North-Holland Mathematical Library Board of Advisory Editors:
M. Aran, H. Bass, J. Eells, W. Feit, P. J. Freyd, F. W. Gehring, H. Halberstam, L. V. Hdrmander, J. H. B. Kemperman, H. A. Lauwerier, W. A. J. Luxemburg F. P. Peterson, I. M. Singer and A. C. Zaanen
VOLUME 24
NORTH-HOLLAND PUBLISHING COMPANY
AMSTERDAM OXFORD NEW YORK
Stochastic Differential Equations and Diffusion Processes Second Edition by Nobuyuki IKEDA Department of Mathematics Osaka University
Shinzo WATANABE Department of Mathematics Kyoto University
NORTH-HOLLAND PUBLISHING COMPANY
AMSTERDAM OXFORD NEW YORK 1989
is
KODANSHA LTD. TOKYO
Qvw Copyright © 1981,1989 by Kodansha Ltd. All rights reserved.
No part of this book may be reproduced in any form, by photostat, microfilm, retrieval system, or any other means, without the written permission of Kodansha Ltd. (except in the case of brief quotation for criticism or review)
ISBNO-444-873783
ISBN4-06-203231-7 (Japan)
Published in co-edition with North-Holland Publishing Company Distributors for Japan:
KODANSHA LTD. 12-21 Otowa 2-chome, Bunkyo-ku, Tokyo 112, Japan Distributors outside Japan, U.S.A. and Canada:
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ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 655 Avenue of the Americas New York, NY 10010 U.S.A. Printed in Japan
It is with our deepest gratitude and warmest affection that we dedicate this book to our teacher
Kiyosi Ito who has been a constant source of knowledge and inspiration.
Preface to the Second Edition
A considerable number of corrections and improvements have been made in this second edition. In particular, major and substantial changes are in Chapter III and Chapter V where the sections treating on excursions of Brownian motion and the Malliavin calculus have been much more expanded and refined. Also, sections discussing complex (conformal) martingales and Kahler diffusions have been inserted.
We would express our sincere thanks to all those who kindly communicated us mistakes in and gave us helpful camments on the first edition of this book. November 1988 N. IKEDA S. WATANABE
Vii
Preface
The theory of stochastic integrals and stochastic differential equations was initiated and developed by K. Ito. In 1942 ([57], cf. also [62]) this theory was first applied to Kolmogorov's problem of determining Markov processes [85]. Let yt be a Markov process on the real line R' and, for each instant to, let Fro,, = Fto,t(yta) be the conditional probability distribution of y, given yto. In almost all interesting cases we may assume that F*a(`-t0)-'3
converges as t j to to a probability distribution on R', which we shall denote by Dyo. (Here [a] is the integer part of a and *k denotes the k-fold convolution.) Dyto is thus an infinitely divisible distribution. Kolmogorov's problem is now formulated as follows: given a system L(t,y) of infinitely divisible distributions, find the Markov process yt with given initial distribution such that (0.1)
Dy, = L(t,y,).
Kolmogorov [85] and Feller [26] both succeeded in obtaining Markov processes by solving Kolmogorov's differential equations (equations equivalent to (0.1) for the transition probabilities), thus establishing an analytical method in probability theory. This method has been further developed in connection with Hille-Yosida's theory of semigroups.
In contrast with these analytical methods, a probabilistic approach suggested by Levy and established by Ito enables one to construct sample functions of the process yt directly. Consider the case when L(t,y) = G (a(t,y),b(t,y)), where G(a,fl) is the Gaussian distribution with mean a and standard deviation fl. The intuitive meaning of (0.1) then is that the infinitesimal change of the conditional distribution given y, = y is G(a(t,y)dt,
b(t,y),/dt). On the other hand, if x, is a Brownian motion*' (Wiener The Brownian motion is a random movement of microscopic particles which was discovered by the English botanist R.Brown. Its physical theory was investigated by Einstein ([24]). The mathematical theory was initiated and .developed by Wiener *
(177] and L4vy [99]. ix
PREFACE
x
process), the "stochastic differential"* dx, = x,+d, - x, conditioned on the past {x,, s < t} satisfies the law G(0, (0.2)
dt). This suggests that we write
dy, = a(t,y,)dt + b(t,y,)dx,;
thus y, should be determined as a solution of the integral equation (0.3)
Y, = Yo + f ' a(s,y,) ds + S b(s,y,)dx,. 0
0
However, Wiener and Levy had already shown that x, is nowhere differentiable for almost all sample paths and so the integral with respect to dx, cannot be defined in the ordinary sense. In order to get around this difficulty, Ito introduced the notion of "stochastic integrals". Using this idea he was then able to obtain y, as the unique solution of (0.3) fora given initial value yo under a Lipschitz condition on a(t,y) and b(t,y); furthermore, he showed that this y, actually satisfies the original equation (0.1). Independently in the Soviet Union, S. Bernstein ([4] and [5]) introduced a stochastic difference equation and showed that the random variable determined by this equation has the fundamental solution of Kolmogorov's equation as its limiting distribution. Gihman ([35], [36] and [37]) carried out Bernstein's program independently of Ito and succeeded in constructing a theory of stochastic differential equations. Today Ito's theory is applied not only to Markov processes (diffusion processes) but also to a large class of stochastic processes. This framework provides us with a powerful tool for describing and analyzing stochastic processes. Since Ito's theory may be considered as an integral-differential calculus for stochastic processes, it is often called Ito's stochastic analysis or stochastic calculus. The main aim of the present book is to give a systematic treatment of the modem theory of stochastic integrals and stochastic differential equations. As is customary nowadays, we will develop this theory within the martingale framework. The theory of martingales, which was initiated and developed by J.L. Doob, plays an indispensable role in the modern theory of stochastic analysis. The class of stochastic processes to which Ito's theory can be applied (usually called ItO processes or locally infinitely divisible processes) is now extended to a class of stochastic processes called semimartingales. Such processes appear to be the most gen*
The notion of stochastic differentials was considered by Levy ([99], [100] and [101]);
he used a suggestive notation c./dt to denote dx, where g is a random variable with distribution G(0,1).
PREFACE
xi
eral for which a unified theory of stochastic calculus can be developed.* A somewhat different type of stochastic calculus has been introduced by Stroock and Varadhan under the name of martingale problems (cf. [160] for this elegant and powerful approach). In this book, however, we prefer to follow Ito's original approach although an influence of Stroock and Varadhan will be seen in many places. Chapter I contains some preliminary materials that are necessary for the development and understanding of our book. In particular, we review the theory of martingales. The notion of stochastic integrals and Ito's formula are discussed in Chapter II. These constitute the core of stochastic analysis. In Chapter III, the results of Chapter II are reformulated in a more convenient form for applications. In particular, a type of stochastic integral introduced by Stratonovich [153] and Fisk [27] will be formulated as an operation (called symmetric multiplication) in the framework of the theory developed there. It will play an important role in Chapters V and VI. The general theory of stochastic differential equations is discussed in Chapter IV. The solutions of these equations will not necessarily be nonanticipative functionals of the accompanying Brownian paths; we distinguish those solutions having this property as strong solutions. Stochastic differential equations are then used to construct diffusion processes for given differential operators and, in the case of a state space with boundary, for given boundary conditions. In discussing stochastic differential equations with boundary conditions, we are naturally led to stochastic differential equations which are based on semimartingales more general than Brownian motions and on Poisson random measures over general state spaces. The main objects to be studied in Chapter V are the flows of diffeomorphisms defined on a differentiable manifold which are associated with a given system of vector fields. Stochastic calculus enables us to transfer many
important operations in analysis and differential geometry which are defined on smooth curves through ordinary differential-integral calculus to a class of stochastic curves. Flows of diffeomorphisms are one such example. By making use of affine connections or Riemannian connections we can construct flows of diffeomorphisms over the space of frame bundles. The most general non-singular diffusion can then be obtained by projection. We can also realize Ito's stochastic parallel displacement of tensor fields by this method. These ideas are due to Eells, Elworthy and Malliavin. In discussing a similar problem in the case of a manifold with boundary, we * The modern theory of semimartingales and the stochastic calculus based on them have been extensively developed in France by Meyer, Dellacherie, Jacod etc. (cf.
Jacod [77]).
xii
PREFACE
exhibit a probabilistic condition which characterizes the reflecting diffusion processes in the normal direction among all reflecting diffusion processes in oblique directions. Many of the mathematical objects obtained through stochastic analysis,e.g.,strong solutions of stochastic differential equations, are functionals of the accompanying Brownian paths, i.e., Brownian functionals or Wiener functionals. In Sections 7 and 8 of Chapter V we introduce a recent work of Malliavin which analyzes solutions of stochastic differential equations
as Wiener functionals. There he obtains the remarkable result that the smoothness problem for the heat equation can be treated by this probabilistic approach. Some miscellaneous materials are presented in Chapter VI. In the first half we discuss some topics related to the comparison theorem. Although the
results obtained by this method are usually a little weaker than those obtained via partial differential equations, the technique is simpler and it sometimes produces sharp results. The topics discussed in the second half are more or less related to the notion of stochastic area as introduced by Levy. The approximation theorems considered in Section 7 are concerned with the transfer of concepts defined on smooth curves to stochastic curves. There are many topics in the area of stochastic differential equations and their applications which are not considered in this book: e.g., the theory of stochastic control, filtering and stability, applications to limit theorems, applications to partial differential equations including nonlinear equations, etc. We especially regret that we could not include the important works by N.V. Krylov on the estimates for the distributions of stochastic integrals and their applications (cf. e.g. [91] and [92]). We wish to express our gratitude to G. Maruyama for his constant interest and encouragement during the writing of this book. We are also indebted to H. Kunita for his critical comments and constructive suggestions and to S. Nakao who greatly assisted us in the proofs of Theorems VI-7.1 and VI-7.2. Among the many others who have made contributions, we would particularly thank H. Asano, S. Kotani, S. Kusuoka, S. Manabe, Y. Okabe, H. Okura and I. Shigekawa for their useful comments, suggestions and corrections. Finally, we also express our appreciation to T.H. Savits who read the manuscript and suggested grammatical improvements, and to the editorial staff of Kodansha Scientific who provided much help in the publication of this book.
Osaka and Kyoto, February 1980 Nobuyuki IKEDA Shinzo WATANABE
Contents Preface to the Second Edition .................................... vii
Preface ............................................................ix General notation ................................................ ..xv Chapter I. Preliminaries ........................................ I 1.
Basic notions and notations,
1
2. Probability measures on a metric space, 2 3. Expectations, conditional expectations and regular conditional probabil4. 5. 6. 7. 8. 9.
ities, 11 Continuous stochastic processes, 16 Stochastic processes adapted to an increasing family of sub a-fields, 20 Martingales, 25 Brownian motions, 40 Poisson random measure, 42 Point processes and Poisson point processes, 43
Chapter II. Stochastic integrals and Ito's formula ............. 45 1.
2. 3. 4. 5. 6.
Ito's definition of stochastic integrals, 45 Stochastic integrals with respect to martingales, 53 Stochastic integrals with respect to point processes, 59 Semi-martingales, 63 Ito's formula, 66
Martingale characterization of Brownian motions and Poisson point processes,
7.
73
Representation theorem for semi-martingales,
Chapter III. Stochastic calculus
84
............................... 97
The space of stochastic differentials, 97 2. Stochastic differential equations with respect to quasimartingales, 103 3. Moment inequalities for martingales, 110 4. Some applications of stochastic calculus to Brownian motions, 113 4.1. Brownian local time, 113 4.2. Reflecting Brownian motion and the Skorohod equation, 119 4.3. Excursions of Brownian motion, 123 1.
xiii
xiv
CONTENTS
5. 6.
4.4 Some limit theorems for occupation times of Brownian motion, Exponential martingales, 149 Conformal martingales, 155
Chapter IV.
146
Stochastic differential equations .................. 159
1. Definition of solutions, 159 2. Existence theorem, 167 3. Uniqueness theorem, 178 4. Solution by transformation of drift and by time change, 190 5. Diffusion processes, 202 6. Diffusion processes generated by differential operators and stochastic differential equations, 212 7. Stochastic differential equations with boundary conditions, 217 8. Examples, 232 9. Stochastic differential equations with respect to Poisson point processes, 244
Chapter V. Diffusion process on manifolds 1.
2. 3.
4.
....................247
Stochastic differential equations on manifolds, 247 Flow of diffeomorphisms, 254 Heat equation on a manifold, 269 Non-degenerate diffusions on a manifold and their horizontal lifts, 275 Stochastic parallel displacement and heat equation for tensor fields, 297 The case with boundary conditions, 308 Kahler diffusions, 341 Malliavin's stochastic calculus of variation for Wiener functionals, 349
5. 6. 7. 8. 9. Pull-back of Schwartz distributions under Wiener mappings and the regularity of induced measures (probability laws), 375 10. The case of stochastic differential equations: Applications to heat
kernels,
391
Chapter VI. Theorems on comparison and approximation and their
applications ..............................................437
1. A comparison theorem for one-dimensional ltd processes, 437 An application to an optimal control problem, 441 Some results on one-dimensional diffusion processes, 446 4. Comparison theorem for one-dimensional projection of diffusion pro2. 3.
cesses, 452 5. Applications to diffusions on Riemannian manifolds, 460 6. Stochastic line integrals along the paths of diffusion processes, 467 7. Approximation theorems for stochastic integrals and stochastic differen8. 9.
tial equations, 480 The support of diffusion processes,
517
Asymptotic evaluation of the diffusion measure for tubes around a smooth curve, -532
Bibliography .................................................... 541
Index .......................................................... 561
General Notation
Theorem IV-3.2, for example, means Theorem 3.2 (the second theorem in Section 3) in Chapter IV. If this theorem is quoted in Chapter IV, it is written as Theorem 3.2 only. The following notations are frequently used.
A: = B means that A is defined by B or A is denoted by B. A(x) = B(x) : A(x) and B(x) are identically equal, i.e. A(x) = B(x) for all x. : the restriction (of a function f) to a subset A of the 114 domain of definition off. : the symmetric difference of A and B, i.e., (A\B) U (B\A). AoB : the indicator function of A, i.e., I.4(x) = I or 0 accordingI4
ly as x e A or x aVb aAb E(X:B)
: : :
A.
the maximum of a and b. the minimum of a and b. the expectation of a random variable X on an event B, i.e., f 2X(co)P(da)) = E(XIB).
8,j, 6", b 8M
Li.p. a.s.
the Kronecker's J. : the unit (Dirac) measure at a. : limit in probability. : almost surely. :
t -- X(t), It - XON R = R'
:
C
:
Rd
N
Z Z+
: the mapping t --- X(t). the real line. the complex plane. : the d-dimensional Euclidean space. : the set of all positive integers. : the set of all integers. : the set of all non-negative integers. xv
xvi GENERAL NOTATION
: the set of all rational numbers. : the largest integer not exceeding x. "smooth" usually means C"( i.e., infinitely differentiable); "sufficiently smooth" is the same as "sufficiently differentiable". Other notations will be explained where they first appear.
CHAPTER I
Preliminaries
1. Basic notions and notations The reader is assumed to be familiar with some of the basic notions in measure theory (cf. [45] and [129]), especially the notions of a a-field (called also a a-algebra or a Borel field), a measurable space (a pair consisting of an abstract space and a a-field on it) and a measurable mapping. When S is a topological space, the smallest a-field .4'(S) on S which contains all open sets is called the topological afield and an element B E -q(S) is called a Borel set in S. A mapping f from a topological space
into another topological space S' which is P%(S)/.°.erl(S')-measurable (i.e., f -'(B) = {x;f(x) e B} E .'(S) for all B E -q(S')) is called Borel measurable. Any a-additive non-negative measure P on a measurable space such that P(Q) = 1 is called a probability on (S2, ) and the triple
(Q,F P) is called a probability space. If P is a probability on (52,.x`-), then _F" r = {A e 52; 3B1, B2 E Jr such that B, c A c B2 and P(B1) = P(B2)} is a a-field on Q containing .F-:The probability P can be naturally extended to .' a and the space (0.. i', P) is called the completion of
(52, .9 P). A probability P on (Q, F-) such that .9' = J F r is called complete and in this case (0, $; P) is called a complete probability space. If S is a topological space, we set (1.1)
g(s) = n _q(sr, 4
where p runs over all probabilities on (S,.,'(S)). An element in '(S) is called a universally measurable set in S and a mapping f from S into
another topological space S' which is '(S)/.(S')-measurable is called } be a family of a-fields on Q. We denote by V -F-. the smallest a-field on Q which contains all Y;. If IF is a class universally measurable. Let {. a
of subsets of 9, we denote by a[W] the smallest a-field on 12 which contains W. Also, if {Xa} aeA is a family of mappings from 52 into a measurable 1
PRELIMINARIES
2
then the smallest a-field ..Y on 0 such that each X. is 977 '-measurable is denoted by a[Xa; a E A]. In particular, a[X] _ X-'(.Y-') for every mapping X: 0 - 0'. space
Let P) be a probability space and (S,:,13(S)) be a topological space S with the topological a-field R(S). A mapping X from 0 into S which is F-/R(S)-measurable is called an S-valued random variable.*' If, in particular, S = R, S = C, or S = Rd, then X is called a real random
variable, a complex random variable, or a d-dimensional random variable, respectively. If X is an S-valued random variable, then
PX(B) = P[X-'(B)] = P[cv; X(c)) E B] = P[X E B],*z B E °.2(S),
(1.2)
defines a probability on (S,P(S)). PX is called the probability law (or probability distribution) of the random variable X. PX is nothing but the induced measure, or the image measure of the measurable mapping X.
2. Probability measures on a metric space Let S be a separable metric space with the metric p and .2(S) be the topological a-field.
Proposition 2.1. Let P be a probability on (S,P(S)). Then, for every
B E .2(S), (2.1)
P(B) =
P(F) =
sup
FcB. F: closed
inf
P(G).
8 c G,G : open
Proof. Set ' = {B E P2(S); (2.1) holds}. If B E W, then clearly B° (the complement of B) E IF. If B. E B, n = 1, 2, ..., then U Bn E IF. Indeed, for given e > 0, we can choose an open set G. and a closed set F. such that F. c B. c G. and P(Gn\Fn) S a/2n+'
n = 1, 2, .... Ao
Set G = U G. and F = U F. where no is chosen so that n-3
n-+1
no
P Ul Fn\ U n. ni
--/2l
" More generally, a random variable is any measurable mapping from (52,9) into a measurable space. *2 We often suppress the argument co in this way.
PROBABILITY MEASURES ON A METRIC SPACE
3
Then G is open, Fis closed, F c U B,, c G and n-1
P(G\F) <
P (U Fn\F) G E/2 + E/2 = E.
Thus, U B. E lB. Therefore, '? is a Q-field. If G is open, then F. _ na 1
{x; p(x, G') > l/n} * is closed, F. c Fn+, and U F,, = G. Since P(G) = n lim P(F,), it follows that G E V. Therefore, `F = .Ow(S). We denote, by Cb(S), the set of all bounded, continuous real-valued functions on S. Cb(S) is a Banach space with the usual norm 11 f I I = sup I f(x)1, .f E Cb(S). XES
Proposition 2.2. Let P and Q be probabilities on (S,.OJ(S)). If $8f(x)P(dx) = f f(x)Q(dx) for every f E Cb(S), then P = Q. Proof. By Proposition 2.1, it is sufficient to show that P(F) = Q(F) for every closed set F. If we set fn(x) = 6(np(x, F)) where 0(t) is the function defined by
,
t<0, O
,
t> 1,
1
0(t)= 1-t 0
then lim fn(x) = IP(x) for every x E S. Consequently, by the dominated m»m
convergence theorem,
P(F) = lim j' fn(x)P(dx)
= lim
f
Sfn(x)Q(dx) = Q(F).
Proposition 2.3. Suppose that S is complete with respect to the metric p, that is, every p-Cauchy sequence is a convergent sequence. Then any probability P on (S, . '(S)) is inner regular in the sense that for every B
0(S) (2.2)
P(B) =
sup
P(K).
KGB, K : compact
Proof. We first prove that for every a > 0 there exists a compact set " p(x, A) = inf p(x, y). YEA
PRELIMINARIES
4
K c S such that P(K) > 1-c. Since S is separable, S can be covered by a countable number of balls of radius S for any given 0 > 0. Let J. 10, and, for each n, let ak', k = 1, 2, ... be a sequence of closed balls of radius Sn covering S. Then 1 = P(S) = Jim P (k ak ))
:--
s
and consequently we can find In such that (2.3)
ak )) > 1 - a/2n.
P(k 1
Set K = (1 U ak'. Clearly, for every b > 0, K can be covered by a finite n-1 k
number of balls with radius S and hence K is totally bounded. Since S is complete, this implies that K is compact. From (2.3), we conclude that
P(K) > I - e. Next, let B E .HI(S). By Proposition 2.1, we can choose a closed set
F c B such that P(B) < P(F) + e. Now F' = F fl K is compact and P(F) - P(F) < P(KC) < c. Hence P(B) < P(F') + 2c which implies that (2.2) holds. Definition 2.1. A sequence {P.} of probabilities on (S,..W(S)) is said
to be /weakly convergent to a probability P on (S,R(S)) if for every fJ
CMS)
,_f Jim f f(x)Pn(dx) = f f(x)P(dx). s
S
P is uniquely determined from {Pn} by Proposition 2.2, and we write
P = w - lim Pn, or Pn -w Pas Pas
n -- oo.
Proposition 2.4. The following five conditions are equivalent: (i) P. P.
(ii) urn f f(x)Pn(dx) = f f(x)P(dx) for every uniformly continuous s
s
.f E Cb(S).
(iii) lim Pn(F) < P(F) (iv) 1 m Pn(G) > P(G) n--
for every closed set F.
for every open set G.
PROBABILITY MEASURES ON A METRIC SPACE
(v) lira P (A) = P(A) nom
5
for every A E.6.d' (S) such that P(8A) = 0.*
Proof. "(i):=> (ii)" is obvious. To show "(ii) ' (iii)", note that the functions fk(x) used in the proof of Proposition 2.2 are uniformly continuous. Then, (ii) implies that lim P»(F) < limn f fk(x)P.(dx) =
J'fk(x)P(dx)
and, letting k -- oo, we have (iii). "(iii) q (iv)" follows by taking complements. Next, we show "(iii) (i)". Let f E Cb(S). By a linear transformation, we may assume, without loss of generality that 0
(i - 1)/k P {x; (i - 1)/k < f(x) < i/k}
(2.4)
< fs f(x)P(dx) < 1-1 E ilk P {x; (i - 1)/k < f(x) < ilk). If we set F1= {x; ilk
1-o
(iii), it follows that
lim f f(x)P.(dx)
k-1
k_2
Fn 1-0 E P (F1)/k < 1-0 E P(F,)/k
< 1/k + f f(x)P(dx). Since k was arbitrary, we deduce that
RE f sf(x)P.(dx) <- f f(x)P(dx). s
By replacing f with 1 -f in the above argument, we obtain lim f
sf(x)P,(dx) Z f f(x)P(dx)
Consequently, lim f f(x)P (dx) = f f(x)P(dx).
-- s
* aA = A.
s
is the boundary of the set A, A is the closure of A and .4 is the interior of
PRELIMINARIES
6
Finally, we show "(iii) . (v)". If P(6A) = 0, then, assuming (iii) (iv)), we have
P(A) = P(A)S li m P.(-!) < lim P(A)
P(A)
P(A). Conversely, assume (v). Let F be a closed
showing that
set and set F6 = {x; p(x,F) < 61. Then aF6 c {x; p(x,F) = 8} := A8 and, since the As are disjoint for different 8, the set of b such that P(A6) > 0 is at most countable. Therefore, we can choose d1 10 such that P(Aa1) _ 0 and hence P(aF8,) = 0. Consequently
P(F) = lim P(F81) = lim lim P (F8) > lim P (F). I...
I-eo 0.00
n__
Example 2.1. If S = R, there is a one-to-one correspondence between
a probability P on (S,.$(S)) and its distribution function F(x) = P((- co, x]). Then "P -=- P" is equivalent to "F (x) ---- F(x) for every continuity point x of F". The former implies the latter by (v) of the above proposition and the converse implication is proved easily by approximating the integrals !-° f(x)dF (x) and f f(x)dF(x) with the Riemann sums.
Proposition 2.5. Weak convergence of probabilities is a metric concept. To be precise, we can define a metric d on the totality '151(S) of probabilities on (S, .q(S)) such that
P. ' - P
is equivalent to
d(P,,, P) --- 0
as n --- oo.
Proof. The well known Prohorov metric, which is a generalization of that of Levy in the case S = R, is such a metric (cf. [142]). Here we give an equivalent metric in the following way (cf. [165]). If S is a separable
metric space, we can choose an equivalent metric so that S is totally bounded under this metric.* Then, the set of all uniformly continuous functions has a countable dense subset If.) with respect to the uniform norm and we set d(P, Q) = E 2-1 {1 A
(I f fi(x)P(dx) -- f s
,f,(x)Q(dx) J )J
* Indeed, it is well known that a separable metric space is homeomorphic to a subset of the Hilbert cube [0, I]'r, [98].
PROBABILITY MEASURES ON A METRIC SPACE
7
It is easy to see, by using Proposition 2.4 (ii), that d is a metric satisfying the condition of the theorem. Thus, the totality .9(S) of probabilities on (S, .-'(S)) is a metric space under weak convergence. We now want to characterize a relatively compact' set in .9(S). For this, we give the following definition:
Definition 2.2. A family A c .9'(S) is called tight if for every s > 0 there exists a compact subset K c S such that P(K) > 1 - E for every P E A; i.e., inf P(K) > 1 - a. PEA
Example 2.2. If S is complete under the metric p, then every finite set A is tight by Proposition 2.3. Generally, if A, and AZ are tight, then so is A, U A2.
Theorem 2.6. Let A c .9'(S). (1) If A is tight, then A is relatively compact in .9'(S). (2) When S is complete under the metric p we have the converse of (1): namely, if A is relatively compact in .9'(S), then A is tight. Proof. For the proof of (1), we first note that, if S is a compact metric space then .9'(S) is compact and hence every A c .9'(S) is relatively compact. Indeed, by Riesz's theorem, .9'(S) = {p E C*(S);,u(f) > 0 forf > 0 and p(1) = 1} *2 and, since C(S) = Cb(S), weak convergence is equivalent to convergence in the weak *-topology on C*(S). Thus .9'(S) is compact since it is a weak *-closed subset of the unit ball in C*(S) and, as is well known, the unit ball is weak *-compact. In the general case, we note that S is homeomorphic to a subset of a
compact metric space (actually, a subset of [0,l]r') and hence we may assume that S is a subset of a compact metric space S. We want to show that for every sequence
from a tight family A we can always choose
a convergent subsequence. For a probability u on (SO(S)), we define a probability on (S, J(S)) by i(A) = p(A n s), A e .9'(S). Note that A c S is in O(S) if and only if it is expressed as A = A n S for some A E 0(9). Now, is a sequence in and hence, by the above remark, we can choose a subsequence, which we denote again by converging weakly to a probability v on (S, `(S)). We now show that there is a probability u on (S,3'(S)) such that = v and p converges *' i.e., its closure is compact, or equivalently, from every infinite sequence contained in this set we can choose a convergent subsequence. * 2 C(S) is the Banach lattice under the natural order of all real continuous functions on S and C*(S) is the dual space. 1 denotes the function f(x) = 1.
PRELIMINARIES
8
weakly to p. Indeed, for every r = 1, 2, ... , there exists a compact subset K, of S (and hence a compact subset in S) such that #.(K,) > 1 - 11r
for all n. Clearly K, E a (s) n-9(9) and 4.(K,) = pn(K,). Since
v
weakly, we have
v(K,) > lim
1 - 11r.
Therefore U K, = E C S is both in .°,a`(S) and R(S) and v(E) = 1. If A E .9J(S), then A fl E E .4(9) since A fl E = A" (1 S fl E = A fl E for some
A E R(9). (S). We set p(A) = v(A fl E) for every A E R (S). Now it is easy to see that p is a probability on (S, .'(S)) and 12 = v. Finally we show that
p - p weakly in 0'(S). Let A be closed in S. Then A = A fl s for some closed set in S and
p (A). Consequently
lim p (A) = lim /2.(A) < /2(A) = p(A)
and the assertion follows from Proposition 2.4, (iii). We omit the proof of (2) (cf. [6], [138]).
Consider an S-valued random variable X; i.e., an ..Y'/.'(S)-measurable mapping from a probability space (12,f,P) into S. The probability law PX of X is the image measure of the mapping X: 0 -- S.
Definition 2.3. Let X,,, n = 1, 2, ... , and X be S-valued random variables.* We say X. converges to X in law if PX --- PX weakly. Suppose X,,, n = 1, 2, ... and X are defined on the same probability space P). Then X is said to converge to X almost everywhere (or almost surely) if P (w; p(X (o)), X(w)) --- 0 as n -- oo} = 1 and X. is said to converge to X in probability, if, for every a > 0, P {co;
X(co)) > a} ---- 0
as n --- oo.
It is well known that almost everywhere convergence implies convergence in probability and convergence in probability implies convergence in law. The following theorem due to Skorohod asserts that, if S * They may be defined on different probability spaces.
PROBABILITY MEASURES ON A METRIC SPACE
9
is complete under the metric p, a converse of the above implications holds in a certain sense.
Theorem 2.7. Let (S, p) be a complete separable metric space and
P,,, n = 1, 2, ... , and P be probabilities on (S, ,%(S)) such that P -" P as n -- co. Then, on a probability space (L, 2, P), we can construct S-valued random variables X,,, n = 1, 2, ..., and X such that ( i ) P = PXn, n = 1, 2, ... and P= PX. (ii) X. converges to X almost everywhere. Thus, in particular, convergence in law of random variables X. can be realized by an almost everywhere convergence without changing the law of any X..
