Markov Processes from K. Itô's Perspective (AM-155) 9781400835577, 9780691115436

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Annals of Mathematics Studies Number 155

Markov Processes from K. Ito's Perspective

by

Daniel W. Stroock

PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD 2003

Copyright © 2003 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 3 Market Place, Woodstock, Oxfordshire OX20 1SY All Rights Reserved The Annals of Mathematics Studies are edited by John N. Mather and Elias M. Stein ISBN 0-691-11542-7 (cloth) ISBN 0-691-11543-5 (pbk.) British Library Cataloging-in-Publication Data is available. The publisher would like to acknowledge the author of this volume for providing the camera-ready copy from which this book was printed Printed on acid-free paper oo www.pup.princeton.edu Printed in the United States of America 3579108642 3 5 7 9 10 8 6 4 2 (Pbk.)

This book is dedicated to the only person to whom it could be:

Kiyosi Ito

Contents

Preface Chapter 1 Finite State Space, a Trial Run

xi 1

1.1 An Extrinsic Perspective 1.1.1. The Structure of 8, 1.1.2. Back to M1 (Zn) 1.2 A More Intrinsic Approach 1.2.1. The Semigroup Structure on M 1 (Z,) 1.2.2. Infinitely Divisible Flows 1.2.3. An Intrinsic Description of T8x (M1 (Zn)) 1.2.4. An Intrinsic Approach to (1.1.6) 1.2.5. Exercises 1.3 Vector Fields and Integral Curves on M 1 (Zn) 1.3.1. Affine and Translation Invariant Vector Fields 1.3.2. Existence of an Integral Curve 1.3.3. Uniqueness for Affine Vector Fields 1.3.4. The Markov Property and Kolmogorov's Equations 1.3.5. Exercises 1.4 Pathspace Realization 1.4.1. Kolmogorov's Approach 1.4.2. Levy Processes on Z, 1.4.3. Exercises 1.5 Ito's Idea 1.5.1. Ito's Construction 1.5.2. Exercises 1.6 Another Approach 1.6.1. Ito's Approximation Scheme 1.6.2. Exercises

1 1 4 5 5 6 8 9 9 10 10 11 12 14 16 17 18 21 24 26 26 31 32 33 34

Chapter 2 Moving to Euclidean Space, the Real Thing

35

2.1 Tangent Vectors to M 1 (JR") 2.1.1. Differentiable Curves on M 1 (IR") 2.1.2. Infinitely Divisible Flows on M 1 (IR") 2.1.3. The Tangent Space at Ox 2.1.4. The Tangent Space at General JL E M 1 (IR") 2.1.5. Exercises

35 35 36 44 46 48

viii

Contents

2.2 Vector Fields and Integral Curves on M 1 (IRn) 2.2.1. Existence of Integral Curves 2.2.2. Uniqueness for Affine Vector Fields 2.2.3. The Markov Property and Kolmogorov's Equations 2.2.4. Exercises 2.3 Pathspace Realization, Preliminary Version 2.3.1. Kolmogorov's Construction 2.3.2. Path Regularity 2.3.3. Exercises 2.4 The Structure of Levy Processes on IRn 2.4.1. Construction 2.4.2. Structure 2.4.3. Exercises

Chapter 3 Ito's Approach in the Euclidean Setting 3.1 Ito's Basic Construction 3.1.1. Transforming Levy Processes 3.1.2. Hypotheses and Goals 3.1.3. Important Preliminary Observations 3.1.4. The Proof of Convergence 3.1.5. Verifying the Martingale Property in (G2) 3.1.6. Exercises 3.2 When Does Ito's Theory Work? 3.2.1. The Diffusion Coefficients 3.2.2. The Levy Measure 3.2.3. Exercises 3.3 Some Examples to Keep in Mind 3.3.1. The Ornstein-Uhlenbeck Process 3.3.2. Bachelier's Model 3.3.3. A Geometric Example 3.3.4. Exercises

Chapter 4 Further Considerations 4.1 Continuity, Measurability, and the Markov Property 4.1.1. Continuity and Measurability 4.1.2. The Markov Property 4.1.3. Exercises 4.2 Differentiability 4.2.1. First Derivatives 4.2.2. Second Derivatives and Uniqueness

49 50 52 53 54 56 56 57 59 59 60 65 69 73 73 74 76 79 84 89 94 96 96 99 104 104 105 106 108 110 111 111 111 113 115 116 116 122

Contents

ix

Chapter 5 Ito's Theory of Stochastic Integration 5.1 Brownian Stochastic Integrals 5.1.1. A Review of the Paley-Wiener Integral 5.1.2. Ito's Extension 5.1.3. Stopping Stochastic Integrals and a Further Extension 5.1.4. Exercises 5.2 Ito's Integral Applied to ItO's Construction Method 5.2.1. Existence and Uniqueness 5.2.2. Subordination 5.2.3. Exercises 5.3 Ito's Formula 5.3.1. Exercises

125 125 126 128 132 134 137 137 142 144 144 150

Chapter 6 Applications of Stochastic Integration to Brownian Motion 6.1 Tanaka's Formula for Local Time 6.1.1. Tanaka's Construction 6.1.2. Some Properties of Local Time 6.1.3. Exercises 6.2 An Extension of the Cameron-Martin Formula 6.2.1. Introduction of a Random Drift 6.2.2. An Application to Pinned Brownian Motion 6.2.3. Exercises 6.3 Homogeneous Chaos 6.3.1. Multiple Stochastic Integrals 6.3.2. The Spaces of Homogeneous Chaos 6.3.3. Exercises

151 151 152 156 160 160 161 167 171 174 175 177 181

Chapter 7 The Kunita-Watanabe Extension 7.1 Doob-Meyer for Continuous Martingales 7.1.1. Uniqueness 7.1.2. Existence 7.1.3. Exercises 7.2 Kunita-Watanabe Stochastic Integration 7.2.1. The Hilbert Structure of M 1oc(IP';IR) 7.2.2. The Kunita-Watanabe Stochastic Integral 7.2.3. General Ito's Formula 7.2.4. Exercises 7.3 Representations of Continuous Martingales 7.3.1. Representation via Random Time Change 7.3.2. Representation via Stochastic Integration 7.3.3. Skorohod's Representation Theorem 7.3.4. Exercises

189 189 190 192 194 195 196 198 201 203 205 206 209 213 217

Contents

X

Chapter 8 Stratonovich's Theory

8.1 Semimartingales and Stratonovich Integrals 8.1.1. Semimartingales 8.1.2. Stratonovich's Integral 8.1.3. Ito's Formula and Stratonovich Integration 8.1.4. Exercises 8.2 Stratonovich Stochastic Differential Equations 8.2.1. Commuting Vector Fields 8.2.2. General Vector Fields 8.2.3. Another Interpretation 8.2.4. Exercises 8.3 The Support Theorem 8.3.1. The Support Theorem, Part I 8.3.2. The Support Theorem, Part II 8.3.3. The Support Theorem, Part III 8.3.4. The Support Theorem, Part IV 8.3.5. The Support Theorem, Part V 8.3.6. Exercises

221 221 221 223 225 227 230 232 234 237 239 240 242 243 246 248 252 257

Notation

260

References

263

Index

265

Preface

In spite of (or, maybe, because of) his having devoted much of his life to the study of probability theory, Kiyosi Ito is not a man to leave anything to chance. Thus, when Springer-Verlag asked S.R.S. Varadhan and me to edit a volume [14] of selected papers by him, Ito wanted to make sure that we would get it right. In particular, when he learned that I was to be the author of the introduction to that volume, Ito , who had spent the preceding academic year at the University of Minnesota, decided to interrupt his return to Japan with a stop at my summer place in Colorado so that he could spend a week tutoring me. In general, preparing a volume of selected papers is a pretty thankless task, but the opportunity to spend a week being tutored by Ito more than compensated for whatever drudgery the preparation of his volume cost me. In fact, that tutorial is the origin of this book. Before turning to an explanation of what the present book contains, I cannot, and will not, resist the temptation to tell my readers a little more about Ito the man. Specifically, I hope that the following, somewhat trivial, anecdote from his visit to me in Colorado will help to convey a sense of Ito's enormous curiosity and his determination to understand the world in which he lives. On a day about midway through the week which Ito spent in Colorado, I informed him that one of my horses was to be shod on the next day and that, as a consequence, it might be best if we planned to suspend my tutorial for a day. Considering that it was he who had taken the trouble to visit me, I was somewhat embarrassed about asking Ito to waste a day. However, Ito's response was immediate and completely positive. He wanted to learn how we Americans put shoes on our horses and asked if I would pick him up at his hotel in time for him to watch. I have a vivid memory of Ito bending over right next to the farrier, all the time asking a stream of questions about the details of the procedure. It would not surprise me to learn that, after returning home, Ito not only explained what he had seen but also suggested a cleverer way of doing it. Nor would it surprise me to learn that the farrier has avoided such interrogation ever since. The relevance of this anecdote is that it highlights a characteristic of Ito from which I, and all other probabilists in my generation, have profited immeasurably. Namely, Ito's incurable pedagogic itch. No matter what the topic, Ito is driven to master it in a way that enables him to share his insights with the

Xll

Preface

rest of us. Certainly the most renowned example of Ito's skill is his introduction of stochastic differential equations to explain the Kolmogorov-Feller theory of Markov processes. However, the virtues of his theory were not immediately recognized. In fact, sometime during the week of my tutorial, Ito confided in me his disappointment with the initial reception which his theory received. It seems that J.L. Doob was the first person to fully appreciate what Ito had done. To wit, Doob not only played a crucial role in arranging for Ito's memoirs [11] to be published by the AMS, he devoted §5 of the ninth chatper in his own famous book [6] to explaining, extending, and improving Ito's theory. However, Doob's book is sufficiently challenging that hardly anyone who made it to §5 of Chapter IX was ready to absorb its content. Thus, Ito's theory did not receive the attention that it deserves until H.P. Me Kean, Jr., published his lovely little book [23]. Like Lerey's theory of spectral sequences, Ito's theory was developed as apedagogic device, and, like Leray's theory, Ito's theory eventually took on a life of its own. Both Ito's and Me Kean's treatments of the theory concentrate on practical matters: the application of stochastic integration to the path-wise construction of Markov processes. However, around the same time that Me Kean's book appeared, P.A. Meyer and his school completed the program which Doob had initiated in Chapter IX. In particular, Meyer realized that his own extension of Doob's decomposition theorem allowed him to carry out for essentially general martingales the program which Doob had begun in his book, and, as a result, Ito's theory became an indispensable tool for the Strasbourg School of Probability Theory. Finally, for those who found the Strasbourg version too arcane, the theory was brought back to earth in the beautiful article [21], where Kunita and Watanabe showed that, if one takes the best ideas out of the Strasbourg version and tempers them with a little reality, the resulting theory is easy, Gesthetically pleasing, and remarkably useful. In the years since the publication of Me Kean's book, there have been lots of books written about various aspects of Ito's theory of stochastic integration. Among the most mathematically influential of these are the book by Delacherei and P.A. Meyer's [5], which delves more deeply than most will care into the intimate secrets of stochastic processes, N. Ikeda and S. Watanabe's [16], which is remarkable for both the breadth and depth of the material covered, and the book by D. Revuz and M. Yor's [27], which demonstrates the power of stochastic integration by showing how it can be used to elucidate the contents Ito's book with Me Kean [15], which, ironically, itself contains no stochastic integration. Besides these, there are several excellent, more instruction oriented texts: the

Preface

Xlll

ones by B. 0ksendal [25] and by K.L. Chung and R. Williams [3] each has devoted followers. In addition, the lure of mammon has encouraged several authors to explain stochastic integration to economists and to explain economics to stochastic integrationists. 1 In view of the number of books which already exist on the subject, one can ask, with reason, whether yet another book about stochastic integration is really needed. In particular, what is my excuse for writing this one? My answer is that, whether or not it is needed, I have written this book to redress a distortion which has resulted from the success of Ito's theory of stochastic integration. Namely, Ito's stochastic integration theory is a secondary theory, the primary theory being the one which grew out of Ito's ideas about the structure of Markov processes. Because his primary theory is the topic of Chapters 1 through 4, I will restrict myself here to a few superficial comments. Namely, as is explained in the introduction to [14], when, as a student at Tokyo University, Ito was assigned the task of explaining the theory of Markov processes to his peers, he had the insight that Kolmogorov's equations arise naturally if one thinks of a Markov process as the integral curve of a vector field on the space M 1 (JRn) of probability measures on ffi.n. Of course, in order to think about vector fields on M 1 (JRn), one has to specify a coordinate system, and ItO realized that the one which not only leads to Kolmogorov's equations but also explains their connection to infinitely divisible laws is the coordinate system determined by integrating p, E M 1 (JRn) against smooth functions. When these coordinates are adopted, the infinitely divisible flows play the role of "rays," and Kolmogorov's equations arise as the equations which determine the integral curve of a vector field composed of these rays. 2 From a conventional standpoint, Chapter 1 is a somewhat peculiar introduction to the theory of continuous-time Markov processes on a finite state space. Namely, because it is the setting in which all technical difficulties disappear, Chapter 1 is devoted to the development of Ito's ideas in the setting of the n-point space Zn {0, ... , n- 1}. To get started, I first give M 1 (Zn) the differentiable structure that it inherits as a simplex in JRn. I then think of Zn as an

=

1 Whatever the other economic benefits of this exercise have been, it certainly provided (at least for a while) a new niche in the job market for a generation of mathematicians. 2 It should be mentioned that, although this perspective is enormously successful, it also accounts for one of the most serious shortcomings of Ito's theory. Namely, Ito's ideas apply only to Markov processes which are smooth when viewed in the coordinate system which he chose. Thus, for example, they cannot handle diffusions corresponding to divergence-form, strictly elliptic operators with rough coefficients, even though the distribution of such a diffusion is known to be, in many ways, remarkably like that of Brownian motion.

XIV

Preface

Abelian group (under addition modulo n), develop the corresponding theory of infinitely divisible laws, and show that the inherited differentiable structure has a natural description in terms of infinitely divisible flows, which now play the role that rays play in Euclidean differential geometry. Having put a differentiable structure on M 1 (Zn), one knows what vector fields are there, and so I can discuss their integral curves. Because it most clearly highlights the analogy between the situation here and the one which is encountered in the classical theory of ordinary differential equations, the procedure which I use to integrate vector fields on M 1 (Zn) is the Euler scheme: the one in which the integral curve is constructed as the limit of "polygonal" curves. Of course, in this setting, "polygonal" means "piecewise infinitely divisible." In any case, the integral curve of a general vector field on M 1 (Zn) gives rise to a nonlinear Markov flow: one for which the transition mechanism depends not only on where the process presently is but also on what its present distribution is. In order to get usual Markov flows (i.e., linear ones with nice transition probability functions), it is necessary to restrict one's attention to vector fields which are affine with respect to the obvious convex structure of M 1 (Zn), at which point one recovers the classical structure in terms of Kolmogorov's forward and backward equations. The second half of Chapter 1 moves the considerations of the first half to a pathspace setting. That is, in the second half I show that the flows produced by integrating vector fields on M 1 (Zn) can be realized in a canonical way as the one-time marginal distributions of a probability measure lP' on the space D ([0, oo); Zn) of right-continuous paths, piecewise constant p : [0, oo) ---+ Zn. After outlining Kolmogorov's completely general approach to the construction of such measures, the procedure which I adopt and emphasize here is Ito's. Namely, I begin by constructing the JID, known as the Levy process, which corresponds to an infinitely divisible flow. I then show how the paths of a Levy process can be massaged into the paths of a general Markov process on Zn. More precisely, given a transition probability function on Zn, I show how to produce a mapping, the "Ito map," of D([O, oo); Zn) into itself in such a way that the lP' corresponding to the given transition probability function is the image under the Ito map of an appropriate Levy process. The chapter ends by showing that the construction of the Ito map can be accomplished by the pathspace analog of the Euler approximation scheme used earlier to integrate vector fields on M1 (Zn). The second and third chapters recycle the ideas introduced in Chapter 1, only this time when the state space is JR.n instead of Zn. Thus, Chapter 2 develops the requisite machinery, especially the Levy~Khinchine formula, 3 about 3

In the hope that it will help to explain why these techniques apply to elliptic, parabolic,

Preface

XV

infinitely divisible flows, and then shows4 that, once again, the infinitely divisible flows are the rays in this differentiable structure. Once this structure has been put in place, I integrate vector fields on M 1 (JR.n) using Euler's approximation scheme and, after specializing to the affine case, show how this leads back to Kolmogorov's description of Markov transition probability functions via his forward and backward equations. Before I can proceed to the pathspace setting, I have to develop the basic theory of Levy processes on JR.n. That is, I have to construct the pathspace measures corresponding to infinitely divisible flows in M 1 (JR.n), and, for the sake of completeness, I carry this out in some detail in the last part of Chapter 2. Having laid the groundwork in Chapter 2, I introduce in Chapter 3 Ito's procedure for converting Levy processes into more general Markov processes. Although the basic ideas are the same as those in the second half of Chapter 1, everything is more technically complicated here and demands greater care. In fact, it does not seem possible to carry out Ito's procedure in complete generality, and so Chapter 3 includes an inquiry into the circumstances in which his procedure does work. Finally, Chapter 3 concludes with a discussion of some examples which display both the virtues and the potential misinterpretation of Ito's theory. Chapter 4 treats a number of matters which are connected with Ito's construction. In particular, it is shown that, under suitable conditions, his construction yields measures on pathspace which vary smoothly as a function of the starting point, and this observation leads to a statement of uniqueness which demonstrates that the pathspace measures at which he arrives are canonically connected to the affine vector fields from which they arise. In some sense, Chapter 4 could, and maybe should, be the end of this book. Indeed, by the end of Chapter 4, the essence of Ito's theory has been introduced, and all that remains is to convince afficionados of Ito's theory of stochastic integration that the present book is about their subject. Thus, Chapter 5 develops Ito's theory of stochastic integration for Brownian motion and applies it to the constructions in Chapter 3 when Brownian motion is the underlying Levy process. Of course, Ito's famous formula is the centerpiece of all this. Chapter 6 showcases several applications of Ito's stochastic integral theory, especially his formula. Included are Tanaka's treatment of local time, the CameronMartin formula and Girsanov's variation thereof, pinned Brownian motion, and but not hyperbolic equations, I have chosen to base my proof of their famous formula on the minimum principle. 4 Unfortunately, certain technical difficulties prevented me from making the the connection here as airtight as it is in the case of M1 (Zn)·

xvi

Preface

Ito's treatment of Wiener's theory of homogeneous chaos. Kunita and Watanabe's extension of Ito's theory to semimartingales is introduced in Chapter 7 and is used there to prove various representation theorems involving continuous martingales. Finally, Chapter 8 deals with Stratonovich's variant on Ito's integration theory. The approach which I have taken to this topic emphasizes the interpretation of Stratonovich's integral, as opposed to Ito's , as the one at which someone schooled in L. Schwartz's distribution theory would have arrived. In particular, I stress the fact that Stratonovich's integral enjoys coordinate inveriance properties which, as is most dramatically demonstrated by Ito's formula, Ito's integral does not. In any case, once I have introduced the basic ideas, the treatment here follows closely the one developed in [38]. Finally, the second half of this chapter is devoted to a proof of the "support theorem" (cf. [39] and [40]) for degenerate diffusions along the lines suggested in that article. Although applications to differential geometry are among the most compelling reasons for introducing Stratonovich's theory, I have chosen to not develop them in this book, partly in the hope that those who are interested will take a look at

[37]. Before closing, I should say a few words about possible ways in which this book might be read. Of course, I hope that it will eventually be read by at least one person, besides myself, from cover to cover. However, I suspect that most readers will not to do so. In particular, because this book is really an introduction to continuous time stochastic processes, the reader who is looking for the most efficient way to learn how to do stochastic integration (or price an option) is going to be annoyed by Chapters 1 and 2. In fact, a reader who is already comfortable with Brownian motion and is seeking a no frills introduction to the most frequently used aspects of Ito's theory should probably start with Chapter 5 and dip into the earlier parts of the book only as needed for notation. On the other hand, for someone who already knows the nuts and bolts of stochastic integration theory and is looking to acquire a little "culture," a reasonable selection of material would be the contents of Chapter 1, the first half of Chapter 2, and Chapter 3. In addition, all readers may find some topics of interest in the later chapters. Whatever is the course which they choose, I would be gratified to learn that some of my readers have derived as much pleasure out of reading this book as I have had writing it. Daniel W. Stroock, August 2002

Markov Processess from K. Ito's Perspective

CHAPTER 1

Finite State Space, a Trial Run

In his famous article [19], Kolmogorov based his theory of Markov processes on what became known as Kolmogorov's forward 1 and backward equations. In an attempt to explain Kolmogorov's ideas, K. Ito [14] took a crucial step when he suggested that Kolmogorov's forward equation can be thought of as describing the "flow of a vector field on the space of probability measures." The purpose of this chapter is to develop Ito's suggestion in the particularly easy case when the state space E is finite. The rest of the book is devoted to carrying out the analogous program when E = JR.n. 1.1 An Extrinsic Perspective Suppose that Zn = {0, ... , n- 1} where n ~ 1. Then M1 (Zn) is the set of probability measures J-Lo = 2::~-:!o Om8m, where each 15m is the unit point mass at m and (J is an element of the simplex en ~ JR.n consisting of vectors whose coordinates are non-negative and add to 1. Furthermore, if en is given the topology which it inherits as a subset of JR.n and M1 (Zn) is given the topology of weak convergence (i.e., the topology in which convergence is tested in terms of integration against bounded, continuous functions), then the map (J E en f-----.> J-Lo E Ml(Zn) is a homeomorphism. Thus it is reasonable to attempt using this homeomorphism to gain an understanding of the differentiable structure on M 1 (Zn). §1.1.1. The Structure of en: It is important to recognize that the differentiable structure which en inherits as a subset of JR.n is not entirely trivial. Indeed, even when n = 2, en is a submanifold with boundary, and it is far worse when n ~ 3. More precisely, its interior is a nice (n - I)dimensional submanifold of JR.n' but the boundary aen of en breaks into faces which are disconnected submanifolds of dimensions (n- 2) through 0. For example,

en

1 In the physics literature, Kolmogorov's forward equation is usually called the FokkerPlanck equation.

2

1 FINITE STATE SPACE

is a diffeomorphism, and

ae3 = io U J1 U J2 U {(1, 0, 0)} u {(0, 1, 0)} u {(0, 0, 1)} where each of the maps (hE (0, 1) f-------7 (O,B1, 1- B1) E I 0 , (:1 0 E (0, 1) f-------7 (Bo, 0, 1- Bo) E h, and Bo E (0, 1) f-------7 (Bo, 1-00 , 0) E h is a diffeomorphism. In particular, after a little thought it becomes clear that although, starting at each () E en, motion in en can be performed in (n- 1) independent directions, movement in some, or all, those directions may have restricted parity. For example, if() E Io, then movement from () will require that the Oth coordinate be initially nondecreasing. Similarly, if() = (1, 0, 0), at least initially, the Oth coordinate will have to be nonincreasing while both the 1st and 2nd coordinates will have to be nondecreasing. For the reasons just discussed, it is best to describe the notion of a differentiable path in en in terms of one-sided paths. That is, we will say that t E [0, oo) f-------7 O(t) E en is differentiable if 2 the derivative of

t E [0, oo)

f-------7

. O(t)

=lim O(t+h)-O(t) E IRn h h""O

is a smooth curve,

in which case we will call 0(0) a tangent to en at 0(0). Further, given () E en, we will use To( en) to denote the set of all vectors 0(0) which arise as the (right) derivative at t = 0 of a smooth t E [0, oo) f-------7 O(t) E en with 9(0) = 9. The following statement is essentially obvious. 1.1.1. For each m E Zn, let em E IRn be the vector whose mth coordinate is 1 and whose other coordinates are 0. Then en is the convex hull of {eo, ... , en-d· Moreover, if 9 E en, then bE IRn is a tangent to en 10 (b,em)JRn = 0, (9,em)IR" = 0 ===? (b,em)JRn 2:0, at 9 if and only and (9,em)JRn = 1 ===? (b,em)IR" ::::; 0. In particular, Te(en) is a closed, convex cone in JRn. LEMMA

iiz=:-==

As we anticipated, the shape of Te(en) undergoes radical changes as () moves from to the faces making up In particular, the preceding description makes it difficult to spot any structure which plays here the role of model space the way IRn does in the study of n-dimensional differentiable manifolds. On the other hand, as yet, we have not taken advantage of the basic property enjoyed by en: it is a simplex. In particular, we should expect that the structure of To( en) for all 9 E en can be understood in terms of the structure of spaces Te(en) as 9 runs over the vertices eo, ... , en-1 of en· This expectation is given substance by the following simple result.

en

2 When differentiating a function derivative.

oen.

f with respect to time, we will often use j to denote the

3

1.1 AN EXTRINSIC PERSPECTIVE

LEMMA 1.1.2. For each 0 ::; m ::; n- 1, a E Tem (8n) if and only if

(a, ee)ffi.n ~ 0 for each£ =F m and (a, em)ffi.n =- L (a, ee)ffi.n' e-1-m Moreover, for any (} E 8n, b E To(8n) if and only if there exist ae E Tee(8n), 0::; £::; n- 1, such that n-1

b = L(6,ee)ffi.nae. e=O

(1.1.3)

PROOF: The first assertion is a special case of Lemma 1.1.1. In proving the second part, we use the notation Be = (6, ee)ffi.n, bm = (b, em)ffi.n, and

ae,m

(ae,em)ffi.n' = 2:;~~ Beae, where ae E Tee8n for each £ E Zn, then bm = 2:;~~ Beae,m, and so =

If b

~ bm =~Be(~ ae,m) = 0, e=O

m=O

Bk = 0

===?

m=O

bk = L Beae,k ~ 0,

and Bk = 1

===?

bk = ak,k ::; 0.

e#

Hence, bE To(8n)· To go the other way, we work by induction on n ~ 2. Thus, suppose that (} E 8 2 and bE To8 2 are given. Clearly, either b = 0, and there is nothing to do, or bo > 0 and 1- Bo > 0, or b1 > 0 and 1- B1 > 0. Thus, we may and will assume that b1 > 0 and 1 - B1 > 0, in which case we take ao = 1! 111 b and a 1 = 0. Next, let n ~ 3 and assume the result for n- 1. Reasoning as before, we reduce to the case when bn-1 > 0 and 1- Bn-1 > 0. Set {J = --'-(B_o,_._·.-,-'B_n_-2-'-) 1 - Bn-1

and

b= A

(bo + Bobn-1, ... , bn-2 + Bn-2bn-1) 1 - Bn-1

~----------------------~

and observe that 0 E Bn-1 and bE T98n-1· Using the induction hypothesis, find ae E TeeBn-1 so that b = I:;~g Beae. Next, take ae = 0 if£= n-1 or Be = 0, and, when 0::; £ ::; n- 2 and Be > 0, take

It is then an easy matter to check that ae E Tee 8n for each 0 ::; £ ::; n - 1 and b = 2:;::-~ Beae. 0

4

1 FINITE STATE SPACE

§1.1.2. Back to MI(Zn): We now transfer the above analysis back to MI(Zn). Namely, using p E M1 (Zn) ~------* (}ll E en to denote the inverse of the map (} E en ~------* Po E M1 (Zn), we say that the curve t E [0, oo) ~------* PtE MI(Zn) is differentiable if t E [O,oo) ~------* O(t) = 01,, E en is differentiable. In order to describe the tangent corresponding to such a curve, first note that, since (}ll = (p({O}), ... ,p({n -1})), an equivalent formulation of the preceding is to say that that, for each 'P E C(Zn; JR), the map 3 t E [0, oo) ~------* ('fJ, Pt) E lR is differentiable. Thus, it is reasonable to identify the corresponding tangent /1 0 as the linear functional on C(Zn; JR) given by

.

