of the integrable RM K . It is often necessary to work with RM’s which are not integrable, and we shall make no finiteness assumptions on u, unless it is essential to do so. (36.2) REMARK.Recall that the lifetime is being supposed identically infinite throughout this chapter. Without this hypothesis, the potential function of an AF A would not necessarily be excessive because the potential function does not vanish at A unless A is a s . constant on jC,mI). It is however easy to adapt the results of this chapter to the general case. See $48 for the discussion of AF’s vanishing on [IC, 00 I), and chapters VII and VIIJ for the more general AF’s of (X, S) with S a terminal time.
(36.3) THEOREM. Let u, be the potential function of a H R M K not charging [ O n . If {u, = m} is polar, then IC is a-integrable on P,and its dual predictable projection is uniquely determined by u,.
PROOF:Fix a sequence t, 11 0 and let
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176
It is clear that Y is a predictable process, strictly positive on 10, 00[ and finite off an evanescent set since {un = 00) is polar. One has, almost surely
Therefore
Thus, IE is a-integrable on P. Its dual predictable projection, K P , is an HRM by (35.8), and its potential function is also u,. To complete the proof, it suffices to prove that if K and y are predictable RM's, both not charging [On and homogeneous on R++, having the same potential function u with B := { u = m} polar, then K = y. (If B were empty, this would be an immediate consequence of (33.4).) As we showed in (12.30), the process XI obtained from X by restriction to El := E \ B is also a right process. The sample space can be taken to be QI = { w E R : Xt(w) E E' for all t 2 0). The restrictions of K and y to $21 are then predictable relative to XI and their potentials are just U ~ E IBy . the finite case mentioned above, ~ ( w. ),= y(w, . ) P " a s . for all x E El. On the other hand, homogeneity of K and y implies that for any t > 0 and x E B ,
P " { K ( ~.), = y(w, . ) o n ] t , m [ } = P"PX'{n = y on R++}, since P"{Xt 4 El} = 0. Letting t 11 0 one obtains ~ ( w. ),= y(w, .) on R++as., concluding the proof. The class (D) property of potentials of integrable RM's gives one the following class (D) property of potential functions.
(36.5) PROPOSITION. Let u, be the potential function of a H R M (36.6) Tn T 00 in T, u,(x) < 00 & p " u , ( X ~ , ) -+ 0.
K.
Then:
The condition (36.6) is the analogue of a "zero boundary" condition in ordinary potential theory.
(36.7) DEFINITION. A function u on E is a potential function of class
(D) for X if u is excessive, finite valued and satisfies (36.6). If, in addition, u ( X ) is regular (33.1), then u is a regular potential function for X . If the underlying process X is recurrent, so that every excessive function is constant, then every potential function of class (D) is identically zero. In
IV: Random Measures
177
this case, the notion of potential function is not useful, and one resorts to artifices which render the process transient. Possibilities include killing the process at terminal times or passing to the space-time process. Another approach, which amounts to killing the process at an exponentially distributed time independent of X, is to introduce the a-potential function for n. In case a = 0, we obtain just the potential function defined in (36.1). (36.8) DEFINITION.The a-potential function u: for a H R M n is (36.9)
The following analogues of the properties of potential functions are easy. (36.10) THEOREM. Let n be a H R M . Then: (i) u: E So; (ii) if uz < 0;) except on a polar set, then n is a-integrable on P, and its dual predictable projection is completely determined by uz; (iii) if u: < 00 on E , then e-atuE(Xt) is the potential, in the sense of (33.21, of the R M e-atn(dt); (iv) if T, E T and T, 0;) as., then
(v) if 0 5 a (36.12)
< p, then = u!
+ ( p - a)UP u:
= 21:
+ ( p - a)U%$
The resolvent equation (36.12) for a-potential functions is easily obtained from the formula
and the fact that t + &,oo[e-aS n(ds) has optional projection e-atua(Xt). Finiteness conditions are not needed. (36.13) DEFINITION. A function u on E is an a-potential function of class(D) in case u is a-excessive, finite valued and satisfies (36.11).
The notion of the a-potential function has an extension to that of the a-potential operator, which is particularly useful in the study of optional
HRM's.
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178
(36.14) DEFINITION. The a-potential operator UE for a HRM K is defined by (36.15)
u: f (I):= P"
e - " t f ( X t ) tc(dt), f E pE".
For f E pE", since f(Xt) is homogeneous and in Ot, f(Xt)K(dt) = (f (X)* ~ ) ( d t is, ) by (29.3), homogeneous on R ' + . Therefore UEf (x)is the a-potential function of f ( X ) * IE. The following facts follow at once from (36.10).
Let IC be a HRM. Then, for f E pE", (36.16) THEOREM. 6) U E f E S"; (ii) ifUE f < 00 everywhere on E , then UEf is an a-potential function of class (D); (iii) if 0 5 a < p then
U,"f = u,pf
+ ( p - cy)u~u:f = u,pf + ( p - a)U"U,p f.
There is also a useful analogue of (36.3). (36.17) THEOREM. Let a 2 0 and let K be a H R M and not charging [On. If there exists a strictly positive function f E pE" such that UE f is finite except on a polar set, then tc is o-integrable on 0,and its dual opt-
ional projection KO has the same potential operator U,O. The dual optional projection of tc is completely determined by the operator U z .
PROOF:Let y := f (X)* tc. Then u: = UE f is finite except on a polar set, so y is o-integrable on P by (36.3). Therefore IC = (1/ f ( X ) )* y is ointegrable on 0, using (28.5). The first assertion now follows from (32.4). It remains to prove that if K' and tc2 are optional HRM's not charging 101, then if K' and tc2 have a-potential operator UE, one must have I C ~= tc2. For j = 1 , 2 let y j := f ( X ) * KIt~suffices'to . prove that y' = y2. For every g E bpE", the a-potential functions of g(X)* yj are identical and finite off a polar set. By (36.3), their dual predictable projections are identical. Hence, for all I E E,g E bpE" and Y E bpP,
Jr
If UE f (I) < 00, M -+ P" e-atMt y j ( d t ) defines a finite measure on M , and the above equality states that the two measures agree on products Mt := ytg(Xt)with Y E bpP and g E bpE". As we saw in (23.4), such products generate 0 and so the measures agree on 0. It follows that y' and y2 are P"-indistinguishable if U; f (z)< 00. Exactly the same kind of argument as was used at the end of the proof of (36.3) now shows that y' and y2 are indistinguishable.
IV: Random Measures
179
Let X be uniform motion to the right on R+ and let (36.18) EXERCISE. f (x) := z-2 lp,m[(x). Then &(dt) := f ( X , ) dt defines a predictable RM which is homogeneous on R++but which is not determined by an AF. The potential function for IE is u,(z) := 2-l ~ p ~ [ ( z ) . We point out in the next exercises some consequences of the results of this paragraph for the potential theory of the resolvent (U"). Recall from $10 that for f, g E E", f = g a.e. means {f # g} is null. (That is, U" l{f#s) is identically zero for all a 2 0.) (36.19) EXERCISE.Conclude from (36.10ii) that if f,g E pE" and if, for some a 2 0, U"f < 00 a.e. and U" f = U"g a.e., then f = g a.e. (36.20) EXERCISE.Show that if a 2 0 then for all f,g E pE" (with finite values) such that U"f < 00 a.e. and U a g < 00 a.e., the equality f pU" f = g p U Q g for some p > 0 implies that f = g a.e. (Hint: put f' = f- f A g, g' = g- f A g so that f ' Ag' = 0 and f'+pU" f' = g'+pU"g' except on the polar set B := {U"f = co} U {U"g = co}. Deleting B from theprocess, this implies Pg'(x) 2 U" f'(z) ifg'(z) = 0, hence if f ' ( x ) > 0. By (10.27), P g ' 1 U" f' except on B. Symmetrically, U" f' 2 U"g' except on B so U" f' = P g ' a.e.1
+
+
(36.21) EXERCISE.Let a 2 0 and suppose there exists a strictly positive f E E" such that U" f < 00 a.e. (This is no restriction if a > 0.) Suppose (Vp)is also the resolvent of a right process on the same state space, and that U" = V" as kernels on E . Show that U p = V p for all p > a and hence, by (4.9), the associated semigroups are identical. (Hint: since Ua = V",a set is null for ( U p ) if and only if it is null for (VO). If p > a and g E Ee with 0 5 g 5 f, two applications of the resolvent equation give
[I + ( p - " ) u a ] u p g = [I+ ( p - a!)v"]Vpg = [I+ ( p - a)U"]Vpg which is finite a.e., so by (36.201, U p g = V p g a.e., hence everywhere.) (36.22) EXERCISE. Show that if a 2 0 and A is a continuous AF, the kernel U z satisfies the complete maximum principle (10.27). (36.23) EXERCISE. Let (Xt) be a Poisson process on the integers E , and let At = X t - X o . Evaluate UA and show that UA does not satisfy the complete maximum principle. (36.24) EXERCISE. Let A be a continuous AF and suppose that uQAis bounded. Define the process CFtAby the formula rt
(36.25)
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180
Then C r 7 Asatisfies the ('a-additive" condition
(36.26)
c;:
=
c;qA+ e-atC;fAoet.
The limit C z A exists a s . and PxCzA= 0 for all x. Then C r > Ais a right continuous, uniformly integrable martingale relative to every P". Writing e-as dA, e-at dAt as a sum of integrals over the wedges 0 5 s 5 t and 0 5 t 5 s (the diagonal has zero measure), we obtain the formula
(36.27) a A )2 ) + ."Ax)' = 2uj%;(x). P" ((Cd By polarizing (36.27), if B is another continuous A F with ug bounded,
+
(36.28) P " [ C z A C z B ]= U ~ U ~ ( UB X'a)u aA ( z ) - u ~ ( ~ ) u $ ( z ) . Observe that the formulas above hold for all A, B in the class A of differences of continuous AF's having bounded a-potential for some, and hence all, a > 0. For all A E A, u ( z ) := P"(CzA)' is a regular (2a)-potential, so u(x)is the (2a)-potential function for some continuous A F A'. Given A', A' E A Jet B := 1/2[(A' A')' (A1 - A')']. Then (36.29) u2a B - uzuz2 u ~- uz1uz2. ~ ~ ,
+
+
+
Conclude that every product uZ1uz2(Al, A2 E A) is of the form ui: for some A3 E A, and hence the space A of functions which are, for some a > 0, the a-potential of some A E A, is an algebra. Compare with (8.7).
(36.30) EXERCISE.Let K. be homogeneous on R+, with P"n(R+) < 00 for all x E E . Define the potential operator U, for K. as in (36.15), except that the integral is extended over [0, co[in lieu of 10, m[. Show that U, satisfies the complete maximum principle (10.27). This generalizes (36.22). (36.31) EXERCISE.Let L be a co-optional time. As noted in $35, the RM , d t ) := l { o < ~ < ~ ) c ~ is( dhomogeneous t) on R++. Define A := K O , an AF. Then, for f E bpE" and a 2 0 Px{e-aLf(X,); o < L < oo) = U A Q ~ ( Z ) . In case L is the last exit time from a nearly optional set B , A is called the optional capacitary AF for B . (Recall that in classical potential theory, the equilibrium putential (25.15) for B is defined to be the potential of the capacitary measure for B. In probabilistic potential theory, AF's are the correct analogues of measures on the state space.) K(.
(36.32) EXERCISE.Let A, B be predictable AF's of X such that P"At = P"Bt < 00 for all t 2 0, x E E. Then A and B are indistinguishable. (Hint: show that ( t , z )+ P"At is excessive for the backward space-time process (k, X ) of $16, and use (36.3) relative to (k,X ) . )
IV: Random Measures
181
37. Left Potential Functions
In the study of LAF's, the proper analogue of the potential function is the left potential function defined below. We discuss only the case a = 0, the extension to the case a > 0 being quite straightforward. (37.1) DEFINITION.Let A be a RLAF. Then potential function for A.
VA(I)
:= P"A, is the left
Observe that if n is the RM generated by A, so that n is homogeneous on R+, V A ( Z ) = P"n([O,GO]) and therefore
As t 11 0, n([t,oo]) increases to n(]O,oo)) 5 A,, so P ~ v A ( z5) VA(I). The excessive regularization of V A is, clearly, PZn(]O,GO]), which is just the ordinary potential function for the RAF Bt := At+ -Ao+. Because Ao+ > 0 in general, V A is not necessarily excessive. It does not seem possible even to show that V A is necessarily an optional function. Of course, V A E E", and for T E T, on {T < oo},
That is, if 2 is the left potential process (33.8) generated by n, then (37.2)
YA(XT) = ZT as. on {T < m},
T E T.
If V A were known to be nearly optional, the section theorem would imply that V A ( X )- 2 E 2, hence in particular that V A would be an optional function. The following definition is due to Mertens [Mer73].See also [Az73]. (37.3) DEFINITION.A strongly supermedian function is a positive, optional function v on E such that for all x and all bounded S 5 T E T,
From (37.2) and the properties (33.9) of left potential processes, the following result is obvious. (37.4) THEOREM. Let A be a RLAF with finite left potentiai function V A . Then: (i) V A is regular-that is, V A ( X )is regular in the sense of (33.1); (ii) V A ( X )has the class (0)property (36.6).
Markov Processes
182 If V A is nearly optional, then, in addition: (iii) V A is a strongly supermedian function.
(37.5) DEFINITION.A finite valued positive function v on E having the properties (i), (ii) and (iii) of (37.4) is an optional left potential function for X. s
We may now describe the structure of a RLAF. If A is a W A F and E T, 8.5. on (2' < m},
> 0 is fixed, then for T (At+, - At)
OT
= (At+s+T - A T ) - (At+T - A T ) = At+s+T - AT
for all t 1 0. That is, t + At+, -At is homogeneous on R+. Letting s 110 sequentially, we see that AAt := At+ - At is homogeneous on R+. (37.6) LEMMA.Let Y E p M be homogeneous on R+, and suppose that At := CoSs
Therefore, A is additive. The measurability of A is a consequence of (28.7). From (37.6) one sees that if A is a RLAF, then At := Coss
- A, I 3,)= Pp{A,
- At+ I F t } ,
and therefore Zt+ is the potential of the AF At+ - Ao+. Because of the computation following (37.1), it follows that Zt+ is indistinguishable from u ( X t ) ,where u is the excessive regularization of v. Thus (37.2) shows that for T E T,
(37.7)
AAT = ~ ( X T-) ~ ( X Ta.s. ) on (2' < 00).
In case v is nearly optional, (37.7) implies that AAt and v(Xt) - u ( X t ) are indistinguishable, hence that A may be written as the sum of a continuous AF and a LAF of the form
Cola
:'?I
Random Measures
183
(37.8) THEOREM.Let A , B be RLAF's having the same (finite) left potential function v. Then A0 = B".
PROOF:If v were optional, (37.8) would follow directly from (33.11). In any case, since A and A" have the same left potential function v , we may assume that A and B are optional. It follows then from (37.7) that AAt and ABt are indistinguishable, hence that the continuous parts At - ~ o l s < tAA, and Bt - CO<,
+
+
(37.11) EXERCISE.The limit of a decreasing sequence of strongly supermedian functions is also strongly supermedian. (37.12) EXERCISE.If v is an optional left potential function (37.51,then v differs from its excessive regularization only on a semipolar set.
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184
38. Generating Homogeneous Random Measures
The results of this section, originating with the work of Sur [SuSl] and Volkonskii [VoSO], and culminating with the definitive thesis of Meyer [Me62a] were historically the prelude to the Doob-Meyer decomposition of supermartingales of class (D) in the general theory of processes. The first result comes here though as a simple application of (34.1). In fact, (38.1) was the forerunner of (34.1) in the development of the general theory of processes. (38.1) THEOREM. Let u be an a-potential function of class (D)for X . There exists a unique predictable AF A such that u = u s .
PROOF:Let yt := e-%(Xt). Then (Y,) is a right continuous, positive supermartingale relative to every P", and the condition (36.11) implies that Y is of class (D), in view of (33.3). According to (34.5), there exists a unique integrable increasing predictable process B with Bo = 0, such that B has potential Y . Let At := ear dB,. We shall prove that A is an AF with u; = u. First,
ho,tl
lm
u ( z ) = P"Y0 = P"Bm = P5
e--crtdAt.
Given T E T, the potential 2 of &A satisfies, by (33.12),
and so, by (33.13),
That is, A is an AF. The corresponding left version holds also. We state and prove the result only in the case a = 0, the extension to a > 0 being routine. (38.2) THEOREM. Let v be an optional left potential function (37.5) on E . Then there exists a unique perfect LAF A having left potential function v.
PROOF:The hypotheses imply that v ( X ) is a regular left potential of class (D) for X , as defined in (33.1). According to (34.5), there exists a unique optional, integrable RM IC having left potential v ( X ) . Given T E T, (33.12) affirms that the left potential 2 of &PIC satisfies
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185
so (33.13) implies that @TK
= l[T,,[
* K.
. At := That is, K is homogeneous on R+, and P"K(R +) = ~ ( z ) Therefore K ( * , [0,t [ ) is a LAF, and A may be perfected then by (37.9). To complete this section, we discuss an extension of (38.1) to the case of a potential function u which is not finite everywhere. The following situation arises sufficiently often in practice to be worth noting. For simplicity of exposition, we consider only excessive functions, leaving the a-excessive case to the reader.
(38.3) DEFINITION. A potential function of class (0)with poles is an excessive function u on E such that:
(i) {z : u(z)= 0 0 ) is polar; (ii) T,, co in T, ~ ( z < ) 03
P z u ( X ~ , )+ 0.
We are going to formulate in (38.8) below an extension of (38.1) for potentials functions with poles. The key element in (38.8) will be the following result, which is of more general import, giving a way of lifting a HRM from a subprocess to the entire process. For its statement recall that < 00) = 0 for a nearly optional set E" c E is absorbing in case P"(TE\Ev all z E E". One may then define the restriction, X", of X to E". It is convenient to suppose that Xv is defined on (38.4)
Rv := {W E R : X,(W)E Ev u A for all t 2 0).
Clearly 0" E 3 is closed under shifts, P"(0") = 1 for z E E", P"(RV) = 0 for z 4 E", and the process X" defined on R V by X p ( w ) := X t ( w ) is a right process with state space E". The augmented o-algebra 3" = F ( X v ) may be identified with the intersection over all initial laws p carried by E" of the Pp-completion of p . The a-ideal N" of null sets in 0" for X" satisfies (38.5)
Nv cn/;
A E N , A c Rv --*.A E N v .
Let K" be an almost perfect HRM of Xv;that is, there exists a set R' C RV with P"(R') = 0 for all x E E" such that & K ( w , - ) = I I ~ , ~ [ K (.W ) for , all t >_ 0 and w E R V \ R'. Without loss of generality, we shall assume that R' is empty. Also we suppose that K" has been extended as a measure to [0, 001 by setting it equal to zero off 10, co[.Finally we suppose is a K" is a countably Radon kernel on 10,03[ : Le., K" = C, K.,", where each K," is a kernel from (O", 3")to (R++,B++) such that for K compact in R++, K:(W, K ) < 00 for each n and w E Rv. However, it is not assumed that the K," are homogeneous. Here is the extension result.
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186
(38.6) THEOREM. Let Ev, Rv, Xv,and lcV be as above. Let T := TEV be the hitting time of E”. Then there exists an almost perfect HRM, K , of X such that ~ ( w ,= ~ ~ ( . w ) for , every w E RV. Moreover, K is countably Radon, carried by IT, <[I, and K = tcV Pz-a.s. for all z E EV.If, in addition, tcV is optional (resp., predictable) over Xv,then K is optional (resp., predictable) over X. a )
PROOF:Fix (t,) C]O,m[ with t, 11 0. A.s., X t E EV U {A} on ]T,m[. Thus if 00:= {w : XT+t(u) f EV U {A} for all t > 0}, then P”(R0) = 1 for all 2 E E . Also, if t 2 0, then etRo c 00,and if w E Ro with T ( w ) < t , then Btw E R V (defined in (38.4)). For B E B(R++),define (38.7) nn(w,B) :=
{
eT+tnW,
(
B - T ( U )- t,),
w
E
a. n {T < 00},
otherwise.
This is well defined since BT+t,u E Rv if w E Ro and T ( w ) < 00. Note carried by ] T ( w ) t,, OO[ for every w. also that ~ ~ ( wis , We first claim that if n < m so tm < t,, then K, and Km agree on IT tn,m[. For, if w E 00n {T < OO} and B C ] t n T ( w ) ,m[, then B - T ( w ) - t m C It, - t,, 0 0 [ , and so a
+
)
+
+
G ( W ,
B)= K V ( 0 T + t n , B - T ( w )- t n ) = K~ (eT+t,+(t,-t,,,)~, B - ~ ( w -) tm - ( t n - tm)) = K~ ( e t n - t , ~ ~ + t ,B~ , ~ ( w) tm - (t, - t m ) )
=
(eT+t,W,
B - ~ ( w -) tm)
= K m ( 4 B), where the fourth equality uses the perfect homogeneity of I C on ~ Rv and the fact that eT+t,U E RV. Consequently we may define ~ ( w ,to be to 10, m[. In particular, ~ ( wB , )= the consistent extension of the K,(w, lirn,rc,(w,B), andK(w, . ) iscarriedby]T(w),oo[.Moreprecisely, K,(w, . ) is carried by a )
7
)
IW)+ t n , C ( b + t , W ) + t n + T ( w ) [= ImJ)+ ha,C ( 4 [ since COeT+t,w = (C-t,-T)+. Therefore ~ ( w , is carried by ]T(w), C(w)[. Next we show that K is homogeneous on 00.Fix t 2 0 and w E Ro n {T < 00). Then Otw E Ro and T(&w) < 00. Suppose B c]T(w), GO[ n]t, 0 0 [ . By considering the cases t < T ( w ) and t 2 T ( w ) separately, one sees that B - t c ] T ( e t w )m[. , (Note we are using t T(Btw) = t if t 2 T ( w ) and w E Ro.) Then for some n, B - t c]T(Otw) t,, OO[ and so a
)
+ +
K(etw,
B - t ) = KV(eT+t,etw, B - t - T(etw) - t,) = K V ( & + T ( O t w ) + t n W , B - (t + T ( 0 t w ) )- tn).
IV: Random Measures If t
187
< T ( w ) ,this becomes tc(&+t,W,
B - T f w ) - t n )= K ( W , B).
If t 2 T ( w ) , let u = T ( w ) . Then t + T ( & w ) = t and so the above becomes
n(etw, B - t ) = tcV ( e t + t n W , B - t - t n )
B - ( U + t n ) - (t - u ) ) = tcv(eu+t,w,B - u - t,) = K ( W , B), = tcv(et-ueu+t,w,
where the third equality uses the perfect homogeneity of rcV on Rv and the fact that 13u+tnw f Rv. If T ( w )= 00, then T ( & w ) = 00 and so both ~ ( wB) , and rC(&w, B - t ) vanish. Because K ( W , .) is carried by ] T ( w ) ,00[, we have shown that tc(w,B) = n(&w, B - t ) for all t 2 0, B c]t,oo[, and w E Ro (actually w E Ro U {T = 0 0 ) ) . Since P”(R0) = 1 for all 2, tc is almost perfectly homogeneous on R++. It remains to check the measurability of K . For later applications the predictable case is the most important. We shall give details only for tcv predictable over X v . The other cases are handled similarly. Write tcv(dt) = qvdA7, where Y” is a positive, everywhere finite process and AV is an increasing process with both Y v and AV predictable over F : := F t ( X v ) . Clearly it suffces to show each K , defined in (38.7) is predictable. To simplify the notation let R := T + t,. Then R E T, XR E EV U {A} as., and Xt+R(w) E EV U {A} for all t 2 0 if w E 0 0 . Then for w E Ro, K,(w,
a
)
=
6 ~ t c .~) =( O~R, Y ~ ( W* d(6RAv)(w), )
and so it suffices to show that if Zv is predictable over
(FT), then
Z t ( w ) := l ~ R ( w ) , m [ ( t ) z ~ R ( w ) ( e R W ) is predictable over (Ft). Note Z is defined on the set Ro of full measure ~ in R since ORRo c Rv. It suffices to consider Zv := l ~ r v , o vwhere T~ and uv are (F7)-optional times. In this case, 2 is the indicator of I]R T~ O e R , R + uv O e R I], and consequently, it is enough to show that S := R T v o e R E T for every (FT)-optional time rv. Now
+
+
{S < t} =
u
{R < t - q , T V O e R < q),
qEQ+
and so, finally, it suffices to show {TvoBR < q, R < 00) E 3 q + R . Given an initial law p, Y := ~ P is R carried by E V . Since { T < ~ q } E Fq(Xv), it
Markov Processes
188
follows readily that there exist HT, H Z E with HF 5 l { 7 ~ < q
(38.8) THEOREM. Let u be a potential function of class @ with I poles. ) There exists a unique predictable RM n, perfectly homogeneous on R++, with potential function u.
PROOF:Let B := {u = m} and E" := E \ B. Since B is polar and universally measurable, the process X" obtained from X by restriction to E" is also a right process, by (12.30). The restriction u" of u to E" is a potential function of class (D) for X" so there exists a unique predictable, almost perfect AF A" of X" such that P"AL = uv(z) for all z E E". Define the perfect, predictable HRM n as the extension of nv(u,dt) := dAT(u) described in (38.5). Once we prove that n has potential function u, uniqueness will follow from (36.3). For z E E",
and if x E B ,
P"n(R++)= lim P"n(]tn,cm[) n-cc
-
lim P ~ P ~ K(R++) ( ~ ~ )
n-+w
= lirn P"u(Xtn) n+cc
= u(x)
since u is excessive. This proves that completes the proof.
K,
has potential function u, and
v
Ray-Knight Methods
The Ray-Knight completion of a resolvent was carried out in $17, but up to this point, we have no serious applications. We shall see in this chapter that there are at least two distinct circumstances where Ray-Knight completions are essential. The first, concerns the description of an entrance boundary for a Markov process. Roughly speaking, given a bounded entrance law vt ($1)relative to (Pd),one wishes to represent qt as q& where 170 is a measure carried by a space which includes E . It is of course necessary that (Pt) be extended to the enlarged space. This occupies sections 39 and 40. Second, as the later part of this chapter shows, the projections and dual projections discussed in the last two chapters bear intimate connections to the process observed in its Ray topology, and permits some remarkable theorems characterizing, for example, predictable times and totally inaccessible times in terms of geometric features of the process at those times. See $44. The predictable projection kernel is discussed in $42 and $43, accessible processes and accessible times in $45. Throughout this chapter, it is assumed that X = ( Q , F Ft, , X t , Ot, P")is a right process on ( E ,E " ) with infinite lifetime. Applications to processes with finite lifetimes must be made considering the dead state A as an ordinary element of E . Unless otherwise mentioned explicitly, E denotes the Bore1 a-algebra generated by the Ray topology on E. The original topology plays almost no part in this chapter, except in 546, where we compare left limits in the Ray topology with left limits in the original topology, and in $47-48 where we discuss standardness hypotheses that depend on the original rather than the Ray topology. The chapter concludes with a brief description of rCduite of an excessive function on a set and the theorems of Hunt and Shih identifying rCduite in terms of the underlying process.
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Markov Processes
39. The Ray Space It was pointed out in 517 that the Ray-Knight completion ( E ,p, U") of the objects ( E ,d, U") relative to a generating family C' C C d ( E ) is not generally the same for all choices of d and C'. We shall prove in this section that by suitable restricting ( E ,p, U"), one may obtain objects depending only with no essential dependence on the topology of E and the resolvent (P), on the particular metric d or the generating family C'. The restriction will turn out to be large enough to describe all probabilistic features of X which could be described in terms of ( E ,p, a"). We begin with some complements to the Ray-Knight theory of 517. In the statements of (39.1)-(39.3) below, suppose given ( E ,d, U", C') as above, and let (B, p, be the associated Ray-Knight completion.
u")
(39.1) PROPOSITION. Let x , y E I?. Then x = y if and only if U"(z,. ) = > 0.
P ( y , - ) for some LY
PROOF:If f a f ( z )= u"f(y) for all f E b€", the resolvent equation shows that @f(x) = Upf(y) for all ,L3 > 0, f E b€". For the rest of the proof, f will denote the unique continuous extension of f E R and 2 := {f : f E 72). Let fl,. .. ,fn E R,PI,. . . ,/?, E Q++ and g := Cy UP3 fj E R SO that g := C: @3 By the assertion in the first sentence, 3 ( x ) = g(y) for hence for all g E A(U(R)).Recall (17.5) that A(U(72))is all g E U(R), separates E , it follows that x = y. uniformly dense in R. Since 'I?
6.
(39.2) PROPOSITION. Fix LY > 0. For x,, x E E , p(xn, x) + 0 if and only if the measures P ( x n , converge pweakly to P ( x , -). a
)
PROOF:If p(x,,z) 0 and f E Cp(E) then U"f E Cp(E)and consequently v"f(x,) + U " f ( z ) .On the other hand, if DQ(xn, converges consider any plimit point y of the sequence {x,}. pweakly to U " ( x , It is clear that for every f E C,(E), U"f(z)= a"f(y) and so, by (39.1), x = y. Since (I?, p) is compact this proves that p(z,, x) -+ 0. --f
a )
m),
We examine next the Ray topology on E -that is, the topology on E generated by p. Recall that the notation bC(E,p ) stands for the bounded (real) functions on E which are continuous relative to the Ray topology, and that C,(E) denotes the puniformly continuous functions on E . While the metric p and the space C J E ) are the more useful objects for the purpose of proving theorems, it is the topology alone and the associated space bC(E,p ) which are more natural descriptively, as later results in this section will bear out.
(39.3) PROPOSITION. For every a > 0, U" maps both bC(E,d)and
bC(E,p ) into bC(E,p ) .
V: Ray-Knight Methods
191
PROOF:According to (A2.1), given f E C ( E , d ) , we may choose g,, h, E C d ( E ) with gn T f and h, 1 f. Because U a ( C d ( E ) )C C,(E) it follows that U" f is both upper and lower psemicontinuous, so U"f E bC(E, p ) . The same type of argument works if instead f E bC(E,p ) . (39.4) THEOREM. The Ray topology on E is uniquely determined by the original topology on E and by the resolvent (U"), and not by the particular choice of d and C'.
PROOF:Let d l , d2 be totally bounded metrics compatible with the original topology on E and for j = 1,2, choose a subset Ci C pC,(E) whose linear combinations are uniformly dense in Cdj(E). Let 731, R2 be the corresponding rational Ray cones and p 1 , p 2 the associated Ray metrics. By symmetry, it will suffice to show that R1 C bpC(E, p 2 ) . Now, bpC,,(E) is a convex cone closed under A and, by (39.3), closed under the action of U". In addition, U.(bpC(E,dl)) = U"(bpC(E,dz)) is contained in U"(bpC(E, p 2 ) ) by (39.3) and so R1 C pC(E, p 2 ) by definition of R1. With these preparations in hand, we may now consider the Ray space of (E,d , U " ) and set for X. Fix some Ray-Knight completion ( E ,p,
u")
(39.5)
ER := {x E E : U'(z, . ) is carried by E}.
Give ER the subspace topology it inherits from E . It will turn out that ER together with the resolvent U" restricted to ER is uniquely determined, up to homeomorphism, by the original topology on E and the resolvent (U"). This makes ER a natural object. See (39.7) below. In what follows, E R denotes the Bore1 a-algebra on ER. That is, E R is the trace oft? on ER. (39.6) PROPOSITION.(if E C ER; (ii) ER is a Radon topological space; (iii) for every a > 0, ER = {x E E : U a ( x ,. ) is carried by E}.
u1
PROOF:Statement (i) is an obvious consequence of the fact (17.14) that extends U 1 . Next, as (17.11) and (A2.11) give us E E €", U 1 ~ E \ EE Eu, hence E R = {x E I? : U 1 l ~ \ E ( x = ) 0) E t?. This proves (ii), and (iii) follows easily from the resolvent equation. We turn now to uniqueness of the Ray space. (39.7) THEOREM. Let (El,p1, Or) and ( 8 2 , p 2 , u,O)be Ray-Knight completions of ( E , d l , U a ) and ( E , d z , U " ) respectively, dl and d2 being totally bounded metrics compatible with the original topology of E . Then the corresponding Ray spaces E k , E&are homeomorphic under a mapping 11, : E i + E i satisfying
Q?($J(~), B)= U
~ ( ZB ), ,
B E E , a > 0.
Markov Processes
192
PROOF:Given x E EA, choose (5,) c E with p l ( x n , x ) 0. Passing to a subsequence if necessary, we may assume pz ( 2 , ,y ) + 0 for some y E &. According to (39.2), the probability measures U1(x,, - ) (on E ) converge weakly relative to the Ray topology. By Le Cam's theorem (A2.13ii), there exist (Ray-)compact subsets K , of E such that U ' ( x n , K m )1 1 - l / m for all n 2 1. Since x, + y in E z , (39.2) says that U'(x,, . ) + Ui(y,. ) weakly as measures on E z , so that ---$
It follows that y f E i . The points x E EA and y f E i are connected by
U t h ( z ) = Ui h(y),
(39.8)
h E bC(E).
The relation (39.8) determines a mapping $(x) := y of EA into E i , for y is uniquely determined because of (39.1). Interchanging the roles of p1 and pz, one sees that 1c, is an injective map of EA onto E i . Using (39.2) again, the map $ is easily seen to be a homeomorphism. For the final assertion of the theorem, extend (39.8) to h E bE" and use the resolvent equation to see that for (Y > 0 and h E bl", U F h ( x ) = Vg h(y) if y = @(x). Denote now by (pt)the Ray semigroup corresponding to the Ray resolvent (Va)on E . See $9. It was shown in (17.16) that pt is an extension of Pt to E , in the sense of (19.8).
For x E ER and t 2 0, (39.9) PROPOSITION. PROOF:As kernels on
pt(x, . ) is carried by ER.
(E,c"),one has the obvious equality
PtU"(2, .) = ULZPt(x, - )
v x E E.
If x E ER, U a ( x ,.) is carried by E and so, Pt being an extension of Pt,
o1
For x E ER, l ~ ( x= ) U 1l ~ ( x = ) l ~ ( x )so , PtU1lE(x) = ptlE(x) = 1. On the other hand, (39.10) shows that
PtU'
l E ( 2 ) = U'P, l E ( Z ) =
V'
lE(Z) = 1.
Consequently PtV' l ~ \ ~ (=x 0) for every z E ER,and therefore Pt(x, -) does not charge {y : U1(y, I? \ E ) > 0) = E \ ER.
V: Ray-Knight Methods
193
(39.11) PROPOSITION. For z E ER and t > 0, Pt(z,. ) is carried by E.
PROOF:As Ua is the resolvent of the semigroup Pt, z E ER implies
so pt(z,E \ E ) = 0 for almost all t > 0. We may therefore choose t , with Pt, (z, E \ E ) = 0 for all n. If 0 5 t , 5 t , then
11 0
for if y E E , Pt-t,(y, . ) is the extension of Pt-t,(y,. ) to E . We shall restrict the kernels Ua and Pt to ER. These restrictions are kernels on ( E R ,E R ) . Letting B and D denote respectively the sets of branch points and non-branch points of (Pt)on E , set
ED := D
(39.12)
n ER;
E B := B n ER.
Since D E E, E D , EB E ER. As ED = {z f E R : &(z, . ) = e z ( . ) } and Pt is an extension of Pt, E c ED. We showed in (9.11) that Po(z, . ) is carried by D , so in view of (39.9) we find
For every z E ER, &(z, . ) is carried by ED. (39.13) PROPOSITION. (39.14) PROPOSITION. If z E B , then &(z, . ) is not concentrated a t any point of E . PROOF: If &(z, were point mass at y E E , then for every f E C(E), = f(y), and so for g E C ( E ) , U a g ( z ) = PoUag(z) = D a g ( y ) . Using (39.1) one concludes that z = y, so z 4 B. In general, a Ray semigroup (Pt) may have degenerate branch pointsthat is, points z at which Po(z, . ) is point mass at some y # 2. Thus (39.14) states that a Ray semigroup derived from completion of a right semigroup can have no degenerate branch points. a
)
&f(z)
Therestriction (Qt)of(Pt) toED is aright semigroup, (39.15) THEOREM. and ED \ E is quasi-polar (12.32) for any realization X of (Qt) as a right process. PROOF: According to (9.13), (Qt) is a right semigroup. Let X be a right process on E with semigroup (Pt). Then X is an extension of X (considered in the Ray topology) in the sense of (19.8). We now argue almost exactly as in the proof of (19.9) but omitting most of the details. Using the notation
194
Markov Processes
established in that proof, but with the sets J finite subsets of R++ rather than R+, it follows that if F denotes the set of paths Lz1 for which Xt(Lz1) E E for all t > 0, then for every probability p on ED,the Pp-outer measure of r is the infimum of C , P P ( A n ) as ( A n ) ranges over all sequences in U J with r C U,An. Using (39.11) and the fact that J C R++, the same calculation made in (19.9) shows that the outer measure in question is 1, whence the quasi-polarity of ED\ E relative to X. The most profitable way to think of (39.15) is to observe that the trace of X on {Xt E E for all t > 0) defines an extension of X on E to a right process X on EDsuch that ED\ E is polar for X. Of course, X and X are right processes relative to the Ray topologies on E and ED respectively. If one wishes to study X in the original topology of E (as we shall in (40.10)), the following result is of use. (39.16) THEOREM. Let (Pt) be a right semigroup on E and let ( X t ) be a Ray-right continuous process defined on (0,F,Ft,P) which is Markov with semigroup (Pt) and initial law p . Then P{t + Xt is right continuous in the original topology of E } = 1.
(a,
PROOF:Let X defined on 9,yt,P) be right continuous in the original topology of E and Markov with semigroup (Pt) and initial law p . Deleting a null set from fi, we may assume, by (18.1), that X is also Ray-right continuous. Let W with coordinate functions Yt be the space of Ray-right continuous maps of R+ into E and let @ : 0 + W and 6 : fi -+ W satisfy respectively yt 0 = Xt and yt 0 6 = Xt. It is clear by (19.6) that the image Q of P under @ is the same as the image of P under 6, for Q makes Y Markov with semigroup (Pt) and initial law p. Let Wo := {w E W : w is right continuous in the original topology of E } and r := { ( t ,w) : Ys(w) does not converge to Y,(w) in the original topology of E as s 11t } ,so that WOis the projection of onto W . For f E bC(E), foY is optional over Y, and it follows from (A5.3) that { ( t ,w) : f(Y,(w)) does not converge to f(yt(w)) as s 11 t } = {liminf,iit f(Y,(w>>< lim,lit f(Ya))}U {liminfalLt f(Y,(w)) # .f(x(w))} is progressive over Y. Letting f run through a sequence uniformly dense in Cd(E),it follows that r is progressive over Y . It follows from (A5.2) that WOis Q-measurable over Y . Since Q = P 0 6-l and P is carried by &-'(WO), Q must be carried by WO.On the other hand, Q = P0CP-l shows that P is carried by @-l(Wo)= {w E 0 : t Xt(w)is right continuous in the original topology of E } . --$
(39.17) EXERCISE. Show that the map 1/, of (39.7) takes E i to E i , E& to E;, and maps the semigroup correctly. (39.18) EXERCISE. Let X be uniform motion to the right on 10, m[. Show
~
V: Ray-Knight Methods
195
that, up to homeomorphism, E = [0, CQ], ER = [0,w[= ED and Pt(0, - ) = . ) for all t 2 0.
ct(
(39.19) EXERCISE.Let X be uniform motion to the right on ]0,1[ with killing as it approaches 1. Show that, u p to homeomorphism, E = [0,1] with A identified with 1, ER = [0,1] = ED and Pt(0, = c t A l ( a )
a ) .
(39.20) EXERCISE.Let E c R2 consist of the x-axis together with the dead point identified with ( 0 , l ) . Let X be uniform motion to the right at unit speed except that when X hits (O,O), it continues to the right with probability 1/2 or it jumps to A with probability 1/2. Show that ER is homeomorphic to the space pictured in (45.4), with O+ identified with (0,O) E ( ~ , - ~ ] (- ) so that 0- is a branch point. and &(O-, . ) = ql,l)(- )
4
+
(39.21) EXERCISE.Let K(y, d z ) be a probability kernel on (R,I?). Let X be uniform motion to the right along horizontal lines in E := R2,except that as X approaches the y-axis along ((2,y) : 2 < 0 } , it jumps to another location on the y-axis according to the law K(y, .) and then continues its uniform motion to the right. Show that ER is homeomorphic to the space constructed as follows. Separate the left half plane {(a,y) : x < 0) from the right half plane { (x,y) : x 2 0}, and adjoin a line to the left half plane, labelling the new points (0-, y), y E R. Then identify (0-, y) with (0, z ) if and only if K(y, = c z ( .). The branch points are the points (0-,y) which were not so identified, and for such y, &((O-,y), .) = €0 8 K(y, a
)
a ) .
40. The Entrance Space
The set ED = {x E ER : &(z, . ) = e Z ( . ) } defined in (39.12) will be called the entrance space for X . It was observed in 539 that E C E D . As justification for this terminology, observe that for every x E ER, the restriction qt( .) of pt(s, to E is a probability entrance law for Pt on E. That is, qt(E) = 1 for every t > 0 by (39.11), and vtPS = qt+s for every t > 0 and s 2 0. (Recall that we are assuming throughout this chapter that Pi1 = 1.) The integral representation theorem (40.2) will display the role of ED as a boundary for the integral representation of entrance laws. We have first a simple uniqueness of representation result which should be compared with (39.1). a
)
(40.1) PROPOSITION. Let A, p be finite measures on ED and suppose that for some CY > 0, X U a = pU". Then X = p. In addition, every finite measure Y on ED is the weak limit (in the Ray topology) of the measures up,, which are carried by E .
PROOF:As in the proof of (39.1), X U a = pU" for some a > 0 implies that the same equality holds for every a > 0. For every f E C(E),t + p t f ( x )
Markov Processes
196
is right continuous on R+ for every x E E , and for x E E D , Pof(z) = f(x). Because X and p are carried by E D , this implies that t + Apt(f) and t -+ p P t ( f ) are right continuous on R+ and their values at t = 0 are respectively X(f) and p(f). Our initial observation shows that these functions o f t have the same Laplace transform, hence A(f) = p ( f ) for every f E C(B). The last assertion is an obvious consequence of the argument above. (Note that the first assertion is weaker than (10.40).) (40.2) THEOREM. Let ( v ~ ) ~ > beoa probability entrance law for ( P t )on E . Then there exists a unique probability measure 770 on ( E D €&) , such that for every t > 0, vt = qoPt.
PROOF: Let ( E ,p, 0") be a Ray-Knight completion of ( E ,V"). Each measure r]t on E extends to a measure ijt on by setting ijt(E\E) := 0. Then ( a t ) is an entrance law for (Pt),for if t > 0 and s 2 0, ijt being carried by E allows one to write, for any B E €",
= 77t+s(B n E ) = fit+,(%
By weak compactness, we may select t , 11 0 such that ijt, converges weakly on ?!, to a probability measure v. For arbitrary a > 0, since U"C(E) c C(@, ijt,U* converges weakly to vU". But for f E C(E),
1 lm 00
ijt,U"(f) =
e-atijt,Ptfdt
0
= -
+
It follows that
Jdm
e-atijt,+t(f) dt
LmW
e-""ijs(f) ds
e-"'ij,(f) d s as n + 00.
e-"'vP,(f) d s =
Right continuity of s conclude that vP, =
+
Irn
e-"'fj,(f) ds.
v P s ( f ) and s + jj8(f) on R+ allows one to for every s > 0. Now set 770 := vP0. Then
V: Ray-Knight Methods
197
q0Pt = Y P o P ~ = Y P = ~ qt, and since PO is carried by D (39.11), 70 is carried by D. In fact, qo is carried by ED,for if a > 0,
qODa(E \ E ) =
s,"
e-"tfjoPt(E \ E ) dt =
e-"tfjt(E
\ E ) d t = 0,
so fjo does not charge {x : U a ( x , E \ E) > 0) = E \ ER. Uniqueness comes from (40.1). There is an interesting probabilistic version of this representation. Suppose given a right semigroup (Pt) in a Radon space E , and suppose that the topology on E is sufficiently rich that (40.3)
every a-excessive function for ( P t ) is nearly Bore1 on E .
See (18.4), (18.5) and (19.2). Suppose given also an E-valued process (Xt)t>O(defined only for t > 0) on a filtered probability space (W,6 ,Q t , P) satisfying the usual hypotheses, and such that, in the language of $1, ( X t ) t > has ~ the E-Markov property with semigroup (Pt)and entrance law ( q t ) . Suppose also that t -+ X t is a s . right continuous on R++.Let 70 be constructed on ED as in (40.2) so that qt = qoPt. (40.4) THEOREM. With the above assumptions in force, let WO:= {XO+ exists in the Ray topology of E D } . Then: (i) P(W0) = 1; (ii) setting Xo := Xo+ on Wo, the process t -+ Xt is a s . right continuous in the Ray topology of ED,and has the E-Markov property with semigroup ( P t ) (restricted to ED) and initial law 710 relative to (W,6 ,6t, P); (iii) i f f is a-excessive for (Pt)on E D , then t -, f(xt)is a.s. right continuous on R+.
PROOF:According to (19.3), the condition (40.3) permits us to construct a canonical realization Y = (a',7 ,F t ,yt, B t , P") on the space R' of right continuous maps of R+ into E . Let R c Cl' denote the space of paths in R' which are right continuous in the Ray topology of E . Fix s > 0 and let as: W -, R' be determined by
Because t + Xt+s has the E-Markov property relative to (W,$7, 6t+3,P) with semigroup (Pi) and initial law q3, the image of P under as is the canonical Markov measure P q s for Y , and according to (18.l), P O a is carried by R. It follows that P{t + Xt is Ray-right continuous on R+} = 1. A
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similar argument shows that if f is a-excessive for (Pt) then P{t + Xt is right continuous on R+} = 1. More particularly, if f is bounded and aexcessive, t + e - " t f ( X t ) is a right continuous supermartingale with index set R++relative to (W,0, G t , P). The reverse supermartingale convergence theorem shows then that P{f(Xt) converges as t 11 0) = 1. Letting f vary through the rational Ray cone, it follows that P{Xo+ exists in E } = 1, and since the distribution of XO+is the weak limit of qt in the Ray topology, (40.1) shows that XO+ has distribution ~ 0 .In particular, P{Xo+ E E D } = 1. Delete from W those w for which t -P Xt(w) is not a Ray-right continuous map of R+ into E D . Let i denote the space of Ray-right continuous and let Pxdenote the meamaps of R+ into ED with coordinate maps sures making a right process with semigroup pt, the construction being justified by (18.5) and (19.3). Let @ : W + ?i satisfy
Yt,
y,(@(w)) = Xt(w),
(40.6)
If 0 5 tl <
*
< t,
and
fl,
w E W.
.. . ,f,, E C,(ED) then
Pf1(Xtl) * . .fn(Xt,) = !~~Pfl(Xt,+s)...f,(xt,+s) = lim P q a ello
. . f,(yt,)
f1(Yt1).
because of the observation following (40.5). However,
Pqsf l ( Y t 1 )
*
- -fn(Yt,) = Pqof l ( Y t l + a ) - *
fn(Yt,+s)
converges to P q O f1(cl). . . f,(En)as s 11 0, and we conclude that PqO is the image of P under a. Now, for f E b&k and t > 0, (7.4) shows that s -P is as. right continuous on [0,t [ . Therefore
Pt-,f(x)
P{w : lim &sf(X,(w)) # Ptf(Xo(w))I 410
(
= P @-l{w : lim Pt-,f(Y,(w)) # ello
Ptf(Yo(w))}
We now have, because GO = GO+ by hypothesis,
P{f(Xt) I G o ) = l $ ) P { f ( X t )I 081
= Ptf(X0).
This proves the Markov property of (Xt)t>o. Finally, if f is a-excessive for pt, s + f ( Y e ) is a s . right continuous, aGd a comparison similar to that is 8.5. right continuous on [0,t [ . above implies that s -, f(X,)
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(40.7) EXERCISE. Show that if (Xt)t>o satisfies the hypotheses of (40.4), then X obeys the following form of the strong Markov property: if T is an optional time over ( G t ) , then for t > 0 and f E bE'"
P{f(Xt) I G T } = P t - ~ f ( x T )a s . on {0 < T < t}. (40.8) DEFINITION.An entrance law ( q t ) for ( P t ) is minimal if every other entrance law (771) dominated by (qt) (i.e., q; _< qt ils measures on E for every t > 0) is proportional to Pt(x,. ). (40.9) PROPOSITION.Let (Pt) be a Markovian right semigroup. Then an entrance law ( q t ) for (Pt) is minimal if and only if qt is proportional to Pt(x,. ) for some x E ED. PROOF:As (Pt) is Markovian by assumption, every entrance law ( q t )has constant mass and so, by (40.2), qt = q0pt for some measure qo on ED.If q o ( E ~= ) 1 and if q0Pt 5 E , P ~ for every t > 0, then using (40.1), it follows Taking such an f vanishing that vo(f) = cZ(f) for every f E $,(ED). only at x,one sees that v o is concentrated at x and so qo = E ~ This . proves .) is a minimal entrance law. Conversely, if qt = qopt and if that Pt(x, qo is not concentrated at a point, then there exists a compact set K with 0 < qo(K) < ~ o ( E D ) Set . 76 := 1 ~ q and o q: := q&pt.Then (q:) is an entrance law dominated by ( q t ) . If (771) were proportional to ( q t ) we would have to have qo(E)q: = qo(K)qt. Choose f E p c ( E ~ , p )vanishing only on K . Then from (40.1) we get qo(E)qL(f) = qo(K)qo(f),a contradiction since qk(f) = 0 but qo(f) > 0. Our next result shows that, under mild conditions, one may construct a measure on path space making ( X t ) t > o Markov with a given semigroup and given bounded entrance law. This strengthens the force of the hypothesis HD1. To be specific, we shall suppose that (Pi) is a right semigroup on E satisfying (40.3), and that (qt) is a probability entrance law for (Pt). Let 0 denote the space of right continuous maps of R+ into E (using the original, not the Ray topology on E ) and set Xt(w):= w ( t ) for t > 0, 3;:= a{f(X,) : 0 < s 5 t , f E E"}. (40.10) THEOREM. In the situation described above there exists a unique such that (Xt)t>o is Markov with semiprobability measure P on (Q,3'") group ( P t ) and entrance law ( v t ) . The entrance law ( v t ) is minimal if and only if q+ := f l t > o 3 ? is trivial under P.
PROOF:Let W denote the space of Ray-right continuous maps of R+ into ED and let Y,(w) := w ( t ) , 6; := a{f(Y,) : 0 5 s 5 t , f E EL}. Thanks to (9.13) and (39.15), we may construct a measure P q o on (W,G") making := { w E Y Markov with semigroup (pt) and initial law qo. Now set
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W : Y,(w) E E for all t > 0). Because (39.15) shows that ED \ E is quasi-polar relative to Y , the Pq0-outer measure of is 1. Therefore, the trace, P q O , of P Q O on W is a probability measure making Y Markov in E with semigroup (Pt) and entrance law (qt). Now applying (39.16) to t ---* Yt+T with T > 0 arbitrary, we see that for Pqo-a.a. w E WI is right continuous relative to the original topology of E . Now we may simply transfer P q o to P on R by the natural injection map @ of WO:= {w E W : w ~ ~is right ~ , continuous ~ [ } into R. We think of WOas a subset on of R. According to (40.4), P ( R \ WO)= 0. However, the trace of WOis just a{f(K) : 0 < s 5 t,f E €&}, and because of (40.9) and the Blumenthal 0-1 law, the latter a-algebra is P q o trivial if and only if (qt) is minimal. Throughout the following exercises, we drop the condition Ptl = 1.
w
m,
(40.11) EXERCISE. An entrance law (qt)is called locally integrable provided J; r]t(l) dt < m. Given a locally integrable entrance law (qt) for (Pt) show that the finite measures c"( = 7j"(. ) := e-"tqt( dt satisfy a )
(40.12)
ca - cp = ( p - a)caUp,
a )
a , @> 0,
ca(l) + 0 as Q +. m.
(40.13)
Show that (7j" : a > 0) uniquely determines {qt : t sup,qt(l) < m if and onlyifsup,afj"(l) < 00.
> 0}, and that
(40.14) EXERCISE. Let {c" : a > 0) be a family of finite measures satisfying (40.121, (40.13) and sup,aca(l) < 00. Show that there exists a bounded entrance law (qt) with ca = 7j". (Hint: assume first that (Pt) is Markovian, and define qo to be any weak limit on E of ac" as Q + m. Show that qo is carried by ER and let qt := qoPt. To get the sub-Markovian case, extend the measures C" on E to measures E" on EA by setting E" := C" (a-'limg,, ,f?cg(l)- c a ( l ) )EA.)
+
(40.15) REMARK. The condition supac"(1) < m in (40.14) may be removed, the entrance law ( q t ) then being only assumed locally integrable in the sense of (40.11). See [GSh73a]. (40.16) EXERCISE. An s-finite measure p on E is called excessive for (Pt) if pPt 5 p for every t 2 0, invariant if pPt = p for every t > 0. (i) A a-finite measure p is excessive if and only if a p U " 5 p, or equivalently a p U" increases to p as (Y + 00, or, again equivalently, if p Pt increases to p as t 1 0. (ii) If p is a a-finite excessive measure, then pLco:= limt+m p Pt exists and defines a a-finite measure which is invariant
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under (Pi). Using the Radon-Nikodym theorem, show that there exists a unique a-finite measure v such that p = p m v, and show that v is purely excessive in the sense that v is excessive and v, = 0. (iii) Let p be a finite excessive measure for (Pi). Accepting the result mentioned in (40.15), prove that p - a p P = fja for some locally integrable entrance law. Conclude that if p is finite and purely excessive, then p = qt d t .
+
fr
41. Meager Sets and Predictable Functions
Due to the fact that E and ER are in general only universally measurable in E , there are technical difficulties in discussing predictability of processes of the form f ( X - ) , even if f is Bore1 on E or ER. (Recall from (20.4ii) that Xt-(w), the Ray-left limit process taken in El exists for all t > 0.) The results of this paragraph are the technical preliminaries to describing the class of predictable functions on E -that is, those functions f on E such that f ( X - ) E P. The best result will be proved in (43.2), to the effect that if f E E" and if f l is~ an optional function, then f is a predictable function. The methods described here are of importance in later sections. (41.1) DEFINITION.A set
r
E M is meager if there exists a sequence [ R, I] , u p to evanescence.
{h} of F-measurable times such that I' C U,
Obvious modifications of the sequence {R,} would allow one to assume that the {R,} had disjoint graphs and that r = U, [IR,I]. In the general theory of processes, a set I' obeying (41.1) is usually called thin, but because that term has a different (but related) meaning in Markov process theory, we change the name here. It is shown in [DM75, IV-1171 that if all w-sections of a measurable set are countable, then that set may be exhausted, up to evanescence, by a countable union of graphs of measurable times. However, that construction depends on the particular probability measure, and we cannot conclude from this result that such a set is meager in the sense of (41.1), the independence of initial law being the critical issue. (41.2) DEFINITION.A set r c R+ x R is optionally meager (resp., predictably meager) if there exists a sequence {T,} in T (resp., predictable times) with = U, IT, 1 u p to evanescence.
Once again, the independence of initial law in (41.2) is an important implicit feature. It may be assumed in (41.2) that the T, have disjoint graphs. Here is a simple criterion for a set to be meager. (41.3) PROPOSITION. Let rc be a RM in M (resp., &,$). Then := { ( t ,w) : ~ ( w{,t } ) > 0) is meager (resp., optionally meager, predictably meager).
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Markov Processes
PROOF:If K. E M ,there exists a strictly positive 2 E M such that 2 * K. is integrable. We may therefore assume, without loss of generality, that K. itself is integrable. Let At := ~ . ( [ O , t l ) so that A is right continuous, increasing and finite valued. (If K. E 0, Z may be chosen in 0 and A is optional. Similarly, if K. f P we may assume A predictable.) It follows P) and for a.a. w,its that for every /3 > 0, {AA > p} E M (resp., 0, w-section {t 2 0 : AAt(w) > p } is a finite subset of R+.Let TI := inf{t 1 0 : AAt > ,8}, Tz := inf{t > TI : AAt > p}, and so on. It is easily seen that {AA > p } = U [ T n l and that every T, is F-measurable (resp., an optional time, predictable time). Letting 0 decrease to zero through some sequence, the proof is complete. (41.4) EXERCISE.Let A be a RAF (resp., AF, predictable AF), and let > 0. Recalling (35.121, show that {AA > p} = U,>l[TmIj where T m are the iterates of the random time (resp., terminal time, predictable terminal time) T := inf{t > 0 : AAt > p}. Show that if the A F A has bounded a-potential, then T is totally thin in the sense of (12.8). ,f?
(41.5) LEMMA.Let R : R + [O,m] be 7-measurable. Then {q[Rl is optionally meager and {q[Rl > 0) is predictably meager.
> 0)
PROOF:Let At := l ~ j ~ , ~ n (Then t ) . A' determines an integrable, optional increasing process. According to (31.18), AA" = qAA) = ql[Rn), so (41.3) implies that {o(luRl) > 0) is optionally meager. The predictable case is similar. (41.6) THEOREM. Let r E 0 (resp., P) be meager. Then (resp., predictably) meager.
I' is optionally
PROOF:Let {&} be F-measurable times with c U [I&!. If I? E 0, we obtain 1r = q r 5 x n q [ R , n and (41.5) shows that { z n q [ R n ] > 0) = U, [IT,I] for some sequence {T,} c T. Consequently, r c U ITn I]. For each n, let I?, := {w : Tn(w) E I?(w)} and let S, := (T,)rn.(Recall the notation of (5.6).) Because I?, E FT,,S, E T and I? = U[S,I]. The predictable case is entirely analogous. There is another simple setting in which meagerness appears. First of all, if g E bpB(R+), it is easy to see that if we set, for t > 0, -g(t) := limsupg(u); -g(t) := liminfg(u) uttt uttt -g"(t) := sup{e -a(t-8) g(s) : 0 5 s < t } , then t -+ "g"(t) is left continuous, and as a 3 XI, " g " ( t ) 1 "g(t). If (yt) is a bounded, positive and progressive relative to (7;)for every initial
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203
law p, define " Y , -Y and "Y" in the obvious way, and observe that measurability of debuts (A5.1) implies that if D := inf{s : e-"(t-S)Y8> p}, then D E T, so {'yt" > P } = {D < t} E 3t.Consequently ' Y a E P , and thus "Y E P. A similar argument shows that -Y E P. The condition that Y be bounded and positive can of course be lifted by composition with a bounded monotone function. Having disposed of this measurability difficulty, we show (41.7) THEOREM. If Y is right continuous and adapted to (3t), then {-Y # Y } and {"Y # Y } are optionally meager, and {-Y < ' Y } is predictably meager.
PROOF:We may assume Y bounded and positive. Given 6 > 0, the set {t : 'yt 2 "yt +6} can contain no strictly decreasing sequence, for if tn 11 t were such a sequence, the oscillation of s -+ Y, in any right neighborhood o f t would exceed 6, violating right continuity of Y at t. For T a positive rational, set TT,6:= inf{t 2 T : '& 2 "& + 6) so that TT,6E T and [ITT,61]C {"Y 2 -Y + 6 ). w e may then write {"Y 2 -Y + 6) as the countable union of graphs of the TT,6,as T ranges over &+ and 6 runs through the sequence l/n. That is, {'Y > - Y } is meager, and since {'Y > " Y } E P , (41.6) shows that {"Y > - Y } is predictably meager. We show next that {'Y 2 Y } is optionally meager. Since "Y" 1" Y as a 00, given 6 2 0 we may write
If we show that {"Yk < Y - 6 ) can contain no strictly decreasing sequence, the same construction that was used above will show that {"Y < Y } is optionally meager. To this effect, suppose tn 11 t and 'ytk, < yt, - 6 for every n. Setting 2, := e k t y t , we have yt, > 6 + Zt, > > n6 + Zt,,,, violating boundedness. To complete the proof, observe that {-Y > Y} is also optionally meager and that {-Y # Y } U {"Y # Y } = {"Y < Y } U {'Y < Y } U {-Y > Y } is therefore meager. As both {"Y # Y } and {"Y # Y } are meager and belong to 0, (41.6) applies. (41.8) PROPOSITION. The set { X -
# X } is optionally meager.
PROOF:Though (41.7) could be applied here, a more direct argument seems better. Let p be a Ray metric on E and let ,f3 > 0. Since t + X t ( w ) is rcll in E , {t : p(xt-,X t ) > 0) can contain no finite accumulation points, and so the same argument as given in the proof of (41.3) is valid here. We turn now to predictable functions with an analogue of (32.6).
Markov Processes
204 (41.9) PROPOSITION. I f f E &", then f(X-) E Pt.
PROOF:Given p and an integrable RM K , p, g + P" J g ( X t - ) ~ ( d t de) fines a finite measure on (I?, €). Given f E &u, there exist g1 5 f 5 g2 with g1,g2 E & and P"J(g2 - gz)(Xt-) ~ ( d t = ) 0. Since g l ( X - ) and g2(X-) are obviously in P , the result follows. (41.10) THEOREM. Let 2 E P t , and suppose that { Z # 0) is contained in an optionally meager set. Then 2 E P.