Proof. We prove this theorem by taking £ = [0, 1), .1 = 0 ([0, 1)) and P(d(o) = dco: the Lebesgue measure. To every finite sequence (i1, i2, ...,ik) (k = 1, 2, ...) of natural numbers, we associate a set S(,1,12.---.1k) (E r (S) as follows:
(1) If (il, i2, ... , ik) # (h, i2, -
,Jk), then
S(11,12,..-. tk) n S(11.J2....,Jk)
(2) U SJ = S J
1
w
and J
U S(11,12.....tk.n = S01,12--1k) 1
k) < 2-k *1; (3) diam S(,,,,2 (4) Pn(aSU1.t2,...,tk)) = 0, n =
1, 2, ... and P(aS(11.12..... 1k)) = 0. Thus, by (1) and (2), {S(,1,12..... k)} forms for each fixed k a disjoint covering of S which is a refinement of that corresponding to k' < k. We can construct such a system of subsets as follows. For each k, let am), m = 1, 2, ... be balls with radius < 2-(k+l) covering the whole space S and satisfying 0, P(ao ) = 0 for every n, k, m. Set, for each k, DIk) = all'), D Z) = 6?)\a ), ... , Dr) = on )\(ollx" U ... U 6)1), ... and D111) n Diz) n ... n D,(,'). It is easy to S(,1.1..... 1k) = verify that the system of sets so defined possesses the above properties. Fixing k, we order all (il, i2, ..., ik) lexicographically. Define intervals*2 '4 ,')-12...... k) in [0, 1) with the following properties: l 1) I d U1.12.....'k) I- P`SU1,:2.....1k)1' I d (11.12.....1k) I=
U1,12,.._.tk),
(ii) if (i1, i2,
... , ik) < (jl, j2, ... , jk), then the interval
is located to the left of the interval
d(i1.12...... k)
d(,1,J2,...,Jk) (resp.
diam A = sup p(x, y). ..YEA * 2 Here, by intervals in (0, 1), we mean intervals of the form [a, b) (a 5 b) only. I'd stands for the length of the interval A.
PRELIMINARIES
10
(iii)
U
du,r,....t) 2 k _ [0, 1),
U
1
0,
1).
U1.t2..... tk>
U,.12. ..tk)
Clearly these intervals are uniquely determined by these properties. For each (it, i2, ... , ik) such that S(1i.12.... tk) # ' we choose a point xt,.t2,....tk E Sty, 12). For co E [0, 1), we set Xn(w) = Xt1.t2..... tk
if
Xk((,O) - Xt].t2.....ik
if wE d(77.12..... 1101
for k = 1, 2,
41).12
'k)
and
... , n = 1, 2, ....* Clearly, P(X k(w), X'
p( n\w), X n+P(w)) S 1/2 k'
(co))
1/2 k
and hence XX(w) = lim Xk,(co), X(c)) = lim Xk(w) exist by the complete-
k-'
k-.-
ness of (S, p). Since Pn(SUt,12..... 1k>) = Id(ri.t2.....tk) I
---
Id(tt,t2.... 'k> I =
if co dul,t2,_...tk) then there exists nk such that co E for all n Z nk. Then Xn(co) = Xk(w) and hence
P(S(11.12.---.1k) ), tk)
P(X ((O), X(w)) < P(Xn((O), Xn(w)) + P(X n(co), X k((O))
+ P(X k(w), X(w)) < 2/21,
Therefore, if we set 00 = (1 (
U k-t (1112.....1k)
if n Z nk.
401.12..._.tk)), XX(w) -- X(w) for
co E Do as n --- no and clearly P(f2,) = 1. Finally, wee show that fix- = P. and PX = P. Since P{w;
Xn+D(w) E S(11.12..... tk)} = Pt();
Xn+D(w) E SUi.t2.....tk)}
= Pn(S(i1.t2.....'k))
and since every open set in S is expressed as a disjoint countable union of tk)'S, we have, by Fatou's lemma, lim PXn(O) > P,(O)
for every open set 0 in S. Then, by Proposition 2.4, PX-P converges weakly
to P. asp -- no showing that PX, = P. Similarly, we have -fix = P. *
4a.12 ...,rk, # ¢ or
defined.
# 0, then S(11,11,.,,tk) 4 0 and hence this is well
EXPECTATIONS, CONDITIONAL EXPECTATIONS
11
3. Expectations, conditional expectations and regular conditional probabilities Let X be a real (or complex) random variable defined on a probability
space (92,Y,P). Two random variables X and Y are identified if P[co;X((o)
f
Y(c))] = 0. X is called integrable if
I X(co) I P(dcv) < co. Q
More generally, if
f
a
I X(c)) I °P(dco) < co,
(p > 0),
it is called p-th integrable.*l Let p > 1. The totality of p-th integrable random variables, denoted by ,(S2, ,P) or simply by 2D(Q) or . (P), forms a Banach space with the norm IIXII, =
(f
IX(w)IVP(dw))'/'.
(Sl, P) is the Banach space of essentially bounded random variables with the norm IIXII. = ess.sup I X(co) 1. For an integrable random variable X, E(X) = f aX(c))P(du)) is called the expectation of X. For a square-integrable random variable X, V(X) _ E(X2) - E(X)2 ( = E((X - E(X))2)) is called the variance of X.
A family {Y } aGA of sub cr-fields of -9- is called mutually independent
if for every distinct choice of al, a2, ... , ak E A and A, e
1,2,...,k,
i=
P(A,fl A2 fl ... fl Ak) = P(A1)P(A2) ... P(Ak) A family of random variables {Xa} aaA is called mutually independent*2 if {6[Xa]} aE,i is mutually independent. A family {Xa, a e Al of random
variables is called independent of a afield 9 c .F if u[Xa; a E A] and 9 are mutually independent. Let Xbe an integrable random variable and 9e-9- be a sub a-field of .F. Then p(B) = E(X:B): = f BX(co)P(d(9), B E 9, defines a If p = 2, it is called square-integrable. *= This definition applies for general S-valued random variables.
PRELIMINARIES
12
a-additive set function on W with finite total variation and is clearly absolutely continuous with respect to v = P I,,. The Radon-Nikodym derivative dp/dv(w) is denoted by E(X 19)(co); thus E(X I') is the unique (up to identification) '-measurable integrable random variable Y such that
E(Y:B) = E(X:B) for every B e 3?. Definition 3.1. E(X I ')(co) is called the conditional expectation of X given IN. The following properties of conditional expectations are easily proved.
(X, Y, X,,, below are integrable real random variables and a, b are real numbers.)
(E.1) E(aX+bYI9)=aE(XI')+bE(YI') a.s. (E.2) If X > 0 a.s., then E(X I ') > 0 a.s. (E.3) E(1 I ') = 1 a.s. (E.4) If X is '-measurable, then E(X I') = X, a.s., more generally,
if XY is integrable and X is '-measurable E(XYI ') = XE(YI ')
a.s.
(E.5) If X is a sub a-field of ', then
°)
E(E(X I 9) I °) = E(X I
a.s.
(E.6) If X. --- X in .91(Q), then E(X. I') --- E(X I ') in g1(92). (E.7) (Jensen's inequality) If yr: R' - R' is convex and V(X) integrable then 1V(E(X I g')) S E(yr(X) I ')
a.s.
In particular, I E(X I c9) I < E(I X I ') and, if X is square-integrable, IE(XI W)I2 < E(IXIZI,y). (E.8) X is independent of ' if and only if for every Borel measurable function f such thatf(X) is integrable, E(f(X) I ') = E(f(X)) a.s. Let 1; be a mapping from Q into a measurable space (S, R) such that it is _9-/R-measurable. Then p(B) = E(X: {co; ((o) e B}) is a a-additive I
set function on .h9 which is absolutely continuous with respect to the image measure v = Pf. The Radon-Nikodym density dp/dv(x) is denoted by
E(X 11; = x) and is called the conditional expectation of X given = x. It possesses similar properties as above.
Definition 3.2. Let (Q,,F,P) be a probability space and 9 be a
EXPECTATIONS, CONDITIONAL EXPECTATIONS
13
sub a-field of J. A system {p(co, A)} cc{a,AEJ is called a regular conditional probability given ' if it satisfies the following conditions: (i) for fixed co, A --- p(co, A) is a probability on (Q, -F-);
(ii) for fixed A E F-, co - p(c), A) is 19-measurable; (iii) for every A E-= Y and B E 3?, P(A fl B) = f p(co, A)P(dco). Clearly property (iii) is equivalent to (iii)' for every non-negative random variable X and B E 9
E(X: B) = f sa
(IB(c))
f a X(c)')p(co, dco')} P(dco),
that is, Ja X(co')p(co, dc)') coincides with E(XI ')((o) a.s. We say that the regular conditional probability is unique if whenever {p(c), A)} and {p'(co, A)} possess the above properties, then there exists
a set N E ' of P-measure 0 such that, if co 0 N then p(c), A) = p'(c), A) for all AE -. Definition 3.3. A measurable space (0, .F-) is called a standard measurable space if it is Borel isomorphic* to one of the following measurable spaces: (<1, n>, , (
)), (N,,;i(N)) or (M,.q(M)), where <1, n> = {1, 2,
... , n} with the discrete topology, N
{1, 2,
discrete topology and M = {0,1 } N = {w = (a),, co2, 1} with the product topology.
...
}
with the
... ), co, = 0 or
It is well known that a Polish space (a complete separable metric space)
with the topological a-field is a standard measurable space and every measurable subset of a standard measurable space with the induced a-field is a standard measurable space (cf. [98], [138]).
Theorem 3.1. Let (Q, .,1-) be a standard measurable space and P be a probability on (0, Let ' be a sub a-field of .F-. Then a regular conditional probability {p(co, A)} given
' exists uniquely.
Proof. We consider the case where (52,.x) is isomorphic to (M, .'(M)) and hence we may assume that 0 = M and -9-= R (M). Let r": 52 E) co '-. (co,,co2i ... , con) E {0,1}" be the projection and * (92,.Y ) and are Borel isomorphic if there exstis a bijection f: 92 --> S2' such that f is F/.F'-measurable and f-' is .7'/.F-measurable, i.e. f(. -) =.F'.
PRELIMINARIES
14
F- = nnl.od" [{O,1}n]. Clearly {JIF"n} is an increasing family of finite = Jr. If Pn, n = 1, 2, ... are probabilities on (S2, .9) a-fields and V . n
and {Pn} is consistent in the sense that Pn+, j Tn = Pn, n = 1, 2, ... ,
then there exists a unique probability P on (Q, ,Y') such that P I,-. = Pn.
Indeed, by consistency, P is well defined on U . n-I
and if B. E U F -k, k-3
n = 1, 2, ..., is such that B. D Bn+, and lim P(Bn) > 0 then (1Bn = nom
n
since {Bn} is a system of closed sets in a compact space 12 having the finite
intersection property. P is then extended to a[ U .
]=V. n
= jr by
Hopf's extension theorem. We set pn(co, A) = E(1,, j (9)(co), A E fin. Clearly, there exists a set N. E ' of P-measure 0 such that if co (4 Nn, then pn(w, A) is a proband pn(co, A) = pn_,(c), A) for A E n = 1, 2, ... . ability on , If we set N = U Nn, then for each co ij N, {pn(w, .)} is a consistent family n
and hence determines a unique probability p(w, ) on (S2, 7-). Take a prob-
ability v on (Q, Y) and define p(co, ) = v if co E N. Then the system is a regular conditional probability given
'. Indeed, properties
(ii) and (iii) are obvious for A E U 5 and extend to F by a standard monotone lemma. If {p(co, )} and {p'(co, .)} are two regular conditional
probabilities, then the set N = {w; p(co, A) a p'(w, A) for some A E U -9-.1 has P-measure 0 and if co
n-1
N, then p(co, A) = p'((o, A) for all
A E 9" again by the monotone lemma. This proves the uniqueness of the regular conditional probability. Definition 3.4. Let (Q, .9'") be a measurable space. We say that -Vis countably determined if there exists a countable subset 7-0 c JV such that whenever any two probabilities agree on -9-0 they must coincide. Clearly, if (12, Y) is a standard measurable space then Y is countably determined.
Theorem 3.2. Let (Q, .F) be a standard measurable space and P be a probability on (0, .,Y'"). Let cy be a sub a-field of Y and p(co, dco') be a regular conditional probability given '. If 2' is a countably determined of P-measure 0 such sub a-field of ', then there exists a set N E that co e N implies p(co, A) = IA((o) for every A E X.
Proof. Let X, c 2 be a countable set in Definition 3.4 for X. Clearly if A E °o, then there exists NA E g' of P-measure 0 such that N4. Set N = U NA, then p(co, A) = I4(w) holds p(co, A) = 1,,(w) if co
for all AE 2' ifcoc N.
AeXO
EXPECTATIONS, CONDITIONAL EXPECTATIONS
15
Corollary. Let (0, .Y") be a standard measurable space and P be a probability on (92, .-). Let ' be a sub a-field of 5 and p(co, ) be a regular conditional probability given '. Let (w) be a mapping from Q into a measurable space (S, R) such that it is '/.g-measurable. Suppose
further that 12 is countably determined and {x} ER for every x E S (this is true, for example, if (S, . ) is a standard measurable space). Then (3.1)
p(co, {co'; c(cv') = c(co)}) = 1,
a.a. co.
Proof. Since .' is countably determined, there exists a countable subset :/r 0 c _q with the property of Definition 3.4. Hence, if we set B e .°01} , 2° is a countably determined sub a-field of ' 0= with 2°o = B E .qo} . By Theorem 3.2, there exists a set N E <9 of P-measure 0 such that if Co (Z N, p(co, A) = IA(w) for every A E_ 2'.
By taking A. = {co'; (w') = c(co)} E 2', we have (3.1) for co
N.
In the same way as Theorem 3.1, we can prove the following result.
Theorem 3.3. Let (Q, 91-) be a standard measurable space and P be a probability on (Q, .7-). Let (co) be a mapping from Q into a measurable space (S, _q) such that it is -9-/-T-measurable and let Pt be the induced measure on (S, R) by t. Then there exists a system {p(x, A)) XES,AE,,- such
that
(i) for fixed x E S, A - p(x, A) is a probability on (0, -57-); (ii) for fixed A E_ -V-, x ' --- p(x, A) is P'-measurable; (iii) for every A E Y" and B E R, we have P(A fl {w; (co) E= B}) = f p(x, A)PF(dx). Furthermore, if {p'(x, A)) is any other such system, then there exists a set N E_ . of Pe-measure 0 such that x e N implies
p(x, A) = p'(x, A)
for all
A E -q
Thus, for every integrable random variable X, f a X(w)p(x, do)) coincides with E(X I = x), PC-a.e. x. p(x, A) is called the regular conditional probability given = x. In the same way as Theorem 3.2 and its corollary we can prove the following:
Corollary. We assume further in Theorem 3.3 that .,J is countably determined and {x} E.. ' for every x E S. Then there exists a set N E °,1} of Pc-measure 0 such that if x N, p(x, {co; g(w) E B}) = IB(x) for every
B E R. In particular, if x (4 N then
PRELIMINARIES
16
(3.2)
p(x, {co; c(co) = x}) = 1.
4. Continuous stochastic processes
Let Wd = C([0, oo) --- Rd) be the set of all continuous functions w: (0, oo) F) t s- w(t) a R. We define a metric p on Wd by (4.1)
p(w,, w2) = En-r2-"[(max I W1(t) - w2(t) I) A 1], o
w1, w2 E W.
It is easy to see that Wd is complete and separable under this metric. Clearly, w" ---- w with respect to the metric p if and only if wn(t) converges to w(t) uniformly in t on each bounded interval. Let . '(Wd) be the topological v-field. By a Borel cylinder set we mean a set B e Wd of the following form B = {w; (w(t,), w(t2),
... , w(t")) E E}
for some sequence 0 < t, < t2 < . . . < t" and E e .ail'
(Rnd).
The
totality of Borel cylinder sets is denoted by IF. Since the mapping w c Wd i--- (w(t), w(t2), ... , w(t")) E Rnd is continuous, it is clear that
IF c .(Wd). Proposition 4.1. Q[(8] = R(Wd).
Proof. It is only necessary to show that a[ '] D .,4( Wd). The totality of sets of the form {w; max I w(t) - wo(t) I < e} , w0 E Wd, e > 0, n = Ostsn
1, 2,
... , forms a basis of neighborhoods in Wd and we have {w; max I w(t) - wo(t) I < e} = OSt$n
n
{w; w(r) e U(wo(r), e)}
rcQ,Osr
where U(a, e) = {x E Rd; I x - a I < e} . Thus, such a set is representable as a countable intersection of sets in V. Now -q(W4) c a[W] is obvious. Corollary. Every probability on (Wd, . (Wd)) is uniquely determined by its values on IF.
Definition 4.1. By a d-dimensional continuous process X we mean a W4-valued random variable defined on a probability space (Sl, ..V, P), i.e.,
a mapping X: 0 - Wd which is Y7.'( Wd)-measurable. Thus, if X is a d-dimensional continuous process, then for each w,
CONTINUOUS STOCHASTIC PROCESSES
17
Wd. The value of X(w) at t E [0, oo) * is denoted by X,(w) or XX(c)) is a d-dimensional random variable. X(t, w). For fixed t, co X(co)
Conversely, a collection {Xt(w)},n[o m] of d-dimensional random variables X(t) is continuous determines a d-dimensional continuous process if t with probability one.
We say that two d-dimensional processes X and X' have the same Y
law, denoted by X .;: X', if their probability laws PX and PX' coincide. Since Px and PX are determined by their values on W , X X' if and only if all their finite dimensional distributions coincide : a finite dimensional
distribution of X for a given sequence of times 0 < tl < t2 < . . . < tn, is the probability law of nd-dimensional random variable (X,,, Xtz, Xt).
Theorem 4.2. Let X. = {XX(t)}, n = 1,2, ... , be a sequence of ddimensional continuous processes satisfying the following two conditions: (4.2)
(4.3)
lim sup P { I X.(0) I > N} = 0; N--
n
for every T > O and e > 0, lim sup P { max I Xn(t) - Xn(s) I > e} = 0. h10
t,se[O,T]
n
It-sI$h
Then there exists a subsequence n, < n2 < -- oo, a < nk probability space and d-dimensional continuous processes ±nk = (Xnk(t)), k = 1, 2, ... and X = (X(t)) defind on it such that
f
k = 1,2, ...
(4.4)
rink
(4.5)
xnk converges to X almost everywhere as k - oo, i.e.,
Xnk,
,
P {w; P(,?nk(w), ±(c )) - 0 as k - co} = 1. Furthermore, if every finite dimensional distribution of PXn converges as
n -- oo, then we need not take a subsequence: we can construct
n = 1, 2, ... and X such that (4.4) and (4.5) hold for n = 1, 2, .. . and as
n - oo
respectively.
Proof. First we show that {PXn} is tight (cf. Definition 2.2) if (4.2) and (4.3) are satisfied. By the Ascoli-Arzela theorem, a subset A C Wd is relatively compact in Wd if and only if it is both (i) uniformly bounded, i.e., for every T> 0, sup max I w(t) I < no, .EA te[O,T]
* t e [0, oo) is considered as time.
PRELIMINARIES
18
and
(ii) equi-continuous, i.e., for every T > 0, Jim sup Vh(w) = 0, where h10 wcA
Vh(w) = max
I w(s) - w(t)1.
i.SEtO,T]
It-SIsh
We see by (4.2) that for every e > 0 there exists a > 0 such that PXn (w; I w(0) I < a) > 1 - a/2 for all n. Also, by (4.3), for every e > 0 and k =
1, 2, ... there exists hk > 0 such that hk 10 and PXn {w; Vhk(w) > 1/k) < a/2k+1 for all n. Thus PXa[ n {w; Vhk(w) < 1/k}] > 1 - e/2. Set k-1
K. = {w; I w(o) I < a} n (n {w;Vhk(w) < 1/k} ). Then clearly k-1
K8 satisfies
(i) and (ii) and hence it is compact. Now PXn(K8) > 1 - a showing that {PX8} is tight. So, by Theorem 2.6 (1), a subsequence Ink) exists such
that PXnk * - P for some probability P on (Wa, .q(W')). We now apply Theorem 2.7 to construct JYnk and t with properties as above. If every finite
dimensional distribution of PXn converges, then clearly a limit point of {PX8} is unique and hence PXn itself converges weakly to P as n --- oo.
Theorem 4.3. Let X. = (X8(t)), n = 1, 2,
... , be a sequence of
d-dimensional continuous processes satisfying the following two conditions: (4.6)
there exist positive constants M and y such that E { I X.(0) I Y} <
M for every n = 1, 2, ... , there exist positive constants a, fi, Mk, k = 1,2, ... , such that E { I X8(t) - X (s) I a} < Mk I t - s for every n and t,s E [0,k], (4.7)
(k=1,2,...). Then
satisfies the conditions (4.2) and (4.3) of Theorem 4.2.
Proof. By Chebyshev's inequality,
n = 1, 2, ... ,
P { I X.(0) I > N} < M/NY,
and (4.2) is clearly satisfied. We now prove (4.3). We may take T to be an
integer. By (4.7), Y(t) = X8(t), n = 1, 2, ... , satisfies
E{IY(t)- Y(s)Ia} <MTIt-sI'+P,
t,sE[0,T].
By Chebyshev's inequality, (4.8)
P { I Y((i + 1)/2m) - Y(i/2m) I > 1/2m°} < MT27,n a+s> 2maa
=
M7-2-m a+R-aa)
,
i = 0,1,2, ... ,2-T - 1.
CONTINUOUS STOCHASTIC PROCESSES
19
We choose a such that 0 < a < fl/a. By (4.8), we have
P{ max I Y((i + 1)/2m) - Y(i/2m)I > 1/2m°} < MT2-m(P-°a'
(4.9)
0Si52m T-11
Let e > 0 and 6 > 0 be given. Choose v = v(8, s) such that (1 + 2/(2° - 1))/2v° < e and P[ U (4.10)
mv
{
max
05i52m T- i
I Y((i + 1)/2m) - Y(i/2m) I > 1/2-a)]
< MTT 57,2-m
(s-°°) < S.
m-v
Set Sly = U { max
m-v 05r52mr-1
Y((i + 1)/2m) - Y(i/2m) I > 1/2-41. Then
P(Qy) < 6 and if co o Sly Y((i + 1)/2m) - Y(i/2m) I < 1/2m°
for all m > v and i such that (i + 1)/2m < T. Let DT be the set of all binary rationals in the interval [0, T]. If s r= DT is in the interval [i/211,(i + 1)/2y), then s is written as s = i/211 + ± a1/2y+r, where a, is 0 r-1
or 1, and hence, if co (4 Sly,
Y(s) - Y(i/2y) I < E I
Y(i/2y + rE a11211+')
- Y(i/2y + E
ar/2v+r) I
i
E 1/2(y+k)° < E 1/2(y+k)° = 1/(20 - 1)2"°.
k-1
k-1
Therefore, ifs, t E DT, I s - t I < 1/2v and co (4.11)
Qy, then
I Y(s) - Y(t) I < (1 + 2/(24 - 1))/2y° < e.
Indeed, if t e [(i - 1)/211, i/2y) and s E [i/2y, (i + 1)/2y), then
I Y(t) - Y(s) I < I Y(s) - Y(i/21) I + I Y(t) - Y((i - 1)/2v) I + I Y(i/2y) - Y((i - 1)/2y) I < (1 + 2/(2° - 1))/2y° and if t, s E [i/2y, (i + 1)/2y), then
I Y(t) - Y(s) I < I Y(s) - Y(i/2v) I + I Y(t) - Y(i/2v) I < 2/(20 - 1)2y°.
Since DT is dense in [0, T], (4.11) holds for every s, t E [0, T) such that
Is - t I< 1/2y. Therefore,
PRELIMINARIES
20
P{
max
t.sE [U. T]
I Y(t) - Y(s) I > E} < P(Q) < S.
It-sI51/2v
Since v = v(8, 6) is independent of n, we have proved (4.3).
Corollary. Let {X(t)} re[o,,,] be a system of d-dimensional random variables such that for some positive constants a, fl and Mk, k = 1, 2, ... , the following condition (4.12) holds: E(I X(t) - X(s) I ") < Mk I t - s I'+s
(4.12)
for every t, s E [0, k], k = 1, 2, ... . Then there exists a d-dimensional continuous process t = (X(t)) such that for every t E [0, oo), P[X(t) = X(t)] = 1.
Proof. Just as in the above proof, (4.9) holds for Y(t) = X(t) and so, by Borel-Cantelli's lemma ([129]), we have for almost all m I X((i + 1)/2m) - X(i/2m) I < 1/2'"°,
i = 0, 1, 2, ... , 2mT - 1, for all m > v = v(w).
This implies, as in the above proof, that I X(t) - X(s) I < (I + 2/(2" 1))/2m4 if t,sEDT, It - sI < 1/2m and m > v. Consequently, t E DT F--- X(t) is uniformly continuous almost surely. Let X(t) be the continuous extension of X(t) I Then X = (X(t)) is a continuous process km ° .
and it is easy to prove that P[X(t) = X(t)] = I for every t e [0, oo).
5. Stochastic processes adapted to an increasing family of sub a-fields
Let (0, _50, P) be a probability space and (,9i ),>o be an increasing family of sub a-fields of 57-: i.e., (5.1)
(
.1",C.
) is called
if 0
e>0
+E = .
for every
t e [0, oo). In the following, unless otherwise stated, (.) is assumed to be right continuous. Such a family (,) is called a reference family. Suppose we are given (Q, ..", P) and (. ). By a d-dimensional stochastic process* * We consider here the d-dimensional case (i.e., the state space is R°) but the generalization to an arbitrary state space is obvious.
STOCHASTIC PROCESSES ADAPTED TO SUB a-FIELDS
21
we mean a family of d-dimensional random variables X = (X,). In this section, a d-dimensional stochastic process is simply called a process.
Definition 5.1. A process X = (X,),Z0 is called adapted to (. ) *' if X, is .7-measurable for every t. Generally, a process X = (X,)rzo is called measurable if the mapping
(t, co) E [0, oo) x 0
X,(c)) E Rd
is .,d([0, oo)) x _F"/,q(Rd)-measurable.
Let So be the smallest v-field on [0, oo) x 0 such that all left-continuous (.7)-adapted processes Y: [O,oo) x 92 D (t,co) i -- Y,(co) E Rd are measurable. If we replace left-continuous Y by right-continuous Y in the above, the corresponding cr-field is denoted by -F".
Definition 5.2. A process X = (X,) is called predictable*z if the mapping (t,co)'---- X,(co) is So/.(Rd)-measurable; it is called well measurable*s if the corresponding mapping is F-1-9(R d)_measurable. Clearly, both predictable and well measurable processes are measura-
ble and adapted to (.7). Remark 5.1. A predictable process is a well measurable process. Indeed,
if X = (X,) is left-continuous, i.e., t -- X, is left-continuous for every co, then X. = (X(7)) defined by X tr> = Xk,zn for t E [k/2", (k + 1)/2") is right continuous and X(,n)(w) --- X,(w) as n --- oo. Thus X is .9-measurable. This implies that S'(-_,57-.
Proposition 5.1.*4 Let 0 be a linear space of real and bounded` measurable processes satisfying the following two conditions: (i) 0 contains all bounded, left (resp. right)-continuous (,7;)-adapted processes; (ii) if {0"} is a monotone increasing sequence of processes in 0 such that 45 = sup 0n is bounded, then 0 E 0. Then 0 contains all bounded predictable (resp. well measurable) processes.
Proof. A bounded predictable process, i.e., a bounded .7-measurable We also say (.F)-adapted. We also say (Y)-predictable. *2 To be precise, predictable with respect to *3 To be precise, (J57;)-well measurable. It is also called optional.
*4 In this proposition, we assume d = 1. "That is, it is bounded as a function from [0, oo) XSl into R.
PRELIMINARIES
22
function, is a monotone increasing limit of 5' measurable simple functions. Hence, it is enough to prove IB e 0 for B E Y. Set
S°' _ {B;B C [0, oo) x Q such that IB E 01. Then, since 0 is a linear space and 1 E 0, S°' has the following properties: (i) [0, oo) x S 2 E S°' ;
(ii) A,BES°',ACB B\AEY';
(iii) n
Let Y, (i = 1, 2, ... , k) be a left-continuous (.)-adapted process k and E1 (i = 1, 2, ... , k) be an open set in R'. Then f1 Y,-'(E,) E .91. k 1-I Indeed, Ik (t, co), = rj IE,(Y,(t, w)) and since there exists a sere 2
(Er)
I
1
quence of bounded continuous functions 0.' on R such that 51(x) t IE,(x) as n -- oo, k
k
II ¢n(I 1(t, co)) t II'E,(Y,(t, co)). 1-1
1-1
The left-hand side being a bounded left-continuous --adapted process is in 0 and hence so is the right-hand side. flkYi 1(E,) is closed under the But the totality Id' of sets of the form (-1 finite intersection and a[ ' by the definition. Then 9' C .7' follows from the lemma below.
Lemma 5.1.*' Generally, if a family IF of subsets of [0, oo) X 0 satisfies (i), (ii) and (iii) above, it is called a d-system. If ' is closed under finite intersections, it is called a rt-system. Let the smallest d-system con-
taining IF be denoted by d[ IF). Then for every n-system r', d[am] _ The proof is standard and is left to the reader. Definition 5.3. Let (Q,_V-,P) and (. )rzo be given. A mapping a: Sl --w [0, oo] is called a stopping time*Z (with respect to (.Y)), if, for every t Z 0,
{co; a(cv) < t} E .Y. For a stopping time a, we set (5.2)
.f; = {A E f; vt e [0, oo), An {a(w) < t } E r} .
Clearly, .9 is a sub a-field of ..r and if a(w) = t, then .w; = _501, Due to Dynkin (cf. [21]). *Z Sometimes, it is called a Markov time.
STOCHASTIC PROCESSES ADAPTED TO SUB a-FIELDS
23
Example 5.1. If X = (Xt) is a right-continuous, (.Y)-adapted process and E c Rd is an open set in Rd, then the first hitting time aE to the set E, defined by crE(c)) = inf It > 0; XX(co) E E} *'
(5.3)
is a stopping time. Indeed, {aE(co) < t} = n {aE(co) < t + 1/n} n
= fln reU {X,((v) E E} E .1'+0 =. r
It is known that if (. ,) satisfies .7/' = .Y for every t or, more generally, if for some system {Pa} of probabilities on (Q, J,), n . per _ .ate for a
every t, then aE is a stopping time for every Borel subset E e Rd for every well measurable X(cf. [15] and [117]).
Proposition 5.2. Let a, z, an, n = 1, 2, ... be stopping times. Then (i) a V r,
a n z,
(ii) a = Jim an, when ant or an j , n
are all stopping times.
Proof. Since {u v r < t} *, = {a < t} n {r < ti, a V r is a stopping time. Similarly, am is also a stopping time. If ant a then
{a < t} = n {an < t} and hence a is a stopping time. If an j a, then n
{a < t} = U {an < t} and by the following lemma, a is a stopping time.
Lemma 5.2. a is a stopping time if and only if {a < t} E .
for
every t.
Proof. If a is a stopping time then {a < t} = U {a(w) < t - 1/n} n E -57;. Conversely, if {a < t j E 9 for every t, then {a(co) < t j _
n {a(m) < t + l/n} E 9+o =
.
A
Proposition 5.3. Let a, a, an, n = 1, 2, ... , be stopping times. (1) If a(co) < r(co) for all co, then . e .9. (2) If an(co) 1 a(w) for all co, then n . n = . ;. n
We define always inf ¢ = oo unless otherwise stated. *Z We often write IX n A} for [co; X(co) e A].