(1.1.4)

Po'P

.

11m = h"-,0

('fJ, Ph) - ('fJ, Po) h

.

Clearly, the relationship between fi, 0 and 0(0) is

n-1 Mo'P

=

L

'fJ(m)(O(O),ern)JRn"

m=O

Hence All is an element of the space Til (M 1 (Zn)) of tangent vectors to MI(Zn) at p E M1(Zn) if and only if (cf. Lemma 1.1.1)

n-1 (1.1.5)

A 1,'{J =

L

'{J(m)(b,em)JRn'

'{J

E C(Zn;lR), for some bE To"(en)·

rn=O

Of course, this means that, like To" (en), TIL (M1 (Zn)) is a closed, convex cone, this time of linear functionals on C(Zn;lR). It also means (cf. Lemma 1.1.2) that each All E Til (M1 (Zn)) admits the representation (1.1.6)

All= {

Jzn

A6x p(dx),

where x E Zn

~------* A6x

E T6x(Ml(Zn)).

On the other hand, except when p is a point mass, the representation in (1.1.6) is far from unique! In fact, as the proof of Lemma 1.1.2 makes abundantly clear, except at vertices there are innumerable ways of decomposing b E Teen into ae's, and each decomposition leads to a different representation in (1.1.6). REMARKS 1.1. 7. Our derivation of (1.1.6) is a straight forward computation which fails to reveal the reason why we should have expected its validity. To understand its intuitive origin, one needs to know the interpretation of 3

(.~)t to denote the infinitely divisible flow with (.\~)o

M (p(O) = 0) = 1 and, for any m ~ 1 and 0 :::; to < · · · < tm, the ]p>M_ distribution of p -v-+ (p(t1)- p(to), ... ,p(tm)- p(tm-d) is TI~ 1 Are, where Te = te -te-l· Equivalently, under lP'M, p(O) = 0 almost surely, increments of p over disjoint time intervals are mutually independent, and the distribution of the increment p -v-+ p(s + t) - p(s) is At. In the jargon of probability theory, ]p>M is the distribution of the process starting at 0 with homogeneous, independent increments for which At is the distribution of an increment over a time interval of length t. In accordance with more recent terminology, we will call such a process a Levy process. In order to develop a better understanding of ]p>M, we will now give a hands-on construction. Let (n, F, IQ) be a probability space on which there exists a family { Te,y, (£, y) E z+ x (Zn \ {0})} of mutually independent, unit exponential random variables. That is, the Te,y's are mutually independent, and IQ(Te,y > t) = e-t for all t ~ 0. Without loss in generality, we will assume that enough sets of IQ-measure 0 have been eliminated from n so that, for all w, 00

Te,y(w) > 0,

LTe,y(w) = oo, £=1 m

m'

and (m, y) =/:- (m', y') :::::::::} LTe,y(w) =/:- LTf',y'(w). i'=l

f=l

22

1 FINITE STATE SPACE

Next, define the random variables

Tm,y(w) = {

~m L...C=l

ifm = 0 (

T£,y W

)

if mE

z+,

and

Ny(t,w) = max{m: Tm,y(w):::; tM({y})}. Obviously, a({Ny(t): t::::: 0}) is independent of a({Ny'(t): t::::: 0}) when y' "I- y. In addition, for each y and w, Ny( · ,w) is anN-valued, right continuous, nondecreasing path which starts at 0; and (cf. Lemma 3.3.4 in [36] or Exercise 1.4.11 below)

(1.4.8)

for all k ::::: 1, 0 = to < t 1 < · · · tk, and 0 = mo :::; m1 :::; · · · :::; mk. Equivalently, Ny(s + t)- Ny(s) is Q-independent of a({N(a): a E [O,s]}) and has Q-distribution given by Q(Ny(t) = m) = e-tM({y}) (tM(~}))"' That is, however one states it, {Ny(t) : t ::::: 0} is a simple Poisson process with intensity M( {y} ). Now set N(t,w) = LyEZn\{O} Ny(t,w). Then {N(t) : t :::0: 0} is again a simple Poisson process, this time with intensity A1(1Zn)· In particular, if T 0 (w) 0 and

=

Tm(w) =min{ t::::: Trn-l(w): N(t, w) > N(Tm_I(w), w) }, then the T m - T m-l 's are mutually independent and Q (T m - T m-1 > t) = e-tM(Zn) form::::: 1 and t::::: 0. Furthermore, for each m::::: 1 and x E Zn,

(1.4.9)

Q(Nx(Tm)- Nx(Trn-d =

11 Fm) =

M({x})

M(Zn)

where Fm =a( {Tm} U {Ny(t): t E [0, Tm) & Y E Zn \ {0}} ). To check (1.4.9), let 0 < h < s and A E a( {Ny(a) : a E [0, s- h] andy E Zn \ {0}}) with Q(A) > 0 be given, and observe that, because Q(N(s)-

N(s- h)::::: 2) = O(h 2 ) ash'\. 0, Q(Nx(Tm)- Nx(Tm-d

= 1 & s- h < Tm :S s & A)

= Q(Nx(s)- N, (s- h)= 1

1 & Tm

> s- h &A)+ O(h 2 )

= hM({x})Q(Tm > s- h & A)+ O(h 2 ),

1.4

23

PATHSPACE REALIZATION

and, similarly, Ql(s- h

< Tm:::; s & A) = hM(Zn)Ql(Tm > s- h & A)+ O(h 2 ).

We are, at last, ready to introduce the advertised understanding of JP>M. Namely, set X(t, w) = yNy(t, w),

L

yEZn \{0}

where here the sum is taken in Zn (i.e., modulo n). Then, Ql(X(O) and

EQ[e~,n(X(s+t) -X(s))

= 0) = 1

la({X(a): a E [O,s]})]

~ IJJEO[eE,n(YN,(t))j ~ IJ (e-M provides us with lots of information about !P'M. For one thing, the paths X(·, w) are much better than generic paths in (Zn)[O,oo). In fact, for each w, X(· ,w) is an element of the space D([O,oo);Zn) of piecewise constant paths p : [0, oo) --+ Zn which are right continuousY Hence, JP>M lives on the space D([O, oo); Zn), 12 and from now on we will always think of it living there. Something else which our construction reveals is the mechanism underlying the evolution of paths under JP>M. Namely, the paths are piecewise constant, the times between jumps are independently and exponentially distributed with intensity M(Zn), and (cf. (1.4.9)) when a jump occurs, the size of the jump is independent of everything that happened before and has distribution All this and more is summarized in the following.

M7:L).

THEOREM 1.4.10. Given p E D([O,oo);Zn), define ry(t,f,p) fort 2: 0 and r ~ Zn so that ry(O,f,p) = 0 and 13

ry(t, r,p)

=

L

lrn(Zn \{0}) (p(T)- p(T- ))

ift E (0, oo).

O 0, and define To(p) = 0 and Trn(P) = inf{t 2 Tm-1(P) : p(t) "1p(Tm-d}. Then, for all mE z+, IP'M(Tm < oo) = 1,

IP'M (ry(t

+ Trn-1,r)- ry(Tm-1,r) = 0 I Tm-1) = e-tM(r),

and

IP'

M(p(Tm)- p(Trn-d

= y

ITm ) =

r ~ Zn,

M({y}) M(Z,),

where Trn and Tm_ 1 are the a--algebras generated, respectively, by {T m} U {p(t): t E [0, Tm(p))} and by {Tm-d U {p(t): t E [0, Tm-1(p)]}. §1.4.3. Exercises 1.4.11. The purpose of this exercise is to verify (1.4.8). Thus, let {T£ : f! 2 1} be a sequence of mutually independent, unit exponential random variables, set T 0 = 0 and Tm = 2::~ Tp form 2 1, and take N(t) = sup{m: Trn::; At}, where A E [0, oo). If A= 0, then it is clear that N(t) = 0 almost surely for all t 2 0. Thus, assume that A > 0, and justify: EXERCISE

IP'(N(ti)

= m1, ... , N(tc) =me)

= 1P'(Tm 1 =

Amt+ 1

::;

Afi < Tm 1 +1, ... , Tm, ::; Ate< Tm,+1)

J···J (-A f exp

1

17

J=1

A

~j) d6 · · · d~m,+1

where

mj+1

mj

A= B

{

~ E (O,oo)m'+ 1 : ~~k::;

= { "1

b.,j =

}

< ~ ~k for 1::; j::; I! ,

E (0, oot''+ 1 : 7/j < f/J+1 & 7/m.i ::; t1 < 7/rnj+1 for 1::; j::; !!},

{u E ]RrnJ-mJ-

and to= 0.

tj

1

:

tj-1::; U1

< · · · < Um 1 -mJ-1::;

tj},

25

1.4 PATHSPACE REALIZATION

EXERCISE 1.4.12. A probability measure IP' on (Zn)[O,oo) is said to be time homogeneous Markov with transition probability (t,x) "'"'* Pt(x, ·)if (1.4.1) holds for all m ~ 0, 0 =to < h · · · < tm, and {r£}1)' ~BE. (i) Show that IP' satisfies (1.4.1) if and only if IP'(p(O) E r) = t-to(r) and, for each s E [0, oo) and t > 0: (1.4.13)

I

IP'(p(s + t) E r a({p(a): a E [0, s]}) = Pt(p(s),r).

(ii) The goal here 14 is to show that (1.4.1) is not implied if (1.4.13) is replaced by the weaker condition

(*) To this end, take E = Z 2, and let (0, F, Q) be a probability space on which there exists a sequence {X£}]"' of mutually independent, Z2-valued random variables with Q(Xt = 0) = = Q(Xt = 1). Further, assume that {N(t) : t ~ 0} is a Poisson process with intensity 1 on (0, F, Q) and that a({Xt: ~ 1}) is Q-independent of a({N(t): t ~ 0}). Finally, set X 0 = X 1 + X 2 (addition in the sense of Z2) and Y(t) = E~g) Xt. Show that (*) holds with Pt(x, ·) = e-t8x + (1- e-t)Az 2 , but that (1.4.13) is false.

!

e

EXERCISE 1.4.14. It will be useful to know that Theorem 1.4.10 readily generalizes to time dependent, translation invariant vector fields. That is, suppose that t "'"'* M (t, · ) is a continuous (i.e., t "'"'* M (t, {y}) is continuous for each y) map from [0, oo) into Levy measures, and determine the time dependent, translation invariant vector field t "'"'* A(t) by

A(t)iix'P= { (cp(x+y)-cp(x))M(t,dy)

Jzn

for x E Zn and 'P E C(Zn;lR.). Finally, for each s E [O,oo), let (cf. Exercise 1.3.16) t "'"'* As,t be the integral curve oft"'"'* A(t) starting from 80 at time s. (i) Proceeding in exactly the same way as we did above and using the notation introduced in Theorem 1.4.10, show that there is a unique probability measure JP>M. on D([O,oo);Zn) such that JP>M. (p(O) = 0) = 1, the processes p "'"'* 17( ·, {y},p) for y E Zn \ {0} are mutually JP>M. -independent and, for any fixed y E Zn \ {0},

I

IP'M. ( 17(s + t, {y}) -17(s, {y}) = m a(p(a): a E [0, s]}))

( t

=exp - Jo M(s+T,{y})dT 14

) (J;M(s+T,{y})dT)m m!

I got the key idea for this exercise from Richard Dudley.

26

1 FINITE STATE SPACE

(ii) Show that the measure JP>M described in (i) satisfies (1.4.4) when J.Lo =Do and Pt((s,x), ·) = (Tx)*As,s+t· In addition, show that JP>M -almost surely {Tm-1+t

IP'~ (ry(Tm-1+t,r)-ry(Tm-1,r) =OITm-1) =exp ( - }Tm-l

M(T,f)dT

)

on {Tm-1 < oo} and that

M·(( ) ( ) I") M(Tm,{Y}) IP'o pTm -pTm-1 =y .lm = M(T z) m,

n

on {Tm

< oo}.

1.5 Ito's Idea Theorem 1.4.10 provides us with a rather complete description of the way in which the paths of an independent increment process evolve on Zn· In particular, it gives us the path decomposition

r yry(t,dy,p);

p(t) =

(1.5.1)

Jzn

of p in terms of the family { ry( ·, {y}) : y E Zn} of mutually independent, simple Poisson processes. Ito's method allows us to extend this type of description to the sort of process described in Theorem 1.4.3 corresponding to a general affine vector field. §1.5.1. Ito's Construction: In order to explain Ito's idea in the present context, we must begin by examining what happens when we replace the integrand "y" in (1.5.1) by "F(y)", where F: Zn' ------> Zn maps 0 E Zn' to 0 E Zn· That is, for p' E D([O,oo);Zn'), set

Xp(t,p') =

(1.5.2)

1

zn,

F(y') ry(t, dy',p').

Clearly, Xp(-,p') E D([O,oo);Zn)· In addition, for each s E [O,oo), p'...,... Xp(s,p') is a({p'(a) : a E [O,s]})-measurable and, for every t 2: 0 and M' E 9Jt(Zn' ), the increment Xp(s + t) - Xp(s) is JP>M' -independent of a({p'(a) : a E [O,s]}) and has the same distribution asp' ...,... Xp(t,p'). Finally, for each ~ E Zn, !Eli'M'

[e~,n(Xp(t))]

=

IT

Eli'M'

[e~,n(ry(t,{y'})F(y'))]

y'EZn'

=

IT

y'EZn'

exp(tM'({y'})(ee,n(F(y'))-1))

27

1.5 ITO'S IDEA

Hence, we have now shown that the lP'M'-distribution of p' ""* Xp( · ,p') is that of the independent increment process on Zn determined by the Levy measure F*M' given by (1.5.3)

1

F*M ({y}) =

{

0

M'({y' E Zn': F(y') = y})

if y = 0 if y E Zn \ {0}.

Next, suppose that we are given a map F : Zn X Zn' ---+ Zn which takes Zn' X {0} to 0 E Zn, define x ""* M(x, ·) by (cf. (1.5.3)) M(x, ·) = (F(x, · )LM', and let A be the affine vector field on M 1 (Zn) determined by x ""* M(x, · ). Ito's idea is to construct the process corresponding to A by modifying the construction in (1.5.2) to reflect the dependence of F(x, y) on x. To be precise, refer to the notation in Theorem 1.4.10 and define

(x, p') E Zn x D([O, oo ); Zn')

f----)

X(·, x,p') E D([O, oo ); Zn)

so that

X(O,x,p') (1.5.4)

=

x

X(Tc(p'),x,p')- X(Tc-l(p'),x,p')

= F(X(Tc-l(p'), x,p'),p'(Tc)- p'(Tc-d) X(t, x, p') = X (Tc_l(p'), x,p')

if Tc-1 (p') :S:: t

< Tc(p').

It should be evident that, for each (t, x) E [0, oo) x Zn, the map p' ""* X(t, x,p') is IT( {p'(T)) : T E [0, t]} )-measurable. THEOREM 1.5.5. Referring to the preceding, let t ""* /Lt be the integral curve of A starting from J.L, and let lP' be the associated measure described in Theorem (1.3.9). If (t, x,p') ""* X(t, x,p') is defined as in (1.5.4), then lP' is the J.L x lP'M' distribution of(x,p') ""*X(· ,x,p'). In particular, lP' lives on D([O,oo);Zn)· PROOF: The first step is to observe that lP' is characterized by the facts that p ""* p(O) and that, for any s 2 0, A E IT( {p(IT) : IT E [O,s]}), and 'P E C(Zn;lR),

J.L is the lP'-distribution of

(1.5.6)

lEJP['P(p(s + t))- 'P(p(s)), A]=

1t

lEIP'[L'P(p(s + T)), A] dT,

where

Indeed, because A is affine, this is precisely the characterization given in the final part of Theorem 1.4.3.

28

1

FINITE STATE SPACE

In view of the preceding paragraph and the remark following (1.5.4), it suffices for us to check that for ( s, x) E [0, oo) x Zn and A' E a ({p' (a-) : a E

[O,s]}),

lEIP

(*)

AI'

[cp(X(s + t, x)) - cp(X(s, x)), A']

=fat JEIPM' [Lcp(X(s + T, x)), A'] dT.

To this end, note that

JEIP

M'

[cp(X(s+t+h,x)) -cp(X(s+t,x)), A']

L

lEn>.u'['P(X(s+t+h,x)) -cp(X(s+t,x)), A'nJ'(h,y')] u'

+lEIP'

[cp(X(s+t+h,x)) -cp(X(s+t,x)), A'nB'(h)]

where

J'(h,y')::::::: {p': ry(s+t+h,zn,,p') -ry(s+t,zn,,p') = 1

ry(s + t + h, {y'},p')- ry(s + t, {y'},p')} B'(h)::::::: {p': ry(s+t+h,Zn,,P') -ry(s+t,zn,,p') 2 2}]. =

Because IP'M' (B'(h)) tends to 0 like h2 ash"" 0, while

JEn>M'[cp(X(s+t+h,x)) -cp(X(s+t,x)), A'nJ'(y')] =

JEIPM' [cp(X(s + t, x)) + F(X(s + t, x), y') - cp(X(s + t, x)), A'] X

JP'~!' ( J' (h, y'))

and JP'M'

(J'(h, y')) = IP'M' (ry(h, {y'}) = 1 = ry(h, Zn' ))

=

e~hM'( {y'}) hM' ({y'}) - IP'M' ( ry(h, {y'})

= 1 < ry(h, Zn')),

we conclude that

.

hm

JEIP"r'[cp(X(s+t+h,x)) -cp(X(s+t,x)), A']

h"'O

= JEn>M'

h

[! (cp(X(s + t, x) + y) - cp(X(s + t, x))) M(X(s + t, x), dy), A'] ,

29

1.5 lT 0, Me(dy) = l[e,ooJ(Iyi)M(dy), then, for each E > 0, there is no question that L(a,b,M')

(a,b,M)

66

2

MOVING TO EUCLIDEAN SPACE

is JP(a,b,M)_independent of a({p(t)- w(t,p): t 2: 0}). Although the idea is simple, its implementation is complicated by the fact that, in general, a path p E D([O,oo);!Rn) does not admit a clean decomposition into a continuous part and purely jump part. The point is that p can have infinitely many very small jumps which, in composite, cannot be canonically distinguished from a continuous path. 13 In the next lemma, and elsewhere, p( t-) = lims/t p( s) will be used to denote the left limit at t E (0, oo) of a path p E D([O, oo); IRn). LEMMA 2.4.8. For any Levy system (a, b, M), jp(a,b,M)

In addition,

if~

(p(t)

= p(t-)) = 1

for all t E (0, oo ).

is a Borel subset of!Rn \ {0} and

M(~) =

0, then

jp(a,b,M)(::Jt E (O,oo) p(t)- p(t-) E ~) = 0. PROOF:

To prove the first part, note that, by Fatou's Lemma, for any jp(a,b,M)

(lp(t)- p(t- )I

>E) :::;

limlP(a,b.M) (lp(t)- p(s)l

E

> 0,

>E)

s/t

=lim A~~~,M) (BRn (0, E)C) = 0. s/t

To prove the second assertion, remember that JP(a,b.M) is the jp(a,b,O) x jp(O,O,M)_distribution of (p 1 , p 2 ) 'V'-7 p 1 + p 2 . Thus, since JP(a,b.o) lives on C([O,oo);!Rn), we may and will restrict our attention to JP(o,o.M). Second, observe that when M(IRn) < oo, the result follows immediately from the representation of JP(o,o,M) in Lemma 2.4.3. Finally, to handle general Levy measures M, we use the representation of JP(o.o,M) given in Lemma 2.4.5. Namely, that lemma says that JP(o.o,M) is the Ql-distribution of w 'V'-7 q( ·, w), where, for Ql-almost every w = (p 1 , ... , prn, ... ) , q( · , w) = .Z:::::~o pp and the sum converges uniformly on compact subsets. Further, since .Me(~) :::; M(~) = 0 and MP(!Rn) < oo, we know that Ql(::Jt E (0, oo) l(t)- pp(t-) E ~) = jp(O,O,Me)(::Jt E (O,oo) p(t)- p(t-) E ~) = 0. Thus, we will be done once we show that, Ql-almost surely, J (q( · , w)) (a,b,M)_distribution of p

D([O,oo);~n) 2

is

]p>(O,O,ML>)

X

L1+11YI2

y M(dy),

E D([O,oo);~n)

t-+

(pt:::..,p- pt:::..)

E

]p>(a,b,M-ML>).

PROOF: Like the analogous results in §1.4.2, these results are most easily proved by making a construction and using uniqueness. Here is an outline. A more detailed account of a similar argument can be found in the proof of Corollary 3.3.17 in [36]. Note that JP>(a,b,M) is the distribution of (p 1 ,p2 ) E D([O, oo); ~n) 2 t - + p 1 + p 2 E D([O, oo ); ~n) under Q = JP>(o,o,ML>) x JP>(a,b,M-ML>). In particular, we know (cf. Exercise 2.4.14) that p 1 and p 2 never jump at the same time. Hence, by the second part of Lemma 2.4.8,

ry( · ,~,p 1

+ p 2 ) = ry(t,~n,p 1 )

Q-almost surely,

which, by the first part of Lemma 2.4.3, means that the JP>(a,b,M)_distribution of p""' ry( ·, ~,p) is that of a simple Poisson process with intensity M(~). At the same time, by that same lemma, P 1 (t)

+t

L

1 1 + IYI 2 yM(dy) =

L

yry(t,dy,p 1

+ p2)

Q-almost surely,

and sop""' (pt:::.., p- pt:::..) has the same distribution under ]p>(a,b,M) as (p 1 , p 2 ) have under Q, which is the assertion that we wanted to prove. 0 We can now prove that the paths of an ~n-valued independent process admit a decomposition which mimics the components making up the associated Levy system.

68

2 MOVING TO EUCLIDEAN SPACE

THEOREM 2.4.11. Let (a, b, M) be a Levy system, and set

ij(t, dy,p) = 77(t, dy,p)-

tl l2 M(dy).