PROOF:Let A be optionally meager with ( 2 # 0) C A, and let {&) c T have disjoint graphs with A = U Rn 1. As 2 E Ot, Z R , , ~ { R , , < E~ }FR,, for every n by (32.5), and consequently 2 = 2 1 =~ C , Z R , ~ ~ R 1 ,is, in 0 and hence in M . In view of (23.1) it will suffice to show that 2 E nPp. Take the RM IE := En2-n E R , , ~ { R , , < ~ and ) choose V, W E P with V 5 2 I W and P p s(Wt - Vt) ~ ( d t= ) 0. Then V1A 5 2 1 6~ W ~ Aand , the outer members of these inequalities are Pp-indistinguishable. Therefore, up to Pp-evanescence,
But if V, > 0, 2, > 0 so t E A. Consequently Z+ - V+ E 2". Similarly, 2- - W - E P,and it follows that Z - [V+ - W - ] E P,hence 2 E Pp. (41.11) THEOREM. Let f E E" or €; Then f is a predictable function.
and suppose that f := J I E E E e .
PROOF:If f is defined only on ER,extend it to E by J := 0 on E \ ER. Let A := {X- # X} so that A is optionally meager. Because 1 ~E &" \ ~ and { l E \ E ( x - ) # 0) c A, (41.9) and (41.10) imply that lE\E(X-) E P . Consequently ~ E ( X - )= 1 - l~\~(x-) E P. Suppose we show that for every a > 0 and every g E bE", U a g ( X - ) (= lE(X-)Uag(X-)) is in P. It will follow then from (8.7) and the MCT that h(X-) E P for every h E Ee and so, i f f E E" and f = J I E E E", f ( X - ) = f(X-) (fl~\~)(X-) belongs t o P , using (41.10) again on the second summand. To complete E P. Set Y := Uag(X)- E P. Then the argument, we show that Uag(X-) (41.7) shows that {Uag(X) # Y ) is optionally meager. Since
+
is therefore also optionally meager, another application of (41.10), this time E P. to Uag(X-) - Y ,proves that Uag(X-)
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42. Left Limits and Predictable Projections
The result of this section are fundamental to all calculations involving predictable projections over a right process. We emphasize that Xt- denotes the left limit taken in the Ray topology, and has values in a Ray compactification E of E . It will be shown shortly that Xt- is in fact in the Ray space ER and so, in view of (39.7), there is no ambiguity in the meaning of Xt- when different Ray compactifications are used. (42.1) THEOREM. Let f E bE" and let f E €" with f l ~= f . Then Pof(X-) is the predictable projection of f ( X ) .
PROOF:Since ( p o f ) = ~~ f E bE", &f(X-) E P by (41.11). We must show that for every initial law p and every predictable time T , (42.2)
P p { f ( X ~ )T; < 00) = PP{Pof(X~-) ; T < CO}.
As each side of (42.2) defines a finite measure on ( E , E u ) , it is enough to prove that (42.2) holds for f E C(E). Suppose we prove that for f E C ( E ) and a > 0, (42.3)
P p { U a f ( X ~;)T < C O } = P p { U a f ( X ~ - )T; < KI}.
Then, multiplying by a and letting a 00, the fact that t ptf(z) is e-atPtf(z) d t converges right continuous (9.8) shows that a u " f ( z ) = boundedly to Pof(z),hence (42.2) holds. (Observe that ( p o f )=~~~ I soE that P o ~ ( X T=) ~ ( X T ) .We ) shall prove (42.3) in the equivalent form -+
-+
Fix a sequence (T,) of finite optional times announcing T relative to Pp. Set H := e-*'ff(Xt) d t so that
Conditioning relative to FGn gives FT"
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But V 3 F n = Fg-,and so (A5.29) yields, letting n + 00,
In the last summand, we may replace U a f ( X ~ ) by - U a f ( X ~ - on ) {T < m}, since Uaf E C(E).Because s + S,"ematf(Xt)dtit continuous and adapted to ( F f ) ,s,' e - a t f ( X t ) dt E F&-,and we obtain therefore
which proves (42.4), after first conditioning its left side relative to FF. (42.5) COROLLARIES. (i) If T is predictable relative to Pp and if FF = FF-,then Pp-a.s. on {T < OO), XT- = XT; (ii) as., X t - E {x E ER : Po(x,E) = 1) C E U EB C ER for all t > 0.
PROOF:(i): The identity (42.3) shows that for every f E C(B),
Therefore P'{~(XT) # ~ ( X T - )T; < m) = 0,
g E R,
and since R is countable and separates E , the result follows. (ii): Let F := {x E E : Po(x,E) = 1) c ER. Obviously F E €u. The function f := E bEu vanishes on E so (42.1) implies that Pof(X-) is evanescent. However, Pof = f since FOP,= Po, and f being strictly is -evanescent. ) positive on E \ F implies that l ~ , ~ ( x The corollary (42.5ii) permits us to write (42.1) in the more convenient form (42.6)
'(f
0
X ) =Pof(X-),
f E bE".
The point here is that, knowing that X - stays in the set where POis carried by E , it is not necessary to extend f to E . Moreover, as we pointed out at the beginning of this section, the form (42.6) does not depend on the particular Ray-Knight completion used.
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Let A be a predictable AF with finite a-potential (42.7) PROPOSITION. function u:. Then AAt is indistinguishable from u:(Xt)- - &$(Xt-).
PROOF:The RM e-at dAt has potential process yt := e F a t u A ( X t ) .Now simply use (33.14) and (42.6). Show using (42.2) that for every predictable time T, (42.8) EXERCISE.
(42.9) EXERCISE. Let f E p€%. Using the methods of (32.7) and keeping in mind (41.9), show that for every IE E P
43. The Predictable Projection Kernel
We describe now a kernel fi which realizes a version of the predictable projection, at least for processes in B+ @ F’. Recall the construction of the optional projection kernel ll in 522. For simplicity we assume that R has a splicing map (w,t,w’) -+ w/t/w’ (22.2), though just as in the case of the optional projection operator, this serves merely to give a compact expression for I?. See (22.10). We assume also that, after deleting a shift invariant null set from R (42.5), all w E !2 have the property (43.1)
Xt-(w) exists in {z E ER : P o ( z , E )= 1)
(C
EUEB).
Setting X o - ( w ) := Xo(w), define I?M for M E b(B+ @F”) by (43.2)
ir M (t ,w ) :=
1
Mt (w / t / w’) PPO (X,-
I
. ) (clw ) .
In (43.2) the integration is relative to P p ( d d ) where p is the measure &(Xt-(w), .), for a fixed w. Note that (43.1) guarantees that p is carried by E for every ( t , w ) . Just as in the case of the optional projection kernel ll, I? is linear, positive and respects bounded monotone convergence. The exact specification of its action as a kernel is described below. Recall the meaning of the a-algebras Po and Pe defined in $23.
Markov Processes
208 (43.3) PROPOSITION. I f f E bE (resp., bE“) then belongs to Po (resp., P e V 2).
fI(f
0
X)=
Pof X 0
PROOF:Setting Mt := f(Xt) in (43.2) and letting p ( . ) denote the measure & ( X t - ( w ) , - ) , we obtain
If f E bE, there exists f E bER such that f = f l ~ , and because of (43.1), P o f ( X - ) is identical to P o f ( X - ) . But since Pof E ER and X - is left continuous with values in ER, P o f ( X - ) E Po. For the case f E bEe, it is enough to suppose f := Uag with Q > 0 and g E b p P . Now, (42.7) applied to At := s,’g(X,) ds shows that U a g ( X t ) - and &Uag(Xt-) are and has left limits indistinguishable, and since U a g ( X t )is adapted to (3;) almost surely, it follows that poUag(X-) E Pe V I.
In connection with showing in (22.21) that JIM was well defined for M E b M n , we needed to observe that for fixed ( t ,w ) , w’ + Mt(w/t/w’) is 3-measurable. This same observation shows also that fIM is well defined for M E b M n . The next result is a little stronger than saying that fi leaves Pe and the u-algebra ff of constant adapted processes invariant. (43.4) PROPOSITION. If M E b(Pe V f f ) and if N E b M n , then
I? ( M N ) = MI? N
identically on R++ x
a.
PROOF:By monotone classes, one may suppose that Mt = f ( X 0 ) Z t where f E bEUand Z is left continous and adapted to (3;). Then for every t > 0, Zt(w/t/w’) = lim,Ttt Z,(w/t/w’), and for every s < t , there is a countable set C c [O,s] such that 2, E 3; is a function only of { X , : T E C}. Consequently, if t > 0, Zt(w/t/w’) does not depend on w’. Obviously f ( X o ( w / t / w ’ ) ) = f ( X o ( w ) )does not depend on w’ if t > 0. Thus (43.4) follows by inspection of (43.2). For every M E b M n , f i I M = f I ( E M ) identically, (43.5) PROPOSITION. and for every t > 0, f I ( O t M )= O t ( f I M )identically on I]t,oo[.
V: Ray-Knight Methods
PROOF:Fix t
> 0 and M
209 as above. Then I!I (II M ) ( t ,w ) expands as
where, to obtain the last equality, we used the fact that
J
p y d w ’ ) p X O ( w ‘ ) ( .) = p q . )
for every initial law /I on E. A routine calculation using (43.2) again establishes the second identity. (43.6) THEOREM. For every M E b M n , f i M is a predictable projection of M . If M E b(B+ 8.30)(resp., t3+ @ F e )then I!I M E Po (resp., Pe V 2).
PROOF:If M E b M n , IIIM E bC3 and (23.4) shows that there exist M’, M 2 E b(R V ?LO) with M’ 5 I I M 5 M 2 and M 2 - M1 E bZ. Applying fI and using (43.5), one obtains fiM1 5 f i M < I!IM2. A monotone class argument based on (43.3) and (43.5) shows that for j = 1,2, fi M j is a predictable projection of M j . In particular, M 2 - fi M1 E bZ, so fi M is a predictable projection of M . If M E b(B+ @ F”), (23.6) shows that I I M E b(PO V Xo). One obtains then f i M E P using the MCT together with (43.3). The case M E B+ 18 Fe is treated in the same manner. (43.7) EXERCISE. Let f E b P and s > 0. Setting Mt := Ps-tf(Xt)l[o,s[(t)) + f(XS)l[S,..[(t), show that (43.2) gives (lno,mu
fi ~t
= P . - t f ( ~ t - ) l p , ~ ] (+ t )f ( - ~ ) 1 l ~ , ~ [ ( t ) .
(43.8) EXERCISE. Show that for all M E b M n ,
{Ti M # rI M } c { X - # X}. (43.9) EXERCISE. T h e predictable analogue of (23.4iii) is P = R V Po V 2. Verify this using (23.4iii) and (43.3). (43.10) EXERCISE. Check the validity of the results mentioned in (31.131, (34.8) and (34.9) which depend on the existence of the predictable projection kernel Ti.
Markov Processes
210 44. Topological Characterizations of Projections
We are now in a position to describe precisely the relationships between projections and continuity of sample paths. Deleting from 52 a null set if necessary, we assume (43.1) is satisfied. In addition, for compactness of calculations, we assume that 52 admits a splicing map (22.2). First of all, we decompose R++ x 52 as J U J B U K where
J := { ( t , W ) : t > O,X,-(W) # X ~ ( W ) , X ~ - E( UE )} ; JB := { ( t , w ) : t > O , X , - ( W )4 E } ; K := { ( t , ~: t) > O,Xt-(w) = X~(U)}.
(44.1)
The set J B is in
P because of
(41.11), and we observed in (41.8) that E 0 and K = R++ x R \ {X-# X} E 0. It is also immediate that the sets J , J B and K are perfectly homogeneous on R++. The following facts about J , JB and K are a little more refined.
{X-# X} E 0 so J = {X- # X} n J$
(44.2) PROPOSITION.
(i) (ii) (iii) (iv)
The sets J , J B and K belong to 49; J B is predictable over J and K are optional over ( E ) ; J U J B is meager in the sense of (41.2).
(e);
PROOF:For any (Ray-)Bore1function g on E , tj(X-) is predictable over and in 49, by monotone classes. Letting (gn) be an enumeration of a rational Ray cone for X, (43.1) and (9.11) show that J B = n,{Pogn(X-) < gn(X-)} is predictable over and in fig. Another monotone class is optional over argument shows that for every f E b(€ @ E), f(X-,X) and in 49. Assertion (iv) is a direct consequence of (41.8), (41.9) and (41.10). The remaining assertions are now obvious.
(c)
(e)
(e)
(44.3) THEOREM.
(i) If T E T is PP-predictable, then [TI p(1J) = 0. (ii) For every Z E b M , ("2 # PZ} c J U J B . Let K be a RM that is a-integrable on P. Then:
nJ
E
P,and hence
(iii) if K is carried by J , KP is diffuse; (iv) if K is carried by K , KO = KP.
PROOF:In order to prove (i), we show first that PIJ = 0. As (44.2) implies that J E B+ 8 30 C M n , it is enough to show that fi 15 = 0, where fi is
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211
the predictable projection kernel of 943. Setting M t ( w ) := l j ( t , w ) , observe that Mt(w/t/w’) = l { X t - ( w ) E E } l{Xo(w’)#Xt-(w)}.
Thus, for any law p,
Substituting p( . ) = & ( X t - ( w ) , . ) and observing that X t - ( w ) E E implies that p( is point mass at X t - ( w ) , it follows that f i l ~ is identically zero. For (ii), note (43.8) that if M E b M n then l l M ( t , w ) = f I M ( t , w ) provided X t - ( w ) = X t ( w ) . Consequently l~ II M = 1z(fi M for every M E b M n , and sandwiching shows 1 ~ =~1 ~2p Zfor every Z E b M . Turning to (iii), if R is a predictable time, then since [IRInJ is evanescent, PI”.P([IRn) = P”K(URI]) = 0, so KP is diffuse. Finally, if IE is carried by K, K. does not charge JB E P so d‘does not charge JB. Moreover, d’does not charge J because PIJ = 0. Therefore IEP is carried by K. For every Z E pc?, using (ii) in the second step, a )
1”
P’”
J;I
00
2, K O ( & ) = P’”
(1KZ)t
1
KO(&)
03
= P’”
( I K P Z ) , nO(dt)
/d” zt
= P’”
dJ(dt).
Now (31.2) implies KO = d,completing the proof. One consequence of (44.3) is a useful technical improvement of (41.11). (44.4) PROPOSITION. Let f E bEu and suppose that f = f l is ~ an optional function. Then: (i) J(x-)EP; ( 4 Yf(X)) = Pof(X-).
PROOF:Let Z := f ( X ) and Y := PZ. Then { f ( X - ) # Y } C {X-# X }U { Z # Y } C {X-# X} because of (44.3ii). Since { X - # X } is optionally meager by (41.8), (41.10) shows that f ( X - ) E P and &,bf(X-) E P . Thus (ii) follows directly from the equality (42.2) in the proof of (42.1).
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212
Theorem (44.3) contains the following classification of optional times. Recall the following notation from 56: if R : R + [0, 001 and A C R then
(44.5) THEOREM. Let p be an initial law and let T be an optional time Set over (3;).
A p := { W : XT-(W) = XT(W), T ( w )< 0 0 ) ; Ai := { W : XT-(W) E E , X,-(W) # X T ( W ) ; T(W)< OO}, A, := { W : XT-(W)4 E , T ( w )< w}. Then Tp := TnPis PP-predictable, Ti := TA,is Pp-totally inaccessible and T, := TA,is Pp-accessible.
PROOF:Since A p , hi and A, belong to 3 T , the random times T A ~TA, , and T A are ~ optional. Since JB E P and J B c {X- # X} is meager, J B is predictably meager by (41.6). Thus [Tan satisfies the definition of accessibility. See A5 and §45. Take now IE to be the RM putting unit mass at Tp. Since [ITpDc [Ion U K, (44.3) shows that IC is predictable, hence that [IT,] E P. Finally, if IC is point mass at Ti, IEP is diffuse by (44.3) and so ~ ( A I c=) 0. This implies that [TiI] n [IS] E Z for every predictable time S , so Ti is PP-totally inaccessible. (44.6) COROLLARY. Any optional subset of J may be expressed as a countable union of graphs of totally inaccessible times.
PROOF:The set J is optionally meager by (41.8), and so we may write J = U n [ I T n ] with each T, E T. Theorem (44.5) shows that each T, is totally inaccessible. The same argument applies to any optional subset of J.
The decomposition [ITJ= [ITpJU [TanU surprise.
[Ti] does contain one little
(44.7) PROPOSITION. Each of the sets {T = T p } ,{ T = T,} and {T = Ti} belongs to 3 T - . If T is a weak terminal time, then each of the above sets belongs to the left germ field 3 [ T - ] defined in (14.2).
PROOF: Because Tp is predictable, lnTp] E P yields l[Tpn(T)E 3 T - . That is, {T = T p } E 3 ~ -Since . J B E P , l { ~ ==~~ ~ J , i( T E) FT-, so {T = Ti} E P by complementation. Suppose now that T is a weak terminal
V: Ray-Knight Methods
213
time. For every t 2 0, a.s. on { t < T } , t + T o Ot = T and
Consequently {To& = TpOBt} a.s. on {t < T } , so { T = T p } E Similar arguments establish the other cases.
g[T--].
(44.8) EXERCISE.Using 0 = P V Xo (23.4) together with (31.2), show that T E T is Pp-predictable if and only if, for every f E C,(E),
45. Accessibility
In early expositions of the general theory of processes, the accessible (Talgebra of processes was accorded a fundamental role comparable to that of the predictable and optional cr-algebras. Accessibility is a cross between predictability and optionality, and in applications it has so far not displayed the vigor demanded of successful hybrids. See the discussion in 5A5, and especially (A5.28).
If T E T is Pp-accessible for every initial law p, (45.1) PROPOSITION. then there exists a unique predictably meager set A such that [IT] c A and A is minimal u p to evanescence. PROOF:Replacing T by T{T>O)we may assume that T > 0, the remaining part T{T=o) being trivial to handle. Set At := lnT,mo(t), the RM putting unit mass at T if T < 00, and let A denote the set { t : AAf > 0) so that A is predictably meager (41.3). For every initial law p , Pp-accessibility of T shows that there is a Pp-predictably meager set Ap such that [IT]c Ap up to Pp-evanescence. It follows that l ~ *&Ap = AP up to Pp-evanescence. Consequently, AP is purely discontinuous, and therefore l a * AP = Ap. But ( 1 ~t.A)P = l ~ * cA p since A E P, so l ~ *c A = 0. That is, [ T I c A. If r is predictably meager and [TI]c r then
implies that A C r up to evanescence. We shall say that a process 2 E M is accessible if Z is Pp-accessible for every initial law p. The following results are easily checked.
Markov Processes
214
A process Z is accessible if and only if one may (45.2) PROPOSITION. write Z = Y WlA with Y E P, W E 0 and A predictably meager.
+
A RM K is accessible if and only if one may write (45.3) PROPOSITION. = W * y with W E p 0 and y a predictable RM.
K
Because of (44.3) the predictably meager set A in (45.2) may be chosen to be JB. That same result also shows that K is accessible if and only if 15 * K = 0. If one knew that JB were evanescent, then accessibility would be identical to predictability. The following simple example illustrates the difference between accessibility and predictability. Let X be uniform motion to the right at unit speed on R, killed with probability p (0 < p < 1) as X passes through 0. (That X is a right process is easily seen directly. It is also a consequence of later results concerning killing with a MF.) The Ray topology for X cuts R at 0 and introduces a branch point 0- as illustrated below.
0-
0
0
Figure (45.4)
+
One checks that P o ( O - , . ) = PEA (1- P ) E O . Thus JB = { t > 0 : Xt- = 0-}. The hitting time TOof the state (0) is then accessible because [Ton c J B . Because luT,n = lno+,U l{o}(X),the predictable projection of lnTon is equal to
so TOis not predictable. In this case, the set JB is the minimal predictably meager set containing [TOI]. The results of $44 lead to a decomposition of a RM into parts having distinct dual projection properties. Suppose that K is a RM not charging 101. We shall say that K is totally inaccessible in case K is carried by J . If At := n([O,t]) is finite valued, K is totally inaccessible if and only if A is quasi-left-continuous in the sense of (9.21)-that is, if T, E T are bounded and increase to T , then AT,, AT a s . as n + 00. Given now an optional RM K not charging 0, write --f
(45.5)
IE
= 1K * K
+ 15, *
K+
15 * K .
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215
According to (44.3), 1~* K is predictable, 15, * n is accessible and purely atomic, and 15 * K. is totally inaccessible. The part 1~ * K decomposes further, by means of (29.6), into the sum of a diffuse part &' and a predictable n K * K . Because K , J B and J are homogeneous purely atomic part l{aK>o} on R++,it is immediate that for every time T , the decomposition (45.5) commutes with 6 ~It . is also easy to see that the diffuse part of 6g-~ is just ~ T K ' (see (29.6)). In particular, if n is homogeneous on R++, so are 1~ * n, IJ, * 6 , 15 * IE, )cc and l{A,>o}nK * K . We shall see in Chapter VIII that these parts of n have quite distinct properties.
ua)
(45.6) EXERCISE.Suppose that (I?, p, is a Ray-Knight completion of ( E ,d, U"). Show that if g E C , ( E ) and if g := ~ Iis Ea natural @-potential - (g(XS-) - &g(X,-))defines a perfect, function for X ,then At := Co<s 0. (45.8) EXERCISE.Using (45.6) and (45.7), show that if R = {gn}, then := {an > Pogn} C E B , and EB = UEB, gives a decomposition of Eg into subsets Eg, with the property that R, := inf { t > 0 : Xt-E EB,} is a predictable, totally thin terminal time (12.8). This gives a '(left" analogue in this particular case of the decomposition of a semi-polar set into totally thin subsets (10.16).
EB,
(45.9) EXERCISE.Let R = {gn} where, for each n, gn is a natural potential. Set
p,-
Show that A is a perfect, predictable AF with bounded I-potential and that {AA > 0) and J B are indistinguishable. (Hint: if @ 2 1 and if B is an AF with u$ 5 c then ub = u$ (@- l)U1u$. 5 @c.) Conclude from this exercise and (41.3) that JB is predictably meager.
+
(45.10) EXERCISE.Using the sense of accessible projection and dual projection described in p e 7 2 , V-T14], show that for every initial law p, the OZ1J , , and the dual accessible projection of Z E bM is given by PZ15; accessible projection of a RM )c, assumed to be o-integrable in P , is given
+
216 by
Markov Processes
*
~ J S , KP
+ l ~ *,no. In particular, the accessible projection of f ( X ) for
f E E" is f ( X ) l ~ ,+ f ( X - ) l ~ ; .
(45.11) DEFINITION. An AF, A, is a reference AF for J B provided: * (i) A is perfect and predictable; (ii) {AA > 0) = J B up to evanescence; (iii) A has bounded I-potential. i -
Exercise (45.9) gives one particular example of a reference AF for J B . In $73 we shall encounter a reference AF for the random set J which will permit us to construct a so-called LCvy system detailing its structure. The remaining results in this paragraph will show some typical uses of a reference AF. Observe first of all that if K is an RM (resp., optional RM) carried by J B , then one may write K = Y * A, where A is a reference AF for J B and Y E M (resp., O), taking Y := (&/AA)1JB with 0/0 := 0, and recalling (28.8). In addition, if D is a RAF (resp., AF) carried by JB then since AD E 49 (resp., Onfig) by (35.12) and J B E Pnfig c Onfig, one may write D = Y * A with Y E f i g (resp., 0 n 49). Largely because the process of jumps of an arbitrary RAF belongs to 49, it is of interest to characterize that u-algebra in a manner directly related to the sample paths of X. Recall that it was shown in (24.30) that P n f i g is generated by Z and by processes of the form t -+ f(Xt)-lp,wi(t) with f E Sa for some a 2 0. Because 0 = P V Xo (23.4) and P n f i g = X? V Z (24.32), one would suspect that (45.12)
o n f i g = x:
v x0vz.
(The second equality comes from (24.32).) This result is in fact true, but the proof is far from direct. See the remarks (45.16). It may be stated informally in the following way. The u-algebra of processes optional relative to the past and to the future is essentially that generated by f (X) and g ( X ) - with f (Ray) continuous and g a-excessive for some a > 0. One might also suspect that f i g = x: vqd, which is the reverse-time analogue of 0 = P V Xo, but nothing seems to be known on the subject. The following results are steps on the way to the proof of (45.12). (45.13) PROPOSITION. Let Z E M (resp., 0 ) satisfy the weak homogeneity condition (45.14)
for every t 2 0,
2, 0 Bt = Z,,,
Vs
> 0, a s .
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217
Then there exists Y E f i g (resp., 0 rl fig) such that 21J , - Y1 J , E 1. PROOF: Without loss of generality we may assume that 0 5 Z 5 1. Let A be a reference AF for J B and set B := 2 * A. If t 2 0, (45.14) and (29.3) give 6tB = lnt,wn * B and so, by (35.77, B is a RAF (resp., AF) of X. However, as we showed above, we may write B = Y * A with Y E 49 (resp., 0 rl fig), and the conclusion of (45.13) follows immediately. (45.15) THEOREM. Define BdyOto be the a-algebra of processes of the form ( t , w ) -t F(Btw), F E p , so that f i d v o C f i d . Let 2 E f i g (resp., 0 r l f i g ) . Then Z ~ J ,E 3E5 V E j d > O V Z (resp., X5 V JoV 2).
PROOF:Fix 2 E fig with 0 5 Z 5 1. For F E b p , let FOB denote the process ( t , w ) -, F(Otw). Then ( F o O ) Z E fig and therefore, by (24.32), P(F0OZ) E P r l fig. Use (22.10) to select Z’ E B+ 8 3’with 0 < Z’ 5 1 and 2 - 2’ E 2. For all f E b€, f ( X ) 2’ E B+ 8 F’, so we may realize P((FoO)2’) as l?((FoO)Z’),where l? is the predictable projection kernel of $43. The map F -+ fi((F06)Z‘)is a kernel from (R+ x R , P n f i g ) to ( R , p ) which is dominated by the kernel F --t fI(Fo0) = Pof(X-), f(x) := P”F. Since (R, 9) is separable, Doob’s lemma (A3.2) shows that we may select @ E ( P nfig) 8 30 with
l? [(FoB)Z’](t,w)=
s
P‘o(xt-(w)l’)(d~’) $((t,W),w‘)F(w’)
for all F E b p . Define Wt(w) := @ ( ( t , w ) , & ( w ) ) E ( P nfi9) V fid3’. We shall prove that ( Z - W ) ~ JE ,2. To this effect it is enough to show that for every integrable RM D carried by J B , which we noted above is necessarily of the form D = M * A with M E bpM, we have
P“
s,”
03
Zt dDt = P”
Wt dDt.
Because M = P V E j d > O (23.12), it is enough to show that
for all 2 E E , Y E b P and F E b3’. Let y denote the left side of the last display. On one hand,
y = P”
I”;
e-txP((FoO)Z‘)tdAt
218
Markov Processes
On the other hand, it is easy to see by monotone classes, starting with the case $ ( ( t , w ) , w ' ) = M(t,w)G(Btw')with M E P n 49 and G E bF", that ( t ,w ) 4 P" ( X t - ( w f v . (dw')F(w')$ ((t ,w ) ,w ' )
J
is the predictable projection of (FoB)W E ( P n fig) V y = P"
1"
fjd?O.
Since A E P ,
e-tYtF(et) wt d ~ t ,
and this shows (2 - W ) ~ JE, 2, as noted above. This proves the first assertion. In case 2 E 0 n fig, it follows that "2 - "W E 2, and since "W E ( P nag) V 9by a monotone class argument, the second assertion is also proved. (45.16) REMARK. With (45.15) in hand, it is easy to indicate the proof of (45.12). By arguments similar to those used in (45.15), we shall show (73.17) that if 2 E 0 satisfies (45.14), then there exists f E E" 8 E such that (2 - f(X-, X ) ) ~ JE Z.It will then follow that 2 E 0 f l 4 g implies 2 1 E~ X5 V Xo.Finally, if 2 E 0 n fig, 2 1 E~0 n 4 9 and by (44.3), 2 1 =~P 2 l E~ X5 V Xo.This proves (45.12), modulo (73.17). 46. Left Limits in the Original Topology
Recall that Xf- denotes the left limit of X at t > 0 taken in the original topology of E , provided that limit exists. In this paragraph, we compare X:- with the left limit X+ taken in the Ray topology. In this discussion, it is helpful to embed E relative to its old topology in a compact metrizable space 8. By (A2.11), E E &". Let Xt- denote the left limit of X at t > 0 in h, if that limit exists. (46.1) PROPOSITION. (i) IfF := {X- exists in 81, then ?i is predictably meager. (ii) For every f E €IL, lpf(X-) E Pt.
PROOF: (i): If f E C(&), 2 := f ( X ) is adapted and right continuous. Let ^Zt := liminf,Ttt 2, and "2, := limsup,TTt2,. Then (-2 < " 2 ) is predictably meager by (41.7). Let {fn} be uniformly dense in C(h), 2, := f n ( X ) . Then (i) follows since ?I = U,{-Z, < -2,). (ii): Given f E C(@, yt := f(Xt)E 0 and so, by the argument following (41.6), -Y E P. In view of (i), this implies l p f ( X - ) E P. Consequently l p f ( X - ) E P for every f E bk. Given p and an integrable RM K , g 4 Pp lp(t)g(Xt) ~ ( d tdefines ) a finite measure on (h,i).Then (ii) follows exactly as in the proof of (41.9).
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219
(46.2) THEOREM. The sets { X o does not exist in E } and { X o exists in E but X!! # X-} are both predictably meager.
PROOF:By (46.1i), is predictably meager, and (46.lii) implies that P n { X - E B \ E } E pt. But P n {X- E E \ E) c F n { X - # X I is optionally meager, by (41.7). Then (41.10) and (41.6) show that f. n {X- E E \ E } is predictably meager, and so therefore is {X! does not exist in E } = i..u (F n { X - E \ E } ) . Setting A := {XO exists in E but X!! # X-},one has A c {X! exists in E , X ! # X } U {X- # X}. The first set on the right is optionally meager, as we observed above, and so is { X - # X } = J U J B , so A is contained in an optionally meager set. It is enough now to prove that A E Pt. But (41.9) and (46.1) imply E Pt, so by the MCT, that for all f,g E bE, t + lp(t)f(Xf-)g(Xt-) l,h(X!,X-)E Pt for every h E b(& 63 &). Taking h ( z ,y) := l{,#y) the second assertion follows. (46.3) REMARK.It is an obvious consequence of (46.2) and (44.5) that a.s., Xf-exists and is equal to Xt- for all t > 0 such that X t - E E and X t - # X t (ie., t E J ) . Moreover, if T E T and if, 8.5. on {T < co}, either Xg- does not exist in E or Xg- = XT,then T is accessible by (44.5). 47. Quasi-Left-Continuity In the early development of Markov process theory, quasi-left-continuity conditions of the type (9.21) played an important role. The Ray-Knight theory provides a clear explanation of the simplifying nature of such hypotheses. Throughout this paragraph we use the expression “T, T T” as an abbreviation for “{T,} is an increasing sequence in T with limit T.” Recall that in the general theory of processes over a system (R, 9,Qt, P), a measurable process Y with values in a metric space E is called quasi-leftcontinuous (qlc) if T, t T implies YT, + YT a.s. on {T < 00). Extending this notion a bit, Y is called qlc on [ O , S [ , where S E T, in case (47.1)
T, T T implies YT,,+ YT a.s. on {T < S}.
Recall that a filtration ( Q t ) is quasi-left-continuous (qlc) provided (47.2)
T, 7 T implies GT =
v
QT,,.
n
(47.3) DEFINITION.A right process is a Hunt process provided it is qlc relative to every Pp.
Observe that the Hunt condition depends on the topology of E , but the lifetime plays no role. In effect, the death point A is considered as just an undistinguished trap in the state space.
220
Markov Processes
(47.4) DEFINITION.A right process with lifetime 6 is a standard process provided it is qlc on 10, <[I relative to every P P . The next condition does not depend on the topology of E nor on the lifetime. (47.5) DEFINITION. A right process is special if T, T T implies X T E VnFT,.
(47.6) THEOREM. the following conditions on a right process X are equivalent: (i) (ii) (iii) (iv) (v)
JB = {X- E E B } is evanescent; X is a Hunt process in its Ray topology; X is special; for every initial law p, (Fr)is qlc; for every p and every Pp-predictable time T , 3$= 3$-.
PROOF:The equivalence of (iv) and (v) is well known [De72,III-T51]. Assume now that (i) holds. If T, t T , then X(T,) -, X ( T ) certainly holds on {Tn = T for some n, T < m}. Letting R, := ( T n ) ( ~ , < T one ) , has R, tT R = T { T , < T ~ (and ~ ) , since R is clearly predictable, (i) and (44.5) imply that X R - = X R as. on {R < m}. Thus (i) implies (ii). By (23.9), 3; = Fg- V ~ ( X T )and , thus (iii) *(v). Finally, (v) +(i) follows from (42.5i) and the fact that JB is predictably meager. (47.7) EXERCISE.Show that under any of the equivalent conditions in (47.61,T, T T implies that 3 T = V 3T,, , and for every predictable time T , 37'= FT--. (47.8) EXERCISE.Let ( X t ) be the deterministic process constructed from a right continuous Aow ( G t ) as in (8.8). Show that X is special, and infer from (47.6) that the Ray topology on E converts (&) into a continuous Aow. Because a Hunt process is obviously special, (47.6) implies that the Hunt property is preserved under the change to the Ray topology. We now compare the left limits X!! and X- for Hunt and standard processes. (47.9) LEMMA.Let S be an optional time over a system ( f l , Q , Q t , P ) satisfying the usual hypotheses. If Z is bounded, real, right continuous, and qlc on 10, S [ , then Zt- exists for all t < S , almost surely.
PROOF:Given c > 0, consider the increasing sequence (T,) of optional times defined by To := 0, and for n 2 1, T, := inf{t > Tn-l : [Zt - Z T , , - ~2~ t}.
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Clearly ~ Z T ,2 E on {T, < m}. If T := IimT,, then T 2 S a s . because liminf, Z(T,) < limsup, Z(T,), violating qlc on [IO,S[ if T < S. However, for every n 2 1, sup{lZt - ZT,_~I;T,-~ 5 t < T,} 5 E and so Zl~o,s(iis approximated to within c by Z(T,+~)l[Tn-l,~,,[l[o,s[, which is rcll on 10, S[I. The result follows then by ar, elementary uniform convergence argument.
c,"==,
(47.10) THEOREM. Let X be a standard process with lifetime 5. Then:
(i) almost surely, X:- exists in E for all t < C; (ii) almost surely, for all t < C, either X:- = Xt- or Xt- $ E and = x,; (iii) if X is a Hunt process, then X! and X - are indistinguishable; (iv) if X is special standard, then X! and X - are indistinguishable on U0,CU.
x:-
PROOF:Embed E topologically in a compact metrizable space k. With the notation of (46.1), it follows directly from (47.9) applied to f n ( X ) ,as f, runs through a uniformly dense sequence in C(k),that Xt- exists in E a.s. for all t < 5. Thus f' := { X - exists in k } contains [IO, <[, and (46.1) shows that { X - E $ , \ E } n I? is predictably meager. Knowing that Xexists on [ I O , C [ , standardness implies that for every predictable time T, (47.11)
XT- =
xT E E a s . on {T < 0.
It follows then that (8-E E \ E } n [O, ([I E 2, proving (i). For (ii) it will suffice to show that [IO,<[I n {X! # X-}nJg and [ O , ( [ I n { X ! # X } f l J B are both in 2. The latter set is contained in the predictably meager set r r l { X ! # X} r l JB fl [IO,C[, which is evanescent by (47.11). On the other hand, the first set is contained in f' n {X! # X-} f l J;, which is predictably meager by (46.2). In conjunction with (44.5), it follows that I? n {X! # X} n J g c f. n { X ! # X - } r l { X - = X } , which is predictably meager, and writing this last set as I? n {X! # X } n { X - = X } , one sees that it is evanescent on [IO, ([I because of (47.11). This proves (ii), and (iii) is obtained by setting = 00 in (i). Finally, if X is special standard, (47.6) states that J B E Z and the proof of (ii) above shows that f' n {X! # X - } = $ n { X ! # X - } n J$ is evanescent on lo,<[. As i? 3 [o,c[, (iv) follows. Part (ii) of (47.10) implies that if X is standard, then {X! # X} n 10, C[I C ( J U J B ) f l [IO, C [ I , so that the Ray topology introduces additional discontinuities on [ I O , C ] . However, (46.3) shows that J c {X! exists in E and X! # X } , so the additional discontinuities are introduced only at
<
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branching times. This phenomenon is illustrated by the example (45.4) which may easily be seen to be standard, but not Hunt. A theorem of Walsh and Meyer [WM71] states that this example is the prototype of the Ray-Knight completion for standard processes, in that essentially all branch points 2 have &(z, .) carried by a two point set {A, y} with y E E . Note also that the above example shows that standardness is not preserved in the passage to the Ray topology. 48. Natural Processes
Before starting on this section, the reader should at least review the material in Appendix A6, where the natural u-algebra PR and the natural and dual natural projections are described at some length in a general nonMarkovian setting, R being some distinguished optional time. Throughout denotes a right process with lifethis section, X = (f2,3,Ft,Xt,f3t,P”) time <. In applications of the Ray theory to X,it will be necessary to use ( E A , € A )as the state space in place of ( E , € ) .We shall define the natural a-algebra Pc for X as the trace of P on [ O , C [ I , so that C now plays the r61e that R played in A6. The a-algebras Mc and Oc denote the respective The entire discussion in A6 may be carried traces of M and 8 on [O, over to the Markovian setting word for word, making use of the special versions of optional and predictable projections available in this case. In what follows, otherwise undefined terms and notation are to interpreted as the Markovian cases of the corresponding objects discussed in A6. For example, a natural time is an optional time T such that [TI] E Pc. As luo,cn = ~ E ( X )the , process Pln_o,cn = p(lE(X)lno,oon) may _be , to (42.1). ( ( P , ) computed as r(Xt-)lDo,mfl(t), r ( z ) := P o l ~ ( z )thanks denotes the Ray-Knight extension of the semigroup (Pt)extended to EA.) Lemma (A6.15) shows that a.s., r ( X , - ) > 0 for 0 < t < C. In fact, since PO has no degenerate branch points, r ( 2 ) > 0 if and only if 2 E E U E g , where EB is the set of branch points for (Pt). Let A := { r ( X - ) > 0) = {X- E E U E B } . Clearly ] O , C [ I c A c I]O,
[I.
We shall call
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The natural projection %Zof a process 2 E M vanishing off I]O,<[i and possessing a meaningful predictable projection is defined (A6.18) as
It is clear from the discussion of the predictable projection kernel fi in $43 that, under the hypothesis (43.1), the natural projection is given by the kernel
for A4 E bMc. The dual natural projection of an increasing process A carried by A (and assumed to have a finite dual predictable projection) is defined (A6.22) by
Observe that A is carried by A if and only if A is constant on ] C , COD and A does not charge the unnatural part of 5 (ie., AAco = 0). The natural and dual natural projections inherit nice properties relative to shifts from the predictable and dual predictable projections. (48.5) PROPOSITION. Let 2 E bMc. Then for T E T,u p to evanescence,
PROOF:This is a direct consequence of (22.15) since l g T , o o ~E P. In the same way, (29.7), (31.7) and (A6.25) lead to (48.6) PROPOSITION. Let A be an increasing process carried by A and having a dual predictable projection. Then for T E T,
In particular, if A is a RAF carried by A, then A" is a natural AF having the same a-potential function as A . Though only increasing processes were discussed in A6, it is clear that the proper domain for the dual natural projection is the class P c of RM's tc carried by A and u-integrable on P. The extension of the dual natural projection to this class is quite routine, and we shall use this extended dual natural projection without further comment.
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For the rest of this section, we shall impose standardness hypotheses on X . Standard processes make up the most important subclass of the class of right processes because most processes of practical importance obey the standardness hypothesis (47.4). This definition (47.4) of a standard process differs slightly from that in [BG68, I], where it is always assumed that the state space E is locally t is a Borel semigroup . The compact with a countable base, and that P following artifice will show that, under the additional hypothesis that X is a Borel right process (ie., E is a Lusinian space and Pt maps Borel functions on E to Borel functions), there is no essential difference between these two definitions. Let F be a compact metriaable space containing E A := EU{A} topologically. Then E is necessarily Borel in F by (A2.11). Extend X on E to a Markov process X on F by adjoining to R the collection of all constant paths in F \ EA and setting
X t ( w ) :=
X t ( w ) if X o ( w ) E EA, X o ( w ) if X o ( w ) E F \ EA.
The process X is obviously standard in the sense of [BG68, I], and results about X are obtainable by simple restriction to E A , In applications it is sometimes interesting to work with X run under a fixed initial measure p, rather than with all p at once. This gives us a little greater generality at no additional cost. (48.7) DEFINITION. A right process X is p-standard provided: (i) Pp-a.s., X f - exists in E on I]O,c[I, and (ii) given an increasing sequence (T,)c T with limit T , X(T,) X ( T ) Pp-a.s. on {T < C } .
+
It is immediate from the main result of this section-Theorem (48.15) below-that condition (i) in the above definition is a consequence of (ii). See also the proof of (47.10). Obviously, if X is p-standard for each initial measure p , then X becomes a standard process after deleting the null subset of R in which X:-(w) fails to exist in E for some t E]O,C(w)[. Some of the results of this section require that every Ray-Bore1 function be nearly Borel relative to the original topology. We have not made this a part of the definition of a p-standard process, however. It is clear from (18.5) that this condition would imply that every a-excessive function for X would be nearly Borel relative to the original topology. The class Pr of p-natural processes consists of those processes in M p which are Pp-ir.distinguishablr: on 10, C[I from a natural process. A process in Pr is (Ff)-adapted. It is easy to check that Y E Pr if and only if there exists W E P p which is Pp-indistinguishable from Y on 10, ([I. An
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(Ff)-optional time is p-natural provided it is equal to a natural time Ppalmost surely. A p-natural increasing process A is an increasing process which is p-natural and satisfies A [ - = A < . We claim that such an A is Pp-indistinguishable from a natural increasing process B. In proving this, one may suppose A uniformly bounded. Since A is (.Ff)-adapted, it is a standard fact that A is Pp-indistinguishable from an (optional) increasing = Ac- for all t 2 <. We may and shall also assume that process A with A is uniformly bounded. Then An exists as a natural increasing process, and in particular, it is a dual natural projection of A, hence of A , relative to P p . Therefore An and A are Pp-indistinguishable, establishing the claim. If A is a p-natural increasing process for every p, then, A being (.Ff)adapted for every p, is (Ft)-adapted. Consequently A is (indistinguishable from) a natural increasing process-namely, A". The next result shows that the natural projection of a function of X has a very simple form when X is standard. Let h be positive or bounded and nearly Borel on E (48.8) THEOREM. relative to the original topology. If X is p-standard, then n(hoX) is PPindistinguishable from h(XO)lio,Cu. (Of course, h(X!)liO,Cl is defined only u p to Pp-evanescence.) PROOF: It suffices, by obvious sandwiching considerations, to assume h Borel in proving (48.8). By a monotone class argument, it then suffices to assume h E bC(E).Let yt := liminf,rtt h(X,) for t > 0. By (41.7), Y E 7'. Now, h being continuous, Y is Pp-indistinguishable from h ( X ! ) on 10, Cl. Let "(h(X)lDo,Cn) = W110,~nwith W E bP. If T is a predictable time, then XT = X $ - PP-a.s. on {T < (} because of (48.7ii), and so P p { W ~ ; O< T
< C}
= P p { h ( X ~ )0;< T < C} = P p { h ( X $ - ) ; 0 < T < C} = P"{Y,; 0 < T < C}.
Therefore P(110,CnW) = P(lj0,CUY)up to Pp-evanescence, and the result follows then from (48.1). We are now in a position to characterize natural increasing processes over a standard process. (48.9) THEOREM. Let X be p-standard, and assume that every Ray-Bore1 function is nearly Borel relative to the original topology. Then an optional, right continuous increasing process A is p-natural if and only if A is carried set of ( t , w ) with by ]O,C[I and AA vanishes Pp-a.s. on {X # X!}-the 0 < t < [ ( w ) such that either X:-(w) does not exist in E or Xf-(w) exists in E but X:-(w) # X t ( w ) .
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PROOF:If A is p-natural, then dA is carried by 10, Cb, and by the discussion preceding (48.8), one may suppose A is natural in showing that A A vanishes Pp-a.s. on { X # X!!}. Let E > 0 and let R := inf{t : AAt > E } . Then R is a natural time by (A5.5~).It follows from (A6.12iii) and (48.7ii) that X i - = X R Pp-a.s. on { R < C}. The same argument applies to the successive times at which A A > E , and by varying E , one sees that A A vanishes on { X # X!} Pp-almost surely. For the converse, we let H
:= {t
> 0 : X:- exists in E and X-:
= Xt},
B := 1~* A.
Then B = A up to Pp-evanescence and B is an optional increasing process with dB carried by 10, <[Iand A B vanishing off { X = X!!} = H . It suffices to show that B is natural. In other words, to complete the proof of (48.9), it is enoygh to show that an optional increasing process A with dA carried by 10, <[I and A A vanishing off { X = X ! } is necessarily natural. We may and shall assume A bounded. Let W := Y h ( X ) with Y E bP and h E bE. (Caution: h is only Borel relative to the Ray topology.) Then from (A6.18) and (48.8), which may be invoked because of the nearly Borel hypothesis on X , ,(Yh(X)) = Y h ( X ! ) l ~ o , cand ~ hence
P"
1"
x h ( X t )dAt = P"
1"
yth(Xg-) dAt = P"
1"
n(YhoX)t d&.
However, (23.4) shows that processes of the form W generate 0,and consequently P" 2, dAt = P" "2, dAt for all 2 E bO. Since A is optional, this last equality holds for all 2 E b M , by an obvious projection argument. Therefore (A6.29) implies that A is p-natural, completing the proof.
s
s
Assume that Ray-Borel functions are nearly Borel (48.10) COROLLARY. relative to the original topology. Let T E T. If X is p-standard, then T is p-natural if and only if X g - = XT Pp-as. on (0 < T < C}. Under the latter condition, there exists an increasing sequence (Tn)in T such that, Pp-as., the following condition holds: (48.11) T, < T on (0 < T < C } , limT, = T on {T < C } , and limTn 2 on {T 2 0.
<
PROOF: Applying (48.9) to the increasing process At := l[Tlm[(t)l{O
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process and if XT = Xg- on (0 < T < ca}, then T is actually predictable because of (44.5) and (47.6). The next result, in which no standardness hypotheses are imposed, is critical for the proof of (48.15) below, but it is also of independent interest. It should be compared with Theorem (23.2).
(48.13) LEMMA.Let T be an exact terminal time for X. (i) Let F, := {z : P”(e-T) 2 1 - l/n} and let T, := TF, be the hitting time of F,. If T is Pp-predictable, then Pp-a.s., T, < T on (0 < T < oo}, and { t : Xt E F,} contains an open interval ]U,T [ ,U < T on T > 0. If T is predictable, then the assertions in the preceding sentence hold a s . rather than Pp-almost surely. (ii) Let F, := {z : Pz(e--TAC)2 1 - l/n} and let T, := TF,. If T is natural, then a.s., T, < T on (0 < T < C } , limT, = T on {T < C } , limT, 2 C on {T 2 C } , and {t : Xi E F,} contains an open interval ]U,T[, U < T if 0 < T < C.
PROOF:Let u ( z ) := P”(1 - eWT)= P” e-t dt, Y, := e-tu(Xt)l[o,T[(t). Since T is exact, Y = (Y,) is 8,s. right continuous by (12.9). A simple calculation then gives T
yt = P”{/
e-8 ds
I Ft},
TAt
SoT A t
and so Y is the potential of the increasing process t + e ds relative to each P”. In particular Y is a supermartingale. Now suppose T is Pppredictable, and let (R,) announce T relative to Pp. The assertions in the remainder of this paragraph hold Pp-almost surely. Since T
YR, = Pp{Lne-’ds
I FR,,},
YR, is strictly positive on {T > 0) and YR, + 0 as n + ca. It follows that Y > 0 on 10,Tn and YT- = 0 on {T > 0). As F, = {u 5 l/n} and reg(T) := {z : P”(T = 0) > 0) is contained in F, for every n, it follows that Tn := TF, has the desired properties. If T is predictable, the above argument is valid for every initial law p, and so (i) is established. T A t -t For (ii), let u ( z ) := P” e dt and Y, := e-tu(Xt)lpT[(t). As before, Y is the potential of the increasing process
so
Bt :=
1’
e-’l[o,TAC[(S) dS.
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Let TO:= inf{t : yt = 0). Since YT, = 0 if To < 03, B must be constant on
[TO, 001. Therefore T A < I TO.In particular Y > 0 on [O,T A ([I. Using (A6.12iii), the remainder of the proof goes exactly as before. If in (48.13ii), T is only a p-natural time, then the con(48.14) REMARK. clusion of (48.13ii) holds Pp-almost surely. Observe also that for a general exact, predictable terminal time T , since each T, is a perfect exact terminal time, it follows from (48.133) that T is equivalent to a perfect, exact, predictable terminal time. In other words (48.13) contains a perfection theorem for predictable exact terminal times. A similar observation is valid for exact, natural terminal times. It also follows from (48.13ii) that if
R :=
limT,
on {limT, < C}, on {limT, 2 C},
then R is a predictable, exact terminal time with [RI n [O,<[
= [Tn n
U01CU-
We come now to the main result of this section, characterizing standardness in terms of hitting times. (48.15) THEOREM.Let X be a right process and p an initial law. Then the following conditions are equivalent: (i) if (F,) is a decreasing sequence of finely closed sets in Ee and if T := limTFn, then ~ ( T F , + , ) X ( T ) Pp-a.s. on {T < C}; (ii) X is p-standard; (iii) Pp-as., X-: exists in E on I]O,C[, and if X t - denotes as usual the Ray left limit in a Ray cornpactification I? of E U {A}, then, u p to Pp-evanescence,
PROOF:In proving (48.15) we may suppose that A is isolated in E A := E U {A}. Obviously (ii) =+ (i). The bulk of the argument is to show that (i) + (iii), so suppose that (i) holds. Let E be a compact metric space containing E A topologically as a universally measurable subset. (We are not using the Ray topology here.) As in 546, let X t - denote the left limit of X in E at time t , if that limit exists. The first step is to show the existence of Xi- in E . To this end, if g is a continuous function on E and if a < b, then the set
I’ := {liminfg(X,) < a, limsupg(X,) > b } sttt
STTt
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is predictable by (41.7). Since t -, g ( X t ) is right continuous, I? does not contain any infinite strictly decreasing sequence. It follows from (A5.5~) that the debut T of r is a predictable, thin terminal time with I T ] c r. We shall show that Pp(T < C) = 0 and, by letting a < b run over the rationals and g run over a countable dense subset of C(,@), this will imply the existence of X- in E on no, (1, Pp-almost surely. Let (F,) be the sets in (48.13i) and T, := TF,. Because X t E F, on an interval IT - E , T [if 0 < T < C and [ T I c r, the hitting times of F,n {g 5 a } and F,n { g 2 b} respectively both increase to T on {T < m}. Using (i) for these sequences we obtain g ( X T ) 5 a and g ( X T ) 2 b Pp-a.s. on {T < C}. Consequently Pp(T < () = 0, proving that 2- exists in E Pp-a.s. on IO,C[. We next claim that, up to Pp-evanescence, {X- # X} n 10, Ci cannot intersect the graph of any thin Pp-predictable terminal time T . For, given such a T ,let F, := {z : P"(e-T) 2 1 - l/n} and T, := TF,. Then according to (48.13i), (T,) increases to T strictly from below Pp-a.s. on {T < C}, and so it follows from (i) that XF- = X T Pp-a.s. on {T < C}, establishing the claim. Now let d be a metric for E and t > 0. Set H := {(t,w) : X+(u) exists in E and d ( X t - ( u ) , X t ( w ) ) > E } . Then W := { X - = X } n H r l DO,([ is a discrete optional set, and its debut T is a thin terminal time which is Pp-predictable by (44.5). Consequently Pp(T < C) = 0. Hence, up to Pp-evanescence, (48.16)
{x-= x }n I]o,C[I c {x! = x }
n Do,[[.
Now let A be a reference AF (45.11) for the set J B of branch times for X on E A , and let T be the debut of {AA 2 E } . Then T is a thin predictable terminal time and, by the argument given in the first part of the proof, the intersection of { X # X-} n lo,<[ and [IT] is Pp-evanescent. Varying t > 0, it follows that {X- 4 E A } n I]O,([I n { X # X-} is Pp-evanescent. Writing this in the form
{x-$ EA}
n
no,cu c { X = 2-}n no,cu
up to Pp-evanescence, and bringing in (48.6), the fact (46.2) that X - = X Pp-a.s. on {X-E E A , X - # X } , and the fact that A is isolated in E A , we see that X!! E E Pp-a.s. on ]O,C[ and therefore (i)+(iii). To complete the proof we show that (iii)+(ii). Let (T,) c T increase to T . We must show that X(T,) + X ( T ) Pp-a.s. on {T < I}. By (44.5), T is totally inaccessible on { X T - E E a , X T - # XT,T < m} and so T, = T and X(T,) = X ( T )for large enough n on this set. On { X g - = XT,T < C}, X(T,) --+ X ( T ) . In view of (iii), this covers all possibilities.
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(48.17) REMARK.Note that condition (48.15iii) is unchanged if we replace
{X-4 E A } by {X-4 E } , because r := {X-= A} n 10, <[ is evanescent. To see this, observe that on one hand r E P, has countable sections, while on the other hand, I? C J implies, by (44.3) applied to X with state space E A , that
is a countable union of graphs of totally inaccessible times. Applications of the following characterization of Hunt processes must be made considering A as an ordinary point of the state space. (48.18) THEOREM. Let X be a right process with infinite lifetime. The following conditions on X are then equivalent: (i) if F, is a decreasing sequence of finely closed sets and if T := limTF,,, then ~ ( T F , --t , ) X ( T ) a.s. on {T < 00); (ii) X is a Hunt process; (iii) a.s., X:- exists in E for all t > 0, and X!! is indistinguishable from X- ; (iv) a.s., Xf- exists in E , and for every f E R, f(X:) is a s . left continuous; (v) as., Xf- exists in El and for every a > 0 and f E C,(E), U"f (X!!) is a.s. left continuous.
PROOF: Assume (i) holds. Then (48.15) implies that X is p-standard for every p, and since C = 00 a.s., X is Hunt. Now assume X Hunt. By (48.6), X is also Hunt relative to the Ray topology, and we get (iii) from (47.10). Assume next that (iii) holds. Obviously (iii) implies (iv) and (iv) implies (v) since R - R is uniformly dense in C,(E). To complete the proof, it suffices to prove that (v) implies (i). Under (v), U" f(X!!) and U" f (X-)are both left continuous, and it9 they agree on the dense set of points of continuity of both X!! and X-,they are indistinguishable. As the Uaf are uniformly dense in C,(E), this proves that X! and X- are indistinguishable. That (i) holds is then an obvious consequence of the implication (iii) + (i) of (48.15).
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49. Balayage of Functions
Throughout this section, it is supposed that the resolvent (Ua)of the right process X satisfies the transience hypotheses discussed in (10.36). Namely, (49.1) HYPOTHESIS. 3h E bpE" with U h bounded and strictlypositive.
Given a > 0, the resolvent V @:= U"+@ satisfies (49.1), and the results of this section will therefore always apply to U" for a > 0. (49.2) DEFINITION.The reduite f~ of an excessive function f on a nearly optional set B C E be defined to be the lower envelope of the family of all excessive functions dominating f on B.
In classical potential theory (uiz, Newtonian potential theory in R3), the rCduite of the constant function 1~ on B is called the equilibrium potential of B. It is the potential of the electric field generated by a charge on the conductor B held at unit potential above ground. As a simple example for the notion of rCduite, let X be uniform motion to the right at unit speed on R. The excessive functions for X are the right continuous decreasing functions on R, and in this case,
For, letting g denote the right side of (49.3), it is obvious that any decreasing function dominating f on B must also dominate g, and on the other hand, for a fixed z, k(y) := ml]-,,,[(y) + g(z)l[,,,L(g) defines an excessive function dominating f on B , hence f ~ ( z5)g(x). For the process X, the hitting operator PB obviously satisfies (49.4)
PB$J(I)= sup{$J(y) : y > 2,y E b } ,
11, decreasing.
Observe therefore that for this particular process X ,~ B ( z )= P " ~ ( X D ~ ) ) PB~(z) except possibly at those identically, and consequently f ~ ( z= z E B which are not points of accumulation of B from the right. It is amazing that essentially the same result holds for a general right process. The description of this fact, proved first under more restrictive hypotheses by Hunt [Hu57],is the main result of this section. Indeed, Hunt's theorem establishes perhaps the deepest connection between Markov process theory and abstract potential theory. We shall not offer a proof, as it is lengthy, Vol. IV]. and has been treated completely in [Ge75a]and [DM75, (49.5) THEOREM (HUNT).Let X be a right process satisfying (49.1), and let B c E be nearly Borel for X , f an excessive function for X . Then f ~ ( z = ) P 5 f ( X ( D s ) ) everywhere, and consequently PBf may be
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identified with the rdduite off on B, except possibly on the semipolar set B \ BT of points in B which are irregular for B. The key tool allowing one to extend the proof of Hunt's theorem to a general right process is the approximation theorem for hitting times due to Shih [Shi7O]. Its proof [Ge75] leans heavily on Choquet's capacitability theorem, and is outside the domain of this book. (49.6) THEOREM (SHIH).Let B C E be nearly Borel for X, and let p be an initial law. Then there exists a decreasing sequence (G,) of Ray-open sets in E with G, 3 B for each n, and DG, t DB a s . P P . In particular, if p ( B \ B') = 0, then TG, 7 TB a.s. P P . ((49.1) is not required.) (49.7) EXERCISE.For B C E nearly Borel, define
RB(w):= inf{t 2 0 : Xt(w)E B or X t - ( w ) E B}. Prove that as., RB = D B . (Hint: obviously, RB 5 D B , and RB = DB if B is Ray-open. (49.1) is not required.) (49.8) EXERCISE.Let X be Brownian motion on E =]O, 1[, killed on reaching an endpoint. Then X satisfies (49.1) and the excessive functions for
X are the concave functions on ]0,1[. Prove directly (ie., using only real variable methods) that for every Borel set B C E and every excessive f , the subgraph ((2,y) : 0 5 y 5 f ~ ( x )0,< x < 1) OffB is the convex hull of {(x,f(x)) : x E B } and {(x,O): 0 < x < 1). In particular, f~ is excessive. (49.9) THEOREM. Let X be an arbitrary right process, not necessarily satisfying (49.1). Then the fine topology for X is the smallest topology on E rendering continuous (in the extended sense of maps into [0,00]) all functions in U,S".
PROOF: Let 7'denote the topology generated by U,S", 7 the fine topology. We showed in (10.8) that every f E S" is finely continuous, so that 7 c 7'.To prove 7 3 7', it suffices to prove that for all nearly Borel B E 7 (which form a base for 7)and 2 E B, there exists G E T' with x E G c B. Apply (49.6) to the set A := BC and the measure p := ex, which obviously satisfies p ( A \ A T ) = 0, to find a decreasing sequence (G,) of Ray-open sets whose hitting times T, satisfy := Pg(e-Tn) E S1. Then Px{T, T T} = 1, T := TA. Let $,(y) $n(x) $(x) := Px(e-T). Because $(x) < 1, there exists n with $,(z) < 1, and therefore x E G := Uk{qn < 1 - l/k}. But $, = 1 on A , so G c A" = B. The set G satisfies the requirements of the proof. It seems curious that the proof of (49.9) lies as deep as the proof above makes it appear, but there seems to be no obvious replacement for Shih's approximation theorem for general right processes. --$
VI
Stochastic Calculus
We study in this chapter the foundations of the stochastic calculus over which we mean the theory of local martingales, semimartingales and stochastic integrals over these processes. The reader is assumed to be familiar with the basic parts of the theory relative to a fixed filtered probability space as laid out in the more elementary sections of the books of Jacod [Ja79],Dellacherie-Meyer [DM75,VIII] or the notes of Meyer [Me76].
(a,F,F t s X t ,O t , P"), by
50. Local Martingales over a Right Process In order to abbreviate terminology, we shall say that a process ( M t ) t l o adapted to (Ft) is a martingale (resp., supermartingale, submartingale, local martingale) over X if for every z E E , (Mt)t>o is a martingale (resp., supermartingale, submartingale, local martingale) relative to the P").Thus ( M t ) t > is~ a local martingale filtered probability space (a, over X if and only if for each z, there exists an increasing sequence {T,} of optional times over ( F i z )such that P"{lim,T, = co} = 1 and such that for each n, the process (Mt,.,Tnl(Tn,O))trO is a right continuous, uniformly integrable martingale relative to (R, G", P"). The reducing sequence {T,} is allowed to depend on x. We aim to provide versions of the classical decompositions and constructions which do not depend on the particular P", and to keep sight of the action of the shift operators Ot. The definition of a local martingale ( M t )given above (relative to a fked P") follows that of Meyer rather than that of Jacod in that no integrability condition is imposed on Mo. In effect, one may always add an arbitrary Fo-measurable random variable to a local martingale and still have a local martingale.
cz,
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We shall use consistently the following terminology: (50.1) T denotes the class of all optional times over (3t); (50.2) if W is any process (ie., function on R+ x n), WT denotes the process W stopped at T : W T ( t , u ):= W ( tA T ( w ) , w ) ; (50.3) if W is a class of processes which is closed under stopping ( i e . , W E W and T.E T imply that W TE W), then Wlocdenotes the class of processes M for which there exists an increasing sequence {T,} C T with T, t 09 a.s. such that MTn E W for all n. Note that the sense of localization in (50.3) is more demanding than that used in the definition of local martingale over X , because no dependence of T, on x is permitted. We shall adopt the following notation for use in this chapter only.