PRELIMINARIES
24
Proof. (1) follows from the definition. If o" j u, then n `T Qn D .' n
by (1). Let A E n -F;,,. Then, as in Lemma 5.2, An {Un < t} E , for every (t, n) E [0, oo) x {1, 2, ... 1. Hence A n {ar < t} = U [A n
{vn
on Q. On the other hand {X," E E} n {o, < t} = U [{Xk,2n E E} n k
{v" = k/2"} n {6" < t}] E ... This implies that Xn is .9 0" I D,,-measurable and hence x, is n "I o, _ 57-,J n,-measurable. Intuitive reasonings on stopping times are often justified by the following proposition. Proposition 5.5.* Let a and z be stopping times and Xbe an integrable random variable. Then the following hold: (5.4)
E(I(a>z, X 1. )
E(X I
nt)
(5.5)
E(I4a:) X 1.) = I,o2tT, E(X I
nZ)
(5.6)
E {E(X I.9i) I ,9;;} = E(XI
>, 41>11
AC).
Proof. First we note that I,,>,, and
are 9;A,-measurable. Indeed
{a>Ti nla Az'r, zt, -c
for every t > 0 and hence {o > r} E
Now, {a > z} = {a
< z} c E ,,F-,A, is obvious. In order to prove (5.4), it is sufficient to show that E(I,,>,r, X 1.9 :) = I,,>,, E(X I .;) is . F;A, measurable. This is true because I,>, E(X I.9 _) I(CA t, = E(X19) I tCSf I (t,>s.Tst
is 9 -measurable for every t > 0. The proof of (5.5) is similar and (5.6) is easily obtained from (5.4) and (5.5). * Communicated by H. Asano.
MARTINGALES
25
So far, we have only considered the continuous time parameter case (i.e., t E [0, oo)). The discrete time parameter case (i.e., t = 0, 1, 2, ... ) can be handled in an obvious way. The notion of adaptedness and stopping times are defined as before. In this case, however, the notion of measurable processes is trivial. The notion of predictable processess is defined as folis predictable with respect to a reference lows. A process X = family (. ) if X is .F_,-measurable for each n = 1, 2, ... .
6. Martingales Let T be the time parameter set: T = {0, 1, 2, ... } in the discrete time case or T = [0, co) in the continuous time case. Let T = T U {oo} be the one-point compactification of T. Let (Q,_V-,P) be a probability space and (Y),ET be a reference family.*'
Definition 6.1. A real stochastic process X = (Xt),ET is called a martingale*z (supermartingale, submartingale) with respect to (.Y) if
(i) X, is integrable for each t e T, (ii) X = (X,) is (J',)-adapted, (iii) E(X,1.7,) = X, (resp. E(X, I.F-,) < X., E(X, I ,.` ",) > X,) a.s. for every t, s e T such that s < t. be a First, we consider the discrete time case. Let f = bounded and non-negative predictable process; i.e., there exists a constant
M > 0 such that 0 < f (W) < M for all n and f is n = 1, 2, .... Let X = be a martingale (supermartingale, submartingale) relative to (.9). We define a stochastic process Y = by YO
(6.1)
Y. =
X0
n=1,2,....
+f..(X, - X.-t),
Then it is easy to see that Y is a martingale (resp., supermartingale, sub-
martingale) relative to (.7:). We denote Y =
and call it the mar-
tingale transform of X by f.
Example 6.1. (Optional stopping). Let a: Q --- T be a stopping time n = 1, 2, .... f = is predictable because In
and set f =
v} = [ U {a = k} ]° and clearly X°: = k-o
is given by Xn
In the continuous time case (-V;) is assumed to be right-continuous. *Z Sometimes, Xis said to be an (.7)-martingale or a (P,.Y)-martingale.
n=
PRELIMINARIES
26
0, 1, 2,
.... Thus the process X° obtained by stopping a process X at
time a is a martingale (supermartingale or submartingale) if X is one.
Theorem 6.1. (Optional sampling theorem).*' Let X = be a martingale (supermartingale, submartingale) relative to (.) and let a and ,r be bounded*-' stopping times such that a(c)) < -r(w) for all w. Then (6.2)
(resp. < , >) a.s.
E(XT 13 ;) = X,
In particular, (6.3)
(resp. < , > ).
E(XT) = E(X,)
Proof. First we show (6.3). Let m E T be such that -r(co) < m for = for n = 1, 2, . _ ... Then f = is predictable as Example 6.1 and (f.X)A - X,, = Xn, - XA,,. In particular, (f. X),,, - Xo = XT - X. and hence (6.3) is obvious. Next we show (6.2). Clearly it is sufficient to prove that if B e 39';
-
all w. Set A =
then (6.4)
E(XT: B) = E(X,: B)
(resp. < , >-).
Set
a(w) aB(w)
it
wE B,
m
,
w E Be,
(w)
,
co e B,
,
weB
and
2B(w)
jt m
Then aB and zB are stopping times. Indeed,
{a. < j} =
0 jl{a<j}fBEY
, ,
if j>m, if j<m,
and similarly for TB. Consequently by (6.3) Due to Doob [18].
*2 a is bounded if a constant m (E T exists such that a(w) S m, for all m.
27
MARTINGALES
E(X,8) = E(X,8),
(resp. < , > ).
That is E(X,: B) + E(Xm: B°) = E(X,: B) + E(Xm: Be) (resp. <, >) proving (6.4).
Corollary. Let X = (X,),ET satisfy the conditions (i) and (ii) of Definition 6.1. Then it is a martingale (supermartingale, submartingale) if and only if (6.3) holds for every bounded stopping times o and T such that
Q
(resp. <, > )
for any bounded stopping times a and T. In the following, we call simply that X = (Xn) is a martingale (supermartingale or submartingale) if X is a martingale (resp. supermartingale or submartingale) with respect to some reference family. Clearly this is equivalent to saying that X is a martingale (supermartingale or submartingale) with respect to the proper reference family (,P"X) of X, where JV. X = Q[Xo, X1, ...
,
Xn], n = 1, 2, ... .
Theorem 6.2.* Let X = (Xn)nET be a submartingale. Then for every
2>0and NE T, (6.5)
AP( max X. > A) < E(XN: max X. > 2) < E(XN) < E(I XNI ). 0
05nsN
and
2P( min X.< -2) < -E(Xo) + E(XN: min X. > -2) (6.5)'
0
0-5n-5N
< E(lXoI) + E(IXNI) Proof. Set
a
( min {n < N ; Xn > A), , if { } _ 6. jl N
* Due to Doob (cf. [18]).
PRELIMINARIES
28
Clearly or is a bounded stopping time such that o < N. By (6.2), we have
E(XQ) = E(Xd: max X. > A) + E(XN: max X. < A)
E(XN)
05n5N
AP( max X. > A) + E(XN: max X. < A). 05n5N
OSn_N
Thus
AP( max X. > A) < E(XN) - E(XN: max X. < A) 05"5N
05n5N
= E(XN: max X. > A). 05n5N
The inequality (6.5)' is obtained from E(X0) < E(X,) where
min {n < N; Xn < - Al N, if { }=0. Corollary. Let X = (Xn) be a martingale such that E(I X. I P) < 00, n = 1, 2, ... for some p > 1. Then, for every N, (6.6)
P( max I X. I > A) < E(I XN I P)lAP 0
and if p > 1, (6.7)
E{ max I XnI P} < (pl(p - 1))P E(I XNI P) OSnSN
The case of p = 2 in (6.6) is known as Doob-Kolmogorov's inequality.
Proof. By Jensen's inequality (Section 3, (E.7)), n , I Xn P is a submartingale and so (6.6) follows from Theorem 6.2. As for (6.7), setting Y = max I X I we have by Theorem 6.2 that O5n5N
AP(Y > A) < f I,nx, D
I XNI dP.
Hence
E(YP) = f dP f 'pAP-'d . = f dP f I,raz pAP-'dA a
0
0
SZ
= p f AP-' P(Y> A)dA < p f 0 f 0
= (P/(P - 1)) f n Y'
0
I XNI dP
AP-z Ii2u A
I XNI dPdA
MARTINGALES
29
and, by Holder's inequality, E(Y') < (pl(p - 1)) E(YD-' I XN1) < (p/(p - 1)) E() XNI a)"aE(Y')
1y
from which (6.7) follows.
For a real (.)-adapted process
and an interval [a, b], we
set*
T1 = min In; X. < a), T2 = min In > T1 ; X. > b), (6.8)
T2k+1 = min In > T2ki X. < a), T2k+2 = min In > T2k+1 , X. > b),
{Tn} is clearly an increasing sequence of stopping times. We set (6.9)
UN(a, b)(w) = max {k; T2k(w) < NJ.
Clearly, UN(a, b) is the number of upcrossings of (X )n 0 for the interval [a, b].
Theorem 6.3. Let X = (X be a submartingale. Then for every NE T and real numbers a and b such that a < b, (6.10)
E(UN(a, b)) < b
1
a (E
{(XN - a)+ - (Xo - a)+} ).
where Y. = (X - a)+, Proof. By Jensen's inequality, Y = n = 0, 1, 2, ... , is also a submartingale and UN(a, b) = UN(0, b - a). Let T1, T2, ... be defined as in (6.8) with X, a and b replaced by Y, 0 and respectively. Set '. = T A N. Then if 2k > N,
b-a
2k
(6.11)
YN-Yo=E(Y,-Y n=1 to-1 ) k
v,
\J
- E-1(3T'U - Y"2n-1
k-1
+ NO(Y"2n+1
* We always set min 0 = + oo unless otherwise stated.
Y
'
lT2n)
PRELIMINARIES
30
and it is not difficult to see that the first term in the right-hand side is greater than or equal to (b - a)UN(0, b - a). Also E(YT ,,+1) > E(Y=zi) since Y is a submartingale. Thus (6.10) is obtained by taking expectations in (6.11).
Theorem 6.4. If X = (6.12)
is a submartingale such that
sup E(Xj,) < co, a
then X = lim X. exists almost surely and X. is integrable.*'
Proof. Since E(I X. 1) = E(X) < 2E(X+) - E(X0), we have supE(I X. 1) < oo by (6.12). Thus, if X = lim X exists, then X n__
4 is integrable by Fatou's lemma. Clearly, if we set U!(a, b) =lim UN(a, b),
N- -
{w; 1 m X(c)) < lim X(co)} =, r, Q
(c o; Um(r, r') = oo}.
But, by Theorem 6.3,
E(U!(r, r')) = lim E(UN(r, r')) N-.w
< r,1
lim E {(XN - r)+ - (Xo - r)+}
rN__
and this is finite by (6.12). Consequently P[lim X. < lim proves that lim X. exists a.s.
0, which
m»m
Theorem 6.5. Let X = be a submartingale satisfying the condition (6.12) and let X = limX,,. In order that X = be a submartingale, i.e., X. < E(X I. ), n = 0, 1, 2, sufficient that {X} nET be equi-integrable.*Z *' In particular, non-positive submartingales possess finite limits lim X. a.s.
... , it
is necessary and
or non-negative supermartingales
*Z The reader is assumed to be familiar with the nation of equi-integrability: a family
A c _?'1(0,.F-,P) is equi-integrable if lim sup E(JX J : IX J > 1) = 0. A is equiZ-= XEA
integrable if and only if A is relatively compact in the weak topology a(2',,_511_), (cf. (129]).
MARTINGALES
31
Proof. If X. < E(X- I Y,), n = 0, 1, 2, ... , then, by Jensen's inequality, Xn < E(Xml.9 ,) and hence E(Xn : X. > A) < E(Xm: Xn > 2). Also, P(X > 2) < E(X+) /A < E(X ;) /A and thus it is clear that {X.1-1,.r is equi-integrable. Conversely, if {X} nEr is equi-integrable then the almost sure convergence lim Xn = XW is clearly a convergence in In the same way,
-m
X V (-a) in the sense of 9'1(Q). Since (XnV (-a)).ET
lim n--m
is a submartingale,
E(X- V (-a) I.7-n) = lim E(XmV (-a) I ` n) -2! XnV (-a), and so, letting a r oo, E(X. I.9 ,) > X.
Theorem 6.6. For Y E
set X. = E(YI i',), n e T.
Then X = (Xn) is an equi-integrable martingale and lim X. = X exists n~.
almost surely and in '1(Q). Furthermore, (6.13)
X = E(YI.
.),
where Y = V .,n n
Proof. Since I X.1 < E(I Y1I-9n), n c- T, IX.) is equi-integrable as in the proof of Theorem 6.5. Also (Xn) is a martingale since E(Xm I E(E(YI.irm) I n) = E(YI.Tgn) = X. for m > n. Thus X = lim X. in rW 2'1(52) and X = (Xn)ncr is a martingale. In order to prove (6.13), consider
the totality ' ' of sets B E 9- such that E(X : B) = E(Y: B). If B E
then E(Y: B) = E(X7 : B) = E(X : B) and hence IF
U .5. Clearly
'B is a d-system and hence, by applying Lemma 5.1, we see IF D .V a[ U -7-J. This proves (6.13). n
We consider, for a moment, martingales with "reversed" time. Let
(Q,,F P) be a probability space and ( .)nmo. -1. -z, ... be a family of sub a-fields of . F- such that -9'o D -1 F--z D .... X =
is called a martingale (supermartingale, submartingale) if X. is integrable, .P ,-measurable random variable such that E (Xn I .-9 m)
= X. (resp. < , > ) for every n, m E {0, -1, -2, ... } satisfyingn > m. Theorem 6.7. Let X = (XX)n-o, _1, -2.... be a submartingale such that (6.14)
inf E(X7) > - oo.
PRELIMINARIES
32
Then X is equi-integrable and limX = X_ exists almost surely and in .71(Q).
Proof. Since is decreasing as n j - oo, (6.14) implies that exists finitely as n j -oo. Let e > 0 and take k such that
lim
E(Xk) - lim
e. Then if n < k,
-.
E(I X. 1:
I X. I > 2) = E(X : X. > d) + E(Xn: X. > -),) - E(XX) <E(Xk: -A)-E(Xk)+e >).)+e.
Also
P(I X. I > 2 ) < + E(I X. 1 )
<
(2E(X n)
- E(X,,))
(2E(X o) - lim E(X )).
Now, we can easily conclude the equi-integrability of The almost sure existence of lim X. is proved similarly as in Theorem 6.4.
Now, we consider the continuous time case, i.e., T = [0, oo). Let X = (X,),ET be a submartingale.
Theorem 6.8. With probability one, t E T n Q i---- X, is bounded and possesses lim
QnT-nslr
X, and lim
X,
QnToss1r
for every t > 0.
Proof. Let T> 0 be given and jr, r2, ... } be an enumeration of is the set [r,, r2, ... , the set Q n [0,T]. For every n, if [s,, s2, ... , arranged according to the natural order, then Y, = X,,, i = 1, 2, ... n , defines a submartingale Y = (Y,);_,. Also Y = (Y1);vo where Yo = Xo and Y,,,, = X, is a submartingale. Therefore by Theorem 6.2 and Theorem 6.3, we have
P( max IY,I >A)< + {E(1101)+E(IXTI)} 1515++
and
MARTINGALES
E(U. (a, b)) < b
1
a E {(Y - a)+} < b
33
1
a E{(XT - a)+} .
Since this holds for every n, we have
>A) < +{E(j X.I)+E(IXrI)} and
E(UmiQn1',7'(a, b)) < b 1 aE{(XT
- a)+}.
By letting ) and a < b run over positive integers and pairs of rationals respectively, the assertion of the theorem follows.
By Theorem 6.8, if X = is a submartingale with respect to a reference family (.9 ,), then i, = lim X t E T, exists a.s. and it is easy ,It, reQ to see that t '-- i, is right-continuous with left-hand limits. is also a submartingale with respect to (. ). Indeed, i, is . +o = F-adapted and since for every sequence e. 10, IX,,,.) is equi-integrable by Theorem 6.7, we have
E(i,: B) = lim E(Xt+,n: B) < lim E(X,+B,: B) = E(i,,: B)
for s > t and B E ,F,. Similarly, E(X,: B) < E(X,: B) for every B E F and hence X, < X, a.s. We summarize these results in the next theorem.
Theorem 6.9. Let X = (X,)teT be a submartingale. Then it _ lim X, exists a.s. and i _ (.tt)teT is a submartingale such that t ---
fit, reQ
1, is right-continuous with left-hand limits a.s. Furthermore, X. a.s. for every t E T. I = (i,) in this theorem is called the right-continuous modification of X = (X,). Clearly P[X, = X,) = I for every t E T if and only if t -E(X,) is right-continuous. In the remainder of this section, we consider right-continuous martingale only. The first theorem is an easy consequence of Theorem 6.2 and Corollary.
Theorem 6.10. If X = (XI)IT is a right-continuous martingale such that E(I X, 1 D) < co, then for every T > 0,
PRELIMINARIES
34
(6.15)
P[ sup X,i > Z] S E[ I XrI']/A°, te[0, T]
(6.16)
E[ sup X, I'] < (pl(P - 1))' E[ I XT I'], : r0, r]
(p > 1),
(p > D.
The next theorem is a consequence of Theorem 6.1: we approximate a stopping time a by stopping times a as in the proof of Proposition 5.4,
apply Theorem 6.1 for a and then take limits as n -- oo by using Theorem 6.7.
Theorem 6.11. (Optional sampling theorem). Let X = (X,),,,T be a right-continuous submartingale with respect to (. ) and (a,),E[o,,.) be a family of bounded stopping times such that P[a, < a,] = 1 if t < s. Set X, = X, and J R7 _ -5F;, for t E T. Then X = (X,) is a submartingale
with respect to (.,). Finally, we will discuss Doob-Meyer's decomposition theorem for submartingales. In the discrete time case, any submartingale X = (X,,) with respect to
is decomposed as
n=0,1,2,...,
(6.17)
where M = is a martingale with respect to (..) and A = is an increasing process adapted to (J ,), i.e., A. < Ai+1 a.s., n = 0, 1, 2, ... . A= is always chosen to be a predictable process (i.e., A. is measurable, n = 1, 2, ... ) such that Ao = 0 a.s. and, under these conditions, the decomposition (6.17) is unique. Indeed, the process defined by (6.18)
AO = 0,
A. = A.-1 + E(X, -
I
-1),
n=1,2,...,
is a predictable increasing process and clearly M = (M.), where M. = X - A,,, is a martingale. If we have two decompositions X. = M. + A. = M; + AR, then M. - M;, = An - A. is -measurable and hence
M. - Mn = E[M - M' I Y-1] = M,_1 - Mn-1. Since Mo - Mo = Xo - Xo = 0, we have M. - M.' = 0 for all n thus proving the uniqueness of such a decomposition. In the continuous time case, the situation is more difficult.* * In the following, we fix a probability space (92, .r, P) with a reference family (.97). Martingales, submartingales, adaptedness, stopping times etc., are all relative to this reference family (J? ).
MARTINGALES
35
Definition 6.2. By an integrable increasing process, we mean a process A = (A,) with the following properties: (i) A is (317;)-adapted, (ii) Ao = 0, t ' --- A, is right continuous and increasing* a.s. (hence A, > 0 a.s.), for every t e [0, co). (iii) E(A,) < oo
Definition 6.3. An integrable increasing process A = (A,) is called natural if for every bounded martingale M = (M,), (6.19)
E[ fo MSdA,,]
E[f o M,._dAS]
holds for every t E [0, 00).
It is known that an integrable increasing process is natural if and only if it is predictable, cf. [15]. If an integrable increasing process is continuous (i.e., t -- A, is continuous a.s.), then it is natural. Indeed, ff
1IM,'*M _, dA, = 0 a.s. and hence f o M,dA, = f o M,_dA, a.s. Note also that (6.19) is equivalent to (6.20)
E(M,A,) = E (f
M,,_dA,) u
for every bounded martingale M = (M,). Indeed, for a partition A : 0 = to < t, < ... < t = t, we define M° = (MA,) by M' = M,k}1, t E (tk, tk+1].Then E[M,A,] _
n
E[M,(A,k - A,k_1)] _
km
j E[MMJ (A,k -
A,k_I)] = E[ f,, MadA,], and letting I A I (: = max (tk - tk-1)) - 0 we have E[M1A,] = E[ f o M,dA,].
For a > 0, let S. be the set of all stopping times o such that a S a a.s.
Definition 6.4. A submartingale X = (X,) is said to be of class (DL) if for every a > 0 the family of random variables {X,,; a E Sa} is equiintegrable.
Every martingale M = (M,) is of class (DL) because, by optional sampling theorem (Theorem 6.11),
f
IMaIdP< f fI Ma l `icl
IMaIdP,
o, ESa,
I I Ma l ic}
* When we say "increasing", we mean increasing in the wide sense, i.e., "non-decreasing". Otherwise, we shall write "strictly increasing".
PRELIMINARIES
36
and
sup P(I Mo I > c) < sup E(I M. I )lc < E(I M. I)/c. ass"
OES,
Therefore, if a submartingale X = (X,) is represented as
X, = M, + A,
(6.21)
where M = (M,) is a martingale and A = (A,) is an integrable increasing process, then X is of class (DL). Conversely, we have the following result.
Theorem 6.12. If X = (X,) is a submartingale of class (DL), then it is expressible as the sum of a martingale M = (M,) and an integrable increasing process A = (A,). Furthermore, A can be chosen to be natural and, under this condition, the decomposition is unique. This theorem is known as Doob-Meyer's decomposition theorem for submartingales. Proof.* First we prove that the decomposition (6.21) with A natural
is unique. Indeed, if X, = M, + A, = M,' + A,' are two such decompositions, then since A, - Af = M, - M, is a martingale, we have, for any bounded martingale m,,
'
E [ f m,_d(A, - As)] =Jim E,'_' m,k {(A, IJI-0
0
0
1 - A,',,,)
- (Ark - A;)} ] = 0, where A is a partition 0 = to < t, < t2 <
< t" = t. Consequently, is a bounded random variable, m, _ E[ I -w;] is a bounded martingale and hence E[m,A,] = E[m,A'] _ E[5A,']. Therefore A, = A, a.s. and so, by the right-continuity of A and A', A, = A' for all t, a.s. Next we prove that a submartingale X = (X,) of class (DL) has the by (6.20), E[m,A,] = E[m,A]. If
decomposition (6.21) with A natural. Because of the uniqueness result, it is sufficient to prove the existence of the decomposition (6.21) on the interval [0, a] for every a > 0. Set Y, = X, - E[X, t e [0, a]. Then
Y, is a non-positive submartingale on [0, a] such that Y. = 0 a.s. For
each n = 1, 2, ...
,
let A. be the partition 0 = to") < t;') <
<
t) = a of [0, a] given by t}"' = ja/2". Define a discrete time increasing process A,W, t E A", by * We follow here the proof of Kunita [94] which was originally given by Rao [143].
MARTINGALES
37
t k)
E/.o{E(Yti+1
,,.
Lemma 6.1. (A;"), n = 1, 2, ... } is equi-integrable.
Proof. Let c > 0 be fixed and set inf {tk)1i A(n) ) > c}, aG,)
-{
a
,
if{
Then aC') E S. Since Y,= -E[A;^) 1Y ] + AP), t E da, we have by the optional sampling theorem that Yak = -E[Aa) I JF;ck) ] + AQy). Hence, noting that A(;) < c, 41n -
E[Aa) : A') > c]
a] + E[A(ac,) : ar) < a]
-E[Y (.) ac :
< -E[Y y): a,') < a] + cP[a,') < a]. ac On the other hand
-E[Y
): o
c < a]
< a] = E[A;) -
72
ac/2
/2
E[Aa) - A
17
(n)
02
al') < a] > (c/2)P
a].
Consequently
E[Aa) : Aa) > c] < -E[Y (,) : ac
a] - 2E[Y y): o ac/2
< a].
By the assumption, {Xa, a E Sa} is equi-integrable and hence {Y,,, a E Sa}
is also equi-integrable. Since P(an) < a) = P(Aa) > c) < E(Aa ))lc <
-E(Y0)/c -- 0 as c t oo, we can now conclude that {Aa) ; n = 1, 2, ... } is equi-integrable. Now we return to the original proof. The equi-integrability of {Aa) ;
n = 1, 2, ... } implies that this set is relatively compact in the weak topology a(-?1i -121-) of the space 2'1(Q). Thus, there exists a subsequence
n1, 1= 1, 2, ... , and A. EE 2 1(Q) such that Aa d -. A. in a(2' i Y-). Define A, by A, = Y, + E[A, I. ], t E [0, a].* Since also APP - A, * E[Aa If;] is taken to be right-continuous.
PRELIMINARIES
38
in Q(.'1,
for each t e U A,,, it is clear that t '--- A, is increasing. Since X, = E[X, - A. 1,7;] + At, what remains to be proven is that A = (A,) is natural. If m, is a bounded martingale, then 2"-1
E[maA1) ] = E
k-0
tk+1
tk
1)]
2"-1
=E E[m tk k-0
tk+1
AIL))] tk
2"-1
E E[m,k1(Y
k+l - Ytk))]
2"-1
= E E[m,k)(A,k+l - Atk)]
Letting n -- oo, we have E[maA,] = E C f ° m,-dA,]
.
0
Replacing m = (m,) by m = (m,,4r) for each t e [0, a] it is easy to conclude
that
E[m,A,]=E[fm,-dA,]. 0
Thus A = (A,) is natural. Definition 6.5. A submartingale X = (X,) is called regular, if for every
a > 0 and ua a Sa such that o, t 6, we have E(X,")
E(X,).
Theorem 6.13. Let X = (X,) be a submartingale of class (DL) and A = (A,) be the natural integrable increasing process in the decomposition (6.21). Then A is continuous if and only if X is regular. Proof. If A is continuous, then clearly X is regular. Indeed, every martingale X is regular because E(X,) = E(Xa) for every o E S,. Conversely, suppose that X = (X,) is regular. Then A = (A,) is regular and it is easy to see that if c', t o', a, a S then A," t A,. Define the sequence J. of partitions of [0, a] as in the proof of Theorem 6.12. Set, for c > 0,
At = E[A (") A c I 9`], tk+1
t E (tk 1, tk+l]
MARTINGALES
39
Since An, is a martingale on the interval (tk ), tk+1], it is easy to see that
E[ f AsdA,]
(6.22)
o
= E[f o A;_dA,],
for every t e [0, a].
We prove now that there exists a subsequence n1 such that Af` converges to A, A c uniformly in t c- [0, a] as 1-- oo. Fore > 0 we define
Qn'r-{
tnf{t E [0, a]; A, -A, A c>e}, a
,
if {
} =g.
Since A; = A. A c for every n, 6,,,, = a implies that A j - A, A c < e for all t E [0, a]. Let On(t) be defined by 5,(t) = tk+, for t E (t,), t) 1]. Then a,,,, and ) belong to Sa. Since An, is decreasing in n, a n,, is = Q,. By increasing in n. Let o, = lim v,,,,. Then v, a S. and lim nrcc
the optional sampling theorem,* E[Aon 0] = E[A6nc,n,r> A c] and hence
E[AO0(,n,,)A c - A,n,e A c] = E[Aon e - A,n,5Ac]
> eP(Qn,, < a).
It is easy to see by the regularity of A,, that the left-hand side tends to zero as n - oc. Hence lim P(vn,, < a) = 0, which implies that nlim P(sup Al - Ar A c > e) = 0. From this it follows that there exists
n-- to Co. a] a subsequence nt such that
as 1- no
sup JA4,` - A,Ac! - 0
rero,a]
a.s.
Thus, by (6.22), we have
E [f r (A. A c)dAS] 0
E [f
r
(A,_ A c)dA,] 0
and so
0 = E [f o
(A, A c - A,_ A c)dA,]
? E[
{(A, A c) - (A,_ A c)} 2].
This clearly implies that t - A, A c is continuous a.s. and hence, since c
is arbitrary, t - A, is continuous as. * If we consider the expectation on the set where 6n,, E (tk"', t k+11, we may then apply the optional sampling theorem since A; is a martingale on (tk', t;,+,]. Now sum over k.
PRELIMINARIES
40
7. Brownian motions Let p(t, x), t > 0, x E Rd, be defined by (7.1)
p(t, x) = (2trt)-d'2exp[- I xI2/2t].
Let X = (Xt)te,0,m, be a d-dimensional continuous process such that
for every 0 < t, < ... < tm and E, E .' (Rd), i = 1, 2, ... , m,
(7.2)
P{X,,EE,,X:2EE2i...,XtmEEm} x, - x)dx, f p(t2 - tl, x2 - xI)dx2 = f xdp(dx)f
(
J
2
p(tm - t., xm - xm-1)dxm.
Em
Here, p is a probability on (Rd, .%(Rd)). This is equivalent to saying that
X = (X,) is a d-dimensional continuous process such that for every . < t,,, Xro, Xt, - Xt0, Xt2 - Xr,, ... , Xtm - Xtm_I are
0 = to < t, <
mutually independent, Pxro = p and P
rt,-%t'-'
= the Gaussian distribution
p(t, - t,_1, x)dx, i = 1, 2, ... , M. Definition 7.1. A process X = (Xt) with the above property is called a d-dimensional Brownian motion (or Wiener process) with the initial distribution (or law) p. The probability law P' on (Wd, R(Wd)) is called the d-dimensional Wiener measure with the initial distribution (or law) p.
Thus, the d-dimensional Wiener measure P with the initial law p is a probability on (Wd,.q(Wd)) characterized by the property that P {w; w(ti) E El,w(t2) E E2i ... ,w(tm) E Em} is given by the righthand side of (7.2). Theorem 7.1. For any probability p on (Rd, .q(R4)) the d-dimensional Wiener measure P. with the initial distribution g exists uniquely.
Proof. The uniqueness is obvious. We will show its existence. First, we consider the case d = 1 and it = b0. On a probability space, we con-
struct a family of real random variables {X(t), t e [0, co)) such that
X(0) = 0 and for every 0 < t, < t2 <
i=1,2,...,m,
P{X(t,) E E,, X(t2) a E2,
=
f
E,
...
< tm
and E, e . (R'),
,X(tm) E Em}
P(ti, x,)dx, f EZ p(t2 - ti, x2 - x,)dx2
BROWNIAN MOTIONS
41
P(tm - tm_1i xm - xm._1)dxm. Em
By Kolmogorov's extension theorem ([45]), such a system exists. Then it is easy to see that
E{IX(t)-X(s)I4} =3It-sI2,
(7.3)
t>s>0.
By the corollary of Theorem 4.3, there exists a continuous process
_
(X(t)) such that for every t E [0, oo), X(t) = t (t) a.s. The probability law P0 of t on (W',.'(W')) is the Wiener measure with the initial law 60. More generally, for x c=- R', the probability law P. of the process Y. = (Y(t)) with Y(t) = x + ±(t) is the Wiener measure with the initial law 8x.