1+ y

Next, choose 1 2 rrn ~ 0 so that JB"n(O,rm) IYI 2 M(dy) S 2-rn. Then, for JP'( a,b,M) -almost every p E D ( [0, oo); ]Rn), the limit (2.4.12)

p(t)

=

lim m~CXJ

1

jyj::>rm

yi)M (t, dy,p)

exists, uniformly with respect to t in finite intervals; and therefore t 'V'-7 jj(t) is JP'(a,b,M)_almost surely an element of D([O, oo); JRn). In fact, the JP'(a,b,M)_distribution of p is JP'(o,o.M). Furthermore, if w( · ,p) p- p, then, w( · ,p) E C([O, oo);JRn) JP'(a,b,M)_almost surely, O'({w(t,p): t 2 0}) is JP'(a,b,M)_independent of O'({p(t) : t 2 0}), and the JP'(a.b.M)_distribution of p 'V'-7 w( · ,p) is IP'6a.b,O). In particular, this means that O'({w( · ,p): t 2 0}) is JP'(a,b,M)_independent of 0'( {7](t, r, p) : t:::: 0 & r E Bxn\{0}}).

=

PROOF: Set Mm(dy) = l[rm,CXJ)(IYI) M(dy), and

pm(t) =

1

jyj::>rm

yij(t,dy,p).

In order to prove that the asserted limit exists in the required sense, set ~(m,rn') = {y E JRn : rm' < IYI S rm}, observe that, for 0 S m < m', (cf. the notation in Lemma 2.4.10)

pm(t)- prn' (t) pt:.. XF(t) in lP'M' -measure; and so, for any A' E B: = u({p'(u): a- E [0, s]}) with lP'M' (A')> 0, JEII'M'

[ev'=l (~,Xp(s+t)-XF(s))IR" IA'] = lim JEII'M' [e

yC1 ( f;.Xpm (s+t)-XFm (s) )R"

(t J

m-+(X/

= exp

(ev'=I(E,F(yJ)R"

IA']

-1) M'(dy)).

Knowing that the lP'M' -distribution of p' -v-+ X F ( · , p') is that of a compound Poisson process, we get (3.1.4) as a particular case of (2.4.4). D Lemma 3.1.2, together with the remark preceding it, already tells us how to transform paths of the Levy process with distribution lP'111 ' into the Levy process with distribution JP'(a,b,M) under the condition that (cf. (3.1.3)) M = F*M' for some F which vanishes near the origin. Furthermore, by mimicking the line of reasoning which we used in the proof of Lemma 2.4.5, we could replace this vanishing condition by the condition that there exists an r > 0 for which JBIJI."' (O,r) IF(y')l 2 dy' < oo. However, because such extensions will be incorporated into the program which follows, we will not bother with them here. §3.1.2. Hypotheses and Goals: Throughout the rest of this section, we will be dealing with the situation described in hypotheses (Hl)-(H4) below.

(Hl) M' will be a Levy measure on !Rn' and

lP'M'

will be the probability

measure lP'(J,O,M') on D([O,oo);!Rn').

(H2) F : !Rn x !Rn' ---.. !Rn will be a Borel measurable map which takes !Rn x {0} to 0 and satisfies the estimates (a)

lim sup

r"'O xEJRn

1

1+

1

X

12

r

} B

JR1l

1

(0 r·) '

IF(x, y')l 2M'(dy') = 0,

77

3.1 ITO'S BASIC CONSTRUCTION

and, for each R E (0, oo):

(b)

sup 1

xElR"

11 12 +X

r

IF(x, y'W M'(dy') < oo,

JB.,n' (O,R)

and

(c)

(H3)

a: JR(n----* Hom(IRn', 1Rn) and

which satisfy the

b: IRn----* IRn will be continuous maps

condition 1

and a : JR(n ----* Hom(IRn, JR(n) will be used to denote the symmetric, non-negative definite matrix-valued function aa T.

(H4) Using (a) of (H2), choose 1;:::: sup xElR"

1112

1+

X

r

JBrAn' (0,1·N)

rN

"""0 so that

IF(x,y'WM'(dy')::;2-N.

Then, for N EN,

(x,p') E JR(n x D([O, oo); 1Rn')

f-----7

is defined inductively so that X in Theorem 2.4.11)

N

XN ( ·, x,p') E D([O, oo); IRn) (0, x, p') = x and (cf. the notation

xN (t, x,p')- XN ([t]N, x,p') =

a(XN ([t]N, x,p')) (w(t,p')- w([t]N,p'))

+

r

jiY'I?JN

where [t]N

ij(t, dy,p') 1

We use

II·IIH.S.

+ b(XN ([t]N, x,p'))(t- [t]N)

F(XN ([t]N, x,p'), y')(ij(t, dy',p')- ij([t]N, dy',p')),

= 2-N[2N t] ([t] being the integer part oft) and (cf. (2.4.9)) = TJ(t, dy,p')- tl[o.l)(IY'I) M(dy').

to denote the Hilbert-Schmidt norm.

78

3 ITO'S APPROACH IN THE EUCLIDEAN SETTING Our main goals in the rest of this chapter will be to prove the following:

( G 1) There exists a measurable map p' E D ([0, oo); JR.n') 1----t X ( . , x, p') E D([O,oo);JR.n) to which {XN( ·,x,p'): N 2:: 0} converges uniformly on finite intervals for JPM' -almost every p'. In particular, for each s 2:: 0, p' ""'X(s,x,p') is measurable with respect to the JPM'-completion B~ (cf. the last part of Remark 3.1.23 below) of B~ = a({p'(a): a E

[O,s]}). {G2) For every 'P E CG(lR.n;JR.) and x E JR.n, lJ'P(x + F(x, y'))- 'P(x)

I

- 1[0,1) (ly'l) (F(x, y'), gradx'P )JRn' M' (dy') < oo, and 1

L'P = 2

(3.1.5)

n

L

i,j=l

n

aijOiOj'fJ +

L biai'P + K 'P i=l

is a continuous where K'{J(x) is given by

~n' ( 'P(x + F( ·, y')) -

'fJ(X) - 1[0,1) (ly'l) (F(x, y'), gradx'P )JRn) M' (dy').

Moreover, if 'P E c;(JR.n; JR.), then L'P is bounded and

('P(X(t,x,p'))

-fat L'fJ(X(r,x,p')) dr,BLlP'M')

is a martingale. That is, the JPM' -distribution of p' ""' X(- , x, p') solves the martingale problem starting at x for the operator L on

c;(JR.n; JR.). REMARK 3.1.6. The reader may well be asking why the L in (3.1.5) does not look more like the L in (3.1.1). The answer is that it has the form in (3.1.1), but in disguise. Namely, define b: JR.n ---+ JR.n so that

b(x) = b(x) _ (3.1.7)

f

(1ro,lJ(Iy'I)IF(x, y')l 2 }JRn' 1 + IF(x, y')l 2 - 1[1,oc) (ly'l)) F(x, y') M'(dy')

79

3.1 ITM' -completion B~ of B~, and this is the reason for our having to introduce these completed O"-algebras. For a different approach to the same problem, see Exercise 4.6.8 in [41], where it was shown that one can avoid completions if one is willing to have X(·, x,p') right continuous everywhere but in D([O,oo);JR.n) only JP>M'-almost everywhere.

=

§3.1.5. Verifying the Martingale Property in (G2): It remains to prove that the p' ~ X ( · , x, p') produced in Theorem 3.1.20 has the property described in {G2). However, before doing so, we should check that Lcp E Cb(JR.n; JR.) when cp E Cc(lR.n; JR.).

LEMMA 3.1.24. There exists a K < oo, depending only on sup 1 +11 12

xEJRn

X

(IIO"(x)ll~.s. v lb(x)l 2 v }Br , IRn

(0 1)

IF(x, y'W M'(dy'))

'

and M'(BJRn'(0,1)C), such that, for each cp E C~(lR.n;lR.), Lcp E C(JR.n;JR.) and

90

3 ITO'S APPROACH IN THE EUCLIDEAN SETTING

Moreover, ifcp

E

c;(JR.n;JR.) and cp is supported in the ball BJRn(O,R), then

IILcpllu :S K(3 + 8R 2 )II'PIIc~(JRn;JR) + K'II'PIIu

K' =sup M'({y': jF(x,y')l2 xEJRn

where

h/1 + lxl 2 }) < oo.

PROOF: First observe that M'({y': IF(x,y')l2~v1+lxl 2 }) :::; 1

~

+

X

121

Bll!.n' (0,1)

IF(x,y'WM'(dy')+M'(BJRn'(0,1)C),

which, by (b) in (H2), means that K' is indeed finite. Assume that cp E C~(JR.n; JR.), and observe that, by (a) of (H2),

Ln' ( cp(x + F(x, y')) - cp(x) - l[o,l) (ly'l) (F(x, y'), gradxcp )JRn) M' (dy') =lim { (cp(x+F(x,y')) -cp(x) r'\.O jiY'I?.r - l(o,1)(1y'l) (F(x, y'), gradxcp )JRn) M' (dy') uniformly for x in compacts. At the same time, it is easy to draw from (H3) and (c) in (H2) the conclusion that

{ ( cp(x + F(x, y')) - cp(x) - l[o, 1) (ly'l) (F(x, y'), gradxcp )JRn) M' (dy') jiY'I?.r is a continuous function of x for each r > 0. Hence, the continuity of Lcp is now proved. Thrning to the asserted bounds, note that the first estimate is an easy consequence of (cf. the notation in (3.1.13)) lin' ( cp(x + F(x, y')) - cp(x) - l[o,l) (ly'l) (F(x, y'), gradxcp )JRn) M' (dy')

: :; ~2 (1r

Bll!.n (0,1)

jF(x,y')l 2 M'(dy'))

I

II~PII&2 ) +2M'(BJRn'(0,1)C)II'PIIu·

Next, assume that

M' -distribution of p' .,.. X(·, x, p') is Markov. COROLLARY 3.1.27. Suppose that for each

0, then

is a martingale. To this end, first notice that it suffices to handle smooth v and then observe that

JE~~'[v(tz,p(tz))- v(h,p(h)), =

JE~~'

[1:

2 (

8tv (t, p( tz))

A]

+ Lv (t1, p( t))) dt, A]

for 0::::; h < t 2 and A E Bt,· At the same time,

JEll'

[lt

2 (

= JE~~'

+

8tv(t,p(t 2 ))

[1:

2

+ Lv(t 1 ,p(t))) dt, A]

(atv(t,p(t))

+ Lv(t,p(t))) dt, A]

E'" [, 0, set v( t, x) = u'P (T -t, x) for (t,x) E [O,T] X !Rn. Then v E C1~' 2 ([0,T] x !Rn;!R), v(T, ·) = r.p, and (at +L)v = 0 on (0, oo) x !Rn. Now choose 'ljJ E Cgo(IRn; [0, 1]) so that 'ljJ =: 1 on B(O, 1), and set VR(t, x) = 'lj;(R- 1 x)v(t, x). By the preceding, we know that

( vn(t AT,p(L AT))

-lAT

(8, + L)vn(r,p(r)) dr, 8 1 ,1!')

is a martingale. Thus, if (R(P)

=inf {t 2 0:

sup lp(t)l 2 R}

TE[O,t]

1\

T,

94

3 IT 0. Thus, since urp is bounded and continuous and (n--+ oo pointwise as R----+ oo, we conclude that

IEil'[cp(p(T)) I.Bs] = urp(T- s,p(s))

lP'-almost surely.

To complete the proof from here, let Pt(x, ·) be the distribution of lP'M'distribution of p' """' X(t, x,p'). Then, by the preceding (cp, Pt(x, · )) = urp(t,x) for cp E Cg"(lRn;lR), and from this it follows that (t,x) """'Pt(x, ·) is continuous. In addition, we now know that, for any lP' which solves the martingale problem for L on (JRn; lR),

c.;

Hence, Pt(p(s), ·)is the lP'-conditional distribution ofp """'p(s+t) given .68 • In particular, this, together with lP' (p( 0) = x) = 1, implies that lP' satisfies (1.4.1) with J.L = 8x, and so we have now proved that Q is indeed the one and only solution lP'~ to the martingale problem for Lon C.?(JRn; JR) starting from x. Finally, to check that Pt(x, ·) satisfies the Chapman-Kolmogorov equation, and is therefore a transition probability function, we use (cp, Ps+t(X,

·))=!Ell'~ [cp(p(s + t)) J

=IEil'q(cp,Pt(P(s), ·))] = j(cp,Pt(Y, ·))Ps(x,dy).

D

§3.1.6. Exercises EXERCISE 3.1.29. Suppose that x """'lP'x is a map taking an x E lRn to a probability measure lP'x on (D([O,oo);lRn),.B), and assume that x"""' JEil'x[] is continuous for all .B-bounded continuous 's. Using part (iii) of Exercise 2.3.5, show that x """' JEil'x [] is measurable for each .B-measurable which is either bounded or non-negative.

95

3.1 ITO'S BASIC CONSTRUCTION

3.1.30. Here is the discrete version of Gronwall's inequality which we used to end the proof of Lemma 3.1.16. Let J E z+, and suppose that { Uj }i\ oo

xEJRn

~ RJ1 + lxl 2 ) = 0.

EXERCISE 3.1.32. There is one case in which (3.1.22) presents no problems and does indeed determine X(· ,x,p'). Namely, consider the case in which n' =nand M' = 0, and take a= I and F = 0. Note, for IP0 -almost every p', p' = w( · , p') and therefore p' is continuous. Thus, we may and will think of IP0 as a probability measure on C([O, oo); ffi.n), in which case p' = w( · ,p') for every p'. Show that, for each p' E C([O, oo); ffi.n), XN ( ·, x,p') converges uniformly on compacts to the one and only solution X ( · , x, p') to the integral equation

X(t,x,p') = x + p'(t)

+lot b(X(T,x,p')) dT.

Further, check that p' ""'X(·, x,p') is continuous as a map of C([O, oo); ffi.n) to itself with the topology of uniform convergence on compacts.

96

3 ITO'S APPROACH IN THE EUCLIDEAN SETTING

3.2 When Does Ito's Theory Work? In §3.1 we assumed that the quantities in (Hl)-(H3) had been handed to us and proceeded to show how Ito's method led to a construction of a solution to the martingale problem for the operator Lin (3.1.5). However, in practice, someone hands you the operator L, and your first problem is to construct functions (J, b, and F and find a Levy measure l'v1' so that they not only give rise to the operator L but also satisfy the conditions in (Hl )-(H3). That is, besides satisfying (Hl)-(H3), we need that a(x) = (J(X)(J(x) T and (3.1.8) holds. The purpose of the section is to shed some light on this problem.

§3.2.1. The Diffusion Coefficients: By far the most satisfactory solution to the problem just raised applies to the diffusion coefficients. Namely, what we are seeking are conditions under which a symmetric, non-negative definite, operator-valued function x E IR" 1------7 a(x) E Hom(!Rn; IR") admits the representation a(x) = (J(X)(J(x)T for some Lipschitz continuous function (J : !Rn ----+ Hom(IRn'; IRn), and this is a question to which the answer is quite well understood. In what follows, a : !Rn ----+ Hom(IRn; IRn) will be a symmetric, nonnegative definite operator-valued function, and we will use a~ (x) to denote the symmetric, non-negative definite square of a(x). LEMMA 3.2.1. Assume that a is positive definite at x, and let Amin(x) be the smallest eigenvalues of a( x). Then a~ is differentiable at x and

for

all~

E !Rn

PROOF: Use a and (3 to denote Amin(x) and ffa(x)ffop, respectively, and set

m(y)

=

a(y) -I, (3

y E !Rn.

Then, because y -v-+ Amax(Y) and y -v-+ Amin(Y) are continuous, we can find an r > 0 such that = sup{ffm(y)flop: fy- xl:::; r} < 1. Thus, for y E BJRn ( x, r),

e

(*) where

0) ={ ~(~-1)···(~-£+1) £!

if£=0 iU

~

1

97

3.2 WHEN DOES ITO'S THEORY WORK?

is the coefficient of r/ in the Taylor's expansion of (1 + 'T])! for 1"71 < 1. To check the validity of (*), first observe that the right hand side is symmetric and then act the right hand side twice on any eigenvector of a(y). Knowing(*), one sees that, for any~ E lRn, (o~)xa! exists and is given by

Hence, since for lxl < 1,

(-1)£- 1 (~) 2 0 for I! 2 1 and 2::~ 1!(~)(-x)£- 1 = d:(1- x)!

IIU7~)xa! llop:::; (3-! ll(8~)xallop f/!0) (-llm(x)ll)£- 1 €=1

(3-! II(8E)xallop 2(1- llm(x)llop)!. Finally, because 0:::; I - a~) estimate follows. 0

:::;

(1- ~)I, llm(x) Ilop :::; 1- ~'and the required

Obviously, Lemma 3.2.1 is very satisfactory as long as a is bounded below by a positive multiple of the identity. That is, in the terminology of partial differential equations, when 2::~:]= 1 aij(x)oioJ is a uniformly elliptic operator. However, one of the virtues of the probabilistic approach to partial differential equations is that it has the potential to handle elliptic operators which are degenerate. Thus, it is important to understand what can be said about the Lipschitz continuity of a~ when a becomes degenerate. The basic fact which allows us to deal with degenerate a's is the following elementary observation about non-negative functions. Namely, if f E C 2 (lR; [0, oo)), then

(3.2.2)

IJ'(t)l:::; V2IIJ"IIu J(t),

t E JR.

To see this, use Taylor's theorem to write 0 :::; f(t +h) :::; f(t)

+ hf'(t) +

~ llf"llu for all hE JR. Hence lf'(t)l:::; + ~llf"llu for all h > 0, and so (3.2.2) results when one minimizes with respect to h. 2

h- 1 f(t)

LEMMA 3.2.3. A?sume that a is twice differentiable and that

Then

98

3 ITO'S APPROACH IN THE EUCLIDEAN SETTING

PROOF: First observe that it suffices to handle a's which are uniformly positive definite. Indeed, given the result in that case, we can prove the result in general by replacing a with a+ El and then letting E ""'0. Thus, we will, from now on, assume that a(x) 2: El for some E > 0 and all x E ffi.n. In particular, by Lemma 3.2.1, this means that a~ is differentiable and that the required estimate will follow once we show that

(*) for all x E ffi.n and ~ E §n~ 1 . To prove (*), let x be given, and work with an orthonormal coordinate system in which a(x) is diagonal. Then, from a= da~ and Leibnitz's rule, one obtains

Hence, because y'a + V/3 2:

va + /3 for all a, f3 2: 0,

To complete the proof of(*), set

and apply (3.2.2) to get

which, in conjunction with the preceding, leads first to

and then to (*).

0

REMARK 3.2.4. The preceding results might incline one to believe that there is little or no reason to take any square root other than the symmetric, nonnegative definite one. Indeed, if either a is uniformly positive definite or all one cares about is Lipschitz continuity, then there really is no reason to consider other square roots. However, if one is dealing with degenerate a's and needs actual derivatives, then one can sometimes do better by looking at other square roots. For example, consider a bounded, smooth a : ffi. 2 -----+

99

3.2 WHEN DOES ITO'S THEORY WORK? Hom(~ 2 ;~ 2 ) such that a(x)

=(xi+

x~)IIF.2 for x E BIF.2(0,1).

Clearly a~ (x) = xi+ x~ I for x E BIF.2 (0, 1), and so, although a~ is Lipschitz, it is not continuously differentiable at the origin. On the other hand, a(x) =

J

a(x)a(x)T on BJF.2(0, 1) if a(x)

= ( -xz XI

Xz ) there, and a is smooth on

-x1

BJF.2(0,1). Finally, it should be recognized that, even if a is real analytic as a function of x, it will not in general be possible to find a smooth a such that a= aa T. The reasons for this have their origins in classical algebraic geometry. Indeed, D. Hilbert showed that it is not possible to express every non-negative polynomial as a finite sum of squares of polynomials. After combining this fact with Taylor's theorem, one realizes that it rules out the existence of a smooth choice of a. Of course, the problems arise only at the places where a degenerates: away from degeneracies, as the proof of Lemma 3.2.1 shows, the entries of a~ are analytic functions of the entries of a. §3.2.2. The Levy Measure: Although, as we have just seen, Ito's theory handles the diffusion term in L with ease, his theory has a lot of problems with the Levy measure term. Namely, although, as we are about to show, it is always possible to represent a measurable map taking x E ~n into a Levy measure M(x, ·) as M(x, ·) = F(x, · )*M' for appropriate choices of Levy measure M' and measurable map F, I know of no general criterion in terms of the smoothness of x -v-? M(x, ·) which guarantees that this representation can be given by an F satisfying (H2). THEOREM 3.2.5. Assume that n' ;::: 2, and let M' be the Levy measure on ~n' given by M'(dy) = lJF.n'\{O}IYI~n~l dy. Given a measurable map x -v-7 M(x, ·) from x E ~n to Levy measures M(x, ·) on ~n, there exists a measurable map F : ~n x ~n' ----+ ~n with the property that M(x, ·) = F(x, · ).M' for each x E ~n. PROOF: Because this result will not be used in the sequel, only an outline of its proof will be given here. The technical basis on which the proof rests is the following. Given a Polish space 0 (i.e., a complete separable metric space), think (cf. §3.1 in [36]) of M 1 (0) as a Polish space with the topology of weak convergence. Next, suppose that X andY are a pair of Polish spaces, and set 0 =X x Y. Then there is a measurable map (P, y) E M 1 (0) x Y f----+ Py E M 1 (X) with the property that, for all En-measurable functions t.p : 0 ----+ [0, oo),

L

t.p(w) P(dw) = [

(L

t.p(x, y) Py(dx)) Py(dy),

3

100

ITO'S APPROACH IN THE EUCLIDEAN SETTING

where Py is the marginal distribution of P on Y (i.e., Py(dy) = P(X x dy)). This result follows rather easily from the well known result which guarantees the existence of regular conditional probability distributions for probability measures on a Polish space. Namely, set I:= {X X r: r E By}. Then Theorem 5.1.15 in [36] says that, for each P E M 1(D), there is a I:measurable map w E n t-----> Pw E Ml(n) with the property that, for each

where

oo, use Taylor's theorem to see

IR(OI::::; 2 1;1 3

for

I~ I : : ; ~·

Hence, for p' E SN(t, E),

xN (t,x,p')

~ xcxp (p'(t)- ~ fo t!.;:,p'(l)' + EN(t,p'))

where IEN(t,p')l ::::;

~E

f

D.t:,p'(t) 2 .

m=O

But (cf. Exercise 4.1.11 in [36]),

L

D.t:,p' (t) 2

---+

t for JP>0 -almost every p1 ,

m=O

and so the preceding leads to the conclusion that lim XN(t,x,p') = xep'(t)-1

N

for JP>M'-almost every p1 •

--->00

In other words, Ito says that the solution to (3.3.4) is X(t, x,p') = xeP'(t)-1, which is markedly different from what Bachelier got, whether or not p' is smooth. Without further input from the model, it is impossible to say which of these two is the "correct" one from an economic standpoint. The reason why Ito's is the one adopted by economists is that, in order to "avoid an arbitrage opportunity," economists want X(t) to be a martingale. Be that as it may, from a purely mathematical standpoint, both interpretations make sense and their difference should be sufficient to give any mathematician pause.

§3.3.3. A Geometric Example: Here is another example which displays the same sort of disturbing characteristics as the preceding. Taken= 2 = n', band F both identically 0, and

When lxl =I 0, cr(x)~ is 1/\ lxl 2 times the orthogonal projection of~ in the direction perpendicular to x. In particular, if p' is smooth and we interpret

(*)

X(t, x,p') = x + 1t cr(XN (T, x,p')) dp'(T)

3.3 SOME EXAMPLES TO KEEP

109

IN MIND

ala Riemann, then d[X(t,x,p')[ 2 = 2(XN(t,x,p'),a(XN(t,x,p')dp'(t))Rn = 0, and so [X (t, x, p') I = [x[ for all t 2': 0. Thus, if Ito's procedure were some sort of extension to non-smooth p''s of the classical solution when p' is smooth, then [X(t,x,p'[ would have to equal[x[ for lP'0 -almost every p', and this is simply not the case. Indeed, for m2-N ::; t::; (m + 1)2-N, because a(XN(m2-N,x,p'))XN(m2-N,x,p') = 0 (cf. (3.3.5)),

[XN (t, x,p')[ 2

-

[XN (mTN, x,p'W

= ( xN (t, x,p') + XN (mTN, x,p'), a(XN (mTN' x,p') )~;;;p'(t) )JRn

= ( XN (t, x,p')- XN (mTN, x,p'), a(XN (mTN, x,p'))~;;;p'(t) )Rn =

la(XN (mTN, x,p') )~;;;p'(t) 12 .

ov-7 [XN(t,x,p')[ is nondecreasing. Hence, if [x[ = 1, then a(XN([t]N,x,p')) is orthogonal projection onto XN([t]N,x,p')..L and so the JP'0 -distribution of YN(t,p') = la(XN([t]N,x,p'))~~p'(t)l 2 is that of the

In particular, t

square of a centered Gaussian whose variance is t - [t]N. Furthermore, if ?N(t,p') = yN(t,p')- (t- [t]N), then JEIP'" [YN(t) I B[t]N] = 0, and so

[2N t]+l

L

JEIP'o [YN(t 1\ mTN) 2 ]

::;

3(2Nt + 2)T 4 N _____, 0

m=O

as N ---+ oo. In other words, for lP'0 -almost every p', [X (t, x, p') [2 = 1 + t, which is very different from what we would get if Ito's interpretation of (*) were an extension of a Riemannian interpretation. REMARK 3.3.6. The last example, and the one preceding it, highlight one of the pitfalls in Ito's theory. Namely, his whole theory derives from the consideration of Levy processes, and Levy processes are inextricably tied to the Euclidean coordinates in which they are described. Anything other than a linear changes of coordinates will render any non-deterministic Levy process unrecognizable. In Chapter 8, we will introduce a variant of Ito's theory which overcomes, at least partially, this problem.