(50.4) For M E M , M ; : = sup{lM,I:O 5 s 5 t } . By measurability of debuts (A5.1), for all t 2 0, M: is in 3. Since M; is increasing and M; = M;- V IMtl, with M:- left continuous, it follows that M* E M . If M is adapted to (3t), so is M‘ and the relation M; = M;- V lMtl shows that M’ E 0 if M E 0 and M* E P if M E P.
t
4
(50.5)
For M E M having trajectories which are rcll, M,#(w) := total variation of s --t M J w ) for 0 5 s 5 t .
+
~ t #
Since = limn &l I M ( ~ 1 / ~ -) ~ ( ~ / 2 n ) l l { k p the < ~ process }, M # is increasing, right continuous and measurable. If M E 0 (resp., P) then M # E 0 (resp., P). (50.6) V:= class of processes (At)t>O with A0 = 0 such that t + At is a s . right continuous, incriming, adapted to 3 t and A,:= supAt < 00.
(50.7)
V#: = V - V = class of right continuous processes (At) adapted to 3tsuch that A-, = 0 and A# E V. For 0 < q < 00,
V#94:= (A E V # : S U ~ P ” ( A<~00); )~
Vq := V#*‘n V;
2
V#?O0:= {A E V#: sup[Ps- esssup A%] < m}. 2
Of course, V#@ is the set of A E V# such that A g 5 c as., for some c E R. In later work, the most important of the above classes will be V#il, consisting of those A E V # whose total variations A g are integrable, uniformly in z. (50.8)
L:= local martingales over X , Lo = { M E L:MO= 0).
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235
Note that M E L if and only if M - MO E LO.As MOcan be any finite , = f (XO) a.s. for some f E E". valued random variable in 7 0 MO
(50.9)
For 1 5 q 5 00 and A4 E LO,l l M l l ~ ~is, =the Lq norm of M& relative to P", and Hq := { M E Lo:sup, IIMIIHP.I< 00).
It is easy to see that if M E Hq (C H1for every q 2 l ) , then M is a martingale over X. Since supz PxM& < 00, P"M& < 00 for every initial law p. In the language of Hardy classes of martingales, H1could be expressed as nrH1(P"),where H1(P") is the class of right continuous ~ ( R , F f , P P ) such that MO= 0 and PPM& < m. martingales ( M t ) t > over To see this, just observe that f(x) := P"M& has a finite integral relative to every p if and only if f is bounded. Note the correspondence between the terminology above and that introduced in $28. If A is adapted and satisfies (27.1) and A takes only finite values, then A E Vtocif and only if, in the sense of (28.2), A is uniformly Plocally integrable. Consequently, (31.16) may be expressed in the following way to give a fundamental "uniformization" result. (50.10) PROPOSITION. If A E Vloc (the class of finite valued, right conand if A is locally integrable tinuous, increasing processes adapted to (3t)) relative to every P", then A E V:oc. The point here is that existence for each x of a sequence TZ t 00 with P"A(TZ) < 00 for all n implies the existence of one sequence T, 7 00 with supx P"A(T,) < 00 for all n. (50.11)
For A E V :,,,
A := AP.
That is, A E VfOcn P denotes the unique right continuous increasing process which is the dual predictable projection (31.17) of A relative to every PI". One reason for using A instead of AP is that in this chapter we focus entirely on processes rather then random measures, but a more important reason is that we wish to avoid confusing the dual predictable projection operation with the superscript notation described earlier in this section. For A E Vl#,:, let A := A+ - A-, where A+ := 1/2(A# A) and A- := 1/2(A# - A). Then A+ and A- belong to VtOcand so (A+fand (A-j are defined in Vtocn P. We define 2 := (A+)-- (Ad)- E VZ: n P and call A the dual predictable projection of A, or the compensator of A. The process A - A belongs to LO.In fact, it is easy to see that A is the unique member of Vg: with this property.
+
(50.12) LEMMA.For A E V f:,
A - A E Hioc.
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PROOF:Writing A = A+ - A- as above, variation process A$ :=
I”
1 dAsl 5
1”
d(A+j8 +
A
I”
= (A+)-- (A-1 has total
d(A-js = (A#L.
E V[oc by (31.16) and (31.17). Thus (A - A)# 5 Since A# E V;,,, A# E V;,,. Since (A - A)* 5 (A - A)#, the result is established. In what follows, it is convenient to make the following convention:
+ A#
(50.13) NOTATION.For an arbitraryprocess (Mt)t>o, - Mo- := 0 and AMt(w):=
{
Mt(w) - Mt-(w)
if M t - ( u ) exists, if Mt-(w) does not exist.
(50.14) L E M M A . For ( M t ) E L, At := Co<si}
PROOF:Since M is rcll, M has only finitely many jumps of size exceeding 1 in each finite time interval. Thus A is right continuous, increasing and finite valued. Since A is adapted to (.Ft), it is enough by (50.10) to show that for each z E E , there exists an increasing sequence {T,} in T with , 00. Since we are Px{lim,Tn = ca} = 1 such that for all n, P x A ~ , < working relative to a fixed P”, we may assume that ( M t ) is uniformly integrable relative to P”, stopping M if necessary. In this case, we take T, := inf{t: IMt( 2 n or At 2 n}. On one hand, lAM(Tn)lS IM(Tn)I + IM(Tn-11
+
I )M(Tn)\+ n.
+
Therefore A(Tn) 5 A(Tn) lAM(Tn)l 5 2n IM(T,)l, so P5A(Tn) 5 2n P51M(T,)J < 00 by uniform integrability of M . Since T, obviously increases to infinity P”-as., the proof is complete.
+
(50.15) LEMMA. I f M E L and Bt := Co<sst A M s l ~ l ~ ~ then s l >N~ := ~, M - (B - B ) E L has jumps uniformly bounded by 2.
PROOF:Fix x E E . In showing that P”{IANtl 5 2 V t 2 0) = 1, we may suppose, by stopping if necessary, that M is uniformly integrable relative to P” and that, by (50.14),
c
p5
IAMtP{,*M,,>1)
< co.
t>O
The latter property shows that P”Bg < 00 so Pz@ of (39.12). We have then ANt = AM, - ABt
< 00,
as in the proof
+ ABt = AMtl{laMtIsl) + A&.
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237
The first term on the right is bounded by 1. On the other hand, A B t is the predictable projection relative to P" of
Since M is a uniformly integrable martingale relative to P", the P"predictable projection of AMt vanishes, hence ABt is the P"-predictable projection of -AMtl{laMtII1). From this, the result is clear. (50.16) THEOREM. Every M E L has a decomposition (not necessarily unique) of the form Mt = f(&) Nt Ct where: (i) f E E"; (ii) N E HEc has uniformly bounded jumps; (iii) c E nH : ~ ~ .
+ +
vE:
PROOF:Since MOE Fo, there exists f E E" with MO = f(X0). Supposing now that MO= 0, form B as in (50.15) and let C := B - B and N := M - C . We showed in (50.15) that ( A N t (5 2 for all t 2 0 a s . and N E L vanishes at 0. Set T, = inf{t: INt[ 2 n}. Then T, t 00 as. and NTn is uniformly bounded by n + 2. Since NT" E LO,it follows that NTn E H", proving (ii). Assertion (iii) is a direct consequence of (39.12). Theorem (50.16) is the Markovian version of the fundamental lemma on local martingales [DM75, VI-851. The notation in (50.16) hides to some extent the force of the theorem. In words, we have shown that if MO = 0 and M is a local martingale relative to every P" (with reducing sequence perhaps depending on x) then M decomposes independently of x into the sum of two local martingales, the first of which can be reduced independently of x to a bounded martingale, and the second of which can be reduced independently of x to a martingale with expected total variation uniformly bounded in x. The following is an obvious consequence of these facts. (50.17) COROLLARY. LO= Htoc. In particular, if M E LO,there exists a reducing sequence for M independent of x, and M is a Pp-local martingale for every initial law p. In (29.1) the dual shift 6t was defined on random measures. It is convenient to extend its action to an arbitrary process ( Z t ) by defining
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238
(50.19) PROPOSITION. If M E H1 (resp., L) and if T is an optional time, then &M E H’ (resp., L).
PROOF:If M E H’ then MO = 0, so &M = O T M and M = ‘2, where Z ( t , w ) := Mm(w). Then (22.15) shows that @TM is the optional projection of @ ~ Z ( t , u=)Mw(&w)lp(w),m((t).Since [ T ,001 E 8,then if N is } ,have OTM = Nl(iT,wi. the martingale with final value M , o t ? ~ l { ~ < ~we But, for all z E E ,
Thus NtllT,w[( t ) = Nt - & A T
is a martingale. In addition
is bounded in z,so &M E H1. If M E L,then N = M - MOE LO = Htoc and &M = O T N . Choose {T,} C T reducing N so that for all n, N stopped at T,, is in H1. Set S, := T +T,o&, so that {Sn} is an increasing sequence in T with P”{S, t co} = 1 for all 2, and
This shows that O T N E H:oc = Lo. REMARK.The last part of the above proof shows the identity (50.20)
@ T ( z R )= (@TZ)T+RoeT
for all functions 2 on R+ x R and all functions R, T :R be useful later.
+
[0, m]. This will
VI: Stochastic Calculus
239
51. Decomposition Theorems The results of $34 on representation of potentials and left potentials lead to two important decomposition theorems. The first of these is the Markovian version of the renowned Doob-Meyer decomposition theorem. Let Y be a right continuous supermartingale over X. (51.1) THEOREM. Then there exists a unique M E LO and a unique A E P r l Vtoc such that
Y=Yo+M-A. PROOF: Let T, := (inf{t:K 2 n } ) A n. Then {T,} is an increasing sequence of bounded optional times and P"{Tn t 00) = 1 for all z E E. Let Y" denote YTn, the process Y stopped at T,. Then y;l" 2 P"{Yn 1 Ft} for all t 2 0. That is, Y" is bounded below by a P"-uniformly integrable V-291 implies martingale. A standard result on supermartingales [DM75, that P"(Y"(T,)I < 00 for all x E E . Since Y" is bounded above by the P"-integrable random variable n V ~YT,1, it follows that Y n is of class (D) relative to every P". Let M: denote a right continuous version, simultaneously for all x, of the martingale P"{Yn 1 Ft}.(Just take M nto be the optional projection of the process ( t , w ) -+ YT~(w).)Obviously Y" - M" is a potential of class (D) in the sense of (33.1). According to (34.1) there exists a unique A" E P n V ' having potential Y n- M". We employ now an uncovering argument to find an A E P n V;,c such that for all n, At = A: for all t 5 T,. We must show first that A:+' = A: for all t 5 T,. This is an immediate consequence of the uniqueness of An and the fact that
yt"
+
+
=T AG: AZ"/\'T', = (Yn+l
+ An+l)tA~,
is a P5-uniformly integrable martingale for all z E E . Obviously then At := supn A: E P n Vkc and Y A - YOE LO.The uniqueness of A is evident from (34.1). (51.2) REMARK. Since no assumption was made concerning the sign of Y in (51.1), the submartingale version of (51.1) is also valid, replacing Y in the above results by -Y. In the next theorem, which is the left version of (51.1), the hypothesis on the paths on Z is in fact automatically satisfied, but the proof that this is so is not simple. See [DM75, VIII(Appendice1)-41. For the application we have in mind, this path regularity hypothesis is obviously satisfied.
+
Let Z be a regular strong supermartingale whose paths (51.3) THEOREM. have left and right limits everywhere on R+. Then there exists a unique M E LO and a unique right continuous optional increasing process (possibly with A0 > 0) such that Z = Zo M - A _ .
+
PROOF:For uniqueness, if Z = ZO + M - A- = ZO+ N - B-, then M - N = A- - B- is necessarily continuous and in P n V# n LO.Since a
240
Markov Processes
continuous martingale of locally bounded variation is necessarily constant in time, it follows that M - N = 0. For existence, let T, := (inf{t: Zt 2 n } ) A n, as in the proof of (51.1). Since T, is a bounded optional time, Pz([Z(Tn) A 01) < XI by the definition (33.1) of strong supermartingales, and then since P”Z(Tn) 5 P”Z0 < 00, it follows that P”IZ(T,)I < 00. But then Z(tAT,) is bounded above by the P”-integrable random variable is obviously bounded below by n V IZ(Tn)l.On the other hand, the Pz-uniformly integrable martingale M” with final value Z(T,). This is a consequence of the strong supermartingale property and the section theorem. Consequently 27 := Z(tAT,) is a strong supermartingale of class (D), and a simple calculation shows that 2” is regular. Since M nis regular, it follows that Zn - M n is a regular left potential of class (D), and (34.1) gives us the unique optional increasing process An having left potential Zn- M”. In other words, Z? - M r A:- is a right continuous version of the martingale over X with final value A,. I claim that A:+’ = A: for all t < T,, for, arguing exactly as in the proof of (51.1), we obtain A”+1 = A:- for all t 2 0. Defining At := supn A:ltt
+
+
(51.6) DEFINITION. A process W has homogeneous increments provided, for every optional time T , &W and (W - W T ) l ~ T , mare ~ indistinguishable. One checks easily that if W,+t - Wt = ( W .- W o ) o & identically, then W has homogeneous increments. For examples of processes with homogeneous
VI: Stochastic Calculus
241
increments, take any process W which is homogeneous on R+,or any AF. Note that if WO= 0, then W has homogeneous increments if and only if, for T E T,W + T 1 ( T < m } - (WT + WtOW1{T
PROOF:By the remarks (51.5), for any optional time T, &Y = &M &A is the Doob-Meyer decomposition of &Y. The homogeneity of increments of Y gives &M - &A = ( M - MT)lfT,mi - ( A - AT)llT,oon. But
t
(At - AT)lUT,cz,U(t)=
J l ] T , m [ WdAu 0
is therefore in P fl vtocsince lnT,,[ E P, and (Mt - MT)ljlT,oou(t)= Mt - MtAT is in LO. By uniqueness of the decomposition, &A = ( A AT)l[T,mu,and since A0 = 0, this proves that A is an AF. In the strong supermartingale case, the analogous argument gives instead &(A-) = ( A - - AT-)laT,mn, which is equivalent to the LAF condition (35.6). As an example of (51.7), consider the case yt = f(Xt) with f a finite valued function in Eu such that f ( X ) is a right continuous supermartingale in the sense defined in $50. (If f 2 0, this condition is equivalent to f being a finite valued excessive function.) Then we may decompose f(X)as (51.8)
f ( X t ) = f(X0) + Mt - At
in which A is a predictable AF of X and M E LO has homogeneous increments. This result permits a decomposition of an excessive function into a potential part and a locally harmonic part. This result is not as fine as the classical Riesz decomposition of a superharmonic function into the potential of a measure and a harmonic function, for in the general setting, the term that corresponds to the potential of a measure carried by a polar set cannot be represented as the potential of an AF. (51.9) DEFINITION.An excessive function f is locally harmonic (with poles) if for all z E E (except for x in a polar set), f(X,) is a P”-local martingale.
(51.10) THEOREM.Let f E S and suppose {f = co} is polar. Then there exists a unique function h, locally harmonic with poles, and a unique potential function u of class (0)with poles (see (38.3)) such that f = h+u.
242
Markov Processes
PROOF: Using the procedures of (38.4), one may delete from E the polar set {f = 00) and prove (51.10) assuming that f is finite. Having achieved a unique decomposition f = h u on { f < oo}, h and u extend to E in a unique way (12.31) to give the desired decomposition of f . Applying (51.8) to f , we obtain f(Xt) - f(X0)= Mt - At where A is a predictable AF and Mt is a P"-local martingale for every z and MO = 0 a s . Let T, := inf{t: f ( X t ) 2 n } A n. As we observed in the proof of (51.1), Mt is reduced by the times T, for all P" simultaneously. Thus
+
P"f(X(T'))
- f(z) = PSM(Tn)- P"A(Tn) = -P"A(T').
) PZA, 5 f(z). Hence, for all n, P"A(T,) 5 f ( z ) so u ( z ) = u ~ ( z = Let h ( z ) := f(z)- u(z). Then h ( X t ) is a.s. right continuous and h(&) h(X0) = Mt -At - u ( X t )+ u ( X o ) is a P"-local martingale for all 2 because At u ( X t ) is, for all z 6 E , a version of the P"-uniformly integrable martingale P"{A, I Ft}.Thus h ( X t ) is a P"-local martingale for all z E E . Since h 2 0, Fatou's lemma implies that h ( X t ) is a P"-supermartingale for all z, hence h is excessive.
+
(51.11)
EXERCISE. Let X be Brownian motion in Rd(d 2 3) and let
JzI-(~-~). Show that f is locally harmonic with a pole at (0). (Hint: the It6 calculus is the most direct way, but a purely Markovian method can be described as follows: use (4.10) without further explicit calculations to show that as E 110, if w ( ~ ) is the volume in Rd of the ball B, with center a t 0 and radius E , then u(1& / v ( E ) ) (t~f(z). ) Fix 6 > 0 and z) let T := inf{t: lXtl 5 6). Show that if E < 6 < 1x1,l ' ~ U ( l ~ , / v ( ~ ) ) (= U(l&/W(€))(z) and conclude that P~f(x) = f(z). The terminal time property of T shows that f ( X t h ~is) a P"-martingale if 1x1 > 6. As 6 1 0 , T = 7 0;) a.s. because (0) is polar.) f(z) :=
Theorems (51.3) and (51.7) have important consequences in the stochastic calculus over X . The following result is the basis for the entire theory. It is possible to give a proof which is shorter but uses more of the general theory and the methods of the next section. This was the approach in [CJPSSO]. The proof given below is based on the same idea used in a proof in [DM75, VII-42 bzs] but is even more direct in that it does not even use the fundamental lemma on local martingales. In what follows, AM: means, as usual, (AMt)2 and not A(M$). (51.12) THEOREM. Let M E L. Then: is locally a regular strong supermart(i) Z t : = AM: - (Mt ingale over X; (ii) there exists a unique process in VlOc,denoted by [ M , M ] ,such that M 2 - [ M ,M ] E L and A[M, MI = AM2;
VI: Stochastic Calculus
243
(iii) for T E T,
[ M T ,M T ] = [ M ,M y ; [ M - M T , M - M T ] = [ M ,M ] - [ M ,M y ;
[&M, &MI = & [ M , MI; (iv) if M E
L has homogeneous increments then [ M ,MI is an AF.
PROOF:Uniqueness of [ M ,MI follows from the observation that M2 - A E L and M2-B E L, with A , B E LlOcand AA = AB = AM2, imply A - B E L is continuous and of locally finite variation, hence A - B = 0. Once (i) has been proven, the remaining assertions are obtained as follows. First of all, (51.3) permits us to write 2 in the form 2 = N - A- with N E L and A right continuous, increasing and adapted to (3t). Since AM = M - Msatisfies (AM)- = (AM)+ = 0, it follows that ( A - AM2)+ = A+ = A. On the other hand, M 2 - ( A - AM2) = - Z - A- = - N belongs to L and consequently AAM2 = ( A - AM2)+ = A. Setting [ M ,MI := A = A- AM2 yields (ii). The first two assertions in (iii) are simple consequences of uniqueness of [ M ,M ] and the fact that MfAT - [ M ,M ] t A T E L. To obtain the shift identity let N := &M and observe that
+
+
+
belongs to L since, by (50.19), &M
+
+
E
L vanishes on [O,T[ and Moo& E
FT.It then follows from (51.7) that [ N ,N ] = & [ M , MI. Assertion (iv) is an obvious consequence of (iii). It remains only to prove (i). For this we may assume MO = 0 and then, by localization, that M E H'. Now let Tn := inf{t:)Mtl 2 n } so that P"{T" T a}= 1 for all 2 E E. Observe that
Z ( t AT") = AM2(t A T " ) - M2(t A 2'") = M ( t A T " - ) [ M ( t A Tn-)- 2M(t A T " ) ]
+
is dominated by n(n 2M:). Fix k 2 1 and define a process Yk, indexed by (0, 1/2k, 2/2k,. . . }, by setting Y$ := 0, and for j 2 1
Markov Processes
244
Set now
j / 2 & on {(j- 1)/2& < T"
TF :=
i
0 00
I j/2'},
on {T" = O } , on{T"=oo}.
Then T; is a discrete parameter optional time over ( 3 ( j / 2 ' ) ) ) j 2 0 .Consider now the discrete parameter process j / 2 & Y k ( j / 2 ' )stopped at TF. Observe that IYk(j.2-kA TF)II n[n 2M&], --$
+
and since M E H1, Y k stopped at TF is uniformly integrable relative to every P".It is clear that Y k ( j / Z k )has a conditional expectation relative to ( P " , F ( ( j- 1)/2')), and
This proves that Y k ( j . 2 - &A TF) is a (P",F ( j / 2 & ) )supermartingale ) of class (D). Therefore if R I S are bounded optional times over (Ft) with kth order dyadic approximants Rk, Sk, then for all x E E ,
P"Yk(Rk A Tt) 2 PzYk(SkA TF). Now let k + 00. By the definitions of Rk,Sk,T$, it follows that Rk 1 R and Rk - 2-& T R, and similarly for Sk and TF. Consequently Y k ( R kAT;) converges to Z ( R A T") and Y k ( & A TF) converges to Z(S A T"). Using the domination properties established earlier, we conclude that
P"Z(R A T") 2 P z Z ( S A 2'"). This implies, in light of (33.10),that ZT" is a strong supermartingale. It remains only to prove that ZT" is regular. To this effect, let ~k T be uniformly bounded optional times and set r := {T& < T V k } . Then since ( A M ) - = 0, as k -, 00 we get
Consequently
+
A M ' ( T ~A Tn) [ M ( TA T") - M ( T AT")]' ~
+ AM'(T A
T")
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and therefore Z(Q A Tn) - Z ( T A T") - ~ M ( Q A T " ) [ M ( TA~T " ) - M ( T A T " ) ]
converges a s . to 0 as k -+ 00. The first two terms above are dominated by n(n 2 M z ) and the last term is also dominated by an integrable random ~ 5 n, and the other factor variable, for on {Q ATn < T A T " } , [ M ( TAT")( is bounded by 2ML. Taking expectations and noting that the expectation ) k + 00, of the last term is zero, we find P Z 2 ( ~ ~ A T n ) - P Z 2 ( ~ A-+ T0nas proving regularity of z ~ " .
+
The process [ M ,M] constructed in (51.12) is not in general locally integrable. It is easy to see that [ M ,MI E Vtoc if and only if M - M OE Hfo,, and in this case one defines ( M ,M ) := [ M ,MI; the compensator of [ M ,MI. Since [ M ,M ]- ( M ,M ) E LO,it follows that for every M E HFocthere exists a unique process ( M ,M ) E P n Vfocsuch that M 2- ( M ,M ) E L. The following result follows immediately from (51.12) and properties of dual predict able project ions. (51.13) EXERCISE.Let M E Hfoc and let T be an optional time. Then &M E Hfo, and (&MI & M ) = & ( M , M ) .
(e+)
(51.14) EXERCISE.Show that i f M E L is adapted to (resp., (Fz+)) then [ M ,MI may be chosen adapted to (resp., (F:+)). [Hint: check that all reducing times involved in the proof of (51.12) are optional times and then use (34.7).] over (Fy+),
(e+)
As usual, [ M ,MI extends by the polar identity 4 [ M , N ]= [M
+ N , M + N ] - [M - N , M - N ]
to a Vt,-valued bilinear functional on L x L. We say that M , N E L are orthogonal if M N E L, and this is the case if and only if [ M , N ]E LO. For M , N E Hf,,, ( M ,N ) is defined by polarization, and it then follows that M , N are orthogonal if and only if ( M ,N ) = 0. It is easy to see that [ M ,N ] is the unique process in Vt, such that (51.15)
A [ M ,N ] = A M A N
and M N - [ M ,N ] E L.
One of the basic properties of [ M ,N ] is that it has a simple expression if one factor is locally of finite variation. The following is proved, for example, VII-371. in [DM75,
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Markov Processes
(51.16) PROPOSITION. If M, N E L and if M - MO E V& then
[ M , N ] t=
AMuANu. o
Both here and in the next section we shall make considerable use of the following lemma. The first result of this type was obtained by DolBansDade [DD67],and the form below is essentially due to Stricker-Yor [SY78], with proof following [Me78b]. (51.17) LEMMA.Let ( Y Z ) z be E~ a family ofprocesses such that: (i) for every x E E, P"{t + yt" is right continuous} = 1; (ii) there exists a sequence ( 2 " ) of processes adapted to ( 3 t ) such that for all t 2 0 and all x E E, 2; converges to Y: in Pxprobability a n + 00. Then there exists a right continuous process Y adapted to ( 3 t ) such that Y - Y" E 1'2 for all x E E .
PROOF:Fix t 2 0. Let no(x) := 1, and define nk(x) recursively by
Because of condition (ii), nk(x) < 00 for all k 2 1 and all x E E. For ease of typography, we now write Zt(n,w)for Zr(w).Observe that for each k, n k is E" measurable, so (2, w) + Zt(nk(z),w) is in E" 8 3 t . It follows that if we let Wt(w) := liminfk Zt(nk(x),w),then ( z , w ) + Wt(w)is in E" 8 3t.Since
the Borel-Cantelli lemma implies that for all x E E,
On the other hand, nk(z) + 00 as k + 00 so by (ii), P"{Wt = yt") = 1 for all x E E and all t 2 0. Now set Wt(w) := Wp'"'(w). By composition, w + Wt(w) is 3t-measurable and P"{Wt = W t ) = 1 for all x E E. Therefore P"{Wt = yt") = 1V x E E,Vt 2 0. Denote by A the set of w E R for which the restriction o f t + Wt(w)to &+ has right limits everywhere on R+.Then (i) implies that P"(A)= 1 for all x. Setting then
yt(w):= I A ( ~ )liminf Ws(w), sllt,sEQ
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we obtain P"{Y, = yi" for all t 2 0) = 1. (51.18) REMARK. Exercising a little more care in the proof above, it can be (resp., (3:+)), shown that if the processes Y" and Zn are adapted to then Y may be chosen adapted to (resp., (F:+)). See [CJPS80]. As the first application of (51.17) we consider the decomposition of a local martingale into a continuous part and a compensated sum of jumps. Specifically, define Lc,Ld by: (51.19i) Lc := { M E LO:t -+Mt is continuous a.s.}; (ii) Ld := { M E LO:[ M ,MI is purely discontinuous}. Recall that [ M , M ]being purely discontinuous means that [M,M] is the sum of its jumps. That is,
(e+)
(e+)
There are many equivalent formulations of the condition (51.19ii) for M to be a compensated sum of jumps (ie., belong to Ld).For example M E Ld if and only if M E LO and [ M ,N ] = 0 for all N E Lc.See [DM75, VIII-43, 441, where it is shown also that for every 5,there exists a decomposition of an M E LO into a sum Y" 2" with Y" a continuous local martingale relative to P" and 2" a compensated sum of jumps relative to P", where 2" is determined as follows. Let
+
(51.20)
A? =
AMal{lAM,I>l/n) O<s
be the sum of jumps of M of size exceeding l/n. Then Mp: = AT - A? is the compensated sum of such jumps. There exists then a sequence {Tk} of optional times with P"{Tk T 00) = 1 such that for all t 2 0, ic 2 1
PZ{IZx(tA Tk) - M n ( t A Tk)I}
--$
0 as n --*
00.
It follows that for every optional time S, (51.21)
22 is the limit in P"-probability of M," as n
4
00.
Seeing that the compensator An of A" is defined without reference to the particular P", we may apply (51.17) to establish the following Markovian reformulation of (51.21).
For every M E L, there exists M d E Ld such that (51.22) PROPOSITION. for S E T,M," converges to M,d in Px-probability for every x E E . This gives the first of the following assertions.
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Markov Processes
For every M E L there is a unique decomposition (51.23) THEOREM. M = Mo + M C+ M d with M C E Lcand M d E Ld. In addition, for T E T,
(&M)" = &(MC)
and ( ~ > T M =& ) (~M d ) .
In particular, if M has homogeneous increments, then so do M" and M d .
PROOF:If M n = A" - A n as after (51.20), it is clear that &A" is the sum of jumps corresponding to &M. From (31.7) it follows that the compensator of &A" is just &A". Therefore (&M)" = &p(M"). The remaining assertions of (51.23) are easy consequences of this equality together with (51.22). (51.24) PROPOSITION. If M E L,then all discontinuities oft + Mt occur at times of (Ray) discontinuity oft + Xt. in symbols, AM C J U J B .
PROOF: Because of (50.17), we may as well suppose M E H'. We prove that A M ~ KE 1,where K:= {X = X-} as in (44.1). By (44.3ii), = P(AM)lK = 0 because p(AM) = PM - M- = 0. AM~K (51.25) COROLLARY. If X is Ray continuous, then every local martingale over X is continuous. Using (34.7) (resp., (34.8)) show that if the process Y (51.26) EXERCISE. in (51.3) (resp., (51.1)) is adapted to (*+), then one may choose M and A to be adapted to (*+). The same result holds for the filtration (3:+).
(c+)
(51.27) EXERCISE. Show that in (51.231, if M is adapted to then M d (and hence M" also) may be chosen adapted to (3$'+), and similarly for (3:+). [Hint: you will need to assume what is mentioned without proof in (51.18), as well as the predictable version of (31.131.1
VI: Stochastic Calculus
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52. Semimartingales and Stochastic Integrals
Given an arbitrary filtered probability space (a,Bt, P ) satisfying the usual hypothesis, a right continuous process Y adapted to (Bt) is called a semimartingale if for some local martingale M , Y - M is locally of finite variation. A semimartingale Y is called special if its maximal process Y: (:= sup {/&I: 0 5 s 5 t } ) is locally integrable. For ease of reference later in this section we recall some facts about these objects. All unidentified references below are to the second volume of [DM75]. (52.1) There exists a unique symmetric bilinear form ( Y , Z ) -+ [Y,Z]on the class of semimartingales over (a,G t , P ) , taking values in the space of right continuous adapted processes of locally finite variation, vanishing at t = 0, such that: (52.2i) if Y, 2 are local martingales then Y Z - [Y,Z] is a local martingale and A[Y,Z] = AYAZ; (ii) if at least one of Y ,2 is of locally finite variation then
[Y,Z]t=
C
AY,AZ,.
O<s
See [VII-441. Actually there is a slight difference between the definition above and that of [VII-441 because we do not count jumps of Y or Z at 0. (52.3) If Y and Z are semimartingales, then by [VIII-201, for every t 2 0 , [Y,Z], is the limit in probability as n + 00 of the variational sums
c
[Y((k+ 1 ) P ) - Y ( k P ) l [ Z ( ( k + 1 ) P )- ~(k/2*)11{k/Z"
k>O
(52.4) According to [VII-251, the following conditions on a semimartingale Y are equivalent: (52.43) Y is special; (ii) (AY)*is locally integrable; (iii) Y has a decomposition Y = M + A with M a local martingale and A predictable and of locally finite variation. The process A is uniquely determined, and the decomposition above is then called canonical. Note that by (51.1), every right continuous supermartingale over X is a special semimartingale. (52.5) Denote by P, the class of elementary predictable processesvit, the class of processes of the form
(52.6)
ct := Ho l]o,tl](t) + . . - + H n l ] t n , w [ ( q ,
250
Markov Processes
-
5 t, < oa and H j E bBt, for every j. Given C of the with 0 = to 5 t l 5 form (52.6) and a semimartingale Y , define the elementary stochastic integral C - Y of C relative to Y by -
a
(52.7) ( C = Y ) t:= HO(%htl
- YO)-k
- ''
-k
Hn-l(%At,
- %At,-l)
+ Ha(% -
%At,).
It is shown in [VIII-31 that for a fixed semimartingale Y ,the mapping C -+ C -Y of P, into the class of semimartingales has a unique extension to a mapping, also denoted C + C . Y , of b P into semimartingales, satisfying the conditions: (52.8) C + C - Y is linear; (52.9) if {Cn}c b P are uniformly bounded and converge pointwise to 0 then ( C n . Y ) t-+ 0 in probability for every t 1 0.
s,"
The process ( C - Y ) t ,also denoted C, dY,, is likewise called the stochastic integral of C relative to Y . There is an obvious extension of the stochastic integral to predictable processes C which are only locally bounded. Extension to even larger classes of integrands are discussed in [VIII] and in [Ja79]. We shall not go into these further extensions here, as the locally bounded case gives sufficient generality for our purposes. If Y = M A with M a local martingale and A of locally finite variation, it can be proved [VIII-31 that for a locally bounded C , C Y = C M C A where ( C - A ) tis the ordinary Stieltjes integral JiC, dA, and C - M is the unique local martingale vanishing at 0 satisfying
+
(52.10)
.
9
+
-
[CmM,N] = C * [ M , N ]
for every local martingale N . We set out now to obtain Markovian versions of the above results, and in this project we shall make repeated use of (51.17).
A process Y is a (special) semimartingale over X (52.11) DEFINITION. provided Y is a (special) semimartingale relative to (a, P") for every x E E . Let S (resp., S,) denote the class of semimartingales (resp., special semimartingales) over X .
e,
(52.12) THEOREM. For all Y,Z E S there exists a unique process [Y,Z ] E V t c such that for every P",[Y,Z] is a version of the bilinear form having P"). the properties described in (52.1) relative to (n, PROOF:Obvious from (52.3) and (51.17). (52.13) REMARK.This result gives a much shorter proof of (51.12ii). It
c",
does, of course, use much heavier machinery. We shall examine the shift properties of (Y,2)+ [Y,Z] in (52.17).
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251
(52.14) LEMMA.Let Y E S p be uniformly bounded and have continuous paths. Suppose in addition that [Y, Y] is uniformly bounded. Then Y has a unique decomposition Y = YO M A with A E P n V& and M E LO both continuous. In addition, supt,+ P5(Atl < 00.
+ +
PROOF: For each x E E , Y has a unique decomposition as M"+A" relative to (52,3~",P"),as in (52.4iii). It follows that PY = M? A" (up to P"evanescence), so PY - Y- = AA", once again up to P"-evanescence. As Y is continuous, this proves that P"-a.s., A" is continuous. By the properties .] discussed in (52.1), it follows that the process [Y, Y ] constructed of in (52.12) is a version of [M",M"] relative to Px. It follows that Px(M,")2 is uniformly bounded and so therefore is P"IAT1. Define now a signed measure Q; on (Q, 3)by
+
[ a ,
(52.15)
Q r ( H ) := P"(AFH),
H E b3.
Because AT E 32, if we let Ht denote the martingale " ( H 8 l 1 0 , ~ " ) t (= P"( H I 3 t ) for every x E E ) , we get
Q r ( H ) = P"(A:Ht) = P"(y,Ht) - P"(Y0Ht) - P"(MTH,) = P"(Y,Ht) - P"(Y0Ht) - P"[Y, H]t since, P"-as., [Y,H ] t = [M",H ] t . Therefore
Q ? ( H ) = P"((& - Yo)H - P"[Y,H]t). It follows from this that 5 --,Q f ( H ) is €"-measurable for every H E b 3 . By a now familiar Moreover, Q" << P" on the separable a-algebra (a,9). argument based on Doob's lemma, we may choose Bt E Ft such that Q T ( H ) = P"(BtH) for every H E b p . This implies Px(AT = Bt) = 1 for all x and t , hence that with P"-probability 1, the restriction o f t + Bt to &+ has right and left limits everywhere. Define then At := lim,lit,sEQ+B,. Then Px{A; = At 'dt 2 0) = 1, and A is the member of Vlocwe set out to construct.
Let Y E S,. Then: (52.16) THEOREM.
+ +
(i) Y has a unique canonical decomposition Y = YO M A, independent of P", with M E LO and A E P n V c:; (ii) Y is a special semimartingale relative to (R,.Fr,Pp) for every initial law p; (iii) for T E T, &Y E S p and &Y has canonical decomposition &Y = &M +&A.
252
Markov Processes
+
PROOF: We may suppose YO= 0. For each z, let Y = M" A" be the P"-canonical decomposition of Y . Just as in the proof of (52.14), PY - Yis P"-indistinguishable from AA". If we set o<sg
then, using (52.4ii), we find that 2 E S , and so Y - Z E S,. Moreover, : be the continuous part of A t , then Y - Z has P"-canonical if we let 3 BF. Since B" is continuous, this shows also that decomposition Mf AY - AZ (= Y - PY) is Px-indistinguishable from AM". Using the description in (51.20) and (51.21) for the compensated sum of jumps part of M", it follows by an application of (51.17) again that there exists N E Ld such that M" - N is continuous PX-a.s.for every z. Now consider the continuous process Y - Z - N E S,. By an obvious localization argument, it follows from (52.14) that Y - Z - N has a canonical decomposition independent of z, and so therefore does Y . This proves (i). Part (ii) is an obvious consequence of (i), (50.17) and (50.10), and (iii) comes immediately from (50.19) and the fact that 6~ preserves P n V f f .
+
(52.17) THEOREM. Let Y E S. Then: (i) Y has a decomposition (not necessarily unique) independent of P" as a sum Yo+M+A with M E LOand A E VEc. In particular, Y is a semimartingale relative to (fl,F:,Pp) for all p; (ii) the continuous martingale part Y c of Y can be defined independently of the initial law; (iii) for every optional time T, &Y E S , (&Y)' = &(YC) and &[Y, Y ] = [&-Y,.&Y]; (iv) for every locally bounded C E P one may define a process C Y E S which is, for every initial law p, a version of the stochastic integral of C relative to Y over (fl,F;,Pp); (v) for every optional time T and locally bounded C E P,
-
OTC'6)TY = G)T(C'Y). REMARK.The appropriate class of integrands in these stochastic integrals is not really (bP)loc, but (bPt)loc, the locally bounded, integrally predictable processes discussed in $32. Thus, for example, if F E 3*and if fl has killing operators, Ct := F o k t is permitted to be the integrand in a stochastic integral. PROOF:We begin by stripping Y of its large jumps, setting
Zt :=
Ay8
o<eg
'{lAYs1>I}*
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253
It is clear that 2 E VZCand that lA(Y-2)l 5 1. Consequently Y - 2 E S,, and together with (52.16) this gives the first part of (i). The second part of (i) follows then by defining Y c to be M " , where M" E L" as defined in (51.23). With Z as above, it is clear that (Y - 2)' = Y",and because Y - 2 E S,, (52.16) gives us a canonical decomposition Y - 2 = M + B . As we noted in (52.16), &(Y - 2 ) has canonical decomposition &M+&B. The first two parts of (iii) follow directly from this, making use of (51.23). Y]t may be rewritten as [Y",Y"], '&, E} + 0 by bounded convergence. This proves (iv). In order to prove (v), it is convenient to change slightly the basic class of integrands to that of sample predictable process. These are just processes having the form (52.6) with t o 5 ... 5 t, optional times instead of constant times. It is easy to verify that the stochastic integral is still given by (52.7). If C is a simple predictable process, it is easy to see that OTC is also a simple predictable process, and a simple calculation establishes &[C.Y] = @ T C ~ & - Y .A monotone class argument as above completes the proof of (vi).
+
.
-
-
Let Y E S have homogeneous increments (51.6) and (52.18) THEOREM. let C E (bP),,, be homogeneous. Then: (i) [Y,Y ]is an AF; (ii) we may write Y = YO M + A where M E LO and A E VZc have homogeneous increments; (iii) the continuous martingale part Y" of Y has homogeneous increments; (iv) C Y has homogeneous increments;
+
.
254
Markov Processes
+ + A , then M
(v) if Y E Sp has canonical decomposition Y = YO M and A have homogeneous increments.
PROOF:Parts (i), (iii) and (iv) follow immediately from (52.17), as does AY,, which clearly (v) from (52.16). To get (ii) from (v), let Zt := CO<s
(53.1) DEFINITION.A process Y is a natural potential provided it is a potential over X, and (53.2)
T,, E T, T,
t T 2 C implies P'Y,,
+0
for all x E E .
In view of (33.3), every natural potential for X is also a potential of class (D) for X. The converse is not generally true. For example, if 6 is predictable, the potential Y of q is a potential, but (53.1) fails when we take (T,,) to be an announcing sequence for 5. (53.3) DEFINITION. A function u E S" is a natural a-potential function for X provided yt := e-%(Xt) is a natural potential for X. (53.4) THEOREM. Thepotential Y ofan integrableRMK carried by 10, is a natural potential.
PROOF:By (33.9), Y is a potential of class (D) over X. If Tn t T 2 then limn P"YT~= limn P"K(IT,,, ([I ) 5 P"K([T,C[ ) = 0.
c[ C,
(53.5) THEOREM. Let Y be a natural potential for X. Then there exists a unique natural, integrable RM K havingpotential Y . Given such a K and T E T, QTK is the unique natural RM having potential OTY on [ T ,001. in particular, given u,a natural a-potential function for X, there exists a unique natural AF A of X with u = u z . PROOF: This is all fairly straightforward using (34.5), (48.4) and the results of 5A6. As in 548, let ).( := & l ~ ( zand ) A := {(t,u): .(Xi-(w) > 0). Let y denote the unique predictable RM having potential Y. Now, C 6 is predictable for every 6 > 0, and condition (53.2) implies that y 6,00[. Thus y is carried by I] 0, ( J , and if T is a does not charge predictable time with T 2 C, (53.2) shows that y does not charge [ T I ] . As DO,<[ c A c lO,CI1 by (A6.15), it follows that y doesn't charge the predictable set 10, Cl \ A , which must be the graph of a predictable time by (A5.5~).Thus y is carried by A. Set K := yn, the dual natural projection of y. By (A6.25ii), K and yp have the same potential Y.Uniqueness of K is a
+
+
VI: Stochastic Calculus
255
direct consequence of (A6.33). The second assertion is a direct consequence of (33.13) and the first assertion. For the third, let yt := e-%(Xt) and let n be the unique natural integrable RM with potential Y . The second hence that it is generated assertion implies that n is homogeneous on R++, by to a natural AF, as claimed.
(53.6) THEOREM. Let Z be a right continuous supermartingale over X with Zt = 2, a s . for all t 2 ('. Suppose that T, t T, T, E T,T 2 (' P"ZT,, + P"Z,. Then there exists a unique M E Lc vanishing at 0 and a unique A E P,, locally integrable on [O,([I, such that 2 = ZO M - A . For every T E T, OTZ = O T Z ~ OTM - O T A is the Doob-Meyer decomposition of O T Z . In particular, if 2 has homogeneous , ~( Z ) - ZT)luT,cU E 2, increments on [IO,('[I in the sense that O T Z ~ ~ T then M has homogeneous increments on QO,<[I and A is a natural AF of X.
*
+
+
PROOF: The first part follows from the decomposition (51.1) in the same way that (A6.36) follows from the usual Doob-Meyer decomposition, and the second comes just as in (51.5) and (51.7). We shall not go into details. Remark. There is a finite lifetime version of the stochastic integral parallel to the discussion in $51 and $52. The details are tedious but fairly routine, given the discussion at the beginning of this section and the general theory in $A6. In brief, local martingales and semimartingales are defined on [YO, ('[I just as at the end of $A6, but making use of the special versions of conditional expectations and projections available in the Markovian case. For Z a locally L2 martingale on [IO,(' [I, define the unique natural increasing process ( 2 , Z ) such that Z2 - (2,Z ) is a local martingale on [YO, ('[I. Define then [Z, 21 for 2 a semimartingale on 10, ('[I as in $51. The only real difference will be that [Z, Z]t may explode as t C. Proceed then as in $52 with the construction of the stochastic integral. The only changes needed in the modified statements of the results of $52 are that natural replaces predictable everywhere, and assertions of homogeneity must be replaced by homogeneity on [I 0, ('[I. The definitions (36.7) of potential functions of class (D) and of regular potential functions depended on knowledge of their expected values at all optional times. This is not a completely natural state of affairs, particularly as Fitzsimmons' extension of Dynkin's theorem (10.33) shows that excessive functions may be defined just in terms of their expected values at hitting times. We finish this section with a brief discussion of the corresponding situation for potential functions.
(53.7) THEOREM. Let f E S" be finite-valued, p an initial law with 00. Then there exists a natural AF, A, such that f(X) = uz(X)
1f dp <
256
Markov Processes
u p to Pp-evanescence, with u2 5 f everywhere, if and only if: (i) PFf + 0 p-a.e. as t + 00, and (ii) given a decreasing sequence (D,)of finely closed sets in E“ with hitting times T, := T D ~satisfying limT, 2 C Pp-as., one has Pp{e-OTn f (XTn)}-+ o as n -+ 00.
PROOF:The necessity is immediate. For the converse, we construct a natural AF, A, such that yt := e-at f (Xt) is the potential of e-a3 dA, relative to P p , in the sense of the general theory of processes. This gives the result, except that uQA5 f need not hold. However, every a-excessive function dominated by a natural a-potential function is also a natural apotential function, and so f A u; is the a-potential of a natural AF which will satisfy all requirements of the theorem. First of all, we shall show that the right continuous supermartingale Y is of class (D) relative to Pp. Let D, := { f 2 n } and T, := TD,,. Since f is a-excessive, D, is finely closed, D, E E“, and
ho,tl
Therefore, if T := limT,,
and so (T,) satisfies (ii). Given R E T , e-R f (XR)> n implies T, 5 R, and so ~ ” [ Y RYR ; > n] 5 P p [ Y ~Tn ; 5 R] 5 P’(YT~) + 0 as n -+ 00 because of (ii). Consequently, the family {YR : R E T } is P P uniformly integrable, proving that Y is of class (D) relative to Pp. Because f is finite valued, yt := edatf (Xt) is a positive supermartingale relative to P” for every v with j” f du < 00. By (51.1), Y has a Doob-Meyer decomposition of the form yt = Mt - Sot e-as dB, where M is a local martingale relative to each P” with J f du < 00, and B is a predictable A F of X, possibly charging C. But Y is a class (D) potential relative to Pp and, hence, is the Pp potential of a predictable increasing process. From the uniqueness of the Doob-Meyer decomposition, it follows that yt is the potential of s,” e-a3 dB, relative to P P (in the sense of the general theory of processes). As in 548, p l ~ o , , - ~= r(X-)lno,mn where T := pol^. We claim that Pp-a.s., d B is carried by A := {r(X-) > 0) E fig. To this end let C := l ~* B. = Then C is a predictable AF, possibly charging C. According to
VI: Stochastic Calculus
257
[[I.
the discussion in $48, A" n 10, ([I c [ C [ I , and so dC is carried by Let R := inf{t : A c t > 0). Then R is a predictable terminal time with [[RI]C [
pp
J
e-"' dB, = Pp{e-"Tnf(XT,)}-+ 0 as n
+ 00.
]Tn>w[
srR,w[
Hence Pp e-"' dB, = 0 . But if R < 00, then 0 < ACR 5 ABR. Consequently P"(R < 00) = 0 and so [email protected].,B is carried by A. Define now A := ( l *~B)". By the discussion above together with (48.6), A is a natural AF and the Pp-potential of & , t l e--us dA, is Y . The condition that a function be a regular a-potential may also be expressed in terms of hitting times. This is an easy consequence of (48.13) and (53.7). Let f E S" be a finite valued and satisfy the condi(53.8) COROLLARY. tions of (53.7). Suppose, in addition, that given any decreasing sequence (D,) of finely closed sets in Ee with hitting times T, and T := limT,, one has P " [ e [ e - " T n f ( X ~+ n )P ] " [ e - " T f ( X ~ ) ]Then . the A in (53.7) is continuous Pp-almost surely.
PROOF:Of course, the condition in (53.8) contains (53.7ii). Given 6 > 0, let R := inf{t : AAt 2 c}. Then R is a natural thin terminal time, and by (48.13ii), there exists a decreasing sequence (D,) with limT, 2 C on {R 2 C}, limT, = R if R < C and T, < R on { R < C}. But dA does not charge [C,oo[,and so if T := limT,,
J(TD,RI
pp
e--us
dA8 = p p
J(Tn,T1
e-"' dA,
Consequently PP(R < C) = 0, completing the proof of (53.8). We may now extend some of the results of $38. Let K be a natural HRM with a-potential function u:. For every p,
258
Markov Processes
and if $ ti: d p < 00, this expression is the potential process of the increasing process t + &,tl e-”’ K ( & ) relative to P p . Assume now that K is carried by A and K is cr-integrable over P,. Then by (48.6),K. and its dual natural projection K= have the same a-potential function for any a 2 0: that is, ti: = ,.:it . This remark will be used in the proof of the next result, which extends (38.8). Let f E S“,,u an initial law with J f dp < 00. Suppose (53.9) THEOREM. the conditions (i) and (ii) of (53.7) are met. Then there exists a natural HRM, K , such that { f ( X ) # uE(X)}is Pp-evanescent.
PROOF:Let E’ := {f < co}. Since J f d p < 00, p is carried by E’. The set E’ is nearly Bore1 and absorbing for X . Let X’ be the restriction of X to E’. I f f ’ is the restriction of f to E’, then f’ is a finite a-excessive function for X’ satisfying the conditions of (53.5) relative to X ’ and p . (Observe that if D C E’ is finely closed for X ‘ , then since ,u is carried by E’, TD = TF Pp-a.s., F denoting the fine closure of D relative to X ’ . ) Therefore there exists a natural AF A‘ of X’ with f ’ ( X ’ ) = u4,(X’) up to Pp-evanescence relative to X ’ . Apply now (38.6) to obtain a natural HRM, K , carried by [ T ,C[ where T := T E f ,with K extending A’. Clearly f ( X ) and u:(X) are Pp-indistinguishable, establishing (53.9).
VII
Multiplicative Functionals
54. Multiplicative Functionals and Terminal Times
Let m E p M , and let N$ := { w : mt+s ( w ) # mt(w)ms(&w)}.Then m is called a multiplicative functional (MF) provided it satisfies (54.1)
V t l O ,Vs20,
NZEN
and additional regularity hypotheses. The exceptional sets in (54.1) depend on both s and t. The dependence on s may usually be removed if, as will normally be the case, m is assumed continuous from one side. Getting an exceptional set not depending on t is a much more difficult problem of perfection of m. The terminology is a bit cumbersome because of the large number of variants which are necessary and useful in the theory. We attempt to alleviate the problem by using the smallest number of qualifying adjectives to describe the most useful cases. Recall that T denotes the class of optional times for (&). (54.2) As in $24, m E M with a s . rcll paths and satisfying (54.1) is called a weak raw MF or weak RMF. The RMF is perfect (resp., almost perfect) provided U,,tN$ is empty (resp., null). (54.3) The strong multiplicative property of m is
The adjective strong is a suppressed qualifier in that unless m is explicitly called a weak MF, m is to be presumed to satisfy (54.4). This usage is
260
Markov Processes
consistent with earlier definitions of terminal time (12.1), AF (35.5) and homogeneous process (24.1). Note that if m is an almost perfect RMF, then m is necessarily a strong RMF. (54.5) If the qualifier raw does not appear, it is to be inferred that m is adapted to ( F t ) . (54.6) All MF’s are to be either a s . right continuous (the default) or a.s. left continuous, and the term left MF is reserved for the !ztter case. (54.7) m is decreasing provided t + mt is a s . decreasing, supermartingale if ( m t )is a supermartingale relative to every P”. It is an obvious consequence of (54.1) and (54.6) that a weak RMF m is decreasing if and only if 0 5 mt 5 1 a s . for each t 2 0,
and a weak M F m is a supermartingale if and only if (54.8)
P5mt
5 1 for all t 2 0 , x E E .
As an example of the above usage, if m is declared to be a MF, the above conventions mean that ( m t ) is adapted to ( F t ) ,t + mt is a s . right continuous, and (mt)satisfies the strong multiplicative property (54.4). For reasons which will shortly become apparent, an exactness condition analogous to the exact terminal time condition in $12 plays a major part in the theory. A weak RMF, m, is exact provided, for every t (54.9) DEFINITION. and every sequence t, 11 0, (54.10)
mt-t, O O t ,
+ mt
a.s. as n -,
>0
00.
Other forms of exactness will appear later in this section. If the t , are replaced by optional times in (54.10), m could be called strongly exact and if the exceptional set is independent of both t and {t,}, we could call m perfectly exact, though we shall try not to. The following exercises give some classical examples, and show that the use of the terms raw, left, perfect and exact is consistent with previous definitions. Let At be a R A F and let mt := e-At. Then ( m t )is (54.11) EXERCISE. a raw MF. In addition, ( m t ) is exact, and ( m t )is (almost) perfect if and only if ( A t ) is (almost) perfect.
VII: Mu1tiplicative Functionals
261
(54.12) EXERCISE.Let ( A t ) be a RLAF. Then mt := e-At is a raw left MF which is exact if and only if A is continuous. (54.13) EXERCISE.Let T be a terminal time. Then m := 1U0,Tl is a MF which is exact if and only if T is exact, (almost) perfect if and only if T is (almost) perfect. (54.14) EXERCISE.Let h be an a-excessive function, and let S denote the hitting time of the absorbing set { h = 0). Then mt := e-at h(Xt)/h(Xo)l{h(xo)
<
>
t 2 0.) (54.15) EXERCISE.Let R E pF. Then m := 1[O,R] is a raw left MF ifand only if for all T E T , T ROOT= R 8.s. on {R2 T } . If R is a co-optional time (25.1), l[O,R] is a raw left MF and l[O,R[ is a raw MF.
+
(54.16) DEFINITION.A left terminal time is an optional time R such that for every T E T , T ROOT= R a.s. on { R 2 T } .
+
It is evident that every left terminal time is a terminal time. Here are some examples of left terminal times. (54.17) EXERCISE.Let A be a continuous AF and let R := inf{t: At > 0). Then R is a left exact terminal time which is (almost) perfect if A is. (In the special case where At = 1 B ( x , ) d s , B E E", R is called the Lebesgue penetration time of B. See 564 for further discussion.)
s,"
(54.18) EXERCISE.Let T be a terminal time. Then T is a left terminal time if and only if T 2 (:= T + TOOT) = T almost surely. In particular, the hitting time TB of a nearly optional set B is a perfect left terminal time if B \ B' is polar. By contrast, the debut of a nearly optional set is always a perfect left terminal time. (54.19) EXERCISE.Let I' c R++ x fz (resp., R+ x 0) be progressive relative to every P'' and homogeneous on R++ (resp., R+). Then its debut Dr(w) := inf{t:(t,w) E I?) is an exact terminal time (resp., left terminal time). Moreover, its penetration time
Rr(w) := inf{t: (0, t]n r ( w ) is uncountable) is a left exact terminal time. (Consult pM75,IV-111-112] for measurability properties of penetration times, which show in particular that the
Markov Processes
262
penetration time of a progressive set is an optional time. In checking that R r d ( R r )= 0 on {Rr < m}, it may be helpful to note that
Rr = inf{t : V E > 0 , n [t,t + E ] is uncountable}, for I? n [0,Rr] is countable and
n [Rr,Rr + E ] is uncountable if Rr < m.)
(54.20) EXERCISE. Let K be a HRM of X , and let T := inf{t : K is not c-finite on 10, t ] } . Then T is a left terminal time, which is (almost) perfect provided K is. (54.21) EXERCISE. The hitting time of a non-polar thin set is an exact terminal time which is not a left terminal time. (54.22) EXERCISE. Let m be a raw weak MF or a raw weak left MF. If mo+ exists and is equal to 1 as., then m is exact. (54.23) EXERCISE. Let m be a weak MF. Then P2{mo = 1) = 0 or 1 for every x E E.
The result of (54.23) is true but uninteresting for a left MF. If m is a left MF and if mo+ exists a.s., mo+ = f(X0)for some f E bE" but f need not take only the values 0 and 1. See $59 for further discussion of left MF's. We might remark also that it is not generally the case that the right limit of a left MF is a MF, nor that the left limit of a MF is a left MF. The notation in the next exercise will be used consistently throughout this chapter. (54.24) EXERCISE. Let m be either a MF or a left MF, and define S, := inf{t: mt = 0). Then S,,, is a terminal time and, as., mt = 0 for all t > S,. (If m is only a weak MF then S, is only a weak terminal time. See (57.9) for further connections between m and S,.) (54.25) EXERCISE. Let X be Brownian motion in E := R2and let 2 c E , 2 E E" be a set of the type described at the end of $19, so that every Bore1 subset of 2 is polar, but every non-constant path in E hits 2. Let T(w) := 0 or 00 according as X o ( w ) 4 2 or Xo E 2. Show that T is a terminal time,
not exact, and there is no almost perfect terminal time a.s. equal to T.
VII: Multiplicative finctionals
263
55. Exact Perfection of a Weak MF The main theorem (55.19) of this section is much more complicated than the earlier perfection proofs in $24, but yields more in some special cases. The reader is urged to look only at (55.2), which introduces notation used throughout the chapter, and the statements of (55.19) and (58.20). The proof itself is not particularly illuminating. We suppose given a weak RMF m of X. That is, m satisfies (54.1) and t -+ mt is a.s. right continuous. It is not necessary to assume that m is measurable relative to M . Let C denote Lebesgue measure on R+. We , 6 3 3 is augmented in the sense of 56 and assume m E (B+ @ 6 ) * where satisfies: (55.li) V t , V G E 6 , GOOt E 6 ; (ii) Vp, VG E 8, ( t , w ) + G(Otw) is in the CxPp-completion of B+&. By (4.3), the conditions are satisfied if 6 := 3. We shall also assume throughout this section that m is decreasing, and in fact, modifying m on an evanescent set if necessary: (55.2i) t + m t ( w ) is decreasing and right continuous for all w E R; (ii) mo(w) = 0 or 1 for all w ; (iii) if mt(w)= 0 then m,(w) = 0 for all '(I 2 t. In the regularization that follows, it is helpful to describe a MF m in terms of a process indexed by intervals Is,t] C R+,defined by (55.3)
ml,,tl(W) := mt-,(eew),
o IL t < 00.
Note that exactness of m is expressible as a weak form of right continuity ~ ]s, to the effect that for any sequence s, 10, m ~ , ~ , ~ m]~ o ,a~s ]. of m ~ , , in asn+oo. The doubly indexed process m],,t] has the following properties; (55.4i) V s 2 0, V w , t + r n ~ , , ~ l ( wis) right continuous on [s, m[; ) m]r+s,r+t]( w ) identically; (ii) m],,t]( 0 , ~ = (3 v r I s I t , m],,,]m],,t] = m],,t] a-s.; (iv) ( s , t , w ) -+ m ~ , , ~ l ( uis) measurable relative to the C x C x P p completion of B+ @ B+ @ 8 , C being Lebesgue measure on R+. All of the above, except (iv), are obvious consequences of (55.2i,ii), (55.3) G(Otw) is in and (54.1). For (iv), use (55.1) to see that G E b6, ( t , w ) the C x Pp-completion of B+ @ 6 . Using this and right continuity in t , (iv) follows at once. in s by perfection techniques similar We now regularize the process rn~,,~] to but more potent than those of $24. Define, for 0 5 s < t , -+
-+
(55.5)
+,,t(w) := lim esssup(m],,~](w):s< r ell0
< s +6).
264
Markov Processes
Apply (A5.7) with Bt := B to see that for each fixed t > 0, the map (5, w ) -, $ J , , ~ ( w is) in np(B+ 8 @‘). In particular, since S is assumed to be augmented, w + $,,t(w) is 9-measurable for every pair s,t. By (55.4iii), s + is decreasing on 10, t]and t -,m ~ , , ~is]decreasing on ]s,oo[. It follows that $J,,t is decreasing in both s , t for s < t. In particular, $,+,t I for s < t. If $,,t(w) > 6 , then there exists c > 0 such that m~,,~l(u) > 6 for a.a. T E Is,s c[. It follows that $Ju,t(w)> S for all u E]S, s c[. That is
+
+
s < t.
(55.6)
Let Ro c R be the set of w such that the conditions (55.7i) and (55.7ii) below both hold: (55.7i) the map ( s , t ) + m ~ , , ~ l (defined u) on (0 5 s 5 t } C R+ x R+ is C x &measurable; = rn~,,~](w for ) (Lebesgue) a.a. triples ( T , s , t ) (ii) m~,,,l(w)m~,,~l(w) with r 5 s 5 t. It will turn out that $,,t(w) is a good regularization of m ~ , , ~ l ( uas) long as w E 520. The following two properties of Ro are obvious. (55.8i) (55.84
S ,
-
E R ~u ,L o euw E ao. 1 O,e,,w E Ro V n & w E Ro.
In addition, (55.4iv) implies that for every initial law p,
P’”(R\ Go) = 0.
(55.9)
Until further notice, w E f20 is fixed and suppressed in all notation. Let A(= A ( w ) ) denote the set in R4specified by
A := { (r, 8, t ,u ) :0 I r
I I t I 21,
m]r,u]m],,t] # m]r,t] m],,u]}*
Clearly then, C x L x B x C(h)= 0. Recomputing this using Fubini’s theorem, integrating first in u, then in t , then in (r,s), we find that for a.a. ( T , s ) with 0 I T 5 s, for a.a. ( u , t ) with u 2 t 2 s, (55.10)
m],,u]m],$1 = mlr,t ]m]s,u]
VII: Multiplicative Functionals
265
In view of right continuity in t and u,the identity (55.10) holds in fact for all u 2 t 2 s, for a.a. ( T , s) with 0 5 T 5 s. Therefore, for all u > t > 0,
Applying Fubini's theorem again, integrating first relative to (55.12)
for 8.a. s > 0, (55.10) holds for a.a. T
-+
we find
< s, V u > t > s.