Now let x = (x', x2, ... , xd) E Rd be given and consider the one dimensional Wiener measures P.,, i = 1, 2, ... , d. The product measure Qx Px4 on W' X W' x x W' = Wd is denoted by Px. It is easy to verify that P. is the d-dimensional Wiener measure with
Pxl Qx Px2 Qx
the initial law 8x. For any probability p on (Rd,.%(Rd)), P,,(B) = !Rd Py(B)p(dx) defines the d-dimensional Wiener measure with the initial law p.
On the probability space (Wd, _q(Wd), P,), where P, is the d-dimen-
sional Wiener measure with the initial law p, the coordinate process X(t, w) = w(t), w s Wd, defines a d-dimensional Brownian motion with the initial law p. This process is called the canonical realization of a ddimensional Brownian motion. Suppose we are given a probability space (Q,.P) with a reference family ( ) [o m). Definition 7.2. A d-dimensional continuous process X = is called a d-dimensional (9 )-Brownian motion if it is (, )-adapted and satisfies (7.4)
E[exp[i<e, X, - X,)] I ,+ ] = exp[- (t - s) I
12/2],
a.s.,
for every ce Rdand0<s
Brownian motion if (.) is the proper reference family of X; i.e., . n a[Xi; u ::g t + e].
a>o
PRELIMINARIES
42
It is easy to verify the following result.
Theorem 7.2. If X = {X, = (X t, X z:, ... , Xa)} is a d-dimensional (.91)-Brownian motion, then for every t > s Z 0, (7.5)
E(X' - XJJ.,Fs) = 0,
(7.6)
E((X: - X`)(X? - Xs)1,
a.s.,
and
) = (t - s)b,,
a.s.
Thus, if E(I X012) < 00, then each (XD, i = 1, 2, ... , d, is a squareintegrable martingale relative to (. ) such that X;Xf - 8t1t is a martingale
relative to (-7;), i, j = 1, 2, ... d. We shall see in Theorem 11-6.1 that these properties characterize a d-dimensional (.Y)-Brownian motion.
8. Poisson random measure
Let (X, Rx) be a measurable space. Let M be the totality of nonnegative (possibly infinite) integral-valued measures on (X, P. ix) and R. be the smallest a-field on M with respect to which all p e M,--- p(B) E Z+ U fool, B E .fix, are measurable.
Definition 8.1. An (M, -qM)-valued random variable p (i.e., a map-
ping p : Q ---- M defined on a probability space
P) which is
-9-/.qM-measurable) is called a Poisson random measure if (i) for each B E .°OJx, p(B) is Poisson distributed; i.e., P(p(B) = n) _ 2(B)" exp[-2(B)1/n!, n = 0, 1, 2, ... , where )(B) = E(,u(B)), B E .gx;* (ii)
if Bt, B2, ... , B E Ox are disjoint, then p(B1), p(B2),
... ,
are mutually independent.
Theorem 8.1. Given a u-finite measure 2 on (X, fix), there exists a Poisson random measure u with E(p(B)) = 2(B) for every B E dx.
Proof. Let U. E .fix be disjoint, 0 <
on and U U. = X.
On a probability space we construct the following:
(i) for each n = 1, 2, ... and i = 1, 2, ... , ,'> is a U.-valued random variable with P(ly) E du) = (ii) p,,, n = 1, 2, ... , is an integral-valued random variable such that P(p = k) = k = 0, 1, ... ; (iii) yt, p,,, n = 1, 2, ... , i = 1, 2, ... are mutually independent. Set * If .t(B) = oo, then we understand that
u(B) = oo a.s.
POINT PROCESSES AND POISSON POINT PROCESSES
p(B) = net E G. I . ,
43
B E .qX.
It is a simple exercise to prove that, = {,u(B)} is a Poisson random mea-
sure with E(p(B)) = 1(B), B E R x: just verify for every disjoint B,,
B2i...,B,ERXanda,>0,i=1,2,...,m,that m
E(exp {- E a,p(B)]) = exp[ 1= tml
1)2(B1)].
Clearly the probability law of 1u is uniquely determined by the measure
A. 2 is called the mean measure or the intensity measure of the Poisson random measure u.
9. Point processes and Poisson point processes Let (X, OX) be a measurable space. By a point function p on X we mean a mapping p: D, c (0, oo) - X, where the domain D. is a countable subset of (0, oo). p defines a counting measure N,(dtdx) on (0, oo) x X* by (9.1)
N,((0, t]x U) = # {sED,; s 0, U e Ox.
A point process is obtained by randomizing the notion of point functions. Let nX be the totality of point functions on X and A(IIX) be the
smallest a-field on IIX with respect to which all p - N, ((0, tJ x U), t > 0, U e ,56X, are measurable. Definition 9.1. A point process p on X is a (Hx,.9(17X))-valued random
variable, that is, a mapping p: 92
nX defined on a probability space (0, Y,P) which is .7/. '(Hx)-measurable. A point process p is called stationary if for every t > 0, p and Op have
the same probability law, where 9p is defined by DB, = Is E (0,00); s + t e D,} and (8,p)(s) = p(s + t). A point process p is called Poisson if N,(dtdx) is a Poisson random measure on (0, oo) x X. A Poisson point
process is stationary if and only if its intensity measure n,(dt dx) _ E(N, (dtdx)) is of the form (9.2)
n,(dtdx) = dtn(dx)
for some measure n(dx) on (X,X). n(dx) is called the characteristic * We endow (0, oo) xX with the product a-field .i((0, oo)) x.i
.
PRELIMINARIES
44
measure of p. Given a measure n(dx) on (X, Ox), p is a stationary Poisson
point process with the characteristic measure n if and only if for every t > s Z 0, disjoint U1, U2, ... , U. E .eX and i., > 0, m
E(exp [ - E AIN,((s, t] x UU)] I Q[N,((0, s'] x U) ; s' < s, U E RX])
= exp [(t - s) E (e-11 - 1)n(U,)] i-1
a.s.
Theorem 9.1. Given a Q-finite measure n on (X,.%1) there exists a stationary Poisson point process on X with the characteristic measure n. The following construction is essentially the same as in Theorem 8.1; indeed, p may be identified with a Poisson random measure on (0, o) x X having the intensity measure dtn(dx). Let Uk E k = 1, 2, ... , be such that they are disjoint, n(Uk) < oo and X = U Uk. Let ', k,i = 1, 2, ... , be Uk-valued random variak
E dx) = n(dx)/n(Uk) and z(', k,i = 1, 2, ... , be nonbles with negative random variables such that P(z ' > t) = exp [- tn(Uk)] fort > 0. We also want all ', z?' to be mutually independent. After constructing such random variables on a probability space (0, ,,F-,P), we set
D, =
kUl
{z11) ,
zik' + zZ' , ... , z' -}- y' +
...
+ zn', .. .
and
P(z) + T2' -{- ... + zm') _ m),
k, m = 1, 2, ... .
It is easy to see that the point process so defined is what we want.
It is easy to see that this union is disjoint a.s.
CHAPTER II
Stochastic Integrals and Ito's Formula
1. Ito's definition of stochastic integrals Let (0, , -, P) be a complete probability space with a right-continuous increasing family ( of sub 6-fields of _57- each containing all P-null sets. Let B = (B(t)),zo be a one-dimensional (,7-,)-Brownian motion (cf. Definition 1-7.2). Since with probability one the function t '---- B(t) is nowhere differentiable, the integral f' f(s)dB(s) can not be defined in the
ordinary way. However, we can define the integral for a large class of functions by making use of the stochastic nature of Brownian motion. This was first defined by K. Ito [59] and is now known as Ito's stochastic integral.
Definition 1.1. Let -272 be the space of all real measurable processes 0 _ {0(t, w)},zo on 92 adapted to (,F;) *1 such that for every T > 0, (1.1)
II41IIz,r= Elf. 02(S,w)ds] < 00.
We identify 0 and W in -912 if IIO - O'II2.T = 0 for everyT> 0, and
in this case write h = V. For 0 E 22, we set (1.2)
II0I12 = E 2--(110112,. A D.
Clearly I10 - (P'il2 defines a metric on -72i furthermore, 2'2 is complete in this metric. Remark 1.1. For every 0 E 22 there exists a predictable*2 Sly' E -72 Cf. Chapter 1, §5. *2 Cf. Definition 1-5.2. 45
STOCHASTIC INTEGRALS AND ITO'S FORMULA
46
such that (P = W': for example, we may take (t, w) = lira -L
0(s, co)ds.*
hto h
t-h
Thus without loss of generality we may assume that 0 E 22 is predictable.
Definition 1.2. Let 2', be the subcollection of those real processes
0 _ {0(t,w)} E . having the property that there exists a sequence < t < - oo and a sequence of real numbers 0 = to < t1 < of random variables { f (co)} tao such that f is Y-,-measurable, sup 11f ,11I
,
if
t=0,
,(w)
,
if
t E (t1, tt+11,
i = 0, 1, ... .
Clearly such 0 is expressed as 0(t, CO) =.fo(w)l(.-op(t) + E f(w)I ,.t,+,](t) Lemma 1.1. -210 is dense in -92 with respect to the metricIIzI I
Proof. Take ch c--- Y, and set Om(t,co) Then OM E - 9 and I 10
- 0MII2 - 0 as M - oo. Thus it is enough
to show that for any bounded cp E .°2 we can find ch E moo, n = 1,
2, ..., such that III' - 0. 112 -- 0 as n - oo. Let 0 = {ch E 2'2i 0 is bounded and there exists ch E .2°o such that 110 - 45n112 ~ 0 as
n - oo} . 0 is a linear space and it is easy to see that if fi E 0, 10 1 < M for some constant M > 0 and 45. t 0 then 0 E 4$. Suppose 0 is a left-continuous bounded (,7;)-adapted process. Then if we set 0(0, 0(
k T-
co),
t = 0, 1 (L, L+--], 2B 2n
it is clear that . E 2°o and I On - 45I I2 --- 0 as n -- co by the bounded
convergence theorem. Now, by Proposition 1-5.1, we can conclude that * To prove this rigorously, we have to appeal to Theorem 4.6 in [1171, p. 68 to guarantee
that 0 has a modification & progressively measurable with respect to (.7) (i. e., for every T > 0, the mapping (t, w) - fi (t n T, W) is .i ([0, 21) x ,7 .-measurable).
ITO'S DEFINITION OF STOCHASTIC INTEGRALS
47
0 contains all bounded (.)-predictable processes. By Remark 1.1, 0 contains all bounded 0 E 2z. Remark 1.2. A more direct proof may be found in [18] p. 440-441.
Definition 1.3...dlz = {X = (X,),2.0; X is a square integrable martingale*1 on (Q,_7",P) with respect to (. )rz0 and X0 = 0 a.s.}...4 = {X E .illz; t '--- X, is continuous a.s.}.
We identify two X, X' E./z if t - X, and t i--- X' coincide a.s. Definition 1.4. For X e llzi we set (1.3)
IXIT = E[XT]1/z,
T > 0,
and
III = E 2-n(IXI. A 1).
(1.4)
We note that since X is a martingale, I I, is non-decreasing in
t.
Lemma 1.2. 02 is a complete metric space in the metric IX - Yl, X, Y E -4/z, and llz is a closed subspace of .%'2.
Proof. First we note that if I X - YI = 0, then X = Y; indeed,
IX - Y I = 0 implies X,= Y. a.s., n= 1, 2, ..., and so X,= E[X 1,9`j = E[ Y. I ,] = Y, for t < n. By the right-continuity of t - X, and t i--.Y, we conclude that X = Y. Next, let lim
X(°),
n = 1, 2, ... ,
ix(n) - X(m) I = 0.
be a Cauchy sequence;
i.e.,
Then by the Kolmogorov-Doob inequality for martingales*z we have
for every T > 0 and C > 0
P[ sup I XP) - X,('", I > Cl < rz IX O5t5T
X (m) IT.
Therefore, there exists X = (X,) such that sup
O tsT
I X,(n) - X I -0
*1 It is always assumed that t *'Theorem 1-6.10.
in probability X, is right continuous a.s.
STOCHASTIC INTEGRALS AND ITO'S FORMULA
48
as n -- oo for every T > 0. Clearly, for every t > 0, E [I X 1n) - X, I zJ -- 0 as n --- oo and we
X1 -- 0 as n --- co. can conclude from this that X E=-..02 and IX Finally, it is also clear from this proof that if X` E.4 y, then X E -020. We will now define the stochastic integral with respect to an (.V)Brownian motion as a mapping
0E
-.1(0) E -4Z20.
Suppose we are given an (Y)-Brownian motion B(t) on
If 0E-2oand 0(t, co) = f0(w)11r-O) (t) + E f (w)4,,t,+11(t) i-o
then we set n-1
(1.5)
1('k)(t, co) = Vf (w)(B(tr+1, (o) - B(t1, co))
+ff,(w)(B(t, c)) - B(t n, co))
for t < t < (1.6)
n = 0, 1, 2, .... Clearly 1(0) may be expressed as
1(O)(t) = f (B(t A tt+1) - B(t A t1))
(This sum is, in fact, a finite sum.) We can easily verify that for s < t
E[f(B(tA t1+1) - B(tAt,))I
] =.f(B(sAt1+1) - B(sAti)),
and hence 1(0)(t) is a continuous , -martingale, i.e., 1(P) E -W c. Also it is easy to verify that (1.7)
E(I(O)(t)z) _
1-0
E[f 3(t A t1+1 - t A tJ]
= E[f '0 V(s, w)ds].*
Thus, (1.8)
1(fi)I r =
From this formula we see that 1(0) is uniquely determined by 0 and is independent of the particular choice of {t j .
ITO'S DEFINITION OF STOCHASTIC INTEGRALS
49
and
(1.9)
110)1= II0II2
Next, let 0 E 22. Then by Lemma 1.1 we can find 0n E -2% such that 110 - 0.112 -- 0 as n -- oo. Since 11(0,,) - 1(0m)l = 110 - 0mhI2, is a Cauchy sequence in .i/'2 and consequently by Lemma 1.2, it converges to a unique element X = (X,) .% . Clearly X is determined uniquely from 0 and is independent of the particular choice of 0,,. We denote X by I(0).
Definition 1.5. 1(0) e ..4 defined above is called the stochastic integral of 0 X 22 with respect to the Brownian motion B(t). We shall often denote I(0)(t) by f a 0(s, w)dB(s, co) or more simply f o O(s)dB(s). Clearly if 0,1' E -1212 and a, f E R, then (1.10)
1(aO + ffl(t) = aI(0)(t) + fI(Y')(t)
for each t Z 0 a.s.
Remark 1.3. Thus a stochastic integral 1(0) is defined as a stochastic process (in fact, a martingale). But for each fixed t the random variable 1(0)(t) itself is also called a stochastic integral.
Proposition 1.1. The stochastic integral with respect to an Brownian motion has the following properties: (i) 1(0)(0) = 0 a.s. (ii) For each t > s ? 0, (1.11)
E[(1(0)(t) -
0
(S)-
a.s.
and (1.12)
E[(.I(0)(t) -
I(0)(s))21_g;]
= E[f 02(u, w)duI.
]
a.s.
More generally, if a,r are (-F-,)-stopping times such that r > o a.s., then for every constant t > 0 (1.13)
E[(10)(tl\ T) - I(0)(tAa))1. ] = 0
a.s.
and
(1.14)
E[(I(0)(t A z) - I(05)(t A a))2 1. ;] = E[ f rns02(u, w)du rnv
as.
STOCHASTIC INTEGRALS AND ITO'S FORMULA
50
(iii) (1.12) and (1.14) have the following generalization: if 0, Yr E 22,
then
I j",] (1.15)
= E[
f (0 - [)(u, (o)dul.l;] t
a.s.
S
and
E[(I(O)(t Az) - !(0)(tAa))(I(T)(t Ar) - I('E)(tAa))I (1.16)
(iv)
(1.17)
=
E[ ranno (o-T)(u, Jr
]
co)dujJ9;] a.s.
If a is an (.)-stopping time, then
I(0)(tna) = I(-P')(t)
for every t >- 0,
where V(t, w) = Ii,ccn)zn O(t, w).*,
Proof. (i) is obvious. (1.11) is clear since I(0) is a martingale. (1.12) is easily proved first for 0 E -`°0 and then by taking limits. (1.13) and (1.14) are consequences of Doob's optional sampling theorem. Therefore,
we need only prove (iv).*2 First consider the case when 0 E 20. Let {sy)} i_0 (n = 1, 2, ... ) be a refinement of subdivisions {t1} i_(' and {i2-"}l0. Suppose ch has the expression 0(t, co) = fo(w)Iu-(,(t) + * for each n = 1, 2, .... Define S
a"(&)) = si+,
if a(co) E (s)"',
It is easy to see that an is an (Y';)-stopping time for each n = 1, 2, .. . and an 1 a as n --- oo. If Y E (s(,"), s,(" ),], then = I,o>4c",,and therefore, if we set 0n(s, co) = 0(s, co) On' E 2°o. Clearly, for every t > 0, II0, - W1II2', = E[
as n --- oo. Hence
u
02(s, w)I{Q
I(0.) - I(W)
p>o,
] --' 0
in -#f2.
*1 Clearly WE=- -212.
*2 A simpler proof is given in Remark 2.2 below.
Also,
ITO'S DEFINITION OF STOCHASTIC INTEGRALS
51
(B(tASk+1) - B(tASk)))
I(c)(t) =
= if,(a)I(a>s
(B(t A an A sk+)l) - B(t AQn AS ")) k)f
_ Lfk (0))(B(tAa"Ask+)1) - B(tA9"Ask))) k-0 ('
J
w)dB(s) 0
an < sk1
since
if a < sk ). Consequently A
I(W)(t) = 1 m
f to
o (h(s, (o)dB(s) = I(45)(tA6).
The general case can be proved by approximating 0 with 0" E -9',. )Let B(t) = (B1(t),B2(t), ... , B'(t)) be an r-dimensional Brownian motion and let 01(t,w), ... , 0,(t, co) E 2'2. Then the
stochastic integrals f a 0,(s)dB`(s) are defined for i = 1, 2, ... , r.
Proposition 1.2. For t > s > 0, ,
E[ fs Pi(u)dB`(u) f r 0,(u)dBJ(u) I
]
=
(1.18)
= Su E[ I 0,(u)Oj(u)du l Fl, ij = 1,2, ... , r. S
Proof. It is easy to prove this when 0, e .1°0i i = 1, 2, ... , r. The general case follows by taking limits.
So far we have defined the stochastic integral for elements in 22. This can be extended to a more general class of integrands as follows. Definition 1.6. Y 10C = {sls = {fi(t)}
ra0 i
0 is a real measurable pro-
cess on 0 adapted to (,,`) such that for every T > 0
f
(1.19)
o
a.s.}.
02(t, (o)dt < co
We identify 0 and 0' in 17 ° if, for every T > 0,
i
J
T i 0(t, co) - W'(t, (o)12dt = 0 0
a.s.
STOCHASTIC INTEGRALS AND ITO'S FORMULA
52
In this case we write 0 = 0'. Similarly as in Remark 1.1, we may always assume that 0 E _172'°° is a predictable process.
Definition 1.7. A real stochastic process X = (X,),2,0 on (Q, Jv-, P) is called a local (Y )-martingale if it is adapted to (-0';) and if there exists a
sequence of (,F-,)-stopping times a such that a < co, a t oo and X. = (XX(t)) is an (.Y)-martingale for each n = 1, 2, ... where X (t) _ X(t A a.). If, in addition, X. is a square integrable martingale for each n, then we call X a locally square integrable (Y )-martingale. By taking an appropriate modification if necessary, we may and shall always assume for such X that t' -- X, is right continuous a.s.
Definition 1.8..2 ° _ {X = (.
X is a locally square integrable
)-martingale and Xo = 0 a.s.}. 111cil" = {X E .4'2`; t - X, is
continuous a.s.}. Let B = (B(t)),2:0 be an (.Y)-Brownian motion and 45 (E . °2 `. Let o(w) = inf {t; f' 02(s,a>)ds > n} An, n = 1,2, .... Then a is a
sequence of (,Y;)-stopping times such that a t oo a. s. Set sl (s,w) _ P(s, w). Clearly
f
0n(s, w)ds = o
a
02(s, co)ds < n un
and hence 0 e Y2, n = 1, 2, .... By Proposition 1.1 (iv), we have for m < n that I(Pn)(t A am) = I(Pm)(t) Consequently if we define I(45)
1(0)(t) = I('n)(t)
for
by
t
then this definition is well-defined and determines a continuous process I(c1) such that
n = 1, 2, ... .
I(0)(tAa.) = Therefore I(cP) e
4'2
Definition 1.9. I(0) E 4'2'° defined above is called the stochastic integral of 0 E 22'a with respect to the Brownian motion B(t). We shall
STOCHASTIC INTEGRALS WITH RESPECT TO MARTINGALES
53
often denote I(c)(t) by f 45(s,w)dB(s,a) or more simply f o -h(s)dB(s). As in Remark 1.3, we shall also call the random variable I(h)(t), for each fixed t, a stochastic integral.
2. Stochastic integrals with respect to martingales Let (92, , `,P) and (.
)tZo be given as in Section 1. Suppose we are
We shall now define a stochastic integral f's(s)dM(s) which coincides with that in Section 1 when M is an (,)-Brownian given M
motion (Kunita-Watanabe [97]).
Proposition 2.1. (i) Let M = (M,) E .4g. Then there exists a natural integrable increasing process*l A = (A,) such that M2 - A, is an (. )martingale. Furthermore, A is uniquely determined.*2 (ii) Let M = (M,) and N = (N,) be in -.4/2. Then there exists A = (A,) which is expressible as the difference of two natural integrable increasing processes such that MN, - A, is an (Y )-martingale. Furthermore, A is uniquely determined.
Proof. Let M = (M,) E ..2. Then t - M, is a non-negative submartingale of the class (DL) and hence by Doob-Meyer's decomposition theorem*', there exists a unique natural integrable increasing process
A = (A,) such that t F- M2 - A, is an (f )-martingale. If M, N E
.4 i then M, = (M -f- N)/2 E
2
and M2 = (M - N)/2 E 02.
Let Al and A2 be the natural integrable increasing processes corresponding
as above to M, and M2 respectively. Then A = A' - A2 satisfies that t :- M,N, - A, is an (_F;)-martingale. The uniqueness of A is a consequence of the uniqueness of Doob-Meyer's decomposition.
Definition 2.1. (i) A = (At) in Proposition 2.1 (i) is denoted by <M>= (<M>t),Z0.
(ii) A = (At) in Proposition
2.1
(ii)
is denoted by <M, N> =
(<M, N> )rzo.
Note that <M> = <M, M>. We call <M, N> the quadratic variational process corresponding to M and N. By Theorem 1-6.13, we see that <M, N> is continuous if at least one of the following conditions is satisfied: (i) (. )rzo has no time of discontinuity, that is, if o is an increasing *1 Cf. Definition 1-6.2 and Definition 1-6.3.
*2 i.e., if A' is another such process, then t - A, and t - A' , coincide a.s. *1 Theorem 1-6.12.
STOCHASTIC INTEGRALS AND ITO'S FORMULA
54
sequence of (,91-,)-stopping times and 6 = lim a., then _0-, = V 9 n
n
(ii) M,NE .f2.
Indeed, if (i) is satisfied, then MA,. = E[MM I . E[M:I. AQI = Mtn, as Qn T o and hence M(t)2 is regular.
A,n]
Example 2.1. Let B(t) _ (B'(t),B2(t), ... , B!(t)) be a d-dimensional (.F)-Brownian motion such that B(O) = 0 a.s. Then, for each i, B` E=- _,//2 and >(t) = 8,,t, i, j = 1, 2, ... , d. A famous theorem
of Levy asserts that a d-dimensional (.)-Brownian motion is characterized by this property (cf. Theorem 6.1).
Let M e _02 and <M> be the corresponding quadratic variational process.
Definition 2.2. 2'2(M) = {0 =
' is a real
(.
)-pre-
dictable process and, for every T > 0, (2.1)
(II0112T)2=E[f0 V(s,w)d<M>(s)l <-}.
For 0 E -72(M), we set (2.2)
110112 = E 2°(II0II2n A 1). n-1
We identify 0 and W in 2'2(M) if I10 - x'11 T = 0 for every T and write 0 = 0'. If M is an (Y )-Brownian motion, 22(M) coincides with -2 of Definition 1.1. Note that 2'2(M) -2'o,* where 2' is defined by Definition 1.2. In exactly the same way as in Lemma 1.1, we have the following result.
Lemma 2.1. 20 is dense in 2'2(M) with repect to the metric 11 IIz Using this lemma, we can now define stochastic integrals in the same way as in the case of a Brownian motion. First let 0 E _1210. Then 0(t) _
.fo(w)Irt-m(t) + E
and we set
n-1
I M(p)(t) = El w)M+1, co) -M(1, w) + f,(o))(M(t, w) - M(tn, w)) for t :
n=1,2, ... * Every 0 e .1Z°0 is left-continuous and so it is predictable.
STOCHASTIC INTEGRALS WITH RESPECT TO MARTINGALES
55
and 11 M(4)' = 11011m. Using this isometry, 0
As before IM(O) e
vo -- IM(h) E_ /2 is extended to 0 E Y 2(M)
IM(45) a -02 as
in Section 1.
Definition 2.3. IM(0) is called the stochastic integral of 0 e .2(M) with respect to M E=-,02. We shall also denote IM(')(t) by f o 0(s)dM(s).
Thus we have defined the stochastic integral with respect to a martingale M e.,ff2. It is clear that if M E:.4 then IM(45) E`42c and if M is an (Y)-Brownian motion, then IM(c) coincides with I(0) defined in Section 1.
Proposition 2.2. The stochastic integral IM(0), 0 e _91'2(M), M -02i has the following properties: (i) IM(0)(0) = 0, a.s.
(ii) For each t > s > 0, (2.3)
a.s.
E[(IM(P)(t) - IM(O)(s)) I.9s] = 0,
and (2.4)
E[(IM(cl)(t) -
E[ f r 02(u)d<M>(u)I
a.s.
S
More generally, if a, T are .9-,-stopping times such that T > a a.s., then for every t > 0, (2.5)
E[(IM(c)(t AT) - IM(O)(t A a)) I] = 0,
a.s.
and
(2.6)
E[(IM(c)(tAT) - IM((P)(tAa))2I. E[ 'n 02(u)d
f
=
,AQ
] a.s.
<M>(u) I J7;],
(iii) (2.4) and (2.6) can be generalized as follows: if 0, 1 e .172,(M), then (2.7)
and
E[(IM(cP)(t) -
= E[
f
IM('I')(s))I
t
('P!')(u)d<M>(u) (
],
a.s.
]
STOCHASTIC INTEGRALS AND ITO'S FORMULA
56
r'
E[(IM(0)(t AT) - IM(45)(t A o))(IM([)(t A T) - IM( [')(t A a)) I = E(J (cM')(u)d a.s. <M>(u) I Yl,
(2.8)
]
rnq
(iv) If a is an (75;)-stopping time, then
IM(fttAa) = IM(W)(t)
(2.9)
where
for
t >_ 0,
0'(t) =
The proof is same as in Proposition I.I.
If M,N E. 2, 0 E .2(M) and 'I' E 2'2(N), then IM(O) and IN(Y') are defined as elements in -02.
Proposition 2.3. For t > s > 0, (2.10)
E[(IM(c)(t) -
IM(O)(s))(IN(W)(t)
- IN(J')(s))I JF,]
= E[ .'(0W)(u)d<M,N>(u)IY]
The proof is easily obtained first for O,tl' E .o and then by a limiting procedure. (2.10) is another way of saying that (2.11)
<1-(O), IN(V)>(t)
= J o (0W)(u)d<M,N>(u),
or more symbolically, (2.11)'
d, = 0(t)1I'(t)d<MN>,.
Remark 2.1. (2.10) implicitly implies the following result: if M, N e
-02, 0 re Y 2(M) and ¶ E 5f2(N) then E[ J o I `htVI (u)dl <M,N)1(u)] < 00
where 1 <M,N> 1(t) denotes the total variation of s E [0, t] i--- <M, N)(s). In fact, we have the inequality 112
(2.12)
f
o
I OP I (u)dl <M,N>1(u) <_
x jf
IffO(u)2d<M>(u)}
0P(u)'d(u)}112
* Iip2,, is left continuous in t and so it is predictable. Hence Sis' e 9'2(M).
STOCHASTIC INTEGRALS WITH RESPECT TO MARTINGALES
57
(2.12) is easily proved when 45,P E Y. and the general case follows by a limiting procedure. Note that if t - A, is of bounded variation on finite intervals and is continuous, then there exists a predictable process 0, such that 10,1 = 1 a.s. and JAI, = f o (P(u)dA,,. The above definition of stochastic integrals is extended in an obvious
manner to the case of local martingales. Let M, N E .% C. Then there exists a sequence of (_F-,)-stopping times vn such that ant oo a.s. and Man = (M,A,n) and Non = (NA,,) are in .02. It follows from the uniqueness of the quadratic variational process that if m < n then \MQm, Nam>(t) = <Mg", Non>(t A 0m).
Hence there exists a unique predictable process <M,N> such that <M, N> (t A 6n) = <Man, Nan>(t) for all n and t > 0. We write <M, M> as <M>.
Definition 2.4. Let M E 92 °. i °(M) = {s _ (45(t)); 0 is a real (,;r)-predictable process on Q such that there exists a sequence of stopping times vn such that Qn t oo a.s. and (2.13)
E[ f
rAon
02(t, co)d<M>(t)] < 00
o
for every T> Oandn= 1,2, ...}.* Let M E .0? ° and 0 E 2' °(M). Then clearly we may choose a
sequence of (.9)-stopping timesa Qn such that an t oo a.s., Man = (M (t A an)) E and (2.13) is satisfied. Hence for (P,(t, w) = co) and Man, we can define and it is easy to see that IM (On)(t A vm) for m < n. Thus there exists a unique process IM(P)(t) such that n = 1, 2, ... . It is clear that lm(O) E -Wz°C. Definition 2.5. IM(O) is called the stochastic integral of 0 E 2z °(M) with respect to M E -4/210IM(O)(t) is also denoted by f o 0(s)dM(s). It is clear that Propositions 2.2 and 2.3 easily extend to this general case. In particular, we note that (2.11) holds. As a special case of (2.11), we obtain (2.14)
(t) = f '45(u)d<MN)(u) 0
This property completely characterizes the stochastic integral; more precisely, we state the following proposition. * If <M>, is continuous, (2.13) is equivalent to that fo 0z(t,c))d<M)(t) < oo
T> 0, a.s.
for all
STOCHASTIC INTEGRALS AND ITO'S FORMULA
58
Proposition 2.4. Let M E -02 and 0 E 2'2(M) (or M E _,021-2 E _2'2'-(M)). Then X= I M(4?) is characterized as the unique X E (X E -//2 '0') such that (2.15)
2
<X, N)(t) = fo -P(u)d<M, N)(u)
for every
(N E=-_J/100 ) and all t > 0.