110

3 IT0 sE[O,TJ xE!Rn

tE[O,h]

116

4

FURTHER CONSIDERATIONS

4.2 Differentiability In the previous section we showed that Ito's method produces maps x E lRn ~-----+ lP' x E M1 ( D ([0, oo); lRn)) which are continuous and so the transition probability function (t,x) E [O,oo) x lRn ~-----+ Pt(x, ·) E MI(JRn) is continuous. the Markov semigroup {Pt : t -2: 0} given by 3

(4.2.1) Ptcp(x) =

1,

cp(y) Pt(x, dy),

(t, x) E [0, oo) x lRn & cp E B(lRn; JR).

leaves Cb(lRn; JR) invariant. Our goal in this section is to find conditions under which {Pt: t -2: 0} will leave C~(JRn;JR) invariant fork's other than 0. Of particular interest to us will be k = 2. Indeed, when C~(lRn; JR) is {Pt : t -2: 0}-invariant we will be able to show (cf. Corollary 4.2.6 below) that the measure lP'x = lP'~o-,b,F,M') is uniquely determined by the fact that it solves the martingale problem for the operator Lin (3.1.5) on c;(lRn; JR).

§4.2.1. First Derivatives: Throughout this subsection, we will be replacing (H2) and (H3) of §3.1.2 by

(H2) 1 F( ·, y') (a)

(b)

E C 1(JRn; JR) for each

y' E lRn \ {0} and

r IF(x, y'W M'(dy') 0, sup r (w(x, y')l2q +II aaF (x, y')ll2q) M'(dy') < }]Rn lim sup

r""O xEIR"

xEIR"

11 12 1+ X

JB

ll-.e for any ).. E R In particular, this means that

lEIP'[eiie(t)l]::; e~A2t(1EIP'[Ee(t)

+ E_e(t)J) = 2e~A2t,

where A is the uniform upper bound on IB( t, w) I· At the same time, by Taylor's theorem,

and

IE>-.e(t) + ~;>-.e(t)- 2 _

(Ie(t)2 _ Ae(t)) I ::;

~eiie(t)l

for 0 < ).. ::; 1. Hence, by Lebesgue's dominated convergence Theorem, we get the desired conclusion after letting ).. "" 0. D

131

5.1 BROWNIAN STOCHASTIC INTEGRALS

THEOREM 5 .1.11. There is a unique linear, isometric map () E 8 2 (lP'; ffi.n) ~ Ie E M 2 (lP'; IR) with the properties that Ie is given by ( 5.1. 7) for each () E S8 2 (JP>;JR). Moreover, given 01, ()2 E 8 2 (JP>;IR), (5.1.12)

is a martingale. Finally, if() E 8 2 (JP>;IR) and Ee(t) is defined as in (5.1.10), then ( Ee (t), Ft, lP') is always a supermartingale and is a martingale if Ae (T) is bounded for each T E [O,oo). (See Exercises 5.1.24 and 5.3.4 for more refined information.) PROOF: The existence and uniqueness of() -v-+ Ie is immediate from Lemma 5.1.8. Indeed, from that lemma, we know that this map is linear and isometric on 88 2 (JP>; ffi.n) and that 88 2 (lP'; ffi.n) is dense in 8 2 (JP>; ffi.n). Furthermore, Lemma 5.1.8 says that (Ie(t) 2 - Ae(t), Ft, JP>) is a martingale when () E S8 2 (JP>; JR), and so the general case follows from the fact that Iek(t) 2 - Aek(t)---+ Ie(t) 2 - Ae(t) in L 1 (JP>;IR) when ()k---+ ()in 8 2 (JP>;JR). Knowing that (Ie(t) 2 - Ae(t), Ft, lP') is a martingale for each() E 8 2 (JP>; ffi.n), we get (5.1.12) by polarization. That is, one uses the identity

Finally, to prove the last assertion, choose {fh}1 ~ S8 2 (JP>;IR) so that fh ---+()in 8 2 (JP>;ffi.n). Because (Eek(t),Ft,JP>) is a martingale for each k and Eek (t) ---+ Ee(t) in JP>-probability for each t 2 0, Fatou's lemma implies that (Ee(t), Ft, JP>) is a supermartingale. Next suppose that () is uniformly bounded by a constant A < oo, and choose the ()k 's so that they are all uniformly bounded by A as well. Then, for each t E [0, T],

and so Eek(t)---+ Ee(t) in L 1 (JP>;IR) for each t E [O,T]. Hence, we now know that (Ee(t),Ft,lP') is a martingale when() is bounded. Finally, if Ae(t,w) ~ A(t) < oo for each t E [O,oo) and Om(t,w) l[o,mJ(IO(t,w)I)O(t,w), then Eem(t)---+ Ee(t) in JP>-probability and JEif'[Eem(t) 2] ~ eA(t) for each t 2 0, which again is sufficient to show that (Ee(t), Ft, lP') is a martingale. D

=

REMARK 5.1.13. With Theorem 5.1.11, we have completed the basic construction in Ito's theory of Brownian stochastic integration, and, as time

132

5 ITO'S THEORY OF STOCHASTIC INTEGRATION

goes on, we will increasingly often replace the notation Ie by the more conventional notation (5.1.14) Because it recognizes that Ito theory is very like a classical integration theory, (5.1.14) is good notation. On the other hand, it can be misleading. Indeed, one has to keep in mind that, in reality, Ito's "integral" is, like the Fourier transform on IR, defined only up to a set of measure 0 and via an L 2-completion procedure. In addition, for the cautionary reasons discussed in §§3.3.2-3.3.3, it is a serious mistake to put too much credence in the notion that an Ito integral behaves like a Riemann-Stieltjes integral.

§5.1.3. Stopping Stochastic Integrals and a Further Extension: The notation J~ (0( T), d/3( T) )JRn for Ie(t) should make one wonder to what extent it is true that, for (1 ::::; (2,

(5.1.15)

Of course, in order for the right hand side of the preceding to even make sense, it is necessary that 1[.p~~~;->.) = ~· For lfl{fl_l\

2

p E [1,p>.], set !'(A,p) =~'note that Ef 0 = (e'(>.,p)Ia)

1-)., 2

2

E;0 , and

conclude that, for any stopping time(, Ell' [E>.e(T !\ .(1->.)

E~~'[Ee(T)]

).,2

for all A E (0, 1). After letting A/ 1, conclude first that E~~'[Ee(T)] = 1 and then that (Ee(t), :Ft, IP') is a martingale. The fact that (5.3.6) implies (Ee (t), :Ft, lP') is a martingale is known as N ovikov 's criterion.

CHAPTER 6

Applications of Stochastic Integration to Brownian Motion

This chapter contains a highly incomplete selection of ways in which Ito's theory of stochastic integration, especially his formula, has contributed to our understanding of Brownian motion. For a much more complete selection, see Revuz and Yor's book [27].

6.1 Tanaka's Formula for Local Time Perhaps the single most beautiful application of Ito's formula was made by H. Tanaka when (as reported in [22]) he applied it to prove the existence of local time for one-dimensional Brownian motion. Before one can understand Tanaka's idea, it is necessary to know what the preceding terminology means. Thus, let (,B(t),Ft,IP') be a one-dimensional Brownian motion. What we are seeking is a function I! : [0, 00) X lR X n f---+ [0, oo) with the properties that (a) (b) (c) (d)

for for for for

each y E IR, (t,w) ""'f!(t,y,w) each (y,w) E lR x n, f!(O,y,w) each wE f2, (t,y) ""'f!(t,y,w) !P'-almost every wEn and all

is progressively measurable; = 0 and/!(· ,y,w) is nondecreasing; is continuous; t E [0, oo),

fat l[a.b)(,B(r,w)) dr = 1b f!(t,y,w)dy,

for all - oo oo

= il>(,B(t))- ii>(O)- lim if>(,B(t))- if>(O)

-lot

so that

'P(~{'j,)C(t, mTN)

L rt ii>'(mTN)l[(m-1)2-N,m2-N)(,8(T)) d,B(T)

N-->= mEZ

=

L

mEZ

+ 1)2-N)

lo

if>'(,8(T)) d,8(T)

since

and

Ew [

(fo' (' (Mr)) -' ([fi(r)]N))

dfi(r) )']

=JEll> [lot(if>'(,8(T)) -if>'([,8(T)]N)f dT]

~0

as N-+ oo. Thus, we now know that

~

{ tp(y)C(t,y)dy = ii>(,B(t))- ii>(O)- {t ii>'(,B(T)) d,8(T)

h

2}TII.

IP'-almost surely. At the same time, by Ito's formula, the right hand side of the preceding is also IP'-almost surely equal to ~ J~ 'P (,8( T)) dT, and so we also know that, for each 'P E Cgc'(JR; JR)

l

tp(y)C(t,y,w)dy

=~lot tp(,B(T,w)) dT = ~

l

'P(Y)J-Lt,w(dy),

t 2': 0,

for alllP'-almost every w. Starting from this, it is an elementary exercise in measure theory to see that e has the property in (d). We summarize our progress in the following theorem, in which the concluding equality is Tanaka's formula.

156

6 APPLICATIONS TO BROWNIAN MOTION

THEOREM 6.1.2. Given a one dimensional Brownian motion (,B(t),Ft,lP') there exists a lP'-almost surely unique function £ : [0, 00) X IR X n ----+ [0, 00) with the properties described in (a)-( d) above. Moreover, for each y E IR, (6.1.3)

l(t, y)

- 2 - =,B(t)Vy-OVy-

Jot

l[y,oo)(.B(r))d,B(r),

t

E

[0, oo)

lP'-almost surely. REMARK 6.1.4. It is important to realize that the very existence of l(t, . ) is a reflection of just how fuzzy the graph of a Brownian path must be. Indeed, if p : [0, t] ----+ IR is a smooth path, then its occupation time distribution J-Lt will usually have a nontrivial part which is singular to Lebesgue measure. For example, if the derivative p of p vanishes at some T E [0, t], then it is easy to see that this prevents J-Lt from being absolutely continuous with respect to Lebesgue measure. Moreover, even if p never vanishes on [0, t], and therefore J-Lt is absolutely continuous, the Radon-Nikodym derative of J-Lt will have discontinuities at both min[o,t] p( T) and max[o,t] p(T). The way that Brownian paths avoid these problems is that they are changing directions so fast that they, as distinguished from a well behaved path, never slow down. Of course, one might think that their speed could cause other problems. Namely, if they are moving so fast, how do they spend enough time anywhere to have their occupancy time recordable on a Lebesgue scale? The explanation is in the fuzz alluded to above. That is, although a Brownian path exits small intervals too fast to have their presence recorded on a Lebesgue scale, it does not depart when it exits but, instead, dithers back and forth. From a more analytic point of view, the existence of the Brownian local time functional is a stochastic manifestation of the fact that points have positive capacity in one dimension. This connection is an inherent ingredient in Tanaka's derivation. In fact, his argument works only because the the fundamental solution, x Vy, is locally bounded, which is also the reason why points have positive capacity in one dimension. In two and more dimensions, the fundamental solution is too singular for points to have positive capacity.

§6.1.2. Some Properties of Local Time: Having constructed£, we now want to derive a couple of its elementary properties. To begin with, suppose that f : [0, oo) x IR x n ----+ IR is a bounded, measurable function with the properties that (t,y) "'""'f(t,y,w) is continuous for lP'-almost every w. Using Riemann sum approximations, one can easily see that, for lP'-almost every w, (6.1.5)

1t

f(r,,B(r,w),w) dr =

l (1t

f(r,y,w)l(dr,y,w)) dy,

t

~ 0,

6.1

157

TANAKA'S FORMULA FOR LOCAL TIME

where P(dT, y, w) denotes integration with respect to the measure on [0, oo) determined by the nondecreasing function P( · , y, w). In particular, if 77 E C 00 (~; [0, ll) vanishes on [-1, 1] and equals 1 off (2,2), 'lj; E Cg"(~; [O,oo)) vanishes off [-1, 1] and has total integral!, and

then, by (6.1.5), JP>-almost surely,

whenever 0 < f < R- 1 . Hence, after first letting we see that, for each y E ~. (6.1.6)

P({T ~ 0: (3(T)

=f. y},y) = 0

f

0 and then R /' oo,

~

JP>-almost surely.

An interesting consequence of (6.1.6) is that, for each y, P( ·, y) is JP>-almost surely a singular, continuous, nondecreasing function. Indeed, £( ·, y) is JP>almost surely supported on {T : (3( T) = y}, whereas JEIT' [Leb( {T: (3(T)

= y})] = JEIT'

=

[1

1

00

00

l{y} (f3(T))

JP>({J(T) =

dT]

y) dT = 0.

In order to take the next step, notice that, from (6.1.3), f3(t) A 0 = f3(t)- (3(t) V 0 =

1t

lc-oo,o) (f3(T)) d(3(T)-

P(t~ O).

Hence, after subtracting this from (6.1.3), we find that (6.1.7)

j(J(t)j =

1t

sgn(f3(T)) d(3(T)

+ P(t, 0),

where, for the sake of definiteness 1 we will take sgn(O) = 1. Among the many interesting consequences of (6.1. 7) are the two discussed in the next statement, both of which were discovered by P. Levy. For much more information about local time, the reader should consult the books [15] and [27]. 1 It really does not matter how one takes sgn at 0 since, lP-almost surely, f3 spends no time there.

158

6 APPLICATIONS TO BROWNIAN MOTION

THEOREM 6.1.8. With JII'-probability 1, £(t,0) > 0 for all t > 0. Furthermore, if, fort > 0 and E > 0, NE(t) denotes the number of times 1,81 I [0, t] downcrosses 2 the interval [0, E], then (6.1.9)

lim IlENE(·) - £( ·, 0) II [o t] = 0

E

".0

lJI'-almost surely for each t

'

PROOF: To prove the first statement, set B(t) view of (6.1. 7), it suffices for us to prove that

(*)

JII'(Vt > 0 3r E (O,t] B(r)

=

> 0.

J~sgn(,B(r))d,B(r). In

< 0) =

1.

For this purpose, we will first show that (B(t),.Ft,lF') is a Brownian motion. To do this, apply Ito's formula to see that (exp ( >.B(t)-

>.~t) ,.Ft,lF')

is a martingale for every >. E C. In particular, this means that, for all 0::::: s < t,

lEIP'[ev=lE(B(t)-B(s)) I.Fs]

= e-" 2 (~-s)

lJI'-almost surely,

which, together with B(O) = 0, is enough to identify the distribution of w -v-7 B( · , w) as that of a Brownian motion. Returning to (*), there are several ways in which it can be approached. One way is to apply Blumenthal's 0-1 law (cf. Theorem 8.1.20 in [36]) which says that, because B is a Brownian motion under JII', any set from nt>oa({B(r) r E [O,t]}) has JII'-probability 0 or 1. In particular, this means that

JII'(V t > 0 3r E [0, t] B(r) < 0) E {0, 1}

Hence, since JII'(V t > 0 3r E [0, t] B(r) < 0) :::: lim JII'(B(t) < t".O

=

limlJI'(3r E [0, t] B(r) < 0)

t".O

o) = -21 ,

(*) follows. Another, less abstract approach is to use JII'(3r E [0, t] B(r) < 0) 2: lim JII'(3r E [6, t] B(r) < 0) 6".0

2 That is, N,(t) 2: m if and only if there exist 0 :::; 71 \,6(72£-1)\ 2: E and ,6(72£) = 0 for 1:::; P:::; m.

< 72 < · · · < 72rn :::; t

such that

6.1

159

TANAKA'S FORMULA FOR LOCAL TIME

where, in the passage to the last line, we used the reflection principle (cf. (4.3.5) in [36]) when we replaced IP'(:J T E [0, t- J] B(T):::; -x) by 21P'(B(t6) :::; -x) for x < 0. In order to prove (6.1.9), we introduce the stopping times { (e : g :::=: 0} so that (o = 0, (2£+ 1 = inf{t :::=: (2£ : I,B(t)l :::=: E} for g :::=: 0, and (2£ = inf{t :::=: ( 2e_ 1 : ,B(t) = 0} for g :::=: 1. Next, set Ie = [(2e-1, (2e), and observe that

=

I:(I,B(t 1\ (2e) 1-I,B(t 1\ (2£-1) I) £=1

=

= -ENc(t) + (I,B(t)l- E) L l1£(t). £=1

At the same time, because ,B(T) =I= 0 when T E Ie and therefore e(t 1\ (2£) e(t 1\ (2£-1), (6.1.7) tells us that f:(I,B(t 1\ (u)I-I,B(t 1\ (2e-1)1) £=1

= where Je

I,B(t)l-

~

= [(2e, ( 2£+1).

1t

=

f 1t°

=

l1£(T)sgn(,B(T)) d,B(T)

£=1

lh(T)sgn(,B(T)) d,B(T)- f(t, 0),

Hence, we have now shown that

ENc(t)- f(t, 0) = -I,B(t)l

f

1J£(t)- E

£=0

f

1J£(t)

£=0

+

1t

ec(T) d,B(T),

where ec(T) = L~o lJg(T)sgn(,B(T)). Finally, notice that t E Je ==} I,B(t)l :::; E, and so the absolute values of the first two terms on the right are each dominated by E. As for the stochastic integral term on the right, observe that IOc(T)I:::; 1[- 0. But

(m + 1)- 2 :::; E < m- 2 ==? (m + 1)- 2N1jrn2 :SEN< :S m- 2N1/(rn+1)2, and so the asserted result follows.

0

It should be clear that the results in Theorem 6.1.8 are further evidence of the "fuzziness" alluded to in Remark 6.1.4.

160

6 APPLICATIONS TO BROWNIAN MOTION

§6.1.3. Exercises

EXERCISE 6.1.10. Let (,B(t), Ft, JP>) be a one-dimensional Brownian motion, and let £( ·, 0) be its local time at 0.

(i) Show that f(t, 0) is measurable with respect to the IP'-completion Yt of a({I,B(T)I: T E [O,t]}) and that a({sgn(,B(T)): IP'-completion (}of a({I,B(T)I: T ~ 0}).

T

~ 0}) is IP'-independent of

(ii) Set B(t) = J~sgn(,B(T))d,B(T), and show that (B(t),(}t,IP') is a onedimensional Brownian motion. (iii) Refer to part (ii), but think oft""" B(t) as being a Brownian motion relative of the a-algebras t""" Ft, instead oft""" Yt· Show that

,B(t) = 1t sgn(,B(T)) dB(T),

t

~ 0,

but that ,B(t) is not measurable with respect to the IP'-completion of a ({B( T) :T~O}).

The conclusion drawn in (iii) demonstrates that it is important to make a distinction between the notion of strong and weak solutions to a stochastic differential equation. Without going into details, suffice it to say that strong solutions are those which are measurable functions of the driving Brownian motion, whereas weak solutions are those which simply solve the stochastic differential equation relative to some family of a-algebras with respect to which the driving process happens to be a Brownian motion. For more details, see [27]. 6.2 An Extension of the Cameron-Martin Formula

In Exercise 5.1.20 we discussed the famous formula of Cameron and Martin for the Radon-Nikodym derivative Rh of the (Th)*IP' 0 with respect to IP'0 when h(t) = J~TJ(T)dT for some TJ E L 2 ([0,oo);1Rn). Because, on average, Brownian motion has no preferred direction, it is reasonable to think of the addition of h to a Brownian path p as the introduction of a drift. That is, because h does have a velocity, it forces p+ h to have a net drift which p by itself does not possess. Thus, the Cameron-Martin formula can be interpreted as saying that the introduction of a drift into a Brownian motion produces a path whose distribution is equivalent to that of the unperturbed Brownian motion as long as the perturbation has a square integrable derivative. Further, as I. Segal (cf. Exercise 5.2.38 in [36]) showed, only such perturbations leave the measure class unchanged. In fact, ThiP'0 is singular to IP'0 if h fails to have a square integrable derivative. The intuitive reason underlying these results has to do with the "fuzz" alluded to in Remark 6.1.4. Namely,

6.2 AN EXTENSION OF THE

CAMERON~MARTIN

FORMULA

161

because the "fuzz" is manifested in almost sure properties, the perturbation must be smooth enough to not disturb the "fuzz." For example, the drift must not alter JP>0 -almost sure properties like 2N

lim

N--+oo

L

lp(mTN))- p((m- 1)TNI 2 = n.

m=l

In this section, we will give an extension of their formula, one which will allow us to introduce Radon-Nikodym derivatives to produce a random "drift".

§6.2.1. Introduction of a Random Drift: The key result here is the following application of Ito's formula. THEOREM 6.2.1. Suppose that (3: [O,oo) X C([O,oo);lRn) - - t }Rn is a {Bt: ~ 0}-progressively measurable function, and assume that JP> is a probability measure on C([O, oo); JRn) for which ((3(t), B/', 1P)3 is a Brownian motion. If (} E 8foc(JP>;1Rn) and (cf. (5.1.10)) JEIP'[Eo(T)] = 1 for all T ~ 0, then there exists a unique Q E M 1 (C([O,oo);1Rn)) such that t

for all T

~

-IP'

0 and A E Br .

Furthermore, if 'TJ is a {Bt : t ~ 0}-progressively measurable satisfying for \'TJ(T)\ dT < oo JP>-almost surely for all T E [0, oo) and if X(t) = J~ ry(T) dT, then, for each cp E C 1 ' 2 (lRm x }Rn; C), 4 (cp(X(t),(3(t)) -lot H'P(T)dT,BtiP',Q) H'P(t) =lot (('TJ(T),gradxcp)IR"'

with

+~~yep+ (0(T),gradycp)Rn)(X(T),(3(T))dT

is a local martingale.

PROOF: In order to prove the first assertion, note that, by part (ii) of Exercise 5.3.4, (Eo(t), Bt IP', !P) is a martingale. Hence, if the Borel probability measure QT is determined on BriP' by Q(A) = JEIP'[E11 (T), (3 E A] for all

-IP' A E Br , then the family {QT : T ~ 0} is consistent in the sense that, for each 0 :::; T1 < T2, QT1 is the restriction of QT2 to Br1 • Thus, by a minor

F

3 Given a probability measure lP defined on a u-algebra :F, is used to denote the IP-completion of :F. 4 In the following, the subscripts "x" and "y" are used to indicate to which variables the operation is being applied. The "x" refers to the first m coordinates, and the "y" refers to the last n coordinates. Also, Ll denotes the standard Euclidean Laplacian for Rn.

162

6 APPLICATIONS TO BROWNIAN MOTION

variant (cf. Theorem 1.3.5 in [41]) of Kolmogorov's extension theorem, there is a unique Borel probability measure Q whose restriction to Br is QT for each T ~ 0. To prove the second part, define

Because

and Q f Br « lP' f Br for each T ~ 0, we know that (R / oo Qalmost surely as R / oo. Thus, it is enough for us to show that, for each 'P E C 1 •2 (1Rm x !Rn;C), 0 ~ h < t2, A E Bhll'' and R > 0, -II'

-II'

(*) where

M(t)

= T]

=

lEQJ [S(T)VN F, (R > T]

= lEQJ [S(T A (R)VN F,

(R > T].

Using the second expression for V N, we see that, as N ---+ oo, V N ----+ Ee(T ;\ (R) in lP'~-probability. In fact, by taking into account the estimates in (5.1.25), one sees that this convergence is happening in £ 1 (lP'~; JR). At the same time, by using the first expression for V N, the same line of reasoning shows that VNS(T ;\ (R)----+ 1 in L 1 (Q;JR). Thus, we have proved that

lEI\£ [F, (R > T] and, after letting R /

dQ

r Br

=

Ee(T) dlP'~

= JE~~'~ [Ee(T)F, (R > T],

oo, we have proved that JEll'~ [Ee(T)]

r Br.