It follows then from the definition (55.5) of $ that for
(55.14) LEMMA. For every w E 00,t
T,
T
< t < u,
$,,t(w)is right continuous on
IT, oo[,and $ , , t ( ~ )= 0 implies $,,,(w)= 0 for all u > t. PROOF:Fix w
E Ro and drop w from all notation. If q!~,,~= 0 then by (55.13), either rn~,,~] = 0 for a.a. s E ] T , ~or [ q!IT,, = 0 for all u > t. Under the first alternative, (55.2iii) gives rn],,,] = 0 for a.a. s E]T, t [ and therefore +, = 0. That is, $,,t = 0 implies &,, = 0 for all u > t. Suppose next that QT,t > 0. Then (55.13) shows that we may select s ~ ] r , so t [ that rnl,,t]> 0 and &,, = $,,t rn1,,,1/rn1,,~1 for all u > t. From this, right continuity of $ in its second variable is evident. Arguing now exactly as in (55.10)-(55.13), but starting instead with
for all w E Qo, we are led to the fact that for w E Qo, for a.a. s > 0, rn],,,](w)rn~,,~l(~) = rn~,,~l(w) for 8.8. T < s, for all t > s. It follows that for all w E 00, for 8.8. s > 0 (55.15) Since (55.14) gives us right continuity of $ in the second variable, we may take limesssup over s in (55.15) to obtain (under (55.2iii))
(55.17) LEMMA. For all w E 00and t
> 0,
Markov Processes
266
PROOF:Fix w E 510 and suppress it. In view of (55.15) and (55.16) we have, for all T < t and a.a. s €IT, t [ , $T,B
m]S,t]
= $T,S
$B,t*
Therefore m ~ , , = ~] for a.a. s < t such that $J~,,> 0 for some T < s. However, if &,, = 0 for all T < s, then (55.16) shows that qT,t = 0 for all T < s 5 t. If TO were a point of density from the right of the set 2 E. Thus, E > 0 being {T < s : m~,,~ >] E } , (55.5) would give arbitrary, we conclude that m ~ ~= ,0~for] a.a. T < s. As we may choose s < t as close to t as desired, this proves that $,.,t = m ~ , ,for ~ ] 8.8. T < t , as claimed. Define now R ( w ) := inf{t : Otw E no}. Then R = 0 as., and by (55.8i,ii), Btw E Ro for all t E [R(w),oo[.Set then
(55.18)
A ~ ( u:= )
{
1
if w E 00, if w 4 Ro and t < R ( w ) ,
$O,t-R(w)(eR(w)u)
if w
Q0,t ( w )
GO and t 2 f i ( w ) *
(55.19) THEOREM, The process m constructed above is a perfect exact MF of X such that: (i) Vw E R, s + ml,,t~(w)is right continuous and decreasing on [O, t [ ; (ii) Vw E R, t r % ~ ~ , ~ lis ( wright ) continuous and decreasing on [s,4; (iii) mt = m t for all t 2 0 8.9. on {mo = 1); (iv) mt 5 mt for all t 2 0, a.s.; (v) m and m are indistinguishable if m is exact. The MF is called the perfect exact regularization of m.
Remark. Condition (i) is a substantial strengthening of exactness for m. PROOF:The perfect M F property and conditions (i) and (ii) are immediate consequences of the construction of A, especially (55.16), and the remarks concerning R before the statement of the theorem. Since m and m are as. right continuous, it suffices for (iii) to prove mt = f i t 8.8. on {mo = 1) for each fixed t. Obviously = $,,t a s . for s < t. For t fixed, use (55.17) to select sn 11 0 such that m],,,t] = m ~ ~ ,a.s. , ~for ] all n. Then as., mt
= ms, m]sn,t] = ma,
fi]n,,,t].
--t 00, m,, + 1 as. on {mo = l}, and by (i) above, r%lB,,t] -, At. Therefore mt = fit a s . on {mo = 1). Item (iv) is an immediate
As n
consequence of (iii), for on {mo # I}, mt = 0 for all t, almost surely.
VII: Multiplicative Functionals
267
Let T be a weak terminal time. Then there exists a (55.20) COROLLARY. perfect, exact terminal time T such that: (i) T 5 T a.s.; (ii) T = T a.s. on {T > 0); (iii) for all w , t + t T ( 8 t w ) is increasing and right continuous.
+
PROOF:Apply the theorem to m := I[O,T[I and set T := inf{t > 0 : mt = 0). It is clear that T is a perfect terminal time satisfying (i) and (ii), but the strong exactness condition (iii) requires a little argument. By (55.5) and (55.18), r5i takes only the values 0, 1, and by (55.19ii), r5i = l a o , rAs ~. f i ~ . ,= ~ l] p s + ~ ' 0 8 , ~ ( (iii) t ) , follows at once from (55.19i). Theorem (55.19) also applies in an obvious way to get perfect versions of other functionals, but it should be noted that the nice measurability features of §24 are lost. See $57 and 560 for further perfection results. 56. Exactly Subordinate Semigroups
We assume for this paragraph that m is a decreasing weak MF. According to the conventions established in $54,this means that t -, mt is a s . right continuous, m has the weak multiplicative property (54.1) and for every t 2 0, mt is Ft measurable and, as., 0 5 mt 5 1. We shall also assume from now on that mt = 0 for all t 2 C. The set Em of permanent points for m is defined by
Em := {z E E: P"(m0 = 1) = 1).
(56.1)
By the previous hypothesis, Em C E . Since mo = 0 or 1 a s . and mo = mo+ E Fo,PS{mo = 1) = 0 if x E E \ Em. Thus P"m0 = l ~ , ( z )and consequently Em E E". If T is a weak terminal time and m = l [ O , T [ I ,Em is equal to E\reg(T) where reg(T) is the set of regular points for T, defined in (12.7) as ( x E E:P5{T = 0) = 1). Given m, a decreasing weak M F of X, define operators P;, Qt and V Q on pE' by setting, for f E pE", a 2 0 and t 2 0,
:= P"[f (Xt)mt],
(56.3)
Qtf(.)
(56.4)
V af (z) := P"
1
oi)
e-atf(Xt)mt dt.
268
Markov Processes
In (56.2), (-dmt) denotes the RM generated by the increasing process ma - mt. The integrals in (56.2) and (56.4) are meaningful because of the discussion in (4.3). That discussion also shows that the manipulations below using Fubini's theorem are justified. We remark also that if m = 1 ~ 0where , ~ T ~ is a weak terminal time, then
We used these operators in the discussion of killing at a terminal time in $12. The results obtained in this paragraph are natural generalizations of those of $12. (56.5) PROPOSITION. Let m be a decreasing weak MF of X and let P;, Q t , and V" be the operators defined by (56.2-4). Then each of these operators is a kernel on ( E ,E") and the following properties obtain: (56.6)
( Q t ) is a subMarkov semigroup of kernels on ( E ,E") with resolvent (V"). For every x E E , Q t ( x , and V"(x, are carried by Em; for every Q 2 0 and f E bpE", U" f = V" f PgU" f ; for every a 5 0 and f E S", Pg f is a-super-mean-valued (4.11), and if m is exact, then Pz f is a-excessive; ifm is exact, (V")is exactly subordinate to (U") in the sense that for Q > 0 and f E bpE", U"f - V af E S". e )
(56.7) (56.8) (56.9)
e )
+
PROOF:The kernel property comes from (3.4) via (4.3), as the random variables whose expectations define the operators are in 3. The semigroup property of ( Q t ) uses mt+s = mt m R O B t and the fact that rnt E F t , so that
It is clear that (V") is the resolvent generated by ( Q t ) . Property (56.6) is obvious since rnt = 0 P"-a.s. for x q! Em. It suffices to prove (56.7) for cr > 0, the case Q = 0 being obtained therefrom by a simple limit argument. Let f E bpE" and Q > 0. Then for x E E \ Em, V" f (x)= 0 and PZU" f (x)= U" f (x). For x E Em, mo = 1 a.s., so, making repeated
VII: Multiplicative F'unc tionals use of Fubini's theorem,
U " f ( 5 ) - vaf(z)= P"
269
iM
e - a t f ( x t ) ( l - mt) d t
I"
= P"
e-atf(xt)
= PZ lm(-drn8)
1
J
l-v4
= P " 1 " e-"' = P"
JdM
(-drn,) dt
lO,tl
(-dm,)
e-atf(xt) dt
(Jd" e-aUf(xu)
du) "0s
e-aS(-dm,)uaf(X,)
= P;U"f(z). Now let a
> 0 and f E S". I f f
= UQg with g E bpE", then
P,"PZf(z) = P,"P;Uag(z) = P,"(UQg- V"g)(z)
- pse-ot PX' = P"
/I"
1 M
e-a"g(xu)(l - mu)du
e-asg(x,)(l-
ms-toet)du,
using the simple Markov property. Since ms-toOt = m ~ ~2 , m~ ~]o , = ~ ]m, a s . for each s 2 t , it is evident from the above expression that
P,QP;UQg(z) 5 P:Uag(z). Therefore P;U"g is a-super-mean-valued. Finally, if m is exact, mlt,.] + mu a.s. as t 11 0 through any sequence. Using the dominated convergence theorem, one sees that PzU"g is aexcessive. The same two facts hold then for f € Sa-just take a sequence of a-potentials increasing to f . The case a = 0 follows since f E S implies f f S" for all a > 0 and Pgf increases to Pmf as a 11. 0. This proves (56.8). The assertion (56.9) follows at once from (56.7) and (56.8).
Let m be a decreasing weak exact MF. Then Em E (56.10) COROLLARY. E " , Em is finely open, and P;(bE") C bE".
PROOF:Trivially, Em = {V1l > 0). Thus Em = {U1l - V1l < l}, and since U1l - V1l = PkU1l is 1-excessive, the first two assertions follow. A monotone class argument based on (8.6i), (8.7) and (56.8) completes the proof.
270
Markov Processes
(56.11) COROLLARY. I f m is a decreasing weak MF and m is a perfect exact regularization of m (55.191, then: (i) Em C Em; (ii) for every f E S", Pg f is the a-excessive regularization of P z f; (iii) if (V") is exactly subordinate to (U"), then m is an exact MF.
PROOF:We showed in (55.19) that m 5 rsL up to evanescence. This implies item (i). Suppose first that (Y > 0. Let f E S" be of the form f = U"g with g E bp&". We showed in the course of the proof of (56.5) that P:P$Uag(z) = P" As a function oft, the right side increases as t 11 0 since PgU"g is a-supermean-valued. Fix cp E bC(R+)with compact support. Since mlt,.l = f i l t , . ] for a.a. t < s, using Fubini's theorem in the second equality, we find
1
1
00
V ( t )dtP:P$U"g(z) =
=
00
d t ) dt p"
e - " 8 g ( x s > ( l -m]t,a]) ds
locp(t)
dt P:P$U"g(z).
It follows that PyPzU"g(x) = PyP$U"g(z) for a.a. t , and since the term on the right increases to P$U"g(z) as t 11 0, it is the case that P ~ P ~ U * g (T zP$U"g(z) ) as t 11 0. By ( 4 . 1 2 ) , P$U"g is the cr-excessive regularization of PzUQg. The same holds for f E S"(a > 0), considering increasing limits of a-potentials. The case a = 0 follows by letting (Y 41 0 as in the proof of (56.5). In each of these cases the interchange of limit operations is justified since each limit is monotone in the same sense. This proves (5). Suppose now that (V") is exactly subordinate to (U"). By (56.7), for f E bpEU, PgU"f is finely continuous and hence is its own a-excessive regularization. That is, by (ii), P$UQf = PgUaf for all f E b p P . Let (V") denote the resolvent generated by a. The last identity shows that V a = V a for a > 0. As m 5 m,
implies that m and f i are indistinguishable.
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(56.12) EXERCISE.Prove that if m and m' are decreasing weak MF's generating the same resolvent (V"), then m and m' are indistinguishable. (Hint: the semigroup ( Q t ) has the property that t + Qt f (z):= P"[f(Xt)mt] is right continuous for every f E C d ( E ) , so by Laplace tranform inversion, m and m' generate the same semigroup ( Q t ) . The given equality states that P"[J(Xt)mt] = P"[f(Xt)mg] for every f E bE". It suffices to prove that for f1,. . . , fn E bE",
Proceed by induction on n-see
the proof of (60.2).)
(56.13) EXERCISE.Let m be an exact decreasing MF of X,S := inf{t : mt = 0}, and let f be a-excessive relative to the resolvent (V"). (That is, DVa+flf7 f as L,I t 00.) Prove: (i) f is an increasing limit of a sequence of V"-potentials of bounded functions (cf. (4.15vi)); (ii) f E E e ; (iii) e-at f (Xt)mtis a right continuous P"-supermartingale provided f(ZPE,(Z) < 00; (iv) t + f ( X t ) is a s . rcll on 10, S [ . (Hint for (iii): use (i) and (A5.16).) There is a converse to part of (56.5) which gives one a means of constructing a MF from a given semigroup. The theorem is from [Me62a]. We shall not make use of it, and merely quote it, referring to [BG68, 1111 for the proof. See however the related result (62.26). Let (V") bearesolvent on (E,E") exactlysubordinate (56.14) THEOREM. to the resolvent (U")of a right process. Then there exists a unique exact decreasing MF, m, of X so that (56.15)
V af (z) = P"
e-"t
f (Xt)mtdt, f E pE".
10
(56.16) EXERCISE.Let X be uniform motion to the right on R, and let g be a decreasing, strictly positive, right continuous function on R+.Let mt := g(Xt)/g(X~)l~xo>o). Then rn is a MF which is not exact, m ~ , ,= ~] g(Xt) /9(XS11{ x,> O ) 9 and f i t = 9( X t1/9P o ) 1{ xo > O } .
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57. Decreasing MF’s
In this section we discuss three fundamental results about a class of MF’s which will be called right MF’s in (57.1). The name is not meant to suggest right continuity, but rather that such a M F gives us a way to transform by a killing operation one right process into another. In the case of a decreasing right MF, this is the content of (61.5), one of the principal results of this chapter. See also (62.19) for the supermartingale case. First we compare a decreasing right M F with its perfect exact regularization to get a perfection result for decreasing right MF’s. If m is a weak MF, the set Em of permanent points for m was defined (56.1) as { z : P 2 { m=~ 1) = 1 ) . The set Em is in general only in EU. As in the preceding section, all MF’s are assumed to vanish identically on uc,cou. ( 5 7 . 1 ) DEFINITION.A right MF is a decreasing MF m (vanishing on [Icl 001) whose set Em of permanent points is nearly optional for X. See (57.4)for another condition on a decreasing, weak M F implying that it is a right MF. For T the terminal time in exercise (54.25), m := 1 ~ 0 , T ~ is an example of a M F which is not a right MF. Recall (56.10) that if m is an exact decreasing MF, then Em E E“ and so m is a right MF. The first result does not require m to be decreasing. The general case will be of importance in 562. (57.2) PROPOSITION. Let m be a supermartingale MF for which Em is a nearly optional set, and let D be the debut of E \ Em. Then mD = 0 a s . on {D < co}. That is, a.s., Sm 5 D. PROOF: we shall prove that mtlE\E,(Xt) is evanescent by means of the section theorem (A5.8). Given an initial law p and T E T, mT = mTmOOOT a.s. on {T < m} implies m T 1 { m o O ~ T ==O0) a.s. on {T < co}. Therefore
and this shows the evanescence of mlE\E, (X). Since mt > 0 for all t < s,, it follows that a.s., Xt4 E \ E, for all t < S,. This implies that S, 5 D a.s., as claimed. ( 5 7 . 3 ) THEOREM. Let m be a decreasing weak MF such that: (i) Em is nearly optional; (ii) if D is the debut of EA \ Em then mD = 0 a.s. on {D < co}.
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Let m be the almost perfect exact regularization of m (55.19). Then m is indistinguishable from the almost perfect (not necessarily exact) MF ml[O,R[I, where R is the debut of Em \ Em.
PROOF:Let D denote the debut of E \ Em. Then Em is nearly optional (56.10) so, applying (57.2) to m and D we obtain m 5 lao,~n up to evanescence. By hypothesis (ii), m 5 lao,on up to evanescence. Since R 2 D as., m 5 l ( o , R [ up to evanescence. By (55.19), m 5 mlno,Rn up to evanescence. We prove that m and m l n 0 , ~ nare indistinguishable by checking cases. Since Em c Em, for x E E \ Em, m and m are both P"-evanescent. If x E Em, mo = 1 as., so by (55.19), mt = f i t for all t 2 0 as. P". However, since m~ = 0 a s . , m and m 1 1 0 , ~ nare P"-indistinguishable. Finally, if 2 E Eii, \ Em, mt = 0 for all t 2 0 and R = 0 P"-almost surely. (57.4) COROLLARY. If m is a decreasing weak MF satisfying (57.3i,ii), then m is indistinguishable from a perfect MF and hence m is a right MF. The proof is evident in view of the fact that every perfectable weak M F is necessarily a MF. Using (57.2) to verify (57.3ii) we get (57.5) COROLLARY. Every right MF is indistinguishable from a perfect MF. In case m = 1n0,sI with S a weak terminal time, Em = E nearly optional if and only if reg(S) is nearly optional.
\ reg(S) is
(57.6) DEFINITION. A right terminal time is a terminal time S such that reg(S) is nearly optional for X. When we apply (57.3) to m = ln0,sU we obtain the following result. (57.7) THEOREM. (i) A weak terminal time S is a right terminal time provided reg(S) is nearly optional and a.s., S 5 Dreg(s), the debut of reg(S). (ii) Every right terminal time may be perfected. (iii) If S is a right terminal time and if S is its perfect exact regularization, then S is a.s. equal to A R, where R is the debut of r e d s ) \ re&).
s
We can be a little more precise about the connection between the debut R of Em \Em and m in (57.3), and hence also in (57.7). (57.8) PROPOSITION. Let m , r% and R be as in (57.3) and let T be the hitting time of Em \ Em. Then a.s., m~ = 0 so that a.s., T 2 Sm. Consequently the set Eii, \ Em is polar for X killed at Sm.
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PROOF:Let D, T denote the debut and hitting time respectively of Em. By (57.2), m 5 l ~ , ,But ~ D ~ 5. T 5 T , so m 5 l u O , T [ . The assertions of (57.8) are obvious consequences of this inequality. One cautionary remark is in order. If m is an exact MF and S, := inf{t: mt = 0 } , then S, is a terminal time, but in general S, is not exact. For example, if X is uniform translation to the right on R+ and mt := X o / X t l { ~ , , >it ~is}easy , to see that m is an exact M F but S, = D{o} is not an exact terminal time. Observe that reg(S,) \ reg(&) = (0) is not polar for X , though it is for X killed at S,. (57.9) PROPOSITION. A decreasing MF m is a right MF if and only if Sm is a right terminal time.
PROOF:Obvious from the identity reg(Sm) = E \ Em. By (57.7) if S is the perfect exact regularization of a right terminal time S, then S is a.s. equal to S A R where R is the debut of reg(S) \ reg(S). In addition, (57.8) shows that reg(S) \ reg($ is polar for (X, S). As the above example shows, reg(S) \ reg(S) is not in general polar for X . (57.10) EXERCISE. Let S be an almost perfect terminal time and let S be its perfect exact regularization. Define, for t 2 0, St := t So& and St := t So&. Show that a.s., ( S t ) and ( S t ) are increasing and for all t 2 0, s t = st+.
+
+
(57.11) EXERCISE. Use (57.7) and (57.8) to show that if S is a perfect terminal time such that reg(S) is nearly optional, then the subprocess ( X ,S) of (12.20) is a right process. (Hint: compare the resolvent for ( X ,S ) with the resolvent for (X,
s).)
In view of (57.7) and (57.8), ( X , S ) may be obtained by first killing at the exact terminal time 3, deleting after that the polar set (for ( X ,S ) ) reg(S) \reg(S). See (12.26) for this last operation. In comparisons between the subprocess and the original process, the following result is critical. (57.12) PROPOSITION. Let S be a right terminal time with exact perfect regularization S, and let €5 denote the 0-algebra on E \ reg(S) generated by the a-excessive functions for ( X ,S ) . Then E: is the trace of Ee on E \ reg(S)*
PROOF:By (12.29), (12.31) and (57.8), the a-excessive functions for ( X ,S ) are precisely the restrictions of the a-excessive functions for (X, S). Therefore € 5 is the trace of €5 on E \ reg(S). However, because 3 is exact, (56.13ii) shows that E z is the restriction of E" to E \ reg(S).
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58. m-Additivity
In order to look at the correspondence between AF’s and MF’s, we need to enlarge the scope of additivity beyond what was defined in §35 and $49. Given a MF, m, recall (54.24) that S, := inf{t: mt = 0) is a terminal time for X . (58.1) DEFINITION.A process Y is m-additive if for every T E T, Y T +-~
YT is indistinguishable from mT(Y, - YO)o&. In the notation of (50.18) the condition reads (58.2)
mTOTY=Y-YT
on
{T<m}.
+
If Y is rn-additive, then so is Y F for any random variable F . There would therefore be little generally lost if one restricted attention to those Y with Yo = 0, but this is not entirely natural. Observe too that in case m f 1, m-additivity means precisely the same thing as the additivity discussed in (51.6).
If Y is m-additive, then a.s., yt = Ys, V t 2 S,. (58.3) PROPOSITION. PROOF:Set T = S, in (58.2). The most important special cases of m-additivity are seen in connection with the MF’s (i) mt := e-at; (ii) m := lno,sn for a terminal time S. To simplify notation, m-additivity is referred to as a-additivity and Sadditivity in these respective cases. The following result shows that Sadditivity is essentially weaker than ordinary additivity. (58.4) PROPOSITION. Let S be a terminal time and let Y be additive. Then Ys and Y lno,so are S-additive.
PROOF:Given T E T, one has
for all t 2 0. On {T < S } , S = T
(t - T ) a s . for all t
+ SOOT as., so
A SOOT = (t -T)A
( S - T )= t A S - T
2 T. Therefore W Y S P { T < S )=
(&WS1 { T < S ) .
Using (58.2) we get 1[O,SIJ(T)&v
s 1 - (y - y T ) s = y s - yTs ,
so Ys is S-additive. The other case is similar but even easier.
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(58.5) EXERCISE.If m is a MF then m is m-additive. There are variants of the definition (58.1) along the same lines as in $35.
A process Y will be called weakly m-additive if (58.2) holds whenever T is a constant time, (almost) perfectly m-additive if m is (almost) perfect and (58.2) holds identically on R (except on a null set independent of T ) . (58.6) DEFINITION.A right continuous measurable increasing process is a raw additive functional (RAF) of ( X , m ) in case A0 = 0, A is madditive, At < oo a s . for all t < S, and, as., (58.7)
At = As,-
for all t 2 S,.
Similar definitions apply to AF’s of (X,m), LAF’s and RLAF’s of ( X ,m). In case m = 1u0,Su we write (X, S ) in place of (X, m). If the A in (58.6) is only weakly m-additive we call m a weak AF of (X, m), and if A is (almost) perfectly m-additive, then A is called an (almost) perfect AF of ( X ,m). The following simple examples show that the only essential new notion here is S-additivity. (58.8) EXERCISE.Let A be an RAF and let m be a MF. Then Bt := ~ ~ m , d Ais, an RAF of (X,m). (In fact, A need only be an RAF of ( X ,S m ) . ) (58.9) EXERCISE.If A is an RAF of (X,m ) , then Bt := J,’ m i 1 dA, is an RAF of (X, m). (Note that the condition (58.7), which states that dA, doesn’t charge the interval [Sm,GO[ on which m;’ = 00, is essential here.) We assume for the rest of this section that S is an almost perfect terminal time, but we do not insist that S be exact. (58.10) DEFINITION. The process 2 is homogeneous on 10, S [ (resp., 10, S [ ) provided 2 vanishes on 1S, 00 [I and 2 satisfies (58.11) (resp., (58.12)) for every T E T: (58.11) l I T , s I 0 ~ and 2 l~T,2 S ~are indistinguishable; (58.12) lgT,sl 0 ~ and 2 lgT,sn 2 are indistinguishable. The homogeneity on 10, S[I is perfect (resp., almost perfect) in case, for all (resp., a.a.) w , Z,(f3tw) = Zu+t for all t 2 0 and u > 0 with u+t < S(w). (Replace u > 0 with u 2 0 to get perfect homogeneity on [ O , S [ .) The class 49,is the a-algebra on 10, S [ generated by the a s . lcrl measurable processes which are almost perfectly homogeneous on 10, S [ , and fjd, is the
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277
u-algebra on [ 0, S [I generated by the a s . rcll measurable processes which are almost perfectly homogeneous on [0,s[I .
=
Note that if S 00, 2 is homogeneous on [IO,S[ (resp., I]O,S[) if and only if 2 is homogeneous on R+ (resp., R++)in the sense of (24.1). It is easily checked that if 2 is homogeneous on R+ (resp., R++)then Z l g 0 , S ~is homogeneous on 10, S [ (resp., I]O,S[I).By a trivial modification of the proof of (24.34), every process in sj: is indistinguishable from an almost perfect such process, and for Z E B;, there exists F E 3; (the a-algebra on R generated by foXtlIt<S), f a-excessive for ( X , S ) ) with Z t l ~ o , s ~ (-t )Fo&lno,sn(t) E Z. See §70. (58.13) DEFINITION. Let n be a RMin the class M defined in (28.6). Then is homogeneous on [IO, S [ (resp., 10, S [ )provided n( . , [S,m[) = 0 and n satisfies (58.14) (resp., (58.15)) for all T E T:
rc
(58.14) (58.15)
1[T,S[I
lgT,sn
* &K * &K
and lnT,sn and 1[T,S[
* n are indistinguishable;
* n are indistinguishable.
The perfect (resp., almost perfect) cases are defined in a now familiar way, but make sense only if S is a perfect (resp., almost perfect) terminal time. It is clear from the construction in $12 of the subprocess ( X , S ) of X killed at S that n is homogeneous on [O,S[ (resp., lO,S[) if and only if n is homogeneous on R+ (resp., R++) for the shift 6, for the killed process, and K ( W , [S(w),0 0 [ ) = 0. For this reason, we also speak of n as a RM which is homogeneous on [O, S [ or 10, S [ as a homogeneous RM's for ( X , S ) . It is clear from the perfection theorem (55.19) applied to the subprocess (X, S ) with the shift 6, that if A is a weak RAF of (X, S ) , there is a RM n which is almost perfectly homogeneous on 10, S [ such that n is indistinguishable from the RM generated by A. Note that if 2 is homogeneous on [ O , Sn (resp., 10, S [ ) , and if 2 is measurable and either bounded or positive, then its optional projection 'Z is also homogeneous on [O,S[ (resp., I]O,S[). This is an obvious consequence of (22.8). Similarly if n is a-integrable on 0 and homogeneous on [IO,S[ (resp., ! O , S [ ) its dual optional projection no is homogeneous on 10, S[T (resp., 10, S[I) by (31.5). The corresponding statements for predictable and dual predictable projections are in general true only if S is predictable. See 570 for the general case.
Markov Processes
278 59. Left MF’s and Exceptional Sets
Let m be a decreasing left MF. That is, (mt)is adapted to (3t), t + mt is as. decreasing and left continuous, and m enjoys the strong multiplicative property (54.3). Let m bea decreasingleft MFsuch that f(x) := P%o+ (59.1) THEOREM. is a nearly optional function. Then m is indistinguishable from an almost perfect left MF.
PROOF:For any T E T, a s . on {T < m},
Therefore m ~ = + m T f ( X T )a s . on {T < m) since the zero-one law (3.11) implies that mo+ = f ( X 0 ) almost surely. Since f ( X t ) is a nearly optional process, the section theorem (A5.8) proves that the processes mt+ and m t f ( X t ) are indistinguishable. Let S := inf{t:mt = 0). It is obvious that S is a terminal time for X and ms+ = 0 a s . on { S < m}. Let nt := f ( X , ) . Fix w with mt+(w)= f ( X t ( w ) m ) t ( w ) for all t 2 0. Then for all t < S(w),
nOl,,,
It follows that, a.s. on { t < S}, the product defining nt converges and so, as., nt > 0 for all t < S. Because 0 5 f 5 1, n t ( w ) is defined for all t 2 0 and t -.+ n t ( w ) is left continuous on ]O,m[ for all w E 0. It is clear that nt E 5 for all t 2 0 for all t > D, where D is the debut of {f = 0). Thus n is an almost perfect, decreasing left MF. Since reg(S) = {f = 0) is nearly optional, we may suppose that S is a perfect terminal time, by the perfection theorem (55.20). Let qt := mt/nt lno,sn(t). It is clear that q is as. decreasing and continuous on [0, S[. Moreover, qt E Ft and q satisfies the strong multiplicative property (54.3). Thus q is a decreasing MF. The set Eq of permanent points for q is determined by 1E&)
= P”{mo/no l{S>O}I = P”tmo+/no+ l { S > O ) ) = P ” { f ( x o ) / f ( x o l{f(X,)>O}) ) = l{f(S)>O}*
Therefore E, = {f > 0) is nearly optional. By (55.19), q is indistinguishable from a perfect MF. Replacing q by its perfect version, we see
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279
that mtlno,sn(t) is indistinguishable from qtnt luo,sn(t). Since m is left continuous and vanishes on IS, mu, mt is indistinguishable from fit := qt-mtl[o,sn ( t ) . Since qt- = qt except possibly at t = S, fit+s= fit f i s o B t for all (s,t ) except for those w E R such that m ~ + ~ ( #w ms(w)rnt(Qs(w)) ) for some t 2 0, or such that & t ( w ) # rnt(w) for some t 2 0. Hence f i is almost perfect, though in general not perfect. (59.2) COROLLARY. Let S be a left terminal time such that reg(S) is nearly optional. Then there is an almost perfect left termina.1 time S so that a.s., S = S . (59.3) THEOREM. Let R be an exact left terminal time. Then R is a.s. equal to the hitting time of the set F of regular points for R. The set F is finely closed, Ee-measurable and finely perfect.
PROOF:Since R is an exact terminal time, m := l[O,R[ is a decreasing exact MF so Em = E \ F is finely open and belongs to E" by (57.10). Let T := TF. By (59.1), a.s., R 5 T. Since R is a left terminal time, R = R ROQRa.s., and so for all x E E
+
0 = P"{RoQR> 0, R
< CQ}
= P " { P X R { R> 0); R
< m}.
Hence X R E F as. on { R < m}. If x 4 F , X R 4 F P"-a.s.on { R < T}. Thus P"{R < T } = 0 for all z 4 F . Let cp(x) := Pze-T and $(x) = Pze-R. Then since R and T are exact terminal times, cp and $J are both l-excessive. Since R 5 T a.s., cp 5 $J on E , and for x f F , R = T a s . so cp(z) = $(z). If 2 E F T , P Z { T = 0) = 1 so P"{R = 0) = 1. This proves that cp = $J except on the semipolar set F \ F'. By (10.13), cp = $ everywhere on E and therefore, as., R = T . In particular F \ F' is empty so F is finely perfect. Theorem (59.3) will prove to be of importance in connection with the notion of fine support in $64. We shall give below (59.7) an application of (59.3) to a probabilistic analogue of the Cantor-Bendixson theorem which states that every closed set in a Polish space is the disjoint union of a perfect closed set and a countable set. Before stating the result, we need to introduce some new terminology. (59.4) DEFINITION. Let p be an initial law. A subset F
cE
is:
(i) p-polar if {TF < 00) E NP; (ii) left p-polar if { I t > 0 with X t - E F } E NM; (iii) p-semipolar if there exists a nearly Bore1 semipolar set F, such that FAF, is p-polar; (iv) nearly semipolar if F is p-semipolar for every initial law p ;
280
Markov Processes (v) p-temporally countable in casePP{rF < m} = 0, T F denoting the penetration time (54.19) of F ; (vi) temporally countable if it is p-temporally countable for every
P; (vii) p-inessential in case F is p-polar and its complement is quasiabsorbing. Note that a nearly semipolar set is nearly optional and temporally countable by (10.16), and that a p-polar set is p-semipolar. We shall prove in (59.8) below that p-semipolar sets are the same as p-temporally countable sets, hence that nearly semipolar sets are the same as temporally countable sets.. It is as yet unknown whether a nearly semipolar set is necessarily semipolar. See (59.9). The other exceptional sets defined in (59.4) will be discussed following (59.9). The following four results are due to Dellacherie. The second is essentially an old result of Meyer [Me67a,p. 1801, which was proved only under the hypothesis (10.25) of absolute continuity. (59.5) LEMMA.Let F be a nearly optional set with penetration time T F , and let $(x) := P"e-TF, H := reg(rp) = {$ = 1). Then F \ H is temporally countable.
PROOF:The function $Jis obviously l-excessive, and consequently the process $(Xi) is a.s. right continuous. For p < 1, let Fp := F n {$ 5 p}. It suffices to prove that the penetration time R of Fo is a s . infinite. By (59.3), R = TK where K := reg(R). But, if x E K , then x is also regular for T F and thus $J(x)= 1. By definition of regular point, a.s., X , E {$J5 p } for uncountably many t in a neighbourhood o f t = 0, violating right continuity . K = 0, proving that R is a.s. infinite. of $ J ( X t ) Hence Let F C E be finely closed, nearly optional for X , (59.6) THEOREM. and p-temporally countable. Define Fi for every countable ordinal i by and more generally, transfinite recursion as follows. Let Fo := F , Fl := FT, for any countable ordinal i, having defined Fj for all j < i,
Fi :=
{ nj
if i is a successor ordinal, if i is a limit ordinal.
Then there exists a countable ordinal io (depending on p ) such that Fi, is p-polar. The set F is therefore p-semipolar.
PROOF:We may clearly assume that p is a probability on E. Note that every Fj is finely closed and in E" by (10.6). As { X E F } E P , [DM75, IV-1171 applies to show that is PP-indistinguishable from U,[Tn 1 for some
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281
sequence (Tn) C T. Define p n ( d z ) := P ~ { X T , E dx,Tn < oo}, Y := x n p n / 2 n so that v is a probability on E carried by F , and G c F is p-polar if and only if v(G) = 0. Since the function i -+ v(Fi) is obviously decreasing, the existence of an countable ordinal io with v(Fi, = v ( F j ) for 0-81. Note that F \ Fi, = Ui io is assured by [DM75, is semipolar by (10.16). We prove now that K := Fi, is p-polar. The set K is finely closed and in E", and therefore { X E K } is a.s. right closed. In addition, v ( K \ K') = 0 by construction, and so K \ K' is p-polar. Every T E T with graph contained in { X E K } therefore has the property that Ph-a.s., T is not isolated to the right in { X E K } if T < 00. Applying this to the optional times Tt := inf{s > t : X i E K } , we conclude that Pp-a.s., { X E K } has no isolated points. Hence, Pp-a.s., the set {X E K } is either empty or has a closure which is perfect (in the sense of classical real analysis) and must therefore be either empty or uncountable, since it can differ from its closure only by a countable set. It follows that {X E K } must be empty Pp-a.s., and the fact that K is p-semipolar comes from the observation that F = Ui
PROOF:According to (54.19), T F is a left exact terminal time. Since F is finely closed, H C F and by (59.3), H E E" is finely perfect and T F is the hitting time of H . Fix an initial law p. We prove that F \ H is p-semipolar. Let $(z) := P" (e-7F) so that II, E S1.By (59.5), F \ H is temporally countable, and it may be written as the countable union of finely closed sets F n {II,5 p } for p < 1. It follows then from (59.6) that F \ H is p-semipolar. REMARK.The set H defined in (59.7) is sometimes called the perfect kernel of F . The term is unfortunate, as the word kernel here has no connection with the more common object of the form k(z,dy),and the term perfect refers to real variable rather than the Markovian usage. (59.8) THEOREM.A nearly optional set is p-temporally countable if and only if it is p-semipolar.
PROOF:It follows from (10.16) that a p-semipolar set is p-temporally countable, so suppose that F is p-temporally countable. By [DM75, IV-1171, we may write { X E F } = U n u T n l up to Pp-evanescence, the T, being optional times for X . There is no loss of generality assuming that p is a probability. As in the proof of (59.6), define a probability v ( f ):= En2 - n f ( X ~ n ) 1 p n < o o f ~E, pE", so that v is carried by F , and a set G c F ( G f E") is p-polar if and only if v ( G ) = 0. If G c F
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is compact, then G satisfies the conditions of (59.6), and therefore G is p-semipolar. Let 6 denote the class of all p-semipolar sets in the Radon space F , so that 6 is closed under countable unions, and G' c G,G E 6 implies G' E 6. We may therefore decompose v = v1 v2, where vl does not charge sets in 6 and vz is carried by a set in 6. (This is like the Lebesgue decomposition of v relative another measure A, but the class 6 replaces null sets for p. Consider 6 := sup{v(G) : G E 6 } ,and choose d ~ G, E 6 so that v(G,) t 6. Let G := UG, E 6. Let dvz := l ~ and vl := v - vz.) The above argument proves that v l ( G ) = 0 for all compact G c F,and since all finite measures on a Radon space are inner regular, this proves that u1 = 0. Therefore F = F' U F" where F' E 6 carries v and v(F") = 0. The set F" is therefore p-polar, and consequently F is p-semipolar.
+
(59.9) EXERCISE. Suppose X satisfies hypothesis (L) (10.25) and let p be a reference measure for X. Prove that every p-polar set is polar, and every p-semipolar set is semipolar. It should be noted that p-polar sets are not nearly as useful or important as polar sets, because one may not in general delete such a set from the state space with impunity. This is the reason for introducing p-inessential sets. It is clear from (12.30) that a p-inessential set F may be deleted from E so that X on FC is a right process equivalent in law to X under Pj'. (59.10) PROPOSITION. Let f be a nearly Borel function on E. Then there exist Borel functions g , h with g 5 f 5 h such that { g < h } is both p-polar and left p-polar.
PROOF:The set { X - # X} := { ( t , w ) : 0 < t < oo,Xt-(w) exists in ET and X t - # X t ( w ) } is optionally meager for X by (41.8), and thus may be expressed as U,[R,l, where each R, E T. Let v n ( d z ) := PP{X(R,-) E dx,R, < oo}, and set v := p C, v,/2,. Choose then Borel functions g , h with g 5 f 5 h and P " { g ( X t ) < h ( X t ) for some t > 0) = 0. It is then evident that g and h have the claimed properties. (59.11) REMARK.This result is also valid for left limits taken in the original topology, as long as the sense of nearly Borel and left p-polar are changed to reflect the original rather than the Ray topology on E .
+
(59.12) COROLLARY. Let G C E be nearly Bore1 and semipolar. Then there exist GI, G2 E E with G1 C G c G2 and G2 \ Gl both p-polar and left p-polar. (59.13) EXERCISE. Let F C E be nearly Borel, and assume that FC is quasi-absorbing for X and that p is an excessive measure for X . Show that F is p-polar if and only if p ( F ) = 0.
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Let (f,) be a decreasing sequence in Sa and let (59.14) PROPOSITION. f := limn f,. Then { f > 0) is p-polar provided p{ f > 0) = 0.
PROOF:Because Pgf, 5 f n for every n, it follows that P$f 5 f . In particular, P$f(z) = 0 wherever f(z) = 0. If {f > 0) were not p-polar, the section theorem (A5.8) would give us a T E T and an E > 0 such that [ T i C { f ( X ) 2 e } and Pp{T < oo} > 0. Therefore, if f (z) = 0, 0 = PTQf(z) 2 € P S e - a T .
Thus P”{T < oo} = 0 if f(z) = 0. Consequently, p{f > 0) = 0 implies P”{T < 00) = 0, and this contradiction establishes that { f > 0) is p-polar. The next three exercises are valid only under the hypothesis that p is an excessive measure for X, in the sense of (40.16).
(59.15) EXERCISE.Let p be an excessive measure for X . (i) If F C E is finely open, nearly Borel and if p ( F ) = 0, then F is p-polar. (ii) If F E € p (the p-completion of €) and if p U a ( F ) = 0, then p ( F ) = 0. (iii) I f f , g E Sa and if p{f # g} = 0, then {f # g} is p-polar. (Hint: for (i), consider PI” 1F(Xt)d t . For (ii), note that F E E X for X := pPt pU”. Show that p P t ( F ) = 0 for a.a. t , and use p P t ( F ) T p ( F ) as t LO.)
fr
+
(59.16) EXERCISE.Let p be an excessive measure for X, and let f E
Sa. Prove that there exist g , h E S‘I, both Borel, such that g 5 f 5 h and {g < h } is p-polar. (Hint: consider first the case where f is an apotential, making use of (59.15iii). For a general f E Sa with a > 0, take f, E bE” with U af, T f and using the potential case, choose Borel g,, h, sandwiching U af, with {g, < h,} p-polar. Then g := liminf gn, h := liminf h, are a-supermedian, {g < h} is p-polar, and g 5 f 5 h. Set g := limg,, Ug+ag and similarly for h. Use (59.15ii) to show p { g < g} = 0 and p { g < h } = 0. For the case a = 0, take a, 11 0 so that f E Sent/,, use the case above to sandwich f by gn, h, having the asserted properties, and let g := liminf g,, h := liminf h,. Prove that g, h are supermedian and then argue as above.)
(59.17) EXERCISE.Let p be an excessive measure for X, and let F C E be p-polar. Then F is contained in a Borel p-inessential set which may be assumed finely open if F is finely open. In particular, a p-inessential set is contained in a Borel p-inessential set. (Hint: $(z) := P”(TF < 00) is excessive, p ( $ ) = 0 and F C {$ > 0) if F is finely open. Use (59.16) to dominate $ by a Borel excessive h with {$ < h } p-polar. Now use (59.15i), getting H := { h > 0) 3 ,F.) (59.18) THEOREM. Let F C E be p-semipolar. The F is the union of an p-polar set and a countable number of totally thin sets.
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PROOF: We observed earlier that F is the union of a p-polar set and a nearly Borel semipolar set, and it therefore suffices to prove the theorem in case F is thin. Let ‘p(z) := P ” e x p ( - T ~ )so that ‘p < 1 and ‘p is 1excessive. Use (18.5) to get a Borel H c F and a Borel h 2 ‘p with F \ H and {‘p < h } both p-polar. Put
H , := H fl { h 5 1 - l/n},
F, := F fl { ~p 5 1 - l/n}
so that H , C F, for every n. Clearly, H , E E, F = UF ,, and F\U,H, C U,(F, \ H,) is p-polar. Thus it is enough to prove that every H, is totally thin. Observe that TH, 2 TF, and therefore, for x E H,,
P“ eXp(-TH,) 5 P” eXp(-TF) = ‘ p ( X ) 5 h ( Z ) 5 1 - l/n, completing the proof. 60. Measurability of a MF
In some applications of AF’s and MF’s, time reversal and excursion theory being two relevant examples, one uses shifts OR where R is a random time which is not necessarily optional. If m is a perfect MF, there is no problem about the identity mt+R = m ~ r n ~ obut @ ~measurability , of mtOOR is a problem. If mt is only 3-measurable, mtO@R need not be in 3. Of course, if m is a concretely given MF such as exp{ - s,” f ( X B d) s } with f E bEu or l [ O , T [ with T the hitting time of a Borel set, there is no difficulty because mt E 7’. (It is a routine matter to check that H E 7* and R E p 3 imply HOBRE 3 . ) These considerations suggest that one should be prepared to find a version of a given MF having good measurability properties as well as perfect shift properties. See (24.36) for one such result. The first two results come from [BJ73b]. (60.1) LEMMA. Let G E b F t and suppose that for all H E b e , x + Px(GH) is in E (resp., E “ ) . Then there exists G E b e (resp., bFg) such that G = G almost surely.
PROOF: The argument is a simple modification of that used in the proof of (3.10). Define a finite measure Q” on the separable 0-algebra fl by Q ” ( H ) := P”(HG), H E b e . Since z + Q ” ( H ) and x + P”(G) are E (resp., E“) measurable by hypothesis, Doob’s lemma (A3.2) gives us a ‘p E b(E @ (resp., b(Ee €3 such that
e)
e))
P”(HG) = P”(’p(z,* ) . H ) V H E b P . e (resp., b3;). Then P”IG - GI = 0 for all x E E , Set G := ‘ ~ ( X O ,E b just as in the proof of (3.10). a
)
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(60.2) PROPOSITION. Let m be an exact decreasing MF. Then there exists m, an exact, decreasing, but not necessarily perfect, MF adapted to (F,",), indistinguishable from m. Applying this to mt := e-At with A an AF, it
follows that every AF has a version adapted to (3:+).
PROOF:Let ( Q t ) denote the semigroup generated by m. That is, for h E bE", Qth(x) := P"(h(Xt)mt].Since m is right continuous, if h E b(U,S") then for each x E E , t + Qth(x)is bounded and right continuous on R+. The Laplace transform of this right continuous function is V"h(x). Since m is exact, (57.9) tells us that V*h is a difference of a-excessive functions, hence V"h E bEe. By the inversion formula (4.14) for Laplace transforms, Qth E bEe. It follows by the MCT, using (8.7), that Qt(bE")c bE". Given 0 5 tl < . . . 5 t, = t and f l , . . . ,fn E bE", the multiplicative property of m gives
-
(60.3) P " [ f l ( X t , ) . -fn(Xt,)mt,l = Q t l [ f l . Q t z - t l [ * * *
[Qt,--t,-lfnI
... ]I*
As a function of x , this is in bEe. By the MCT, since E c E", x + P"[Hmt]E bEe for all H E b z . By (60.1) there exists a random variable mi E bE" such that a.s., mt = m:. One may assume 0 5 mi 5 1. Let my := inf { r n : : ~E Q+, T 5 t } . Then t -+ my is decreasing and for each fixed t , my 2 mt a.s. In addition, my E bF:. Finally, set mt := my+ so that t + mt is decreasing and right continuous. Given E > 0 and t 2 0, mt+e5 my++,5 mt a.s., so letting E decrease to zero through some sequence, right continuity of m proves that {mt # mt for some t 2 0) is null. Since riz and m are therefore indistinguishable, 7iz is also an exact decreasing MF. (60.4) REMARK.Under the stronger condition Va(bE)c bE, the proof above is easily modified to give kt E b e + for all t 2 0. In particular, if the 1-potential operator U i generated by the AF A maps E into itself, then A has a version adapted to (*+). The best result along the direction of simultaneous good measurability and perfection comes from [Me74a, p.1851, showing that given m, a decreasing exact MF, there exists a MF 7% indistinguishable from m and enjoying the properties: (60.5i) 7$1 is perfect MF; (ii) t * m t ( W ) is right continuous with values in [0,1]; (iii) Qt 2 0, E b3*; ( uall ) t 2 0 and w E 0. (iv) mt(u) = lim,llo & ~ s , t ~for In addition, if 0 has killing operators ( k t ) and if m = m 1 ~ o , c7iz~ ,may be chosen so that, in addition to (i)-(iv):
mtoks = mt l{s>t}.
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286 61. Subprocess Generated by a Decreasing MF
Let m be a decreasing MF of X vanishing on [ < , 0 0 [ . The semigroup ( Q t ) and the resolvent ( V u ) associated with m by (57.4) generate another Markov process. We give a construction of this process, called a subprocess of X ,and investigate conditions on m which imply that it is a right process with state space Em U {A}. The discussion here extends the procedure of ~ la suitable killing at a terminal time ($12) and reduces to it if m = 1 ~ 0 , for terminal time T . The construction below formalizes the intuitive notion of killing X at the rate -dmt/mt. We shall make the following hypothesis on m throughout this section. (61.1) HYPOTHESIS.m is a decreasing right MF of X .
In particular, (61.1) holds if m is exact. By (57.3), if m denotes the perfect exact regularization of m, then the debut D of E \ Em satisfies mD = 0 as., and m is indistinguishable from m t l i o , R n ( t ) , where R denotes the debut of G := Em \Em. According to (57.8), G is polar for (X,Sm). That is, R = 0 as. on {R < Sm}. In what follows, it is important that we assume that t -+ X t ( w ) is Ray-right continuous for every w E $2. In particular, X satisfies the refined right hypotheses of 820. For ease of manipulation, we suppose that R admits killing operators (kt) satisfying (11.3). See the discussion at the end of 511. We define for each x f Em a probability measure Pz on ( Q , Pby) (61.2)
P ” ( H ):= P”
io
H o k t (-dmt),
H Ebp.
rml
The integral, we emphasize, is taken over ]0,00], and the mass of -dmt at mt. The map ( t , w ) --* H ( k t w ) is in B+ 8 30, as one sees by a monotone class argument which reduces to the evident measurability of ( t , w ) --* f ( X , ( k t w ) ) for f E bE. The measures Ps extend automatically to 3’3 3”and the formula (61.2) remains valid, by sandwiching, for all H E bF*. Observe that if H E b q , (61.2) yields
t = 00 is defined to be mm := limt,,
= P’”
H o k s l{t
since < o k s = s A 5 s 5 t for all s 5 t . Moreover, H o k , = H for all s > t by a monotone class argument, and for s > t , t < C o k , if and only if t < C.
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As the measure (-dmt) lives on 10, (1, it follows that for all H E b e ,
PP{H l{t
(61.3)
It is clear by a simple limiting argument that (61.3) holds also for all H E b32;. Equation (61.3) obviously implies that (61.4)
Pp
5 Pp on the trace of
fl on {t < C } .
< t , and f l , . . . , f a f b€”, apply (61.3) with H := Letting 0 5 t l < fl(Xt,). . .fn(Xtn),noting that H o k , = 0 for all s 5 t,, to get
= P”
lt
I..,
[fl(Xt,) * * fn(Xt,)Iokt (-dmt)
n
= P”{f1(Xtl). . . fn(Xt,) 7 %
1.
Using (60.3), this last expression reduces to Qtl [ f i Q t z - t l
[a
* . [fn-1Qt,-t~-1fnI
]](z)-
This proves that under each measure Pz,(Xt)t>~ - has the Markov property with transition semigroup (Qt) and lifetime C. Let 3denote the completion of .P relative to the measures P p , where p is a measure on ( E m ,€),: and let 3, denote the usual augmentation of with respect to the P-null sets in 3. (61.5) THEOREM. Let m be a MF satisfying (61.1). Then the restriction of (0,.F, 3t,x,, et, Px) to E , is a right process having transition semigroup Qtf(z) := P”{f(Xt)mt} (z E E m ,f E bEk). Every a-excessive function for (Qt) is the restriction to E, of a nearly optional function for X.
PROOF:Let iiz, Q t , Va,G and R be defined as in the preceding paragraphs, and let F, 3 t and P“ denote the objects corresponding to 3, yt, P” but relative to the MF m in place of m. As we noted above, (0,$, yt,Xt, at, P”)has the simple Markov property with transition semigroup ( Q t ) , and similarly (0,f,Ft, Xt, &, P”)has the simple Markov property with transition semigroup ( Q t ) . By the earlier remarks about the relation between m and R , we have
Markov Processes
288 It follows at once from (61.7) that
P”H = P ” ( H ~ E \ G ) , H E bF*. The content of this last equality is that Pz is obtained from Pz by deleting the polar set G, an operation known (12.30) to preserve the right process property. In addition, (12.29) shows then that the a-excessive functions for P” are the restrictions of a-excessive functions for Pz. It suffices therefore to prove that V a satisfies the condition (7.4ii), which asserts that for all f E Cd(Em),f i ~:= {t -+ V a f ( X t )is not right continuous} is null for the P p with p carried by E*. We may in fact assume f E Cd(E), since the d-uniformly continuous functions on Em are simply the restrictions to Em of the d-uniformly continuous functions on E . The set is certainly null for X. In fact, since right continuity must fail before f if it fails at all, !=lo = U,,Q+& n {t < C } , and the set !=lo n {t < C } is null in Ft by the refined right hypotheses (20.4iii), since exactness of f i implies that Vaf is a difference of a-excessive functions for X . That is, for each rational t , there exists Rt E with Pp(Rt) = 0 and f i n ~ {t < C } c Rt. By (61.4), P ” ( R t n {t < C } ) = 0, hence fi0 is null for P p . Now let fi := {w : Xt(w)E E,Vt < C}. It remains only to prove that Pp(fi2) = 1 for all probabilities p carried by E m . Fix such a p. By (56.10), Em E E“, and is therefore nearly (Ray-)Bore1 by (18.5). Hence we may choose Bore1 (for the Ray topology) sets FI C E f i c F2 such that Pp{Xt E F2 \ FI for some t < f } = 0. By (57.2), f i g ! = 0, where D’ is the debut of E \ E f i . It follows that PpliiL~= 0, where D := DF;.The set {D < f} is in F* by (A5.2), and so by (61.2),
= Pp{mo; D
< f}
= 0,
where the third equality used the fact that D and s > D.
< s A 5 if and only if D < f
(61.8) EXERCISE. Suppose Ptl = 1 for all t. If mt e-at so that ( Q t ) = (PF), then Em = E and for any initial law p, the Pp-distribution of C is exponential with parameter a , and for all t , s 2 0,
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(61.9) EXERCISE.If m satisfies (61.1), then for t 2 0 and s 2 0, (61.5) implies that for any initial law ,u on Em
The infinitesimal version of this formula states that, provided mt is replaced by a version adapted to (Fy+),
This justifies the statement that X is killed at rate -dmt/mt. (61.10) EXERCISE.Let m be an exact decreasing MF. A function f on E is called a - ( X , m)-excessive in case f vanishes on E \ Em and f IE, is a-excessive for the m-subprocess of X constructed in (61.5). Using (57.7), show that every a - ( X ,m)-excessive function is in E". Use this and (8.5) to prove that iff E bEe and a 2 0, then Pz f E bE". (61.11) EXERCISE.Following the proof of (61.51, show that if m satisfies (61.1), then every a - ( X ,m)-excessive function is nearly optional for X .
(61.12) EXERCISE.Let n be a RM homogeneous for ( X , S ) . Prove that its a-potential function v,"(z) := PzJ ,e-at n(dt) is a - ( X , S)-excessive. Using the results of $48, prove that n is uniquely determined in the class of natural HRM's of ( X ,S) by v:, provided that function is finite valued. (61.13) EXERCISE.Let X be uniform motion to the right on R and let T := TO,the hitting time of 0, and m := 1 ~ 0 , T+ ~(1 - ,O)loT,mi, where 0 < /3 < 1. Show that m is an exact M F of X , and that the corresponding subprocess is X killed with probability as it passes through 0. (61.14) EXERCISE.Let X be uniform motion counterclockwise around the unit circle, T the first time X t - = x, with x = ( l , O ) , and let 0 < p < 1. Let T" denote the nth iterate of T (To := 0) and let m := p" on IT", T"+' I.Prove that m is an exact MF of X corresponding to killing X with probability p each time X passes through x. Show that the lifetime S for the subprocess is accessible but not predictable. (Hint: each T" is predictable, and for the last assertion, it suffices to prove that relative to the subprocess, { P l ~ > s ~0) = U,[IT"I.)
290
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62. Subprocess Generated by a Supermartingale MF The main construction in this section will require that R be of path type as described in (23.10). Recall that ( a t ) denote the stopping operators (23.9) for X. (62.1) NOTATION. T*denotes the class of optional times T over (F;),
For T E T*, (23.16) shows that T ( w ) 5 t and atw = atw' imply T ( w ) = T ( w ' ) . Recall (23.20ii) that for every optional time T , there exists T' E T* with T' = T almost surely.
-
(62.2) LEMMA.Let T E T*. I f a T ( w ) w= a q w ) w ' , then T ( w ) = T(w'). Let mean U T ( ~ ) W= aT(w)w'. Then is an equivalence relation on R.
WNW'
T
T
PROOF: By (23.16), T ( w ' ) = T ( w ) . In checking the conditions for an equivalence relation, the only non-obvious item is symmetry, and this is in fact the content of the first assertion. Given T E T*, F$ was defined in (23.15),and shown in (23.16) to be identical to the universal completion of 3$ A sandwiching argument shows that ~ ( X TE) F$ for every f E P . By (23.16), if aT(")w = a q w ) W , then T ( w ) = T ( a ) ,~ T W= ~ T Gand , X T ( W )= X T ( ~ ) .
(62.3) LEMMA.Let T E T*. Then F E FTand w-w' T
F ( w ) = F(w').
PROOF:Follows at once from (62.2) and (23.16). (62.4) DEFINITION.Let R be of path type. A sequence (w,) in R is projective relative to an increasing sequence (t,) in cme at, w,+l = at, w, for all n. The space (R, X t , Bt, k t , a t ) is projective provided, given any sequence (w,) in R which is projective relative to some sequence (t,), there exists w E 52 such that at, w = at, w, for all n.
For example, the space of all right continuous maps of R+ into EU {A} admitting A as a death state is clearly projective. Likewise, given any right resolvent (V") on E , the space R of paths admitting A as a death is also state, and rcll on 10, C[1 in a Ray completion of EA relative to (P), projective. At the final stage of our construction, we shall require the following result. (62.5) THEOREM. Let (a,X t , B t , kt, a t ) be projective. Fix an increasing sequence (Tn)in T*with limit T . Suppose given, for every n, a probability kernel QE from ( E ,E") to (R, F*)such that Q;(Xo = x) = 1 and for every F E bF*,
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(The existence of the integral in (62.6) is justified by (23.16ix).) Then there exists a probability kernel Q” from ( E ,E ” ) to (R, Vn3$n) such that for every n and x , Q” and Qt have the same trace on 3$n.
REMARK.The application we have in mind says the following, modulo details. If it is possible to define ( X s ) o s s < ~as, a simple Markov process as a on each time interval [O,T, [, then it is possible to define (Xs)s
Q:+,,W
=
J Qi(dw)F(wlTn(w)/[~~,(w)l)QW). =
It follows that the QZ are consistently defined on the algebra P. Therefore, we may define set functions Q“ on the algebra G so that Q”(B)= Q t ( B ) for B E 3$n. In order to prove the theorem, it suffices to show that the Q” are countably additive on 4. Formula (62.6) states that Kn(w,dw’), a kernel from (0,3$n) to (R, F*) defined by
is a regular conditional probability for QE+l given 3Gn. That is,
Qz+iF =
J
Qi(dw)KnF(w),
F E b3*.
By induction, this leads to the formula
Q& = QzKnKn+l.. . Km-l,
m > n,
using the usual notation for the composition of kernels. For F E b3$n of the form F ( w ) = J ( w ) f ( X , , ( w ) ) ,
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It follows that KnF(w)= F(w)for all F E bF;,. For m > n, the composition KnKn+l . . . K,-1 is a kernel from (0,Fgn)to (0,F). In addition, for B E F;,,Kklg = Kk+11g = .. = 1 ~ It. follows that for every n 2 1 and w E 0, there is a consistently defined positive, additive set function kn(w, . ) on G such that for B E Fgn,
kn(w,B) = K,. ..Km-l(w,B) V m 2 n. From the definition of the (62.7)
kn(w,B) =
s
K,,the K n satisfy
Kn+l(w/T,(w)/w',B)Qn+l(XT,(W),dw'),B E 4;
(62.8)
Let us write w
N
k
W in case UT,(,)W =
If w y 0, then Tk(w)= See (62.2). For F E b 3 * ,
UT,(,)~.
Tk(W),k~,w= ~ T and ~ W XT~(W) = X,(W). w-3 implies by (62.3) that k
Kk(w,F ) = =
J F(b,(,)w/Tk(w)/w')
Q;'"'"'(dW')
F (ICT, (&)W/Tk(a)/W') Q f T k (dw').
-
It follows that kk(W, = kk(W, ) on G if w y W . Suppose now that B, E Q and Bn 1. Countable additivity of Q" will nnBn # 8. Fix such an 2. follow once we prove that Q"(B,)2 6 > 0 Vn By (62.8), there exists w1 E 0 with Xo(w)= 2 and limn Kl(wl,B,)3 6. Suppose inductively that for all j 5 k, there exists wj E 0 such that Kj(wj,Bn) 2 6 for all n and wj ,m wj-1. Then by (62.7), a )
*
3-1
65 limKk(wk,Bn) =
I
limKk+i(Wk/Tk(Wk)/W',Bn)Qk-(-l(XTk(Wk),dW'), n
so for some w' E 0, limnKk+l(wk/Tk(wk)/w',Bn) 2 6 and x~(w') = x~,(Wk). Fix such an W' and let wk+l := Wk/Tk(Wk)/W'. By COWtrUCtiOn Of Wk+l,aT,(,,)(Wk)= aTk(W,)(Wk+l), and therefore wk+l y Wk. Hence, by projectivity, there exists w E R with w Wk for all k. This implies by N
k
(62.8) that limn kk(w,B,) 2 6 for all k. But B, E 3$,for some k, and so kk(w,Bn)= lg,(w),hence w E n,B,, completing the proof. The following will remain in force for the rest of this section.