Proof We need only show uniqueness. If X' E.///2 also satisfies (2.15), then <X - X', N) = 0 for every N E .2 and so, by taking N =
X- X', <X-X')=0. Hence X= X. Remark 2.2. For example, we can prove (2.9) using this characterization. Denoting generally by Xa the stopped process {X(a At)), by Proposition 1-5.5 and Doob's optional sampling theorem, <(IM)(q) N)(t) = <(JM)(-) N°>(t) =
=f where W(t) =
Q
o
-P(u)d <M, N)(u) = fo W(u)d <M, N)(u),
Consequently IM(W) = IM(O)" by Proposition
2.4.
Proposition 2.5. (i) Let M, N E .4 '20C and 4; E _,;,2 °(M) (12' a (N).
Then 0 E 2 120W + N) and
f
(2.16)
o
0(u)d(M+N)(u) = f o 0(u)dM(u)
+foc
(u)dN(u).
(ii) Let M E .4 ° and 0, P E -7 '2-M. Then (2.17)
f o (d5 + 1')(u)dM(u)
(iii) Let M E 4'20c and
= fo
d5(u)dM(u)
PE2
2(M).
+f
P(u)dM(u).
Set N = IM(0) and let
V E=- Y° i C(N). Then OYr E 2'(M) and
f
(0fl(u)dM(u) = f o'Y(u)dN(u). o
(iv) Let M E .4d a °. Let 4P E 2''2'(M) be a stochastic step function
STOCHASTIC INTEGRALS WITH RESPECT TO POINT PROCESSES 59
in the sense that there exists a sequence of increasing (_F;)-stopping times
c, such that
I(t, co) = E.f
(w)I111n"+il(t),
where fn is . .measurable. Then 1M(t)(t) = Efn(w)(MA,,,+,-MfA0. ) The proof is easy and so we omit it. It is proved most simply by using Proposition 2.4.
3. Stochastic integrals with respect to point processes The notion of point processes was given in Section 9 of Chapter I. Here we discuss point processes adapted to an increasing family of 6-fields and their stochastic calculus. Let (Q, _57-,P) and (.. ;),zo be given as above.
A point process p = (p(t)) on X defined on 0 is called (,F)-adapted if for every t > 0 and U E ,9(X), N,(t, U) = E La(p(s)) is , -measurasED0,ssl
ble. p is called a -finite if there exist U. E O (X), n = 1, 2, ... such that U. r X (i.e. U UU = X) and E[N,(t, no for all t > 0 and n = 1, 2,
....
n
For a given (.)-adapted, Q-finite point process p, let f', _
{U E .:3`(X); E[N,(t,U)] < co for all
t > 0). If U E F,, then t -
N,(t, U) is an adapted, integrable increasing process and hence there exists
a natural integrable increasing process IV,(t,U) such that R,: t -9,(t, U) = N,(t, U) - N,(t, U) is a martingale. In general, t -1V,(t,U) is not continuous but it seems reasonable (at least in applications discussed
in this book) to assume that t
*,(t,U) is continuous for every U.
U '---- N,(t, U) may not be a-additive, but if the space X is a nice space e.g., a standard measurable space (Definition 1-3.3), it is well known that a modification of N, exists such that it is a measure with respect to U. Keeping these considerations in mind, we give the following definition:
Definition 3.1. An (Y)-adapted point process p on to be of the class (QL)* (with respect to (.
is said )) if it is a-finite and there exists
= (*,(t, U)) such that (i) for U E T t - N,(t, U) is a continuous (Y';)-adapted increasing process, * Quasi left-continuous.
STOCHASTIC INTEGRALS AND ITO'S FORMULA
60
(ii) for each t and a.a. co E 0, U
N,(t,U) is a cr-finite measure on
(X,P(X)),
(iii) for U E T, t , 9,,(t, U) = N,(t,U) - *,(t, U) is an (.9)-
martingale. The random measure {1V,(t, U)} is called the compensator of the point process p (or {N,(t, U)}).
Definition 3.2. A point process p is called an (Y';)-Poisson point process if it is an (.')-adapted, 6-finite Poisson point process such that {N,(t + h, U) - N,(t, U)} h>o. UEa(x), is independent of .-.
An (.)-Poisson point process is of class (QL) if and only if t ,-E(N,(t, U)) is continuous for U E I',; in this case, the compensator 9, is given by *,(t, U) = E[N,(t, U)]. In particular, a stationary )-
Poisson point process is of the class (QL) with compensator *,(t, u) _ tn(U) where n(dx) is the characteristic measure of p (Chapter I, Section 9). This property characterizes a stationary (.)-Poisson point process (cf. Section 6).
Theorem 3.1. Let p be a point process of the class (QL). Then for U E T RP(-, U) E ..12 and we have (3.1)
A,(-, U1),
U2)>(t) = N,(t, U1 (1 U2).
For the proof, we need the following lemma. Lemma 3.1. If U dictable process, then
e r, and f(s) = f(s,co) is a bounded (3)-pre-
X(t) = f f(s)d2,(s,U)* (=
f(s)I.(p(s)) - f f(s)dV,(s,U)) sEDy
is an (,)-martingale. Proof. By Proposition 1-5.1, it is sufficient to assume that s i-- f(s) is a bounded, left continuous adapted process. Then, for every s E [0, oo), MS) =J(0)IIs_01(s) + k-0 f(2- )1(k2^.(kt1)/2^7(s) -As)-
Thus it is sufficient to prove the result for f (s). But * In the sense of the Stieltjes integral.
STOCHASTIC INTEGRALS WITH RESPECT TO POINT PROCESSES 61
k-0
0
f
2 )[ND(k 2"
1 At'U)
At,U)] - N'(2^ T.-
is obviously an (,)-martingale.
Proof of the theorem. It is clear that there exists a sequence of (Y )-stopping times a. such that Qn { oo a. s. and both 9 ;)(t, U,) _ N,(t A a,,, U,) and N;) (t, U2) = N,(t A Q,,, U2) are bounded in t. Consequently it suffices to show for each n 9,14(t, U,)N;^>(t, U2) = a martingale + *,(tAa,,, U3 (1 Uz)
By an integration by parts, Nn)(t, U,)N;)(t, U2)
= fo
U,)N;'(ds, U2) +
J ' l ' (s, U2)NP)(ds, U)
= f N;U,)ND°)(ds, U2) + f 0
+f
r 0
N;'(s-, U2)Np)(ds, U,) o
[1V; (s, U2) - RP(-)(s-, U2)lN;)(ds, Uj).
But the first two terms are martingales by Lemma 3.1 and the last term Iu, n u2(P(s)) = N, (t A Q,,, U, n U2) = N,(t A Q,,, u, f1 U2) + is equal to sSrna SEDp
N(tAQ,,,U, n U2). This proves the theorem. We are now going to discuss stochastic integrals with respect to a given point process of the class (QL). For this it is convenient to generalize the notion of predictable processes.
Definition 3.3. A real function f(t, x, (o) defined on [0, oo) x X x Q is called (-97-,)-predictable if the mapping (t,x,w) -f(t,x,w) is ,9"1.%(R')measurable where .So is the smallest a-field on [0, oo) x X x Q with respect to which all g having the following properties are measurable: g(t, x, co) is . 4'(X) x.7-measurable; (i) for each t > 0, (x, co) (ii) for each (x, co), t F--- g(t, x, co) is left continuous. We introduce the following classes:
F, = { f(t, x, o)); f is (.,)-predictable and for each t > 0, (3.2)
f f x I.f f(s, x, w) I N,(dsdx) < cc 0
a.s.},
STOCHASTIC INTEGRALS AND ITO'S FORMULA
62
Fa = { f(t, x, w); f is (J",)-predictable and for every t > 0, (3.3)
E1$$
OX
I f(s, x, ) I Xt,(dsdx)] < oo} ,
F, = { f(t, x, co); f is (-F' )-predictable and for every t > 0,
'
(3.4)
E[ f ofx I f(s, x, _) 1 2 1G,(dsdx)] < _j,
F;,'- = {f(t, x, co); f is ()-predictable and there exists a sequence of (. )-stopping times a such that a too
(3.5)
a.s. and Ico,,,,j (t)f(t, x, co) E F,, n = 1, 2, ...
For f e F (3
f o+ f Xf(s, x, )N,(dsdx)
is well defined a.s. as the Lebesgue-Stieltjes integral and equals the absolutely convergent sum
Ef(s,p(s), )
(3.7)
SEA
Next let f E F. By the same argument as in the proof of Lemma 3.1, it is easy to see that
E[ f 'J" I f(s, x, ) I N,(dsdx)] = E( f of x 1 f(s, x, ) This implies, in particular, that F,' e F,. Set
J, J f(s, x, )N,(dsdx) + x
(3.8)
=f
f xf(s, x, .)N,(dsdx)
- f f xf(s, x, )*,(dsdx).
Then t - f fx f(s, x, )N,(dsdx) is an (,W;)-martingale by the same o 3.1. proof as in Lemma
If we assume f E F,,(1 F;, then
t'
'
f o+f xf (s' x, ')N,(dsdx)
E llZ
and
(3.9)
< f '+ f xf(s, x, )N,(d dx)> = f
ofxf2(s, x,
)1,(dsdx).
SEMI-MARTINGALES
63
(3.9) is proved by the same argument as in the proof of Theorem 3.1 and Lemma 3.1.
Let f c F. If we set f,(s, x, co) = I(-..»)(f(s, x,
x, (o)*)
then f a Fp fl FP and so f `o f x f (s, x, )N,(dsdx) is defined for each n. By (3.9),
E[ If "
)N,(dsdx) - f o+J
xf.(s, x,
= E[f of x
xf.(s, x, )N,(dsdx)} 2]
x, ) - fm(s, x, )]ZN,(dsdx)]
and thus { f f x f (s, x, )N,(dsdx)} ;,-, is a Cauchy sequence in ..4 . We u denote its limit by $f x f(s, x, )R,(dsdx). Note that (3.8) no longer holds: each term in the right-hand side has no meaning in general. The integral f f xf(s, x, )Sr,(dsdx) may be called a "compensated sum".
o Finally, if f e F2,1", the stochastic integral f$ f(s, x, )N,(dsdx) x
is defined as the unique element X E .4i ° having the property
X(tAa.) = f
+f o
x Iro.
x, )N,(dsdx),
n = 1, 2,
where {a } is a sequence of stopping times in (3.5).
4. Semi-martingales The time evolution of a physical system is usually described by a differ-
ential equation. Besides such a deterministic motion, it is sometimes necessary to consider a random motion and its mathematical model is a stochastic process. Many important stochastic processes have the following common feature: they are expressed as the sum of a mean motion and a fluctuation from the mean motion.*', A typical example is a stochastic process X(t) of the form X(t) = X(0) + f f(s)ds + f o g(s)dB(s) {UJ is chosen so that U. T X and E[N,(t, U,)] < co for every t > 0 and all *2 It may be considered as a noise.
n.
STOCHASTIC INTEGRALS AND ITO'S FORMULA
64
wheref(s) and g(s) are suitable adapted processes and f -dB is the stochas-
tic integral with respect to a Brownian motion B(t). Here the process f of(s)ds is the mean motion and f 0'g(s)dB(s) is the fluctuation. Such a stochastic process is called an It6 process and it constitutes a very important class of stochastic processes. The essential structure of such a process is that it is the sum of a process with sample functions of bounded variation
and a martingale. Generally, a stochastic process X(t) defined on (0, -9-,P) with a reference family (-F',) is called a semi-martingale if
X(t) = X(0) + M(t) + A(t) where M(t) (M(0) = 0, a.s.) is a local (. )-martingale and A(t) (A(0) = 0, a.s.) is a right-continuous (Y)-adapted process whose sample functions
t i-- A(t) are of bounded variation on any finite interval a.s. We will restrict ourselves to a sub-class of semi-martingales which is easier to handle and yet is sufficient for applications discussed in this book.* For simplicity, we will call an element of this sub-class a semi-martingale. Namely we give the following
Definition 4.1. Let (Q,.F P) and (.F )rZo be given as usual. A stochastic process X = (X(t)),ZO defined on this probability space is called a semi-martingale if it is expressed as
X(t) = X(0) + M(t) + A(t) + f + f fi(s, x, -)N,(dsdx) o
(4.1)
+ fo
x
f fz(s, x, -)N,(dsdx)
x
where
(i) X(0) is an J -measurable random variable, (ii) M E 6Cz'°° (so, in particular, M(0) = 0 a.s.),
(iii) A = (A(t)) is a continuous (.)-adapted process such that a.s. A(0) = 0 and t i- A(t) is of bounded variation on each finite interval, (iv) p is an (.Y)-adapted point process of class (QL) on some state space
(X, MX)), fi E F, andfz E Fp" such that (4.2)
.fi fz = 0.
It is not difficult to see that M(t) in the expression (4.1) is uniquely determined from X(t); it is called a continuous martingale part of X(t). The * For a detailed treatment of the general theory of semi-martingales, we refer the reader to Meyer [121] and Jacod [77].
SEMI-MARTINGALES
65
discontinuities of X(t) come from the sum of the last two terms and, by (4.2), these two terms do not have any common discontinuities. Example 4.1. (Levy processes). Let X(t) be a d-dimensional time homogeneous Levy process (i.e. a
right continuous process with stationary independent increments) and be generated by the sample paths X(t). Let D, = {t > 0; X(t) 4 X(t-)} and, for t E D,, let p(t) = X(t) - X(t-). Then p defines a stationary (.)-Poisson point process on X = Rd\ {0} (cf. Definition 3.1). The famous Levy-Ito theorem*' states that there exist a d'-dimensional Brownian motion B(t) = (Bk(t))k ,, 0 < d' < d, a dxd' matrix A = (a,) of the rank d' and a d-dimensional vector B = (b') such that X(t) = (X'(t), X2(t), . . . , Xd(t)) can be represented as r+fRd\fOfxilnxtauNn(
X'(t) = X'(O) + E akBk(t) + bit + f0
)
IC-2
f0i+
+J
fRd\
xiI11X1
i = 1, 2, ... , d.
{0
In this case, the compensator #,(dsdx) of p is of the form 1 (dsdx) = dsn(dx), where n(dx) is the characteristic measure of p. n(dx) is also called the Levy measure of the process X. It is a Q-finite measure on Rd\ {0} such
that fad\ro' {1xIZ/(1 + I xIz)}n(dx) < co. In the above expression, the Brownian motion {B"(t)} and p are automatically independent (cf. Section 6). By (4.3), we get the following LevyKhinchin formula: (4.4)
E[ei<e.xa>-x(a)> ice} = e
a.s.,
t > s > 0,
E Rd,
where i
2 (4.5)
+
fRd\ [0)
(ei<'x> -1 - iI(j=i<1) )n(dx).*2
Let X(t) be a given semi-martingale with respect to (,) and let (.fi) be generated by the sample paths X(t). Clearly .,fix c .9 for each t > 0. It is not trivial that X(t) is also a semi-martingale with respect to Cf. L6vy [99] and Ito [58], [69]. *Z A* stands for the transposed matrix of A. rs Cf. Stricker [155].
STOCHASTIC INTEGRALS AND ITO'S FORMULA
66
The continuous martingale part of X(t) with respect to (-rf) differs from that with respect to and generally it is an important problem to discuss how the martingale part differs under a change of a reference family.*'
5. Ito's formula Ito's formula is one of the most important tools in the study of semimartingales. It provides us with the differential-integral calculus for sample functions of stochastic processes. Let (Q,,x ,,P) with be given as above. Suppose on this probability space the following are given: (i) M'(t)E.42sec, (i 1, 2, ,d); (ii) A'(t) (i = 1, 2, ... , d): a continuous (Y )-adapted process whose almost all sample functions are of bounded variation on each finite interval
..
and A'(0) = 0; (iii) p: a point process of the class (QL) with respect to (.) on some state such that f°(t, x, m)g'(t, x, co) = 0, i, j = 1, 2, , d; furthermore, we assume that g(t, x, co) is bounded, i.e. a constant M > 0 exists such that I g'(t, x, co) I S M
for all i, 1, x, co.
(iv) X°(0) (i = 1, 2, , d): an moo-measurable random variable. Define a d-dimensional semi-martingale X(t) = (X'(t), X2(t), Xd(t)) by
XI(t) = X`(0) + MI(t) + AI(t) (5.1)
+
f',
x
f'(s, x,
)N, (dsdx) + f ,`f
Denote also f = (f', f 2, ... J11) d) and g = (g', g2,
x, )N,(dsdx),
i= 1,2,
.. d.
... , gd).
Theorem 5.1. (Ito's formula).*2 Let F be a function of class C2 on Rd and X(t) a d-dimensional semi-martingale given above. Then the stochastic process F(X(t)) is also a semi-martingale (with respect to and the following formula holds:*'
F(X(t)) - F(X(0)) *1 Cf. e. g. Jeulin [78] for a general theory and applications. In filtering theory,it is related
to the notion of "innovation", Fujisaki-Kallianpur-Kunita [29]. *2 Cf. Ito [63], Kunita-Watanabe [97] and Doleans-Dade-Meyer [17]. *3
:=aFFi=a f, i.l=1,2,....
ITO'S FORMULA
67
i
= 1-1E f F,(X(s))dM`(s) + E f 1 Fr(X(s))dA`(s) 0
+ (5.2)
2
Jo
Fti(X(s))d<M`, M'>(s)
+ f o+ fX {F(X(s-) + f(s, x, )) - F(X(s-))}N,(dsdx)
+ f o+ fx {F(X(s-) + g(s, x, )) - F(X(s-))}N,(dsdx)
+ f(,I {F(X(s) + g(s, x, )) - F(X(s))
-
t
g`(s, x, )FF(X(s))}1G,(dsdx).
Proof. To avoid notational complexity, we shall assume d = 1; there is no essential change in the multi-dimensional case.
First, we will prove the result in the case of continuous semimartingale: (5.3)
X(t) = X(0) + M(t) + A(t).
Formula (5.2) then reduces to F(X(t)) -- F(X(0)) = f 'F'(X(s))dM(s) + f0` F'(X(s))dA(s) (5.4)
+ 2 Jo Let 0
,
if I X(0) I > n,
zn= inf{t;IM(t)I >norIAI(t)>nor <M>(t) I >n}, if
I X(0) I < n.
Clearly z r oo a.s. Consequently, if we can prove (5.4) for X(t A on the set {zn > 01, then by letting n r oo we see immediately that (5.4) holds. Therefore, we may assume that X(0), M(t), IAI(t), <M>(t) are bounded in (t, co) and also that F(x) is a C2-function with compact support.
Fix t > 0 and let A be a division of [0, t] given by 0 = to < t1
<
< to = t. By the mean value theorem,
F(X(t)) - F(X(0)) _
k-1 k-1
{F(X(tk)) - F(X(tk_3))}
F(X(tk-1)) {X(tk) - X(tk-1)}
STOCHASTIC INTEGRALS AND ITO'S FORMULA
68
+ 2 kE F"(Sk) {X(tk) - X(tk-I)} 2, where & satisfies X(tk)
AX(tk_I) S bk <X(tk)V X(tk_I). Firstly
E F'(X(tk-I)) {X(tk) - X(tk-I)} k-I n
=E k-IF'(X(tk-I)) {M(t k) - M(tk-I)}
+ Ek-IF'(X(tk-I)) {A(tk) - AN-01 say.
=1'd+ 1.2,
It is clear that, as I J J = max I tk -I*-,] - 0, I? ---. f' F'(X(s))dM(s) k
a.s. Also, if we set
V(s, co) =1.'-o)(s)F'(X(0)) +
(Xk_I..kJ(s)F'(X(tk-I))
and
0(s, c)) = F'(X(s)),
then 0" E 20 and II0° - 0II = E[
I45°(s, c)) - 0(s, (v)I zd<M>(s)]'12 0
as 1.4 1 -- 0. Hence I, = f' 0°(s,co)dM(s) .P2(Q) as
-0
f o F'(X(s))dM(s)
0.
IAI
Secondly,
2 kE F"(k) {X(tk)
=
-
X(tk-I)} 2
1
E F"(k) {A(tk) - A(tk-I )} 2 2 k-I
+E
{M(tk) - M(tk_I)}{A(tk) - A(tk_I)}
+ .l kE F"(Yk) {M(tk) - M(tk-1)} 2
=I;+I;+ISi say.
0
in
ITO'S FORMULA
69
It is not hard to show that I; and I; tend to 0 a.s. as I A I - 0; for example,
I$ I < sup I F"(x) I max I M(tk) - M(tk_,) I JA 1(t) - 0 a.s. as "ER
15kS
d -- 0. We now show that IS -- fo F"(X(s))d<M>(s) in .I(Q). For this we need the following lemma.
Lemma 5.1. Let C > 0 be a constant such that I M(s) 1 < C, s E [0, t]. Set Vd = E {M(tk) - M(tk_1)} 2,1= 1, 2, ... , n. Then k-1
12 C4.
Proof. It is easy to see that
(Vn)2 = E {M(tk) - M(tk.i)}4 k-1
n
+ 2 E (Vn - V k)(M(tk) - M(tk-1))2 and n E[(Va - Vk)I. k] = E[1-E1 WOO - M(t,_1)}2I.
k]
= E[(M(t) - M(tk))2I.k] < (2C)2. Hence,
E[
k-1
{(V n
- V)(M(tk) - M(tk-1))2} ] (2C)2E(M(t)2) < 4C4.
Finally, n
E[ E {M(tk) - M(tk_1)}4] <
4C4,
k_1
and the proof is complete.
Now, returning to the proof, we set
I6 = 2 kE F"(X(tk-i)) {M(tk) Then
M(tk-1)12
STOCHASTIC INTEGRALS AND ITO'S FORMULA
70
{ max I F'(&) - F'(X(tk-,)) I z} )1iz(E {(V)JZ} )1 iz E(I Is° - I°s 1) < 1(E 2 1
<
I
max I F"(ck) - F"(X(tk-,)) 12}
)112 - 0
as Id 1 -0 by the dominated convergence theorem. If we set
I, =
2 k-1
F"(X(tk-1)) {<M)(tk)
- M(tk-1)} ,
then clearly
E[ I I,° - 2 fo F (X(s))d<M>(s) I]- 0
(5.6)
as
1,41 -> 0
by the dominated convergence theorem. Finally,
E{II6 -
=
1
4
I;12}
E {[
E F (X(tk-1)) {(M(tk) - M(tk-1))2
k-1
- (<M>(tk) - <M)(tk-1))} ]2} . Noting that E {[(M(tk) - M(tk-1))2 - (<M)(tk) - <M) N-1))] 1-97,j = 0
the above equals E{ n [F"(X(tk 4
Z
for all k, -1))2 ))2
((M(tk) - M(tk-1))2
- (<M)(tk) - <M)(tk-1))} 2]}
<2 + (5.7)
I F"(x) 12E 2
max
I
(M(tk) - M(tk-1))4}
F"(x) 12 E {
kE
(<M>(tk) -' <M)(tk-1))2}
< 1 max I F"(x) 12E[ max ((M(tk) - M(tk-1))2Vn] n
I F"(x) 12E[m5x(<M)(tk) - <M)(tk-1))<M)(t)]
+2 <_
2 max I F"(x) I
+
max I .Rl
2(E[(Vn)2])1 /2(E[
maSx I M(tk) - M(tk-1)
I4])1/2
F"(x) 12E[ max (<M>(tk) - <M)(tk-1))<M)(t)]-
1-5n
This last expression tends to zero as 1,J I --- 0 by Lemma 5.1 and the dominated convergence theorem.* By (5.5), (5.6) and (5.7), we have * Note that <M>(t) is bounded.
ITO'S FORMULA
71
proved that IS - 4$ F"(X(s))d <M>(s) in 2'1(Q). Thus (5.4) is true for a fixed time t but, since both sides are continuous in t a.s., (5.4) holds for
all t>0a.s. Next we will prove (5.2) for the general X(t) in the case d = 1: o+
X(t) = X(0) + M(t) + A(t) + f f Xf(s, x, )N,(dsdx) + f 'J.' g(s, x, )N,(dsdx). 0 x First we note that the proof is easily reduced to the case that I f (s, x, )
is bounded; i.e. there exists M > 0 such that
If(s,x,cv)I < M
for all (s, x, co);
because the set {s; I f (s, p(s), co) I > M } is discrete in (0, oo) almost surely. Then, by the usual method of truncation, the proof can be reduced to the case where g E F,2 and F is such that F, F' and F" are
bounded. For the point process p, let U. E .'(X), n = 1, 2, be such that Un c U U. = X and E(N,((0, t] x Un)) < oo for all t > 0. For each n set X,,(t) = X(0) + M(t) + A(t) + f+ fXfcn>(s, x, )N,(dsdx)
+ f f +g(n) X (s, x, )N,(dsdx), 0
x, w)lun(x). co) where f cn> (s, x, co) = f(s, x, First we prove (5.2) for the semi-martingale XX(t). The point process pn
defined by D,n = is E D,; p(s) E Un} and pn(s) = p(s) for s E D,n is discrete in the sense that # {s; s < t, s E D,n} is finite a.s. for each t > 0. If we order the set D,n according to magnitude, say 0 < 6, < a2 <
< 6m <
it is easy to see that Qm is an (.)-stopping time. Now
XX(t) is represented as
XX(t) = X(0) + M(t) + A(t) + Ef(r,,,P(6m), ) o,n5t
r
+ E g(cm, P(Q,), ) = 0X, g `n' (s, x, )1V,(dsdx). c,n5t Then, setting vo = 0,
F(Xn(t)) - F(X(0)) = F. {F(X, Jm A t)) - F(Xn(7m A t -))}
STOCHASTIC INTEGRALS AND ITO'S FORMULA
72
+ E {F(Xn(tlm A 0) - F(XA(am-1 A t)))
=11(t) + 12(t) say. We can apply (5.4) to obtain F(Xn(am A t )) - F(Xn(am-1 A t)) m_i
om_tAt
F'(X.(s)) dM(s) +
f
At
P'(X (s))dA(s)
om_iAt
emu
+ f QmAt m-1 F"(Xn(s))d<M>(s) - f am_1 F'(Xn(s))dA'n(s), where Agn(t) = f t f gn)(s, y, )NP(dsdy).
Thus t
12(t) = f 0F'(XX(s))dM(s) + f 0 F'(X4(s))dA(s) (5.9)
f
+ 2 fo
F'(Xn(s))dAgn(s). o
Noting the assumption f(s,x,c))g(s,x,co) = 0, we have 11(t) _
m
{F(X,,((crm)) - F(X,(Qm ))} cmo
o)
+ E {F(Xn(Qm)) - F(Xn(Cm))} Ilam5(.8(om.P(am). )+r0)
= f to f x (5.10)
{F(Xn(s-) + f (n) (s, x, )) - F(XX(s-))l NP(dsdx)
+ f fx = f+f
{F(Xn(s) + g (n) (s, x,
)) - F(Xn(s-))l N,(dsdx)
o
o
x
{F(X (s--) + f n) (s, x,
+ fo
)) -
F(Xn(s-))} NP(dsdx)
fx{F(XX(s-) + g(n)(s, x, )) - F(Xn(s-))} N,(dsdx)
+ f ofx
{F(X,,(s) + g(n)(s, x, )) - F(XX(s))} R,(dsdx).
By (5.9) and (5.10), we see that (5.2) holds for the process XX(t). The forTo be precise, F(Xn(am At-)) =
{F(Xa(t))
if if
Qm S t
t < am.
MARTINGALE CHARACTERIZATION OF BROWNIAN MOTIONS
73
mula (5.2) for the process X(t) is obtained by letting n -- oo. First, fo fxg`"'(s, x, )N,(dsdx) converges to fa fxg(s, x, .)R,(dsdx) in -4-, as n- oo and hence, by taking a subsequence if necessary, we may assume that this convergence is uniform on every finite interval a.s. Also f f xf " "' (s, x, )N,(dsdx) converges to f fx f(s, x, )N,(dsdx) as n --- 00 o uniformly in t on each finite interval a.s.oConsequently X"(t) converges
to X(t) as n - oo uniformly in t on each finite interval a.s. and hence F(X"(t)) - F(X(0)) -- F(X(t)) - F(X(0)) a.s. Also, it is easy to see, by the dominated convergence theorem, that f o F'(X"(s))dM(s) 0
-
f ` F'(X"(s))dA(s) 0
f'0 F"(X"(s))d<M>(s) j
0x
f t F'(X(s))dM(s)
in -Az,
0
f ' F'(X(s))dA(s)
a.s.,
0
-f
F"(X(s))d<11I>(s)
a.s.,
o
{F(X n(.) + g (n> (s, x, )) - F(X n(s)) g(n)(s, x, )F'(X.(s))}LV,(dsdx)
- f of x {F(X(s) + g(s, x, )) - F(X(s))
- g(s, x, )F'(X(s))}lQ',(dsdx)
0
x
a.s.,
{F(X"(s-) + f , c"> (s,x, )) - F(X"(s-))}N,(dsdx)
f o+
f x {F(X(s-) +.f(s,x, )) - F(X(s-))}N,(dsdx)
a.s.,
and
f f x {F(X"(s-) + g t"' (s, x, o+
)) - F(X"(s-))}17,(dsdx)
f f x {F(X(s-) + &,x, )) - F(X(s-))}N,(dsdx)
in Al,.
o
Thus the proof of (5.2) for X(t) is now complete.
6. Martingale characterization of Brownian motions and Poisson point processes It is a remarkable fact that many interesting stochastic processes are
STOCHASTIC INTEGRALS AND ITO'S FORMULA
74
characterized as semi-martingales whose characteristics (e.g., the quadratic variational process of the continuous martingale part, the compensator of the point process describing discontinuities of the sample path) are given functionals of sample paths.*' Martingale problems (first introduced by Stroock and Varadhan [157]) are just such ways of determining stochastic processes. They are based on the fact that the basic stochastic processes such as Brownian motions and Poisson point processes are characterized in terms of the characteristics of semi-martingales. This fact itself may be considered as a typical martingale problem.
Theorem 6.1. Let X(t) = (X'(t),X2(t), ... , Xd(t)) be a d-dimensional (,F;)-semi-martingale such that (6.1)
MI(t) = X`(t) - X1(0) E _0J-
and (6.2)
1,2, ... , d.