=

1 and that

0

COROLLARY 6.2.3. Let b: [0, oo) x JR" ----+ JRn be a locally bounded, measurable function. Then there exists a Q~ E M 1 (C([O,oo);1Rn)) with the property that

is a Brownian motion if and only if (cf the notation in Corollary 6.2.2)

where

R''(T,p)

=exp ( [ (b(r,p(r)), dp(r)) •• - ~ [ib(r,p(r))i' dr),

in which case dQ~ r Br = Rb(T) dlP'~ there is at most one such Q~.

r Br

for each T ;::: 0. In particular,

167

6.2 AN EXTENSION OF THE CAMERON-MARTIN FORMULA

REMARK 6.2.4. There is a subtlety about the result in Corollary 6.2.3. Namely, it might lead one to believe that, for any bounded, measurable b : !Rn ---+ IRn and JID0 almost every p E C ([0, oo); IRn), one can solve the equation X(t,p) = p(t)

(*)

+

1t

b(X(T,p)) dT,

t;::: 0,

and that the solution is unique. However, this is not what it says! Instead, it says that there is a unique QS E M1(C([O,oo);IRn)) with the property that JID0 is the QS-distribution of

There is no implication that p can be recovered from /3( · ,p). In other words, we are, in general, dealing here with the kind of weak solutions alluded to at the end of Exercise 6.1.10. Of course, if b is locally Lipschitz continuous and therefore the solution X(· ,p) to(*) can be constructed (e.g., by Picard iteration) "p by p," then the JID0 -distribution of p 'V'7 X(· ,p) is QS.

§6.2.2. An Application to Pinned Brownian Motion: A remarkable property of Wiener measure is that ifT E (O,oo), x,y E !Rn, and (6.2.5)

Pr,y(t) =: p(t AT)+

tAT

---r-(Y- p(T))

for p E C([O, oo);!R.n),

then the JID~ -distribution of p 'V'7 PT.y I [0, T] is the JID~ -distribution of p 'V'7 I [0, T] conditioned on the event p(T) = y. To be more precise, if F : C([O,oo);!Rn) ---+IRis a bounded, Br-measurable function and f E BJRn, then (cf. Theorem 4.2.18 in [36] or Lemma 8.3.6 below)

p

IEIP'o [F, p(T) E X

1 { r] = (27rT)~ }JRn

(

j

F(pr,y) JID 0 (dp)

) e-~ ly-xl 2

dy.

C([O,oo);JRn)

For this reason, p 'V'7 PT,y I [0, T] is sometimes called pinned Brownian motion, or, in greater detail, Brownian motion pinned to y at time T. The fact that (6.2.5) gives a representation of Brownian motion conditioned to be at y at time T is one of the many peculiarities (especially its Gaussian nature) of Brownian motion which sets it apart from all other diffusions. For this reason, it is interesting to find another representation, one

6 APPLICATIONS TO BROWNIAN MOTION

168

that has a better chance of admitting a generalization (cf. Remark 6.2.8 below). With this in mind, let t E [0, T) be given, notice that, by definition, for any bounded, Brmeasurable F,

lEil'~[Fjp(T) =y]

=IEI!'o [F'Yr-t(y-p(t))l, 'Yr(y-x)

where 'Yr ( ~ ) = ( 21fT ) -2 e- ""27 denotes the centered Gauss kernel on JRn with covariance T I~n. Next, recall that 1~1 2

n

•

8r'YT-r(Y- . ) + ~~'YT-r(Y- . )

= 0 on [0, T)

X ]Rn,

where ~ denotes the Euclidean Laplacian for lRn; and apply Ito's formula to conclude that "fr-t(y-p(t)) 'Yr(Y- x)

=

1-

t'Yr-r(y-p(r)) (p(r)-y,dp(r)) 'YT(Y- x) T- T ~n

Jo

fortE [0, T); and so, by Exercise 5.3.4, 'YT-t(Y- p(t)) 'YT(Y- x)

(6.2.6)

= exp(- t

Jo

(p(T~- y, dp(r)) T

T

~n

-

~

2

rt

Jo

lp(T~- y T

2 1

T

dT)

fortE [0, T). Furthermore, by the Chapman-Kolmogorov equation,

!Ell'~

['Yr-t(Y- p(t))l = { 'YT(Y- x) }~n

'Yt(~ _

x) 'YT-t(Y- 0 'YT(Y- x)

d~ =

1.

Hence, by Corollary 6.2.3, we have already proved a good deal of the following statement. THEOREM

> 0, there a continuous map (y,p) E lRn x Xy( · ,p) E C([O, T); JRn) such that

6.2.7. Given T

C([O, oo); JRn)

c--+

Xy(t,p) = p(t)- t Xy(r,p)- y dr Jo T-T

fortE [0, T).

Furthermore, for each x E JRn and 1P'~-almost every p, limt/T Xy(t,p) = y, and so there exists a unique 1P'~,x,y E M 1 (C([O,T];1Rn)) with the prop0 erty that the lP'T -distribution of p ---+ p I [0, T) is the same as the ~,x,y distribution of p -..-. Xy ( · , p) I [0, T). In fact, 1P'~,x,y (p(T) = Y) = 1,

169

6.2 AN EXTENSION OF THE CAMERON-MARTIN FORMULA

= (Tx)*IP'~,O,y-x' (x, y) .,...JP'~,x,y is continuous, and, if p(t) = p(T- t) for p E C([O, T]; lRn) and t E [0, T], then 1P'~,y,x is the 1P'~,x,y-distribution of p .,... p. Finally, for any BT-measurable F which is either bounded or nonnegative, 1EIP~,x,y [F] is the 1P'~-conditional expectation value JEIP~ [F p(T) = y] ofF given p(T) =yin the sense that

1P'~,x,y

I

IEIP~ [F I a (p(T))]

= IEIPT,x,v(T) [F] 1P'~-almost surely. PROOF: The existence and continuity of (y,p) .,... Xy( · ,p) f [0, T) follow easily from the standard Picard iteration procedure. Moreover, by uniqueness, it is easy to see that Xy( ·, TxP) = x + Xy-x( · ,p), and so the distribution of p .,... Xy( · ,p) under IP'x coincides with the IP'0 -distribution of p.,... TxXy-x(. ,p). Thus, if they exist, then ~,x,y = (Tx)*IP'~,O,y-x· By Corollary 6.2.3, (6.2.6), and the preceding discussion, we know that, for any t E [0, T) and A E Bt,

IEIP~

(*)

['YT-t (p(t) -

'YT(Y- x)

y) '

A] = IP'o (X (-) E A). x

Y

Our first application of (*) will be to show that the IP'~ distribution of p .,... Xy( · ,p) I (0, T) coincides with the 1P'g-distribution of p .,... Xx(T· ,p) I (0, T); and for this purpose, it suffices to observe that, by repeated application of(*), for any m ~ 1 and 0 < t1 < · · · < tm, IP'~ (Xy(it) E df;t, ... Xy(tm) E d~m) = 'Yt, (~t- x) · · ·"ftm-tm-1 (~m- ~m-th(Y- ~m) d6 'YT(Y- x) = IP'~ (Xx(T-

... d~m

tm) E dem, ... Xx(T- t1) E d6).

In particular, we have shown that

IP'~ (lim Xy(t) = y) = IP'~ (lim Xx(t) = y) t/'T t'\.0 and so we now know that 1P'~,x,y exists, IP'~,x,y(p(T) ~,y,x is the 1P'~,x,y-distribution of p.,... p.

= 1,

= y) = 1, and that

0 We next want to check the continuity of (x, y) .,... IP'T . To this end ' ,x,y remember (cf. the end of the first paragraph of this proof) that we know ~,x,y = (Tx)*IP'~,O,y-x· Hence, it is enough for us to show that y.,... 1P'~,O,y is continuous. Next, observe that, for any 8 E (0, 1),

IIXy'( · ,p)- Xy( · ,p)ll[o,T)

:S

(ly'- Yl + IIXy'( · ,p)- Xy( · ,p)ll[o,(t-8)TJ) log~ +iT (1-8)T

IXy,(T,p)T-

T

y'l dT +iT (1-8)T

IXy(T,p)T-

T

Yl

dT.

170

6 APPLICATIONS TO BROWNIAN MOTION

Since, for any

oE (0, 1) and

E

> 0, we know that

J,i~)Xy'(. ,p)- Xy(. ,p)ll[o,(1-oJTJ = o for all p, we will know that (**) once we show that, for each R lim sup

0 ~ 0 IYisR

But

iT

(1-o)T

i

> 0, JEIPo [IX (r)

T

Y

T-

(1-o)T

JEIPo [IXy (r)

-

Yl]

_--=-:_____::__:__:______::__:_;_

T-

dT

r

=

-

Yl]

r

dr

= 0.

1"T lEIP~ [IXo (r) - Yl d o

T,

r

and it is an easy matter to check that sup

sup

IYISRTE(O,~J

lEIP~ [IXo (r) - Yl] I r;;

yT

< oo.

After combining these, we know that (**) holds and therefore that (x, y) ""'* o . COnt'1nUOUS. lP'T,x,y 1S All that remains is to identify 1P'~,x,y as the conditional distribution of lP'~ I BT given p(T) = y. However, from(*), we know that, for any r E BJRn,

lEIP~ [An {p(T) E r}] = llP'T,x,y(A)'yT(Y- x) dy for all A E Bt when t E [0, T). Since the set of A E BT for which this relation holds is a 0'-algebra, it follows that it holds for all A E BT. D REMARK 6. 2. 8. Certainly the most intriguing aspect of the preceding result is the conclusion that limt/'T Xy(t,p) = y for 1P'~-almost every p. Intuitively, one knows that the drift term- X(}r:_}t-y is "punishing" X(t,p) for not being close to y, and that, as t / T, the strength of this penalty becomes infinite. On the other hand, this intuition reveals far less than the whole story. Namely, it completely fails to explain why the penalization takes this particular form instead of for example -2X(t,p)-y or- (X(t,p)-y) 3 • As a careful ' ' T-t T-t examination of the preceding reveals, the reason why - X(}r:_}t-y is precisely the correct drift is that it is equal to grad x (t,p) log 9T-t ( · , y) where gT ( · , y) is the fundamental solution (in this case 'YT ( · - y)) to the heat equation. It is this observation which allows one to generalize the considerations to diffusions other than Brownian motion.

171

6.2 AN EXTENSION OF THE CAMERON-MARTIN FORMULA §6.2.3. Exercises

EXERCISE 6. 2. 9. The purpose of this exercise is to interpret the meaning of (cf. Corollary 6.2.3) JElP'0 [Rb(T)] failing to be 1. Let b : JRn ~ }Rn be a locally Lipschitz continuous function. Choose 'ljJ E C 00 (1Rn;[0,1l) so that 'ljJ =: 1 on BJRn(0,1) and 'ljJ =: 0 off BJRn(0,2), and set bR(x) = 'ljJ(R- 1 x)b(x) for R > 0 and x E JRn. Given R > 0 and p E C([O,oo);lRn), define (R(P) = inf{t ~ 0: lp(t)l ~ R} and determine XR( · ,p) E C([O, oo); JRn) by the equation

XR(t,p) = p(t)

+

1t

bR(X(r,p)) dr t

E

[0, oo).

{i) Given 0 < R1 < Rz, show that XR 2 (t,p) = XR 1 (t,p) for 0 :S t :S (R 1 o XR 1 ( • ,p), and conclude that

Next, define the explosion time e(p) = limR/oo (R oXR( · ,p), and determine t E [0, e(p)) f - t X(t,p) E ]Rn so that X(t,p) = XR(t,p) for 0 :S t :S (R(p).

(ii) Define Rb as in Corollary 6.2.3, show that lElP'0 [Rb(T), (R > T] = IP'0 ((R o XR( ·) > T), and conclude that JElP'0 [Rb(T)] = JP>O(e > T). That is, JElP'0 [Rb(T)] is equal to the probability that X(· ,p) has not exploded by timeT. ExERCISE 6.2.10. When the drift b in Corollary 6.2.3 is a gradient, the Radon-Nikodym Rb(T) becomes much more tractable. Namely, apply Ito's formula to show that if b( t, x) = gradx U (t, · ) , then

(6.2.11)

Rb(T,p) = exp (u(T,p(T))- U(O,p(O))where V(t,x)

= lgradxU(t, · )1 2 +(at+ ~6-)U(t,x).

{i) Recall the Ornstein-Uhlenbeck process p'

and let «J!x denote the distribution of p ~ preceding, show that

lEQ"' [

..)

= eMt(t)- A22 \llto,tJfii~ 2 CI 0 ,ool;11!.nl. Using Ito's formula, as

in part (i) of Exercise 5.3.4, one sees that, for each ).. E C,

Thus, if (cf. (6.3.1)) J(ml(t)

R 0(t,>..)

= E(t,.A)

and

= Ij';;L(t) =

,;,Jj';;L and

RM+l(t,.A) =A

lot RM(T,>..)(f(T),dp(T))Jlf.n

for M ?: 0, then, by induction, one sees that M

E(t, .A)= 1 +LAm J(m)(t)

+ RM+l(t, )..)

m=l

for all M?: 0. Finally, if o: = 9\e(>..) and F(t) = J~ if(T)i 2 dT, then

JE.lP'o[JRo(t,>..)J2] = ei>-I2F(t)JE.lP'o[E(t,2o:)] = ei>-I2F(tl, and

JE.lP'o [JRM+l(t, .A)J2] = I.AI21t JE.lP'o [JRM (T, .A)J2]F(T) dT. Hence, the asserted estimate follows by induction on

}.,1

?: 0.

0

THEOREM 6.3.8. The span of lREB {J}~'L(oo): m?: 1 &

f

E L 2 ([0,oo);1Rn)}

is dense in L 2 (1P'0 ;JR). In particular, (6.3.6) holds. PROOF: Let lHI denote smallest closed subspace of L 2 (lP'0 ; JR) containing all constants and all the functions oo). By the preceding, we know that cosoiJ(oo) and sinolt(oo) are in lHI for all f E L 2 ([0,oo);1Rn). Next, observe that the space offunctions : C([O,oo);lRn) ___, lR which have the form = F(Ih (oo), ... , ltL (oo))

lj";;L (

for some L ?: 1, Schwartz class test function F : lR L ___, JR, and h, ... , h E L 2 ([0, oo); lRn) is dense in L 2 (1P'; lRn). Indeed, this follows immediately from the density in L 2 (1P'0 ; JR) of the space of functions of the form

p -v-7 F(p(tl), ... ,p(tL)),

180

6

where L ::0: 1, F : (!Rn)L

0:::; toD;JR) =

d~

J

if!(p)elsgi(oo,p)-4\ll(p- shj) JP'O(dp)ls=O'

where hj(t) = J~ gj(r) dr.

(ii) Define DP- : V---> V inductively so that DP-(+j) = Dj o D~'. When if! E V, use part {i) in Exercise 6.3.11 and (iv) above to show that

10

The terminology here comes from Euclidean quantum field theory.

184

6 APPLICATIONS TO BROWNIAN MOTION

and conclude that

+"' 00

o[ 0, set (R(w) = sup{t ~ 0: IMI(t,w):::; R}, observe 2 that (R is a stopping time, and set MR(t) = M(t A (R)· By Doob's stopping time theorem, (MR(t), :Ft, lP') is a continuous martingale. At the same time, IMRI(t, w) :::; R. Hence, by the preceding theorem,

is a continuous martingale. In particular, this means that

2 Remember that we have adopted { ( whether ( is a stopping time.

< t}

E Ft as the condition which determines

192

7

THE KUNITA-WATANABE EXTENSION

On the other hand, because A1R( · ,w) is continuous, as well as of bounded variation, integration by parts leads to the pathwise identity

MR(t,w) 2

=

21t

MR(r,w) MR(dr,w)

for IP'-almost every w.

Hence, after combining this with the above, we conclude that JEIP'[A1R(t) 2 ] = 0. Finally, suppose that IP'(::Jt 2: 0 0 < IMI(t) < oo) > 0. Then there would exist an R > 0 and t E (0, oo) such that IP'( (R :::; t) > 0, which would lead to the contradiction JEIP'[MR(t) 2 ] 2: iJEIP'[IIMII[o,r]J > 0. To complete the proof, suppose that A and A' are two functions with the described properties. Then (A (t) - A' (t), Ft, IP') is a continuous local martingales whose paths are of locally bounded variation. Hence, by what we have just proved, this means that A = A' IP'-almost surely. D

§7.1.2. Existence: In this subsection, we will show that if (M(t),Ft,IP') is a continuous, JR-valued local martingale, then there exists a IP'-almost surely unique progressively measurable A : [0, oo) x n --+ [0, oo) such that A(O) = 0, t-v-> A(t) is IP'-almost surely continuous and nondecreasing, and (M(t) 2 A(t), Ft, IP') is a local martingale. To begin, notice that Corollary 7.1.2 provides us with the required uniqueness. Next, observe that it suffices to prove existence in the case when (M(t), Ft, IP') is a bounded martingale with M(O) = 0. Indeed, if this is not already the case, we can take (m(w) = inf{t 2: 0: IM(t,w)l 2: m} and set A1m(t) = A1(t 1\ (m)· Assuming that Am exists for each mE z+, we would know, by Doob's stopping time theorem and uniqueness, that Am+l I [0, (m) = Am I [0, (m) IP'-almost surely for all m 2: 1. Hence, we could construct A by taking A( t) = sup{ Am (t) : m with (m 2: t}. Now assume that (A1(t),Ft,IP') is a bounded, continuous martingale with 1\4(0) = 0. For convenience, we will assume that lv!( · ,w) is continuous for every w E n. The idea behind Ito's construction of A is to realization that, if A exists, then Ito's formula would hold when t is systematically replaced by A(t). In particular, one would have M(t) 2 = 2 J~ M(r) dM(r) + A(t). Thus, it is reasonable to see what happens when we take A(t) = M(t) 2 2 J~ M(r) M(dr). Of course, this line of reasoning might seem circular since we want A in order to construct stochastic integrals with respect to A1, but entry into the circle turns out to be easy. Set (m,o(w) = m for m E N. Next, proceeding by induction, define {(m,N} ~=0 for N E z+ so that (o,N = 0 and, for m E z+' (rn.N (w) lS equal to (£,N- 1 (w) 1\ inf { t 2: (m-l,N(w): IM(t, w)- M((m-l.N(w),w) I 2: for the£ E z+ with (£-l,N-l(w):::; (m-l.N(w)

< (£,N-l(w).

fr}

7.1

193

DOOB-MEYER FOR CONTINUOUS MARTINGALES

For each N E N, {(m,N : m ~ 0} is a nondecreasing sequence of bounded stopping times which tend to oo as m ~ oo. Further, these sequences are nested in the sense that {(m,N -1 : m ~ 0} c;;;; { (m,N : m ~ 0} for every NEZ+. Now set

Mm,N(w) = M((m,N(w), w)

and

6.m,N(t,w) = M(t 1\ (m,N(w),w)- M(t

1\

(m-l,N(w),w),

and observe that

M(t, w) 2

-

M(O,w) 2 = 2YN(t,w)

+ AN(t, w),

where 00

00

m=l

m=l

Furthermore, (YN(t),:Ft,P) is a continuous martingale, and AN: [O,oo) x n ___, [0, 00) is a progressively measurable function with the properties that, for each wE 0: AN(O, w) = 0, AN(·, w) is continuous, and AN(t, w) + J2 ~ AN(s,w) whenever 0:::; s < t. Thus, we will be done if we can prove that, for each T E [0, oo), {AN : N ~ 0} converges in L 2 (lP; C([O, T]; lR), which is equivalent to showing that {YN : N ~ 0} converges there. With this in mind, for each 0 :::; N < N' and m E z+, define

and note that

L 00

YN'(t,w)- YN(t,w) =

m=l

(Mm,N'(w)- M~~1,(w))6.m,N'(t,w).

Because jMm,N'(w)- M,~1,(w)j :::; thogonal,

i1

and the terms in the series are or-

In particular, as an application of Doob's inequality, we see first that, for each T ~ 0,

194

7 THE KUNITA-WATANABE EXTENSION

and then that there exists a continuous martingale (Y(t), :Ft, lP') with the property that

J~oo IEP[IIYN- Yll~o,r]]

= 0

for each T E [O,oo).

To complete the proof at this point, define the function A : [0, 00) X n ~ [0, oo) so that A(t, w)

= 0 V sup { M(s, w) 2

-

2Y(s, w) : s E [0, t] },

and check that A has the required properties. Hence, we have now proved the following version of the Doob-Meyer decomposition theorem. THEOREM 7.1.3. IE(M(t),:Ft,lP') is a continuous, JR-valuedlocalmartingale, then there exists a lP'-almost surely unique progressively measurable function A: [O,oo) X n---+ [O,oo) with the properties that A(O) = 0, t"""' A(t) is lP'-almost surely continuous and nondecreasing, and (M(t) 2 - A(t), :Ft, lP') is a local martingale. From now on, we will use the notation (M) to denote the process A described in Theorem 7.1.3.

§7.1.3. Exercises: EXERCISE 7.1.4. Because we have not considered martingales with discontinuities, the most subtle aspects of Meyer's theorem are not apparent in our treatment. To get a feeling for what these subtleties are, consider a simple Poisson process (cf. §1.4.2) N(t) on some probability space (n, :F, lP'), let :Ft be the lP'-completion of u(N(r) : T E [0, t]), set M(t) = N(t)- t, and check that (M(t),:Ft,lP') is a non-constant martingale. At the same time, t """' M(t) lP'-almost surely has locally bounded variation. Hence, the first part of Corollary 7.1.2 is, in general, false unless one imposes some condition on the paths t """' M(t). The condition which we imposed was continuity. However, a look at the proof reveals that the only place where we used continuity was when we integrated by parts to get MR(t) = 2 J~ MR(r) MR(dr). This is the point alluded to in the rather cryptic remark preceding Theorem 7.1.1. EXERCISE 7.1.5. Let (M(t),:Ft,lP') be a continuous local martingale, (a stopping time, and set MC.(t) = M(t 1\ ().

(i) Show that (MC.)(t) = (M)(t 1\ (). (ii) If (M)(() E L 1 (1P';JR), show that (MC.(t)- M(O),:Ft,lP')

are martingales.

and

( (MC.(t)- M(0)) 2

-

(Mc,)(t),:Ft,lP')

7.2 KUNITA-WATANABE STOCHASTIC INTEGRATION

195

(iii) Suppose a: n----> JR. is an F-aJmost surely, for all T ~ 0.

II(M2)!- (M1)! ll[o,T] :S (M2- M1)(T)

PROOF: The first assertion requiring comment is the inequality in (7.2.2). To prove it, first note that it suffices to show that for each 0 :S h < t2 and a

>0, 21(M1, M2)(t2)- (M1, M2)(t1)l 1

(*)

:S a((M1)(t2)- (M1)(h)) + -((M2)(t2)- (M2)(tl))

a JP>-almost surely. Indeed, given(*), one can easily argue that, JP>-almost surely, the same inequality holds simultaneously for all a > 0 and 0 :S t1 < t2; and once this is known, (7.2.2) follows by the usual minimization procedure with which one proves Schwartz's inequality. But (*) is a trivial consequence of non-negative bilinearity. Namely, for any a> 0,

0 :S (a! M1 ±a-!M2, a! M1 ±a-!M2)(t2) - (a!M1 ±a-!M2,a!M1 ±a-!M2)(t1)

=a( (M1)(t2)- (M1)(t1)) ± 2( (M1, M2)(t2)- (M1, M2)(t2)) + a- 1( (M2)(t2)- (M2)(t1)) JP>-almost surely. Knowing the Schwarz inequality for (M1, M2), the triangle inequality iJ(M2)(t)- V(M1)(t)l :S (M2- M1)(t) :S (M2- M1)(T)

, 0 :S t :S T,

JP>-almost surely follows immediately. Hence, completing the proof from here comes down to showing that if p, 1 and p, 2 are finite, non-negative, non-atomic Borel measures on [0, T] and vis a signed Borel measure on [0, T] satisfying lv(I)I :S VJ.L1(I)p,2(I) for all half-open intervals I= [a,b) 0, 2Jp,1(I)p,2(I) :S ap,1(I) + a- 1p, 2(I), the absolute continuity statement is clear. In addition, we have

ii.

21T c.pgdp, :Sa foT1'Pifidp,+a-1foT1'Pihdp, first for c.p's which are indicator functions of intervals [a, b), next for linear combinations of such functions, then for continuous c.p's, and finally for all Borel bounded measurable c.p's. But this means that, p,-almost everywhere, 2lgl :S afi + a- 1h for all a > 0, and therefore that IYI :S Vllf2. 0 In the following, and throughout, we will say that a sequence {Mk}]"' in Mloc(JP>; JR.) converges in Mloc(JP>; JR.) to M E Mloc(JP>; IR) if, for each T ~ 0, (Mk- M)(T) ---> 0 in JP>-probability.