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293
(62.9) HYPOTHESIS.m is a right continuous supermartingale MF of X adapted to ( F t ) ,vanishing on [I[, 00[, with Em := {z E E : P"(m0) = 1) nearly optional for X; (Qt) is the subMarkov semigroup generated by m from (Pi)and ( V a )is the corresponding resolvent. By an m-subprocess of X, we shall mean a right continuous Markov process with state space Em and semigroup (Qt). Since mt = 0 for all t 2 0 P"-as. if z 4 Em, we may and shall assume in the following construction that Em = E , by a preliminary killing of X at the debut of E&, if necessary. (This uses the fact that the killed subprocess is also a right process by (61.5).) It follows then from (24.36) and the subsequent remarks that m may be assumed adapted to (FF+). It follows then that Qtf E Ee for every f E bEe. The Doob-Meyer decomposition (51.1) gives a decomposition of m independent of the initial law: (62.10)
mt = m0
+ Lt - At
with ( L t ) a local martingale over X, A a predictable increasing process with PzA,5 1 for all z E E. As in §52, we shall use the notation 2: := supslt IZ,I for the maximal process associated with a process 2. (62.11) LEMMA.For any optional time T, supz P"L$/2 5 supz Pzm$ 5 1+sup, PxL$, and there are absolute constants c, C such that for all such T, cP"L$ 5 P"[L, L];'2 5 CPxL$.
+
PROOF:As m 2 0, m: 5 mo LF, which proves the second inequality. According to (50.17), there exist optional times T, t 00, not depending on the initial law, such that S U ~ , P ~ L<$00~ for every n. Then for every n, supz Pzm$n < 00, and therefore supz P"AT, < 00. By the Doob-Meyer decomposition of m stopped at T,, P X A ~ , , ~=, P z m ( T A T,) - P5mo I P"m*(T A T,) - P"m0. It follows that PzL*(T A T,) 5 Pzm*(TA Tn)Pzmo P"A(T A T,) 5 2Pxm*(TA T,). Letting n + ca completes the proof of the first inequality. The last inequalities are those of Davis (Ja791. An optional time T E T* with T 5 C is called a reducing time for m provided T 5 C and supx P"m$ < 00.
+
(62.12) PROPOSITION. There exists an increasing sequence (S,) of reducing times for m with limn S, = 6.
PROOF:By (51.1) and (50.17), there is a sequence (T,) of reducing times for m increasing a s . to infinity. In view of (23.21) and the remarks at the beginning of this section, we may assume T, E T*,and then it suffices to set S, := T, A C.
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Fix a reducing time T for m. Define then measures Qg on (0,3*) by
The first integral in (62.13) is supposed to be a stochastic integral as constructed in 552, independently of the initial law. We are making use of the remark following (52.17), permitting us to work with integrands such as F o k t in Pi instead of P. Note that Q$ is indeed a measure because, the integrand in (62.13) being in b’Pt and m t A ~being in the Hardy class H’ of (50.9), (-dmt) may be replaced by the predictable integrable increasing process A. Observe that if m is decreasing, Q$ agrees with the measure P” constructed in 561, a t least on F$-. (62.14) PROPOSITION. (i) QgW) = 1; (ii) Q+ (XO= r ) = 1; (iii) for F E bF,*,Q$(F l { , < ~ ) )= P” (Fm, I{,
+
PROOF: By (62.13), Q$(ln) = P” ((1- m ~ ) m ~ =) 1. In addition,
5 t, = s so that (F1{,,q)okt = Let F := fl(Xt,). . . f,(Xt,), 0 5 t l 5 Fl{s<~}l{e
(J Fl{s
- mT)1{3
+ F1{3
= P” ( F m u q , < T ) ) *
This proves (iii) for F E b3:, and the case F E b3: follows by sandwiching. Finally, the right side of (iii) is in E” for F E b F by a monotone class argument, hence for F E b F by sandwiching, and this proves (iv).
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295
(62.15) LEMMA.Let S, T be reducing times form, and set R := S+To0s. Then R is also a reducing time for m. PROOF:Assume sup, P+m$ and sup+P”m; are bounded by c < 00. By (23.16ix), R E T’. Now,
Thus R reduces m. (62.16) PROPOSITION. Let S, T be reducing times for m, and set R := S Toes as in (62.15). Then
+
(62.17)
Q E ( F )=
I
Qg((dw) F ( w / S ( W ) / W ’ ) Q ~ ” ‘ “ ’ ( ~F~E’ )b3*. ,
PROOF: As each side of (62.17) is a measure in F , it suffices to prove it for F E b p . Let (p(s,w,w’) := F(w/s/w’) E Po @I 30 by (23.11). Thus (62.17) is equivalent to the condition that for cp E b(PO8 p ) ,
By monotone classes, it suffices to prove (62.18) in case (p(s,w,w’) := K(w)H(w’) with Y E bPO and H E bF’. Let J ( w ) := Ys(,)(w) E b$so that F := v ( S ,ks,0s) = J Hoes. Then by definition of QE,
where
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Now, for t 5 S ( w ) , Os(ktw) = [A], so 11 = 0 In the second integral, substitute t = u 5’ (0 < ti 5 Toes) and observe that
+
( H O B S ) ( k i + S ( w ) 4=
H(@s(w)(k,+s(w)w)) = fwu(esw)).
Therefore, using ( 5 2 . 1 7 ~ and ) the SMP for the third equality,
In addition,
Hence
= Pz(Jms Q ; ” ) ( H ) ) = Q: =
1
( J Q ,x ( s )( H ) )
Qg((dw) F ( w / S ( w ) / w ’ )Q$”’“’(dw’),
completing the proof. Suppose that R is projective. Then there exists a (62.19) THEOREM. rendering X Markov unique Markov kernel Q” from (E,&”) to (QP) with semigroup ( Q t ) and Q“(X0 = x) = 1. In addition: (i) for every optional time T over (F,*), (62.20)
Q ” ( F ~ { T < c= > )P ” ( F ~ T ) , F E b-G;
in particular, Q“ << P” on the trace of F$ on ( T (ii) X is a right process relative to the Q”.
< C};
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297
PROOF:Choose reducing times S, for m with Sn T C. Let T~ := 0, T~:= sl,. . . , T, := T,-~
+ sn0eT,-,, ...
<
so that Tn 7 a.s. and T, reduces m for every n. By (62.15), each Tn is a reducing time for m. We shall use the construction described in (62.5) with QE := Q$,. Equation (62.17) shows inductively that (62.6) holds for every n 2 1. An application of (62.5) then produces a measure Q“ whose restriction to F$, is QE. Fix an optional time T over (F;). In proving (62.20), it suffices by the usual sandwiching to give a proof only in the case F E bpT. By (23.4), the latter is generated by products J f ox,, J E b e - , f E bE. Then J~OXT~{T<E T ,bF$, } and so
By ( l l . l l ) , for t 5 T n , Tokt
< Tnokt
%TA
t < Tn h t
In addition, f(XT)Okt = f ( X ~ ) l i ~ , Tand }
Jokt
T
< t.
= J for t 2 T. Therefore
But P ” ~
+ 00
Item (i) then follows from the MCT. Note that (62.20) yields Q”(f0Xt) = P”(f0Xtmt) = Qtf(z). Another application of (62.20) with T := t and F := Fn-lfnoXtn, with Fn-l :=
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f l o X t l ...fn-loXtn-l, f j E bE", 0 5 tl 5 Markov property of P" gives
Q"(F)= P"(Fmt, ) - P"(Fn-lmtn-t
+
+.5 t,
= t , together with the
(fnoXt,-tn_lmt,-tn_l>oet,-l)
= P"( Fn-1 Qt, -tn-
f n ~ x t ,mt,-l -~
),
and this proves that X under Q" is Markov with transition semigroup ( Q t ) . As we remarked in the discussion following (62.9), ( Q t ) preserves E". Thus, i f f E bE, Qtf E Ee is by (18.5) nearly Bore1 in the strict sense for X relative to P". We shall prove that Qtf is nearly optional for X relative to the Q", so that X is a right process relative to the Q" by (7.4). Given an initial law p on E , for every T E &+, there exist h;, hi E bE with and AT := { h l ( X , ) < hz(X,) for some s < r } E N f . Thus tf there hi exists < rh,' E with Pp(r') = 0 and A' c I". Then
&"(A'
n { T < C } ) 5 Qp(rr n { T < C } ) = P p ( m , l p ) = 0.
Let hl := sup,hi, h2 := inf,hi. Then A := { h l ( X , ) < hz(X,)} = U , . E Q + , T ~ t AnT { r < C } , and the argument above shows that each of the sets A' n { T < C } is Qp-null, so QP(A) = 0, proving (ii). Under the conditions of (62.19), (62.20) also holds (62.21) COROLLARY. and F E b3;+, where 3;+:= n,.>03;+~. in case T is optional over (3:+)
PROOF:Let S,, := T + 1/n so that the S, are optional times for (T:),and F E bF.& for all n. By (62.20), QZ(F1{S,
Q"{T < C } = P " ~ T .
The most important special case of the above construction is the socalled h-transform of Doob, in which m is the supermartingale MF specified by (62.23)
mt := h ( X t ) l h ( X o )l{O
with h E S. Since the sets { h < 00} and { h = 0) are absorbing for X , m vanishes a.s. on [S,,oo[I, where S,, := inf{t : mt = 0) = D, D := inf{t : h ( X t ) = 0 or h ( X t ) = 00). In th'is case,
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299
As a consequence of (62.19), the h-transform of X is a right process on the state space (0 < h < ca} with transition semigroup Q t ( x , d ~ := ) p t ( x ,d y ) h ( y ) l h ( z ) .
Jz
For example, if h = ‘ ~ l gwith B a RAF having finite potential, then for F E b 3 * , P”:J F o k t ( - d h ( X t ) ) = Fokt dBf for every reducing time T for m, and the latter term reduces to $ . Fokt d B t by definition of dual predictable projection. It follows that for every reducing time T for m, Q 5 ( F )= P”
(l*
Fokt dBt
1
+ Fh(XI-1)
lh(z).
Since h is of class (D) on 10, C [ I , we may take a sequence T, T C so that P Z h ( X ~ ,-+ ) 0, hence substituting T, for T in the last identity and letting n + 00 gives (62.24)
Q z ( F ) = P”
1
c Fokt d B t ,
F E b3*.
0
This example is of particular interest in case B := luL,,u l { o < ~ < c with } L a co-optional time, in which case Q ” ( F ) = P ” ( F o k ~ l { o < L < c } ) . In other words, X killed at L if 0 < L < C is an h-transform of X corresponding to the excessive function c(x) := P”{ 0 < L < C } . The general h-transform is of first rate importance in many investigations in Markov process theory, especially in the presence of strong duality hypotheses which permit X to be conditioned to die at a given point. (62.25) EXERCISE. Let X be a process having a transition demitypt(z, y) satisfying (16.8) relative to some measure 5. Let X := (R,B ) denote the forward space-time process over X . Fix (t, y) E R+ x E and let f(r, x) := pt--T(x,y)lrO,t[(T). Then f is excessive for X,and the corresponding htransform of X is a right process. Show that the space component Y of this h-transform is an inhomogeneous Markov process with transition density function relative to E given by #T,9(2,
2)
:= P s - - r k , Z)Pt--s(.z,Y)lPt-T(z,Y).
Under strong duality conditions, it can be shown that Y, + y a.s. as s TT t . If X is Brownian motion, then the temporally inhomogeneous Markov process (Y,)o~,
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300
(ii) V t 2 0, (62.27)
3:
E E , f E bEx, and every optional time T over
(e+),
Q “ { f ( x ~ + tI3;+) ) = QX(T’f(Xt).
Let F := {x E E : Q x { X o = z} = 1). Suppose that for every x E F and every t > 0, the trace of Q” lF; on {t < C } is absolutely continuous relative to the trace of P” lF; on {t < 6). Then there exists a supermartingale MF rn such that for every optional time T over (3:+), (62.28)
Q” ( H ~ { T < o = ) P” ( H ~ T ) , H E b3;+%
(62.29) REMARK. Condition (ii) applied with t = 0 and f = 1pC implies Q”{XT 4 F, T < C } = 0. This that for every optional time T over is weaker than the condition Q x { X t E F for all t < C } = 1, but if F were known to be nearly optional relative to X , the section theorem would guarantee the equivalence of the two conditions. PROOF:Fix t > 0. Note that the absolute continuity condition is satisfied restricted to {t < C } , trivially for x 4 F . On the separable a-algebra apply Doob’s lemma (A3.2) to obtain # ( x , w ) E E” 8 with
(e+),
ee
Q“ ( H l { t < < } )= P“
(dJ(x, * ) H ( .)l)
H E b-6’3 x E E -
Replace #(x,w) if necessary by ~ F ( z ) # (wz),so that we may suppose #(z, w) vanishes for x 4 F . Set then n t ( w ) := d J ( X ~ ( w ) , w ) l { ~ < E ~ ( ~to ) )yield (62.30)
Q” ( H l { t < ~ l=) P” ( H nt),
H E b c ,z E E.
Note that nt(w):= 0 for all t 2 0 if Xo(w) 4 F . Each side of (62.30) defines a finite measure on b e , so (62.30) holds by sandwiching for all H E 3:. Observe then that P”nt = Q x { t < C } so that t -+ P ” q is right continuous and bounded above by 1. Let H E b e and G E b c . Then
using the Markov property relative to Q“, and this reduces to
P”{H nt P x L( G n,)} = P”{H nt G O O t n, o O t } because of the Markov property relative to P”. As products H Go& with H E b e and G E b3: generate 3,”,,, it follows that a.s., nt+, = nt n,oOt. The fact that P”nt 5 1 implies that nt is a supermartingale relative to
VII: Mu1tiplica tive Functionals
30 1
every P". Consequently, the restriction of t + nt to Q+ is a regulated function, and therefore mt := lim,llt nr is a right continuous supermartingale with P"{mt # nt} = 0 for all z E E. See (A5.13). Therefore, (62.30) holds with nt replaced by mt. In addition, mt+s = mtm,oBt a s . for all s, t 2 0. We may and shall replace m by r n l ~ o , cbecause ~, the MF property and (62.28) are not affected by this change. Fix now a n optional time T over (F:+)with dyadic approximants T,, which are then optional times for (F:),and let H E bF$+ C bF$.,. Then, using (62.30) for m ,
because mT,, + mT in a uniformly integrable manner (A5.17ii). The only point left to verify is that mT+t = mT mtoBT for all t 2 0. It is enough to show that for H E b3; and G E b3?+,
and this follows by the same argument used above to establish the multiplicative property of nt , but using instead the strong Markov property relative to both Q" and P".
VIII
Additive Functionals
The connections between additive functionals, potentials and potential operators have been discussed in Chapter IV. This chapter contains a more detailed treatment of the structure of additive functionals (AF’s) and homogeneous random measures (HRM’s). In a rather natural sense, AF’s are the probabilistic analogues of measures on the state space. A function f on the state space E is studied probabilistically by examining the properties of the homogeneous process f(Xt). There is no comparable way of composing a measure on E with the process, so the connection between measures and AF’s is not as direct. The sense of the analogy comes more from potential theoretic ideas. The Riesz decomposition theorem of classical potential theory expresses a positive superharmonic function as the sum of a harmonic function and the potential of a measure. The probabilistic counterpart is (51.8), where the potential of a measure is replaced by the potential of an additive functional. Some of the results of this chapter demonstrate other ways in which AF’s resemble measures on E . Perhaps the most important property in this regard is the absolute continuity theorem (66.2), which is an analogue of the Radon-Nikodym theorem. Throughout this chapter we assume that for all w E a, t -, X t ( w ) is right continuous in both the Ray topology and the origidal topology and that for all t > 0, X t - ( w ) , the left limit in the Ray topology, exists in {z E E g :Po(z, E ) = 1). This is possible by (18.1) and (42.5ii).
V I E Additive Functionals
303
63. Classification of Additive Functionals In (44.1), we defined the random subsets J , J B and K of
R+ by
J := { X - # X , X - E E } ; J B := { X - E E B } ; K := { X - = X } . By convention, Xo-(w) := Xo(w). Then J U J B U K = R+ x R and J , JB and K are respectively the times of totally inaccessible jumps, branching jumps and times of continuity. As X E f i d and X - E fig, J , JB and K are perfectly homogeneous on R++ and in fig. Because we assume that all paths of X are Ray-rcll, p ( X t - , X t ) = limn p(X(t-llnl+,X t ) is strictly optional over hence so are J and K . Similarly, JB E Po. Thus J , K E 1 1 ° n 4 9 ,J g f P o n f i 9 . Let K be an optional RM. The RM's 15 * K and 15, * K are called respectively the totally inaccessible part and discontinuous accessible part of K . The totally inaccessible part of n is called the quasi-left-continuous part of K in the classical literature because, if K = 15 * K , then if T, 7 T in T, then ~(10, T,]} + &{lo,TI} a s . as n. -+ 00. This is obvious from the fact that T { T ~ Jis )a totally inaccessible time (44.5), because it must be that T, = T for sufficiently large n if n{[T]}> 0. Thus, ET is a totally inaccessible RM if and only if T is a totally inaccessible time. The part 15, * K of n carried by JB is accessible. See §45. Let S be a perfect terminal time for X . If the optional RM K is homogeneous on 10, S[, so are the RM's 15 * n, 1J~ * n and 1~ * n. If A is an AF of (X,S), then so are 15 * A, 15, * A and 1~ * A. We write in this case AX:= 15 * A and Ab := 15, * A for the totally inaccessible and discontinuous accessible parts of A. Thus
(e),
The remaining part of A is 1~ * A, which decomposes into a continuous part A" and a discontinuous predictable part Ad, where
A ~ ( u:= )
C
AA,(w)l~(s,w).
O<s
Since AAslK(s) is homogeneous on 10, S[ and predictable, Ad is a predictable AF of ( X , S ) . The continuous part A" is therefore also a predictable AF of ( X ,S ) . Thus every AF of ( X ,S) has the form
A = A" + Ad + A*
+ AX.
The different parts of an AF have quite distinct properties. The totally inaccessible part Ai has the simplest structure. It is shown in $73 that there
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304
exists a positive function f E Ee 8 & vanishing on the diagonal such that AAf and f(X+ , Xt)are indistinguishable. The discontinuous predictable part Ad has no general easy characterization, but in sufficiently nice cases such as under duality hypotheses, there exists a positive function f carried by a semipolar set such that AA,d and f(X,-) are indistinguishable. Aside from the fact that Ab is carried by JB and JB has a nice decomposition into a union of graphs of iterates of predictable terminal times (45.8), there is no general structure theorem available to describe Ab. The continuous part A" is in many respects the most interesting part of A. Even though it does not seem possible to give a simple representation of A" in terms of analytic objects on the state space, the next few sections will provide justification for the assertion that the properties of AC correspond to the properties of measures on E not charging semi-polar sets. Under duality hypotheses, there is in fact a bijective correspondence between these classes. 64. Fine Support We suppose for this paragraph that A is a continuous AF of X. The first measure-like property of A that we discuss is the notion of its fine support, introduced in [Ge64]. Because of (35.10) there is no loss in generality in supposing that A is almost perfect. As we pointed out in (54.17), if RA(w):= inf {t:At(w) > O},
(64.1)
then RA is a perfect, exact left terminal time. The fine support Supp(A) of A is defined to be the set of regular points for RA. According to (59.3), RA is a s . equal to the hitting time of the set Supp(A) , and Supp(A) E Ee is finely perfect. Note that A = 0 if and only if Supp(A) = 0. (64.2) THEOREM. The set Supp(A) is the smallest finely closed, nearly optional set K in E satisfying ~ K ( X*)A = A.
PROOF:If G is finely open and nearly optional, and if ~ G ( X*A ) = 0,then ) 1 for all sufficiently GnSupp(A) = 0, for z E GnSupp(A) implies l ~ ( X t = small t and At > 0 P"-a.s. for all t > 0. This shows that if K is finely closed and ~ K ( X*)A = A, then K 2 Supp(A). On the other hand, for = At A R 00, ~ = At since A R ~ 0. Therefore every t > 0, A ( t RAoB~) A does not charge intervals of the form It, t + R A O ~ The ~ [ .union of such intervals as t range over the rationals is the complement of the closure of { t : X t E Supp(A)} since RA is the hitting time of Supp(A). As Supp(A) is finely closed, {t:Xt E Supp(A)} is a.s. right closed (10.17) and so differs
+
+
VIII: Additive Functionals
305
from its closure in R+ by only a countable set. Since A is continuous, this proves that l E \ S u p p ( A ) ( X ) * A = 0. That is lSupp(A)(X)* A = A. It is a major unsolved problem to prove that every non-empty finely perfect set in E is the fine support of some continuous AF. For the solution in the case that X has a reference measure, see [Ae72b]. In analogy with terminology for ordinary measures, we say that A is carried by F E E" if ~ F ( X*)A = A, and that A does not charge F if ~ F ( X*)A = 0. Because of (10.16) we have: (64.3) PROPOSITION. A continuous AF does not charge any semipolar set.
Let U x be the a-potential operator associated with A. Then A does F ) = 0 for all not charge F E E" if and only if for some a 2 0, Ux(x, x E E . Continuous AF's whose fine supports are singletons in E are called (continuous) local times, and will be discussed briefly in $68. Since a fine support must be finely perfect, Supp(A) = {x} implies that x must be regular for {x}. It turns out that this condition is also sufficient for the existence of a continuous local time at x. 65. Time Change by the Inverse of an AF
We assume given a perfect continuous AF A of (X,C). For this section, let F denote Supp(A), the fine support of A defined in $64. For the following construction, it is important to recall that by convention, At([A]) := 0 for all t 2 0. We do not need to assume that At < 00 for all t. O the right continuous process inverse to At. That is, for Let ( T ~ ) ~ ?be w E Cl and t 2 0, (65.1)
~ t ( w := ) inf
{ u :A , ( w )
> t},
(inf 0 := 00).
See (A4.1). Obviously rt(w)< 00 if and only if t < A,(w). By continuity of A, A(T~)= t provided rt < 00. Observe that TO = RA, in the notation of $64, so TO is a s . equal to the hitting time of F , the fine support of A. ) Let S(w) := inf {t:At(w) = 00). The following properties of ( ~ t are elementary : each ~t is an optional time, and r t ( w ) < C (65.2) t < Ac(w); (65.3) ATt(u)(w)= t A A,(w) since At(w) is continuous in t; ) T ~ ( w provided ) ~ ~ ( <w00; ) if s < t, then T ~ ( w < (65.4) T,(w) = S(w) is a perfect, exact terminal time. (65.5) (65.6) PROPOSITION. For u , t 2 0,
+
+
T ~ + , ( w )= ~
+
( w ) Tu(OTt(ul(w)).
PROOF:Clearly T~ = rt inf {s:A8oOTt> u}. If ASofbt = - ATt = AS+Tt- t, so rt + rU0OTt= inf {s
T~
< 00, then
+ rt:s 2 O,AS+Tt> t + u } = Tt+".
306
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On the other hand, if 7t = 00, then 7t+u = 00. The result is thus proven in both cases. Given X and A its above, define the time changed process (&)t20 on R with values in E A by
Let ( be the lifetime for Y. By (65.2), ( ( w ) = A,(w). For t 2 0, set f&(w) := LJ,,(,)(w). Then (65.6) states that the 8, are shift operators for &. Since t -+ T t ( w ) is increasing and right continuous, t -+ & ( w ) is right continuous in EA. The filtration we use for the process (&) will in general be larger than its natural filtration even if we start with the natural filtration for X. Suppose ( M t ) is a filtration such that (R, M , M t , Xt,O t , P") is a right process. For each t 2 0 let Bt := M , ( t ) . It is easy to see that ( G t ) is a right continuous filtration to which (Yt)is adapted. (65.8) LEMMA.If T is an optional time over (Gt), then TT is an optional time over ( M t ) . If (Wt) is optional over ( M i ) , then (W,(t)l{,(t)<w}) is optional over ( o t ) . Every process optional over (&) which vanishes on I(, 00[ is of this form. In particular, if T is a (Gt)-optiond time such that T = 00 a.s. on {T 2 (}, then GT = M , ( T ) .
PROOF:Let T be an optional time over ( o t ) and let T, be the sequence of dyadic approximants (5.4) to T . For each u 2 0,
However, {T, = k/2n} E G k p n = M,(k/zn),so by definition of M , , {7(Tn) 5 u } E M u . Now let n -+ 00 to obtain {T(T) Iu } E Mu+ = M u . Next, if Wt is right continuous and adapted to ( M t ) , (W,(t)l{,(t)<w}) is right continuous and adapted to ( B t ) . It follows then from the MCT that ( W T ( t ) l ( , ( t ) < wE} ) O(&) for all (Wt) E O ( M t ) . Given an optional time T over (&), let Wt := l ~ , p ) , ~ [ ~ Then ( t ) . WT(t)l{,(t)<w} = 1 ~ , ( T ) , (~~I ~ ) 1 { , ( ~ so, ) < because ~} 7-t is strictly increasing on [0,A,[,
Since the stochastic intervals IT, 001 generate O(Gt), the third assertion is established. The fact that 6~ = M,(T) is an easy consequence of these facts, making use of (A5.12). We now have the notation and preparations necessary to state the main theorem on time changes.
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307
(65.9) THEOREM. Let F be the fine support of A. The process Y = (a,G,G t , Y,,&,P ” ) is a right process with state space FA and lifetime Ac. I f ( M t ) is a strong Markov filtration for X (6.10), then (Gt) is a strong Markov filtration for Y .
PROOF:Since F E E “ , 1E\F(Xt) is optional over ( M t ) and so, by (65.8), l ~ \ p ( Y ,is) optional over ( s t ) . If T is an optional time over ( G t ) , R := T ( T ) is an optional time over ( M t ) ,by (65.8) again, so for all z E F , P z { Y ~E E\ F, T < CO} = P“{Y, E E\F, T < (} 5 P”{XR E E \ F, R < m}. The identity (65.6) shows that That is,
TT
= TT+TOO&,
so T006R = 0 on { R < m}.
) optional, This proves that YT E F a s . on {T < co}, and since ~ E \ F ( Y ,is the section theorem implies that & E FA for all t 2 0, almost surely. We show next that (Y,)has the strong Markov property relative to ( G t ) . Given an optional time T over (Gt), t 2 0, f E bE” and z E F , we show
We may replace T by TiT
establishing (65.10). Since F is finely perfect, P”{Yo = z} = 1 for all 5 E F . To complete the proof of the first assertion, it suffices by (7.4) to prove that the transition semigroup (pt)for (Y,)has the property that for every t 2 0 and every f E C d ( E ) , ktf is nearly optional for (yt), f^ denoting the restriction o f f to F . Since, for x E F
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308
it suffices by (65.8) to show that each function ht(x) := P"f(X,(,)) is optional for X . Since f E Cd(E), t + h t ( x ) is right continuous for each fixed x. By the inversion formula (4.14) for Laplace transforms, it is enough e-"tht(x) d t is in E e . But to prove that x +
s r
In the last equality we used a simple case of the formula for change of variables described in (A4.3). Let mt := exp(-crAt). Then mt is a continuous decreasing M F which is exact because Em = E . Since m is continuous, dmt = dAt, so
Jo"
e p Q t h t ( x dt ) = P"
Jo"
f ( X , ) (-dm,) = Pmf(x).
We pointed out in (56.10) that Pmf E E", completing the first part of the proof. For the second assertion, one must prove a version of (65.10) with f(&) replaced by a random variable F E bG. Using (65.8), the computation reduces to the strong Markov property of ( M t )for X . (65.11) REMARK. Suppose it is known that the AF A is adapted to and that Supp(A) = E . Then the latter part of the proof shows that x + h t ( x )is in E for all t , and consequently Pt maps bE into itself.
(e+)
(65.12) EXERCISE. I f f is excessive for X , then f l is~ excessive for Y . (65.13) EXERCISE. If A is as. finite and if Supp(A) = E , then X and Y have the same excessive functions, the same polar sets, the same semi-polar sets, but not necessarily the same null sets. (65.14) EXERCISE. Let A be a.s. finite and let Supp(A) = E . Prove that Ee(Y)= E e ( X ) . (Hint: it is enough to prove that if V" is the apotential operator for Y ,then f E bpE" implies Vaf € E " ( X ) . Write mt := exp(-adt) and express V" f (x)= Pmf(x)/a.Then use (56.10).) The potential operator for Y is U A I F . (65.15) EXERCISE. (65.16) EXERCISE. Let A be a continuous AF of X . Using (10.40), prove that if p, u are measures on E such that pU2 and uU2 are u-finite, then pU2 = UU: implies p = u on Supp(A).
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66. Absolute Continuity
We come now to the analogue of the Radon-Nikodym theorem for AF's. The result (66.2) for continuous AF's is of great utility. The first version was proved for Hunt processes satisfying hypothesis (L) by Motoo [Mo62]. The proof below is due to Kunita (unpublished), and first appeared in print in [Me7la] after Getoor communicated Kunita's proof to Meyer. Kunita's proof is valid for a general right process. (66.1) DEFINITION.Let A and B be continuous AF's, a.s. finite valued. Then A is absolutely continuous relative to B ( A << B) provided, for every f E bpE, f ( X ) * B = 0 implies f ( X ) * A = 0.
The proof of the following theorem uses the Lebesgue differentiation theorem in a n unconventional form. One form of this theorem states that for a Radon measure X on R+ and C Lebesgue measure on R+, X << C implies that limhlo X ( [ t , t h])/C([t,t h ] )converges l-a.e. to a measurable function X ' ( t ) with A( [a,b ] ) = J : A'@) a(&). The less conventional form asserts that the same is true if C is replaced by any Itadon measure on R+ not charging points and not vanishing on non-empty open intervals. The proof is an exercise in real analysis using (A4.3).
+
+
(66.2) THEOREM. Let A and B be finite valued continuous AF's with A << B. Then there exists g E pEe such that A = g ( X) * B. If A and B are adapted to (*+), then g may be selected in p E .
PROOF:If A << B, if f E pE" and f ( X ) * B = 0, then given an initial law p , use a typical sandwiching argument to produce g, h E pE with g 5 f 5 h and PpJ,"(h - g)(Xt)d(At Bt) = 0. Then g ( X) * B = 0, hence g( X ) *A = 0, hence f ( X ) * A = 0 Pp-a.s., and because p is arbitrary, f ( X ) * A = 0. In other words, the condition in (66.1) holds for f E pE". We assume first that A and B satisfy the stronger conditions: there exists an AF A' such that A A' = B; (66.3) Bt 2 t for all t 2 0. (66.4) From condition (66.4) and the additivity of B, it follows that Bt - t is an AF. We can and will assume that versions of A , B and A' have been chosen to be almost perfect and B+ @ Fe-measurable. Let
+
+
(66.5)
Wt(w):= liminf(At+l/n(W) ,--roo - At(4)/(Bt+l/,bJ)
-
w4).
Obviously 0 5 Wt 5 1 for all t 2 0. Applying the extended Lebesgue differentiation theorem to the measures dAt(w) << dBt(w), we see that for 8.a.w E fllAt(w) =S,"W~(w)dBU(w)andJ~IWt(w)-W,'(w)ldBt(w) =O. Almost perfect additivity of A and B shows that for 8.9. w E 52, Wt(OUw)=
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Wt+"(w)for all t,u 2 0. Moreover, W E 23+ 8 F",and WOE Fo.Let g ( x ) := P"(W0) E E", 0 5 g 5 1. By the Blumenthal 0-1 law, as., WO= g(X0). It follows that for T E T, WT = g ( X T ) a.s., and applying this to the optional times rt := inf{s : B, > t} < 00, the change of variable formula (A4.3) and F'ubini's theorem yield pp
lw
1
00
IWt
- g(xt)l
dBt
= pp
- g ( x T ( t ) ) l dt = O.
IWT(t)
Therefore, as., At = s," g ( X u )dB,. The theorem is therefore proved under the conditions (66.3) and (66.4). Passing to the general case, set Ct := At+Bt+t. By the case above, there exist f , h E pEe such that A = f ( X ) * C and B = h ( X ) * C . Set g := f / h l { h , o ) . I claim that A = g ( X ) * B. Set k := l{h=O}. Then k ( X ) * B - ( k h ) ( X )* B = 0, so by hypothesis, k ( X ) * A = 0. thus
g ( X )*
=f~
~
~
I
~
= f (x)l{h(X)>O}*
~
~
~
* c{
~
h
(
X
)
,
c
-
- l{h(X)>O}* A = A, completing the proof of the first assertion. For the second, note t,.at if A and B are adapted to then obvious changes in the argument allow us to conclude that g E pE. The density g of (66.2) is unique in the following sense.
(e+),
(66.6) DEFINITION. Let A be any finite valued A F and f E E". Then f = 0 A-almost everywhere if and only if If [ ( X )* A = 0. Write f = g A-a.e. (or a.e.(A)) in case f - g = 0 A-a.e.
Note that if At = t A C, then f = 0 A-a.e. if and only if f is null in the sense described in $10. (66.7) THEOREM. Let A be a finite valued A F and let f , g E pE". If f ( X ) * A = g ( X ) * A is a finite valued AF, then f = g a.e.(A).
PROOF: Let k := l { p g ) .Then ( f k ) ( X ) * A= ( g k ) ( X ) * Ais a finite valued AF. Since f k 2 g k and, by our finiteness assumptions, it is legitimate to subtract, [(f - g ) k ] ( X )* A = 0. Hence (f - g ) { f z g )= 0 a.e.(A). Symmetrically, (f - g ) l { f l g ) = 0 A-a.e., so If - g1 = 0 a.e.(A). (66.8) COROLLARY. For any continuous AF A and any f E p€", there exists g E pEe such that f = g a.e.(A). In particular, taking At := t A C, it follows that for every f E E", there exists g E E" such that f = g a.e. PROOF: One may assume f E bpE". Take B := f ( X ) * A so B << A. Apply (66.2) to get g E bpE" with B = g ( X ) *A. By (66.7), f = g a.e.(A).
O
}
~
~
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There is a purely analytic version of the absolute continuity theorem (66.2). The following results are in fact valid for more general resolvents than those of right processes. See [Mok'TOa]. (66.9) THEOREM. Let ( U a ) be the resolvent of a right process. Suppose u := U a f < 00 a.e. (a 2 0, f E pE"). Then for a.a. x, (66.10) P ( u ( x )- pUP'%(x)) + f (x) as P + 00; - e-"'Ptu(x))/t -t f(x) as t 11 0. (66.11) (u(x)
PROOF:Since { U a f = 00) is polar, we may as well delete it and assume U af < ca on E. The resolvent equation gives
In addition, for t
> 0,
Suppose we show that this latter term converges to f(x) as t would have then
11 0. We
The first term on the right converges to zero as ,d + 00 and the second term converges to zero because the integrand is dominated by ae-asPsf (x)and pe-PSlp/~,,r(s) + 0 as /3 + 00. That is, (66.10) holds wherever (66.11) holds. Let At := s,' f (X,) ds and Bt := t. Then A is a continuous AF with finite a-potential function u = U a f , and A << B. Just as in the proof of (66.2), if y t ( w ) := liminf,llt(A,(w) - A t ( w ) ) / ( s- t ) , and Z t ( w ) := limsupSllt (AS(w)- A t ( w ) ) / ( s- t ) , then 12, - ytJds = 0 by Lebesgue's differentiation theorem, and (yt - f ( X t ) ( d s = 0 as., by the proof of (66.2). Given an initial law p, it follows from Fubini's theorem that for almost all t >_ 0, yt E F f . Let
fr
g(x) := liminf tll0
f
I
t
s;;"
'I'
e-*'Ps f (x)ds = liminf tll0 t
P, f (z) ds.
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For a t 2 0 such that yt E F f ,
For any sequence rn 11 0, we obtain by Fatou's lemma
Pp-almost surely. This being true for every sequence rn that yt 5 g ( X t ) PP-almost surely. Now let
11 0, we conclude
e-""P,f(z) ds = limsup tll0
The analogous argument using Fubini's theorem on limsup is valid because for almost all w , the Hardy-Littlewood maximal inequality shows that
O
foX,(u)ds < { ;l+r 1
00
for
8.8.
t 2 0.
One finds then h ( X t ) 5 Zt PP-a.s. for (Lebesgue) a.a. t 2 0. This proves that J r [ h ( X t )- g ( X t ) ] d t = 0 a.s. and l g ( X t ) - f ( X t ) l d t = 0 a s ., completing the proof. (66.12) DEFINITION. Let u, w E S". Then w is strongly dominated by the function u provided there exists w E So with w w = u.
+
Let an a-potential function of class (D) with poles (38.3). If w E S" is dominated by u (ie., w 5 u ) , then w is also an a-potential function of class (D) with poles. If v is strongly dominated by u and w w = u, both w and w are a-potential functions of class (D) with poles. Let u = u g , w = uz and w = u$ where A , B and C are predictable HRM's. The uniqueness theorem (36.10) shows that B = A + C. Delete a polar set so that A, B and C may be viewed as AF's. Then B and C are continuous if A is.
+
(66.13) THEOREM. Let B be a continuous AF with u s < 00 a.e., and Jet w E S" be strongly dominated by uz. Then there exists g E p&e with 0 5 g 5 1 and w = Ugg.
PROOF:By the above remarks, w = uz for some continuous AF A with A C = B , where C is another continuous AF. Now use (66.2). There are other forms of the absolute continuity theorem valid for larger classes of RAF's than continuous AF's, but the conclusions are weaker.
+
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(66.14) DEFINITION.Let A and B be arbitrary RAF's which are a.s. finite valued. Then A << B in case Z E bpfig, Z * B = 0 =$ Z * A = 0. The condition in (66.14) is satisfied if, in particular, there exists a RAF,
C , such that A + C = B . Notice also that if A and B are optional (resp., predictable), it is enough to check the condition in (66.14) just for 2 E b p ( 0 fl f i g ) (resp., bp(P n f i g ) ) . This is a simple consequence of the fact that if B is optional and 2 E f i g , then ( Z * B)" = " Z * B = 0 if and only if Z * B = 0. A similar argument is valid in the predictable case. (66.15) THEOREM. Let A and B be arbitrary RAF's which are a.s. finite valued. If A << B , then there exists Z E pfig such that A = Z * B. If A and B are both optional [resp., predictable), Z may be chosen in 0 n f i g [resp., Pnfig). If Z1and Z2 are two such densities, then IZ1 - 21 ' * B = 0.
PROOF:The uniqueness part is a consequence of (30.10). By (24.36), we may assume that A and B are almost perfectly additive. Then AAt, ABt E p ( 0 n fig) by (35.10). Decompose A and B as in $63. Since the sets J , JB and K of (63.1) belong to 0 r l f i g , A2 = 15 * A << 15 * B =: B i , A ~ K *= BB C + B d . A b - l ~ *, A << 15, * B =: Bb and A " + A d = ~ K * << Let Z := ljaB;,o). Then 2 * Bi = 0 so Z * Ai = 0. This implies that for almost all w , AAf(w) = 0 for all t such that A B ~ ( w=) 0. If one sets Z l ( w ) := AAj(w)/ABe(w)(with 0 / 0 := 0 ) then Z2 is a.s. finite, ZZ E f i g by (35.10), and Ai = Zi * B i . The process Zivanishes off J . The same argument applied to Ab and Bb gives a process Zb with A' = Zb * Bb with Z b vanishing off JB. Now let H := { ( t , w ) : A B , d ( ~ >) 0) so that H E fig. Also, H c K and H has countable w-sections so 1~ * B" = 0 and 1~ * A' = 0. I claim that A" << B" and Ad << B d . For, if Z * Bd = 0, z1H * B = 0 so z1H * A = 0. That is, Z * Ad = 0. By the same argument, 3 Zd E Onfig such that Ad = Zd*Bd. It may be assumed that Zd vanishes off H . Finally, if Z * B" = 0, Z ~ K \ H * I3 = 0 so Z1K\H * A = 0. But ~ K \ *HA = 1 q * A" ~ lK\HZd * Bd = ~ K \ H* A", SO A" << B". This condition certainly implies the condition of (66.1), so by (66.2) there exists g E pE" with A" = g ( X ) * B". Now set
+
Then A = Z * B because the components of A and B are carried by the disjoint sets K\H, H , J and JB . If A and B are predictable, Ai = Bi= 0, H is predictable, Zd and Zb are predictable, and we may replace g ( X ) by its predictable projection & g ( X - ) to obtain a relative density 2 E 2 5 . We shall see in 573 that if A and B are totally inaccessible AF's with A << B , then A = Z * B where Z = f ( X , - , X t ) for an f E p(Ee @ E ) .
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Glover [GI811 shows, assuming hypotheses (L), that in a simultaneous compactification of E relative to X and its reverse, the density Z in the predictable case may be chosen of the form f(X-). See (66.21) though. (66.16) EXERCISE.Suppose that A is an RAF which is not necessarily increasing but whose total variation is finite on finite intervals. Show that & t l IdA,I, its total variation on 30, t ] ,is an RAF. Hence show that the positive and negative variations A + , A - of A are RAF's. Show that if A is optional (resp., predictable), then so are A+ and A - .
(66.17) EXERCISE.Using (66.141, show that if A and B are continuous AF's, there exists a set F E E" and a function g E pE" carried by F such that l p ( X ) * A = g(X) * B and lp(X) * B = B. This is the analogue of the Hahn-Jordan decomposition. The AF's l p ( X ) *A and B are mutually singular in the sense that they are carried by disjoint subsets of E. (66.18) EXERCISE.Let 5 be a reference measure for X , A and B finite valued AF's such that for all t 2 0, Pr(At # B,) = 0. By considering IdA, - dB,I, show that A and B are indistinguishable.
h,,,,
(66.19) EXERCISE.Let A , B be continuous AF's of X. Suppose pU2 is ufinite on E . (Recall pUz(f):= p(dx) U2f (x).)Show that ifpU2 = p U 2 , then A and B are Pp-indistinguishable. (Hint: let C := A B so that A = f (X) * C and B = g(X) * C for some f,g E pE" with values in [0,1]. Let k := l{g,f}and choose k, 7 k with pU;(k,) < 00 for all n. Let D" := k,(X) * (B- A ) , and show pU& = 0 so that D n is Pp-evanescent. Then argue by symmetry as in the proof of (66.7).)
s
+
(66.20) EXERCISE.Let A be a continuous AF and let 2 E p u t be homogeneous on R++.Starting with the bounded case, show that there exists g E p E e such that a.s., 1.2, - g(Xt)l dAt = 0.
sooo
(66.21) EXERCISE.Let X be the deterministic process of uniform motion to be right on the crotch E pictured in (12.36). Let A := l ~ R , M ~ l { R > O } and B := l u T , m ~ ,where R := sup{t:Xt E F } and T := inf{t:Xt = 0). Show that A and B are AF's with A << B , but there is no function f on E such that A = f ( X ) * B or A = f ( X - ) * B. (66.22) EXERCISE. Let A , B , C be LAF's whose left potential functions are nearly optional and let A C = B. Show that there exists an optional function f such that A = f (X) * B. (Hint: (37.7) and (66.2)).
+
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67. Balayage of an AF
Part of the discussion of random subsets of character, which we now describe.
R++is of purely real-variable
(67.1) DEFINITION.A leading function is an increasing function T : [O,m] such that: (67.2i) T ( t ) 1 t for all t 2 0; (ii) T ( t ) > t implies T ( S ) = T(t) for all s E [ t , T ( t ) [ . A lagging function is an increasing function X : R+ -+ R+ such that:
R+ -+
(67.3i) (ii)
X(s) X(s)
5 s for all s 2 0; < s implies X(u) = X(s) for all u E]X(S),
s].
The slight dis-symmetry between leading functions and lagging functions is a consequence of defining them both on R+ rather than on R. The differences are in fact trivial and we shall treat them as interchangeable in ar gurnents. Two prime examples of such functions are given by the formulas (67.4)
~ ( t:=) inf{s > t : s
EM};
X(s) := sup{s
where M c R++.In these formulas, r is a leading function and X is a lagging function, T being right continuous and X left continuous.
___
leading function for M
-
lagging function for M the set M
Figure (67.5)
316
Markov Processes
(67.6) LEMMA.Let r be a leading (resp., lagging) function. Then the right and left limits of r are both leading (resp., lagging) functions. In addition, if r(t+)# T ( t - ) , then T ( t - ) = t and either ~ ( t=)t or T ( t ) = r(t+). PROOF: The first assertion is practically obvious. If ~ ( t + > ) t , (67.2ii) ) s €It,t c[. If, in addition, implies that for some E > 0, T ( S ) = ~ ( t +for r ( t )> t , (67.2ii) would yield ~ ( s = ) ~ ( tfor ) s €It,+)[. These conditions would force r ( t )= r(t+),a conclusion equivalent to the statement of the lemma. We shall abbreviate terminology by referring to a right continuous leading function as a right leading function. Left lagging functions, etc., are defined in an analogous manner. It turns out that the most useful cases are the left leading functions and the right lagging functions. It follows from (67.2ii) and (67.3ii) that if T is left leading and X is right lagging, then
+
(67.7)
T(T(t)) = T(t);
X(X(s)) = X(s).
(67.8) LEMMA.Let M c R++ and let ~ ( t := ) inf{s 2 t : s E M } . Then r is a left leading function if and only if M is left closed. Similarly, X(s) := sup{s 5 t : s E M } is a right lagging function if and only if M is right closed.
PROOF: Only the first assertion requires proof. Suppose M is left closed, and let t, t t. Set L := lim, r(t,). Then either: (i) t < L , in which case ~ ( t , )> t for some n, hence r ( t ) = .r(tn); that is, T ( t ) 5 L ; or (ii) L = t, in which case .r(tn) 5 t , hence t is in the left closure of M , hence in M . Conversely, if r is left continuous, then t, T t , t, E M implies ~ ( t , = ) t,, hence T ( t ) = t , so that t E M . That is, M is left closed. (67.9) LEMMA.Let r be a left continuous increasing function, and let X be its right continuous inverse. Then X is a right lagging function if and only if r is a left leading function. In this case, €or t > 0, r(t) = t if and only if X ( t ) = t , and the set A4 := { t > 0 : X ( t ) = t } = {t > 0 : T ( t ) = t } is a closed set in R++ with X ( t ) = sup{s 5 t : s E M } . I t follows that M is also the range { ~ ( t: t) > 0) of T and M = {X(s) : s > 0) n R++. PROOF: Let T be left leading. Obviously X(s) 5 s, and if X(s) < u 5 s, then r ( u ) := inf{v : T ( W ) 2 u} and left continuity of T imply ~ ( u>)s, hence that r ( u ) > s 2 u.By (67.2ii), T ( S ) = ~(u), and this implies A(=) := inf{v : r ( v ) > u } = X(s). Thus X is right lagging. The converse argument is similar. Now suppose T ( t ) > t. Then X ( t ) := inf{s : r ( s ) > t} < t by left continuity of r. Similarly, X ( t ) < t implies ~ ( t>) t. The set M
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is clearly left closed in R++, and because we may write M = { t > 0 : ~ ( t=) t } , it is also right closed in R++,hence M is closed in R++. Let X ' ( t ) := sup{s 5 t : s E M } so that A' is right lagging by (67.8). If t E M , then X ' ( t ) = t , hence X ' ( t ) = A ( t ) . If t 4 M , X ' ( t ) < t is the last point of M to the left of t , hence X(s) < s for all s E [ X ' ( t ) , t ] and X(X'(t)) = X ' ( t ) . According to (67.7), X ( t ) E M, hence X ( t ) 5 X ' ( t ) . By (67.3ii), X(s) = X ( t ) for s E [ X ( t ) , t ] ,hence X ( t ) = A'@). The last assertion is an obvious consequence of these facts. (67.10) LEMMA.Let M C R++ have closure in R++ and let T ( t ) := inf{s > t : s E M } . Then -r(t-) = inf{s 1 t : s E M } and M = {t > 0 : T ( t - ) = t } . Similarly, if X ( t ) := sup{s < t : s E M } , then X ( t + ) = sup{s 5 t : s E M } and M = {t > 0 : X ( t + ) = t } .
PROOF:If t g & and l t , Tt t , then t, 5 ~ ( t , )5 t for all n, so t 5 ~ ( t - 5) t , hence ~ ( t - = ) t. On the other hand, if t 4 M , there exists E > 0 with It - E , t E [ n M = 0, so that T ( S ) = ~ ( tfor) all s ~ ] -tt, t]. Thus, in this ) T ( t ) > t. This proves that M = { t > 0 : ~ ( t - = ) t } , and so case, ~ ( t - = by (67.91, ~ ( t -=) inf{s 2 t : s E a}.The lagging case is analogous. The discussion above shows that closed subsets of R++,left leading functions and right lagging functions on R+ are equivalent objects. Fix now M c R++, a closed set in R++.(Note: M := { 1,1/2,1/3,. . . } is closed in R++, even though it is not closed in R + . ) The maximal open intervals in R++\ M , excluding that with left endpoint 0 if there is such, are called the intervals contiguous to M . Let At := sup{s 5 t : s E M } and Tt := inf{s 2 t : s E M } so that M = {t > 0 : At = t } = {t > 0 : ~t = t}. The right boundary MDof M is, by definition, the set { t > 0 : A t = t < ~ t + of } left endpoints of open intervals contiguous to M . The left boundary Ma is defined as { t > 0 : At = t > At-} E 0. Given a 2 0 and a measure p on R++ with S r e - " t p ( d t ) < 00, the weak a-transport p(") of p is defined as follows. For 1.1 = E , with s > 0,
+
(67.11)
F'"'
:= exp[-a(s - ~ 8 ) 1 ~ x ( s ) ~ { x ( s ) > o } ~
For a general p , p(") is defined as the unique continuous linear extension of the map (67.11) of point masses. Namely, for f E pB(R+),
This formula takes on a more pleasing appearance when the term f ( s ) is replaced by f ( s ) e W a sto give the formula
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The weak a-transport of p on M could be described as a true sweeping of masses to the left along R++onto M , with mass sticking to the broom at an exponential rate according to the distance swept. Any mass on M is left fixed, and the mass on [0,T O [is entirely lost. In the special case where a = 0, no mass sticks to the broom, so that if ]a,b[ is an interval contiguous to M ( a > 0 ) , then all mass of p on ]a,b[ is swept to a E Mb. There is a corresponding a-transport p ( ( a ) )of p on M , differing from weak a-transport only in that A is replaced in (67.13) by A _ , so that
In effect, the a-transport moves mass of p at a right endpoint of an interval ]a,b[contiguous to M back to a, while the weak a-transport leaves that mass fixed. In Newtonian potential theory, the balayage of a measure p on a set F c Rn ( n 2 3) is defined to be the measure v carried by the fine closure of F and having the property
1
~ v ( z ) ( : =u(z,y)v(dy)) = inf{f(z) : f E S , f 2 up on F } ; z 4 F \ F T . In this formula, u ( z ,y) is the Green function on Rn.The right side is the reduite or reduced function of U p on F , discussed briefly in $49. By Hunt’s theorem (49.5), it follows that
where, as usual, PF denote the hitting operator for the set F . (The formulas above can be shown to be true in the wider setting of classical duality VI].) discussed in [BG68, The principal goal of this section is to give a formulation of balayage generalizing the Newtonian case, but without any special duality hypotheses on X . This will require that we consider balayage of AF’s rather than measures. It is more convenient to work with balayage not just on a subset F of E , but on certain random subsets of R++. (67.16) DEFIN,ITION. A homogeneous random set M is a subset M of
R++x R with M E 8 n 3s. The set M is closed provided almost every section M ( w ) := { t > 0 : ( t , ~E )M } is closed in R++. For example, if F c E is totally thin, its occupation time set M := { X E F } := { ( t ,w ) : t > 0, X t ( w ) E F } is such a random set. If X has continuous
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paths and if F c E is closed in E , then once again {X E F } is a closed, homogeneous random set, and the same is true for a general right process X in case M := { (X,X-) E B } := { ( t , ~: t) > 0,(Xt(w),Xt-(w)) E B } , where B c E x E is Bore1 and bounded away from the diagonal, E denoting a Ray compactification of E . Other interesting examples will involve closing certain homogeneous random sets which are not closed to begin with. For the rest of this section, given any random subset M c R++x R, A? denotes its closure in Rf+ x R. That is, M is the subset of R++x R such that for every w E 52, the w-section M(w) of M is the closure in R++ of the w-section M ( w ) of M . The following result is an immediate consequence of (24.37) and (25.12). Let M c R++ x il with M E npOp almost per(67.17) PROPOSITION. M , is in fectly homogeneous on R++.Then its pathwise closure in R++, 0 n 4 9 , and is therefore a closed homogeneous random set. Another way to generate a closed homogeneous random set is as follows. Suppose given an almost perfect, exact terminal time T . (For example, if M E 0 nfjg, T := inf{t > 0 : t E M } is such a time.) Define then (67.18)
Tt(w):= t + T ( O t ( w ) ) .
The almost perfect terminal time property of T shows that for w not in some null set, T,(w)> t s implies Tt+,(w)= T,(w). Exactness shows that t -+ Tt is a s . right continuous and increasing. Let ilo denote the null is not right continuous and increasing, set of w for which either t + Tt(w) or such that T,+t(w) # TB(w)for some s, t with T,(w) > s t. Then
+
+
(67.19)
+ Tt(e,w) = q o t + , ( w ) )
vt, s
2 0, w 4 no.
The map t + Tt(w)is a right leading function for every w $2 Ro, and (67.19) shows that the process t + Tt - t is in B d . Define the left leading function D t ( w ) := Tt-(w) for t > 0. By (67.19), (67.20)
+ ot(eSw) = ~ ( e , + , ( ~ ) )vt > 0,
2 ow
ao.
Equivalently, the process t + Dt - t is in 49. Then { t > 0 : &(w) = t } E 49,and Tt is a.s. its corresponding right leading function. (In the example, (67.10) shows that { t > 0 : D t ( w ) = t } = M ( U ) . If F C E is nearly optional, it follows from the discussion above that the closure in R++x R of M := { ( t , ~: Xt(w) ) E F } is a closed, homogeneous random set. I n this case, T = T F ,the hitting time of F , and by (10.6), we could substitute F U F', the fine closure of F , for F and not affect the set A? up to evanescence.)
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For the rest of the section, we suppose M is a closed homogeneous random set with right leading process Tt. Let 00denote as before the null set of w for which l ~ ( s , & + ~#ul) ~ (+s t,B,w) for some s > 0, t > 0, T 2 0. Let Dt := Tt-. The right continuous inverse function s + L,(w) of the function t -, Tt(w),is defined by L,(w) := inf{t : T t ( w ) > s} = sup{t : T t ( w ) I s}.
By (A4.4), { L , 2 t } = {Dt 5 s}, and because each Dt E T, { L , 5 t } E 3,. That is, (L,) is adapted to (3,), and because it is a.s. rcll, it is in 0.(A similar argument shows that its left limit 1, := L,- is in P.) According to (67.9), M = { ( t , ~: t)> 0, L t ( w ) = t}. It is easy to check that
(67.23) PROPOSITION. Let M be a closed, homogeneous random set with debut T,and let cp(z) := P"exp(-T). Then, for a.a. w , M ( w ) fl {t : cp(Xt(w))< 1) is a countable union of graphs of optional times.
PROOF:Since T is an almost perfect, exact terminal time, cp E S1.Fix E [0,1[ and let R be the debut of M n {t : cp(Xt) 5 p}. Since M ( w ) is closed in R++and cp(Xt) is as. right continuous, cp(X~)I p a s . on {R < 00) and R(w) E M ( w ) a.s. on (0 < R < m}. Let R1 := R, and define Rk (k 2 2) recursively by Rk := R"-' + R0QRk-i. By induction, for every k 2 1, cp(X(Rk)) 5 0 as. on {Rk < m} and Rk((w)E M ( w ) as. on (0 < R < m}. Using the fact that T ( w ) = inf{t : t E M ( w ) } 2 R ( w ) , one obtains
p
P" exp (-Rk+') = PZ{exp(-Rk)PX(RC) exp(-R)} < P"
exp(-R'
) F J X ( ~exp( ~ -TI
1
= P"{exp(-Rk)cp(X(Rk))}
5 PP"exp(-Rk) + . g k .
This proves that Rk 00 a s . as k 00, hence that M n {cp(X) 5 p } and Uk 1Rk I] are indistinguishable. Take now a sequence Pn TT 1 to complete the proof. --$
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321
(67.24) COROLLARY. Let M and 'p be as in (67.23). Then the decomposition MD = Mg U M,", where M,D := MDn { ' p ( X ) < 1) and M," := MDn {cp(X) = I}, is the unique [up to evanescence) decomposition of MD into an optional part M i and a progressive part M: containing the graph of no optional time.
PROOF: The set Mg is optional by (67.23). Let T denote the debut of M . If R E T and if [RI] c M', then ToBR > 0 a.s. on { R < co}, and so by the strong Markov property, c p ( X ~ < ) 1 a.s. on { R < m}. That is, '(lp) = 1 ~ ; The . corollary follows at once from this observation. Given M , Tt, L t , Dt and lt as defined earlier in this section, define the weak a-transport (resp., a-transport) on M of a RM fc satisfying (67.25)
1
e-at fc(dt) < oo as.
by simply letting d a ) ( w , . ) (resp., (ii((a))(w,.)) be the real-variable weak a-transport (resp., a-transport) on M ( w ) of K ( W , . ) for those w for which e-at ~ ( wd ,t ) < 00. That is,
(67.28) PROPOSITION. Let K. be a RM satisfying (67.25). Then RM's and, for every u 2 0,
d a )and
it((a))are
(67.29)
(67.30) PROOF:For every u 2 0 and s > 0, (67.22) shows t h t outside some fixed null set, (67.31) L,oBu > 0 if and only if L,+, > u; (67.32) if L,OB, > 0, then u L,OB, = L,+,.
+
322
Markov Processes
For a fixed w , L R O @ ~ ( = W )L~(e,,)(@,w) and, letting s := R(8,w) if R(0,w) < 00, the equality of &(ida)) and lgu,..U * OUn -(a)follows
1
from (67.31) and (67.32). It also follows for this particular K that d")is a RM since LR is 3-measurable. For a general n, let At(w) := n(w,]O,t]) so that At is right continuous, increasing, finite valued and measurable. By (A4.3),it is the case that for every Z E p M ,
That is, the RM dAt is the mixture relative to Lebesgue measure of RM's of the form E R ~ { ~ < R The < ~ general ). result for weak a-transport follows at once because both weak a-transport and shifts respect such mixing. The a-transport case is similar. (67.33) COROLLARY. Let n be a HRM not charging { 0 } , and let M be a closed homogeneous random set with debut T . Then the a-transport and weak a-transport of K onto M are also homogeneous on R++.If n has a-potential function uz, then da)) has a-potential function Psu:. K. on R++gives bun = * n. To prove that lnU,oob* & ( E ( ( ~ ) ) ) = lnu,ooo* do)), it is enough to prove
PROOF:Homogeneity of (67.34)
lgU,oon *(
~ n ~ *, K J~ ( ( ~n ) ) = lnU,OOn * ~((~1).
The identity (67.34) holds as a real variable identity for every fked w , as one may verify by letting ~ ( w.,) be point mass at s > 0, each side of (67.34) reducing in that case to e x p [ - a ( s - l ( ~ ) ) ] l { ~ ( ~ ) ,Thus ~ ~ ~ do) ~ ~ ~is)homoge. neous on R++. The a-potential function of is P"fOm e-at d a ) ) ( d t ) , and using (67.271, this evaluates to P" f? lfl(t)>o)e-atnfdt). However, l ( t ) > 0 if and only if t > T,so the last expression is equal to
There are four balayage operations one can define now corresponding to the dual optional and dual predictable a-transport and weak a-transport of n onto M .
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323
(67.35) DEFINITION.Let n be a H R M and let M be a closed homogeneous random set. The predictable (resp., optional) a-balayage of n OR M is the dual predictable (resp., optional) projection of The corresponding dual projections of ii(Q) are called weak predictable and optional a-balayages. According to (67.33), the predictable a-balayage of n on M is a predictable AF with a-potential function P$uc, where T is the debut of M . In case M is the closure of {X E F } , so that T = TF,this corresponds to the classical balayage of (67.15). The other cases of balayage are also interesting. See [GS74]. The following result is a simple consequence of (67.26) and (67.27). (67.36) LEMMA.Let K: be a HRM and let M be a closed homogeneous random set with debut T. Then (67.37)
+
i i ( " ) { t }= l ~ ( t ) n { t } I MD (~)[/ e--as ~ ( d s ) ] o e t ; ] O m
(67.38)
dca)){t) = lM\Ma(t)n{t}+ l ~ ~ ( t ) [ / e-Os
rc(ds)]oOt.
10,TI
Using the fact that if y is a RM , then y'{t} is the optional projection of t + y { t } , the fact (proved in (67.24)) that O ( ~ M M =D l~ ) ;and , the obvious fact that Ma E 0 , (67.36) leads immediately to (67.39) PROPOSITION. For K:, M and T as above, let y (resp., 7 ) denote the optional a-balayage (resp., weak optional a-balayage) of n on M . Let f(x) := P" &,Tr e--us n(ds) and g(x) := P" ePas n(ds). Then
hO,Tl
(67.42) COROLLARY. If ME is evanescent, then the optional and predictable a-balayage and weak a-balayage on M of any diffuse HRM, n, all agree and are diffuse. PROOF: Direct consequence of (67.39). (67.43) COROLLARY. Let n, M , and T be as above, and suppose no does not charge A4 and that M,D c J u p to evanescence. Then the predictable a-balayage of n on M is diffuse and is carried by reg(T).