<M`, M'>(t) = 3, jt,
Then X(t) is a d-dimensional (.9)-Brownian motion*z
Proof. It is enough to prove that a.s.
e
for every E Rd and t > s > 0. Let F(x) = mula. Then we have e'<', x(t)>
(6.4)
-
and apply Ito's for-
('
_ Z J ,1
ke`qx(u)>dMk(u) + 1 E f r(2 k-1
Clearly (6.2) implies that Mk Fall? and hence (6.5)
E[ f et<1,1("udM'(u)I J] = 0
a.s.
Take any A E .T. Then, multiplying both sides of (6.4) by e-t14,x(s)>I4 and taking the expectation, *' Grigelionis [44]. *Z Cf. Definition 1-7.2.
MARTINGALE CHARACTERIZATION OF BROWNIAN MOTIONS E[e'
- P(A)
I J 2 $'
75
E[e`q-x(")-xu >I4]du.
From this integral equation we see at once
= P(A)e4'V2('-$).
E[erq,xa>-x(s)>: A]
This proves (6.3). In particular, this theorem implies that a continuous process X(t) is a
one-dimensional Brownian motion if and only if both t i--- X(t) and t r-- X(t)2 - t are martingales. This result is known as a theorem of P. Levy (cf. Doob [18], Chapter VII, Theorem 11.9).
Example 6.1. Let X(t) = (X1(t),X2(t), ... , Xd(t)) be a d-dimensional (J57;)-Brownian motion and p = (p,(t, w)) be a process with values
in orthogonal d x d-matrices such that each component pl (t,cv) is an (.)-predictable process. Set
MI(t) = X`(t) - X'(0) and
Xk(t) = Xk(0)
+
E ' pi(t, cv)dM'(s), 1-1 J 0
k = 1, 2,
... , d
where Xk(0) is an ,9"o measurable random variable. Then X(t) 12(t), ... , Xa(t)) is a d-dimensional (., )-Brownian motion. Indeed, setting Mk(t) = X"(t) - X'`(0),
<Mk, M`)(t) =
ft
E pm(s, w)p(s, w)d<Mm, Mn)(s)
Om, n-1 t
= J 0 m-1 pm(s, w)Pm(s, w)ds (Skit.
Theorem 6.2. Let p be a point process of class (QL) with respect to (,`) on some state space (X,.%(X)) such that its compensator 1V,(dtdx) is a non-random a-finite measure on (0, oo) x X. Then p is an (_9F-,)-Poisson
point process. If, in particular, N,(dtdx) = dtn(dx) where n(dx) is a nonrandom o-finite measure on X, p is a stationary (.9;)-Poisson point process with n as its characteristic measure.
STOCHASTIC INTEGRALS AND ITO'S FORMULA
76
Proof. The idea of the proof is essentially same as in Theorem 6.1.
Let t > s > 0 and let U1, U2, ... , U. E -q(X) be disjoint sets such that 1Q,((0, t] x Uk) < oo, k = 1, 2, ... , m. It is enough to prove for m
E[exp(-Zt1AIN,((S, t] x U,))1 ,7;1
exp[ - m (e ak - 1)2,((s, t] x Uk)] k-I In fact, one can easily deduce from (6.6) that N,(E1),N,(EZ), ... are independent if El, E2, ...E .°. f((0, oo)) x .9'(X) are disjoint. Let F(xl, x2, ... , xm) m
7
,kxk] = exp [-E k-1
and
fk(t,x,w) = IUk(x),
/{ m). f = (.f',f 2, ... J-).
Then
f
f xf k(S' x, )N,(dsdx) = N,((O,t] X Uk)
0
and by Ito's formula (setting N(t) = (N,((0,t] x U),
...
,N,((0,t]
x Um))
F(N(t)) - F(N(s)) = f s+f x
[F(N(u-) +f(u, x, ))
- F(N(u-))]N,(dudx)
= a martingale + f t f [F(N(u) + flu, x, )) - F(N(u))11Q,(dudx). x
But
F(N(u) +f(u, x, )) - F(N(u)) =
ekE1AkN((0.s]xUk)(e-k2:1xkUk'x'
- 1).
Consequently, as in the proof of Theorem 6.1 we have from this
E[e
kEIZkN((s.t],tUk): A]
t
- P(A)
= fSfXE[exp(- k-1 1kN((s, u] x Uk)): for any A E ... Hence, E[e-kElxkN,((s.t]'Uk) : A]
-1)1Q,(dudx)
MARTINGALE CHARACTERIZATION OF BROWNIAN MOTIONS 77
= P(A) exp [k-l E (e -1k - 1)*,((s, t] x Uk)]. This proves (6.6). If, in particular, N,(dtdx) = dtn(dx), we have that in
m
(6.7)
E[e k=_,. JN,,((S.:]XUk) I -F;] = exp[(t - s) E (e-1k - 1)n(Uk)j
and sop is a stationary Poisson point process with n(dx) as its characteristic measure.
Theorem 6.1 and Theorem 6.2 can be combined as in Theorem 6.3: an interesting point is that we have automatically the independence of Brownian motion and Poisson point process.
Theorem 6.3. Let X(t) = (X'(t),X2(t), ... , Xd(t)) be a d-dimensional (Y) semi-martingale and p,, p2, ... , p. be point processes of class (QL) with respect to (.) on state spaces X1, X2, ... , X. respectively. Suppose that (6.8)
MI(t) = X'(t) - X'(0) E
(6.9)
<M', M'>(t) = 5,jt,
(6.10)
i,.1 = 1,2,
... , d,
the compensator P,,(dtdx) of p, is a non-random Q-finite measure
on [0, oo) x X,, i= 1,2, ... , n and (6.11) with probability one, the domains D,, are mutually disjoint. Then X(t) is a d-dimensional (.)-Brownian motion and p, (i = 1, 2, ... , n) is an (Y)-Poisson point process such that they are mutually independent. n
Proof. Let D, = UD,, and set, p(t) = pt(t) if t e: D,,. Then we have n
a point process p on the sum U X, * which is clearly a point process of the 1=t
class (QL) with compensator Xt,(dtdx)
Therefore,
p is an ()-Poisson point process. This clearly implies that p, i = 1,
2, ... , n, are mutually independent Poisson point processes since the X, * The sum of X1, X2, ... , X. is a set H such that there exists a family of subsets H,,
H2, ... , H with the following property: H, are mutually disjoint, r=t UH1 = H and there exists a bijection between X, and Ht for each i. Identifying H, with Xt, we often n denote the sum as tUX
STOCHASTIC INTEGRALS AND ITO'S FORMULA
78
are supposed to be disjoint (by the definition of sum U X,). Thus it is suf-
ficient to prove the independence of X(t) and p and this will be accomplished if we show that for every t > s > 0, E[ef(4'.x(t)-x(a')) a k=1
(6.12)
=
e-Ti4i2(t-S)
exp [ E (e-1k - 1)1Crp((s, t] x Uk)], k-1
where , 2 = (2k), { U1} have the same meaning as in the proof of Theorems m
6.1 and 6.2. Let F(x1, ... , xd, y', ... , ym) = and apply Ito's formula to the (d + m)-dimensional semi-martingale (XX(t), N,((0, t]x Uk))r-1,2....,d,k-1,2,....m (6.12) then follows as in the proof of previous theorems.
Finally we note that the strong Markov property of Brownian motions and Poisson point processes are simple consequences of Theorems 6.1 and 6.2.
Theorem 6.4. Let X(t) = (X'(t),X2(t), ... , V(t)) be a d-dimenmotion and a be an (,l)-stopping time such that a < oo a.s.*' Let X*(t) = X(t + a) and J w-,* _ . + , t e [0, oo). Then X* = {X*(t)} is a d-dimensional (9 *)-Brownian motion. In particular, B*(t) = X(t + a) - X(6) is a d-dimensional Brownian motion which is independent of . * _ -F-,. sional
Proof. By Doob's optional sampling theorem, M*t(t) = Xt(t + u) X'(a) is a local martingale with respect to (J0",-*) and also <M*1, M*t)(t) = 6,1(t + a - a) = btjt, i,j = 1, 2, ... , d. Then the assertion of the theorem follows from Theorem 6.1. In the same way, we have
Theorem 6.5.*2 Let p be a stationary (.)-Poisson point process on some space X with the characteristic measure n(dx) and a be an (.F)stopping time such that a < no a.s. Let a point process p* on X be defined by
D,* = {t; t + o c- D} *1 If we only assume that P(a < oo) > 0, we have the same conclusion by restricting S2 to s2 fl {Q < oo) and by substituting P by P(-)= P( fl {a < ool)/P(o- < co). *2 Cf. It8 [70], Theorem 5.1 where it is called the strong renewal property.
MARTINGALE CHARACTERIZATION OF BROWNIAN MOTIONS 79
and p*(t) = p(t ± o), t E D,*. Let ,
* = . +,. Then p* is a stationary (.*)-Poisson point process with the characteristic measure n.
Let X(t) = (X'(t), X2(t), ... , Xd(t)) be a d-dimensional Brownian motion on a complete probability space and let (-9-,X) be the family of o fields generated by the sample paths X(t):.F'tx = 6 {X(s); s < t} V if/. Here .,V stands for the totality of P-null sets.
Lemma 6.1..E o = 7-,X. Proof. Let p(t, x) be given by (1-7.1) and set
(H1f)(x) =
f
Rd
f E C0(R"").*
p(t, x - y)f(y)dy,
{H,} constitutes a strongly continuous semigroup of operators on C0(Rd). We can rewrite (1-7.2) in the following form EUi(X(t1))f2(X(t2)) - ...fn(X(tn))J
f
RdU(dx)H(t1,
t2,
.
-
, tn; {fl,f2, ... ,fn)(x),
where 0 < t1 < t2 < .. < tn,f,,f2,... ,fn E C0(Rd) and Hn(t1, t2, ... , tn; f ,f2, ... ,fn) E Co(Rd) is defined inductively by Hn(tu t2, ... , tn; fl, f2, ... , f,,) = Hn-1(t1, t2, ... , tn-1;f1,f2) ... J.-21f.-1A1,-,.-,f.)
H1(t;f) = Hj.. Consequently, if tk_1 < t < tk we have EL f, (X (t l))f2(X (t2)) ... f. Wt.)) 1 1-1
1 XI
t, tk+1 - t, ..., to - t;fk,fk+l,.. ,fn)(X(t))
and hence
E( 1(X(t1))f2(X(t2)) ... fn(X(tn)) I -0t+-03 * C0(Rd) is the Banach space of all continuous functions on Rd such that lim I f(x) I ixI= 0 with the maximum norm.
STOCHASTIC INTEGRALS AND ITO'S FORMULA
80
= lim ELJ 1(X(t1))f2(X(t2)) ... f,,(X(tn)) 1-9U] hto
This
= EW 1(X(t1))f2(X(t2)) ... fn(X(tn)) I proves ,-+ = x
tXl
Lemma 6.2. For any increasing sequence aA of (.Y )-stopping times,
where a = lim n.--
Proof. By the strong Markov property, E[f (X(t1))f2(X(t2)) ... fn(X(tn)) I
E Itrk_15T
fk,fk+ll ... ,f)(X(T)) + 1-1 for any (,,`7-,X)-stopping time T. Using this, the lemma can be proved as in Lemma 6.1.
Let B'(t) = XI(t) - X'(0), i = 1, 2, ... , d. Then B' E The following theorem, first proved by Ito as an application of the multiple Wiener-Ito integrals (c.f. [64]), is very useful and will be used often in this book. The proof which we give is based on Theorem 6.1 and is due to Dellacherie [16].
Theorem 6.6. Let M = (Mr) E .4(5) (_02 (. exist o, E 2' (.9"r'r) (P (6.13)
X)). Then there
i = 1, 2, ... , d, such that
M(t) _ Z f r ',(s)dB'(s). 1-1
0
That is, every martingale with respect to the proper reference family of X(t) can be represented as a sum of stochastic integrals with respect to the basic martingales B', i = 1, 2, ... , d. Proof. We assume that X(O) is constant. The proof is easily reduced * , llZ (9x) is the space .,K2 with respect to (.9X).
MARTINGALE CHARACTERIZATION OF BROWNIAN MOTIONS
81
to this case by a standard argument. For simplicity of notations, we assume d = 1; i.e., X(t) is a one-dimensional Brownian motion and B(t) X(t) - X(0). It sufficies to prove (6.13) on each finite interval [0, T]; indeed, it is easy to see that '(s) is determined consistently on different intervals and thus defines the expression (6.13) on [0, oo). The spaces -02i .4Li, .2°2, etc. all refer to the proper reference family (,99-f) and time is
restricted on [0, T]. Let .% 2 c .4' be defined by 2
.,Lz = {M(t) = u -P(s)dB(s); 0 E _92} . J
The theorem then asserts that _dV2 = -,//*. To prove this, we first show that every M E 2 can be expressed as (6.14)
M(t) = MI(t) + M2(t)
where M, E.'2* and M2 E.4LZ satisfies <M2i N> = 0 Clearly, such a decomposition is unique if it exists.
for all N E 02*,
Let 2' = {MI(T); M, E.4l z } . It is easy to see that W is a closed subspace of .f2(Q,P). Let 21 be the orthogonal complement of X. Now, let M E. 2 be given. Then, since M(T) E Y2(Q,P), we have the orthogonal decomposition
M(T) = H, + H2, where HI E A and H2 E 2'-. By definition, H, is of the form H1(c)) f 45(s)dB(s) for some 0 E -°2. Let M2(t) be the right-continuous modification of E[H2 1Yr 9. Then clearly
M(t) = MI(t) + M2(t), t E [0, T], where MI(t) = f 0(s)dB(s). It remains to show that <M2, N>(t) = 0 on [0, T] for every N E .4' : that is, t p--- M2(t)N(t) is an (. X)-martingale on [0, T]. For this it is sufficient to show that, for every (-97-,x)-stopping time or such that a < T, E[M2((7)N(a)] = 0.*1
But if N(t) = f o P(s)dB(s), then N°(t) _ !o F(s)I,SS,,dB(s)E. g *2 and hence Cf. Corollary of Theorem 1-6.1 and its continuous time version.
*2 For X = (X(t)), X° = (X°(t)) is defined by X°(t) = X(tAa).
STOCHASTIC INTEGRALS AND ITO'S FORMULA
82
E[N(a)M2(a)] = E[N(a)E[M2(T) 13`a]]
= E[N(0r)M2(T)] = E[N°(T)H2] = 0. Thus, we have now completed the proof of the decomposition (6.14). To prove the theorem, it is sufficient to show that, for a dense subspace ..,Y c sz2i the M2-part in the decompositions (6.14) of M E -N' vanishes: indeed, we then have that -4'c .0,* and, since .4 is closed, we
have .4 -0,*. Let Y = {M e .4'2; M is bounded} . It is easy to see that Y is dense in 02 because, in the space Ae = {M(T); M E.4} =_22(Q,Jrr, P),* ,, = {F e X; bounded} is dense and the norm I.1,. of .4; (restricted to the interval [0, T]) was defined by the -272-norm of g°. Let M E _,T 'and M = M, + M2 be the decomposition of (6.14). Since M, is a continuous martingale, there exists a sequence a (= an(M,)) of (. x)-
stopping times such that a a [0,T], an t T and M;n = (M,(tAan)) is a bounded martingale, n = 1, 2,
....
As we know, Min E _0*2 and
Man = Min + Man is the decomposition of (6.14) for Man since = = an = 0 for every N e.4' y. Set
Y= Then by Lemma 6.2 it is easy to see that Y is dense in 12 and if M = M, + M2 is the decomposition of (6.14) for M E.AY then both M, and M2 are bounded. It is sufficient to show M2 = 0. This follows from the next lemma.
Lemma 6.3. Let M e 2 be bounded and suppose that <M, N> = 0 for every N E .4 2. Then M = 0. Remark 6.1. The condition <M, N> = 0 for every N c--,02* is equivalent to the condition <M,B> = 0 since <M, N>(t) = f o P(s)d<M, B>(s) if N(t) = f o h(s)dB(s).
Proof. Assume I M(t) I < a where a is a positive constant and set D(c)) = 1 + M(T, w)/2a. Then D(co) > 1/2 and E[D(w)] = 1. Define a new probability measure P on YX by 13(B) = E[D(w)IE(w)], B E,97. *
P) = IF e -2'2(Q, P); F is ,Y-measurable}.
MARTINGALE CHARACTERIZATION OF BROWNIAN MOTIONS
83
Then for every J r' xX stopping time a E [0, T],
E[B(6)] = E[D(w)B(a)] = E[E[D((o) I Y,]B(a)] = E[B(o')] + 2 - E[M(c)B(c)] = E[B(a)] = 0
because <M, B> = 0. Similarly, E[B(o)2 - cr] = 0 because B(t)2 - t = 2 fo B(s)dB(s) E _'// *2
and hence = 0. That is, both t'--- B(t) and t F--B(t)2 - t are continuous,,` 7-martingales with respect to the probability P. By Theorem 6.1, t i--- B(t) is an (.Y )-Brownian motion with respect
to P. This clearly implies that P = P on Y and hence we must have D = 1 a.s., i.e., M = 0 a.s. Corollary 1..112(17;X) =
X).
Corollary 2. Let F E 2'2(Q,-9'T P) for a positive constant T > 0. Then there exists an (."X)-predictable processf(s) (0 < s< T) such that E[ f T f 2(s)ds] < oo 0
and (6.15)
F = E[F I oX] + f
Tf(3)dB(S).* 0
Similar proof (based on Theorem 6.2) applies for Poisson point processes.
Theorem 6.7. Let p be a Poisson point process on some state space
(X,- (X)) and M E=- A2( (6.16)
fl c[N,(s,E);s < t + e, E e.g(X)]. Then every
,>0
p) (.%08(Y9)) can be expressed in the form
M(t) = f f f(s,
x, )1V,(dsdx)
(f E Fa
o for some f E F,'(_P)
P))
As we saw above, the proof of representation theorems for martingales
(like Theorem 6.6 and Theorem 6.7) is based on the martingale characterization of the basic processes. Generally, we can say that a martingale
representation theorem holds if the process is determined as a unique * This corollary is also true for T = oo if we set .3m = V .F"x. r
STOCHASTIC INTEGRALS AND ITO'S FORMULA
84
solution of a martingale problem. More generally Jacod [77] showed that
the validity of such a representation theorem is equivalent to the extremality of the basic probability in the convex set of all solutions of a martingale problem.
7. Representation theorem for semi-martingales In this section, we will see how semi-martingales are represented in terms of Brownian motions and Poisson point processes. The results of this section will play an important role in the study of stochastic differential equations. Let (52,,;'-,P) be a given probability space with (.l)tZ0 as usual.
Theorem 7.1. Let M' E .00y'°°, i = 1, 2, ... , d. Suppose that rhil(s) E 9 ;°` * and Y'fk(s) E 2' 120% i, j. k = 1, 2, ... , d, exist such that (7.1)
<M', MJ>(t) 'Pil/
(7.2)
= Jo
Orl(s,
(o)ds,
/
d
(s) = Ek-IWik(s)WJk(s)
and (7.3)
det(Y'1k(s)) ZV 0
a.s.
for every s.
Then there exists a d-dimensional (.P)-Brownian motion B(t) = (B'(t), B2(t), ... , Bd(t)) with B(0) = 0 a.s. such that d
(7.4)
Mi(t) = kZ f'0
r 1k(s)dB'(s),
i = 1, 2, ... , d.
Proof. We will consider the case where M'E ,.4
i O,j E-= 121, and
zik E .172; the general case is easily reduced to this case. For N > 0 we set (7.5)
0 "'1 (s, (o)
j(W-');k(s, w), if I (Y'-'),k(s, w) I < N for all 0, otherwise,
where Yr-' denotes the inverse of 1 _ (Yr k(s)). Then clearly O and for every i and j
i, k,
E -2'z
10 = ('(t))120i ' is a real (.5;)-predictable process and f' IP(s)I ds < °° a.s. for every t > 01. -17, = (0 e 2';°`; E[ f o 0(s)I ds] < oo for every t > 01.
REPRESENTATION THEOREM FOR SEMI-MARTINGALES
85
t
(7.6)
f
0
E[ Ikk/-1 E e),M(s, w)ef9(s, w)0kk/(s, w) - ati I Zlds -- 0 as
N- co.
If we set
f' Bi`k''"(s, w)dMk(s, CO)
Bu(t) = k-1 Z
0
then B('N) E=- -002 and :
(7.7)
(t) = f
E OIk )(s, w)9Jk)(s, (O) kkk,(S, co)ds.
0 k,k/ot
converges to some B' in .4' as N -- co and (t) = d 1t. By Theorem 6.1, B(t) = (B'(t), B2(t), ... , Bd(t)) is a d-dimensional (_F)-Brownian motion. Since From (7.6) and (7.7) we see that
Ef
k-1
t 0
t
'I'ik(s)dB V)(s) = f0 IN(s)d P(s),
where IN(s, co) =
if I ('P-');k(s, (A) I S N
1,
for all i, k,
otherwise,
0,
we have, by letting N - oo,
M'(t) = km1 E f'0
W1k(s)dBk(s)
Theorem 7.2. Let M e .,ff '00 such that lim<M>(t) = oo a.s. Then, /t_ if we set (7.8)
T, = inf {u; <M>(u) > t),
and .. = JF,-,, the time changed process B(t) = M(z,) is an (.
')-
Brownian motion. Consequently, we can represent M by an (.P)-Brownian motion B(t) and an (A)-stopping time <M>(t):* (7.9)
M(t) = B(<M>(t)).
* Note that <M)(t) is an (.P,)-stopping time for each t
{T.Zt) E,5{y=,
.
0. Indeed, f<M)(t) S u)
STOCHASTIC INTEGRALS AND ITO'S FORMULA
86
Proof. First, we remark that, with probability one,
t - B(t) =
M(r) is continuous. It is sufficient to show that, for any fixed r < r', we have, except a set of probability zero, (7.10)
{<M>(r') = <M>(r)} C {M(u) - M(r), due [r, r']).
Indeed, if (7.10) holds, we can conclude by a standard argument that the following is true with probability one: for any r < r', <M)'(r') = <M>(r) implies that M(u) = M(r) on the interval [r, r']. This clearly implies that, with probability one, t '--- M(T,) is continuous. To prove (7.10), set a = inf Is > r; <M>(s) > <M>(r)l. Then a is an
(J;)-stopping time and hence N(s) = M(ar A (r + s)) - M(r) is a localmartingale with respect to (,9,) where .9 = .F;A(r+,). Since (s) = <M> (a A (r + s)) - <M> (r) = 0, N = 0. This implies that M(a A (r + s)) = M(r) for all s > 0 a.s., and hence (7.10) holds. By Doob's optional sampling theorem, E[M(r, A n)2] = E[<M>(T, An)] < E[<M>(T,)] = t. Letting n -- oo, E[M(T,)2] = t. Then we can conclude by the same theorem that B(t) = M(T,) is .4'2 with respect to fit and (t) = <M>(T,) = t. By Theorem 6.1 B(t) is an (.9)-Brownian motion. This theorem was generalized by Knight [84] (cf. also [119]) as follows:
Theorem 7.3. Let Mt e ' '", i = 1, 2, ... , d, such that <M', MJ> i j and lim <M'>(t) = oo a.s. Set
= 0 if (7.11)
tt«
T,' = inf {u; <M'>(u) > t} ,
i = 1,2, ... , d.
Then if we set B'(t) = M'(r), i = 1, 2, ... , d, B(t) = (B'(t), B2(t),
... ,
Bd(t)) is a d-dimensional Brownian motion.
Proof. By the previous theorem, B'(t) is a one-dimensional Brownian motion for each i = 1, 2, ... , d. Consequently, we only need to prove that the processes B'(t), B2(t), ... , Bd(t) are mutually independent. We shall show this by induction. Suppose that B'(t), BZ(t), ... , B'(t) are mutually independent and we will show that (B'(t), B2(t), ... , B'(t)) and B'+1(t) are mutually independent. Let ' = a[B'(t), B2(t), ... , B'(t), t e [0, oo)] and ', = (1 a[B'(s), B2(s), ... , B'(s); s < t + e]; let e>o
= o[B'+'(t), t E [0, oo)] and or, = fl c[B'+'(s); s < t + e]. We 8>0 may assume, without loss of generality, that our probability space (9,
REPRESENTATION THEOREM FOR SEMI-MARTINGALES
87
J ,,P) is such that (0, ..F") is a standard measurable space (cf. Chapter 1, Section 3). Let P( I ') be the regular conditional probability given gam. Clearly (B'(t), B2(t), ... , Bt(t)) and B'+1(t) are mutually independent if and only if B'+1(t) is a one-dimensional Brownian motion with respect to
P(. 19) a.s. Hence, noting Theorem 6.1, it is sufficient to prove that for every t > s and 2°S measurable bounded function F1(co) (7.12)
E[(Bt+'(t) - Bt+1(s))F)(co) I'] = 0
a.s.
and (7.13)
E{[(B'+'(t) - B'+'(s))z - (t - s)]F,(co) I'} = 0
a.s.
Thus, it is sufficient to prove the following: for every t > s, every bounded
X, measurable function F,(a) and every bounded '-measurable function F2(co), (7.14)
E[(B'+1(t) - B'+'(s))Fj(w)Fz(w)) = 0
and (7.15)
E{[(Bt+1(t) - Bi+)(s))z - (t - s)]F,(w)F2((o)) = 0.
We will prove (7.14) only: the proof of (7.15) can be given similarly. Let 9 (k) = a[B"(s); s e [0, co)], k = 1, 2, ... , i. Since F2(w) can be approxit
mated by a linear combination of functions of the form II Gk(c)) where k-1
Gk(c)) is <9 (k) -measurable, we may assume from the beginning that F2(co) _ t
jI Gk((O). By Corollary 2 of Theorem 6.6, F1(co) and Gk(co), k = 1, 2, ... ,
k-1
i, can be expressed as
$ F1((o) = c + s ch(u)dB' '(u), 0
Gk(w) = Ck + f:0¶k(u)dBk(u),
where c and ck are constants and 0 is an (,Y,)-predictable process and
is a (,X ))-predictable process. Here v) = fl a[Bk(s);s < t + s], a>0 k = 1, 2, ... , i. By setting a, = zs+', we can write 'Pk
STOCHASTIC INTEGRALS AND 1TO'S FORMULA
88
F1(w) = c +
f 0s,
s3(u)dM1+1(u)
and
Gk(w) = Ck +
0
f:Tk(u)dMk(u),
where q3(u) = 0(<M°+1>(u)) and YPk(u) = ilk(<Mk)(u)) are (.)-predictable processes. Since <M1,Mj>(t) = 0 for i # j, we have by Ito's formula 1
FZ(w) = H Gk((O) k-l
= d c2 ... C1 +
x-i
I
o Il
i (c, + J Vl(u)dM1(u)) Pk(t)dMk(t)
1#k
= c' + kml J 00
Bk(t)dMk(t).
Then the left-hand side of (7.14) becomes E{(B1+1(t) - B'+l(s))F1(w)FZ(w)}
= E{(M1+1(Tr) - M1+1(T,))(c +
J u` $(u)dM'+'(u))
x (c' + k-l E f ek(u)dMk(u))} 0
E {(M1+1(T,) - M1+1(T,))
k-l
x (c + J oT
$(u)dM1t1(u)) f ok(u)dMk(u)} o
1
= E E {(M1+1(T,) k-1
M1+1(T,))
ek(u)dMk(U)(C + f T S q5(u)dMI+l(u))}
l
0
+ k-1 E
//
//
M1+1(T,)) f'0 ' ek(u)dMklu)
x (c + f o' q5(u)dM1+l(u))} . Now the first term vanishes because <M'+1, M"> = 0 and hence m
0 E {(M1+1(Tj) - M1+1(T,)) f Ok (U)dMk(U) 1,9"=j =
REPRESENTATION THEOREM FOR SEMI-MARTINGALES
89
and the second term vanishes because E{(M'+1(T,) - M:+'(TS))I ` _,} = 0.
Theorems 7.1, 7.2 and 7.3 also hold under weaker assumptions: however, it is generally necessary to extend the given probability space in order to guarantee the existence of Brownian motion. Before stating these results we shall first make precise the notion of extension of a probability space.
with a reference Definition 7.1. We say a probability space family (f) an extension of a probability space (0, .F, P) with a reference family (,, ) if there exists a mapping 7L: S2 - 0 which is _V'/_F-measurable such that ,D (i) (ii) P = 7r(P) (: = Po -1) and (iii) for every X(c)) E (S2, ;P)
E(X(i) I
) = E(X I
where we set 1(c) = X(7cw)
)(nci),
P-a.s.,
for Co E A
Definition 7.2. Let (Q,J P) be a probability space with a reference family (Y). Let (S2',-',P') be another probability space and set
P=PxP' and
nci = (o
for
Co = (o), co') E f2.
If (.l) is a reference family on such that ..x . ' n , D ,fix {Q', }, then (52,.7',P) with (,) is called a standard extension of (0,511-,P) with (.
).
It is easy to see that a standard extension is an extension in the sense of Definition 7.1. Let (52,. `,P) with (,F) be an extension of (.Q,.9",P) with (. ). then M = (Mt(w)), If M E .412 with respect to (Q,-F-,P) and
where 9o(m) = M,(rc ), belongs to ..1l2, i.e., the space d,Z with respect to (52,.9,P) and (.5). Also, if M,N E l4' then <M,9),(W) = <MN>,(niv) holds. This is an easy consequence of the property (iii) in Definition 7.1. Therefore, the space og (.,1c2, A/1 20c etc.) is naturally imbedded into the space
-Wi , etc.) and M E ..112 maybe regarded as a martingale
STOCHASTIC INTEGRALS AND ITO'S FORMULA
90
on the extension i by identifying M and R. The following three theorems are natural extensions of the above three theorems. Theorem 7.1'. Let (Q, .,g", P) be a probability space with a reference
family(,;-,) and M'E.4fi-,i=1,2,...,d.Let 45tJ,ij=1,2,..., d and 1',k, i = 1, 2, ... , d, k = 1, 2, ... r, be (.;)-predictable processes such that f' I P,J(s) I ds < oo and f' I t1'k(s) I gds < oo for all
t > 0 a.s., (7.16)
<M', MJ>(t) = J' O,J(s)ds
and
(7.17)
O;J(s) _ k-1
T'1k(s)'I'Jk($)
Then on an extension (L, 7, P) and (. ) of (0, _01'',P) and
there
exists an r-dimensional (.5';)-Brownian motion B(t) = (B'(t), B2(t), B'(t)) such that
... ,
(7.18)
M'(t) =
k-1 J 0
tk(s)dBk(s),
i = 1,2, ... , d.