198

7 THE KUNITA-WATANABE EXTENSION

COROLLARY 7.2.3. If Mk ----+Min Mloc(lP; IR), then li(Mk- Mk(O))- (M- M(O))II[o,T] V II(Mk)- (M)II[o.T]----+ 0 in JP-probability for each T

2: 0.

Moreover, if {Mk}j"' ~ Mloc(lP; IR) and

lim sup(Mc - Mk)(T)

k->oo £?_k

then there exists a M E M Min Mloc(lP; IR).

=0

lac (JP;

in lP-probability for each T

2: 0,

lR) to which { M k - M (0)} \"' converges to

PROOF: Without loss in generality, we will assume that Mk(O) = 0 = M(O) fork 2: 1. In view of the triangle inequality proved in Theorem 7.2.1, the only part of the first assertion which requires comment is the proof that II.Jtfk Mli[o,T] ----+ 0 in lP-probability for all T 2: 0. However, if (R

= inf {t 2: 0: sup(Mk)(t) 2: R} , k?.l

then (R / oo JP-almost surely as R-+ oo and, by Exercise 7.1.5 and Doob's inequality,

as k -+ oo for each R > 0. Turning to the Cauchy criterion in the second assertion, define (R as in the preceding paragraph, and observe that the argument given there also shows that lim sup lEI!' Jl;h 11 2[0 TA( 1] = 0 k->oo £?_k

[liMe -

·

R

for each R > 0. Hence, there exists an M E Mloc(lP; IR) such that IIMk Mll[o,T] ----+ 0 in lP-probability for all T > 0. At the same time, we know that, for each R > 0 and T > 0,

as k

-+

oo. Hence, for each T > 0, (Mk - M) (T) ----+ 0 in JP-probability.

D

§7.2.2. The Kunita-Watanabe Stochastic Integral: The idea of Kunita and Watanabe is to base the definition of stochastic integration on

7.2

199

KUNITA~WATANABE STOCHASTIC INTEGRATION

the Hilbert structure described in the preceding subsection. Namely, given (} E 8foc((M),JID;IR), they say that Ift should be the element of Mloc(IP';IR) with the properties that

Ift (0) = 0 and (Ifi, M')(t)

(7.2.4)

=fat (}(T)(M, M')(dT)

for all M' E Mloc(IP'; IR). Before adopting this definition, one must check that (7.2.4) makes sense and that, up to a JID-null set, it determines a unique element of Mloc(IP'; IR). To handle the first of these, observe that, by Theorem 7.2.1,

Hence, (} E 8foc ( (M), JID; IR) implies that, JID-almost surely, (} is locally integrable with respect to the signed measure (M, M'). As for the uniqueness question, suppose that I and J both satisfy (7.2.4), and set 6. =I- J. Then (6.) = 0, and so there exists a nondecreasing sequence { (m}1 of stopping times such that (m / oo and ( 6.( · 1\ (m) 2, Ft, IP') is a bounded martingale for each m, which, since 6.(0) 0, means that JEll'[fl(t 1\ (m?] = 0 for all m 2: 1 and t 2: 0. Having verified that (7.2.4) makes sense and uniquely determines Ift, what remains is for us to prove that 1 always exists, and, as should come as no surprise, this requires us to return (cf. §5.1.2) to Ito's technique for constructing his integral. Namely, given M E Mloc(IP'; IR) and a bounded, progressively measurable(}: 0--> IR with the property that (}(t) = (}([t]N) for some N EN, set

=

It

It1 (t) =

L 00

e(mTN)(M(t 1\ (m

+ 1)TN)- M(t 1\ m2-N)).

m=O

Clearly (cf. part (ii) of Exercise 7.1.5), if ( is a stopping time for which (M)(() E L 1 (JID;IR), then Ift(t 1\ ()is JID-square integrable for all t 2: 0 and, for all mEN and m2-N:::; t 1 < t 2 :::; (m + 1)2-N, lEll'[Ift (t2 1\ ( ) - Ift (t1/\

()1Ft,]

= (}(mTN)lEll'[M(t2/\ ()- M(h 1\

()1Ft,]

= 0.

Thus Ift E Mloc(IP';IR). In addition, if M' E Mloc(IP';IR) and (M')(() is also

200

7

THE KUNITA-WATANABE EXTENSION

JP>-integrable, then JEll' [IJ"f (t2 1\ ()M' (t2 1\ () -

IJ"f (h 1\ ()M' (h 1\ () IFtl]

= 8(m2-N)1Ell'[M(t2/\ ()M'(t2/\ ()- M(t1 1\ ()M'(t1 1\

= 8(m2-N)1Ell' [(M, M')(t2 1\ () - (M, M')(t1 1\ () IFt 1 ]

~IE'

[t:'

O(r)(M, M')(dr)

()I FtJ

IF,,] ,

which proves that (IJW", M')(dt) = O(t)(M, M')(dt). Hence, we now know that IJ"f exists for all bounded, progressively measurable I) : [0, 00) Xn ----4 JR. with the property that B(t) = B([t]N) for some N E N. Furthermore, by Corollary 7.2.3, we know that if {ON }1 U {8} ~ 8foc( (M), lP; JR.) and (7.2.5) foTIBN(T) -8(T)I 2 (M)(dT) then

IJ"f

----4

0 in JP-probability for all T > 0,

exists. Hence, we will be done once we prove the following lemma.

7.2.6. For each 8 E 8foc( (M), JP>; JR.) there exists a sequence {ON }1 of bounded, JR.- valued, progressively measurable functions such that BN(t) = BN([t]N) and ( 7.2.5) holds. LEMMA

Clearly, it suffices to handle O's which are bounded and vanish off [0, T] for some T > 0. In addition, we may assume that t """ (M)(t, w) is bounded, continuous, and nondecreasing for each w. Thus, we will make these assumptions. There is no problem if t""" O(t,w) is continuous for all wEn, since we can then take BN(t) = B([t]N ). Hence, what must be shown is that for each bounded, progressively measurable 8 there exists a sequence {ON }1 of bounded, progressively, IR.-valued functions with the properties that t """ B(t,w) is continuous for each wand (7.2.5) holds. To this end, set A(t,w) = t + (M)(t,w), and, for each s E (O,oo), determine w En~------> ((s,w) E (O,oo) so that A(((s,w),w) = s. Clearly, for each w, t """ A(t,w) is a homeomorphism from (0, oo) onto itself, and so, an elementary change of variables yields PROOF:

(*)

{ J[o,oo)

f(t)A(dt,w)= {

fo((s,w)ds

J[o,oo)

for any non-negative, Borel measurable f on [0, oo) .. To take the next step, notice that, for each s, w """ ( (s, w) is a stopping time, and set F~ equal to the JP-completion of F((s)· Thus, if B'(s,w) =

7.2

KUNITA-WATANABE STOCHASTIC INTEGRATION

201

O(((s,w),w), then e': [O,oo) X S1 ---7 lR is a bounded function which vanishes off of [0, A(T)] x S1 and is progressively measurable with respect to the filtration {F~ : s ~ 0}. Hence, by the argument given to prove the density statement in Lemma 5.1.8, we can find a sequence {8~}1" of {F~: s ~ 0}progressively measurable such that t "'""' e~ (t, w) is bounded, continuous, and supported on [0, A(l + T, w)] for each w, and lim JEll'

N...c,oo

[1

A(l+T)

0

l

IO~(s)- e'(s)i 2 ds = 0.

Finally, set eN(t,w) = e~(A(t,w),w), and note that each eN is a bounded, {Ft : t ~ 0}-progressively measurable function with the properties that t "'""'ON(t,w) is continuous and vanishes off [0, l+T] x S1 for each w. Further, by(*)

0

Hence, (7.2.5) holds.

Summarizing the results proved in this subsection, we state the following theorem. THEOREM 7.2.7. For each M E Mloc(IP';JR) there is a linear map 9foc((M),lP';JR) ~-----+If/ E Mloc(IP';JR) such that (7.2.4) holds.

eE

Just as we did in the case treated in Chapter 5, we will use the notation

J~O(r)dM(r) interchangeably with It'1 (t). More generally, given stopping times ( 1

:::::; ( 2 ,

we define

l

t/\(

2

O(r) dM(r) =Itt (t A (2)- Itt (t A ( 1).

t/\(1

Starting from Exercise 7.1.5, it is an easy matter to check that (7.2.8)

§7.2.3. General Ito's Formula: Because our proof oflto 's formula in §5.3 was modeled on the argument given by Kunita and Watanabe, its adaptation to their stochastic integral defined in §7.2.3 requires no substantive changes. Indeed, because, by part (i) of Exercise 7.1.5, we already know that, for bounded stopping times (1 :::::; ( 2

1( 2

(1

O(r) dM(r) =

1

00

l[o,()(r)O(r) dM(r),

0

the same argument as we used to prove Theorem 5.3.1 allows us to prove the following extension.

202

7 THE KUNITA-WATANABE EXTENSION

THEOREM 7.2.9. Let X = (X1, ... , Xk) : [0, oo) x S1 ----> ]Rk and Y = (Y1, ... , }/;) : [0, 00) X S1 ----> JRP be progressively measurable maps with the properties that, for each 1 ::::; i ::::; k and w, t""" Xi(t, w) is continuous and of locally bounded variation and, for each 1::::; j::::; £, Yj E Mloc(lP';lR); and set Z =(X, Y). Then, for each FE C 1·2 (JRk x JRP;JR), F(Z(t))- F(Z(O))

lP'-almost surely. Here, the dXi-integrals are taken in the sense of RiemannStieltjes and the dlj-integrals are taken in the sense of Ito, as described in Theorem 7.2. 7.

We will again refer to this extension as Ito's formula, and, not surprisingly, there are myriad applications of it. For example, as Kunita and Watanabe pointed out, it gives an elegant proof of the following famous theorem of Paul Levy. COROLLARY 7.2.10. Suppose that {M1 : 1 ::::; j ::::; n} 0, (E~, Ft, lP') is a continuous martingale. In particular, EIP'[exp( H

(~,M(s + t)- M(s))JFEn) IFs]

=

e-~1~1 2 ,

which, together with M(O) = 0, is enough to see that (M(t),Ft.lP') is a Brownian motion. 0

203

7.2 KUNITA-WATANABE STOCHASTIC INTEGRATION §7.2.4. Exercises

EXERCISE 7.2.11. Suppose that M1, M2 E Mioc(lP';IR), and assume that a({M1 (r) : T:::: 0}) is lP'-independent of a({M2(r) : T:::: 0}). Show that the product M 1M2 is again an element of Mioc(lP'; IR), and conclude that (M1, M2) 0 lP'-almost everywhere.

=

EXERCISE 7.2.12. Given M E Mioc(lP';IR), () E 8foc((M),lP';IR), and rt E 8foc((Ir),lP';IR), check that rtB E 8foc((M),lP';IR) and that

I~(t) =

1t

ry(r)dlr(r) lP'-almost surely.

EXERCISE 7.2.13. When ME Mioc(lP';IR) is a Brownian motion, and therefore (M)(t) = t, it is an elementary exercise to check that (M)(t) is lP'-almost everywhere equal to the square variation

=

L (M(t 1\ (m + 1)TN,w)- M(t 1\ mTN,w))

lim

N-+=

2

m=O

of M( · ,w) f [0, t]. The purpose to this exercise is to show that if lP'-almost everywhere convergence is replaced by convergence in lP'-probability, then the analogous result is easy to derive in general. In fact, show that, for each pair M1,M2 E Mioc(lP';IR) and all T E [O,oo), lim

sup jf((Ml(t/\(m+1)2-N)-M1(t/\mTN))

N-+= tE[O,T] m-O -

X

(M2(t 1\ (m + 1)TN)- M2(t 1\ mTN))) - (M1, M2)(t)l = 0

in lP'-probability. Hint: First, use polarization to reduce to the case when M 1 = M = M 2. Next, do a little algebraic manipulation, and apply Ito 's formula to see that

L= (M(t 1\ (m + 1)TN,w)- M(t 1\ m2-N,w))

2

- (M)(t)

m=O

=

21t (M(r)- M([r]N)) dM(r).

Finally, check that

1T

(M(r)- M([r)N)) 2 (M)(dr)--+ 0

lP'-almost surely, and use this, together with Exercise 7.1. 7, to get the desired conclusion.

204

7 THE KUNITA-WATANABE EXTENSION

EXERCISE 7.2.14. The following should be comforting to those who worry about such niceties. Namely, given M E M 1oc(lP'; JR), set At equal to the lP'-completion of a({M(r): T E [O,t]}), and use the preceding exercise to see that (M) is progressively measurable with respect to {At : t 2: 0}. Conclude, in particular, that no matter which filtration {Ft : t 2: 0} is the one with respect to which M was introduced, the (M) relative {Ft : t 2: 0} is the same as it is relative to {At: t 2: 0}. EXERCISE 7.2.15. In Exercise 5.1.27, we gave a rather clumsy, and incomplete, derivation of Burkholder's inequality. The full statement, including the extensions due to Burkholder and Gundy, is that, for each q E (0, oo), there exist 0 < Cq < Cq < oo such that, for any M E Mloc(lP'; JR) with M(O) = 0 and any stopping time(, (7.2.16) Here, following A. Garsia (as recorded by Getoor and Sharpe), we will outline steps which lead to a proof (7.2.16) for q E [2, oo). As explained in Theorem 3.1 of [16], Garsia's line of reasoning can be applied to handle the general case, but trickier arguments are required.

(i) The first step is to show that it suffices to treat the case in which both M and (M) are uniformly bounded and (is equal to some constant T. (ii) Let q E [2, oo) be given, and set Cq

=

(q~q1~~

1 .

Prove that the right

hand side of (7.2.16) holds with this choice of Cq·

Hint: Begin by making the reductions in (i). Given E > 0, set Fc(x) q (x 2 + E2 ) 2 , and apply Doob's inequality plus Ito's formula to see that

lEJI>[IIMII[o.rJJ ::; (q')qlEJI>[Fc(M(T))]

~ (q')'q;q -l)E' [ [ F;'(M(r)) (M)(dr)] ::; (q')qq;q- 1)1EJI>[(iiMII[o.TJ

+E 2 )%- 1 (M)(T)].

E""

Now let 0, and apply Holder's inequality. (iii) Assume that M and (M) are bounded, set 8(t) = (M)(t)%-~, and take M' = If/. After noting that

(M')(T)

=

1 T

0

(M)%- 1 (r) (M)(dr)

=

2 q -(M)(T)2, q

205

7.3 REPRESENTATIONS OF CONTINUOUS MARTINGALES

(iv) Continuing part (iii), apply Ito's formula to see that 1

M(T)(M)(T)%-2 = M'(T) +

1 T

0

q

1

M(T) d(M)(T) = lP' X !P'0 , and :is equal to the P-completion of F~ x Bs. Then, if B( s, w) = p( s) and M(s,w) = M'(s,w) for s 2': 0 and w = (w,p), (B(s),i,,P) is a Brownian motion, (Nf(s),i"JP>) is a continuous, local martingale, (M)(s,w) = s 1\ (M)(oo, w), and, by Exercise 7.2.11, (B, Nf) = 0. Finally, for w = (w,p), we take A(t,w) = (M)(t,w) fort 2': 0 and

n

n

(3 (s

')

'

w

=

when 0:::; s < (M)(oo,w) when (M)(oo,w):::; s < oo.

{ M(s,w) B(s,w)- B((M)(oo,w),w)

By the reasoning in the first paragraph, we know that A(t) is an {:is : s 2': 0}-stopping time for each t. Furthermore, because an equivalent description of (3(s, w) is to say

(3(s,w) = N!(s,w)

+

1s

l[A(oo).oo)(a)

dB(a),

we see that (f3(s),F8 ,P) is a continuous, local martingale with ((3)(s) = (M)(s)

+2

+los

los

l[A(oo),oc)(a)

l[A(oc),oc)(a)

(111, B)(da)

(B)(da) = s 1\ A(oo) + (s- s 1\ A(oo)).

Hence, (f3(s),Fs,P) is a Brownian motion. Finally, by the result in the first paragraph,

f3(A(. ,w),w)

= M'((M)( · ,w),w) =

M( · ,w)

for P-almost every w= (w,p), and so we have completed the proof. 0 One of the most important implications of Theorem 7.3.2 is the content of the following corollary. COROLLARY

7.3.3. If ME Mloc(IP'; lR), then"'

lim

M(t)

t->oc

J2(M)(t)log( 2 )(M)(t)

= 1 = - lim t-Hxo

M(t) 2(M)(t)log( 2 )(M)(t)

!P'-almost surely on the set { (M) (oo) = oo}. In particular, the set { (M) (oo) < oo} is !P'-almost surely equal to the set of w such that limt_,oc M (t, w) exists in R 4

Here we use log( 2 ) r to denote log(log r) for r

> e.

7.3 REPRESENTATIONS OF CONTINUOUS MARTINGALES

209

PROOF: Using the notation in Theorem 7.3.2, what we have to do to prove the first assertion is to show that

lim t->rxo

,B(A(t)) = 1 = - lim ,B(A(t)) J2A(t) log( 2 ) A(t) t->= J2A(t) log( 2 ) A(t)

P-almost surely on the set {A( oo) = oo}. But, by the law of the iterated logarithm for Brownian motion (cf. Theorem 4.1.6 in [36]), this is obvious. Given the first assertion, the second assertion follows immediately from either Exercise 7.1.8 or by another application of the representation given by Theorem 7.3.2. D

§7.3.2. Representation via Stochastic Integration: Except for special cases (cf. F. Knight's theorem in Chapter V of [27]), representation via random time change does not work when dealing with more than one M E Mloc(JID; JR) at a time. By contrast, Brownian stochastic integral representations have no dimension restriction, although they do require that the (M) 's be absolutely conditions. To make all of this precise, we will prove the following statement. THEOREM 7.3.4. Suppose that M = (M1 , ... ,Mn) E (Mtoc(JID;JR)r and that, for each 1 :::; i :::; n, t """* (Mi)(t) is JID-almost surely absolutely continuous. Then there exists progressively measurable map o: : [0, oo) x n ----+ Hom(JRn; JRn) with the properties that: o:( t, w) is symmetric and non-negative definite for each (t, w), and

Furthermore, there exist an 1Rn-valued Brownian motion (,B(t), Ft, Ji») on some probability space (D, J:, P) and an a E E>foc (Ji»; Hom(lRn; ]Rn)) such that the P-distribution of

w """*(a(. ,w),Ia(- ,w)) is the same as the JID-distribution of

w """* (o:( · ,w),M( · ,w)- M(O,w)) when

210

7 THE KUNITA-WATANABE EXTENSION

PROOF: The first step is to notice that, by Theorem 7.2.1, t-v-+ (M;, Mj)(t) is lP'-almost surely absolutely continuous for all 1 ::; i, j ::; n. Hence, we can find (cf. Theorem 5.2.26 in [36]) a progressively measurable a : [0, oo) x n ----+ Hom(!Rn; !Rn) such that (M;, Mj)(dt) = a;j(t) dt. Obviously, there is no reason to not take a( t, w) to be symmetric. In addition, because

a( t, w) is non-negative definite for A[o,=) x lP'-almost every (t, w) E [0, oo). Hence, without loss in generality, we will take a( t, w) to be symmetric and non-negative definite for all (t, w ). The next step is to take o:(t,w) to be the non-negative definite, symmetric square root of a( t, w). To see that a is progressively measurable, we need only apply Lemma 3.2.1 to see that the non-negative, symmetric square root o:c (t, w) of a( t, w) + El is progressively measurable for all E > 0 and then use o:( t, w) = limc~O o:c (t, w). Obviously, o:(t, w) will not, in general, be invertible. Thus, we take n(t, w) to denote orthogonal projection onto the null space N(t,w) of a(t,w) and o:- 1 (t,w) : IRn ----+ N(t,w)j_ to be the symmetric, linear map for which N( t, w) is the null space and o:- 1 ( t, w) I N( t, w)j_ is the inverse of o:( t, w) I N (t, w) j_. Again, both these maps are progressively measurable:

n(t, w) =lim a(t,w) (a(t, w)+Elr 1 & o:- 1 (t,w) =lim a(t,w)(a(t, w)+El)- 1 . c~O

E~O

Because (cf. Exercise 7.2.17) o:- 1 (t,w)o:(t,w) = n(t,w)j_, and therefore

we can take B(t) = J~o:- 1 (T)dM(T), in which case (B(t),Ft,lP') is an JRn_ valued, continuous local martingale and, if BE(t) (~, B(t) )JRn, then

=

Furthermore, if X(t) = J~ o:(T) dB(T), then, by Exercise 7.2.12,

211

7.3 REPRESENTATIONS OF CONTINUOUS MARTINGALES and so, if Mr;.(t)

= (~,M(t))JRn'

then

(Xr;.- Mr;.)(t) =lot (~,n(T)a(T)n(T)~)JR.n dT

= 0.

Clearly, in the case when a(t,w) > 0 for A[o,oo) x IP'-almost every (t,w), we are done. Indeed, in this case (B(t), Ft, IP') is an 1Rn-valued Brownian motion, and so we can take (0, F, JP) = (n, F, IP'), Ft = Ft, 0: = o:, and (3 =B. To ·handle the general case, taken= n X C([O,oo);!Rn), j" and Ft to be the IP' x !P'0 -completions ofF x B of Ft x Bt. lP = IP' x IP'0 , &(t,w) = o:(t,w), and

(3(t,w) = B(t,w) +lot n(T,w)dp(T)

fort~

0 and w = (w,p). Because (Br;.)(dt) = ni(t)_i~l 2 dt and (cf. Exercise 7.2.11) (Br;_,pr;_)(dt) = Odt when pr;_ (~,p( · ))JRn' it is easy to check that

=

these choices work. 0 By combining the ideas in this section with those in the preceding, we arrive at the following structure theorem, which, in a somewhat different form, was anticipated by A.V. Skorohod in [31]. COROLLARY 7.3.5. LetM= (M1, ... ,Mn) E (Mloc(IP';!R)r begiven, and set A(t) = E~(Mi)(t). Then there is a probability space (O,F,JP) on which there exists a Brownian motion (f3(t),Ft,W) and {Ft: t ~ 0}-progressively measurable maps 0: : [0, 00) X n ~ Hom(!Rn; !Rn) and A : [0, 00) X n ~ [O,oo) such that: &(t,w) is symmetric, OIJR.n ~ &(t,w) ~ IRn for all (t,w), A(t) is stopping time for each t ~ 0, and, the P-distribution of

w-

(

{A(t)

A(t), 10

a(T,w)d(3(T,w)

)

,

is the same as the IP'-distribution of w- (A(·, w), M( ·, w)- M(O, w)). There are essentially no new ideas here. Namely, define ((s,w) = inf{t ~ 0 : A( t, w) ~ s}. By the techniques used in the preceding section, we can define M': [0, oo) x n ~ !Rn so that M'( · ,w) = M((( · ,w),w) IP'-almost surely and can show that (M'(s),F;,IP') is an 1Rn-valued continuous martingale when F; is the IP'-completion of F((s) and that E~(MI)(dt,w) ~ dt IP'-almost surely. In addition, those same techniques show that, for each t ~ 0, A(t) is an {F;: s ~ 0}-stopping time and M(t,w) = M'(A(t,w),w) IP'-almost surely. Hence, all that remains is to apply Theorem 7.3.4 to (M'(s),F;,IP'). A more practical reason for wanting Theorem 7.3.4 is that it enables us to prove the following sort of uniqueness theorem.

212

7 THE KUNITA-WATANABE EXTENSION

COROLLARY 7.3.6. Let CJ: [O,oo) x lRn----+ Hom(IFF';lR") and b: [O,oo) x IR" ----+ IRn be measurable functions with the properties that CJ(t, x) is symmetric and non-negative definite for each (t, x), t '-"' IICJ(t, O)IIH.S. V lb(t, O)l is locally bounded, and ~ tE[O,TJ

IICJ(t, x2)- CJ(t,xl)IIH.s. V lb(t, x2)- b(t, xl)l

lx2 -XII

x2#x 1

0. Set a( t, x) = CJ 2 ( t, x), and define the time dependent operator t '-"' Lt on C 2 (IR"; IR) so that Ltoo

sup JEII'o [ucp(t,p(t)) 2 , lp(t)l;::: tE(O,l)

R] = 0.

But

JE~~'o [ucp(t,p(t))

:; L

2

,

lp(t)l;:::

R] = Jlxi?_R { ( { '¢x(Yhl-t(Y- x)dy) }'R.

2

rt(dx)

'¢x(y) 2 !R(t,y)rl(dy),

where

!R(t,y)

=

e y22 V27rt(1 - t)

and S(t,R)

=

1 lxi:2':R

e

(y-x)2 2 0 -almost

surely fortE [0, 1).

Hence, since ux(t,p(t))---. ¢x(p(1))in L 2 (JP>0 ;JR) as t /' 1, we conclude that ¢x(p(1)) =

1u~(T,p(T))dp(T) 1

JP>0 -almost

surely.