Markov Processes
324
PROOF:Let y denote the optional a-balayage of K on M . By (67.41), y{t} vanishes outside J. It follows from (44.3) that the predictable a-balayage of K on M , which is the same as yp,is diffuse. Obviously yp is carried by M , and by (67.23), it is also carried by {'p = l}, where 'p(x) := P"e-T. That is, yP is carried by reg(T). 68. Local Times
Let x be a non-polar point for X. The random set { ( t , w ) : t > 0, Xt(w) = x} is right closed in R++, and its closure M in R++is an example of a homogeneous closed random set considered in the preceding section. There are two distinct cases to consider, according as x is regular for itself or x is thin for itself. Suppose first that x is regular for itself, and let T denote the hitting time of x. Let M denote the closure in R++ of { t > 0 : Xt = x} so that M has debut T . As in (67.23), let p(y) := P"exp(-T). According to (67.33) and (67.43), the predictable a-balayage on M of any AF A not charging {X = x} defines a continuous AF, B , with a-potential function u$(y) = p;)uQ,(y) = PYe-"TuQ,(XT) = G(xM;(y),
where @(y) := PY(e-QT). That is all B arising in this manner are the same up to a multiplicative factor. The particular B normalized to have 1-potential 4; = 'p is called the normalized local t i m e L" for X at x. By (67.43), the continuous AF Lz has fine support {x}. (68.1) PROPOSITION. Suppose x is regular for itself Then every continuous AF C with fine support {x} is a scalar multiple of L".
PROOF:For any f E pE, f ( X ) * L" = 0 if and only if f(x) = 0, and thus f ( X ) * L" = 0 implies f ( X ) * C = 0. That is, C << L". By (66.2), there exists g E pEe with C = g(X)* L". The scalar c := g(x) then satisfies c = CL". (68.2) PROPOSITION. Let x be regular for itself Then, for any right terminal time R for X , the PY-distribution of L s is exponential for all y E E .
PROOF:We may assume R = C by a preliminary killing of X in the manner of $61. It suffices to prove that PYexp[-pLT] = c/(p a) for some c = c(x,y), a = a(x,y). But, T denoting the hitting time of x,
+
PY
e x p [ - ~ ~ ;= ] PY exp [-,D
(LT
- L$)]
= PY exp [ - ~ L ? o & ]
< S) = PY(T< 5 ) ~ exp " [-PL?] . = Py (P"exp [-,l?L?] ; T
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325
Thus it suffices to consider only the case y = x. Let rt denote the right continuous inverse of L”. According to the time-change theorem (65.9), the process X ( 7 t ) with state space {.,A} is a right process. According then to (2.10), the lifetime of X ( T ~has ) an exponential distribution. But = LT, and this finishes the argument. A subordinator is an increasing process Zt taking values in [O,m] satisfying, for S := inf{t : Zt = m}, (68.3) for every t 2 0, the conditional distribution of Zt+s - 2, given t < S is the same as the distribution of Z,, and Zt+s - Zt is conditionally independent of o{Zu : u 5 t } , given t < S.
<
(68.4) PROPOSITION. Let x be regular for itself. The inverse T t of a local time L” is a subordinator relative to P”.
PROOF:According to (65.6), T t + s = T t p“
{
e-P(7t+s-7t)
; Tt
< m}
+ T S o B T s . For ,8 > 0,
{
= p“p”
e--13(7t+s-7t)
1
{7t<m)
I T,,}
I 37t} p”pX(7t)e-PTs l { T i <w) and because X ( T t ) = x for all t such that T t < 00, the last expression is = pZp2{(e-87s)0~rt 1{,,<,) 7
equal to P”exp [ - , 8 ~ ~ P]” { T< ~ m}. The condition (68.3) is a n elementary consequence of this calculation. Suppose every point in E is regular for itself. In the cone of continuous AF’s of X , the local times at points are the extremal elements, and it is natural to seek an integral representation formula representing every continuous AF of X as an integral mixture of local times at points. We do not have the tools available to give a general result in this section, and defer such a representation to $75. The following result is one basic ingredient. (68.5) LEMMA.Suppose that every point of E is regular for itself, and that the map 4i(y) := PY(e-T{z))is jointly measurable in (x,y). Then the local times L” may be chosen so that the map ( t , w , x ) Lt (w) is measurable. --$
PROOF:This is a direct corollary of (34.9) via (43.10). In the situation described in (68.5), given any measure p satisfying Sp(dx)di(y) < 00 for all y E E , ~ c ( w , d t ):= J p ( d x ) d L t ( w ) defines a HRM of X , and by Fubini’s theorem,
That is,
K
corresponds to the continuous AF Ct := p(dx)Lf
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69. Homogeneity and Subprocesses We consider in this section the relationship between ordinary homogeneity S),with S a terminal time for X ,as defined and homogeneity relative to (X, in 558. We assume throughout this section that S is perfect but we do not insist that S be exact. Form the random set M C R++ x R whose w-section is the closure in R++ of the set {t S(Otw):t > 0) n]O,m[ and let St := t So&. See $67 for a discussion of the following properties of M and St: (69.li) St+%(w) = & ( w ) Vu 2 0, t 2 0, w E R such that Su(w)> u t; , = 0 or St(w) = S,(w); (ii) for 0 5 t 5 u,either It, St(w)] n ] ~Su(w)] the w-section of I]O,oo[ \ M is equal to Ut,e++ It, St(w)[; (iii) M is optional and perfectly homogeneous on R++. (iv) Let M Cdenote the complement of M in l0,m [ = R++x a. Recall that &$ denotes the u-algebra on E \ reg(S) generated by the a-excessive functions for (X,S). As in $58, let 3.5denote the u-algebra on R generated by f ~ X ~ l ( ~
+
+
+
(69.2) THEOREM. Let Z E 3: (resp., fig). There exists a unique (up to evanescence) process 2 E fig vanishing on M (resp., M U [ O n ) such that lro,snZ = 2 (resp., lno,snZ = ln0,snZ). Moreover, if Z is either bounded or positive, then 72)= ("2): In particular, if Z E 0,then 2 E 0. PROOF:For uniqueness of 2, it is enough to remark that if W is perfectly homogeneous on R++ and W vanishes on both 10, S [ and on M then since M C= ut,e++ It, St [ and
W vanishes identically off an evanescent set. Suppose first that 2 E 33;. In view of (24.34) relative to (X,S), there exists F E 35 with Ztlno,sn(t) = FoOtlno,sn(t) up to evanescence. By the remarks preceding (69.2), there exists F E 3ewith F ( O t w ) = F(8tw) for all t < S(w). By the discussion following (67.19), M \ M b = {t : Tt = t } is in Set Zt := F o O t ( 1 - lM\Mp)(t)to get 2 E E j d with the correct restriction. Let f(x) := P"Z0 so thatf(x) = P"F for x E E \ reg(S). Then "2 = foXlno,su and "(2)= f o x , and the commutation of optional projections follows. It remains only to prove existence and commutation of optional projections in the fig case. By (24.34) for (X, S ) , 49, is generated by processes of the form Zt := lim,ttt F O B a , where F E 3.5 has the property t -, Foot is a.s. rcll on 10, S [ . By the MCT, it suffices to give a proof for such 2 1 0.
ad.
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Extend F to F E Fe as in the preceding paragraph so that Foot = F o o t for all t < S. Then ( F - F)ofltoO,. = 0 a.s. for all T > 0, and writing M C = UTEQ++ ]r, S ( r ) [ ,it follows that as., Foet = F o e t for all t E M C , and lim,tttFoOs exists for all t E M". Define Dt := St- so that, as mentioned in $67, Dt - t E f i g and M" = {Dt > t}. Given E > 0, the intervals where D t - t > E are those subintervals of M C distant at least E form the next point to the right in M, and as such they are separated from one is a s . rcll on R+ and perfectly hoanother. Therefore, F o & l { D ( t ) - t > , ] mogeneous on R+.Define 2; := lim,rtt Foe, l { D ( t ) - t > E } E fig, and let Z := supn Z1in E f i g . Then 2 = 2 on 10, SU. By uniqueness, in order to prove that " ( 2 )= ("2);it is enough to prove the equality of their restrictions to 10, S [ . But since 10, S [ E 0 ,
1no,suo(z) = o(lno,s[z) = O ( Z ) = 1Ro,sn("z)(69.3) THEOREM. Let n be a raw, perfect HRM of ( X , S ) , u-integrable on 0. There exists a unique RHRM it of X, u-integrable on 0 , and not chargingMUUOJ, such thatlno,sl*ii = l p o , s n * l c . Inaddition, (no)-= (ii)'. In particular, if n is optional, so is k.
PROOF:We may assume that n does not charge 101. Let yl and 7 2 be perfectly homogeneous on R++and lno,sn * y1 = lgo,su * 7 2 . For t 1 0, Q)t{lno,sn
* 71)
=Wln0,sn
* 721,
l R t , S ( t ) [ * 7 1 = l J t , S ( t ) [ * 7 2 . As M" = U t E Q + + It, S t [ , it follows that 1~~ * 7 1 = ~ A . I C * 7 2 . This proves the uniqueness of R. Turning to existence of R , let V denote supremum in the sense of measures and set
so
R(w, * ) := VtEQ++&(w,
-).
It is obvious that iidoes not charge M U [ O n . To prove the first assertion we prove the following points: ii is a kernel from (a,.?=) to (R+,B+); (69.4) R is a-integrable on 0 ; (69.5) ii is perfectly homogeneous on R++. (69.6) Since the limit of an increasing sequence of kernels is a kernel, (69.4) will follow once we prove that the supremum (in the sense of measures) of two B+) is also such a kernel. Let y := 6 1 l c 2 . kernels from (R, F)to (R+, Since (R+,B+) is countably generated, Doob's lemma shows that there t )y ( w , d t ) exists f(w,t ) E F@B+ with 0 5 f 5 1 such that n1(w,d t ) = f(w, and K ~ ( w d, t ) = (1 - f ( w , t ) )y ( w , d t ) . Then
+
(n1 v
nz)(w, d t ) = l { f ( w , t ) > l / 2 } m ( w , d t ) + l { f ( w , t ) < l / 2 } na(w1 d t )
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Markov Processes
a+). This proves (69.4). Since K is is clearly a kernel from (Q, 3)to (R+, a-integrable on 0 , there exists (28.1) a strictly positive Z E 0 such that supz P" J Zt d'(dt) < 00. Let At(w) := Z,(w)~ " ( wd,s ) so that A E (3 is right continuous, increasing, and A0 = 0. Moreover, sup,PzA, < 00. Let { r n } be an enumeration of Q+ and let
ho,tl
It is clear that Y is optional and strictly positive on I]O,m[I.One has then
However, for all t 2 0 and u 2 0, by definition of homogeneity on 10, S [ ,
from which it follows using (69.1) that
By (69.1ii), this implies that for a.a. w , & K ( w , . ) and 6 ) t + u ~ ( w.,) agree on It + u,St(w)[= It, St(w)[n ] t ti, St+%(w)[. Therefore
+
(69.8)
~ ( w .,) = &(w,
- ) as measures on It, St(w)[.
Returning to (69.7), we have
= C2-"Pz{eTn[Z*~](.,R+)} n
=
C 2-nP5Px(rn){[Z * .I(.
,R ' ) }
n
This proves (69.5). From (69.8), we obtain for u 2 0,
< 00.
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329
Since iiis carried by M C we have
- lnt+u,s(t+u)u E ,
*
proving (69.6). We prove now the last assertion, assuming that no is perfectly homogeneous on I]O,S[I. To prove that ( E ) O = (no); it suffices by the uniqueness result to prove that their restrictions to 10, S [ are indistinguishable. This is easy, for since 10, S [ E 0 , Ig0,sn * ( R ) O = (1g0,su * 6)' = no = Igo,su
* (no)-
(69.9) EXERCISE.Prove an analogue of (69.2) for Z perfectly homogeneous on 10, S [ . ( M must be replaced by M \ M,D, where ML is the optional part of the set of points in M isolated to the right. See (67.24).) (69.10) EXERCISE. Prove an analogue of (69.3) for n perfectly homogeneous on "0, S [ . (As in (69.9), M must be replaced by M \ M,D.)
70. Relative Predictable Projections We pointed out in 569 that the analogues of (69.2) and (69.3) are not valid for predictable projections unless S is predictable. The notion of relative predictability introduced below will allow us to obtain correct analogues of (69.2) and (69.3). It is a generalization of the notion of relative predictability (naturality) discussed in $48 and in A6. The difference is that we treat here predictability on a rather general random set rather than just an interval of the form [O, S [ . Most but not all features of the theory go through with minor changes. Let A c R+ x R be in M . Throughout this section, we shall suppose that A satisfies the hypothesis (70.1)
{ P l ~> 0) 3 A up to evanescence.
The hypothesis (70.1) is satisfied trivially if A E P,but there are other examples of much greater interest that we shall develop below. Throughout this section, given a random set A c R+ x 52, A ( w ) denotes its w-section, A ( w ) := {t 2 0 : ( t , w ) E A}. We showed in §48 that (70.1) holds in case A := [IO,S[.
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(70.2) THEOREM. Let A E 0 and suppose all of its w-sections are open in
R++.Then A satisfies (70.1). PROOF:For each t 2 0 let Rt(w) := inf{u > t : u E A“(w)}. Since A is optional, each Rt E T, and because A(w) is open, A = U{ It, Rt [I: t E @}. It suffices to prove that { p l ~> 0) 3 I t , Rt[ up to evanescence, for in this case, { p l ~> 0) 2 {P1it,Rtn > 0) 3 It, Rt[ for all t E Q+,up to evanescence. But, for R E T, 10, R[ satisfies (70.1) by (A6.15), and the proof of (70.2) is completed by observing that
(70.3) EXERCISE.Let R E T. Then A := [RI] satisfies (70.1) if and only if R is accessible. (Hint:if R is accessible, one may reduce to the case [RI] C JB. Set A := { P l ~ R ~ ( R =) 0) and replace R by RA to obtain [Rl C JB withPluRD(R) = 0 afmost surely. Enumerate [ J B ] by predictable times {T,} and show that P”{R = T,} = 0 for all n.) Suppose for the rest of this section that A c R+ x S2 satisfies (70.1), and let := { p l ~> 0). By hypothesis, 3 A and ii E P. (70.4) LEMMA.The set set containing A. PROOF: Let I? 3 A and
is the minimal (up to evanescence) predictable
r E P. Then for any predictable time T,
which implies by the section theorem that P l A l p is evanescent, hence that ii c r, up to evanescence. Let M A , OA and P A denote the respective traces of M , 0 and P on A. We interpret Y E P A to mean that Y = 2 1 for ~ some 2 E P. That is, Y is automatically extended to be zero outside A. Similarly, a RM K is said to be predictable on A if there exists a predictable RM y such that K = * 7. (70.5) DEFINITION. An optional time T is predictable on A provided there exists a predictable time R with A n [TI] = A n [RI].
It is clear that if T is predictable on A, then the set A n [TI is in PA. The converse is not true in general-see (70.11) below. The relative predictable projection of 2 E b M on A is defined by (70.6)
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331
where 010 := 0. Clearly XZ vanishes outside 6. The relative dual predictable projection on A of a RM o-integrable on P is the RM K.: given by
K.
which is
Clearly, IE: is carried by 6. Similar definitions could be made for the opt> 0 on A, but our applications ional projections with A E M satisfying ‘ 1 ~ will involve only the predictable case. (70.8) LEMMA.Let Z E p P vanish on A. Then XZ = 0 u p to evanescence. Similarly, if IE is a RM which does not charge and which is o-integrable = 0. on P, then K.;
PROOF:Since { Z = 0) E P contains A, it contains 6 up to evanescence, by (70.5). The first assertion follows at once from the definition (70.6). The second assertion is equally easy. The assertions of the following theorem are proved by a routine application of the definitions of predictable and dual predictable projections. See the corresponding cases in $48 and $A6. (70.9) THEOREM. ; is the unique member of P A such that for (i) For Z E bMA, Z every initial law p and every predictable time T over (Ft),
P ” { : Z ( T ) ; T < W} = P p { Z ( T ) ;T < m}. (It suffices that this identity hold only for times T which are predictable on A.) (ii) If K. is a R M carried by A which is o-integrable on P, K.; is the unique RM which is relativelypredictable on A such that for every initial law p and all Y E pP,, PP
jo Yi K.X(dt) = jo yt K.(dt). PP
(70.10) COROLLARY (SECTIONTHEOREM). Let Z E bPA and suppose that P P Z T ~ { T < ~=) 0 for every time T predictable on A. Then Z is evanescent.
PROOF:Direct consequence of (70.9i).
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Let X be process on the unit circle killed as in (61.14). (70.11) EXERCISE. L e t T : = C a n d A : = I T , o o [ . ProvethatAsatisfies(70.1), that [TI]€ P A , but there is no predictable time R with [RI] = [TI] fl A. (Hint: if there were such an R, then P l [ T ] 5 P l n R n implies that {PlnTn > 0) C [RI]. Use now the second part of (61.14).) Let S be a perfect terminal time and let M be the random closed set defined above (69.1). Define, as usual, S ( t ) := St := t S o & . By (70.2), the random sets 10,SI and M" satisfy the hypothesis (70.1).
+
o , homogeneous s~ on 10, S [ . (70.12) LEMMA.The process l ~ o , s ~ / P l ~ is
PROOF: Let T E T.By (22.13), 1nT,su @T('1no,su) = 1nT,su (P(@rlno,su) and @ T l g o , s ~= l j o , q T ) u . Since S(T) = S on {T < S}, the result follows. Relative predictable and dual predictable projections on 10, S [ commute with shifts, in the following sense.
(70.13) THEOREM. Let A := I]O,S[,Z E bM and let is a-integrable on P. Then for T E T, (70.14)
(70.15)
K,
be a R M which
I"[, = 1nT,S[Ix [ @ T z ] ; l ] T , S [ * &[4 = 11T,S[ * [6TK,I;. 1nT,S[I@T
In particular, if 2 (resp., K,) is homogeneous on R++ or on 10, S[I, then ! Z (resp., K,;) is homogeneous on 10, S [ .
PROOF:Let yt := l ~ o , s ~ / P l ~ o We , s ~shall . prove only (70.14), the proof of (70.15) being entirely analogous. It must be shown that, with equality meaning indistinguishability, 1jT,su @ T [ Y ~ Z =I 1 n ~ , s u Y "[@TZ]. By (70.12), lnT,sn@TY= llT,suY. But (22.13) states that lnT,m[i "[@rZ] = lnT,mii
@T[~Z]. The equality (70.14) is therefore evident. If Z is homogeneous on R++or on I]O, S [ , 1nT,sn 0 ~ =21nT,su 2 , and it is then clear from (70.5) that 1]T,S[Ixz = x[1nT,SI[Z].
Thus (70.14) states that = InT,suXz, ~IIT,SU@T[P,Z] which shows that x.27 is homogeneous on JO, SR. The following results are the predictable analogues of the homogeneous extension theorems (69.2) and (69.3). For these next two results, let A := M".
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333
2,w
Let Z , W E 49, and let E fig be their homo(70.16) THEOREM. geneous extensions (69.2). If W := : ( Z ) , then = ;((z). That is, ( ~ S Z ) - = ;((z). In particular, i f 2 E ~ sthen , Z E PA. (70.17) THEOREM. Let n be a raw perfect HRM of ( X ,S) with hornoge(ii);. neous extension it, (69.3). I f y := K:, then 7 = ( E ) ; . That is, (n:)-= In particular, if n is predictable on 10, S [ , then it, is predictable on A.
PROOF:We prove only (70.17), the proof of (70.16) being similar but easier. Observe first that A E 49, so l A / ’ l A is homogeneous on R++ by so is 3,and there(22.11). Since (69.3) shows iiis homogeneous on R++, fore = 1 # 1 ~ * izp is also. (Note that the latter is defined because (69.3) shows that it, is carried by A.) Because of the uniqueness part of = y. Since (69.3), the result will follow once we show that lno,su * t%i no,su = j o , s n n A E P,,
If we show that l h / P l ~is the homogeneous extension of l ~ o , s n / P l ~ o , s ~ , the last equality will yield the claimed equality, since n is carried by 10, S [ . The extension property is equivalent to the equality of P l a and f’lj0,sn on I]O,S[. Since nO,S[ nA = 1O,S[ and 1O,S[ E P A ,
from which the equality sought is obvious. (70.18) EXERCISE.Let X be uniform motion to the right on R killed with probability p E ] O , l [ a t 0. See (61.13). Let S := C so that M = [I(, 001. Show that P l ~ c = i / ? l , c , - , ~ ~ ~Use . this to show that if n is the RMputting mass 1 at C if < 00, then the relative dual predictable projection of n on lo,([ puts mass p/(l - p) at To ifT0 < 00. (70.19) EXERCISE.Let X be uniform motion counterclockwise around [ time it reaches x := the unit circle, killed with probability p ~ ] 0 , 1each (1,O). See (61.14). Let S := C. Show that Pluci = P‘&l l u p i where T := inf{t:Xt- = z}. (Hint: [<1 is contained in the predictable set {Xt- = x} which can be expressed as the union of the graphs of the iterates T n of T . Use this to show that if n puts unit mass at C, then ‘E50,cu = En21 1 1-6 E p .)
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71. Exponential Formulas Appendix A4 gives an account of the Stieltjes logarithm slog and the Stieltjes exponential sexp. Observe that if m is a decreasing almost perfect M F with S := inf {t:mt = 0}, then for almost all w € 0 such that mo(w) = 1, t -, mt(w) is an M-function in the sense of §A4 so a t @ ) := slogmt(w) = J;o,tAs(w)](-dm,(w))/m,-(~) is an A-function in the sense of 5A4.
(71.1) THEOREM. Let m be a decreasing almost perfect RMF of X and Jet S := inf {t:mt = 0). If one defines At(w) := 0 if mo(w) = 0 and A t ( w ) := (slogm)t(w) if mo(w) = 1, then A" := lgo,si * A is an almost S). perfect RAF of (X,
PROOF:It suffices to show that, after excluding the exceptional set for the multiplicative property of m, At+,(w) = A t ( w ) + A,(&w) provided t + u < S(w). If t + u < S(w ), then since S ( w ) > 0,
(71.2) THEOREM. Let S be an almost perfect terminal time and let A be a almost perfect RAF of (X, S) such that a.s., At < 00 ift < S and such that AAt < 1 for all t < S. Then m t := l[o,s[(t)exp(-A~)no,,5t(l - AA,) is an almost perfect RMF of X. PROOF:If t + u < S, mt+, = mfm;oBt is obvious from the fact that each factor in the product defining m is multiplicative over that range. If t + u 1 S, mf+, = 0. If m e > 0 so that t < S, then
hence m;o& = 0. This proves the result, taking into account the obvious exceptional set.
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335
(71.3) PROPOSITION. Let S , A and mA be as in (71.2) and let K: be the homogeneous extension of A constructed in 569. If S is the exact regularization of S, then mA, the exact regularization of mA, is given by (71.4)
r$
:= l n o , (qt ) exp(-~:~(10, t])
n
(1- K:{u)).
O
In particular, mA is exact if and only if for all x E reg(S) \ reg(S), PZ{ti{]O, t]} = 00 for all t > 0) = 1. Thus if S is exact, mA is exact.
PROOF:By definition of the homogeneous extension, tcC(]0,t])= At if t < S and ~ { u=}AA, if u < S. Thus mt = l[O,S[(t)exP(-K:c(lO,tl))
n
(1 - .{.I).
O
Hence for 0 < T 5 t , m]r,t]= mt--roer =
l[r,r+soe,[(t)exp( - K : ~ ( I T , ~ I ) >
n
(1 - .{.I>.
r
+
+
Since T Sod,. 1 S as T 11 0 and T So& = S for sufficiently small T > 0 if S > 0, (71.4) follows by letting T 11 0. The remaining assertions are obvious consequences of this identity. The transformations m (71.5) PROPOSITION. mA of (71.2) are inverse to one another.
+
A" of (71.1) and A
+
PROOF:By (A4.17), for all w E R , the transformations m.( w ) I [O,S(w)[
+
I [ O , S ( w ) [ and A . (w)I [O,S(w)[ .-+m A ( 4 I [O,S(w)[
are inverse to one another. By definition of mA and Am, only the restrictions to [0, S(w)[ are relevant to the transformations. Following Meyer [Me66b], we derive some useful relationships between the kernels Uz,V a and Pg defined below for S an almost perfect terminal time, m an almost perfect decreasing MF and A an AF of (X,S). For f E pE", set FS
(71.6)
(71.8)
U $ f ( x ) := P"
lo
e-"'f(Xt) dAt;
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Markov Processes
The operators V" and PE were introduced in $56 in connection with the identity
U Z f = V"f
(71.9)
+ P;U"f,
f E pE".
The operator U z is the same as that introduced in (36.14) in the special case S = 00. We continue to refer to U z as the a-potential operator for A though it would be more precise to call it the a - ( X ,S)-potential operator for A . For f E pE", U zf is the a-potential function of the HRM K of ( X , S) given by ~ ( w , d t:= ) f o X t ( w ) d A t ( w ) . It follows then by (61.12) that U Z f is a - ( X ,S)-excessive. (71.10) THEOREM. Let rn be an almost perfect decreasing MF, S := inf {t:mt= 0) and A := Am as in (71.1). Then for a 2 0 and f E pE", (71.11)
P"
Jds
e-Ot f ( X t )dt = V" f (z)
+ UzV" f (z).
PROOF:By (A4.17), (71.12)
l[O,S[(t)( - d m t ) = l[o,s[(t)mt- dAt = mt- dAt
since A does not jump at S. Consider the identity (Fubini!) e-at dAt
(71.13)
lt,s[ e-"(u-t)
f (xu)mu-to& du
1
e-"u f (Xu)
=LS[
By (71.12), for all u
1,4
mum;' dA,.
10,Ul
< S,
mum;' dAt =
rnum;'rnF. ( - d m t )
l0,4
lo..,
( by (A4.11))
d(m,')
= mu = mu(% - l - 1)
=l-rnu. Substituting this in (71.13), we see that if a
> 0 and f
E
bpE", then
rS
= P"
j0
e-"u
f (Xu) du - V" f (z).
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337
However,
for all t
< S so the optional projection of this term is
Substituting this in (71.14) gives (71.11) for a > 0 and f E bpE". The general case follows by a monotone increasing passage to the limit. (71.15) EXERCISE. By similar manipulations, prove that for all a 2 0 and f E p€", under the hypotheses of (71.10),
UZf =PSU,f
+ P Z f =U,P;f
+PSf.
Theorem (71.10) has some important consequences in connection with localization of an AF to achieve integrability properties. (71.16) DEFINITION.Let S be a terminal time for X. An increasing sequence {G,} of subsets of E \ reg(S) is a localizing sequence for ( X ,S ) provided G, E E", G, is finely open, and if T, is the hitting time of G i , then a s . , limn T, 2 S. (71.17) THEOREM.Let S be an exact perfect terminal time and Jet A be an A F of ( X , S ) with uniformly bounded jumps. Then there exists a localizing sequence (G,) for ( X ,S ) such that for all n, supPx X
/
ePt dAt < 0;).
10,TnI
If U1 is bounded on E , then the localizing sequence may be chosen so that supx PxA(T,) < 0;) for all n.
PROOF:We may assume that AAt 5 p < 1 for all t. Let m := mA as in (71.2). Then m is an exact MF by (71.2) and (71.3). With U s , Va and P$ defined as in (71.6-8), (71.10) states that P x J : e - t f ( X t ) d t = V1f(x) UAVl f (x) for all z E E and f E pE". Let q5 := V1l so that q5 E bE" takes values in [0,1] and q5 > 0 on E \ reg(S). Setting 4 in the formula above gives
+
1
S
q5(x) = P"
e-lQ(Xt) d t - U ; ~ ( Z ) .
Markov Processes
338
Since S is assumed exact, P":J e-tQ(Xt) dt is, by (56.9), a difference of two bounded l-excessive functions for X . By the remarks preceding the theorem, UA#(z) is 1-(X, S)-excessive, since it is the 1-(X, S)-potential of Q(X) * A, an AF of (X,S). In particular, Q E E" is finely continuous. Let G, := { x : Q ( x ) > l/n}. Obviously each Gn E E" is finely open, and G, T {Q > 0) = E \ reg(S). Since G: is finely closed, ~ ( X T , , ) 1/71a.s. on {T, < m}. By definition of Q, for all z E E ,
<
Let T := lim,Tn. Then hTR,iXlle-tmtdt hT,m[e-tmtdt boundedly as n --+ 00. Take expectations in the inequality above to conclude that PzhT,mIe-trntdt = 0. Since e-tmt > 0 for all t < S, T 2 S almost surely. Finally, as AAt ,O < 1 for all t, --f
<
P"
/
10,TnI
e-t dAt
/
< ,O + P" e%Q(Xt) IO,Tn[ < c +nUiQ(x)
dAt
<
because UAQ Pz:J e-tQ(Xt) dt 5 1. In case U l is bounded, we may take Q := V1, and dropping all the e-t terms, essentially the same proof yields the last assertion. There is a useful variant of (71.17) due to RRvuz [Re7Oa]. (71.18) THEOREM. Let S and A satisfy the hypotheses of (71.1 7). Suppose
< is a u-finite measure on E which is excessive (45.10). for (X,S ) . Then
localizing sequence G, T E \ reg(S) of (71.17) may be chosen having the additional properties <(Gn) < m and Pc j fe-tlG,(Xt) dAt < m for all n.
PROOF:Choose f E pE" with values in [0,1] such that f > 0 on E\reg(S) and <(f) < 00. This is possible since E is u-finite. Let W abe the resolvent for (X, S ) and let Q := V'f 5 W l f . Then Q has all the properties listed in [(W'f) 5 ((f) < 00 the proof of (71.17), and in addition, <(Q) = <(V'f) since is excessive for (X,S). It follows that if G , = {Q > 1/71},then <(Gn) n<(r#~)< 00, and the proof of (71.17) carries through exactly as before. Finally, '
<
<
<
1'
ps
e-tk',(Xt) dAt = {(UAIGn)
using (71.10) in the form UAV'f
<(f) < 00, completing the proof.
=
W'f
< n<(UAQ)5 nc(W'f), -
V'f.
As before, <(W1f) 5
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339
(71.19) THEOREM.Let f E E", and suppose t -+ f oXt is a.s. rcll. Given > 0, there exists a localizing sequence G, such that for every n, there exists g E bS' - bS" with SUP,^^, - g(z)1 5 E (cfFJ841).
E
If(.)
PROOF: We may replace X by its l-subprocess without affecting the con< 00 a s . and U1 5 1. By a preliminary localizaclusions, so that tion, we may assume f bounded. Fix E > 0, and let LO := <, L.1 := sup{t < Lo : I f ( X t ) - f(XL*-)I 2 E } , . . . , L, := sup{t < L,-1 : I f ( X , )f ( X ~ ~ - , -2) lE } , and so on. Then each L, is a co-optional time for X . Because fox is a.s. rcll, L, = 0 for all sufficiently large n. Let Y := C, l ~ ~ ~ , ~ ~ _ , " f o X ( LThen , - 1 -I )f o. x -YI 5 E , up to evanescence, and hence Ifox - "YJ 5 E . But Y E Sjd has paths of finite variation, so At := Yo -yt defines a signed RAF of X putting mass f o X ( L , - l - ) - f o X ( L , - ) at L , if L, > 0. By (71.17), we may choose a localizing sequence G, so that if T, is the hitting time of Gk, then sup, P" ldAtI < 00. Fix n and let T := T,. Then Y l ~ o , TE~ fig. Thus there exists a signed RAF B of ( X ,S) with Bt = Yo - yt for t < T. Then sup, P5 IdBtI = sup, Pz Idytl < 00, hence B is the difference between two RAF's C,D of (X,T), each of bounded potential. It follows that (uc - u ~ ) ( z := ) P5Y0 is a difference of bounded ( X ,T)-excessive functions, hence a difference of bounded excessive functions on G,, because T is exact.
<
ha,Tl
72. Two Motivating Examples
Suppose X is a regular step Markov process on E. This type of process was constructed in (14.18) by means of (i) a Markov kernel Q(z,dy) on ( E ,E " ) which describes the distribution of the location of the jump away from z, and (ii) a function X on E , X E E", with values in [O,m[ such that if T := inf { t : X t # Xo},then P"{T > t } = e-x(")t. That is, the holding time at z is exponentially distributed with parameter X(z). Let To := 0 and for n 2 1 define T n recursively by T" := Tn-l + T o O p - I . ) X t ( u ) } . Let At := Then limT, 2 C a.s. and U,[IT"n = { ( t , u ) : X t - ( u # s,' X(X,) du. Then A is a continuous AF of X with finite values on "0, 0 while if X(z) = 0, P"A(T) = 0. This points out that A is flat after it reaches a trap.
El=,
For every f E pE", the dual optional projection of (72.1) PROPOSITION. the RM rc(dt) := f o X ( T n )E T n ( d t ) is Q f ( X )* A.
PROOF:It suffices to give a proof assuming f E bpE", for the general case will follow by a monotone passage to the limit. We show first that
340
Markov Processes
, ~ dual ~ predictable projection the increasing process f ( X p ) l ~ T( t~)has
SotAT’
Qf(X,) dA, = Qf(Xo)X(Xo)(t A T1). This is equivalent to showing t) (t A T) is a P”that for all z E E,Mt := f ( X T ) l ~ T , o o ~-( Qf(z)X(z) martingale. This is trivial if X(z) = 0, for in this case T = 00 P”-almost surely. If X(x) > 0, T < 00 a s . and T has an exponential distribution with parameter X(z). Since M stops at T and MT = ~ ( X T-) Qf(z)X(c)T, it suffices to show that
Since T = t
+ Toot and XT = XTo& on ( t < T } ,
But P z f ( X ~ = ) &I(.)and P5T = l/X(z) so the right side reduces to Qf(o)l{t,T}[l - X(z)t - 11 = M t 1 f t < ~ )Applying . 6 %to~~ ( X TET) gives f(XTn+1)E T n + l , which must, by (31.5), have dual optional projection
Summing over all n 2 0 gives the desired result. The property (En,,foX(Tn)ETn ( d t ) ) P= Qf(Xt)X(X,)d t may be interpreted in terms ofthe generator of the semigroup ( P t ) . It is easy to derive from the above relation that i f f f b€” and if X is bounded, then
Suppose next that X is a Levy process in Rd. That is, there exists a vaguely continuous convolution semigroup ( p t ) t l o on Rd such that P t f ( z ) = J f(z y) pt(dy). The resolvent U” maps CO into CO and so the Ray topology is the same as the Euclidean topology of Rd. Since X is qlc (9.21), branch points play no role. The LCvy-Khintchin formula describes ( p t ) in terms of a positive definition symmetric d x d matrix B and a LBvy measure v on Rd satisfying: (72.3) v ( B ) < 00 if B E B(Rd)is bounded away from 0; (72.4) ~ l z 1 5 1 v(dz) < 00 and v{O} = 0.
+
VIII: Additive Functionals
34 1
Let ( , ) denote the usual inner product in Rd. There exists a centering function <: Rd Rd in bC with [<(x)- x]/1zI2 0 as x -, 0 and c E Rd such that the characteristic function fit of ,ut is e-tq(u), where -+
-+
The measure v determines the rate of jumps of X , for by the formulas above, for f E bC vanishing in a neighborhood of zero,
f(Ax,) = t l f d v ,
P”
(72.6)
x
E Rd.
O<sSt
A monotone class argument shows that (72.6) remains valid for all f
E
bB(Rd) vanishing in a neighborhood of zero. Taking limits then yields (72.6) for every f E pB(Rd) vanishing at zero. Thus: (72.7) PROPOSITION. As in (48.1), let J := ( ( t , w ) : X t - ( w ) # Xt(w)}. Then for f E bpB(Rd), the dual predictable projection of ~ ( w , d s:= ) Ct>Of(xt(w))lJ(t,W)f t ( d S ) is g o X s ( w ) ds, whereg(z) := J f ( z + y ) 4 d y ) .
PROOF:It suffices to give a proof for f E bpB(Rd). For g E bpB(Rd) vanishing in a neighborhood of zero, (72.6) implies g(AXs)lJ(s)= t
P” O<s
s
gdv
for all
X E
E.
Since At := CO<s
( h ( X - ) * A) =
I”
h(X,-) ds (
s
gdv).
Fix a neighborhood G of 0. By the MCT, for F ( x ,y) E b[B(Rd)8 B(Rd)],
Fix F 2 0 and let G shrink to (0). It follows then that the same equality holds for all F E bpB(Rd x Rd ), G = (0). Applying this fact to F ( x ,y) := f ( g X) with f E bpZ3(Rd), gives (72.7).
+
342
Markov Processes
The LCvy measure u also has an interpretation in terms of the generator of (Pt),for it can be shown that i f f E bC2(Rd),then
In fact, if f(z):= eC("J), (72.5) implies (72.8) directly. It follows that i f f is constant in a neighborhood of x,then
Hence u determines the non-local part of the generator of (Pt). 73. LBvy Systems Throughout his section, the lifetime is unimportant, so we consider A as an ordinary point in E and = 00. Let J := { ( t , w ) : X t - ( W ) # X t ( w ) , X t - ( W ) E E } , the set of totally inaccessible jump times for X . We prove that (72.1) and (72.7) can be set in a general framework.
<
(73.1) THEOREM. There exists a continuous A F H of X having bounded 1-potential and a kernel N on (E,E") such that (i) N ( x , {z}) = 0 for all z E E; (ii) for every f E p€", N f ( X )* dH (= N f ( X - ) * dH by continuity of H) is the dual predictable projection of the RM
As in the proof of (72.7), the dual projection property above is equivalent to the identity (73.2)
O<s
Jot
VIII: Additive finctionals
343
process with LCvy measure u , (72.7) shows that one may take Ht := t and N ( s ,dy) := v(dy - x). The idea of a LCvy system is due to Ikeda and Watanabe [IW62]whose approach seems to have been motivated by the study of the non-local part of the generator of a Hunt process. Watanabe [Wa64]proved (73.1) for a Hunt process satisfying hypothesis (L) by a rather complicated method involving stochastic integration relative to square integrable martingales. Benveniste and Jacod [BJ73b], whose method we describe below, showed that no special hypothesis are required. The idea of the proof is clever but simple, and the technical problems that arise are minor, given the measurability results now available to us. First one constructs a continuous AF H such that K? << H for every HRM K carried by J . Using the absolute continuity theorem (66.2), one represents IEP as g ( X ) * H for some g E pEe, and then for K. of the form ~,,ofoX,(w)l~(s,w)~,(dt), one shows that the map which sends f to g may be realized by a kernel N . The same style of proof is useful in a number of other settings. We shall obtain a slightly more refined version of (73.1), as detailed in the following assertions. (It is important to remember here that Bore1 and Lusinian refer to the the Ray topology on E.) (73.3) The AF H may be chosen to be adapted to
(e+).
(73.4) For f E p(E" 8 E"), define
For such an f, define the HRM ~ ( wd ,t ) := C8,0f(X,-(w), X,(w))E,(dt). Then ~ P ( d t= ) N f ( X t )d H t . (73.5) If E is a Lusinian, then N may be chosen to be a kernel on (ElE ) . (73.6) There exists 9 E p(E @ E ) with 9 > 0 except on the diagonal D of E x E such that N 9 5 1. (73.7) Let E be a Ray compactification of E . There exists a kernel N from ( E ,E ) to (I?, €) such that for all f E bt?,
We begin by constructing H . Let p be a Ray metric for X and let
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Markov Processes
Set J n ( w ) := { ( t , w ) : ( X t - ( w ) , X t ( w ) E ) r,}. Then U r n = E x E \D and for all n 2 1, r, E E @ E . Since t + X t ( w ) is prcll in E , J,(w) is a discrete subset of R++for all w . But J , E 0 n fig, so its debut T, is a thin terminal time for X. The iterates T," of T, are optional times with T: t 00 since J, is discrete, and J, = U k > l [Tk I]. Note that for all n, Jn is because the process p(&-, X t ) is adapted to Since adapted to J , is discrete, A: := ~ k 2 1 1 ~ T ~ , 0 0 isn (finite t ) and adapted to so {T, < t } = {A: 2 1) E Thus T, is an optional time over Let &(z) := P"e-*n. As T, is thin, qL(z) < 1 for all n, so h, := 1 - 4, E E , h, > 0, and
(c)
(c). (e), (c+).
c.
lyn(z,y)hn(y)/2,. Then Q E E @ E and D ! Let Q(x,y) := on D. The calculation above shows that
P"
c t
eCtQ(Xt-, X t ) = P"
2-, n
c
exp (-T')h,oX(T,k)
> 0 except 5 1.
k
This proves that At := '&<,st Q ( X 8 - , X,) is an AF carried by J with u i bounded. Moreover, AAt := Q ( X t - , X t ) > 0 for all t E J. Define H := A*. By (44.3), H is a continuous AF, and u& = is bounded. It is evident h,(X) * An, so A is adapted to From the remarks that A := in (31.13), H may be chosen to be adapted to
(c). (c+).
In order to construct the kernel N we shall use the following lemma, which is a simple consequence of (A3.3). (73.8) LEMMA.Let H be an AF of X with supz u&(z) 5 00. Let bE'/H denote the family of equivalence classes of functions in bE" which agree H-almost everywhere with some function in bE', where E c E' c E". Let @: bpE + bE'/H satisfy: (73.93) Wclf1 + C Z ~ Z =)cl@.(fl)+c2@(f2) for all c1, cz 2 0, f1, f~ E ~ P E ; (ii) @(f) 2 0 for all f E bp€; (iii) @ ( 1 ~5) 1.
VHJ: Additive Functionals
345
Then there exists a subMarkov kernel N o from (E,&")to ( E , E ) such that for all f E bpE, N of E bE" is in the equivalence class @(f ) . If E is a Lusin space, then N o may be chosen to be a kernel from ( E ,E') to ( E ,E ) . There is in any case a kernel N from ( E ,E') to (I?, E ) such that for every f E bg,
N(f) E @'(flE). PROOF:We point out that the equalities and inequalities in (73.9), are interpreted in the usual equivalence class sense. In order to reduce (73.8) to (A3.3), we let A denote the family of measures on (E,E") having the form ,uUk for some p. Then a set B E E" is H-null if and only if X(B) = 0 for all X E A. It is clear from the conditions (73.9), that @ extends to a linear, positive map 8~of bE into bE'/H. For all f E bE, write f = f+ -fto see that I&(f)l = I@(f+) - @(f-)l 5 @(f+) + Q(f-1 = @(lfl)5 Ilfll. Applying (A3.3) to 6 gives a kernel N o with the asserted properties, and (A3.4) gives the kernel N of the last sentence. (73.10) REMARK.The a-algebra E' will be E if H is adapted to (*+), E" otherwise, reflecting the choice of density in (66.2). Construct a map CP: bpE + b E / H as follows: given f E bpE, f ( X )* A is an AF carried by J such that f ( X ) * A << A in the sense of (66.14). It follows that ( f ( X )* A)P << Ap = H , and since H and ( f ( X )* A ) P are continuous and may be chosen adapted to (G+) by (31.13), (66.2) shows that there exists gf E pE with 0 5 gf 5 llfll such that
(f(X> * Alp = Sf(X)* H.
(73.11)
The uniqueness of derivative (66.7) implies that (73.11) uniquely determines gf up to equivalence H-almost everywhere. Thus @(f) € b&/H may be defined as the equivalence class of gf determined by (73.11). It is clear that CP possesses the properties (73.9). Thus, by (73.8), there is a subMarkov kernel N o on (E,E") such that
f E bpE. ( f ( X )* A ) p = N o f ( X )* H , (Under the hypothesis of (73.5), (73.8) implies that N o may be chosen to be a kernel on ( E , E ) . ) For f E bpE", (73.12) remains valid, because N o f f bpE" and, given an initial law p, one may sandwich f between fi and f 2 E bpE relative to the measure p U i to find that N o f is sandwiched between N o f l and N o f 2 , and (73.12)
pp
I"
/nm
e-t(Nof2 - N o f l ) o X tdHt = Pp
e-"f2
- f l ) o X t d A t = 0.
By the meaning of dual predictable projection, for f E bE" and Y E bP, (73.13)
PP
Lrn
e - t & f ( x t ) d~~ = PP
e-t&NOf(Xt-)d
~
~
.
Markov Processes
346
In fact, by (32.4), (73.13) holds also if Y E bPt (the a-algebra of integrally predictable processes of $32). I claim that for every Z E p(Pt 8 E"),
To see this, note that in both integrands of (73.14) one may insert the factor e-t, replacing Z((t,w ) , . ) by etZ((t,w ) , -). With this factor in place, both sides of (73.14) as functionals in 2 define finite measures on ((R+x R) x E , Pt B E") which agree, by (73.13), on product sets. They agree therefore for every Z E p(Pt B E " ) , proving (73.14). I f f E p(E" B E " ) and Y E pPt, Z ( ( t , w ) , x ):= &(w)f(Xt-(w),z) is in P t BE" because of (41.9), and substituting this in (73.14) gives (73.15)
Pp
loo 1"
Ytf(Xt-,Xt) dAt = Pp
Taking H = 1 and f = 1 0 yields
0 = P'"
NOl0(Xt_) dHt = P p
I"
Jdm
ytNof(X+) dHt.
NOlO(Xt)dHt.
Hence N o ( x ,{x}) = 0 H-a.e., and N'(x, dy) := ~{No(,,(,.)>,,}N~(z, dy) is also a subMarkov kernel on (E,E") with N o f = N'f H-a.e. for all f E p(E" @ E"). If N o is a kernel on ( E , E ) ,so is N'. Finally, define N by
(73.16) Then for f E p(E" 8 E") vanishing on D,
N f ( z ) = N ' ( f / Q ) ( x )= N o ( f / Q ) H-a.e. By (73.15), for Y E pPt and f E p(E" B E") vanishing on D,
roo
= P'"J, yt Nf(Xt) dHt.
The content of this last equality is exactly (73.4). Under the hypothesis of (73.5), since Q E E E , N is a kernel on ( E , E ) . Finally, the last assertion in (73.8) can be used to modify the proof above to produce a kernel from ( E , E ) to (,&?,€) satisfying (73.7). The details are easy to fill in. See the proof of (73.17), where the same argument will be applied.
VIII: Additive Functionals
347
(73.17) THEOREM.Let 2 E 0 be homogeneous on R++.Then there exJ to evanescence. ists a function $ E €" @ € such that Z1J = $ ( X - , X ) ~ up PROOF: Let A be the reference AF for J constructed above (73.8), so that A has bounded 1-potential and J = { A A > 0). We may assume that 0 5 2 5 1 and, replacing 2 if necessary by 2 1 { p ~ ~ - , ~ ) ~that 6}, Bt := Co<slt 2, < 00 for every t . According to (60.2), we may replace B by an indistinguishable process adapted to (3:+), hence 2 may be assumed Proceed now almost exactly as in the proof of to be adapted to (F;+). (73.1) as modified to give (73.7). Let E denote a Ray compactification of E. For each f E bp€, ( ( f ( X ) Z )* A)P << AP = H , so by (66.2), there exists g(= gr) E bpE" such that ( ( f ( X ) Z )* A)P = gf(X-) * AP. The map f -+ gf satisfies the conditions (73.9) with 'E := E", and there exists therefore a kernel NZ(x,dy)from ( E , € " )to ( E , € )such that N z f = g f H-a.e. for every f E bp€. As 0 5 2 5 1, N Z f 5 Nf H-a.e. for every f E bpE, so, replacing N Z ( z ,' ) by its minimum (in the lattice of measures) with N ( x , .), we may assume N Z ( x , . ) 5 N ( x , . ) for all x E E . By Doob's lemma (A3.2), there exists $ E E" @ € such that N Z ( qdy) = $(x, y)N(x,dy). We may clearly assume that 0 5 $ 5 1 and that $ vanishes on the diagonal of E x E. Then, for Y E b P and h E bc, properties of dual predictable projections yield
P"
Som
e-t&h(Xt)2t dAt = P"
1"
e-t&Nzh(Xt-) dAf
By (23.4iii), processes of the form & h ( X t ) generate 0.It follows from the last display and the uniqueness of dual optional projection (A5.23) that 2, dAt and $ ( X t - , X t ) dAt are indistinguishable, completing the proof. (73.18) COROLLARY. Let K be an optional HRM carried by J . Then there exists $ E E" @ E vanishing on the diagonal of E x E such that K ( . ,{ t } ) and $(Xt-, X t ) are indistinguishable. (73.19) COROLLARY. Let T be a totally inaccessible terminal time. Then there exists A E E" @ € , A c E x E , with T = inf{t > 0 : ( X , - , X t ) E A } .
PROOF:For 6 > 0, let
Markov Processes
348
The T6 is also a totally inaccessible terminal time, and its iterates TZ can have no finite points of accumulation, for if there were such a point, the path would have to have an oscillatory discontinuity there. Thus U, [iTZ1 is homogeneous and vanishes a.s. outside J . By the theorem, there exists A6 E E" 8 & such that U,[T,"n = { ( X - , X ) E Ah} up to evanescence. It may be assumed that p ( z , y ) 2 6 for all (z,y) E Ag. Clearly then, as., 2'6 = inf{t > 0 : ( X t - , X t ) E As}. Fix a sequence 6, 11 0. It may be assumed that h b , increases with n. Let A := UnA6, and let R := inf{t : ( X t - , X t ) E A} = inf, T6,. Then R = T a.s. since T = T6 for all sufficiently small 6 > 0. The next results involve an extension of the L6vy kernel E , which is a kernel from ( E , E ) to ( E , E ) , to a kernel K from ( E , & )to (fi,p), where fi denotes the space of all rcll paths in with co-ordinate maps X t and 9 the a-algebra on generated by f ( X t ) for t 2 0 and f E &.It will be essential to realize X on the subspace R of i=l consisting of the (Ray) right continuous paths with values in E which are also right continuous relative to the original topology on E . Since E is only universally measurable in E , 52 is not in general measurable in fi.
e
(73.20) THEOREM. Let ( N ,H ) be a LCvy system for X satisfying (73.3)p )such that for (73.7). There exists a kernel K ( x , dw) from ( E ,E ) to every F E b p ,
(a,
(73.21)
PROOF:The proof is almost identical to that of (73.1), and it will be is separable and Lusinian by only sketched in outline. The space (fl,p) (A2.16). Let At and Ht be as in the proof of (73.1). For every F E b p p , the raw AF F o o t dAt is absolutely continuous relative to dAt, and therefore ( F o e t dAt)p << d H t , ( F O B t dAt)P = g p ( X t )dHt for some g p E bp&. The map F + g p satisfies the conditions (73.9), and consequently there exists such that g p = RoF H-a.e. for every a kernel Ko from ( E , & )to (fi,p) F E b p . Then R ( x , d w ) := I?O(x,dw)/Q(Xt-(w),Xt(w))satisfies (73.21). (73.22) THEOREM. Let 2 E f j g . Then there exists r$(s, w ) E E" 8.F" such ~ r$(Xt-,0,) are indistinguishable. that 2 1 and
PROOF:The proof is parallel to that of (73.17), and only the changes will be described. Fix 2 E 4 9 with 0 5 2 5 1. It may be assumed that 2 E D(R+) @ 3e.For each F E b p p , there exists g p E &" such that ((FoBtZt)dAt)P= g p ( X t ) dHt. There exists, by the same argument as in the last theorem, a kernel K Z from ( E ,E " ) to (fi, 9)such that g p = K Z F
VIII: Additive Func tionals
349
H-a.e. for every F E b p . Use Doob's lemma to produce $ E E e @ p with K z ( z , d w ) = $ j ( z , w ) K ( z , d w ) For . Y E b P and F E b p , computing just as in the proof of (73.17) gives
By (23.12), processes of the form &FOBt with F E p generate M . But F E 9 if and only if there exists F E p with F = Fin. It then follows from the last display that Zt dAt and $ ( X t - , 0,) dAt are indistinguishable. The restriction of $ to E x 52 has the asserted properties. The clever trick of Weil [We'Tla]in the next result has been mimicked in many other situations resembling LCvy systems. ( 7 3 . 2 3 ) THEOREM. Let A c E x E be disjoint from the diagonal, A E E" @ E , T := inf{t : ( X t - , X t ) E A}, and suppose ( X T - , X T ) E A as. on { T < m}. Then for f E pE",
PROOF: We may assume f 2 0. As the trace of ~ F Ton - {T < co} is { Y T l { T < w } : Y E pp}, it suffices to prove that for all Y E p P ,
t ) P and Under the given hypotheses on T , yt l n o , ~ n ( E
It follows from the Lkvy formula (73.4) applied twice that
350
Markov Processes
completing the proof.
74. Excursions from a Homogeneous Set Excursion theory concerns the decomposition of a right process according to (a) its behavior for t E M , a closed homogeneous random set, and (b) its behavior during the intervals in which t 4 M , the latter being called the set of excursions of X outside M . For example, let X be regular step Markov process and let M denote the closure in R++of { t : X , = XO} with xo a fixed state. Let R = R1 := inf{t > 0 : Xt # go}, D = D 1:= inf{t : Xt = XO}, D2 := R' D O ~ RR2 I , := D2 R o e ~ aand , so on, so that
+
+
M = ( no1,
1 u [02, R2I] u . . .) n go,
Following the terminology of 867, the right boundary of M is MD = U,[IR"I, and the intervals contiguous to M are those having the form I]R j , Dj+l [I. These contiguous intervals are also called the excursion intervals away from M . The excursion theory is quite trivial in this case for, as we discussed in (14.18), the X i := X ( R j t ) , t < Dj+lof?(Rj),which describe X during the excursion intervals 1Rj ,Dj+l[, are independent copies of X ' , all Xj being independent of all the holding times Rjoe(Dj) at XO. Thus the only information needed to describe the decomposition completely is (i) the transition mechanism for the X j , which is precisely that of X killed at D ;(ii) the initial law for the Xj,which is just K(z0, the distribution of the position of the first jump away from X O ; (iii) the parameter in the exponential law governing the holding time at 20. Excursion theory for more complex processes cannot be expected to have such a simple structure. For one thing, the excursion intervals in the example above can be enumerated successively, but if we consider, say, the excursions of linear Brownian motion X away from the state 0 (ie., M := { t : X t = 0}), then the countable set of excursion intervals with left to right ordering is not well-ordered, so we cannot simply examine the first
+
a ) ,
VIII: Additive F’unctionals
351
excursion and then iterate. We wish also to consider homogeneous random sets more general than closures of occupation times of singletons. The following general description of the excursions away from a general closed homogeneous random set M was introduced by Maisonneuve [Ma751 under the name exit system for the pair ( X , M ) . It is quite similar in nature to the LCvy system of the last section. (The two theories can be made very close. See [MM74] for a discussion of the excursion theory as a LCvy system for a process of paths, and [Pi811 for a general approach covering both, using the methods of point process theory.) Throughout the rest of this section, we adopt the notation and terminology of 567 for a given closed, perfectly homogeneous random set M 3 [ C , o o [ , so that, for example, T := inf{t > 0 : t E M } 5 C is the debut of M , Tt := t+ToBt = inf{r > t : T E M } , Lt := sup{s 5 t : s E M } is the right continuous inverse o f t -t T,, and MD:= { t : t = Lt < T t } is the set of left endpoints of the intervals contiguous to M . Let cp(z) := Pze-T and F := {‘p = 1) = reg(T). Then M D splits into an optional part M,D := MDn {X E F“} and a progressive part M,” := MDn { X E F } which is totally non-optional in the sense that its intersection with the graph of every optional time is evanescent. Let A , B denote respectively the 1-transport and optional 1-balayage of dCt := lM.(t)dt on M . By (67.33), ub(z)= u i ( z ) = P$u&(z),and since u&(z) := e - t l M = ( t ) dt 5 1, A is an RAF and B is an AF, both with the same bounded 1-potential. By definition of 1-transport,
s,’
Thus l~ * B = B , and by (67.39), the mass of B at a point t of Mg is P X f ( l - eCTf. For example, in the regular step case above, dBt = 3 1 1 PX(R3)(l- e-”) c~~ ( d t ) , and in the Brownian example, MD= M I because ’p(0) = 1, hence B is, by (67.42) and (68.1), a multiple of local time at 0. The AF B will serve as a reference AF for excursions from M in the same way that the AF H of 573 served as a reference AF for the totally inaccessible jumps of X. We are assuming only that the state space E is a Radon space, and the space of all rcll paths in a Radon space does not seem to be Radonian in general. For the purpose of constructing nice kernels, it will therefore be necessary to embed E in a Ray compactification E . Let fi denote the collection of all rcll maps of R+ into E , so that with the a-algebra generated by the coordinate maps X t ( i j ) := a(t),fi is a Lusin measurable space by (A2.16). For z E El let Pz denote the probability on (fi,p)
c.
Markov Processes
352
rendering X Markov with semigroup (pt) and initial position x. Define $ : fl + i=l by $ ( w ) ~:= Xt(w). Because it is one of our blanket hypotheses that t + Xt(w) is Ray-rcll in E for every w E a, $ does indeed map $2 into a. Obviously, $ E p / p ,and so by sandwiching, $ E 3*/p.For every G E b P , P"(G) = P"(Go$). (74.1) PROPOSITION. There exists f' E p p so that T o $ o 0 8 = T O O , for all s > 0, a.s. on R.
PROOF:For every f E Sa,extend f to f ' on E by setting f' := 00 on \ E . The function f' is a-supermedian relative to (u*),and its aexcessive regularization f relative to (@) has the property ~ I =Ef. By there exists a monotone class argument, it follows that for every G E 3e, GE with G o $ = G. Because t -, T O O t is a.s. rcll, (24.34) implies that there exists T' E 3ewith {T # T ' } E Ne. Let T be the extension of T' to i=l with T o $ = TI. Then as., T o $ o O , = T O O , for all 9 2 0, by definition of N 9 . (74.2) LEMMA.For G E b p p ,
dAf := G o $ o O t dAt =
C Go$oO,(l - e-Toe8)c8(dt) 8EMD
defines an raw HRM of X with bounded I-potential. PROOF: Since A is a RAF and G o $ o O t is homogeneous on R+, AG is a raw HRM, and as A has bounded 1-potential and G is bounded, AG has bounded 1-potential. Let BG := (AG)",the dual optional projection of A G , so that BG is also an AF with bounded 1-potential. (74.3) PROPOSITION. For F , B , G as above,
(i) M n {X E F C }= M,D up to evanescence; * BG is a continuous AF, and (ii) CG := 1 ~ o X (74.4)
1FCox
* BG =
C Pxt (Go$ (1 - e - T ) )
€8.
sEM:
PROOF:(i) comes from (67.23). Using the fact (A5.25) that A ( B G ) = " ( A A G )the , total non-optionality of M," shows that
A ( @ ) , = lM:(t)PX1 (GO$ (1 - e - T ) )
+
vanishes for Xt E F . But BG = 1~ * BG = ~ F ( X*)BG l M l p ( X ) * B G , and by (i), the last term is the discontinuous part of BG.
VIII: Additive finctionals
353
+
If G E p p and 0 5 G 5 1, A = AG implies B = BG+ B1-G. By the absolute continuity theorem (66.2), there exists hG E pE", 0 5 hG 5 1, vanishing off F such that ~ F ( X* )dBG = ( lF h G)(X )* dB. In addition, (74.4) shows that if we set k G ( z ):= P"(Go$ (1- e -T ))/P z (l e-T)lFc(x) E pE", then 0 5 kG 5 1, kG vanishes off F C ,and by (74.3ii), l p ( X ) * dBG = ( l p k G ) ( X )* dB. Let f G := hG k G . Then 0 5 f G 5 1, and for G E p p ,
+
dBG = f G ( X )* dB.
(74.5)
By (66.7), the function f G is determined B-a.e. by (74.2). The map G -+ then satisfies the conditions (73.9) relative to the AF H := B and E* := E e , and since ( Q , p is ) Lusinian, (73.8) gives ii kernel N = (N"(dCj)) from ( E , E e )to (n,Fo)such that fG(z)= N " ( G ) B-a.e. for every G E b p p . That is, f G
BG = Nx (G) * B,
(74.6)
G E bpP.
As 0 5 f G 5 1 B-a.e. if 0 5 G 5 1, N may be modified so as to be a subprobability measure for every z E E , for E' := {z : N " ( E ) 5 1) E E" is 1 without affecting B-negligible, and we could replace N by 1 ~ (x)N"(dCj) formula (74.6). The term (Go$(1- e-T))oOs may be replaced by (G(1- e - P ) ) o $ o B s , and by definition of BG as the dual optional projection of AG, (74.6) yields for 2 E p 0 and G E b p p , (74.7)
P"
e-"Z, (G(1- e-'))o$oOs
sEMD
I"
= P"
e-"Z,NX(s)(G) dB,.
As both sides of (74.7) are finite measures in the pair (Z, G), and by (23.11), 2 E 0 of the form $ ( k t , t , X t ) , 4 E p @ B + @ € ,a monotone class argument leads to the more general form (74.8)
1
P"(dw)
e-sp(k,w, s, X , ( w ) , $of3,w)(l- e-T)o$oO,w = sEMD( w )
for 'p E p ( P @ B + @ € @ p ) *(The . notation N x u ( ~ ) ~ ( k s w , s , X 8 ( won ), X8(w),Cj).) , the right side of (74.8) is shorthand for f N X E ( w ) ( d i j ) ~ ( ks,s w a )
354
Markov Processes
} (74.8), so that Now set cp(w, s,x,G ) := l p ( G ) = Oin
P"
C e-31poe,=0)(1-e
-Toes)
1"
e-'NXa( { G : T(iCI)= 0)) dB,.