Proof. We may assume d = r by setting M'(t) - 0 or T,k(t) = 0 if necessary. Since 0 = is for each (s, co) a d x d symmetric nonnegative matrix, 012 is uniquely determined as a d x d symmetric nonnegative matrix such that 01 20112 = 0; moreover, s - 01'2(s) is predictable and f o II01 /2(s)!!2 ds < oo a.s. for every t > 0. We may assume without loss of generality that t' = 0112. Indeed in the general case there
exists a (d x d orthogonal matrices-valued) predictable process P = (P,J(s)) such that 0112 = P.P. Consequently, if we have a representation
M'(t) = E f
(V 12),k(s)dBk(s), we can write M'(t) = E f Y',k(s)dB" (s), 0 ' where Bk(t) = E f Pk,(s)dB1(s) is another d-dimensional (.;)-Brownian 0
1-1
0
motion by Example 6.1. Therefore, we assume tl' = 012. Set tI'(u)
=
Jim P1/2(u)(,p(u) + sI)-' $40
where I is the identity matrix. Let ER(u) be the matrix corresponding to the orthogonal projection onto range 0(u)Rd and set E,,,(u) = I - ER(u). Then clearly tl-f(u)Tl(u) = tPJ(u) P(u) = ER(u). We prepare, on a probability space
REPRESENTATION THEOREM FOR SEMI-MARTINGALES
91
a d-dimensional (Y ')-Brownian (Q', ,F'', P') with a reference family B'(t) = (B",B'2(t), ... , B'd(t)) and construct a standard motion
extension (S2,.f,P) and (.l) of (Q,-9",P) and (, ) by S = 0 x Q', = .F'x .,F'', P = P x P' and -12' , = , 7 x .I','. On this extension, M', B'i E -0 cyI- such that
(7.19)
<M', M>>(t) = J'
j(u)du'
<M', B'>>(t) = 0
and
(t) = 6,,t for i,j = 1, 2, ... , d. Now set
BI(t) =k-1E f 0Itk(u)dMk(u) +k-3E
0
(EN),k(u)dB'k(u).
Then, by (7.19), d
(t) = fOrk,1-I E
f r (EN), (u)du 0
= f `0 (ER),,(u)du + f 0(EN),j(u)du = al,t, and hence {B'(t)} is a d-dimensional (.f)-Brownian motion. Also, noting that f'(u)EN(u) = EN(u) Vf(u) = 0,
E f t Wtk(u)dB"(u) =k.1EI k-I 0
f,
P,k(u)P(u)k,dM'(u)
0
+E f r 'tk(u)(EN)kl(u)dB"(u) k,le1 0
= M'(t) -Z f (EN)u(u)dM'(u) + E1-1f' (P(u)EN(u))udB"(u) /-1 0 0
= M'(t). The second term in the middle line is zero because of
f
0
(EN(u)O(u)EN(u)),,du = 0.
Theorem 7.2'. Let (Q,.9,P) be a probability space with a reference
family (.) and M E
Set
STOCHASTIC INTEGRALS AND ITO'S FORMULA
92
nnf {u; <M)(u) > t} ,
(7.20)
co,
if
t > <M)(oo) = lim<M)(t), ,tom
=V
and
s>o
,A,. Then on an extension
and (_597) of
(Q,.r,P) and (.r) there exists an (.R` )-Brownian motion B(t) such that B(t) = M(;), t E [0, <M)(oo)). Consequently we can represent M by an (. )-Brownian motion B(t) and an (.P;)-stopping time <M)(t): (7.21)
M(t) = B((M)(t)).*
Proof. By the optional sampling theorem (Theorem E(M:,.A
= M=,,,,,, and E((M=4A, - MT'/.,)2 I
:,,,
1-6. 11),
,) =
for every s > s' and u > v. Hence, B(u) = lim Mt.,
st.
exists a.s. and (7.22)
E(B(u) 1,P;) = B(v),
(7.23)
E((B(u) - B(v))2 I frc) = E(<M) A u- M. A v I
)
for every u > v. We prepare, on a probability space (S2',. ',P') with a reference family an (.9')-Brownian motion B'(t). We construct a standard extension (S2, ,P, P) and (. ) of (9, . ", P) and (.,) by setting
S2 = 0x92',
_Fx J'', P = P x P' and ., _ . , x . '. On this
extension let
B(t) = B'(t) - B'(t A <M)(00)) + h(t). Then B(t) is a continuous (.
)-martingale such that
hence it is an (f)-Brownian motion. The rest of the proof is obvious.
Theorem 7.3'. Let (Q,.9,P) be a probability space with a reference (.9). Let M' E LY Z'°5(, t), i = 1, 2, ... , d, such that <M',MJ)(t) = 0, i j. Then, on an extension (I ,.l`,P) of (Q,.9;P), there exists a d-dimensional Brownian motion B(t) _ (B'(t), B2(t), ... , family
Bd(t)) such that (7.24)
B'(t) = M'(rD
t E [0, <M`)(oo)),
where * As in Theorem 7.2, <M>(t) is an (,07)-stopping time for each
t Z 0.
REPRESENTATION THEOREM FOR SEMI-MARTINGALES
(7.25)
zt
93
mf {u; <M'>(u) > t}, {
oo
,t>
<M`)(-)-Consequently,
(M'(t), M2(t), ... ,M,(t)) can be obtained from a d-dimensional Brownian motion B(t) = (B'(t),B2(t), ... , Bd(t)) as
i= 1, 2, ... , d.
MI(t) = Br(<M1>(t)),
The proof is similar to that of Theorem 7.2' and therefore omitted. Finally, we will discuss a similar representation theorem for a class of point processes by means of Poisson point processes.
Theorem 7.4.*' Let (Q,-F-,P) be a probability space with a reference family (,7). Let (X,,%x) be a measurable space and p be an (.F)-point process of class (QL) on X with the compensator .N,(dtdx) = q(t, dx, w)dt. Suppose that there exists a Q-finite measure m on a standard measurable and a predictable X* = X U (Al *'-valued process space (Z,
9(t, z, w): [0, oo)XZX0- X* such that (7.26)
m({z; 0(t, z, w) E E}) = q(t, E, w)
for every E E .5WX.
Then, on an extension (6,J9°,P) and (.l°,) of (Q,_F',P) and (9 ), there exists a stationary (.l)-Poisson point process q on Z with the characteristic measure m such that N,((0, t] (7.27)
x E) =
f" f z 1.(9(s, z, w))NQ(dsdz)
_ # Is E.D is
Proof. First we prove several lemmas. Lemma 7.1. There exists a predictable probability kernel Q(t,x,dz,w) on [0, oo) X XX 0z X 12 (i.e., for a fixed A E .9z, (t, x, co) -
Q(t, x, A, co) is predictable and for fixed (t, x, co) E [0, co) x X x 0, A E .qz ' -- Q(t, x, A, co) is a probability on R z), such that for every non-negative Ox x Oz-measurable function f(x,z) *1 Cf. Grigelionis [43], Karoui-Lepeltier [81] and Tanaka [161]. Proof given here was suggested by [161].
*2 d is an extra point attached to X and 0x* is the v-field generated by Rx and [A} .
STOCHASTIC INTEGRALS AND ITO'S FORMULA
94
(7.28)
fz
{I[ect, z.
e: f(0(t,
z, w), z)} m(dz)
f x { f zf(x, z)Q(t, x, dz, co)) q(t, dx, co). The proof of this lemma is standard and is left to the reader (cf. Chapter 1, Section 3).
and Lemma 7.2. On an extension (Q,5',P) and (A) of (,F;), there exists an (J',)-point process p of the class (QL) on X x
[0,l] such that
(i) DD = D, and 7c(p(s)) = p(s) for s e Do, where rc(x, a) = x, (x, a) a XX [0, 1]; (ii) the compensator JQ (dt, dxda) of p is given by (7.29)
AD(dt, dxda) = q(t, dx, co)dtda,
where da is the Lebesgue measure on [0, 1].
Proof. We prepare a sequence of independent identically distributed n,k = 1, 2, ... on a probability space (0', .r', random variables
P') such that 0<,, < 1 a.s. and are uniformly distributed. Set SZ = S2 x 0', _0' = -9"x -g-', P = P x P'. p may be regarded as a point process defined on this product space. There exist disjoint U. a .fix, n = 1, oo for every t E 2, ... such that U U. = X and E(N,((0, t] x n
[0, oo) and n. Let D,n = Is a D,; p(s) E U.}. We can order each D,, according to magnitude: i.e., D,. = {s; < s2 < < sk < . . . }. Since U D,n = D there exists for each s a D. a unique pair (n, k) such Then the point process p with the that s = sk. We set p(s) = (p(s), domain D. = D, and fl', = (1 a[p(s), s < t + e] is what we want. 8>0
Proof of the theorem. Let Q(t, x, dz, w) be the probability kernel in Lemma 7.1. Then there exists a predictable processf(t, x, a, (0): [0, oo) X
X x [0, 1] x 9 --- Z such that the Lebesgue measure of {a; f(t, x, a, co) e Al = Q(t, x, A, w) for every A e ;qz.* Let p be the point process on Xx [0, 1] given in Lemma 7.2. We write p(s) = (p,(s), p2(s)) for s E DD, where p1(s) = p(s) e X and p2(s) a [0, 1]. Define a point process q2 on Z by Dq2 = Dr, = D, and q2(s) = f(s, p1(s), p2(s), co) for s E Dq2. Thus N112((0, t] X
A) = $j' Xx[0.11 I,,(f(s, x, a, w))Ne(ds, dxda) 0
* Cf. [81].
REPRESENTATION THEOREM FOR SEMI-MARTINGALES
95
and the compensator is given by Ng2((0, t] X A) = f
t o
f xx[o.1]I,,(f(s, x, a, c)))q(s, dx, o. )dads
=f' f xQ(s, x, A, co)q(s, dx, co)ds
(7.30)
o
=f f
A
I[ec3.z..),a3m(dz))ds
for all A E ,z. On an extension (52,.,P) and (.3',) of
and (,F`,) we construct a stationary (,P;)-Poisson point process q, on Z with the characteristic measure m(dz) such that q, and q2 are mutually independent. Clearly such a construction is possible by taking a standard ex-
tension. Define a point process q3 on Z by Dq,= Is E D,,; 0(s, q,(s), c)) = Al and q3(s) = q1(s) for s e Dq,. Then Nq,((0, t] X A) = f o
f .,I[OCS.z.m>-d] Nq,(dsdz),
A E _q z
and its compensator is (7.31)
*,,,((0, t] x A)
= f of 4I[ecs,z,w>-Xm(dz)ds.
Finally, a point process q on Z is defined by Dq = Dq2 U Dq, and
q(t) =
j q2(t)
,
t E D117,1
q3(t)
,
t E Dq3.
By the independence of q, and q2, Dq2 and Dq3 are disjoint a.s., and hence q is well defined; moreover q is a stationary (Y;)-Poisson point process
on Z with the characteristic measure m(dz) by (7.30) and (7.31) (c.f. Theorem 6.2). Thus the only thing remaining to be proven is
N,((0, t] x E) = f o
fz 1 (O(s, z, co))Nq(ds dz), E E ' x.
Let p be the point process on X whose counting measure N,((0, t] x E) coincides with the right-hand side. Then N,((0, t ] X E) = f
0
f
xx [a, 13
IEx[o, i (x, a)Np(ds, dxda)
STOCHASTIC INTEGRALS AND ITO'S FORMULA
96 and
fz IE(0(s, z, a'))Nq(dsdz)
N,((0, t] X E) = f 0
f fz IE.(9(s, z, w))NQ2(dsdz) o
f
o
xxro,17IE(0(s, f(s, x, a, co), (o))N9(dsdxda).
Hence the compensator AD((0, t] x E) is given as
N,0((0, t] x E) = f ds f o
xxto,l]
IE(O(s, f(s, x, a, cv),co))q(s, dx, c))da
z, co))Q(s, x, dz, w)q(s, dx, co) = f ods f x f IE(O(s, z
= f'ods f IE(9(s, z, (o))m(dz) z
=
fo
q(s, E, co)ds = N,((0, t] X E).
Finally, E( {N,((0, t] X E) - ND((0, t ] x E)} 2)
= E( [9,((0, t]XE) - 95((0, t) xE)}2) E {[ f o
f xx[o, ,
(IExro,13(x, a) - IE(e(s, f(s,x,a,w),w))
x Nods, dxda)]2}
= E{ f ' ds f 0
xx[0,1]
[IE(x) - IE(0(s, f(s, x, a, co), co)]2q(s, dx, w)da}
f
= E { ` ds fzf [IE(x) - IE(O(s, z, co))]2Q(s, x, dz, cv)q(s, dx, w)) o x
=E( f
o
fz
= 0,
and this concludes the proof.
e [IE(9(s, z, w)) - IE(0(s, z, w))]2m(dz)}
CHAPTER III
Stochastic Calculus
1. The space of stochastic differentials In Chapter II, we introduced the notion of stochastic integrals and derived Ito's formula. These are fundamental notions in stochastic calculus and its applications. Based on these results, we will now give a systematic
treatment of stochastic differentials for continuous semi-martingales.*' One of the important notions to be introduced here is that of symmetric multiplication (Y'..Z) which corresponds to the so-called Stratonovich integral or Fisk integral ([153], [27] and [133]). Under this multiplication,
the chain rule (Ito's formula) takes the same form as in the ordinary calculus. Let (0, -F-, P) be a probability space and (.
),,, be a reference family (assumed to be right continuous as usual). We introduce the following notations. (previously denoted by _0Z 1OC in Chapter II)
= the family of all continuous locally square integrable martingales M = (M,) relative to (.) such that Mo = 0 a.s.;
.sa ', = the family of all continuous (,')-adapted processes A = (A,) such that Ao = 0 and t --- A, is non-decreasing a.s.;
.sad = the family of all continuous (.Y)-adapted processes A = A, is of bounded variation (A,) such that Aa = 0 and t on every finite interval a.s.;
0 = the family of all (.7 )-predictable processes 0 _ (0,) such that, with probability one, t ' -- 0, is bounded on each bounded interval. It is easy to show that A = (A,) E sad if and only if there exist A`'' _ (A,°) E .sa i. Q = 1, 2) such that A, = Af') - Au), t > 0.*2 *' The material in this section is adapted from Ita [71].
02 We simply write A = AM - AM. 97
STOCHASTIC CALCULUS
98
Let X = (X,) be a continuous semimartingale i.e., a process represented in the form
X,=Xo+M,+A,
(1.1)
where Xo is an Y;-measurable random variable, M = (M,) E-///and A E W. Following Ito [71] we also call it a quasimartingale.
Definition 1.1. We denote by G2 the totality of quasimartingales. Every X E 6l is expressed uniquely as (1.1): M = M1 is called the martingale part and A = AX is called the bounded variation part, respectively. This decomposition is called the canonical decomposition of X E 62. The uniqueness of the canonical decomposition is seen as follows. Let
X(0) + M1 +
X(0) + M'(2) + Aft
be two such decompositions. Then
M,=Mtn -M(2) =A(2) -Ail -=A,. As we saw in Chapter II, Section 5, <M>, is given as the limit in proba-
bility of E (M,, - M,,_1)2 as 1d - 0, where .4 is a partition 0 = t0 <
t1 < .. < t = t and IJ I = max It, - t1_1 I. But E (M,, - Mrr_1)2 t-1
3
n
_ 57, (A,, - At,_,)2 < V,(A) max I A,, - A,,_, - 0 t-1 IS1Sn
where
V,(A)
the total variation of s E (0, t] i--- As. Therefore, <M;1> - Ml2)> = 0 which implies that M'> = MM. The space 62 is closed under addition and multiplication; more generally, if f(x', x2, ... , xn) E C2(Rn - R) and X', X2, ... , Xn E a, then Y = f(X', X2, ... , Xn) 1= a (c.f. Theorem II-5.1).
Definition 1.2. For X, Y E 6Z, we say that X and Y are equivalent and write X Y if, with probability one, (1.2)
X(t) - X(s) = Y(t) - Y(s)
for every 0 < s < t.
Clearly this is an equivalence relation. The equivalence class containing X is denoted by dX and is called the stochastic differential of X. f ; dX(u) is, by definition, the process X(t) - X(s). Let d62 = {dX; X E 4C}, d.,ff _ {dM; M E .ill'} and d V= {dA; A E .sa'}. We introduce the following operations in d a.
THE SPACE OF STOCHASTIC DIFFERENTIALS
99
.V. Addition: (1.3)
dX + dY = d(X + Y),
for X, Y E C.
.91. Product : (1.4)
dX dY = d<Mx, My>,
for X, Y e 67,
where Mx and My are the martingale parts of X and Y respectively. Next we define a multiplication between ., ' 4and 62.
.4'..-Multiplication: If 0 E .9 and X E 62, then (1.5)
(45-X) = X(0) + f' 0(s, w)dMx(s) o
+ f' 0(s, ro)dAx(s), t > 0, a
is defined as an element in Gel. Hence d(Q X) is uniquely defined from fi and dX. Now we define an element 0 dX of de by (1.6)
s .dX =
Sometimes we write 0 dX simply as OdX.
Theorem 1.1. The space dc2 with the operations V, .0 and _0X is a commutative algebra over .09, i.e., a commutative ring with the operations .sad and -17/ satisfying the relations
0.(dX + dY) = (1.7)
0
for 4$,! E . (1.8)
(4
(c +
0 .dX + P.dX 0.(Vf.dX) and dX, dY E d,2 . We also have that
d62. d& c d-V, d 7.d = 0
and
2'.dti2 = 0.
Proof. It follows almost immediately from the property of stochastic integrals established in Chapter II that d62 is a commutative algebra over .° F. (1.8) follows at once because <Mx, M,.> E sad for X, Y E C. Now Theorem 11-5.1 in continuous case can be rephrased in this
context as follows: if X', X2, ... , Xd E a' and f e CZ(Rd - R), then Y = f(X', Xz, ... , Xd) E 40 and (1.9)
dY = rE -1
2,
J-
STOCHASTIC CALCULUS
100
where a, f and ataf are elements in °.d defined by af- (X',X2, and
a2
(X', X2,
... , Xd)
... , Xd) respectively. Also Theorem II-6.1 can be 8f dt, ... , dXd e d..and
ax'axj rephrased as follows: if dX', dX2,
if = 1, 2, ... , d, then (X'(t), X2(t), ... , Xd(t)) is a d-dimensional Wiener process. Such a system of martingales (X',X2, d-dimensional Wiener martingale. Now we introduce the fourth operation.
... , Xd) is called a
Symmetric 47-Multiplication: (1.10)
YodX =
d X e M and Y
2
al.
Theorem 1.2. The space d&? with the operations -V, .7..4. and -91 is a commutative algebra over C; we have, for X, Y, Z E 62, Xo(dY + dZ) = XodY + XodZ, (1.11)
(X+Y)odZ=XodZ+YodZ. X.(dY.dZ), (XY)odZ = Xo(YodZ).
0 and
Proof. We note that since have that (1.12)
0, we
X or Y E sad,
XodY =
and (1.13)
Then, for example,
Xo(YodZ) =
X (Y dZ) +
X Z
2
_ XYodZ.
(Y dZ)
THE SPACE OF STOCHASTIC DIFFERENTIALS
101
The other properties are also easily proved and so we omit the details. is that the chain rule takes A remarkable fact for the operation S°.. the same form as in the ordinary calculus. Namely we have
R), then
Theorem 1.3. If X1, X7., ... , X d E G2 and f E C3(Ra for Y = f(XI, X2, ... , Xd) a 62 we have dY = 7, a1f odX'.
(1.14)
t1
Proof. By Theorem 1.2 a
Z at f odX' t_1
a
_ E (atf dX' + 2
=
atf . dX' +
+
dX')
2 tE (E
dX'
2 ,.EI
2 tE1
(by (1.8)),
= dY. The stochastic integral f u YodX is called the Stratonovich integral or the Fisk integral or sometimes the Fisk-Stratonovich symmetric integral. Indeed, we have the following:
Theorem 1.4. For every X and Yin 12,
I YodX
(1.15)
1.i.p. 1'd1-0
o
At') + t -1
X01-0)
where d denotes a partition 0 = to < t, < . . . < t = t
and J A I
_
max (t, - tt-1). Istsn
Proof.
E
At') +2
=
Y(tt-1)
X(tt-1)) + 4 tLj1
Y(t!-1))(X(tt)
-X(t-1)).
STOCHASTIC CALCULUS
102
Thus the assertion follows by the same arguments as in the proof of Theorem II-5.1. Finally, we discuss the notion of stochastic time change. This is an important operation on quasimartingales.
Definition 1.3. By a process of time change c we mean any process is strictly in_ (fl,) E .sa7+ such that, with probability one, t creasing and lim
oo.
For a given 't process of time change 0, we set (1.16)
r, = inf {u; 0 > t).
Then, with probability one, ro = 0, t i- r, is strictly increasing and continuous and lim r, = oo. Furthermore, r, is an (Y)-stopping time =rW
because {r, < u} = It < (1.17)
.
e -9-.. Set
_ .,.
Thus (.l) is a reference family on (Q, -9-, P). Let X = (X,) be an (,
)well measurable process and define TOX = ((TOX,)) by (TOX), = X. Then TOX is an (fl')-well measurable process by Proposition 1-5.4. TOX is called the time change of X by 0. It is easy to see that if 67 and are the space
of quasimartingales relative to (Y) and (.7) respectively, where
.
is defined by (1.17), then TO: 67 -- is a bijection which preserves all structures on the space of quasimartingales: TO(MX) = MTpx, TO(AX) = ATOX, X Y if and only if TOX -r TOY. Thus TO induces a bijection between dQ and d67. Also, the space . with respect to (.) coincides with To(. 6) and TO is an isomorphism between the .91-algebra d&' and
the .,d-algebra de. In particular, TO commutes with the operation S°..
: T#(XodY) _ (TOX)eTO(dY).
The proof of these facts proceeds as follows. By Doob's optional sampling theorem (Theorem 1-6.11), M e. if and only if TO(M) E ..fF, where ..i is defined with respect to (. 't) and furthermore TO<M,N> = . Also, A E=--V if and only if TOA E &T. Consequently TO(f OdM) = f TO(O)d(TOM) for 45 e .g and M E .4'because N = f cdM is characterized as the unique N e ' such that = f Od <M,L>
for all L E..'. Since all operations in a are defined in terms of addition (which is clearly preserved by TO), the operation <M, N> and stochastic integration, the assertion is obvious.
STOCHASTIC DIFFERENTIAL EQUATIONS
103
2. Stochastic differential equations with respect to quasimartingales Let (0, ,F',P) and (,fir) be given as in section 1. Suppose that we are
given X',X2,
... Xr E 62 and a system
of real locally bounded Borel measurable functions on Rd. We want to find Y', Y2, , Ya e a such that (2.1)
dY'(t) = E
i = 1, 2, ... , d,
J-1
where Y(t) = (Y'(t), Y2(t), ... , Ya(t)).*
Theorem 2.1. Suppose that aji(x), i = 1, 2, ... , d, j = 1, 2, ... , r, satisfy a Lipschitz condition; i.e., there exists a constant K > 0 such that (2.2)
1 a,' (x) - ai(y) I - K I x - y I ,
for all
x, y E Rd.
Then for each given y = (y', y2,
, yd) E Rd, there exists a unique Y = (Y', Y2, ... , Yd) such that Y' E C, Y'(0) = y' and (2.1) is satisfied.
Proof. Since there is no essential change, we assume that d = 1 and that (2.1) is of the form (2.3)
dY = a,(Y(t)) dM + az(Y(t)) dA
where M E.4', A e.si and a,(x) (i = 1, 2) are given, and the a,(x) satisfy (2.2). The problem is equivalent to finding Y E 62 such that (2.4)
Y(t) = y + J u aj(Y(s))dM, +
az(Y(s))dAs.
Jo
Set ¢(t) = t + <M>, + JAI, where 1Air denotes the total variation of [0, t] D s 1- A,. Then 5 is a process of time change. By applying the time change TO to (2.4) we have (2.5)
where
?(t) = y +
a1(?(s))dd, +
Jo
Jo
az(?(s))dA,,
TOY, M = TOM and A = TOA. It is sufficient to show the
unique existence of ? satisfying (2.5). Since <M>, = <M> 0-1 (r) and Ar = * Clearly each ci (Y(t)) E ,dd'.
STOCHASTIC CALCULUS
104
Ao-, a) ,
it is easy to see that t '
t - <M>, - 1.11, is increasing. In
particular we have d<M>, < ds and dill, < ds as Stieltjes measures a.s. Let us construct a solution by successive approximations: Y101(t) = Y,
fu
Y(n)
a,(YcA_,) (s))d2. +
(t) = Y + J
f
a2(Ycn-,> (s))dA.,
o
n=1,2,.... Let T > 0 be given and fixed. Set KT = 2K2(1 + T) where K is the Lipschitz constant in (2.2). Then, if t E [0, T], E{I Y") (t) - Y(o)(t)I2} = E{[a1(Y)M: + a2(y)At]2} S C1,
where C, = 2(a1(y)2T + a2(y)2T2). Suppose now that E{I y(n)(t) - Y(n-,>(t)I2} < C,K7n'1 (ntn 1).
c
for some n
1. Then
E{I y(..+1)(t)-Y(n)(t) 2} <2E({ f [a1(y(n)(s))-a1(Y(n-1)(s))]dM,)2) T
+2E({ fo[a2(y(n)(s))-a2(y(n_1)(s))]d1A1,}2) = I, + I2. But I, = 2E J f 'o [a1(Yn)(s)) - a,(y(n-1)(s))]2d<1ff>,}
< 2E ( E [a1(Y(n)(s)) - a1(y(n_1>(s))]2ds} o
2K2E { f I y(n)(s) -
2K2 f E { I Y(n)(S) O
and
-
Y(n-1>(s) 12} ds
Y`°-ll (s)' 21d9
STOCHATIC DIFFERENTIAL EQUATIONS
105
I2 S 2E 11,11, f' [a2(Y(n)(s)) - a2(y(n-1)(s))12djA1,} 0
f
5 2E {t o [a2(Y(n)(s)) - a2(y(n-)(s))]2ds} r
2TK2 f
-Y(n-1)(s)
E { I Y(n)(s) 0
12} ds.
Hence
Ell y(n+1)(t) - y(n)(t)I2} < (2K2 + 2TK2) f' E (I Y(n) (s) -
Y(n-1) (s) 12} ds
0
< KT f o C'KT' = C1Krn
(nsn-1)i ds
.
Therefore the inequality (2.6)
E{ I y(n+»(t) - y(n) (t)12) < C1KT n, i
t E [0, T},
follows by induction. Also, by Theorem 1-6.10,
E{ sup I y1,+1)(t) - Y(n)(t)I2} (s)) - a1(y(n-')(s))ld.M(s)I2}
< 2E { stpTI f 0
+ 2E { sup I f 05tST
'
a2(y(n-1)(S))]dA(s)
0
12}
< 8E({ f T [a1(Y(n)(s)) - a1(ycn-1)(s))ldM(s)}2) 0
+ 2E({ f o la, (Y(n)(s)) - a2(Y(n-))(s))I dIAI(s)}2) and by the same calculations as above, this is dominated by 8K2 f o E l l Y(n)(s) -
+ 2TK2
f
Y(n-1)(s) 12} ds
E l l Y(n) (S) o
Y(n-1) (s) 1 2} d
STOCHASTIC CALCULUS
106
< (8K2+2TK2)C,KT'
Tn
ni .
Consequently (4K IT)" P( sup { V..+»(t) - Y'n)(t)) > ") < cont. X
and, by a standard application of Borel-Cantelli's lemma, we see that Y)^) converges uniformly on [0, T], a.s. The limit Y(t) is a continuous
adapted process and, by (2.6), E( I YY(t) - Y(t) 12) _ 0 as n --- oo, t E [0, T]. Now it is easy to see that Y = (Y(t)) satisfies (2.5). To prove the uniqueness, let Y, and Y2 satisfy (2.5). Then, by the same calculations as above, we have E { I Y,(t) - Y2(t)12} < K2J ' (I Y1(s) - Y2(s) 12} ds. E
By truncating Y, and Y2 with stopping times if necessary, we may assume that s '--- E { I Y,(s) - Y2(s) 12} is bounded in [0, T]. Now it is easy to con-
clude that Y, = Y2.
Corollary. Let o (x), i = 1, 2, ... , d, i = 1, 2, ... , r, be real continuous functions on Rd such that they are twice continuously differentiable with bounded derivatives of the first and second orders. Then,
for given dX',dX2, ... , dX' E d&' and y = (y', y2, ... , yd) E Rd, , Yd e a such that
there exist unique Y', Y2, ... (2.7)
i = 1,2, ... , d.
1dY'(t)
= E a r(Y(t))odX'(t) -I Proof. This follows at once if we note that (2.7) is equivalent to
Y'(0) = y'
dY'(t) =
r
E
(2.7)' 2
1.1-1
(a?)(Y(t)).
(dX' dX')(t). This equation is a particular case of (2.1) with respect to dX' and dX' W. The coefficients o and the assumption in the corollary.
satisfy the Lipschitz condition by
STOCHASTIC DIFFERENTIAL EQUATIONS
107
Thus a general theory of existence and uniqueness of the equations for semimartingales is established. Below there are some examples for which solution can be written down explicitly.
Example 2.1.* Let d = 1 and consider, for a given X E G ' 2such that Xo = 0 the following equation dY, = a(Y,)odXX + Yo = Y,
where a E C2 (RI -- R) with bounded d and a" and b is Lipschitz continuous. By the corollary of Theorem 2.1, we know that the solution exists uniquely and it is given in the following way. Let u(x, z) be the solution of az (x, z) = a(u(x, z))
u(x, 0) = X.
Let D, be the solution of dD'`
= exp { - f o a'(u(D,, s))ds} b(u(DD, X,))
Do=y. Then the solution Y is given by Y, = u(D,, X,). Indeed, by the chain rule (1.14),
dY, = a(u(D,, X,))odX, + (ax )(D,, X,) exp {- f o' a'(u(D,, s))ds}
x b(u(D,, X,)) dt. But
z LU x= -2-
a'(u(x, z))
x
and ax (x, 0) = 1 imply that
x(x, z) = exp { fo a'(u(x, s))ds}, * Doss [19).
STOCHASTIC CALCULUS
108
and hence dY, = a(Y,)odX, +
Example 2.2. Let Ak =
d
j-,
Ak(x)
a
axi
be a C°'-vector field*' on Rd,
k = 1, 2, ... , r. We assume that the first and second order derivatives of all coefficients are bounded. For given X', Xz, ... , X' E 62 such that
Xa = 0, i = 1, 2, ... , r, and y = (y', y2, ... , y4) E Rd, consider the equation
(2.8)
JdY°(t) = E A4(Y(t))odXk(t) ,
i=1,2,...,d.