In particular, this means that if

Mx(t,p)

till

= Jo

u~(T,p(T)) dp(T),

then (Mx(t), Bt, JP>0 ) is a square-integrable martingale for which

In conjunction with Theorem 7.3.2, the preceding already leads to Skorohod's representation of X. However, for applications, it is better to carry this line of reasoning another step before formulating it as a theorem. Namely, set u~(t-[t],p(t)-p([t])) fortE[O,oo)\N Ox (t,p ) = { 0 fortE N. Then, because p ""'p( · + s)- p(s) is JP>0 -independent of B.. and again has the same JP>0 -distribution as p itself, we see that the JP>O-distribution of

{1m Ox(T,p) dp(T):

mE

z+}

is the same as the distribution of the partial sums of independent copies of X. Thus, we have now proved the following form of Skorohod's representation theorem.

217

7.3 REPRESENTATIONS OF CONTINUOUS MARTINGALES

THEOREM 7.3.8. Let X be a centered, square-integrable, ~-valued random variable. Then there exists an IR-valued Brownian motion (f3(t), Ft, JP>) and a nondecreasing sequence {(m }8" of finite stopping times such that: (1) (o = 0, the random variables {(m- (m-1 : m ;::: 1} are mutually JP>-independent and identically distributed, and JEll" [( 1 ] = JE[X 2 ]. (2) The random variables {f3((m)- f3((m-1) : m ;::: 1} are mutually JP>independent and the JP>-distribution of each equals the distribution of X. In fact, for each q E [2, oo), (cf. (7.2.16))

PROOF: There is essentially nothing left to do. Indeed, by Theorem 7.3.2, we know that there is a Brownian motion (f3(s),F8 ,JP>) on some probability space (O,F,JP>) and a map A: [O,oo) x 0 ~ [O,oo) such that A(t) is an {Fs : s;::: 0}-stopping time for each t;::: 0, the JP>-distribution of

w~ (f3(A(-,w)),A(-,w)) is the same as the lP'0 -distribution of

(lot Ox(r,p) dp(r), lot Ox(r,p)

p~

Hence, all that remains is to set (m

=

A(m).

2 dr)

.

D

§7.3.4. Exercises EXERCISE 7.3.9. Let ME Mloc(lP'; IR) be given.

(i) Show that the conclusion in Exercise 7.1.6 can be strengthened to say that, for any stopping time(, lim M (t

t--+oo

exists in IR

1\ ()

JP>-almost surely on { (M) (() < oo}

and lim M(t 1\ () = oo = - lim M(t 1\ ()

t__,.oo

t__,.oo

lP'-almost surely on { (M)(() = oo }.

(ii) If M is non-negative, show that (M)(oo) < oo lP'-almost surely. (iii) Refer to part (i) of Exercise 6.3.12, and show that there exists a () E 8foc (JP>; !Rn) such that M(t) = M(O)

+lot (O(r),dp(r))JRn'

Further, show that

f0

00

lP'-almost surely for each t E [O,oo).

jO(r)i2 dr < oo P-almost surely.

218

7

THE KUNITA-WATANABE EXTENSION

7. 3.10. The purpose of this exercise is to see how the considerations in this section can contribute to an understanding of the relationship between dimension and explosion. EXERCISE

(i) Suppose that a : IR ---+ IR is a locally Lipschitz continuous function, and let (/3 (t), Ft, lP') be a one-dimensional Brownian motion. Show that, without any further conditions, the solution to the 1-dimensional stochastic differential equation

dX(t)

=

a(X(r)) d;3(r)

exists for all time whenever lP'(IX(O)I < oo) = 1. That is, no matter what the distribution of X(O) is or how fast a grows, X(·) willlP'-almost surely not explode.

Hint: The key observation is that, because its unparameterized trajectories follow those of a 1-dimensional Brownian motion, (X)(oo)

Thus, if ( 0 (rn ==

1

=

00

a 2 (X(r)) dr = oo

==}

X(·) returns to 0 infinitely often.

= 0 and if, form 2: 1,

{

00

.

mf{t 2: (m-1: :lr E [(m-1,t]IX(r,x)l = 1 & X(t,x) = 0},

depending on whether (m-1 = oo or (m-l < oo, then, by the Markov property, { (m+1- (m : m 2: 1} is a family of lP'-mutually independent, identically distributed, strictly positive random variables. Hence, lP'-almost surely, either (X)(oo) < oo or for all s 2: 0 there exists at 2: s such that X(t) = 0. In either case, (X) (T) < oo lP'-almost surely for all T 2: 0.

(ii) The analogous result holds for a diffusion in IR 2 associated with L = ~a 2 (x)6., where a : IR2 ---+ IR is a continuously differentiable function. Namely, such a diffusion never explodes. The reason is that such a diffusion is obtained from a two-dimensional Brownian motion by a randomtime change and that two-dimensional Brownian motion is recurrent. Only the fact that two-dimensional Brownian motion never actually returns to the origin complicates the argument a little. To see that recurrence is the essential property here, show that, for any E > 0, the three-dimensional diffusion corresponding to L = ~(1 + lxl 2 )1+c6_ does explode.

Hint: Take u(x)

1

r

=;: }K/1.3

1 1 lx- Yl (1 + IYI 2 )1+c dy,

show that Lu = -1, and conclude that, for R > lxl, u(x) 2: JE.IP'~ [(R], where (R is the first exit time of BK/1.3 (0, R).

219

7.3 REPRESENTATIONS OF CONTINUOUS MARTINGALES

EXERCISE 7 .3.11. V. Strassen made one of the most remarkable applications of Skorohod's representation theorem when he used it in [33] to prove a function space version of the law of the iterated logarithm. Here we will aim for much less. Namely, assume the law of the iterated logarithm for Brownian motion, as stated in the proof of Corollary 7.3.3, 5 and prove that for any sequence {Xm}o of mutually independent, identically distributed JR-valued random variables with variance 1,

-.-

Sm

n->oo

J2mlog(2) m

hm

where Sm

. =1=- 11m n->oo

Sm v2mlog(2) m

,

=L~ xk.

EXERCISE 7.3.12. Another direction in which Skorohod's representation theorem can be useful is in applications to central limit theory. Indeed, it can be seen as providing an ingenious coupling procedure for such results. The purpose of this exercise is to give examples of this sort of application. Throughout, {Xm}i"' will denote a sequence of mutually independent, centered JR-valued random variables with variance 1, So = 0, and Sm = .L:~ 1 Xt for m 2: 1. In addition, (f3(t), :Ft, P) will be the Brownian motion and {(m: m 2: 0} will be the stopping times described in Theorem 7.3.8.

{i) Set j3m(t)

=

m-!j3(mt), and note that (f3m(t),:Fmt,P) is again a

Brownian motion and that the P-distribution of 13m ( ~) is the same as the distribution of m-! Sm. As an application of the weak laws of large numbers and the continuity of Brownian paths, conclude that

and use this to derive the central limit theorem, which is that statement of the distribution of m-! Sm tends to the standard normal distribution.

(ii) The preceding line of reasoning can be improved to give Donsker's invariance principal. That is, define t""" sm(t) so that sm(O) = 0, sm( 1..) = 1 m m-2 Sm and sm f [£;;;_1 , ~] is linear for each fEz+. Donsker's invariance principal is the statement that the distribution on c ([0, 00); ]Rn) of sm (- ) tends to Wiener measure. That is, IE[oSm( · )] ~ JEIP'0 [] for all bounded, continuous : C([O, oo);JR) ~JR. 5 It should be recognized that the law of the iterated logarithm for Brownian motion is essentially the same as the law of the iterated logarithm for centered Gaussian random variables with variance 1 and, as such, is much easier than the statement for general centered random variables with variance 1.

220

7 THE KUNITA-WATANABE EXTENSION

To prove this, define ym(t) so that ym

(£) =(3m (~)and ym f [£, R~l J

is linear for fEz+. Note that the distribution of ym( ·)is the same as the distribution of sm( ·),and show that, for any L E z+, 6 > 0, and m 2

i,

llym- f3mll[o ' L] S 2 o::;soo

I(e -

!:._ m

l::;R::;mL m

!2 s)

=

0

for each 6 > 0.

(iii) To complete the program begun in (ii), note that it suffices to show that if { Zm : m 2 1} is a sequence of independent, identically distributed, centered, integrable random variables, then (*)

lim lP' (

m-->oo

I_!_

sup

o::;f::;mL m

t

k=l

Zk

12 E) = 0

for each E > 0. When zl is square integrable, (*) follows from Kolmogorov's inequality (cf. (1.4.4) in [36]):

lP'

1

sup ( o 0

in IP'-probability.

At the same time, by (8.1.3),

in IP'-probability uniformly for t in compacts. Hence, for computational purposes, it is best to present the Stratonovich integral as (8.1.4)

1 t

0

Y(T)

0

dX(T) =

1t 0

1

Y(T) dX(T) +-(X, Y)(t),

2

where the integral on the right is taken in the sense of Ito. So far as I know, the formula in (8.1.4) was first given by Ito in [12]. In particular, Ito seems to have been the one who realized that the problems posed by Stratonovich's theory could be overcome by insisting that the integrands be semimartingales, in which case, as (8.1.4) makes obvious, the Stratonovich integral ofY with respect to X is again a continuous semimartingale. In fact, if X = M + B is the decomposition of X into its martingale and bounded variation parts, then

1 t

0

Y(T) dB(T)

1

+ -(Y, M)(t) 2

are the martingale and bounded variation parts of I(t) = J~ Y(T) o dX(T), and so (Z, I)(dt) = Y(t)(Z, X)(dt) for all Z E S(IP'; ~). To appreciate how clever Ito's observation is, notice that Stratonovich's integral is not really an integral at all. Indeed, in order to deserve being

225

8.1 SEMIMARTINGALES AND STRATONOVICH INTEGRALS

called an integral, an operation should result in a quantity which can be estimated in terms of zeroth order properties of the integrand. On the other hand, as (8.1.4) shows, no such estimate is possible. To see this, take Y(t) = f(Z(t)), where Z E S(IP;IR) and f E C 2 (1R;IR). Then, because (X, Y)(dt) = f' (Z(t)) (X, Z)(dt),

1t

Y(r) o dX(r)

0

=

1t 0

f(Z(r)) dX(r)

11t f' 2

+-

0

(Z(r)) (X, Z)(dr),

which demonstrates that there is, in general, no estimate of the Stratonovich integral in terms of the zeroth order properties of the integrand. Another important application of (8.1.4) is to the behavior of Stratonovich integrals under iteration. That is, suppose that X, Y E S(IP; JR), and set I(t) = f~Y(r) o dX(r). Then (Z,I)(dt) = Y(t)(Z,X)(dt) for all Z E S(IP; JR), and so

1t 0

Z(r) o dl(r) =

rt

1t 0

Z(r) dl(r)

1

11t

+-

t

2

0

Y(r) (Z,X)(dr)

= Jo Z(r)Y(r) dX(r) + 2 Jo ( Z(r)(Y,X)(dr) + Y(r)(Z,X)(dr))

=

1t 0

Z(r)Y(r) dX(r)

since, by Ito's formula, ZY(t)- ZY(O)

=

1t 0

1 + -(ZY,X)(t) =

2

Z(r) dY(r)

+

1t 0

1t

Z(r) o d

(for Y(u) o X(u))

0

ZY(r)

Y(r) dZ(r)

and therefore (ZY,X)(dt) = Z(t)(Y,X)(dt) words, we have now proved that (8.1.5)

1t

=

o

dX(r),

1 + -(Z, Y)(t),

2

+ Y(t)(Z,X)(dt).

1t

In other

Z(r)Y(r) o dX(r).

§8.1.3. Ito's Formula and Stratonovich Integration: Because the origin of Stratonovich's integral is in Riemann's theory, it should come as no surprise that Ito's formula looks deceptively like the fundamental theorem of calculus when Stratonovich's integral is used. Namely, let z = (Z1, ... , Zn) E S(IP; JR)n and f E C 3 (1Rn; JR) be given, and set Yi = od o Z for 1 :::; i :::; n. Then Yi E S(IP; JR) and, by Ito's formula applied to od, the local martingale part of Yi is given by the sum over 1 :::; j :::; n of the

226

8 STRATONOVICH'S THEORY

Ito stochastic integrals of fMJ1 f o Z with respect to the local martingale part of z1 . Hence, n

(Y;, Z;) (dt) =

L 8;8jf

0

Z( T) (Z;, Zj )(dt).

j=l

But, by Ito's formula applied to j, this means that n 1 df( Z(t)) = Y; (T) dZ;( T) + 2(Y;, Z;)(dt))'

8(

and so we have now shown that, in terms of Stratonovich integrals, Ito's formula does look like the "the fundamental theorem of calculus" (8.1.6)

f(Z(t))- f(Z(O))

=

t 1t

8;J(Z(T))

o

dZ;(T).

As I warned above, (8.1.6) is deceptive. For one thing, as its derivation makes clear, it, in spite of its attractive form, is really just Ito's formula (8.1.2) in disguise. In fact, if, as we will, one adopts Ito's approach to Stratonovich's integral, then it is not even that. Indeed, one cannot write (8.1.6) unless one knows that 8d o Z is a semimartingale. Thus, in general, (8.1.6) requires us to assume that f is three times continuously differentiable, not just twice, as in the case with (8.1.2). Ironically, we have arrived at a first order fundamental theorem of calculus which applies only to functions with three derivatives. In view of these remarks, it is significant that, at least in the case of Brownian motion, Ito found a way (cf. [13] or Exercise 8.1.8 below) to make (8.1.6) closer to a true fundamental theorem of calculus, at least in the sense that it applies to all f E C 1 (JR."; 1Ft). REMARK 8.1.7. Putting (8.1.6) together with footnote 2 about the notation for Stratonovich integrals, one might be inclined to think the right notation should be J~ Y (T )X (T) dT. For one thing, this notation recognizes that the Stratonovich integral is closely related to the notion of generalized derivatives a la Schwartz's distribution theory. Secondly, (8.1.6) can be summarized in differential form by the expression n

df(Z(t))

=

L 8d(Z(t))

0

dZi(t),

i=l

which would take the appealing form

!:_ f (z (t)) dt

=

t

i=l

8d ( z (t))

z(t)

0

Of course, the preceding discussion should also make one cautious about being too credulous about all this.

8.1 SEMIMARTINGALES AND STRATONOVICH INTEGRALS

227

§8.1.4. Exercises EXERCISE 8.1.8. In this exercise we will describe Ito's approach to extending the validity of (8.1.6).

(i) The first step is to give another description of Stratonovich integrals. Namely, given X, Y E S(lP'; ~), show that the Stratonovich integral of Y with respect to X over the interval [0, T] is almost surely equal to (*)

2~1 Y(~) + Y(~)

.

J~oo ~

2

m=O

(x ((m+ 1)T) -X (mT)). 2N

2N

Thus, even if Y is not a semimartingale, we will say that Y : [0, T] x n - - t ~ is Stratonvich integrable on [0, T] with respect to X if the limit in (*) exists in lP'-measure, in which case we will use 0T Y(t) odX(t) to denote this limit. It must be emphasized that the definition here is "T by T" and not simultaneous for all T's in an interval.

J

(ii) Given an X E S(lP';~) and aT> 0, set _kT(t) = X((T- t)+), and suppose that (.kT(t),JT,lP') is a semimartingale relative to some filtration {iT : t ~ 0}. Given a Y : [0, T] x 0 - - t ~ with the properties that Y( · ,w) E C([O, T]; ~) for lP'-almost every wand that w...,.. Y(t, w) is Ft for each t E [0, T], show that Y is Stratonovich integrable on [0, T] with respect to X. In fact, show that

nJT

{T

Jo

Y(t)

o

dX(t) =

1 {T

2 Jo

Y(t) dX(t)-

1 {T

2 Jo

Y(t) dXT(t),

where each of the integrals on the right is the taken in the sense of Ito .

(iii) Let (f3(t), Ft, IP') be an ~n-valued Brownian motion. Given T E (O,oo), set {JT(t) = f3((T- t)+), f:[ = a({{JT(r) : r E [O,t]}), and show that, for each~ E ~n, ((~,{JT(t))JRn,f:[,lP') is a semimartingale with

!E,

T and bounded variation part t ...,.. - J~ dr. In particular, show that, for each ~ E ~n and g E C(~n; ~), t ...,.. g(f3(t)) is Stratonovich integrable on [0, T] with respect tot...,.. (~, f3(t))JRn. Further, if {gn}l' ~ C(~n; ~) and 9n - - t g uniformly on compacts, show that

({JT, {JT)( t) = t

{T

Jo

in lP'-measure.

1\

9n(f3(t))

od(~,{3(t))Rn

1 T

--t

g(f3(t))

od(~,{3(t))Rn

228

8 STRATONOVICH'S THEORY

{iv) Continuing with the notation in (iii), show that f(f3(T))- /(0))

=

t

loT 8d(f3(t)) o df3i(t)

for every f E C 1 (1Rn;JR). In keeping with the comment at the end of (i), it is important to recognize that although this form of Ito 's formula holds for all continuously differentiable functions, it is, in many ways less useful than forms which we obtained previously. In particular, when f is no better than once differentiable, the right hand side is defined only up to a lP'-null set for each T and not for all T's simultaneously. (v) Let (/3(t), Ft, lP') be an JR-valued Brownian motion, show that, for each T E (O,oo), t-v-; sgn(f3(t)) is Stratonovich integrable on [O,T] with respect to /3, and arrive at I/3(T)I =loT sgn(f3(t)) o df3(t). After comparing this with the result in (6.1.7), conclude that the local time f(T, 0) of /3 at 0 satisfies {T if3(t)i dt- f(T,O) -I/3(T)I = {T sgn(f3(T- t)) dMT(t),

lo

t lo where MT is the martingale part of (3T. In particular, the expression on the left hand side is a centered Gaussian random variable with variance T. EXERCISE 8.1.9. Ito's formula proves that f o Z E S(lP'; JR) whenever Z = (Z1, ... , Zn) E S(lP'; JR)n and f E C 2 (1Rn; JR). On the other hand, as Tanaka's treatment (cf. §6.1) of local time makes clear, it is not always necessary to know that f has two continuous derivatives. Indeed, both (6.1.3) and (6.1.7) provide examples in which the composition of a martingale with a continuous function leads to a continuous semimartingale even though the derivative of the function is discontinuous. More generally, as a corollary of the Doob-Meyer decomposition theorem (alluded to at the beginning of §7.1) one can show that f o Z will be a continuous semimartingale whenever Z E M 1oc(lP'; JRn) and f is a continuous, convex function. Here is a more pedestrian approach to this result.

(i) Following Tanaka's procedure, prove that f o Z E S(lP'; JR) whenever Z E S(lP'; JR)n and f E C 1 (1Rn; JR) is convex. That is, if Mi denotes the martingale part of zi' show that f(Z(t))-

t

lot aif(Z(T)) dM;(T)

is the bounded variation part of f o Z.

229

8.1 SEMIMARTINGALES AND STRATONOVICH INTEGRALS

=((

Hint: Begin by showing that when A(t) (Zi, Zj)(t))) lS:i,jS:n' A(t)-A(s) is IP'-almost surely non-negative definite for all 0 S s S t, and conclude that iff E C 2 (JR.n; JR.) is convex then

is IP'-almost surely nondecreasing.

(ii) By taking advantage of more refined properties of convex functions, see if you can prove that f o Z E S(IP'; JR.) when f is a continuous, convex function. EXERCISE 8 .1.1 0. If one goes back to the original way in which we described Stratonovich in terms of Riemann integration, it becomes clear that the only reason why we needed Y to be a semimartingale is that we needed to know that

L

(6-;;:Y) (6-;;:x)

ms2Nt

converges in IP'-probability to a continuous function of locally bounded variation uniformly for t in compacts.

{i) Let Z = (Z1, ... ,Zn) E S(IP';JR.)n, and set Y C 1 (JR.n;JR.). Show that, for any X E S(IP';JR.),

:L

(6.;;:Y)(6.;;:x)

~

ms2N t

f: 1t i=l

=f

o Z, where

f

E

Bd(Z(T))(zi,X)(dT)

0

in IP'-probability uniformly for t compacts.

(ii) Continuing with the notation in (i), show that

{t Y (T) o dX (T)

h

:= lim

tY

N~ooh

=

N ( T)

dX N ( T)

t Y(T)dX(T) + 21 t; Jo{t od(Z(T))(Zi,X)(dT),

Jo

n

where the convergence is in IP'-probability uniformly for t in compacts.

(iii) Show that (8.1.6) continues to hold for f E C 2 (JR.n; JR.) when the integral on the right hand side is interpreted using the extension of Stratonovich integration developed in (ii).

230

8 STRATONOVICH'S THEORY

EXERCISE 8.1.11. Let X, Y E S(lP'; JR.). In connection with Remark 8.1.7, it is interesting to examine whether it is sufficient to mollify only X when defining the Stratonovich integral of Y with respect to X.

(i) Show that I 01 Y([r]N) dXN (r) tends in lP'-probability to I 01 Y( r) dX( r). (ii) Define '1/JN(t) show that

= 1- 2N (t-

[t]N ), set ZN(t)

= I~ '1/JN(r) dY(r),

and

(iii) Show that 2N-l

fo

(~~X){~~ZN)

-1

1

'1/JN(t)(X, Y)(dt)

--+

0 in lP'-probability.

(iv) Show that for any Lebesgue integrable function a : [0, 1] dr tends to ~ I 01 a(r) dr.

--+

JR.,

I 01 '1/JN(r)a(r)

(v) Under the condition that (X, Y)(dt) = (3(t) dt, where (3 : [0, oo) x

n --+ [0, oo) is a progressively measurable function, that

I; Y(r) dXN(r) tends in lP'-probability to

Io1

use the preceding to see Y(r) o dX(r).

EXERCISE 8.1.12. One place where Stratonovich's theory really comes into its own is when it comes to computing the determinant of the solution to a linear stochastic differential equation. Namely, suppose that A = ((Aij)h:s:i,j:S:n E S(lP';Hom(IR.n;IR.n)) (i.e., Aij E S(lP';IR) for each 1 ~ i,j ~ n), and assume that X E S(lP'; Hom(IR.n; IR.n)) satisfies dX(t) = X(t) o dA(t) in the sense that n

dXi1(t) = L:xik(t) o dAk1(t)

for all1 ~ i,j ~ n.

k=l

Show that det(X(t)) = det(X(O))eTrace(A(t)-A(o)).

8.2 Stratonovich Stochastic Differential Equations

Since every Stratonovich integral can be converted into an Ito integral, it might seem unnecessary to develop a separate theory of Stratonovich stochastic differential equations. On the other hand, the replacement of

8.2

231

STRATONOVICH STOCHASTIC DIFFERENTIAL EQUATIONS

a Stratonovich equation by the equivalent Ito equation removes the advantages, especially the coordinate invariance, of Stratonovich's theory. With this in mind, we devote the present section to a non-Ito analysis of Stratonovich stochastic differential equations. Let 1P'0 denote the standard Wiener measure for r-dimensional paths p = (Pb ... ,pr) E C([O, oo ); ll~.r). In order to emphasize the coordinate invariance of the theory, we will write our Stratonovich stochastic differential equations in the form r

dX(t, x,p) = Vo(X(t, x,p)) dt + L Vk(X(t, x,p))

(8.2.1)

o

dpk(t)

k=l

with X(O, x,p)

= x,

where, for each 0 ::; k ::; r, Vk : JRn ---+ JRn is a smooth function. To see what the equivalent Ito equation is, notice that n

( (Vk)i (X(·, x,p)) ,pk )(dt) = L Oj (Vk)i (X(t, x,p)) (Xj( ·, x,p),pk)(dt) j=l n

r

= L L(Vi)j8j(Vk)i(X(t, x,p))(Pt,Pk)(dt) = Dvk (Vk)i (X(t, x,p)) dt, j=ll'=l

where Dvk = I.:~=l (Vk)j8j denotes the directional derivative operator determined by Vk. Hence, if we think of each Vk, and therefore X(t, x,p), as a column vector, then the Ito equivalent to (8.2.1) is (8.2.2)

dX(t, x,p) = a(X(t, x,p))dp(t) + b(X(t,x,p)) dt with X(O,x,p) = x,

where a(x) = (V1(x), ... , Vr(x)) is then x r-matrix whose kth column is Vk(x) and b = Vo +! 2:.:~ Dvk Vk. In particular, if

L=

1

n

2L

n

aij8i8j

i,j=l

+ L bioi,

where a= aa T,

i=l

then L is the operator associated with (8.2.1) in the sense that for any f E C1 •2 ([0,T] x JRn)

(t(t AT, X(t, x, p))

-!,'AT (8d + Lf)(X(r,

x, p)) dr, 8.,

pO)

232

8

STRATONOVICH'S THEORY

is a local martingale. However, a better way to write this operator is directly in terms of the directional derivative operators Dvk. Namely, (8.2.3)

Hormander's famous paper [10] was the first to demonstrate the advantage of writing a second order elliptic operator in the form given in (8.2.3), and, for this reason, (8.2.3) is often said to be the Hormander form expression for L. The most obvious advantage is the same as the advantage that Stratonovich's theory has over Ito's: it behaves well under change of variables. To wit, if F : JR." ----) JR." is a diffeomorphism, then

where is the "pushforward" under F of

vk.