= P"
0
SEMD
As the left side vanishes identically, we conclude that r := {x : N"(T = 0)) E E" is B-null. Clearly, r c F , and so, replacing N" by lr(x)N", which becomes then a kernel from ( E ,E") to (0,F ) ,we may assume N"(T = 0) = 0 for all x E E. We may of course consider N automatically extended to Define for each x a a-finite measure be a kernel from (E,E") to (fi,p). P" on
(n,p) by P"(dG) := N " ( d G ) / ( l - e-'($))*
(74.9)
Then P = P"(dQ) is a proper kernel from (E,E") to formula (74.8) now in the form (74.10) P"
(n,p). Rewrite
C ~ ( k e , X.9,$ 0 0 , ) = 8,
3EMD
P"
1"
PX.({a :X&J) # X,-})dB,.
The left side of (74.11) vanishes because the discontinuity set J U J B may be expressed as a countable union of graphs of optional times, all of which have an evanescent intersection with Mn{X E F } = ME. However, Pxn(")({G : X , ( ~ C# I ) X,-(W)) = P ~ S ( U ) (: {X~~C( IG#) x,(w)} a.e. relative to d ~ , ( w ) , because dB,(w) does not charge (J U J s ) n {X E F } . Replacing X,by X, on the right side of (74.11), we conclude that := {z E F : P"(X0 # z)} E €" is B-null. Replacing P by lr,(x)P", we may then assume P " ( X 0 # x) = o for all x E E . Next set cp(w,s,x,G) := ~ { C = ~ ~ ( Because G). MD C no,<[, (74.10) vields
VIII: Additive finctionals
355
Thus a further modification of P shows that one may assume P5(<= 0) = 0 for all x E E. Recall now that D denotes the set of non-branch points for Pt in E . Define S(W) := inf{t > 0 : X t ( G ) E E \ D. Since D is Bore1 in E , S is a n optional time for (.Ft+).Define $ ( w , s , z , W ) := l{g(o)<,). Then $(ksw, s,X , ( w ) , $ ~ o O , w ) = l ~ ~ o ~ o ~But , wthe
showing that P may be modified yet again so that i)z(S < co) = 0 for all x E E . (Note that if E was known to be Lusinian, the same argument could have been applied to E instead of D to show that the Pz all live on the set of paths which never leave E at a strictly positive time. Even in the general case, much the same argument shows that the Pz can be assumed to make D \ E quasi-polar for X.) The discussion above proves the assertions in the first sentence of the following theorem, as well as the points (74.13). There exists a proper kernel P = (Pz(dG)) from (74.12) THEOREM. (E,E") to (fi,P) such that (74.10) holds for all cp E p (F " '@ B +@ E @ p )* . Define $(a) := inf{t > 0 : Xt(W)$ D}. The Pz satisfy for each x E E : (74.13i) Pz is a-finite; (ii) P z ( < = 0) = 0; (iii) Pz(T = 0) = 0; (iv) o 5 P z ( 1 - e-') 5 1; (v) P z = P z i f z F ; (vi) Pz is carried by {X,= x}. (vii) Pz is carried by { S = XI}. For every x, (Xt),,,is strong Markov with semigroup ( P t ) relative to Px: that is, for S an optional time over with P5(S = 0) = 0, H E pFs+ and J E pQ*,
(e+)
(74.14)
P'"(HJ0Os;S
< co) = P"(HPX(')J; S < co).
(74.15) REMARK.The content of formula (74.10) may be expressed as follows: for G E p p , the dual optional projection of the random measure CsEMD G o + o O , c,(dt) is Pxs(G) dB,.
PROOF:It remains only to prove that the P5can be further modified so that the assertions in the last sentence hold. By (74.13iv), the measure
356
Markov Processes
l{p2T}P" is of finite total mass for each z. By monotone convergence, it will suffice to prove (74.14) in the form (74.16) Pz(l{sAF>p}HJoos; s
< m) = P " ( ~ { ~ A ~ > T ) H P ' ( ~s) < J ;00)
for each rational r > 0. We consider first the case of a constant time S := u > T . For cp(w, s, x,a)of the form +(w,3,z)G(ij)with G := HKoOU, H E puand K E plthen setting Z,(w):= 4 ( k s w ,s,X,(w)) E 0 , I claim (74.17)
P"
c
c
Z , ( H K o ~ , ) o $ o O s = P"
,EMD
Z,(HPx'KoOU)o?)08s.
sEMD
For, if we select E < u and denote by I]Gn,O n [ the nth interval contiguous to M of length exceeding E , then G, E is an optional time, for G, E 5 t if and only if M n [O,t] contains at least n contiguous intervals of length > E . Thus G, u = G, E (u- 6) is also an optional time. Define Y, := cp(ks,s, X,)Ho$oO,l~(X,).Then Y ( G n )E FG,+%, and by the SMP at Gn U ,
+
+
+
+
P"
c n
+ +
Y(G,)Ko$08(Gn
c
+ u ) = P"
Y(Gn)PX(G"+U) (K)*
n
As E + 0, the terms in the last display increase to the respective terms in (74.17), so (74.17) follows by monotone convergence. Now apply (74.10) twice, first with cp(w, s, z, a):= 4(w,3,z)HKoO,, then with cp(w, s, 2,W) := 4(wls, z ) H P X ( " ) ( K )From . (74.16) we obtain
As Z is by (23.11) the generic element of O0, this implies that for fixed u > T , the set r U , H , K of z for which (74.16) fails, is B-null for all H E pF:, K E p p . Letting H and K run through countable generating systems in and p respectively, we may then modify @ so that (74.16) holds with S := u > 0 for all H E b p U ,K E b p . (The finiteness of l{p2rlP" is vital here to enable application of the MCT.) By further modification of P" on a B-null set, we may assume (74.16) holds for all S := u E &++, T E Q++ with u > T . This proves that, under the finite measure l { ~ ~ , . j P " , ( X t ) t > r , t Eis~ simple Markov with semigroup (Ft). As X is right continuous it follows that for all and t -, Ptf is right continuous for every f E T E &++, under ~ { F ~ , ,(Xt),?, } P ~ is simple Markov with semigroup (pt). := {S = oo}, the set of paths in E never leaving D at a strictly Let positive time. As all the Pzare carried by fro, the strong Markov property
c,
VIII: Additive Functionals
357
in the form (74.16) will follow once we verify that it holds on fro, and for this it is enough to notice that since the restriction of (Pt)to D is a right semigroup by (9.13), then every right continuous realization of (pt) on D is strong Markov by (19.2). Though (74.12) is very useful for doing calculations similar in spirit to those in the last part of $73, it is somewhat lacking in intuitive content as an means of describing excursions away from M . Indeed, it really describes not excursions away from M , but rather the ensemble of paths starting at times at which they leave M . It is not difficult to make a minor adjustment which brings excursions to the fore in (75.12). Keeping the same notation developed earlier in this section, the idea is that a path 3 leaving M at time t is mapped to a n excursion leaving M at time t under the map 3 + e t ( 3 > := k T O B t 3 , where the kt are killing operators on G. (That is, first shift the time origin to t and then kill the path the next time it returns to M . ) The collection {et(c;r) : t E MD}is called the collection of excursions of 3 away from M . The following satisfying description of the collection of excursions may then be read off (74.12). so that (74.18) THEOREM.Let ( Q t ) denote the semigroup for (X,T) = P"(f(Xt)ok~), f E pE". For z E E , define measures Q", Q" on 3*) by
&if(.)
(a,
Q"(G) := P"(Go+okT),
Qz(G) := P z ( G 0 k ~ ) , G E pQ*.
Define also the kernel Q t ( x ,dy) from ( E ,E") to (I?, &) by Q t g ( x ) := l y g o x , ;
t
g E pE".
Then: (i) for Z E pc3 and G E pQ*,
(ii) for B-a.a. x E E , (Qt(z,. )) is a finite entrance law for the semigroup (Q,) with e-,Qt(rc, 1)dt = 1; (iii) for B-a.a. x E E , Qz is carried by { C > 0, XO = z, S = oo}, and under @, ( X t ) , > O is strong Markov with semigroup ( Q t ) and entrance law (Qt(x,* ),.
;s
REMARK. For x
4 F , Q" = Q" and Q ~ ( z ,
a )
= Q ~ ( z ., ) since P" = P".
Markov Processes
358
75. Characteristic Measures of HRM's
<
Fix throughout this section a u-finite measure on ( E ,E), and assume that
<
(75.1) HYPOTHESIS. is excessive for (Pt) : that is, (Pt
5 ( for all t 2 0.
The condition (75.1) is of course equivalent to the condition (75.2) P W t ) 5 PWXo), f E PE",t 10. It is clear that the left side of (75.2) is a decreasing function oft. We shall show below how to associate with any HRM, K , satisfying certain secondary hypotheses, a a-finite measure v, on f i g characterizing K up to PE-evanescence. Before starting the development in full generality, we point out the following special instances. For both examples, we shall suppose that is invariant for (Pt): that is,
<
(75.3) cp(t+s) = p(t)+PEr{]O,+et
<
<
= cp(t)+PtPty{]O,s]} = cp(t)+cp(s).
That is, when is invariant for (Pt),cp is an additive increasing function, hence it is proportional to t , and so takes the form cp(t) = t v i ( Z ) , where the multiplier v $ ( Z ) E [ O , o o ] depends on all three parameters K , <, and 2. It is not difficult to see that for fixed IG and ( satisfying some finiteness assumptions, v i ( Z ) defines a measure in 2. As the first example, let X be Brownian motion in Rd and let be Lebesgue measure on Rd. Then is invariant for X . The HRM K(dt) := d t has the property
<
<
That is, for Zt := f(Xt), v i ( Z ) = <(f). As a second simple example, take X to be uniform motion to the right on R, take to be Lebesgue measure on R, and let At := f ( X t ) - f ( X o ) , where f is a right continuous increasing function on R. Denote by v the Radon measure on R generated by f. Given an arbitrary g E pB(R), set Y s ( w ):= g ( X , ( w ) ) so that Y E 0 n f j d .Then
<
c
VIII: Additive finctionals
359
That is, v$(Y)= v(g). If we had supposed instead that g was rcll on R so that 2 := g ( X ) - E P n w , then a similar computation would give vi(Z)= v(g-). In the general case, where E is only excessive, the final equality in (75.3) must be replaced with an inequality to give instead
cp(t + 8 ) I cp(t)+ cp(8).
(75.4)
That is, cp is an increasing subadditive function on R+.Actually, cp is much better than subadditive. (75.5) PROPOSITION. The function cp(t) := Pe~(10, t]}is concave on R+. In particular, p(t)/t increases to a limit L E [0, m]. If L = 0, then cp(t)= 0 e-at n(dt) t L. for all t 2 0. As a t 00, aPc
PROOF:If cp(t0) = 00 for some t o > 0, the subadditivity gives p(to/n) = 00 for all positive integers n, and since cp increases, this shows that cp(t)= m for all t > 0. We suppose now that cp(t) < 00 for all t. Fix s > 0, and consider the differences $ ( t ) := cp(t + s) - cp(t). By additivity of K , if 0I t l 5 t z , we have
+ t1])o@t2-t1
$ ( t Z ) = PEK{]tl, s
= ptPtz-t1K { ] t l , S
+ tl]}
5 $(tl).
$ ( t ) is decreasing. In particular, if the right hand derivative p'(t2) = 00, then cp'(tl) = m for every tl < t z . But, since p is increasing, it has a finite derivative at a.a. t 2 0 and p(t) 2 1,cp'(s) ds. If cp'(t) = 00, we would have ~ ' ( u=) 00 for all u c t , which would imply cp(t)= 00 for all t > 0. This contradiction establishes that the right derivative p'(t) exists That is, t
-+
and is finite and decreasing. It is also clear by the Lebesgue dominated convergence theorem that cp is continuous. It is then a matter of elementary analysis to see that cp is concave. The existence of L := limt+ocp(t)/t is assured, and if L = 0, then cp'(t) 5 0 for all t > 0 and consequently p(t) = 0 for all t. Let Q be a probability measure on a space carrying an exponential random variable Y with parameter 1. Then
increases to p'(0) = L as a t 00, completing the proof. Fix now a HRM K . It is clear that for every fixed t > 0, the functional 2 t-'Pt Z, ~ ( d s defines ) an s-finite measure on 9 9 . As t 1 0, these measures increase, by (75.5), to the functional --$
(75.6)
s,"
v i ( ~:=) lim t-lpt t+O
1 t
Z, n(ds),
z E pag,
Markov Processes
360
which defines therefore an s-finite measure on $ 9 , called the characteristic measure for K . Special cases of this measure have been called Revuz measures in the literature because the original contributions of [Re701were the first to establish in reasonable generality a correspondence between measures on E and AF's. The last assertion in (75.5) gives us a simple way to compute some characteristic measures. For example, let n(dt) := h ( X t )d t , h E bpE", so that U F ( z , .) = U U ( z ,.)h(-), and for f E pE", (75.5) applied to the HRM f ( X ) * K yields nt
vi(f O
X )
= -
since a U a t t 5 as (Y -+ 00. That is, v i ( f 0 X ) = hE(f) in this case. An HRM K is called &integrable provided v$(l) < 00, +integrable provided v$ is o-finite on fig. This condition is quite different from the condition of o-integrability on 0 and P used in Chapter IV. If K is homogeneous on R+ rather than R++, then it is more natural to define v$ on B d instead of $ 9 , calling vi in this case the right characteristic measure. The definitions must be modified slightly to cover this situationthe details are left to the reader. The following facts are obvious consequences of the properties of dual projections.
(75.7) PROPOSITION. (i) Y ~ ( Y=)o for every PF-evanescent Y E fig; (ii) if K is optional and in d, then v$(Z) = ui("2);ie., v$ is determined by its action on 0 n fig; (iii) if K is predictable and in 9, then v $ ( Z ) = vi(P2); ie., u$ is determined by its action on P n $9; (iv) if K E d is optional and homogeneous on R+, then ufi. is determined by its action on 0 n fidl which is equivalent to {f(X) : f E E e } ; ie., vi should in this case be replaced by the measure p i on ( E , E " )given by the formula rt
&f)
:= ;it-'PS
Jo-f ( X t ) K ( d t ) .
VIII: Additive finctionals
361
<
When is fixed, as in most of the discussion below, we shall simplify notation by writing u, in place of u,$, and pK in place of p i . (75.8) LEMMA. Let
In particular, u:
K.
be [-integrable and let a
> 0.
Then
< 00 8.e. relative to <.
+
PROOF:By (75.5), P€K.{]S, t s]} 5 tv,(1) for all t, s > 0. Consequently,
The following result shows that many HRM's of practical interest are @-integrable. (75.9) PROPOSITION. (i) Let B be a <-integrable AF, and let A be an A F such that, for some a > 0, uQA 5 u;. Then A is also [-integrable. (ii) Let A be predictable AF with ua bounded, and with jumps uniformly bounded by some constant c < 00. Then A is <-aintegrable.
PROOF:The resolvent identity for potentials of AF's (36.12) implies uQA- upA = ( p - a)U"upA.
Therefore, using excessiveness of
< in the second step,
That is, p(<,up,)2 P ( < , u { ) for all p. By (75.5), as p + 00, (<, u i ) increases to v i ( l ) , and (i) follows immediately. For (ii), we may suppose c < 1. Define the MF m by the exponential formula m = sexp(A) studied in 571. Then S, 15'a.s., and as we showed in (71.10), the formula
U1f -V'f
= U p f ,
f EpE",
362
Markov Processes
holds, (V") being the resolvent generated by the MF m. Fix f E b&nZ1(E) with f strictly positive. Define E,, := {z : Vlf 2 l/n}, so that U i l ~ ,5 nUAV'f 5 nU1f , the latter function being bounded and in L'(J). That is, B := ~ E , ( X )* A is an AF with 1-potential bounded above by the 1-potential of the AF dCt := nf(Xt) dt, which is finite because
by excessiveness of E. The AF C is thus also &integrable, by (i). If A is a u-finite measure on an arbitrary measurable space, then the relation f dA = g dA, f , g E pE", implies If - g1 dA = 0. Observe also that characteristic measures have the obvious properties (75.10i) (75.10ii)
dvZ*, = Zdv,, 2 E pag; u, = o implies n is Pc-evanescent.
These are the building blocks for the following uniqueness theorem. Let the HRM n be 6-o-integrable, and n' be another (75.11) THEOREM. HRM with the same characteristic measure u$ as n. Then n and n' are Pc-indistinguishable. If n, n' are homogeneous on R+,then it suffices that they have the same right characteristic measure. If u, = U,I on Onfig, and if n, n' are u-integrable on 0, then no = do.A corresponding assertion holds with 0 replaced by P, as well as the two corresponding cases with n, K' homogeneous on R+,in which cases 4 9 is replaced by 4id.
+
PROOF:Let n'l := n n', so that n and n' are both dominated by n". By the absolute continuity theorem (66.15), we may choose 2 E 49 with 0 5 Z 5 1 so that n = 2 * n" and n' = (1- 2 ) * ic". Thus du, = Zdu,tt and dv,, = (1 - Z ) du,,!. By hypothesis, 2 du,ll = (1 - 2 )du,tt, and consequently 2 = 1 - 2 a.e. relative to u,tt. Hence 12 - 1/21 du,tl = 0, and so, by (75.10), 12 - 1/21 * n' is Pc-evanescent. Therefore, up to Peevanescence, n = n'. This proves the first case. The other cases follow from the corresponding cases of the absolute continuity theorem, taking into account the obvious fact that the restrictions of the characteristic measures of IC and no to Onfig are identical, and making the corresponding changes for the other assertions. REMARK.The standard characteristic measures in the literature axe defined on E rather than f j g or ad.For example, if A , B are 6-u-integrable LAF's of X , the last theorem shows that A = B up to Pe-evanescence if and only if p ~ ( f )= p g ( f ) , f E pE. This result applies, for example, to continuous AF's. Similarly, if we impose hypotheses on X (duality hypotheses,
VIII: Additive finctionals
363
for example) which allow us to identify P n f j g with {f(X-) : f E E e } , then two <-a-integrable AF’s A, B are Pt-indistinguishable if and only if U A = ug on all processes of the form f o x - . That is, the characteristic measures should be thought of as measures on E in the latter case. (75.12) EXERCISE. Assume that the conditions of (68.5) are in force, and let A be a continuous AF. Prove that up to Pt-evanescence, At = pA(dx) L:. That is, every continuous AF is essentially a mixture of local times at points.
s
Under duality hypotheses, the relation (75.6) may be inverted so as to allow the potential of a HRM to be expressed directly in terms of its characteristic measure. The interested reader is referred to [GS84] for a full account.
Appendices AO. Monotone Class Theorems Given a class 31 of real functions on a set M , the notations b'H, p7-l stand respectively for: (AO.l) b7f := {f E 'H : f is bounded}; (A0.2) p7f := {f E 31 : f is positive}. An MVS H ' on a set M is defined to be a collection of bounded, real functions on M satisfying the conditions: (A0.3) 31 is a vector space over R; (A0.4) 7 i contains the constant function 1 ~ ; (A0.5) (fn) C 'H, 0 I f~ 5 . 5 fn t f, f bounded + f E 'H. Given a measurable space (M,M), the class b M of bounded real Mmeasurable functions on M is clearly an MVS. We write f E M in case f is a real M-measurable function on M . Thus, M stands for the collection of all real M-measurable functions on M, as well a a-algebra of sets. The notation a { J } , with 3 an arbitrary collection of real functions on M, stands for the smallest a-algebra M on A4 with J c M . A collection K of real functions on a set M is multiplicative provided f g E K whenever f , g E K. The following result is basic to many of the proofs in the text. In the form given below, it is due to Dynkin. (A0.6) MONOTONECLASSTHEOREM ( M C T ) . Let K be a multiplicative class of bounded real functions on a set M , and let M := a { K } . If 'H is an MVS containing K, then 7-1 3 b M .
The reader is referred to [DM75, 1-21], where a proof is given under the additional hypothesis that 7f is closed under uniform convergence. The
Appendices
365
latter hypothesis is not necessary, as the following lemma verifies. (I thank Bruce Atkinson for pointing out this argument.) (A0.7) LEMMA.Every MVS 3-1 is closed under uniform convergence.
PROOF:Take an arbitrary (f,) C 'FI with fn + f uniformly on M . Let llfll := sup{lf(x)l : x E M } denote the uniform norm. We may suppose that en := llf, - f,+~ll is summable, passing to a subsequence if necessary. ..., s o t h a t a n L O a s n + ~ .Letg, : = f n - a n + 2 a l . Seta, : = E , + C , + ~ + Then gn E 3-1 since 3-1 is a vector space containing the constants, and (gn) is obviously uniformly bounded. Moreover, gn+l - g, = f,+l - f, E, 1 0, and g1 = f 1 + a1 2 0, so 0 5 g1 5 92 5 .... By (A0.5), g := lim, gn E X, which implies that f = g - 2al is also in 3-1. The form (A0.6) covers most of our monotone class needs, but at one point, we shall require the following form involving lattice operations instead of multiplication.
+
(A0.8) LATTICEFORM OF T H E MCT. Let 3-1 be an MVS on M , L a cone of positive bounded functions on M closed under the operation f A g := min(f,g) and containing 1 ~ If3-1 . 3 L,then 3-1 3 ba{L}. PROOF: Let K' := L - L denote the vector space generated by L. As 3-1 is a vector space, 3-1 3 K'. Let K denote the uniform closure of K',so that K is a vector space of bounded functions closed under uniform convergence. Observe that Ic' is a vector lattice. This is a consequence of the identity
As the uniform closure of a vector lattice is also a vector lattice, by trivial limit arguments, it follows that K is a vector lattice containing the - cI E Ic, for every constant c. constants. Therefore, f E Ic implies Given a finite interval [a,b], the function z + x2 on [ q b ] may be approximated uniformly by linear combinations of functions z --t Ix It follows that f E K implies f 2 E K. Given f , g E K , one finds that f g = ((f g)2 - (f - g)2)/4 E K. That is, K is a multiplicative class. We noted above that an MVS is necessarily closed under uniform convergence, and consequently 3-1 3 K. Thus (A0.8) follows from (A0.6).
If
cI.
+
(A0.9) NOTATION.Let ( M , M )and ( Y , Y )be measurable spaces. A map I$ : M + Y is in M / Y provided {I$ E A } := +-'(A) E M for all A E Y . It is easily verified that, under the conditions of (A0.9), if 3-1 is a multiplicative class on Y generating Y , then (f.4 : f e 3-1) is a multiplicative class generating M .
366
Markov Processes
The most common setting for applications of (A0.6) throughout this work is the following. Let M be a set, and let {& : a E d} be a family of maps of M into measurable spaces (Wa,Wa). Let M denote the aalgebra generated by the $,-that is, M is the least a-algebra with the property & E M/W, for every a E A. For each a,suppose H ' , C bW, is a multiplicative class generating W,. Then M is generated by the multiplicative class f i o & , ...fnO$a,, where n 2 1, a l l . . .,an E A , and fj E 'Haj for j = 1 , . . . ,n. The 4, will usually be the random variables defining a stochastic process with state space W,. One of the most frequently used consequences of (A0.6) is that if X,p are finite measures on ( M , M ) and if X(f) (:= J f d X ) and p ( f ) agree for all f in a multiplicative class K: c b M generating M, then X = p. A measure X on ( M ,M) is a-finite provided X(f) < 00 for some f E M with f > 0 a.e. relative to A. We shall say that X is s-finite provided X = EnAn, with every A, a finite measure. (This term was suggested by then p is a finite measure and A, << p Getoor.) If p := C X,/(2"Xn(M)), for each n. Write dX, = fn dp, where fn E bpM. Then dX = f dp, with f := C fn E pM. That is, an s-finite measure always has a representation in the form f dp, with f E M and p a finite measure. The converse is obvious. (The u-finite measures are those with such a representation, but with f taking only finite values.) Note that the image of an s-finite measure under a measurable map is also an s-finite measure, but the image of a u-finite measure is in general only s-finite. The most important observation about s-finite measures is that Fubini's theorem for product measures is valid for this class, as is the Radon-Nikodym theorem. For a simple example of a measure X which is s-finite but not a-finite, let X(B) := 00 or 0 according as B C R has Lebesgue measure strictly positive or 0.
367
Appendices
A l . Universal Completion and Trace Given an arbitrary measurable space ( M ,M ) , the universal completion of M is the a-algebra M u (also denoted by M * ) defined to be the intersection of the A-completions, M A ,of M as X runs over all finite positive measures on ( M ,M ) . The following result is elementary.
( A l . l ) PROPOSITION. Every finite measure A on ( M , M ) extends in a unique way to a measure on (M, M u ) ,and every finite measure on (M, M u ) is the unique extension of its restriction to M . One way to show that a set A c M is in M u is to show that, given an arbitrary finite measure A on ( M , M ) ,the A-outer measure of A is equal to the A-inner measure of A. It is more convenient in many respects to use an equivalent formulation of this in terms of a device called the trace.
(A1.2) DEFINITION. Let A be an arbitrary subset of M and let X be a finite measure on ( M , M ) . The trace of M on A is defined to be the a-algebra M A := {B n A : B E M } . The trace of A on A is defined to be the set function XA on ( A , M A )given by A A ( r ) := inf{A(B) : I? = B n A , B E M } .
It would be more precise to call these objects the outer traces, aa there are entirely analogous "inner" traces. That M A is a a-algebra is easy to verify.
(A1.3) LEMMA. The trace XA of A on A is a measure on ( A , M A )which may be realized as follows. Choose A0 E M with A0 3 A having minimal A-measure. Then for r E M A of the form l? = B f l A with B E M , A A ( r ) = X(B n Ao). PROOF:To obtain a set A0 as described above is simple. Take A, E M with A, 3 A and infA(A,) = inf{A(B) : B E M , B 3 A } , and let A0 := nA,. Fix such a set A0 and observe that if B1,Bz E M and B1 n A = B2 n A then (B1A B2) n A = 0 and so, by minimality of A(Ao), A(Aon(B1 A&}} = 0. Therefore X(AonB1)= A(AonBz),and this shows that if one puts v ( r ) := A(A0 n B) for E M A of the form I' = B n A with B E M , then Y is well defined on M A , and it is clearly a measure on ( A , M A ) .It is obvious from the definition of AA that for all r E M A , X A ( r ) 5 v(r).On the other hand, by the choice of Ao, XA(A) = v ( A ) ,and since XA is clearly subadditive on M A , AA and u are identical on M A . The characterization of subsets of M that belong to M u may now be stated in terms of the trace. In particular, (A1.4) implies that Mu" = M u .
368
Markov Processes
(A1.4) PROPOSITION. Let A c M and let M A be the trace of M on A. Then A E M u if and only if for every finite memure X on ( M , M ) ,there exists B E M with B c A such that X(B) = XA(A).
PROOF:If A E M u , there exist sets B , A0 in M with B c A c A0 and X(B) = X(A0). The set A0 clearly has minimal X measure among all sets in M containing A. Hence XA(B)= X(B n Ao) = X(B) = X(A0) = XA(A). The converse is just as obvious. Another elementary but useful fact about the trace of a cT-algebra is provided by the following proposition, which relates the families of measures on M A and M . (A1.5) PROPOSITION. Let A c M and Jet M A be the trace of M on A. Then: (i) given a finite measure X on ( A ,M A ) ,the formula X(B) := X(B n A ) for B E M defines a finite measure on ( M ,M ) whose trace on A is A; (ii) MU)^ c MA)^, and equality obtains i f A E Mu. PROOF:It is clear that h is a measure on ( M , M ) . Choose A0 3 A with A0 E M having minimal measure. For all B E M , (X)A(Bn A ) = X(B n A o ) = X((B n Ao)n A ) = X(B n A ) , and so X = (X)A on M A , establishing (i). Now let r E MU)^ and let X be a take B1 C B C BZ finite measure on ( A , M A ) .If r = B n A with B E M u , with B1, BZ E M and = A(&). Let rj = Bj r l A E M A ( j = 1,2). Then rl c r c r2and by (i),
x(rl)=
=X
( B ~ ) = X(rz).
Consequently r E MA)^, and this shows that mu)^ C MA)^. Suppose next that A E M u . Given a finite measure A on ( M , M ) ,choose sets Ao, A1 E M such that A1 c A C A0 and X(Ao) = X(A1). Since A0 has minimal A measure, XA(BnA)= X(BnA0)for all B E M . For r E ( M A ) %choose , sets rl,rZ E M A with rl c r c rZ,X A ( r 1 ) = X A ( r 2 ) . If rj = Bj n A with Bj E M ( j = 1,2) the above choice of A0 and A1 leads to
X(B1 n A1) = X(B1 n Ao) = X(B2 n Ao) = X(B2 n A l ) . We may assume B1 c Bz, replacing B1 by B1 n BZ and Bz by B1 U B2 if necessary. Since B1 n Al c r c B1 n Ao, this proves that I' E M A and so r E M u . Thus r E MU)^. Note the following special case of the above facts about the trace of a measure. The formula comes from (A1.3) via a monotone class argument starting with f := lg,B E M .
Appendices
369
(A1.6) PROPOSITION. Let A every f E L1(X), L
~
cM ~
have full A-outer measure. Then, for
s,
A = ~ A
fdA. A
If ( M 1 , M l )and (M2,M z ) are measurable spaces, it is easy to see that ( M I 8 M2)" 3 MY 8 MZ, but these a-algebras are not in general equal. It is also easy to show that if 4 : M I -, Mz is in M 1 / M z -that is, $-l(B) E M I for all B E M z - then 4 E MY/MZ. The same holds if one assumes only the weaker condition $ E M y / M 2 . A2. Radon Spaces Throughout this section, E will denote a separable, metrizable space, and d will denote a metric compatible with its topology. Here, and throughout the entire work, we set: C ( E )= C(E,d) := real continuous functions on E ; C d ( E ) := real d-uniformly continuous functions on E. Thus, using the notation of SAO, bC(E,d) denotes the bounded continuous functions on E. If f is a real lower semicontinuous function on ( E ,d) and if f is bounded from below, it is a matter of elementary analysis to see that the sequence fn(z) := inf{f(y) nd(z,y) : y E E } increases to f(z),and each fn is d-uniformly continuous.
+
(A2.1) PROPOSITION. For each f E bC(E,d), there exist monotone sequences {fn), {gn) c w@> with fn t f and gn 1 f.
PROOF:Let f E bC(E). Then f is lower semicontinuous and bounded below. The sequence (fn) C Cd(E) defined above increases to f . To get (gn),apply the same argument to -f. The Borel a-algebra B ( E ) on E is defined to be the 0-algebra generated by bC(E).In view of (A2.1), this is the same as a(bCd(E)). The advantage of bCd(E)is that if ( E ,d) is totally bounded, bCd(E)is separable in uniform norm 11 . 11 whereas bC(E)is in general not so. For a complete discussion of the properties of the Borel a-algebra of a metric space, consult [DM75, Vol. 11, [Bo56], [Pa67],or [Bi68]. Let B"(E) denote, as in A l , the universal completion of the Borel aalgebra B ( E ) on E . The sets in B"(E) are called the universally measurable sets in E. A topological space E is called a Lusin (resp., Souslin, co-Souslin, Radon) topological space if it is homeomorphic to a Borel (resp., analytic, co-analytic, universally measurable) subset of a compact metric space.
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Our The first three types are discussed in Bourbaki [Bo56]and in [DM75]. definition of a Radon space is more restrictive than that used by Bourbaki and by Schwartz [Sc73,11.531, where E is called Radon if every finite measure on (E,B ( E ) ) is inner regular. In view of (A2.3) below, the definitions agree in case E is separable and metrizable. We note that since every Borel subset of a compact metric space is analytic and every analytic set is universally measurable, every Lusin, Souslin or co-Souslin space is Radonian. If E is a locally compact Hausdorff space with countable base (abbreviated LCCB) then E is a Lusin (hence Radon) space, as the one point compactification of E is compact and metrizable. More generally, if E is a Polish (i.e., complete separable metric) space, a well known theorem of Alexandroff states that E is homeomorphic to a G6 subset of a compact metrizable space. It follows that such an E is Lusinian. A measurable isomorphism : M + N of measurable spaces ( M , M ) , ( N , N ) is a bijection Q : M + N with @ E M j N and Q-l E N / M . One a measurable space (M,M) to be a Lusinian then defines, as in [DM75], measurable space if it is measurably isomorphic to (E,B ( E ) ) ,where E is some Lusin topological space. We shall use also the corresponding obvious definitions of Souslin, co-Souslin and Radon measurable spaces. A Radon measurable space is also known as an absolute Borel space (Mackey's terminology.) Let E be a subset of a separable metrizable space F. (A2.2) PROPOSITION. Then: (i) B ( E ) is the trace of B ( F ) on E; (ii) if E E B"(F), then B"(E) is the trace of B"(F) on E. PROOF: (i): Let d be a metric compatible with the topology of F. Then C d ( E ) = { f l ~ : f E bCd(F)} and so bB(E) = { f l ~ : f E bB(F)} by the MCT (A0.6). (ii): By (A1.5), ( f 3 " ( F ) )=~ ( B ( F ) E )and ~ so by (i), B" (E) = (B"( F ) ) E . A finite measure X on the Borel 8-algebra of a metrizable space E is said to be tight provided
X(E) = sup{X(K) : K compact in E}. (A2.3) THEOREM. Let E be a separable metrizable space. Then E is Radonirtn if and only if every finite measure on (E,B ( E ) ) is tight. PROOF: Suppose that E is Radonian and that E is topologically embedded in a compact metric space F, with E € B"(F). A finite measure A on ( E , B " ( E ) )may be regarded aa a finite measure on ( F , P ( F ) )that is
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carried by E . Since E E B"(F), there exists B E B ( F ) with B C E and X(B) = X(E) = X(F). Since every finite measure on a compact metric space is inner regular, X(B) = sup{X(K) : K compact, K c B}. Suppose next that every finite measure on ( E , B ( E ) )is tight. Let d be a totally bounded metric on E compatible with its topology and let F be the dcompletion of E . Then F is compact metric and by (A2.2), B ( E ) is the trace of B ( F ) on E. We show that E E B U ( F ) .Given a finite measure X on F , let XE be its trace on E . By hypothesis, there exists a K , set H c E with X E ( H ) = XE(E). Since H E B ( F ) , (A1.4) is satisfied, so E E B"(F). It is, perhaps, worthwhile to remark that (A2.3) implies that Kolmogorov's extension theorem holds for products of Radon spaces. This is a consequence of its statement in the form given by Neveu [Ne65,p.821. We may now list some permanence properties of Radon spaces.
(A2.4) PROPOSITION. (i) A universally measurable subspace of a Radon topological space is a Radon space in the subspace topology. (ii) The topological union of two Radon spaces is a Radon space. (iii) The topological product of a finite or countable collection of Radon spaces is also a Radon space. PROOF: Using (A1.5) we obtain (i) directly, and (ii) is obvious. Suppose that I is a finite or countable index set and that for each i E I, Ei is embedded topologically in a compact metrizable space Fi, and Ei E B"(Fi). Then F := Fi is compact and metrizable. We show that n E i E B"(F). Let X be a finite measure on F and let Xi be its projection on Fi. For each i E I, choose sets A*,Bi E B(F,) with Ai c E ~C Bi and Xi(Bi - Ai) = 0. Then A, C E, C B, and
n
fl
n
n
Since X is arbitrary, it follows that
n Ei E B U ( F ) .
The next group of results contains the analogues for Radon spaces of the following well known facts about Lusin spaces. Let E be a Lusin space and Jet F be a separable (A2.5) PROPOSITION. metrizable space. If cp : E + F is B(E)/B(F)-measurable, then the graph of cp is Borel measurable in E x F .
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PROOF:Let d be a separable metric for F. The function (y1,yz) + d(y1,yz) is jointly continuous, hence Borel on F x F. It follows that the diagonal D in F x F is Borel in F x F . The function (p : E x F + F x F defined by @(z,y) := (cp(z), y) is B(E) @B(F)/B(F)-measurablesince each coordinate is Borel measurable, and using the fact that the graph of 'p is (p-l(D), the proof is complete. The following result of Lusin is quite nontrivial. For a nice proof see [DM75, 111-211. See also Parathasarathy [Pa67,p. 221. (LUSIN).Let E be a Lusin topological space and let (A2.6) THEOREM : E + F be injective and B(E)/B(F)-measurabI. Then 4 ( E ) is Borel in F.
F be a separable metrizable space. Let 4
We generalize (A2.5) in (A2.7). In its statement, the condition 4 E B"(E)/D(F) is equivalent to the condition 4 E D"(E)/D"(F), which looks stronger, by the remarks at the end of SAl. Following that, (A2.8) gives an analogue of (A2.6) which is, however, much more elementary in nature. The corollaries (A2.10) and (A2.11) are important for several reasons. The general form of (A2.10) is crucial to the Ray theory in Chapter V, and (A2.11) states an important invariance property in Radon spaces. Let E and F be Radon spaces and let cp : E (A2.7) THEOREM. B"(E)/B(F)measurable. Then the graph of cp is in D"(E x F ) .
+F
be
PROOF: Let?! , and F be compact metric spaces containing (topologically) E and F respectively as universally measurable sets. Fix a sequence (u,) uniformly dense in C ( F ) , and let K := u, I), 11 u, 111 with the product topology. The map : z + (un(z))of F into X is a homeomorphism of F onto [ ( F ) C K . Let (p := [04 and let cp, := u,ocp. Each cp, E bB"(E), and the map (p is a map of E into t ( F ) which is B"(E)/B"([(F))-measurable. It suffices to show that the graph Q of (p is universally measurable in E x K , for then Q is universally measurable in E x [ ( F ) , hence in E x F ( since homeomorphisms preserve universally measurable sets), hence in E x F , making repeated use of (Al.5ii). ' denote the trace of Let X be a finite mesure on E x K and let H B ( E ) @ D ( K) on a. That is, 'H := { A 11 : A E D(E) 8 B ( K ) } . Let Xo denote the trace (A1.2) of X on (Q,'H). We shall establish the fact that Q is universally measurable on E x K by producing a Borel subset A of E x K such that A C Q and Xo(A) = A'(@). To this end, let T I denote the projection map of E x K on E . Since T I is Borel measurable, by definition of 'H, X I ~ q ,is 'H/B(E)measurable. Let p be the image of Xo under T Ilq,. Since E is universally measurable in E there exists a Borel set Eo in E such that EO C E and ~ ( E o=) p ( E ) = Xo(Q). Let (p, be defined on E by (p,l~ := cp, and (p, := 0 on E \ E. Then each (p, is a universally
n,[-Il
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373
measurable function on E . Choose bounded Borel functions gn,hn on E with gn 5 Cp, 5 hn and p(hn - gn) = 0. Let El := (nn{gn = h,}) n Eo. Then El E B ( E ) , El c E and p(E1) = p ( E ) = AD(@), and for all n, ( O , I E ~ is a Borel function on E l . Therefore C ~ I Eis~ a Borel mapping of El into K and its graph A, considered as a subset of E x K is Borel by (A2.5). However, A C @ and
proving the theorem. (A2.8) THEOREM. Let E and F be Radon spaces and let 4 : E + F be an injective B"(E)/B(F)-measurable map. Suppose 4 satisfies the condition (A2.9)
V A E B ( E ) , 3B E B"(F) such that 4 ( A ) = B n +(E).
Then $ ( E ) E B"(F), and in fact 4 ( A ) E D"(F) for all A E B"(E).
PROOF:Once we prove that +(E)E B"(F), the last assertion follows by . we proved in the preceding theorem, the applying the theorem to 4 1 ~ As graph of 4 is Radonian. Let G := 4 ( E ) and let 7r2 denote the projection map fromf E x F to F . Then 7r2(@) = G. The map 7r2lip is injective by @ denote its inverse. Now let 6 denote the trace hypothesis. Let q : G of B"(F) on G. The condition (A2.9) implies that q is E/B(@)-measurable, for if A E B(@) has the form ( A x B ) n @ with A E B ( E ) and B E B ( F ) , ---f
q - l ( A ) = {y E G : y E B and for some 2 E A , +(z) = y} = B n + ( A ) . By (A2.9), there exists C E B"(F) with 4 ( A ) = C n # ( E )= CflG, and this proves that 4-'(A) E 4. Now let v be a finite measure on F and let Y O be its trace on (G, 6 ) . Let X be the image of vo under q. Since @ is Radonian, there exists by (A2.3) a K, subset A of @ such that A(A) = A(@) = vo(G). But H = 7r2(A) is a K , subset of G with v o ( H ) = A(A) = vo(G),so G E B"(F) by (A1.4). (A2.10) COROLLARY. Let dl and d2 be separable metrics on a set E such that the respective Borel a-algebras B l ( E ) , & ( E ) satisfy & ( E ) C & ( E ) c B y ( E ) . If ( E ,d l ) is Radonian, then ( E ,d 2 ) is also Radonian. In particular, if & ( E ) = & ( E ) then ( E ,d l ) is Radonian if and only if ( E ,d 2 ) is Radonian.
PROOF:The last assertion is an obvious consequence of the first. Let F denote the completion of E relative to d2. Since F is Polish, it is Radonian. Let i : E 4 F denote the injection map. Since & ( E ) is the trace on E of the Borel a-algebra &(F) on F, the hypothesis & ( E ) C Dy(E) implies
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that i is B~(E)/B2(F)-measurable. On the other hand the hypothesis &(E) c &(E) implies that the condition (A2.9) holds for i, for if A E Bl(E), i ( A ) E B z ( E > is of the form C i l E with C E B2(F), once again because &(E) is the trace of & ( F ) on E. Now by the theorem, E = i(E) is universally measurable in (F,dz). (A2.11) COROLLARY. Let E be Radonian and let F be a separable metrizable space. If E is embedded topologically in F, then E E Bu(F).
PROOF:Let F' be a Radon topological space containing F topologicallyfor example, the completion of F relative to some metric on F. According to (A2.2), B(E) is the trace of B(F') on E. The injection map i : E + F' is therefore B(E)/B(F')-measurable. It follows also that i satisfies (A2.9) in a trivial way. By (A2.8), E E P ( F ' ) . Then (A2.2ii) shows that E = E fI F E ( B u ( F ' ) )C~ B"(F). Many of the preceding results have versions in terms of measurable rather than topological spaces. For example, (A2.8) implies easily that Let ( M , M ) be a Radon measurable space and let d (A2.12) THEOREM. be a metric on M having the property M u= Bu(M,d). Then (M, d) is a Radon topological space.
A sequence { pn} of probability measures on a Radon space (E, d) converges weakly to a probability measure p provided
If -,If dpn
dp for every f E
bC(E,d).
It is well known [Bi68, p.111 that if pn (n 2 1) and p are probability measures on (E, B(E)), then the following conditions are equivalent: (A2.13). (i) pn converges weakly to p; (ii) J f d p n -+ J f dp for all f E Cd(E); (iii) limsupp,(K) 5 p ( K ) for all closed subsets K of E; (iv) liminf pn(G) 2 p(G) for all open subsets G of E. Recall that a family A of probability measures on (E, B(.E)) is said to be tight if, given e > 0, there exists a compact subset K of E such that X(K) > 1 - t for every X E A. According to (A2.3), every singleton {A} is tight. One has then the following criterion for weak compactness. (A2.14) THEOREM. Let E be a Radon topological space. (i) If A is a tight family of probability measures on E, then every sequence in A has a weakly convergent subsequence. (ii) If pn converges weakly to p on E , then the family { p , 1-11, pz, .. .} is tight .
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Item (i) is usually called the direct part of Prohorov’s theorem. It is 111-591. valid in an arbitrary metric space. See [Bi68, p.371 or [DM75, Item (ii) is a form of the converse part of Prohorov’s theorem. See [Bit& p. 2411. In the next section, another important property of Radon spaces will appear in connection with the construction of kernels, a special case of which is the existence of regular conditional probabilities. IV-18 We conclude this section with some important examples [DM75, and IV-191 of Radon spaces used several times in the text. Let E be a (metrizable) co-Souslinian topological (A2.15) THEOREM. space, D a countable dense subset of R+, and let R = E D with a-algebra generated by the coordinate mappings. Let W = {w E R : w is the restriction to D of a right continuous map w of R+ into E } . Then W , with a-algebra generated by the coordinate mappings, is co-analytic in R, and hence the measurable space W is co-Souslinian. If the term ‘(right continuous” is replaced everywhere by “right continuous with left limits”, and if E is Polish, then W is in fact Lusinian. (A2.16) THEOREM. The meawrabfe space ofrcll maps ofR+ into a Polish space with a-algebra generated by the coordinate maps is Lusinian. A3. Kernels Let ( E , E )and (M,M) be measurable spaces. A kernel K from (M,M) to ( E ,E ) is a function K ( z ,A ) defined for z E M and A E E , having values in [0,00], such that: (A3.li) V A E E , z -+ K ( z ,A) is M-measurable; (ii) V x E M , A -+ K ( z ,A ) is a measure on ( E ,E ) . A kernel K is called finite (resp., bounded, Markov, sub-Markov) if z -, K ( z ,M ) is finite (resp., bounded, = 1,s1). A kernel from ( M , M ) to ( M , M ) is called simply a kernel on ( M , M ) . Being given a kernel K from ( M , M ) to ( E , E ) ,define K f for f E pE, the space of positive E-measurable functions on E , by
The kernel K is proper in case there is a strictly positive f E pE such that Kf(z) < 00 for all z E M . If K is a finite kernel, Kf may be defined for all f E bE, the space of bounded E-measurable functions. A finite kernel K thus induces a mapping of bE into M-measurable functions on M which is linear, positive and which
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respects positive monotone convergence. By this last phrase, we mean that if 0 5 f n 1 f in bE then Kfn t Kf pointwise on M. Conversely, it is obvious that a mapping of bE into the space of M-measurable functions enjoying the above properties is necessarily induced by a finite kernel. A bounded kernel K from ( M ,M) to ( E ,E ) extends automatically and uniquely to a kernel from ( M ,M u ) to (ElE"). (For x E M , just extend K(x, - ) in the only possible way to be a measure on (El&").For p a measure on ( M , M ) and f E bE", choose f i , f2 cz bE with f l 5 f 5 f 2 and pK(f2 - f l ) = 0. Then p(Kf2 - Kfl) = 0, hence K f E Mu.) A measurable space ( E ,E ) is called separable if there exists a countable family of functions on (or sets in) E which generate the a-algebra E. The following lemma is a basic tool in a number of important constructions.
(A3.2) LEMMA(DooB). Let (E,E) and (M,M) be measurable spaces, (El&)being separable. Let K and L be finite kernels from ( M , M ) to ( E , € ) such that for all x E M , K(x, - ) << L(z, -). Then there exists $ E p(M c 3 E ) such that for all x E M,
A proof is given in [DM75, V-581, based on martingale convergence of measure ratios to the Radon-Nikodym derivative. Given a bounded kernel K from ( M ,M ) to ( E ,E ) and a finite measure p on ( M , M ) , pK denotes the measure on (El&)specified by
The following result gives a means of constructing kernels in a manner which is very useful in a number of situations in Markov process theory. It is used in the theory of LCvy systems, representation of excursions, and it is closely connected with the choice of regular conditional probabilities. A more complete discussion is given in [Ge75b]. In the statement of the theorem below we adopt the following terminology. Given a family A of finite measures on a measurable space ( M ,M), then f = g a.e. (A) means X{f # 9) = 0 for all X E A, where f and g are functions on M.
(A3.3) THEOREM. Let (M,M) be a measurable space and let A be a family of finite measures on (M,M). Let (G, 6 ) be a Radon measurable space, and assume given a mapping T : bE + b M with the properties: (i) f E bpE implies Tf 1 0 a.e. (A); (ii) for f , g E bB and a , b E R, T(af bg) = aTf (iii) 0 5 f n T f in bE implies Tfn T Tf a.e. (A).
+
+ bTg a.e. (A);
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377
There exists then a bounded kernel K from ( M ,M") to (G, G ) such that for every f E bG, Tf = Kf a.e. (A). If A is a countable family, or if (G, G ) is Lusinian, then K may be constructed as a kernel from ( M ,M ) to (G, G).
PROOF:It is clearly sufficient to suppose G := E , a Radon topological space, and Q := B ( E ) . Let E be a compact metric space containing E as a universally measurable set, and define T : bB(@ + b M by
Tf := T ( f l ~ ) , f^ E bB(E). Then T satisfies (i) and (ii) with G replaced by E . We shall construct a kernel k from ( M , M )to (&,a(.@))such that Tf^= kfa.e. (A) for all f^ E bB(E), then without changing notation, extend K to a kernel from ( M ,M u )to ( k ,Bu(k)) satisfying ff := T(fl~ for all f^ E bBu(E). Then k l ~=\Tlfi\E ~ = 0 a.e. (A). Set A := {z E M : k l h \ E ( s ) = 0) E Mu and define a kernel K from ( M ,Mu) to ( E ,B ( E ) ) by
K(z,* )
:= l A ( z ) k ( z , ' ) I E .
That is, for x E A , K ( s , . ) is the restriction of the measure k ( z , . ) to E . Then for all f E bB(E), i f f E bB(@ is an extension o f f ,
~f = 1 A k f
=
kf a.e. (A) = ~f
a.e. (A),
and so K will be the desired kernel. It is clear that if E is Lusinian so that E is Bore1 in E, then A E M and K is a kernel from ( M , M ) to (.E,B ( E ) ) . In the special case where A is countable, say A = {An : n 2 l}, let X := Cnr1(2")1 A, ll)-lAQ so that one may as well assume A = {A}. Then one constructs K in a slightly different way from k. Let p := Ak, a finite measure on (&,Bu(h)) with p ( E ) = A(k1~) = A(k1k) = p ( @ . Choose Bo E B ( k ) with p ( B 0 ) = p ( E ) and Bo C E. Then let A0 := {z E M : klfi\Bo(x)= 0) E M . Because A f i ' l ~ \ , ~= p ( k \ Bo) = 0, one finds X(M \ Ao) = 0. Setting then
K ( z , ' ) := 1 A o ( z ) k ( x ', )
~ E I
we obtain a kernel from ( M ,M ) to ( E ,B ( E ) )with the asserted properties. All that remains is to construct the kernel I?. Let 'H be a countable vector space over the rationals &, containing the constant function 1~ and dense inC(k). Let c := Il?'lEII. For h E 'H and z E M , let & ( h ) := Fh(z). For each cu,P E Q and f,g E 'H, let
W % P , f,9) := {.
: &(af
+ P g ) # &(f) + P4z(S)l,
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and for h E H and k E p?i, let
N ( h ) := {Z : l&(h)l
> cII h II},
N+(k):= {Z : &(k) < 0).
By hypothesis, N ( a , /3, f , g ) , N ( h ) and N+(k)are A-null for all X E A. Let N be their union over all f , g, h E H,k E pH, a, /3 E Q. Then X(N) = 0 for all X E A. Define then
.tlr,(h) := 1NC(.)dz(h).
For each z E M , h -+ &(h) is positive, Q-linear, and I$,(h)l 5 cllhll for all h E 3-1. In addition, X { z : $,(h) # f h ( z ) }= 0 for all X E A, h E H. Since H is uniformly dense in C ( f i ) , the map h + & ( h ) extends by continuity to a map f + $,(f) of C(&) into b M that is linear, positive and bounded by c. For all f E C(&) and X E A, X{z : $,(f) # ff(z)} = 0, because if h, f uniformly, f h n -t Tf a.e. (A). By the Riesz representation theorem, for each z E M there exists a finite measure k(z, .) on (&,a(&))such that $,(f)= kf(z)for all f E C(@. Since z -+ kf(z)is M-measurable for all f E C ( f i ) , and hence for all f E bB(&) by the MCT, k is a kernel from (M,M) to (k,B(&)) which is bounded because kl(z)= $,(1) 5 c. The kernel K has the property -+
kf= Tf
a.e. (A) for all f E B(@).
The set K: o f f E bB(&) having this property is a vector space containing B(&). If 0 5 f, t f (bounded) with f, E K: for all n, then f E K: because of the hypothesis (iii). By the MCT,K: contains bB(&), and this completes the proof. The first corollary is a direct consequence of the construction of K in the proof of (A3.3) Let (G,G)= ( E , B ( E ) )with E a Radon topological (A3.4) COROLLARY. space embedded in a Lusin topological space E’. Then there exists a kernel K‘ from ( M ,M ) to (E‘,B(E’))such that for every f’ E bB(E’), T ( f ’ 1 ~=) K‘f’ a.e. (A). The following well known “disintegration” follows at once from (A3.3).
(A3.5) COROLLARY. Let (E,E), ( F , F ) be Radon (resp., Lusin) measurand let X denote the able spaces, p a finite measure on ( E x F,E 63 3), of p on E . Then there is a kernel K from (E,E”) to ( F , P ) projection (resp., ( E ,E ) to (F,F))such that
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A4. Lebesgue-Stieltjes Integrals We set down here some of the properties of Lebesgue-Stieltjes integrals which are useful in the text. Let f be an increasing function with finite values defined on ]a,b[ C R. The right (resp., left) continuous inverse of f is defined to be the function p (resp., A) where for s E R,
(A4.1) (A4.2)
p ( s ) := inf{t € ] a ,b [ : f ( t ) > s} A b,
X(S) := sup{t € ] a ,b [ : f ( t ) < S} V a
(inf 0 := 00); (SUP@ := -w).
It is obvious that p and X are increasing functions taking values in [a,b]. It is also easy to see that for all s E R, p ( s ) = X(s+) and X(s) = p ( s - ) . Thus X(s) = p ( s ) for all but countably many s E R.
(A4.3) THEOREM (CHANGE O F VARIABLE FORMULA). Let ( T t ) be a finite valued right continuous increasing function defined on ] a ,b[ and let (C,) be ) r(b) := ~ ( b - ) . its left continuous inverse function. Set .(a) := ~ ( a + and Then: (A4.4) Vt € ] a ,b[ and s E R, t < C, if and only if rt < s ; (A4.5) ( r t ) is the restriction to ]a,b[of the right continuous inverse of
(&I;
(A4.6)
, for if m is right continuous and increasing on [ ~ ( a )~,( b ) ]then every positive Borel function g on ]a,b[,
PROOF:If t E [a,b[ and r ( t ) < s, right continuity of T implies that r ( t + ~< ) s for all sufficiently small E > 0. By definition of C, C(s) 2 t + E for some E > 0 so C(s) > t. On the other hand, if t < l ( s ) ,the definition of C implies that r ( t ) < s. This proves (A4.4), and (A4.5) is a trivial consequence. It where a 5 u < w < b. By (A4.4), suffices to prove (A4.7) in case g = llu,u~ g(a(s)) l{a
= 1]r(u),r(tJ)](s),
since C(s) 5 w implies l(s) < b. Thus, in this case, the right side of (A4.7) reduces to m(r(w))- m ( r ( u ) )which is equal to the left side of (A4.7). If ]a, b[ =]O, oo[ and m(t) = t , we obtain a more familiar form of the change of variable formula, stating that for all g E bB( 10, m[)
(A4.8)
I.4
380
Markov Processes
Since l ( t ) = l ( t + ) for all but countably many t , the right side of (A4.8) can be written in the form &(o),r(w)[ g(&) l{O
ha,b1ha,bl
(A4.9)
F(b)G(b)- F(a)G(a) =
r
F(t-) dG(t)
or, more compactly, F- denoting the function t d( F G ) = F- dG
If F and F- are strictly positive on
P
G ( t )d F ( t ) , h,bl
da,bl
(A4.10)
+
F(t-),
-+
+ G dF.
[a, b ] , taking
0 = d ( F G ) = F- dG
G := 1 / F , one gets
+GdF
so (A4.11)
d ( l / F ) = -dF/(FF-).
A right continuous function F of bounded variation on [a,b] is called purely discontinuous if, for all a 5 u 5 v 5 b, F(v) - F ( u ) = Cu
PROOF:Using the Hahn-Jordan decomposition theorem, every purely discontinuous function on [a, b] may be expressed as a difference of positive increasing purely discontinuous functions. By the above remarks it suffices to prove the lemma in case F is positive and increasing. By induction, Fn
Appendices
381
is purely discontinuous for every n 2 0. For a 5 u < v 5 b,
u
the interchange of order of summation being justified because the summands are positive. We come now to the Stieltjes versions of exponentials and logarithms. Following Meyer [Me66b] with slight modifications, we reserve the term M-function for a right continuous decreasing function m : [0, 00[+ [0,1] such that m(0) = 1 and an A-function a right continuous increasing function a : [0,00[+ [0,00] such that a ( 0 ) = 0 and Aa(t) < 1 for all t with a ( t ) < a(00), with Aa(t) = 1 possible if a ( t ) = a ( m ) < 00. ( Here a ( w ) := sup a ( t ) = lirnttooa ( t ) . ) For m an M-function, let Crn := inf { t : m(t) = 0) and for a an Afunction, let Ca := inf {t : a ( t ) = 00 or A a ( t ) = 1) The Stieltjes logarithm of an M-function m is defined to be the function slog m, where (A4.13)
(slogm), :=
lo,tAcml (-dmr)/mT--.
If a is an A-function, we may express a in a unique way as a = ac + ad where a' and ad are A-functions which stop at ac being continuous and ad purely discontinuous. The Stieltjes exponential sexpa is defined by
ca,
(A4.14)
(sexp a)t :=
n
[l - AaU].
O
Before looking at these operations in detail we consider some simple ex[. amples. If m := l[o,c[, 0 < 5 00, then slog m = 1 [ ~ , ~Conversely, ~ [ sexp a = l[O,c[. Next, if m is continuous, applying if a := 1 [ ~ ,then (A4.8) to compute the integral in (A4.13), one obtains slog m = -log m, so slog reduces to the negative of the ordinary logarithm. If a is continuous, (A4.14) reduces to sexp a = e-', the ordinary negative exponential of a. We examine now the properties of the Stieltjes logarithm.
<
Markov Processes
382
(A4.15) LEMMA.Let m be an M-function and Jet a = slog(m). Then mC- = 0 if and only if aC = 00. PROOF:If rnf- = 0 one may choose a sequence t, mt,+l < mt,/2. Then
-dm,/m,-
2
c/+ n
tt C such that for all n,
-dm,/m,-
n , n+l
n
= 00.
Conversely, if mC- = P > 0, aC = JI O S 1 (-dm,)/m,-
5 1 / P < 00.
(A4.16) LEMMA.If m and n are M-functions with no common discontinuity then slog(mn)t = (slofdm) + SIOg(n))tACmACn
-
PROOF:For t
< Cm
A
Cn,
mt-
> 0 and nt- > 0, so by (A4.9),
Since m and n have no common discontinuity on [0, Cm A Cn[,nu = nu- a.e. (drn,) so the latter integral reduces to dm,/m,-. Thus slog(mn)= slog(m) slog(n)on [0,Cm A Cn[. By definition, slog(mn)stops at Cmn = Cm ACn and its value at Cmn is slog(mn)(~mn-)-A(mn)(~mn)/mn(~mn-) where the latter summand is zero if (mn)(Cmn-)= 0. If (mn)(C,,-) = 0, either m(Cmn-)= 0 or n(Cmn-)= 0 so that (A4.15) gives slog(m)(Cmn) slog(n)(C,,) = 00. The equality is therefore verified if mn(Cmn-)= 0. If mn(Cmn-)> 0, since m and n cannot both jump at Cmn, either J-, < Cn or Cn < Cm. We argue the case Cm < Cn. One has Cmn = Cm so, since n is continuous at Cm,
+
ho,tl
+
On the other hand,
Appendices
383
so the sought equality follows from slog n(Cm)= slog n(Crn-). Turning t o the Stieltjes exponential of an A-function u = ac+ud, observe that since Au, < 1 for all u < Ca and CO
no
no,,,t[l
If m is an M-function, then u = slog(m) is an A(A4.17) THEOREM. function. The map m + slog(m) is a bijective map of the class of Mfunctions onto the class of A-functions, and u = slog(m) if and only if sexp(a) = m.
PROOF:Because of the results already obtained, it is enough to demonstrate the following points. (i) If rn is an M-function, then slog(m) is an A-function. (ii) If slog(m) = slog(n) then m = n. (iii) slog(sexp u ) = a for every A-function a. If m and n are M-functions, if u = slog(m) = slog(n), and if Crn 5 Cn, then (A4.9) gives, for t < Cm,
However,
J(o.,
mu- dnunu- = -
lo.,
mu- da, =
J(O.,
mu- dm,/m,-
= mt - 1.
It follows that & t l nu d(m,/n,) = 0, and since nu > 0 for all u E I O , ~ ] , m,/n, = 1 for all u 5 t. Thus ml[o,Cm[ = nl[o,cm[.If m(C,-) = 0 then n(Cm-) = 0 so m = n. If m(Cm-) > 0,
so
Markov Processes
384
This proves that n(Cm) = 0, so that in all cases, m = n. The calculations made above show that Aa, < 1 for all u except possibly at the first time m vanishes. We have now verified (i) and (ii). Turning to (iii), if mt := and nt := n,,,,,[l-Aa,], then m and n are M-functions without common discontinuitiesso, applying (A4.16),
We pointed out earlier that slogm = a'. We argue next that n is purely discontinuous. If t < Cn,
shows that log n is purely discontinuous and so, by (A4.12), R. is also purely discontinuous. One has, for u < Cn -An,/n,-
= 1 - nu/nu- = 1 - [l - Aa,] = Aa,.
-dn,/n,for all t < Cn. If n(Cn-) = 0, Ud(Cn-) = 00 Therefore a$ = by (A4.15) so in this case a: = (slog n)t for all t 1 0. On the other hand, if n(Cn-) > 0,
This proves that slog n = ad in all cases. Going back to (A4.18) we have
but since Cm A Cn =
Appendices
385
A5. Sketch of the General Theory of Processes This section gives the briefest possible reference guide to the parts of the socalled general theory of processes which are of greatest utility in the text. Most of the details may be found in the first two volumes of [DM75]. This summary is not intended to be as thorough as Meyer’s “Guide Gris” [Me68b].It is more akin to a two-hour self-guided tour of Strasbourg. Let (fl, 8,P) denote a probability space. A filtration of (R, 6 ,P) is a collection (Bt) of sub-a-algebras of 6 indexed by some subset of R, such that 6, c Bt whenever s 5 t in the index set. We shall suppose from now on that the index set is R+. The filtration (Gt) of ( R , Q , P ) is said to satisfy the usual hypotheses of the general theory in case ( R , Q , P ) is complete, 60contains all P-null sets in 9, and the a-algebras have the A filtration right continuity property that, for every t 2 0, Et = n,&,. is supposed to be the mathematical object specifying for each time t the totality of information observable by time t . Right continuity builds in a form of infinitesimal prescience which simplifies the theory. Unless explicitly stated otherwise, it is always assumed in this section that (R, G, Bt, P) satifies the usual hypotheses. A real function Z t ( w ) defined on R+x fl is called a stochastic process. A stochastic process 2 is measurable provided it is measurable relative to M := f3+ @I G. A stochastic process 2 is adapted to (Gt) in case 2, E Gt for every t 2 0. Given 2 E M and T E pQ, the function Z ~ l p < ~ } (:= w) Z T ( ~ ) ( W ) ~ ( Tis( ~ in )9. <~ The } process 2 is progressive, or equivalently, progressively measurable, if, for every t 2 0, the map ( T , U ) -+ ZT(w)of [0,t ]x R is in f3( [0, t ] )@IGt. Being adapted to (Gt) is the weakest condition on Z reflecting its lack of functional dependence on future information, but without the additional structure imposed by progressive measurability, it is not possible to do much of interest with such a Z . A function T : R + [O,w] is an optional time or equivalently, a stopping time, for the filtration (Gt) provided {T 5 t } := { w : T ( w ) 5 t } E Bt for every t 3 0. Given any S : fl -, [0, 001, define the stochastic interval 10, Sg by
lo,si := { ( T , w ) : o I T I s ( w ) } . Similar definitions apply to definitions of, for example, [ O , S [ , US, w[I, and so on. It is then easy to see that T : R -+ [O,w] is optional for ( G t ) if and only if UT,001 is adapted to (Bt). In view of the assumed right continuity of (Gt), we could equally well have used IT’, 0;) [ . The debut Dr of a subset I? of R+ x f2 is defined by
IS, S’u,
D r ( w ) := inf{t : ( t , w ) E I’}. We come to the first major result in the general theory of processes.
386
Markov Processes
(A5.1) THEOREM (MEASURABILITY OF DEBUTS). If I‘ is progressive over (Gt), then Dr is optional over (Gt). The proof [DM75, IV-501 is based on the following result, which demands no hypotheses of any kind on G.
(A5.2) THEOREM (MEASURABILITY OF PROJECTIONS). Let E M := B(R+) €3 9. Then r is analytic over G (and therefore in the universal completion of G), and consequently Dr is measurable over the universal completion of 8. In other words, the projection .Ir(I’) of I? on R is in the universal completion of 0. The complete proof is discussed in [DM75, IV-441. The first results of this kind were theorems of Doob [Do541 and Hunt [Hu57], the latter showing that the debut of {t : Xt(w)E B} is an optional time in the case B Bore1 in Rd, X a Brownian motion in Rd. His use of the methods of capacity theory was one of the principal spurs toward the development of the general theory by Meyer and Dellacherie. The following application of (A5.2) comes from [DM75, IV-331.
(A5.3) THEOREM. Let 2 be a real progressive process over (&)- Then the following are also progressive: (i) (t, w) + supsst Zs(w);(ii) (t,w) -, limsupsLlt Zs(w); (iii) ( t , w ) -+ limsupsttt Zs(w); etc. See also (A5.6) for related cases. The next result [DM75, IV-341 is of more elementary nature.
(A5.4) PROPOSITION. Let (R,G) be an arbitrary measurable space and let
(X,)be a measurable stochastic process with values in a separable, metriz-
R, able space E which is co-Souslinian in the sense of 5A2. Let Rrcl1,arc, denote the set of w E R such that the map t + Xt(w) is respectively right continuous with left limits, right continuous, continuous. Then each of Rrcll, R,,, fl, is the complement of a &analytic set in R. In particular, each set in in the universal completion of 0. The last result clearly implies that if (Xt) is progressive over an arbitrary filtration (Gt), and if Rt denotes the set of w E R such that the map s --* Xs(w) of [0, t] into E is rcll, then Rt is in the universal completion of Gt. This is of importance in 518. A set A c R is null in case P(A) = 0, and a process Z is evanescent in case sup, IZt(w)l vanishes outside some null set. Two processes Z, 2’ are called indistinguishable if 2 - 2’ is evanescent. The fundamental a-algebras of processes associated with a filtered probability space are the optional and predictable a-algebras. Let V , t denote respectively the spaces of all maps of R+ into R that are right continuous with left limits (rcll, or, en l+un~$s, cridldg), left continuous
Appendices
387
with right limits (lcrl, ou chgldd). The optional a-algebra 0 on R+ x R is defined to be the a-algebra generated by the class of processes Z adapted to (Gt), having all sample paths t -+ Zt(w) in D. The predictable aalgebra P is defined similarly, replacing 2) above by L. As left continuity at t = 0 should be considered automatic by default, the trace of P on [ O n = ( ( 0 , ~ :) w E R} is precisely the product a-algebra 7 @ 00,where 7 is the trivial a-algebra on (0). The trace of P on I]O,m[ is of quite different character, as (A5.5)below explains. A random time is a G-measurable function T : R [O,m]. A random time is a predictable time provided its graph [ T I := { ( T ( w ) , w ) : T ( w ) < m} is in P. This is equivalent to the condition [IT,m[IE P, and to the announcability of T , which is defined to mean that there is an announcing sequence (T,) of optional times such that limn Tn = T and T, < T on {T > 0). See [DM75, IV-711. -+
(A5.5) PROPOSITION. (i) 0 is generated by the multiplicative class of indicators of stochastic intervals of the form [ S , 00 [I (or alternatively [IO, S[I) with S optional over (Gt). (ii) The trace of P on I]O,m[ is generated by each of the following multiplicative classes: (a) indicators of stochastic intervals 10, S ] (or alternatively [ S , mu) with S optional; (b) indicators of JO, S [ (or alternatively ]S,oo[I)with S predictable over (Gt); (c) simple predictable step processes of the form
c
n-1
Z t ( w ) :=
--
~j(41]tj,tj+l](t),
j=O
where 0 5 to 5 - 5 t,, H j E bGtj for j = 0, . . .,n - 1. (iii) P c 0 C {progressive processes} c M . (iv) Let 2 be adapted to ( G t ) and suppose all sample paths of 2 are right continuous. Then Z E 0.( The corresponding result asserting that left continuous adapted processes are predictable is also true, but trivial.) (v) Let f be a predictable subset of R+ x R containing the graph of its debut D r . Then Dr is a predictable time. Parts (i)-(iv) are discussed in [DM75, IV-64-67]. Part (v) is a consequence of the fact that [ D r n = [IO,RI]n I?, and each of the latter sets belongs to P . In the text itself, we shall modify the definitions of 0 and P a little by adjoining the class of evanescent processes. This requires a few minor rewordings in the statements of results, but leads to a setting more convenient for Markov processes.
388
Markov Processes
IV-651. Theorem (A5.3) may be improved considerably in See [DM75, IV-901, the certain cases. According to [DM75, (A5.6) PROPOSITION. Let (Zt) be progressive. Then liminf,tp 2, is predictable , and liminf,,t 2, is optional over ( G t ) . The last result is very interesting when applied to the indicator of a progressive set I?. By the closure of r, we mean the random set whose w-section is the ordinary closure of r ( w ) in R+. As ly(t,w) = limsup,,, lr(s,u), it follows from (A5.6) that E 0. Recall that the essential supremum esssup f of a function f on an interval 1 C R is the infimum of the ?- such that C{z € I : f(z) > ?-}= 0, where !. denotes Lebesgue measure on R. Given a real process 2, one may then define the essential limsup etc., and the analogue to (A5.3) and (A5.6) is IV-381. the following [DM75,
r
r
(A5.7) THEOREM. Let Z be adapted to (Gt), and suppose that Z is in the C x P-completion of B+ @ 8 . Then Ut(w) := lim,llt esssupt
# Z t ( w ) } = 0)
E 8.
We come now to Meyer’s Section Theorems, one of the prime achieve111-44, IV-84-85]. ments in the subject. See [DM75, (SECTIONTHEOREMS). Let ?r denote the projection (A5.8) THEOREM map from R+ x R to R. (i) Let r be a set in M . Then there exists T : R + [O,co]with T E p8, [ T I c l7 and P { T < m} = P{,(I’)}. (ii) Let I’be a set in 0 and E > 0. Then there exists an optional time T with [TI] c I? and P{T < m} > P{.lr(r)} - E . (iii) Let be a set in P and E > 0. Then there exists a predictable time T with [TI]c and P{T < 00) > P { ? r ( r ) }- E . The section theorems are often used in the following manner. Let Z,Z’ E b 0 , and suppose P Z T ~ { T < = ~ )PZ$lp-,,) for all optional T . Then 2,Z‘ are indistinguishable, as one sees applying (ii) above to the set { Z 2 a 6,Z’ 5 a } E 0 for an arbitrary a E R, 6 > 0. Similar results are valid for Z,Z’ E PO. Just apply (A5.9) to the random set { Z _> 2’ + 6,Z _< c} to get Z 5 2’ up to evanescence, then reverse the roles of Z and 2‘. Analogous results are valid in the predictable case. In applications to the foundations of Markov process theory, the followVI-48-49]. ing result is of great utility. See [DM75,
+
Appendices
389
(A5.9) THEOREM.
(i) Let 2 E bO and suppose that, for every decreasing bounded sequence {T,} of optional times with limit T , PZ(T,) + P Z ( T ) as n -t 00. Then Z is a s . right continuous. (ii) Let 2 E b P and suppose that, for every increasing bounded sequence {T,} ofpredictable times with limit T , PZ(T,) + P Z ( T ) as n + 00. Then 2 is a s . left continuous. In both (i) and (ii) above, if one assumes instead only that the respective limits of PZ(T,) exist, then 2 a.s. has right (resp., left) limits everywhere on [O,00[ (resp., lo,.[). Given an optional time T , the u-algebras GT, &- of events prior to T and events strictly prior to T are defined respectively by: (A5.10) a random variable H is in GT if and only if, for every t 2 0, H l { T < t } E 4t; (A5.11) GT- is generated by sets of the form A n {t < T}, with A E Gt, t 2 0. If Z is a progressive process, and if T is an optional time, then [DM75, IV-641 2 ~ 1 { ~E <&.~ )(This result does not require the filtration to satisfy the usual hypotheses. ) A more convenient but equivalent specification of the a-algebras above is [DM75, IV-671: (A5.12) if H E G, then H E GT (resp., &-) if and only if there exists Z E O (resp., P) with H~{T<.,} = Z T ~ { T < ~ } . A supermartingale over ( G t ) is a stochastic process (&)t>o adapted to (&) such that PlY,l < 00 for every t and P{&+, I G t } 5 & for all t , s 2 0. A process Z is a martingale provided 2 and -Z are both supermartingales. A supermartingale Y is a strong supermartingale over ( G t ) provided (a) Y is progressive; (b) for every bounded optional time T , P ~ Y T<~00; (c) for every pair of bounded optional times S 2 T , P{ Y s I G T } 5 YT. Strong martingales are defined similarly, replacing 5 with = in ( c ) . The Optional Sampling Theorem [DM75, VI-101 asserts that a right continuous (super)martingale is a strong (super)martingale. The path regularity theorem for (super)martingales is [DM75, VI-41: (A5.13) THEOREM.
(i) Let ( Z t ) be a supermartingale over ( G t ) , and suppose that t + PZt is right continuous. Then there exists a right continuous supermartingale Zi over ( B t ) such that P{Zt # Zl} = 0 for every t. The paths of Z' are a s . rcll. (Caution: Z - Z' is not in general evanescent.)
390
Markov Processes (ii) Let (Zt) E 0 be a strong martingale over (Gt).Then there exists a right continuous martingale 2' with Z - 2' evanescent.
Point (ii) above is a consequence of (A5.9i) and optional sampling. It follows from (A5.13) that, for every H E S with PH < 00, there exists a version of the martingale P{H I G t } having rcll paths. The optional projection of a general Z E b M is defined to be the process O Z E 0,unique up to evanescence by the section theorem, such that, for every optional time T,
(A5.14)
PZTl{T<w} = pozTl{T
The existence of O Z is guaranteed by a monotone class argument, starting with & ( w ) := (I[,,$] (8H ) ( t , w ) := l[a,b](t)H(w),in which case " z t ( w ) := l[,,b](t)Ht(w), Ht being the rcll version of the martingale P{H I Gt}. The predictable projection PZ of Z is the unique process in P such that, for every predictable time T,
(A5.15)
p ZTl{T<W} = PPZT1{T
:= The existence is similar to the optional case, starting with P(l[,,b] 63 l[a,b](t)&-. Both optional and predictable projections are positive operators which respect bounded, monotone increasing limits. Both projections therefore extend, by standard arguments and the Section Theorem, to at least the class p M with the characteristic properties (A5.14) and (A5.15) preserved. The next result is important in monotone class arguments involving supermartingales.
.
(A5.16) THEOREM. Let 0 5 Z15 2' 5 . . 1 2, with each 2" a right continuous supermartingale over (Gt). Then Z is a.s. right continuous. This is an easy consequence of (A5.9), noting that it sUaces to prove that Z A k is a.s. right continuous for each k 2 1, and that, if a doubly indexed sequence ( a i , j ) is increasing in each variable (or decreasing in each VI-181. variable), then limi limj a i j = limj l i w a i j . See also [DM75,
(A5.17) PROPOSITION. Let Z be a positive, right continuous supermart, := limt+oo 2, exists a.s. ingale over (Gt). Then 2 (i) Let So := inf{t : Zt = 0 or 2,- = O}, S, := inf{t : Zt = 00 or 2,- = m}. Then P{Sw < m} = 0 and a.s., Zt = 0 V t 2
so. (ii) Let T, be any decreasing sequence of optional times (not necessarily finite). Then the collection {Z(Tn) : n 2 1) is uniformly integrable.
39 1
Appendices
See [DM75, VI-171 for (i), [DM75, VI-10, V-301 for (ii). Let us denote by d the collection of all right continuous increasing measurable processes (At)t>O such that A0 = 0 and At < 00 for all t < 00. Members of A are called raw increasing processes. Set A , := sup At = limt+, At. Let A" denote the optional members of A - the optional increasing processes- and dp the predictable increasing processes. It is an easy consequence of (A4.3) that, for every Z E p M ,
(A5.18) (A5.19)
lo
Pl
ZtdAt = P J0 "ZtdAt,
A E A";
m
00
ZtdAt = P l PZtdAt,
A E AP.
A supermartingale Z is defined to be of class (D) in case the family { Z T ~ I T <:~T )optional time over (&)} is uniformly integrable. (A5.20) THEOREM (DOOB-MEYERDECOMPOSITION). Let 2 be a right continuous supermartingale of class (0) over (&). Then there exists a unique A E d p with PA, < 00 such that Z A is a martingale.
+
A potential is a right continuous positive supermartingale Z such that as., limt-, Zt = 0. According to the preceding results, the most general potential of class (D) is of the form yt = O(lR+ 8 Am)t - A t , with A E d and PA, < 00. In this case, Y is called the potential generated by A. It follows from (A5.18) and (A5.20) that in such a case, A , A" and AP all generate the same potential, and if A , B E d p have the same potential, then A and B are indistinguishable. A bounded, signed measure on (R+x S2,M) is called a P-measure provided it does not charge evanescent sets. See [DM75, VI-64-68]. (A5.21) THEOREM. Let p be a positive P-measure. Then there exists A E A, unique up to evanescence, such that for every Z E b M ,
1
m
p(2)= P
2, dA,.
0
The process A is optional (resp., predictable) if and only if
(A5.22)
p ( Z ) = ~ ( " 2 (resp., ) = ~ ( " 2 ) ) V Z E bM.
A P-measure p is called optional (resp., predictable) provided it satisfies the appropriate case of (A5.22). The method of proof for (A5.21) is the basis for some proofs in Chapter IV, and we shall therefore sketch it briefly. For each t , the map p t : H + p ( l p t ] 8 H ) is a finite positive
392
Markov Processes
measure on b6, and pt << P . Choose Bt E 6 with dpt = BtdP. Modifying the variables ( B t ) (t 2 0, rational) on a null set if necessary, it may be assumed that Bt is a s . increasing on the positive ratonals. Replace now Bt by At(w) := inf{B,(w) : s 2 t , s rational} to get A E A with P{At # Bt} = 0 for every t. Theorem (A5.21) leads immediately to the existence of the dual optional and dual predictable projections of an A E A with P A , < 00. With such an A , associate the P-measure p by setting p ( Z ) := P "2, dA,, Z E bM. Use then (A5.21) to get a unique A" E A" such that p ( 2 ) = P Z, dA: for all Z E b M . Then A" is the unique member of do,called the dual optional projection of A , such that, for every 2 E b(3, 00
(A5.23)
P l
00
Z,dA:=Pi
Z,dA,.
Similarly, there exists a unique Ap E d p , called the dual predictable projection of A , such that, for every 2 E bF, (A5.24)
P
l"
2, dAz = P
iw
Z, dA,.
In case A E A", AP is also called the compensator of A, for it is the unique predictable increasing process to subtract from A in order to get a mart ingale. Fix A E A with PA, < 00, and let Zt := (AA)t denote the process of jumps of A. It is easy to see that, up to indistinguishability, (A5.25)
"2 = A(A");
PZ = A(Ap).
The preceding results extend immediately from the class of integrable A to the larger class of A E A with the local integrability properties: (A5.26) optional case: 3 optional times T, 00 with PAT,- < mVn; (A5.27) predictable case: 3 optional times T, 100 with PAT, < mVn. The domain of the dual projections may be expanded considerably if one is willing, as in the text, to consider a general random measure ~ ( wd ,t ) in place of an the random measure dAt(w) generated by an increasing process A. See the discussion in Chapter IV. There is an obvious connection here with the classification of optional times. An optional time T is totally inaccessible in case P{T = S < co} = 0 for every predictable time S. Let At(w) := l~T,oon(t,w)lqw)<,, so that the random measure generated by A is unit mass at location T ( w ) if T ( w ) < 00. According to (A5.26),
r
A(Ap) = p ( l u T i ) = 0.
Appendices
393
That is, the compensator of A is continuous. This condition on AP is also sufficient for T to be totally inaccesible. An optional time T is accessible provided P{T = S < m} = 0 for every totally inaccessible time S , or, what is equivalent, there exists a sequence {S,} of predictable times with [ T I C Un[TSnIj. The a-algebra Y of accessible processes is that generated by stochastic intervals IT, 00[ with T accessible. It is not terribly interesting from a theoretical viewpoint because of the following, easily established, result. (A5.28) THEOREM.An increasing process A is accessible if and only if there exists W E p 0 and B E d p with dAt =.WtdBt. A process Z is accessible if and only if there exist Y E P, W E 0,and A a countable union of graphs of predictable times such that Z = Y WlA.
+
We have occasional need for the following convergence theorem, usually attributed to Hunt, but due earlier and in more general form to Blackwell and Dubins [BD62]. (A5.29) PROPOSITION. Let G, be an increasing (resp., decreasing) sequence of sub-a-algebras of G, and 60 := VGn (resp., nG,). Suppose (Y,) c L ~ ( o , GP), , sup I Y , ~ E L ~and , Y, + Y a s . as n + 00. Then limP{Y, n
16,)
=P{Y
I GO}.
Given a process ( Z t ) taking values in a space E , the entry time or debut D B of a subset B of E is defined by D B ( w ) := inf{t
2 0 : Zt(w) E B}.
In case Z is optional over a filtration (Gt) satisfying the usual hypotheses, Z optional over ( G t ) by the monotone class theorem, and the process 1 ~ o is consequently the set { ( t , w ) : & ( w ) E B } is optional over (&). It follows then from the theorem on measurability of debuts (A5.1) that DB is an optional time over (&). The following result of Dellacherie [Me711 serves as a fundamental approximation theorem for entry times. (A5.30) THEOREM.Under the hypotheses of the preceding paragraph, and with E Radonian, for every Bore1 subset B c E , there exist compact subsets K , t B with, as., D K , 1 D B .
PROOF:Let A := { (t,w ) : Zt(w) E B } ,so that A is optional. Fix E > 0, and let A, := A n [D B ,DB + E 1. As DB is an optional time, Ae is also optional. By the section theorem, there exists an optional time T with [TI]C A, and P{T < co} > P{x(A,)} - E . (Here, A denotes the projection map from R+ x R to 0.) Since .(A,) = { D B < oo}, this says P{DB < m,T =
394
Markov Processes
m} < E . Define p(&) := P{ZT E dz,T < m}, so that p is carried by B. As every measure on E is tight (A2.3), there is a compact K C B with p ( B \ K) < E . Clearly DB 5 D K , and DK I T on { ZT E K, T < 00). But P{ZT E K , T < GO} = p ( K ) = p ( B ) - p ( B \ K ) 2 P{T < 00) - 6. It follows that P { D K 5 T < 00) 2 P { T < m} - E . In addition,
{ D K > D B + E , T < m} c { D K > T , T < OO}, { D K > D B + E , T= OO} c {DB< W , T < MI}, and therefore P { D K > DB + E } 5 26. Applying this argument successively with E,, := 2-", we obtain a sequence {Kn}of compacts in B with
Letting L, := Uy=1Kj, we obtain an increasing sequence of compacts with P { D L , > Dg 2-"} 5 2-"+l. By Borel-Cantelli, DL, 1 D B as. as
+
n + 00.
A6. Relative Martingales and Projections The material in this section is fairly elementary, given the corresponding theory of martingales, local martingales, projections and dual projections outlined in the last section. It is not well known, and so our discussion will be a little more detailed. The Markovian versions of these results are very useful and natural in the setting of processes with a finite lifetime. Fix a filtered probability space (a,6,&, P ) . Throughout this section, equality of processes signifies indistinguishability and, given an increasing process (At)t>o,A denotes its dual predictable projection or compensator relative to the given filtered probability space.
(A6.1) DEFINITION. Let R be an optional time for the filtration ( G t ) . A process M defined on [O, R [ is called a uniformly integrable martingale on [O, R [ in case there exists a uniformly integrable martingale ( a t ) t > O over (6t) with M l [ O , R [ = M l [ O , R l J . Notice that this definition agrees with the usual notion of a uniformly integrable martingale in case P { R = 00) = 1.
(A6.2) THEOREM. Let M be a uniformly integrable martingale on [O, RU. Then M has a unique extension M as a uniformly integrable martingale on R+ satisfying the following conditions: (i) fi stops at R; (ii) M R E QR-.
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395
PROOF:Let N denote an arbitrary extension of M as a uniformly integrable martingale. We may assume that N stops at R, by the optional stopping theorem. The process N R ~ ~is Ra process , ~ ~ of integrable total variation, and it is absolutely continuous relative to l [ R , m [ . Let p, p' denote the respective P-measures (A5.21) generated by these processes. Clearly p << p' on P. Therefore, there exists a predictable process Y such that dp = Y dpt on P , and it follows that
As (1 R , n )-~doesn't charge 1R, 00 [I, we may assume also that Y stops at R. Thus Y R ~ [ I R= , .Y . ~* liR,mu and
(wR- ~
~ ) l ~ Y~* (,i n ~R~, m )yu-*j(-= l n R , m n ) 0. -=
That is, ( N R - YR)liR,mu is a martingale. Set
at := Mtl[O,R[( t ) -k y R l [ R , m [( t ) . Obviously satisfies (i) and (ii). On the other hand, A& - Nt = ( N R YR)luR,oon(t) is a martingale extending M . As to uniqueness, let M' be another uniformly integrable martingale satisfying (i) and (ii), and let L := M' - A?, so that L is a uniformly integrable martingale which stops at R , and vanishes on [IO,R[. By (ii), we may write L = GlnR,,n, with G = LR E QR-, say G = Z R with Z E P. There is no harm assuming that Z stops at R. Note that = 0 because L is a martingale. It follows that Z * A = 0, where A := (lnR,mn): In other words, Z+ * A = 2- * A , and so by uniqueness of the DolBans-Dade P-measure on P,we find 121* A = 0. That is, IZ(* l ~ R , m=~0, and this shows that, a.s., ZR = 0. Consequently, M' = h%, completing the proof. (A6.3) REMARK.Conditions (A6.2i,ii) imply that for every optional time T , A& E QR-, for i @ = ~ M T ~ { T < R ) I@R~{T>R), and both summands clearly belong to QR-.
+
(A6.4) DEFINITION. A process M is a local martingale on [O,R[I in case there exists a sequence (T,) of optional times with: (i) T, T T 2 R a.s. as n + 00; (ii) (M - M o ) is~ a~uniformly integrable martingale on [ O , R [ for every n. The class of local martingales on [IO, RU is denoted by LR. Our definition of local martingale on [O,R[ does not seem to be the same as that proposed by Maisonneuve [Ma77],the latter appearing to
396
Markov Processes
lead to a broader class. The definition given above is an appropriate one for use in connection with Markov processes having a finite lifetime. Given M and (T,) as in (A6.4) with Mo = 0, let (MTn)-denote the unique extension of MTn described in (A6.2). Then we have the following consistency condition between the different extensions. (A6.5) PROPOSITION.For every n, ((&IT,+')-)
Tn
= (MTn);
PROOF:Replace MTn by M and let T denote Tn. We show then that MT = (AdT): Observe that A?T is constant on [R,001, and that MT E GR- by (A6.2ii). It follows that MT is the canonical extension of its restriction to 10, R[,which is in turn equal to
The proposition above permits one to make routine uncovering arguments in much the same manner i ~ 9for ordinary local martingales. For example, given M E LR, [ M , M ]denotes the unique optional increasing process satisfying: (A6.6i) [ M , M ] t< 00 for all t < R; (ii) [ M ,MIm = [ M ,MIR-; (iii) AM: = A [ M , M ] t for all t < R; (iv) M 2 - [ M , M ]E LR. The existence of [ M ,MI comes from obvious modifications of the argument for ordinary local martingales. (A6.7) DEFINITION.The measurable, optional and predictable processes defined on [O, R[ are the respective traces of M , 0 ,P on [O, R[,denoted respectively by M R , U R , P R . (A6.8) LEMMA. Let Y be adapted to (Gt) and suppose the paths of Y are left continuous on 10, R[. Then Y E PR. PROOF: We may suppose that Y vanishes off I]O,R[.Clearly then, Y is progressive over (&). Define yt' := liminf,rlt Y,, so that Y' E P by (A5.6). Obviously Y = Y' on 10, R [ , proving the lemma. (A6.9) DEFINITION.By an increasing process on 1O,R[ is meant a measurable process ( A , ) such that: (i) (ii) (iii) (iv)
t
+ At is a.s. increasing, right continuous, and vanishes a t 0; a.s., At < 00 for all t < R; At = AR a.s. for all t 2 R; AR- = 00 a.s. on { A R = 00).
Appendices
397
For historical reasons connected with Meyer's original definition of natural increasing processes and their use in the representation of so-called natural potentials in Markov process theory, the class P R of relatively predictable processes on 10, R[I is also called the class of natural processes on [IO,R[I. (A6.10) DEFINITION. Let A be an increasing process on 10, R [ . Then A is a natural increasing process on 10, R[I provided At = AR- a.s. for all t 2 R, and A E P R . (A6.11) DEFINITION. Let T be an optional time. Then T is predictable on 10, R [ , or natural relative to R, provided there exists a predictable time S with T = T{T 0, S' := 0 ifT=0,
{
would define a predictable time with [ITI] = [IS' I]n [10,RE. In other words, we may replace [IO,R[Iby 10, R[I in (A6.11) if desired. (A6.12) PROPOSITION. The following conditions on an optional time T are equivalent: (i) T E T;; (ii) [IT]c 10, RU and [IT1 E P R ; (iii) there exists an increasing sequence (T,) in T with lim, T, = T on {T < R}, lim,T, = 00 on {T = 00) = {T 2 R } , and T, < T for all n on (0 < T < 00); (iv) At := l[T,oo[(t)l{O
PROOF:The implications (i)=+(ii)w(iv)are immediate, as is (i)+(iii) once we take note of the observation following (A6.11). Suppose now that (ii) ~ some Y E P R vanishing holds, so that we may write 1 ~ =~Y 1l n o , ~for off 10, RI]. Thus {Y = 1) E P and each of its w-sections contains at most two points. Its debut S is therefore predictable by (A5.5~)and satisfies the condition in (A6.11). Now let (T,) announce S. The sequence (TA) defined by T, if T, < R, T, := 00 if T, 2 R,
'I
then satisfies (iii). Suppose next that (iii) holds. It follows from (A6.8) that IS,R [ E P R for every S E T. Therefore Y := l [ ~ , , [ l ~ o , ~is[ in P R
398
Markov Processes
to conclude that [TI E PR,and complete the proof of (A6.12). (A6.13) LEMMA.(i) Let Y E p P and let T E T P . Then the increasing process A := YTl[T,w[ ~ { o < T < R }is natural; (ii) Let Z E p P and suppose At := ~ O < s 5 t , s < RZ, is finite for all t < R. Then A is a natural increasing process.
PROOF:(i): The process YTluT,wnl{O 0 and enumerate the times when Zt > S by the sequence (T,) of predictable times. (This is possible by (A5.5v).) It follows from (i) that
is a natural increasing process. Letting S proves the result.
10 through some sequence then
(A6.14) PROPOSITION. An increasing process ( A t ) is natural if and only if At = AR- a s . for all t 2 R and there is a predictable increasing process (Bt)t>O - with At = Bt for all t < R.
PROOF:Only one direction demands proof. Let A be a natural increasing process. According to (A6.8), A- E PR,and consequently A A := A-A- E P R . By hypothesis, there is a predictable process ( K ) t > o extending AA. We may assume that Y 2 0 and that Y stops at R. Let A" denote the continuous part of A. Then AC is a predictable increasing process. In view of (A6.13), A=A"+ Y,
c
{O
is a natural increasing process. (A6.15) LEMMA.Let p := p(lJO,R[ ) and A := { p evanescence, IlO,R[ c A c 10, RIl.
>
0).
Then, u p to
PROOF:As 10, R l E P, the second inclusion is obvious. In addition, p = 1 ~ 0 , R- ~Pl[Ri and so 10, R [ f l { p = 0) C { P l ~ R>~ 0). Choose predictable times T, such that U, IT, I] includes the graph of the accessible part of R. (That is, containing the predictable set {'luRn > O}.) It follows that the predictable set { p = 0) n 10,R ] is contained in U[T,l.
Appendices
399
Consequently, there exists by [DM75, IV-881 a sequence Sn of predictable Foreveryn, timessothat { p = O } n ]O,R]=U[rS,].
< m] = P [l]O,RI[ ( s n ) ; s, < W] = P [o < Sn < R ] . This proves that, up to evanescence, { p = 0) n I]O,R]c [R,oo[,hence that ] O , R [ c { p > 0 } , as claimed. 0 =P
[pS,;sn
The u-algebra PR has a section theorem analogous to (A5.8).
(A6.16) THEOREM. Let Y , 2 E PR be bounded or positive, and suppose
P{Y,; 0 < T < R} = P { Z T ; 0 (A6.17) Then Y and 2 are indistinguishable.
< T < R} VT E TpR.
Choose Y', 2' E P with Y = Y'l]O,R[[ and 2 = z'l]o,R[[, bounded if Y and 2 are bounded, positive if Y and 2 are positive. It follows from (A6.17) and the properties of ordinary predictable projections that P { Y ~ ~ T ; T < ~ ~ } = P { Z ; . ~ TVTETp. ;T<~} PROOF:
For the bounded case, the claimed result is an immediate consequence of (A6.15) and the section theorem (A5.8). In the positive case, apply the section theorem to show that (2' 2 Y' + E } n (2' 5 c } is evanescent for arbitrary E > 0, c E R, then interchange Y' and 2'. The predictable projection takes a slightly different form when relativized to a stochastic interval. (A6.18) DEFINITION.For 2 E bMR, the natural projection of Z relative to 10, R [ is nZ := l]O,RI[ V / p ; This definition makes sense because of (A6.15). (A6.19) PROPOSITION. Let M E MR. Then "M is the unique (up to evanescence on 10, RE) member O f P R satisfying
P ( " M , ; O < T < R ) = P ( M T ; O < T < R ) VTETP,. PROOF:It is enough to check (A6.20) for T E T P and bounded M . Using (A6.18),properties of ordinary predictable projections, and setting as usual (A6.20)
010 := 0,
400
Markov Processes
the third inequality holding becauseP(Mlrp,RU) vanishes on { p = 0). This establishes (A6.20). Uniqueness is an immediate consequence of the section theorem (A6.16). (A6.21) LEMMA.I f M E M vanishes o f f I]O,R[,then% =P("M). PROOF: This is a direct consequence of (A6.15) and (A6.18), since PM vanishes off A := { p > 0). (A6.22) DEFINITION. Let A be an increasing process carried by A and such that AP exists as an increasing process. Define then the dual natural projection A" of A by *t
(A6.23) or equivalently, in terms of random measures, (A6.24)
dA? := l]O,R[( t )dA:lP(t). Since t + p;' dAf is a predictable process agreeing with AT for 0 < t < R, it is clear that A" is a natural increasing process in the sense described in (A6.10). Clearly A" = (Ap) ". Note that we have defined A" only in case A is carried by A = { p > 0). (A6.25) PROPOSITION. Fix A a in (A6.22). (i) For M E bMR,
s,
(A6.26) (ii) Ap = (A")p. In particular, A" = (A")", and A" = (AP)n. PROOF:In view of (A6.18), standard properties of predictable projections, and the fact that p = plno,Ru , one finds for M E bMR,
By hypothesis, A is carried by A, and so therefore is AP. Thus, by (A6.24), d ( ~ " ) f= ~ td(lpo,Rn l *w: = p t ' p t dAf = lA(t) dAf = dAf.
The remaining assertions in (A6.25) are clear.
401
Appendices
I f A is a natural increasingprocess, then A" exists (A6.27) PROPOSITION. and A" = A.
PROOF:As we observed in the proof of (A6.14), we may write AAt = Ytl{t
(A6.28)
+C
Y, = At
+ YRl[Rn(t)
O<s
defines a predictable increasing process. By (A6.28), dA = l n o , *~ d~B and hence dAP = p d B exists as an increasing process. Therefore
dA7 = l]o,R[(t)p,'dA;
= l ] o , ~ [ ( t ) l ~dBt ( t ) = dAt,
establishing (A6.27). Let A be an integrable increasing process. That (A6.29) PROPOSITION. is, PA, < 00. Suppose in addition that, for every M E b M ,
P
(A6.30) Then
lo,,, I Mt dAt = P
"Mt dAt.
* A is a natural increasing process.
110,R~
PROOF:Replace A by 1n0,Ru * A if necessary so that one may assume that A is carried by A. Integrability of A guarantees that Ap exists. Then, by (A6.26), condition (A6.30) implies that
P
s
MtdAt=P/MtdA;
for all M E b M . As these are finite quantities, it follows from uniqueness of the P-measure that At = A: for all t 2 0.
(A6.31) COROLLARY. An optional time T with [TI] C 10, R[ is natural if and only if (A6.32)
P(MT;O < T ) = P ( " M T ; O < T )
PROOF: Let At := l[T,,[(t)l{o
VMEbpU.
so that A is an increasing process on
[O,R[ as described in (A6.9). Because A is optional, (A6.30) holds for all M E bM provided it holds for all M E bpU. This follows from the elementary properties of projections. Note also that "M = %(OM)follows directly from (A6.18). Thus (A6.32) implies that A is natural, and (A6.31) follows using (A6.9). Given an increasing process A, its potential Y generated by A is defined dA,, which always to be the optional projection of the process t + exists but may be infinite. If A is integrable, then Y is a positive, right continuous supermartingale. For an A satisfying the conditions of (A6.22), it follows from (A6.25) that A, AP and A" all have the same potential.
4,
402
Markov Processes
(A6.33) PROPOSITION. Let A, B be integrable, natural increasing processes having the same potential Y. Then A and B are indistinguishable.
PROOF:By (A6.27), (A6.25ii) and the preceding remarks, Ap and BP have the same potential relative to (a,Gt, P), and they are therefore indistinguishable by (A5.208). An appeal to (A6.27) and (A6.25ii) completes the proof. (A6.34) DEFINITION.A supermartingale on NO, R [ is a right continuous process 2 such that:
(i) there exists a right continuous supermartingale 2 over (Gt) such that Z l n o , ~=~Zln0,~nu p to evanescence; (ii) a.s., Zt = ZR for all t 1 R. A local supermartingale on [O, RB is a process Z for which there exists an increasing sequence (T,)C T with T, T 2 R and such that for every n, Z T n is a supermartingale of class (0)on 00, RE. The predictable set jO,RI] \ A has, by (A6.15), w-sections consisting of at most a single time, so by (A5.5v), there exists Ro E T* such that I]o,Rn \ A = [Ron. Obviously [Ronc [Rn. For the rest of this section, we assume that the reader is familiar with the basics of the theory of stochastic integration. (A6.35) LEMMA.Let 2 be a supermartingale on [O,R[[.Then the supermartingale 2 of (A6.34) may be assumed to satisfy AZR, = 0 almost surely.
w
PROOF:We may assume that 2 stops at R. The stochastic integral := 1nO,Rn\nRo 1 2 is a supermartingale because the integrand is positive, and it evaluates to
-
l ~ o , R n . Z - l n R o n ~ Z= ~ - A ~ ~ , l a ~ o , ~ n , which agrees with Z on 10, Rfl and satisfies A
~ =R0. ~
(A6.36) THEOREM. Let 2 be a supermartingale of class (0)on [O,R[. Then Z has a unique decomposition Z = 20 M - A on [ 0, R [ , in which M is a uniformly integrable martingale on 10,R[I and A is an integrable, natural increasing process on [O, R [ , both vanishing at 0.
+
PROOF:Take a supermartingale 2 as in (A6.34) and satisfying AZR, = 0.
+
The latter condition is equivalent to l [ R , 1 = 2 = 0. Let 2 = 20 M - A be its Doob-Meyer decomposition, M a martingale and A a predictable, integrable increasing process. As the Doob-Meyer decomposition is unique, A and M stop at R, and O = I I [ R ~ ~ ~l ~Z R= o n . Z o + + ~ R , n . M + l ~ R , n ~ A
Appendices
403
-
implies 1[I R~ 1 /i = 0, hence that A is carried by A. Similarly, 1[R,,1 A? = 0 so AMR, = 0. By the discussion of dual natural projection, A := ( A ) , exists as an integrable, natural increasing process. By (A6.25ii), ( A A)p = 0, hence A - A is a martingale. Let Mt := (At - A t ) . Then 2 = 20 M - A is a decomposition of the correct type. As to uniqueness, suppose that Z = Zo+M‘-A’ is another such decomposition. Set A := AP, A‘ := (A’)p, := M - ( A - A ) and ?ti := MI - (A’ - A’). Then A = A’ by uniqueness of the usual DoobMeyer decomposition. Applying (A6.25ii) again shows that A - A’ = ( A - A’), = 0, completing the proof.
at+
+
(A6.37) THEOREM (NATURALDOO3-MEYER DECOMPOSITION). Let 2 be a local supermartingale on 10, R[ . Then 2 has a unique decomposition of the form 2 = 20 M - A with M E LR and A a natural increasing process on [ 0, R [ .
+
PROOF:The usual localization argument, similar to that used in (A6.5), applies in this case. The only points to bear in mind are (i) natural increasing processes are closed under stopping; (ii) martingales on [O, R[ are closed under stopping; (iii) if T k is a sequence of optional times increasing to T > R, and if Ak are natural increasing processes such that At:& = A!,.,*, up to evanescence for each k, then At := SUpk A!,,*, defines a natural increasing process; (iv) a construction similar to that in (iii) applies to local martingales on 10, R[. In a similar vein, we define semimartingales over [O,R[ as sums of the form L A, where L E LR and A is a right continuous process adapted to ( B t ) and a s . of finite variation on compact subintervals of [0, R[. Special semimartingales over [O,R[ are defined as sums L + A with L E LR and A of locally integrable variation on 10, RE. It may be proved without difficulty that a special semimartingale 2 over [O, R [ has a unique decomposition of the form 2 = Zo + M + A with M E LR vanishing at 0 and A a natural process over [O,R[ a s . of finite variation on compact subintervals of [0, R[.
+
References [Az72a] J. AzBma, Quelques applications de la thtorie ge’ne‘rale des processus, I, Invent. Math. 18 (1972),293-336. [Az72b],Une remarque sur les temps de retour. h i s applications, in “SBminaire de ProbabilitB VI (Univ. Strasbourg),” Lecture Notes in Math. 258, Springer, Berlin Heidelberg New York, 1972, pp. 35-50. [Az73] ,Thtorie gintrale des processus et retournement du temps, Ann. Scient. de 1’Ecole Norm. Sup., 4e s6rie (1973),454-519. [ADR66] J. AzBma, M. Kaplan-Duflo and D. Revuz, RCcumnce fine des processus de Markov, Ann. Inst. Henri PoincarB B2 (1966), 185-220. [BJ73a] A. Benveniste and J. Jacod, Projections des fonctionelles additives et rep&entation des potentiels d’un processus de Markov, C. R. Acad. Sci. Paris SBr. A 276 (1973),1365-1368. [BJ73b] , Systbmes de Le‘vy des processes de Markov, Invent. Math 21 (1973),183-198. [Bi68] P. Billingsley, “Convergence of Probability Measures,” Wiley , New York, 1968. [BD62] D. Blackwell and L. Dubins, Merging of opinions with increasing information, Ann. Math. Stat. 33 (1962),882-886. [B157] R. M . Blumenthal, A n eztended Markov property, Trans. Amer. Math. SOC.85 (1957),52-72. [BG68] R. G. Blumenthal and R. K. Getoor, “Markov Processes and Potential Theory,” Academic Press, New York, 1968. [Bo56] N. Bourbaki, ‘‘Elements de m a t h h a t i q u e , Livre IX, topologie gBnBrale,” Hermann, Paris, 1956. [Ch73] K. L. Chung, Probabilistic approach to the equilibrium problem in potential theory, Ann. Inst. Fourier 23 (1973),313-322. [Ch82] , “Lectures from Markov Processes t o Brownian Motion,” Springer, Berlin Heidelberg New York Tokyo, 1982. [CD65] K. L. Chung and J. L. Doob, Fields, optionality and measurability, Am. J . Math. 87 (1965),397-424. [CW69] K. L. Chung and J. B. Walsh, To reverse a Markov process, Acta Math. 123 (1969),225-251.
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Markov Processes
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Notation Index
at, stopping operators, 118 At, usually increasing process, 146
&*-Markov, 2
At, compensator of A, 235
Ft, 3f,augmented filtrations, 11 F[T-I, left germ field, 78 3:,universal completion of 14 3;,filtration V 107, 111
dual projections of A, 394,159, 223 A("),A((")),transport of A, 321 b7i, bounded elements of the class H, 364 B ( E ) ,Borel a-algebra on E , 369 a+,f?", Borel a-algebra on R+,R++ B" ( E ) ,universally measurable sets, 369 C d ( E ) , d-uniformly continuous, 369 C(E,d), d-continuous functions, 369 D,space of rcll paths, 20 d, metric for topology of E, 1 D A , debut of the set A, 51, 385 e t , excursion starting at time t, 357 E , state space for X ,1 E, R a y compactification of E, 91 E R ,E B ,ED,191, 193 Em,permanent points for a MF m, 267 &, Borel a-algebra on E , 1 Ray-Bore1 a-algebra on E , 104 Eo, synonym for &, 1, 104 Ee, a-algebra generated by 40 &', (temporary use) Ray-Borels, 92 EU,universally measurable sets in E , 1 Ao(pn),
s",
e,F:, e,
minimal filtrations, 2
e, e,
G t , arbitrary filtration, 3 G t + , 25 HD1, HD2, right hypotheses, 7, 30 HRM, homogeneous random measure, 172 H*, Hardy class, 235 fid,fig,homogeneous processes, 126 fid7',217 fid,, fiz, relatively homogeneous, 276 1,evanescent processes, 19 Inv, invariant c-algebra, 140 J, J B , jump times of X, 210 K, continuity set for X, 210 kt, killing operators for 63 constant adapted processes, 116 L", local time a t 2,324 L,L,local martingales, 234 Lc,L d , 247 LR, relative local martingales, 395 lcrl paths, 20 lcrl, left continuous with right limits, 20
x,
c,
Markov Processes
412
lOC, localization, 234 min, minimum of set, 23 mt, MF, 135, 259 fit,
perfect exact regularization of m, 266
m]a,t] 3 263
M , measurable processes, 106 M n , 115
( N ,H ) , L6vy system for x,342
N, Np, null sets, 19 N o ,shift-null sets, 132 0,optional processes, 20
Ot, integrally optional
processes, 163
@, 116 P",law of X
starting at 2,7 system for X, 355 PB,hitting operator, 68 Pt, transition semigroup, 4 P,, operator generated by MF, 267 PR, operator generated by optional time, 68 p?f, positive elements of the class %, 364 P,predictable processes, 20 pt, integrally predictable processes, 163 P,,elementary predictable processes, 249
( P z , B ) ,exit
PR, Pc,natural processes, 222, 396 Qt , semigroup generated
by a terminal time, 70 by a MF, 267 &, &+, &++, rationals, 2 0, > O &, probability on F) R, R+,R++, reds, 2 0, > 0 rcll, right continuous with left limits, 20 reg, regular points, 67 BT,regular points of set B , 52 R,Ray cone, 90 (R, forward space-time process, 86 (R, backward space-time process, 87 semimartingales, 250 &excessive functions, 17 sexp, Slog, Stieltjes exponential and logarithm, 381 SUpp, fine support, 304
(a,
x), x),
s, s,,
s",
TB,hitting time of set B , 51 T, optional times for (Ft), 120 TO("*'), optional times for
(fl(%*')), 120
7,fine topology, 232 U", resolvent for (Pt),14 Ray completion of V a l92
o",
u,89
v",resolvent generated by a terminal time, 69 by a MF, 267 234 accessible processes, 393 Markov process in E , 2 left limit of 218 3?, 116 Z, generic real process "2, projections of 2, 113, 223 total variation of 2, 234 Z*, maximal process, 234 a,p, 6, generic elements of R+ m)-excessive function, 289 A, dead point, 62 [A], dead path, 63 cz, unit mass at location 2 7,K , random measures on , ' R 146 &, shift operators on a, 8 &, big shift, 109 dual big shift, 151 A, p, V , usually measures on E /.is, V,, characteristic measures, 359, 360 rI, optional projection kernel, 111 rI,predictable projection kernel, 207 p, Ray metric on E , 91, 189 w generic element of w f t f w', splicing operator, 110 lifetime of 62 co, unnatural part of <, 222 closure operation for A, 89 [Y, covariance process, 242, 396 [ , 1, stochastic interval, 22 , stochastic integral, 250
v,v#,
v, x, x!, x,
x,
,z, nz,
z#,
@(x,
&,
c,
A,
.
x,
z],
Subject Index
absolute Borel space, 370 absolute continuity hypothesis of, 56 of AF's, 309-314 of kernels (Doob's lemma), 376 of right processes, 299 of random measures, 155 of resolvents, 271 absorbing set, 72 quasi-absorbing set, 72 accessible process, 213 time, 213 adapted process, 4 additive functional, 172 natural, 254 potential function generated by, 175 potential operator generated by, 178 m),276 of 275 additive (relative to a,m, a.e. relative to an AF, 310 AF-see additive functional A-function, 381 almost surely, 11 announcable time, 387 announcing sequence, 387 independent of initial law, 115 approximation of hitting times, 57, 232, 393 as-see almost surely augmentation of a filtration, 25
(x,
s),
AzBma's a-algebras of homogeneous processes, 123-132, 137 balayage, 315-324 of functions, 231 of additive functionals, 323 Bessel process, 76 big shift (@), 109 and projections, 114 dual big shift, 151 and dual projections, 159 Blumenthal 0-1 law, 13 Borel right process, 105 branch point, 44 Brownian motion, 16, 50, 58, 76-77 with drift, 50, 58 canonical space, 46, 98 canonical decomposition of special semimartingale, 251 capacitary additive functional, 180 carried by (of random measure), 146 Cartesian product, 84-85 change of variable formula, 379 Chapman-Kolmogorov equation, 3 characteristic measure of an AF, 359-360 class (D), 166 classification of AF's, 303 of excessive functions, 57-58
414
Markov Processes
of optional times, 212 closure of homogeneous set, 136 progressive set, 140 compensated sum of jumps, 247 compensator, 235, 392 complete maximum principle, 56 completion of a c-algebra, 11, 14 -see also augmentation and sandwich-
3,
of 11-14 measurability issues, 14-15 of resolvent, 89 concatenation of right processes, 77-82 contiguous intervals, 317 continuous part of a martingale, 247 convolution semigroup, 50 ceoptional time, 138-140 coterminal time, 139-141 Dawson’s formula, 113 dead path([A]), 63 dead point (A), 63 debut, 51, 385 measurability of, 386, 393 diffusion, 55 DolBans-Dade P-measure, 169, 391 Doob’s lemma, 376 Doob’s h-transform, 298 Doob-Meyer decomposition of potential, 391 of special semimartingale, 251 of supermartingale, 239 of regular strong supermartingale, 239 in general theory, 391 natural, 403 dual projections, 157, 392, 400 dyadic approximants, 21 Dynkin’s formula for resolvents, 69 elementary predictable process, 249 stochastic integral, 249 &*-Markov, 2 end (of a measurable set), 134 entrance law, 5 closed, 5 closure of, 196 locally integrable, 200 minimal, 199
entrance space, 195 entry time-see debut equilibrium potential, 140, 231 essential increase, 129 limit, 126 regularization, 126 evanescent, 19 exact multiplicative functional, 260 terminal time, 65 regularization, 68 exactly subordinate, 70, 268, 271 exceptional sets, 54 pinessential, 280 nearly semipolar, 279 null, 54 quasi-polar, 73 polar, 54 potential zero, 54 p-polar, 279 semipolar, 54 p-semipolar, 279 shift-null, 132 temporally countable, 280 thin, 54 totally thin, 54 excessive function, 14, 39-40 &excessive, 17 elementary properties, 18 Q-V-excessive, 271 m)-excessive, 271, 289 products of, 41, 179 excessive measure, 200 purely, 201 invariant, 200 excessive regularization, 17, 18, 55 excursion theory, 350-357 existence theorem for Ray semigroups, 46 for right process, 47, 49 exit system, 351 exit law, 88 exponential distribution of holding time, 10 extension of an AF or HRM, 186, 327 of an excessive function, 72 of homogeneous process, 326
@(x,
Subject Index of kernel, 100 of right process, 83, 101 Feller semigroup, 50 filtration, 385 augmented, 25 generated by regular step process, 108 Markov, 26 strong Markov, 26, 29 finely continuous, 54, 56 open, closed, 53 perfect, 56 fine topology, 53, 55 fine support, 304 flow, 41, 57 future, 2, 141 Gaussian semigroup, 16 germ field left, 78 connection with homogeneity, 144 perfection of, 127 mixed, 144 HD1 (Hypothhse Droite l), 7 HD2 (Hypothhse Droite 2), 30 hitting distribution, 68 operator, 68 times, 51 approximating hitting times, 57, 232 characterization of excessive functions, 57 holding point, 10 homogeneous increments, 240, 243 homogeneous process, 123 (almost) perfectly, 123 perfection of, 124-132 projections of, 125 homogeneous random set, 318, 350 homogeneous random measure, 172 connection with AF's, 172 generating, 184 t-integrable, 360 potential function generated by, 175 potential operator generated by, 178 HRM-see homogeneous random measure h-transform, 298 Hunt process, 219
415 characterization of, 231 Hunt's identification of r a u i t e , 231 hypotheses absolute continuity, 56 droites, 7, 30 right, 7, 30 usual, 385 increasing process, 146 integrable, 147 integrable on, 396 left, 146 locally finite, 146 locally integrable, 147 natural, 397 optional, predictable, 150 predictably bounded jumps, 148 uniformly locally integrable, 148 indistinguishable processes, 19 random measures, 146 inessential set, 280 initial law, 4 instantaneous point, 10 integrable HRM <-integrable, 360 a-integrable, 147 integrally measurable, 163 invariant measure, 200 random variable, 140 inverse of an increasing function, 379 of a continuous AF, 305 inversion formula for Laplace transforms, 17 jump sets
J, JB for X,210
kernels, 3, 375 bounded, 375 countably Radon, 185 finite, 375 Markov, 3 proper, 375 subMarkov, 375 killing operators, 63 killing a right process by a decreasing MF, 286
416
Markov Processes
at a terminal time, 65-75, 274 by a supermartingale MF, 290 L (Meyer’s hypothesis (L)), 56 LAF-see left AF lagging function, 315 Laplace transform inversion, 17 last exit time, 58 LCCB-locally compact, countable base lcrl-left continuous with right limits, 20 leading function, 315 Lebesgue penetration time, 261 Lebesgue-Stieltjes integrals, 379 left additive functional (LAF), 172 U-algebra at T , 144 germ field, 78 limit in original topology, 95,218, 224-
230 limit in Ray topology, 95,2038 multiplicative functional, 278 potential, 166 potential function, 181-182 terminal time, 261 LBvy measure, 340 process, 50, 340 system, 342 LBvy-Khintchin formula, 340 lifetime, 62-65 local time, 324 localizing an AF, 337-338 localizing sequence, 337 I cally harmonic function with poles, 241 1 cal martingales, 395 over 241 compensated sum of jumps, 247 continuous part, 247 fundamental lemma, 237 locally H2, 245 on a stochastic interval, 395 Lusin measurable space, 370 topological space, 369 mappings of, 372
8
x,
mappings of state space, 75 Markov filtration, 27 strong, 27 Markov process
definition, 2-8 Dynkin’s formulation, 7-8 equivalence, 99 normal, 8 right continuous, 7 temporal homogeneity, 4 Markov property, 4 &*-Markov property 2, 3, 8 simple, 4, 8 strong (SMP), 26 Markov semigroup, 4 martingale general theory, 389 on a stochastic interval, 394 over 233 maximal process, 234 MCT-monotone class theorem meager set, 201 optional, predictable, 201 measurability of processes, 20, 106 p-measurability, 20 progressive, 21-22, 385 measure s-finite, 366 trace of, 367 MF-see multiplicative functional M-function, 381 mixed germ field at T,144 monotone class theorem, 364-365 multiplicative class, 364 multiplicative functional, 135,259 almost perfect, 260 decreasing, 260, 272, 286 exact, 260 left, 260, 278 measurability, 135,284 perfect, 260 perfect exact regularization, 266, 270,
x,
285 perfection of, 135,263,273 raw, 131, 260 right, 272-273 strong, 259 supermartingale, 290 weak, 259 MVS, 364 natural
Subject Index filtration, 10 increasing process, 397 potential, 401 potential function, 255 process, 396, 222 p-natural process, 224 projections, 223, 399 dual projections, 223, 400 time, 222 section theorem for, 399 characterization of natural times, 226 nearly Borel, 21, 95 strict sense, 95 nearly optional function, process, 20 nearly semipolar set, 280-281 non-branch points, 44 normal Markov process, 8, 38 not charging (of random measure), 146 null set, 11, 19, 54 shift-null set, 132 optional function, 20 nearly optional function, 20 process, 20 p-optional process, 20 projection, 111-1 15 projection kernel, 111, 117 sampling, 389 time, 21 decomposition of optional time, 123 path type, 118 penetration time, 66, 261 Lebesgue, 261 perfection techniques, 124-132, 266-270 perfect kernel, 281 permanent points for a MF, 267 P-measure, 169, 391 Poisson process, 50 polar set, 54, 55 p-polar set, 279 quasi-polar set, 73 poles of a potential, 185 of a locally harmonic function, 241 potential of an AF, 167 characterization by, 166 class (D), 176, 177, 391
41 7 Doob-Meyer decomposition, 169 equilibrium, 140, 231 function of class (D), 176 function of class (D) with poles, 185 of a function, 16 general theory, 391 of a HRM, 167, 175 left, 166 natural potential function, 254 operator, 16 representation of, 169 for subprocess, 336 zero, 54 predictable function, 201 process, 20 elementary, 249 @-predictable process, 20 projection, 113 projection kernel, 207 relative, 329 time, 115, 387 principle of masses, 18, 60 process of class (D), 166 evanescent, 19 indistinguishability of, 19 measurable, 20, 106 optional, predictable, 20 regular, 166 progressive process, 21, 385 projection accessible, 215 and big shift, 114 dual, 157 of homogeneous process, 125 kernels, 111, 207 measurability of, 386 natural, 329 optional, 111-115, 118 predictable, 113 proper kernel, 375 purely discontinuous function, 380 &-cone, 89 qlc-see quasi-left-continuity quasi-left-continuity of processes, 48, 219 of a filtration, 219
418 on a stochastic interval, 219 quasi-absorbing set, 72 quasi-polar set, 73 Radon space measurable, 370 topological, 369 mappings of, 373 random measure (RM), 146 absolute continuity of, 155 integrable, 147 integrable on, 396 kernel associated with, 152 optional, predictable, 150 rational Ray cone, 90 Ray -Knight completion, 89-94 metric, 91 resolvent, 42 semigroup, 44 space, 191 topology, 91,95 rcll-right continuous with left limits, 20 rc11 function of X, 339 realization of a semigroup, 8,46, 97-100 recurrence, 60 reducing time for a MF, 293 r6duite, 231 reference AF for branching jumps, 216 for non-branching jumps, 342 reference measure, 56-57 refined right process, 38, 105 reflection principle, 29 regular point for set, 52 point for terminal time, 67 potential function, 176,257 step process, 82,339 strong supermartingale, 166 regulated function, 46 relative martingales, 394 projections, 399 dual projections, 400 respect (of kernel), 152 resolvent, 14 equation, 16 exactly subordinate, 70, 268 Ray, 42
Markov Processes subordinate, 271 restriction of process, 73 of semigroup, 73 Revuz measure, 360 return time-see co-optional time Riesz decomposition, 241 right continuous flow, 41,57,220 right process Borel, 105 definition of, 38,62, 105 Hunt, 219 resum6 of properties, 104-105 refined, 38, 105 special, 220 standard, 220 with finite lifetime, 62 right semigroup, 38, 62 subMarkov, 62 right terminal time, 273 multiplicative functional, 272, 286 right topology, 55 closed set, 55 RLAF-see additive functional, left RM--see random measure RMF-see multiplicative functional, raw Salisbury’s example, 102 sandwiching, 11 section theorems, 388 for natural processes, 399 semigroup Markov, 4 subMarkov, 62 right, 38 transition, 4 semimartingale, 249 on a stochastic interval, 403 special, 249-250 semipolar set, 54 p-semipolar set, 279 nearly, 279 separable measurable space, 376 s-finite measure, 366 shift-null set, 132 shift operators, 8 Shih’s theorem, 232 S M P - s e e strong Markov property Souslin space, 370
Subject Index space-time process, 61, 86-88 special Markov process, 220 semimartingale, 249 splicing map, 110 standard process, 220 p-standard, 224 state space, 1 Stieltjes exponential and logarithm, 281 stochastic integral, 250 elementary, 250 on a stochastic interval, 402 over a right process, 250 stochastic interval, 23, 385 stochastic process, 2 general theory of, 385 adapted, 385 stopping operators, 118 stopping time-see optional time strong domination, 312 strong Markov property, 26 strong supermartingale, 166 strongly supermedian function, 181 subordinator, 325 subprocesses killing at terminal time, 65-74 killing with MF, 286-301 supermartingale Doob-Meyer decomposition, 239 general theory, 389 on a stochastic interval, 402 over a Markov process, 233 super-mean-valued function, 17 supermedian function, 19 strongly, 181 support (fine, of a continuous AF), 304 temporally countable set, 280 temporal homogeneity, 4 terminal time, 65 exact, 65 left, 261 perfect, 65 perfection of, 130, 267 right, 273 strong, 65 thin, 67
419 totally thin, 67 weak, 65 thin set, 54 totally thin set, 54 terminal time, 67 totally thin terminal time, 67 thinness at a point, 52 tightness, 370, 374 time wcessible, 213 predictable, 115, 387 optional, 21 terminal, 65 time change of right process, 305 totally inaccessible random measure, 214 time, 214, 392 tot ally thin set, 54-55 terminal time, 67 total variation, 234 trace, 367 transfer kernel, 78 transient set, process, 59 transition function, semigroup, 3, 4 transport of measure, 317 of HRM, 321-322 trap, 10 uniform motion, 13, 18, 55 uniformization of integrability for increasing process, 235 universal completion, 367 universal measurability, 367 unnatural part of lifetime, 222 usual hypotheses (in general theory), 385 weak additive functional, 173 convergence, 374 multiplicative functional, 259 terminal time, 65 zer-one law of Blumenthal, 13
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