That is, D F. vk 'P = Dvk ('P

0

F).

§8.2.1. Commuting Vector Fields: Until further notice, we will be dealing with vector fields V0 , ... , Vr which have two uniformly bounded, continuous derivatives. In particular, these assumptions are more than enough to assure that the equation (8.2.2), and therefore (8.2.1), admits a lP' 0 -almost surely unique solution. In addition, by Corollary 4.2.6 or Corollary 7.3.6, the lP'0 -distribution of the solution is the unique solution lP'~ to the martingale for L starting from x. In this section we will take the first in a sequence of steps leading to an alternative (especially, non-Ito) way of thinking about solutions to (8.2.1). Given~= (~ 0 , ... , ~r) E JR.r+l, set V~ = 2:~= 0 ~k Vk, and determine E(~, x) for x E JR." so that E(O, x) = x and d dt E(tC x)

= V~ (E(t~, x) ).

From elementary facts (cf. §4 of Chapter 2 in [4]) about ordinary differential equations, we know that (~, x) E JR.r+l x JR.n ~-----+ E(~, x) E JR." is a twice continuously differentiable function which satisfies estimates of the form (8.2.4)

iE(Cx)- xi :S Cl~l,

ID.;kE(~,x)l V IDx,E(~,x)i :S CeviEI

l8.;k8~£E(~,x)i V l8~k8xiE(~,x)l V l8xi8xjE(~,x)i :S CeviEI,

8.2 STRATONOVICH STOCHASTIC DIFFERENTIAL EQUATIONS

233

for some C < oo and v E [O,oo). Finally, define (8.2.5)

For continuously differentiable, 1R.n-valued functions V and W on JR.n, we will use the notation [V, W] to denote the 1R.n-valued function which is determined so that D[v, w] is equal to the commutator, [Dv, Dw] = Dv o Dw- Dw o Dv, of Dv and Dw. That is, [V, W] = DvW- DwV. LEMMA 8.2.6. Vk(~,x) = Vk(E(~,x)) for all 0:::; k:::; rand (~,x) E JR.r+l x JR.n if and only if [Vk, V£] = 0 for all 0:::; k, £:::; r.

PROOF: First assume that Vk(~,x) = Vk(E(~,x)) for all k and (~,x). Then, if eg is the element of JR.r+l whose fth coordinate is 1 and whose other coordinates are 0, we see that

and so, by uniqueness, E(~+teg,x) = E(teg,E(~,x)). In particular, this leads first to

and thence to

82

Dvk Vf(x) = 8 s8t E(teg, E(sek, x))

t 8 s E(sek, E(teg, x)) = 882

Is=t=O

I

s=t=o =

Dv£ Vk(x).

To go in the other direction, first observe that Vk(~,x) = Vk(E(~,x)) is implied by E(~+tek,x) = E(tek,E(~,x)). Thus, it suffices to show that E(~ + ry,x) = E(ry,E(~,x)) follows from [V~, V'1] = 0. Second, observe that E(~ + 'TJ, x) = E(ry, E(~, x)) is implied byE(~, E(ry, x)) = E(ry, E(~, x)). Indeed, if the second of these holds and F(t) = E(t~,E(try,x)), then F(t) = (V~ + V'7)(F(t)), and so, by uniqueness, F(t) = E(t(~ + ry),x). In other words, all that remains is to show that E(~,E(ry,x)) = E(ry,E(~,x)) follows ~rom [V~, V'1] 0. To this end, set F(t) = E(~,E(try,x)), and note that F(t) = E(~, · )*V'1(E(try,x)). Hence, by uniqueness, we will know that E(~,E(try,x)) = F(t) = E(try,E(~,x)) once we show that E(~, · )*V'1 = V'1(E(~, · )). But

=

d~E(s~, · );:- 1 V'7(E(s~, · )) = [V~, V'1l(E(s~,. )), and so we are done.

D

234

8 STRATONOVICH'S THEORY

THEOREM 8.2. 7. Assume that the vector fields Vk commute. Then the one and only solution to (8.2.1) is (t,p) "'""'X(t, x,p) = E( (t,p(t)), x ). PROOF: Let X(·, x, p) be given as in the statement. By Ito's formula, T

dX(t,x,p) = Vo ( (t,p(t)), X(t, x,p) )dt+ L vk( (t,p(t)), X(t,x,p)) odpk(t), k=l

and so, by Lemma 8.2.6, X ( · , x, p) is a solution. As for uniqueness, simply rewrite (8.2.1) in its Ito equivalent form, and conclude, from Theorem 5.2.2, that there can be at most one solution to (8.2.1). 0 REMARK 8.2.8. A significant consequence of Theorem 8.2.7 is that the solution to (8.2.1) is a smooth function of p when the Vk's commute. In fact, for such vector fields, p"'""' X(·, x,p) is the unique continuous extension to C ([0, oo); of the solution to the ordinary differential equation

lRn

T

X(t,x,p) = Vo(X(t,x,p))

+ LVk(X(t,x,p))Pk(t) k=l

for p E C 1 ([0,oo);~r). This fact should be compared to the examples given in §3.3. §8.2.2. General Vector Fields: Obviously, the preceding is simply not going to work when the vector fields do not commute. On the other hand, it indicates how to proceed. Namely, the commuting case plays in Stratonovich's theory the role that the constant coefficient case plays in Ito's . In other words, we should suspect that the commuting case is correct locally and that the general case should be handled by perturbation. We continue with the assumption that the Vk 's are smooth and have two bounded continuous derivatives. With the preceding in mind, for each N;:::: 0, set xN (0, x,p) =X and

(8.2.9)

xN (t, x,p) = E(D..N (t,p), xN ([t]N, x,p)) where D._N (t,p)

= (t- [t]N,p(t)- p([t]N )).

Equivalently (cf. (8.2.5)):

dXN(t,x,p) = V0 (D..N(t,p),XN([t]N,x,p)) dt

+ L vk(D..N (t,p), xN ([t]N, x,p)) o dpk(t). T

k=l

8.2 STRATONOVICH STOCHASTIC DIFFERENTIAL EQUATIONS

235

Note that, by Lemma 8.2.6, XN(t,x,p) = E((t,p(t)),x) for each N ~ 0 when the Vk 's commute. In order to prove that {X N ( ·, x, p) : N ~ 0} is JP>0 -almost surely convergent even when the Vk 's do not commute, we proceed as follows. Set DN (t, x,p) X(t, x,p)- XN (t, x,p) and Wk(~, x) Vk(E(~,x))- Vk(~,x), and note that DN(o,x,p) = 0 and

=

=

dDN (t, x,p) =(Vo(X(t,x,p))- Vo(XN (t,x,p))) dt r

+ L(vk(X(t,x,p))- vk(xN (t,x,p)))

o

dpk(t)

k=l

+ Wo(~N (t,p), xN ([t]N, x,p)) dt

(8.2.10)

r

+ LWk(~N(t,p),XN([t]N,x,p)) odpk(t). k=l

LEMMA 8.2.11. There exist a C < oo and v

for

all~, 'TJ

> 0 such that

E JRr+I, x E JRn, and 0 :::::; k :::::; r. In particular, if

Wk(~,x)

= Wk(~,x)- L~e[V£, vk](x), £#

then Wk(O,x) = 0 and ia~Wk(~,x)i:::::; Cl~le"l~l for all (~,x) E JRr+l x JRn and some C < oo and v > 0. PROOF: In view of the estimates in (8.2.4), it suffices for us to check the first statement. To this end, observe that E(ry, E(~, x)) = E(~, x)

+ V71 (E(~, x)) + ~Dv, V71 (E(~, x)) + R 1 (~, ry,x),

where R 1 (~, ry, x) is the remainder term in the second order Taylor expansion ofry""' E(·,E(~,x)) around 0 and is therefore (cf. (8.2.4)) dominated by constant C 1 < oo times lrJI 3e"l711. Similarly,

+ V~(x) + ~Dv~ ~(x) + R2(~,x), V71 (E(~,x)) = V71 (x) + D~V71 (x) + R 3 (~,ry,x), Dv, V71 (E(~,x)) = Dv, V71 (x) + R4 (~,ry,x), E(~ + ry,x) = x + Dv~+" ~+ 71 (x) + ~D~~+" ~+ 71 (x) + R2 (~ + ry,x), E(~,x) = x

where

IR2(~,x)l :S C2l~l 3 e" 1 ~ 1 ,

IR3(~,ry,x)l :S C31~1 2 lrJie"l~l,

IR4(~,ry,x)l :S C41~11"11 2 e"l~l.

236

8

STRATONOVICH'S THEORY

Hence

E(TJ, E(C x))- E(~

+ TJ, x)

Hv~, Vry] (x)

=

where IR 5 (~,TJ,x)l::::; C5 (1~1 and so the required estimate follows.

+

+ R 5 (~, ry, x),

ITJI) 3 ev([~I+I7JI),

0

Returning to (8.2.10), we now write

dDN (t, x,p) = Wo(~N (t,p), XN ([t]N, x,p)) dt

I:

+~

[Vc, vk](XN([t]N,x,p))~e"'(t,p)dpk(t)

l:':_:k#£: 0 and q E [1, oo) there exists a C(T, q) < oo

such that 1

lEll'o[iiX(-,x,p)-XN(-,x,p)jjf;,tJr 5,C(T,q)tTN iftE [O,T]. §8.2.3. Another Interpretation: In order to exploit the result in Theorem 8.2.12, it is best to first derive a simple corollary of it, a corollary containing an important result which was proved, in a somewhat different form and by entirely different methods, originally by Wong and Zakai [42]. Namely, when p is a locally absolutely continuous element of C ([0, oo); JR.n), then it is easy to check that, for each T > 0,

lim

N~~

jjXN(- ,x,p)- X(- ,x,p)jj[o ' TJ = 0,

where here X(· ,x,p) is the unique locally absolutely continuous function such that (8.2.13) X(t,x,p)

~ x+

l(

Vo(X(r,x,p))

8.2.14. For each N;::: 0 andp be determined so that

THEOREM

E

for each m ;::: 0. Then, for all T > 0 and

E

+

t,

V,(X(r,x,p))p,(r)) dr.

C([O, oo); lRr), let pN

E

C([O, oo); lRr)

> 0,

where X(· ,x,pN) is the unique, absolutely continuous solution to (8.2.13).

238

8 STRATONOVICH'S THEORY

PROOF: The key to this result is the observation that XN(m2-N,x,p) = X(m2-N,x,pN) for all m ;::: 0. Indeed, for each m ;::: 0, both the maps t E (m2-N, (m + 1)2-N) f--+ X(t, x,pN) and t E(mTN, (m + 1)TN) ---+

E(t(TN,p((m + l)TN)- p(mTN)), X(mTN, x,p)) E !Rn

are solutions to the same ordinary differential equation. Thus, by induction on m;::: 0, the asserted equality follows from the standard uniqueness theory for the solutions to such equations. In view of the preceding and the result in Theorem 8.2.12, it remains only to prove that lim lP'0 ( sup IX(t, x,pN)- X([t]N, x,pN)i

N->oo

tE[O,TJ

v IXN (t, x,p)- xN ([t]N, x,p)l :::::E) for all T E (0, oo) and

E

= 0

> 0. But

IX(t, x,pN)- X([t]N, x,pN)i :::; CTN v IP([t]N +TN)- p([t]N )I

and

IXN(t,x,p)- XN([t]N,x,p)i:::; CeviiPIIio.rJTN V ip(t)- p([t]N)I,

and so there is nothing more to do.

0

An important dividend of the preceding is best expressed in terms of the support in C ( [0, oo); !Rn) of the solution lP'~ to the martingale problem for L (cf. (8.2.3)) starting from x. Namely, take (cf. Exercise 8.2.16 below for more information) S(x; Vo, ... , V,.) = {X(-, x,p): p E C 1 ([0, oo); IR'") }.

COROLLARY 8.2.15. Let L be the operator described by (8.2.3), and (cf Corollary 4.2.6) let lP'~ be the unique solution to the martingale problem for L starting at x. Then the support oflP'~ in C([O, oo); !Rn) is contained in the closure there of the set S (x; V0 , ... , V,.). PROOF: First observe that lP'~ is the lP'0 -distribution of p ""' X(· ,x,p). Thus, by Theorem 8.2.14, we know that

lP'~(S(x;V0 ,

...

,Vr));::: lim lP' 0 ({p: X(-,x,pN) E S(x;Vo, ... ,V,.)}). N->oo

8.2 STRATONOVICH STOCHASTIC DIFFERENTIAL EQUATIONS

239

But, for each n EN andp E C([O,oo);lR,.), it is easy to construct {p~: 0} 0. Hence, X(·, x,pN) E S(x; V0 , ... , V,.) for all n EN and p E C([O, oo); lRn). 0

§8.2.4. Exercises EXERCISE 8.2.16. Except when span( {V1 (x), ... , Vr(x)}) = lRn for all x E JRn, in which case S(x; V0 , ... , Vr) is the set of p E C([O, oo ); lRn) with p(O) = x, it is a little difficult to get a feeling for what paths are and what paths are not contained in S(x; V0 , ... , Vr ). In this exercise we hope to give at least some insight into this question. For this purpose, it will be helpful to introduce the space V(V0 , ... , Vr) of bounded V E C 1 ([0,oo);1Rn) with the property that for all a E lR and x E lR n the integral curve of V0 +a V starting at xis an element of S(x; Vo, ... , Vr)· Obviously, Vk E V(V0 , ... , Vr) for each 1::;k::;r.

(i) Given V, W E V(Vo, ... , Vr) and (T, x) E (0, oo) x lRn, determine X E C([O, oo); lRn) so that X(t)

=

{

X+ J~ V(X(T)) dT

if t E [0, T]

X(T)

if t

+ J; W(X(T)) dT

> T,

and show that X E S(x; V0 , ... , Vr)·

(ii) Show that if V, WE V(V0 , ... , Vr) and rp,'lj; E C~(lRn;lR), then rpV 1/JW E V(Vo, ... , Vr). Hint: Define X: JR 3 x lRn----+ JRn so that X((O,O,O),x) = x and d dtX(t(a, /3, !), x)

=

(aV0

Given N EN, define YN: [0, oo)

X

+ f3V + rW)(X(t(a,/3, !), x).

lRn----+ lRn so that YN(t, x) equals

X(t(1, 2rp(x), 0), x) { X((t- 2-N- 1 )(1, 0, 21/J(x)), YN(2-N- 1 )

YN(t- [t]N, YJ\t([t]N,x))

+

for 0:::; t:::; 2-N- 1 for 2-N- 1

:::;

t:::; 2-N

fort~ 2-N.

Show that YN( ·, x)----+ Y( ·, x) in C([O, oo); lRn), where Y(., x) is the integral curve of V0 + rpV + 1/JW starting at x.

240

8 STRATONOVICH'S THEORY

(iii) If V, W E V(Vo, ... , Vr) have two bounded continuous derivatives, show that [V, W] E V(Vo, ... , Vr). Hint: Use the notation in the preceding. For N EN, define YN : [0, oo) x ]Rn ---+ JRn so that YN(t, x) is equal to X ( (t, 2N/2+ 2t, 0), X)

X((t- TN-2,0,2N/2+2(t- 2N-2), YN(TN-2,x)) X((t- TN-l, -2N/2+ 2(t- 2-N- 1),0), YN(TN- 1 ,x)) X((t- 3TN- 2,0, -2N/ 2+ 2(t- 32N- 2 )), YN(3TN- 2 ,x)) YN(t- [t]N, YN([t]N,x)) according to whether 0 :::; t :::; 2-N- 2 , 2-N- 2 :::; t :::; 2N-I, 2-N-l :::; t :::; 32-N- 2 , 32-N- 2 :::; t:::; 2-N, or t 2: 2-N. Show that YN( ·, x) ---+ Y( ·, x) in C([O,oo);lRn), where Y( · ,x) is the integral curve of V0 + [V, W] starting at x.

(iv) Suppose that M is a closed submanifold of JRn and that, for each x E M, {V0 (x), ... , Vr(x)} is a subset of the tangent space TxM toM at x. Show that, for each x E M, S(x; Vo, ... , Vr) ~ C([O, oo); M) and therefore that lP'~('v't E [O,oo) p(t) EM)= 1. Moreover, if {V1, ... , Vr} ~ C 00 (lRn;lRn) and Lie(V1, ... , Vr) is the smallest Lie algebra of vector fields on JRn containing {V1, ... , Vr }, show that S(x; Vo, ... , Vr) = {p E C([O, oo); M)

: p(O) = x}

if M :3 x is a submanifold of JRn with the property that

Vo(Y) E TyM = {V(y): V E Lie(V1, ... , Vr)}

for ally EM.

8.3 The Support Theorem Corollary 8.2.15 is the easier half of the following result which characterizes the support of the measure lP'~. In its statement, and elsewhere, we use the notation (cf. Theorem 8.2.14) llqiiM,r = llql\1 IIP(I;IR.') forME N, q E C 1 ([0,oo);1Rr)), and closed intervals I~ [O,oo). THEOREM 8.3.1. The support oflP'~ in C([O,oo);lRn) is the closure there of (cf. Corollary 8.2.15) S(x; V0 , ... , Vr)· In fact, if for each smooth g E C1 ([0, oo); JRn) X(., x, g) is the solution to (8.2.13) with p = g, then (8.3.2)

lP'0

(iiX( ·,x,p)- X(· ,x,g)jj[o,r] < tiiiP- giiM,[o,r]:::; 8) ---+

1

as first 8 ""' 0 and then M

-+

oo.

8.3

241

THE SUPPORT THEOREM

for all T E (0, oo) and

E

> 0. 3

Notice that, because we already know supp(JP~) ~ S(x; Vo, ... , Vr), the first assertion will follow as soon as we prove (8.3.2). Indeed, since

p(O) = 0

===}

liP-

gii~.[O,T]:::; 2M+l

[2MT]+l

L

lp(mTM)- g(mTM)I2

m=l

and, for any£ E

z+,

the lP0 -distribution of

has a smooth, strictly positive density, we know that (8.3.3)

Hence, (8.3.2) is much more than is needed to conclude that

for all E > 0. To begin the proof of (8.3.2), first observe that, by Theorem 8.2.12 and (8.3.3), 1P0 (IIX( · ,x,p)- X(· ,x,g)ll[o,T] > EIIIP- giiM,[O,T]:::;

EIIIP- giiM,[O,T]:::; ~~liP- giiM,[o,T]:::;

~I liP- giiM,[O,T]

:::;

oo

sup N>M

-M)Ii2[o,T] ] -- 0,

IEpo [11D M,N ( ·, x,ph

llh-giiM,[O,T)9

and this turns out to be quite delicate. To get started on the proof of (8.3.12), we again break the computation into parts. Namely, because

8.3

247

THE SUPPORT THEOREM

fJM.N

(t

'

x

'

p-M) h

can be written as

-MN

where wk , (t,x,p) denotes

-MN

and wk,k (t, x,p) is equal to

With the exception of those involving p~1 ( T), none of these terms is very difficult to estimate. Indeed,

lEII""

[111.0 wk,k - M,N ( T, x,ph-M) dT 112[O,T]

l

::;

T

1T 0

lE II""

[I wk,k -

M N ( T,

-M) x,ph

12]

dT,

and

At the same time, by (8.2.4), times

w:t,N (t, x,p~ ) is dominated by a constant 1

(li5~1 ([T]N)- P~1 ([T]M) I + IXN ([T]N, x,p~1 )- XN ([7]M, x,p~1) l)evllt>M( · .vt'lllrm

M,

248

8

STRATONOVICH'S THEORY

and so, in view of (8.3.9) and (8.3.10), all the above terms will have been handled once we check that, for each q E [1, oo), there exists a Cq < oo such that lEll'o

(8.3.13)

[IIXN (. 'x,j5~1)- XN (mTM, x,j5~) ~~~m,M] i :::; Cq (1

+ llhiiMJm,M )T-¥

for all x E ~n, (m, M) E N2 , and N 2: M. To this end, first note that, because XN ( +m2-M' x,p) = xN ( 'xN (m2-M' x,p), 6m2-MP), it suffices to treat the case when m = 0. Second, write XN(t,x,jj~I)- x as 0

0

t

- L r (vk(6.N(T,j5~),x,f5~)Pt1 (T)- Vk(6.N(T,j5~1 ),x,j5f,I)ht1 (T)) dT, r

k=l

Jo

and apply (8.2.4) and standard estimates to conclude from this that

is dominated by an expression of the form on the right hand side of (8.3.13). §8.3.4. The Support Theorem, Part IV: The considerations in §8.3.3 reduce the proof of (8.3.12) to showing that, for each T E [0, oo), (8.3.14)

and for this purpose it is best to begin with yet another small reduction. Namely, dominate

lifo' w~·N (7, x,j5f,1 )p~1 (T) dTII

[O,T]

by

8.3

249

THE SUPPORT THEOREM

Again, the first of these is easy, since, by lines of reasoning which should be familiar by now:

for appropriate, finite constants C and C'. Hence, the proof of (8.3.14), and therefore (8.3.12), is reduced to showing that (8.3.15)

lim

sup lElP' 0

M---+oo N>M

[

max m~2MT

IlornTM w;r·N (7, x,pf!)pf: (T) dTI 2]

= 0

uniformly with respect to h with llh- giiM,[O,T] :::; 1. -MN To carry out the proof of (8.3.15), we decompose wk , (T,x,p) into the sum N

""" L..t

('ML( vk ' T,

L=M+l

X N ([T]M, x,p),p)

+ wkML, (7, X N ([T]M, x,p),p) ) ' A

where vt·L(T,x,p)

=vk(~L(T,p), XL([T]L- [T]M, x, b[r]MP)) - vk(~L(T,p), xL- 1([T]L- [T]M, x, J[TJMP)),

and w;r·L(T,x,p)

= Vk(~L(T,p),xL-l([T]L- [T]M,X,b[T],uP)) - Vk (~L-l(T,p), xL-l([T]L-1 - [T]M, X, J[T]MP)).

After making this decomposition, it is clear that (8.3.15) will be proved once we show that

(8.3.16)

250

8 STRATONOVICH'S THEORY

. g1ven . by where F kM.N( · m, x,p ) 1s

L 2M J N

L=fvf+l

iVkM,L (r, xN (mTM, x,p),p) I dT,

lm,M

and that (8.3.17)

lim

M---+oo

lEiP'0 [max

sup N>M

rn JR,

Further, if then

y

E JRr

>-->

f~1

E C([O,oo);lR 7 ) is given by fj7(t)

= (t 1\ 2-M)y,

while

where 'lt(Y)

= (27rt)-~ exp ( -~~~2 ).

Hence, because W, and therefore also

~ M, is bounded and continuous, the asserted equality follows by choosing a sequence off's which form an approximate identity at h((m + 1)2-M)h(m2-M). Given the preceding, there are two ways in which the proof can be completed. One is to note that, trivially,

252

8

STRATONOVICH'S THEORY

whereas, by the preceding,

Hence interpolation provides the desired estimate. Alternatively, one can apply Holder's inequality to get

perform an elementary Gaussian integration, and arrive at the desired result after making some simple estimates. 0 If we now apply Lemma 8.3.19 with

and q = n, we see from Theorem 8.2.12 that

for an appropriate choice of nondecreasing r "-"+ C (r). In view of the discussion prior to Lemma 8.3.19, (8.3.18), and therefore (8.3.16), is now an easy step.

§8.3.5. The Support Theorem, Part V: What remains is to check (8.3.17). For this purpose, we have to begin by rewriting w:t,L(r,x,p) as vk ( ~L(r,p), E(~L(r,p), xL- 1([r]L-1 - [r]M, x, 8[-r]MP))) - Vk ( ~L(r,p)

+ ~L(r,p), xL- 1((r]L-1- [r]M, X, 8[-r]MP) ),

where ~L(r,p) = ([r]L- [r]L-1,P([r]L)- p([r]L-1)· Having done so, one sees that Lemma 8.2.11 applies and shows that w~·L(r,x,p) can be written as

~ .~::)vt, vkl(xL-l([r]L-1- [r]M, x, 8[-r]MP))~f(r,p) + R~/(r,x,p), if.k

8.3

253

THE SUPPORT THEOREM

where

Thus, since

we are left with showing that, for each 1 :::; k :::; r and £ =/= k, (8.3.20)

lim

M--+OCJ

lEJP0

sup

N>M

llh-giiM,[O,Tj::;l

[

max

If

m 0.

EXERCISE 8.3.22. Without much more effort, it is possible to refine (8.3.2) a little. Namely, given a E (0, ~) and T E (0, oo ), set

IIPII(;~ =

sup

[ ' ]

O