Y`(0) = y`,
[I] *Z If the vector fields A,, A2, ... , A, are commutative, i.e., [A A J = 0, p,q = 1, 2, ... , r, then this implies the integrability of (x, z) = AJ(u(x, z)),
az u(x, 0) = x E
i = 1, 2, ... , d, j = 1, 2, ... , r
Rd
and so we have the solution u(x, z) = (u'(x, z), ... , ud(x, z)). If we set Y, = u'(y, X,), i = 1, 2, ... , d, where X, = (XI, X,, ... , X;), it follows immediately from the chain rule (1.14) that Y = (Y,, r,, Y,') is a solution of (2.8). [II] *3 Next we consider the non-commutative case but we assume that the vector fields A,, A2, ... , A, satisfy (2.9)
4j,k = 1,2, ... , r,
[An [A1, Ak]] = 0,
i.e., the Lie algebra £(A,, A2,
... , A,) generated by A,, A2, ... , A,
is nilpotent in two steps. Then the solution of (2.8) is given as follows. Let
1={(i,j);1
consider the following system of equations 1-1
au,
az` - A',(u(z)) - EP-1z'Ap, (u(z))
(2.10) l
au" _ az = 17
Ah .k(u(z))
1 < j < k < r, h = 1,2, ...
*1 Cf. Chapter V, §1. *2 Doss [19].
*' Additional information on this subject can be found in Yamato [184].
,
d,
STOCHASTIC DIFFERENTIAL EQUATIONS
109
where Ai.k(x) is defined by d
E A(x)
a 7x,, =
[A1, Ak] : = A,,k.
We can prove that (2.9) implies the integrability condition of (2.10) and so, for given x E Rd, we have the solution (u'(x, z))1-1,2,...,d of (2.10) such
that u'(x, 0) = x', i = 1, 2, ... , d. Let
Xj.k= IrXsodX,
,
0
1 <j
and Xr = W&m. Then Y,= u'(y, X,), i = 1, 2, ... , d is a solution of (2.8). Indeed au; (y,Xr)odXr
dYj =
az
a
+ 2sJ
k>
u`(y, X:)odX J,k
Z,E
r
=E Ai(YY)odX, - J-1 J-i
k-1
A'.J(Y,)X,odX,
+ EISJ
= E Ai(YY)odX,. J-1
For example*,
- 2x'x
,
if d=3, r=2 and Ai = aa' + 2x2 aX3 ,
A2 = az2
the solution Y, = (Y,,Y,,Yi) is given by Y,' = y' + X;, y2
= y2 + X, and Y, = y3 + 2(y2X' - y'X?) + 2($ XaodXs - f ,XJodXs). Consider the solution Y(t) = (Y'(t), Y2(t), ... , Yd(t)) of the equation (2.8). If a C2 function f(x) defined on Rd satisfies Ak f = 0, k = 1, 2, ... , r, thenf(Y(t)) = f(y) for all t > 0 a.s. Indeed, a
df(Y(t)) = E t-iatf(Y(t))odY'(t) r
a
=E (Akf)(Y(t))odXj = 0. k-1 * Gaveau [33].
r
= 1-1 E k-1 E Ak(Y(t))atf(Y(t))odXj
STOCHASTIC CALCULUS
110
For example, if Ak f =
Ak(x) az (x), k = 1, 2,
... , d,
with A'k(x) = d
151k - x`xkl I x 12, x = (x',x2, ... ,xd) E Rd\ 101, f(x) =
(x`)2 satisfies
Ak f = 0, k = 1, 2, ... , d. Thus the solution Y(t) always stays on the sphere with center 0 and radius I y I (= I Y(0) I). If we take Xk E ..0; k = 1, 2, ... , d, such that i.e., (X', X2, ... , Xd) is a d-dimensional Wiener process, then the solution d
dY'(t) = E A,(Y(t))odXk
i= 1 2,..., d,
y'(t) = y`, defines the Brownian motion on the sphere with center 0 and radius I y .*'
3. Moment inequalities for martingales Various inequalities for moments of martingales are discussed, e.g., by Meyer [120] and Garsia [32] in connection with a martingale version of the theory of HD-spaces. Here we obtain fundamental inequalities for continuous local martingales as an application of stochastic calculus.
Theorem 3.1. There exist universal constants c C, (0 < p < oo) such that for every M E .0 ( = . ' `z'OP) and t > 0, (3.1)
cE(M72) S E(<M, M>') < C,E(M*2')
where M,* = max I M, I .
Proof. *2 It is sufficient to prove (3.1) for a bounded martingale M = (Me) since the general case follows easily by a truncation argument. In-
deed, setting T. = inf {t; I M, I > n or <M>, > n), we have T. t oo a.s., and if (3.1) holds for Mm = (MT,A,) with c, and C, independent of n, then (3.1) holds for M by letting n - oo. In the proof we shall write A for <M, M>. By the inequality (6.16) of Chapter I, (3.2)
E(M*D) S (p p 1 D E(I M, I '),
P> 1.
This representation of a spherical Brownian motion is due to Stroock [156]. *2 The following proof is adapted from Getoor-Sharpe [34].
MOMENT INEQUALITIES FOR MARTINGALES
111
Case 1. If p = 1, E(<M, M>,) = E(MS ) and hence, noting (3.2), we have (3.1) with
c, = 1/4 and C, = 1.
Case 2. If p > 1, E(M*zp) S (2P/(2P - 1))z'E(I M, I z').
Since I x I Z' is of class Cz we can apply Ito's formula to obtain
sgn(M,)dM, + p(2p
M, I z' = J o 2P I M,
- 1) J o
M,12P-2 dA,.
By taking the expectations, we obtain E(I M, I zp) -< p(2p - 1)E(fo I M, I 2--2d.4,) < p(2p - 1)E(M#zp-2A,)
< p(2p -
1)E(M*2p)1-11,E(AD)11,.
Therefore, E(M=*z')
S (2P/(2P - 1))2pp(2P - 1)E(M*2y)1-11'E(A')11',
from which the left side of (3.1) follows. To prove the other side of (3.1),
Then p = p f' As-' dA, = A° and so
set N, = fro A;P- 1)'2dM,.
E(A') = pE(N?). By Ito's formula, MtA(p-»lz = f ` Acp-`)12dM, + f ` Md(A 0
)1z)
0
N, + f t
M,d(A(-)iz)
0
and hence
IN, I < 2M*
APP- 11/2. Thus
P E(A') = E(N,) < 4E(M#2A,- 1) and so E(AD) < (4P)'E(M*2').
4E(Ms2v)''PE(,gj
STOCHASTIC CALCULUS
112
Case 3. Let 0 < p < 1. Set N, = f o Ate'>'2dM,. Then, as above, E(Af) =
and M, = f o A;'-P>12dN,. By M's formula,
NjA(1-P)12 =
A,1-P)/2 dN, +
I
0
= M, +
f`
J u N,d(A`
-P>'z)
N3d(As1-P)'2),
0
and hence
IM,I < 2N*Aj'-P>/2.
Thus M* < 2N*A(1-P)'2, and by Holder's inequality, E(M*2P) < 22'E(N*2PAPc1-P>) < 22'E(N*2)PE(AP)1-P 22P4PE(N2
<
(16/p)PE(AP)PE(AP)1-P
,)PE(AP,)'-P =
_ (16/p)PE(AP).
Finally we must show that E(AP) < CPE(M*2P). Let a be a positive con stant. By applying Holder's inequality to the identity
AP = [A'(a + M*)-2P"-P)](a + M*)2P"-P>> we obtain E(AP) < {E(A,(a + M*)2 c--1>)) P {E((a + M*)21)} 1-1.
Setting N, = f '(a + M*)P-'dM we have 0
, = f ' (a + M; )2 cP-1> dA, > A,(a + M,)2''. By Ito's formula, :
M,(a + M*)P-' = f `0 (a + M*)P-'dM, + f 0 M,d {(a + M; )P-'}
= N, + (p - 1) f 0M,(a + M,*)P-2dM, and hence
SOME APPLICATIONS OF STOCHASTIC CALCULUS tNNI
Therefore
p M:',
M*p + (1 - p) fo E(N,) < - E(M*2p)
113
for every a> 0,
so
E(AD) < p -1-P {E(M#2p)} ° {E((a + M*)2')} 1-D.
Letting a 10, we conclude that E(Arp) < p-11 E(M#2y)
4. Some applications of stochastic calculus to Brownian motions 4.1. Brownian local time. Let X = (X,) be a one-dimensional Brownian motion defined on a probability space (Q,Y",P).
Definition 4.1. By the local time or the sojourn time density of X we mean a family of non-negative random variables {¢(t, x, c)), t E [0, co),
x e R'} such that, with probability one, the following holds: (i) (t, x) i--- O(t, x) is continuous, (ii) for every Borel subset A of R' and t > 0 f 'o I4(X,)ds = 2
f
0(t, x)dx. A
It is clear that if such a family {5(t, x)} exists, then it is unique and is given by
0(t, x) = lim 4E1 CIO
` u
I (X_ .x+e) (XX)ds.
The notion of the local time of Brownian motion was first introduced by Levy [101] and the following theorem was first established by H. Trotter [163].
Theorem 4.1. The local time
x)} of X exists.
Proof. We will prove this theorem by using stochastic calculus. This idea is due to H. Tanaka (McKean [113] and [114]). Let (.) = (-F-,X) be
STOCHASTIC CALCULUS
114
the proper reference family of X. Then X is an (Y)-Brownian motion and X, - Xa belongs to the space .0. Let ga(x) be a continuous function on
R' such that its support is contained in (- 1/n + a, 1/n + a), ga(x) > 0, gn(a + x) = x) and
fI
1.
Set x
ua(x) = f
d y j ' ga(z)dz.
By Ito's formula, u»(XX) - un(X0) = f o ua(X3)dX1 + 2 f ' u;'(XX)ds
and if the local time {0(t, x)} does exist, then
2 f' u'(X,)ds = 2 f '
f
g.(y)c(t, y)dy - 0(t, a) as
n -. oo.
Also, it is clear that
1, x> a I, x = a,
u(x)
as
n - oo.
0, x
0(t, a) = (X, - a)+ - V. - a)+ - f o
1(a..,) (Xs)dX,.
In the sequel we will show that the family of random variables 6(t,a), t > 0, a E R' defined by (4.1) satisfies the properties (i) and (ii) above. It is obvious that (X, - a)+ - (X0 - a)+ is continuous in (t, a). We will show that there exists a process V(t,a) which is continuous in (t, a), a.s., and for each t, a,
SOME APPLICATIONS OF STOCHASTIC CALCULUS
v(t, a) = fo I (a._) (X,)dX..
115
a.s.
For each a e R' and T > 0, [0, T] D t i -- YY(t) = f oI (a..)(Xs)dXs is a continuous process, i.e., C([0, T] --- R)-valued random variable. If we set
I Ya(t) - Yb(t) 1, II Y. - YbII = max 0--9t :9T
then (4.2)
EIIIYa-Yb1I`}
for some constant K = K(T) > 0. Indeed, if a < b, (Ya - Yb)(t) = f 'O I(a.b](XS)dXS e //
and , = f oI(a.b](XS)ds. Now applying the inequality (3.1)
E(IIY. - Ybli`) < zE({J o
I(a.b](XS)ds}Z)
< C2E(f 0 I(a.b](Xa)ds f j
I(a.b](XX)du)
= C22 f T0 ds f duE(I(a.b](Xs)I(a.b](X,,)) s
_f? T ds C2
fTdu
0
x
f R u(dx)
exp{-(x 1
2ns
- y)2} 2s
fb a
dy f ba
dz
1
12n(u - s)
expI(Y - z)21 2(u - s)
1
2x-.Is(u - s)
= K(T)(b - a)2. Here 4u is the initial distribution of X, i.e., p = PXO. (4.2) is proved. By the R)corollary of Theorem I-4.3,* there exists a family {Vr(a)} of C([0, T j
valued random variables such that * This corollary applies for any system of random variables taking values in a metric space.
STOCHASTIC CALCULUS
116
a' -. VI(a) E C([0, T] ----- R)
is continuous a.s. and for each fixed a, y/(a) =
a.s. Clearly yi(t,a) = yi(a)(t) is what we want. Thus by choosing this modification we see that ¢(t, a) satisfies (i). In order to prove (ii), it is clearly sufficient to show that
f' f(X,)ds = 2fR30(t, a)f(a)da
(4.3)
a.s.
for any continuous functionf(x) with compact support. Set
F(x) = f - f(a)(x - a)+da.
Then F E C2(R), F'(x) = f -f(a)1(a,.,,(x)da = f x f(a)da, F"(x) = f(x), and so by Ito's formula,
F(X`) - F(X°) -
f' F (X,)dX, =
2 f f(X,)ds.
The left-hand side is equal to
f m f(a) {(X. - a)+ - (X o - a)+} da - f o { f
f(a)1 ca.0) (X,)da} dX,.
If we now apply the next lemma to the second term, the above reduces to fm f(a) {(XX
-
- a)+ - (X0 -
a)+
-f
o
1 (a,..) (X,)dX,}
da
=f- f(a)o(t, a)da and hence (4.3) is established.
Lemma 4.1. (A Fubini-type theorem for stochastic integrals). Let (Q, ..,P) be a probability space and (Y) be a reference family. Let M E i (i.e., a continuous square-integrable martingale such that Mo = 0
4'
a.s.). Let {h(t,a,co)}, t E [0, oo), a c- R', be a family of real random variables such that (i) ((t, co), a) ,E ([0, co) x 9) x R' -- O(t, a, w) is 5" x .'(R')-measurable (S° is defined in Chapter 1, p. 21),
SOME APPLICATIONS OF STOCHASTIC CALCULUS
117
(ii) there exists a non-negative Borel measurable functionf(a) such that 10(t, a, co) I < f(a)
for every
t, a, co.
i is well-defined. We assume By (i) and (ii), f o 45(s, a, co)dM, E further that (iii) (a, co) i --- f' 0(s, a, co)dM, is .i(R') x,91--measurable for each t > 0.
Let u(da) be a non-negative Bore] measure on R' such that f R, f(a)p(da) < oo. Then (4.4)
t-
0(t, a, c))u(da) E Yz(<M>) fR1
(i.e., it is predictable and E[ f 'o { f R1 h(s, a, ) ,u(da)) zd<M>,] < oo for every t) and we have (4.5)
f
{ f Rl0(s, a, co)p(da)} dM, = J o
{ f 'O(s, a, (o)dM,} p(da). R1
Proof. It is clear that f R1 0(s, a, co),u(da) is (,)-predictable and bounded. Hence it is obvious that
E[ J o {
f
P(s, a,
R1
co)p(da)} 2d<M>,] < °°.
Thus the left-hand side of (4.5) is well-defined as an element in .02°. On the
other hand, a --- - f, 0(s, a, co)dM, is Borel measurable by assumption (iii), and for every T > 0, E[
f Rl'u(da) s x I f P(s, a, co)dM, I] 0
V
p(da) {E[ maaxT I f 0(s, a, co)dM, 12]11/2 a
< 2 f Rl,u(da)(E[ If 0(s, a, co)dM,} 2])112 o
= 2 f 1u(da)(E[ f o 02(s, a, co)d<M).])1/s _< 2 f Rlf(a)k(da) {E[<M>(T)]} 1/i < 00.
STOCHASTIC CALCULUS
118
Hence
f
p(da) ma I f o 45(s, a, w )dM, I < oo
a.s.
RI
and this implies that
t
- SRI u(da) f
o
O(s, a,
w)dM,
is continuous a.s. Thus the right-hand side of (4.5) is well-defined and defines an (.9)-adapted continuous process. It is square-integrable because E {[ SRI
(f
,P(s, a, CU)dM,),u(da)]2} o
= f u(da,) f R, p(da2)E[ f o 0(s, a,, w)dM, f o P(s, a2, w)dM,] Rl
= fRl u(da,) f Rl u(da2)E[ f o -P(s, a, w)O(s, a2, w)d<M>,] f Rlf(a,)p(da,) f R,ff(a2)p(da2)E[<M>(t)]
(f RIf(a)p(da))2E[<M>(t)] < 00.
It is an (F)-martingale because if t > s > 0 and A E .f;, E[I, f Ri,u(da) f , 0(u, a,
f
Rt u(da)E[I,,
f , O(u, a,
0.
Similarly, if N E.A2i then
u(da) f J -P(u, a,
E[IA {
(N, - N,)]
5R1
= f RI ,u(da)E[IA f : ,P(u, a,
= f Rl u(da)E[1., f
q(u, a, (o)d<M, 8
N:)]
N)]
SOME APPLICATIONS OF STOCHASTIC CALCULUS
= E[re J s
if
R1
119
45(u, a, w)u(da)} d <M, N> J.
Thus t '-. f R, p(da) f o 0(u, a, c))dM. = L1 is an element in -,O such that for every , = J o
if Rl
P(u, a,
w)u(da)} d<M, N%.
Now we can conclude that L, = f") {f R, 0(u, a, co)p(da)) dM by Proposition 11-2.4. This completes the proof of (4.5).
4.2. Reflecting Brownian motion and the Skorohod equation. Let X = (X,) be a one-dimensional Brownian motion and let X+ = (X,+) be a continuous stochastic process on [0, oo) defined by
Xf = I X.
(4.6)
It is easy to see that
(4.7)
P[XX; E A,, Xi E Azi
... , Xn E An]
=J
aP+(t1, x, xl)dx1J
u+(dx)J
AP+(ty - t1,
z
x1, xz)dxz
$AP+(tn - to-1, xn-1, xn)dxn, ,
0
A1E. ([0,co)),
where {4.8)
P+(t, x, Y)
=
/ _ z z I exp I- x 2tY + exp {- (x 2tY)
and ,u+ is the probability law of Xo = I Xo I. The process X+ is called the .one-dimensional reflecting Brownian motion. Reflecting Brownian motion
can be characterized in different ways. We shall now present one such characterization due to Skorohod [149].
We set Wo = If E C([0, oo) - R'); f(O) = 0} and C+ = If E C([0, co) -- R); f(t) > 0 for all t > 0). Lemma 4.2. Given f E Wo and x E R+, there exist unique g E C+ and h E C+ such that
(i) g(t) = x +f(t) + h(t), (ii) h(0) = 0 and t i
h(t) is increasing,
STOCHASTIC CALCULUS
120
(iii) to I io1(g(s))dh(s) = h(t ),
i.e., h(t) increases only on the set oft when g(t) = 0.
Proof. Set
g(t) = x +f(t) - min {(x +f(s)) A 01, OSrSr
(4.9)
(4.10)
h(t)
min {(x + f (s)) A 0} . OSsSr
Then it is easy to verify that g(t) and h(t) satisfy the above conditions (i), (ii) and (iii). We shall prove the uniqueness. Suppose g(t) and h(t) E C+ also satisfy the conditions (i), (ii) and (iii). Then
g(t) - g(t) = h(t) - h(t)
for all
t > 0.
If there exists t, > 0 such that g(t,) - g(t1) > 0, we set t2 = max{t < t,; g(t) - g(t) = 01. Then g(t) > g(t) > 0 for all t E (t2, t1] and hence, by (iii), h(t1) - h(t2) = 0. Since h(t) is increasing, we have 0 < g(t1) - g(t1) = h(t1) - h(t1) 5 h(t2) - fi(t2) = g(t2) - g(t2) = 0.
This is a contradiction. Therefore g(t) < g(t) for all t > 0. By symmetry,
g(t) > g(t) for all t > 0. Hence g(t) = g(t) and so h(t) - h(t).
The mappings (x, f) - g and (x, f) - h given by (4.9) and (4.10) are denoted by g = 1'1(x, f) and h = r2(x,f) respectively.
Theorem 4.2. Let {X(t),B(t),O(t)} be a system of real continuous stochastic processes defined on a probability space such that B(t) is a onedimensional Brownian motion with B(0) = 0, X(0) and the process {B(t)} are independent and with probability one the following holds:
(i) X(t) > 0 for all t > 0 and c(t) is increasing with 0(0) = 0 such that f 'rn (X(s))dO(s) = 0(t); (ii) (4.11)
X(1) = X(0) + B(t) + 0(t).
Then X = (X(t)) is a reflecting Brownian motion on [0, oo).
SOME APPLICATIONS OF STOCHASTIC CALCULUS
121
Equation (4.11) is called the Skorohod equation.
Proof. By Lemma 4.2, X = (X(t)) and 0 = (q(t)) are uniquely determined by X(0) and B = (B(t)): X = 1',(X(0), B) and 0 = F,(X(0), B). In order to prove the theorem we have only to show that if x, is a one-dimensional Brownian motion, then X(t) = I x, I satisfies, with some processes B(t) and c(t), the above properties. Let g (x) be a non-negative continuous 1. Set function on R' with support in (0,1/n) such that f u un(x)
=f
dY f o
o
Then it is easy to see that u E C2(R'), I u. 'I < 1,
un(x) t
IxI
and
sgn x as n -- oo.* By Ito's formula,
f ' un(x,)dx, + f ' u;'(x,)ds = f uu(x)dx, + f g, (-y)O(t, y)dy + f
u (x,) - u (xo) =
0
0
2
0
0
W
0
gn(y)9S(t,y)dy,
where ¢(t, y) is the local time of x,. Letting n -- oo, we have
X(t) - X(0) = f
sgn (x,)dx, + 20(t, 0). o
Set
B(t) = f sgn (x,)dx, and 0 (t) = 20(t, 0). Then, since , = t, B(t) is an (. )-Brownian motion, where (-,g;) _ (,01-,x) is the proper reference family for xt. Since X(0) is . X(0) and {B(t)} are independent. Since -L f 1[0.8)(X(s))ds, fi(t) = l im 2z o
80
it is clear that * sgn x =
1
,
0
,
-1
,
x>0, x = 0, x<0.
-measurable,
STOCHASTIC CALCULUS
122
f t0 Iro,(x(s))dL(s) _ 0(t).
Therefore {X(t), B(t), 0(t)} satisfies all conditions in Theorem 4.2. Thus
X = (X(t)) and _ (¢(t)) are characterized as X = r,(X(0), B) and _ r2(x(o), B). An immediate corollary is the following result due to Levy.
Corollary. Let B(t) be a one-dimensional Brownian motion such that B(O) = 0. Then
(i) the processes { B(t) I } and {B(t) - min B(s)} are equivalent in 0:9s5:
law, (ii)
l lr 2E
f
o
I[o.8)(B(s) -omiin B(u))ds = - m B(s). s:gr
We can give yet another description of the reflecting Brownian motion. Let x(t) be a one-dimensional Brownian motion. Then by (4.1), x(t)+
-
x(0)+
0(t, 0). = J o I (O .,)(x(s))dx(s) +
M(t) = fo I,o,,,,(x(s))dx(s) is a continuous martingale such that <M>(t) =fo (x(s))ds. It is easy to see that lim <M>(t) = co a.s. Indeed, if
Q we set 6,=min {t; x(t) = 0}, r, = min{t> v,; x(t) = min It > z x(t) = -1) , ... and " I (x(s))ds,
an
then by the strong Markov property of x(t) (Theorem 11-6.4), it is easy to see that is independent and identically distributed. By the strong law
of large numbers, , + 2 + lim f t I,o,,,,(x(s))ds = co rtm
+
a.s.
o
Set
r, = inf{u; f Iro.m,(x(s))ds > t}. 0
-- oo a.s. This implies that
SOME APPLICATIONS OF STOCHASTIC CALCULUS
123
By Theorem 11-7.2, M(z,) is a one-dimensional Brownian motion. Also it
is easy to see that X(t) = x(z,) is continuous and X(t) > 0 for all t > 0, as. Therefore
0(t): = Or,, 0) = X(t) - X(O) - M(z,) is continuous in t and satisfies f I,o}(X(s))do(s) = ¢(t),
a.s.
Hence {X(t) = x(z,), B(t) = M(z,), 0(t) = O(z 0)) is a system satisfying the conditions of Theorem 4.2. Thus we have the following result. Theorem 4.3. Let x(t) be a one-dimensional Brownian motion and z, = inf {u; f oI[%.) (x(s))ds > t). Set X(t) = x(a,). Then X(t) is a reflecting Brownian motion. If x(t)- = (-x(t)) V 0, then we have similarly
x(t)- - x(0)-
=-
f I (-..o) (x(s))dx(s) + 0(t, 0).
Let ?7, = inf {u; f o I (-,, o} (x(s))ds > t } , N, = - f o I (-..o) (x(s))dx(s), B(t) _
N(') and Y(t) = -x(ri,). Then Y(t) is also a reflecting Brownian motion. As we saw above, X = T} (X(0), B) and Y = r,(Y(0), B). Since <M, N> = 0, B and .9 are independent by Theorem 11-7.3. Therefore, if X(0) and Y(0) are independent (this is the case if x(O) = x a.s. for some x EE R' or x(O) >
0 a.s.), then the processes X and Y are independent. This fact indicates roughly that the motion of x(t) on the positive half line (0, oo) and that on the negative half line (-oo, 0) are independent. This is seen more clearly by studying the excursions of Brownian motion. 4.3. Excursions of Brownian motion. Let X = (X(t)) be a one-dimensional Brownian motion and let Z_ {t; X(t) = 01. It is well known that,
with probability one, X is a perfect set of Lebesgue measure 0 and [0, oo)\Z = U ea is a countable union of disjoint open intervals ea ([73] and a
[101]). Each interval ea is called an excursion interval; the part of the process {X(t), t ea} is called an excursion of X(t) in R'\ {0} . In order to study the fine structures of Brownian sample paths, it is sometimes necessary to decompose them into excursions. Here we prefer to proceed in an equivalent but opposite direction: we start with the collection of all excursions and then construct Brownian sample functions.
STOCHASTIC CALCULUS
124
First of all, we will formulate and construct the collection of all excursions of Brownian motion as a Poisson point process taking values in
a function space. This idea is due to Ito [70]. Let+ (V-) be the totality of all continuous functions w: [0, oo) -- R such that w(0) = 0 and there exists a(w) > 0 such that if 0 < t < a(w), then w(t) > 0 (resp.
w(t) < 0), and if t > c(w), then w(t) = 0. Let .'(+) and .g'(2'-) be the a-fields on 2v+ and Y- respectively which are generated by
Borel cylinder sets. The spaces+ and 2'- are called spaces of positive and negative excursions respectively. There exist a-finite
and on (2'-,
measures n+ and n on (2'+, respectively such that
n:'({w; w(t1) e A,, w(t2) e A2, (4.12) A,
Kt(t1, xl)dx1J
A.))
P"02 - t1, x1, x2)dx2 f
d2
r X
... ,
J Ay
0
I
n
where 0 < t1 < t2 <
i=1,2,...,n,
< t and
(resp. -g'((-oo, 0)),
K4-(t, x) _ 42- x exp(- 2t),
t > 0, x E [0, oo),
K-(t, x) = J t33 (-x) exp(- 2t),
t > 0, x
2
0]
and
At' X, Y) =,l2nt (exp(
(x 2t
Y)z )
Y)z ))'
(x
- exp
2t
t > O, x, y E [0, oo) or x, y E (-00, 0]. A way of constructing the measures n+ and n is as follows. In Chapter IV, Example 8.4, we shall see that for every T > 0 there exists a probability measure P'' on + fl {o(w) = T} such that P'{w; w(t,) E dxl, w(t2) E dx2, ... (, h(0, 0; t1, xl)h(tl, x1; t2, x2) ... h(tn-1,
where
E tn, x,,)dxldx2 ... dx
SOME APPLICATIONS OF STOCHASTIC CALCULUS
K+(T - t, b)
K+(T-s,a)'
h(s, a; t, b) =
(t - s, a, b),
125
0<s0,
f ,J i T3 K+(t,b)K+(T-t, b), s=0, t>0, a=0, b>0, and 0< t1 < t2 <
< t < T. The a-finite measure n+ on
defined by
n+(B)
J0
PT(B n {a(w) = T) -/T T,,
B E .19(5!1'+)
satisfies (4.12). n can be constructed in a similar way. Let 5Y= V+ U 2Y' be the sum, .q(2Y') = .19(V+) V .19(5Y-) and n be the a-finite measure on { r, .19(x )} such that n 19,.t = nt. By Theorem I-9.1, we can construct a stationary Poisson point process p on 2Y with characteristic measure n. We call it the Poisson point process of Brownian excursions. This Poisson point process is what we intended to be the collection of excursions of Brownian motion. Suppose we are given a Poisson point process of Brownian excursions p on a probability space
(0, -; P). A Brownian motion X(t) is constructed from p by the following steps. Set
A(t) = s5
a(p(s))
= J o+f a(w)N,(dsdw)
where D, is the domain of p and N,(dsdw) is the counting measure of p defined by (9.1) in Chapter I, Section 9. With probability one, t -- A(t) is strictly increasing, right continuous and lim A(t) tT'
00.
Indeed, it is a time homogeneous Levy process with increasing paths (cf. Example. 11-4.1) with
E[exp(- tA(t))] = exp(- tV (2)) where
V(A) = f 0(1 - e-1") n ({w ; a(w) E du}) 2du = 2, =f0(1 - ex_)u3
i. e., it is a one-sided stable process with exponent 1/2. Let c(t) = A-1(t)
STOCHASTIC CALCULUS
126
be the inverse function of A(t). Then, almost surely, 6(t) is continuous. We now define, for any sample point on which the above mentioned
properties of A(t) are satisfied, a function: [0, oo] D t - X(t) E A' as follows: For each t > 0, set s = c(t). If A(s-) < A(s), then s c- D, and we set X(t) = p(s)(t - A(s-)). If A(s-) = A(s), we set X(t) = 0. We claim that with probability one, t - X(t) is continuous. Furthermore, we can identify X(t) with a one-dimensional Brownian motion starting at
0 and ¢(t) with the local time at 0:
fi(t) = lim 810
4e
f
t o
r(-e, E) (X(s))ds.
Proof*. First, we establish the almost-sure continuity of t - X(t). We have, by (4.21) below, n({ w; omaxy) Iw(t)I > e}) - 2e ,
e > 0.
Since
E[# {s E D, ; s S T, max (p(s)[t]I > e]J ostso[y(s)]
< oo = T x n({w; max lw(t)I > s)) = 2T C Ostso(,)
for every T > 0 and e > 0, it holds with probability one that
#{sED,; sS T, Osrso(p(s)] max p(s)[t]>e] 0 and a > 0. This implies, in particular, that with probability one, for any sequence s E D, converging to a finite time point to such that sn = to for every n, max
oSt:r
p(sn)[t] - 0
as n - oo.
Now the continuity of t - X(t) is easily concluded: It is obvious that t X(t) is continuous on each interval (A(s-), A(s)), s E D,. If A(s-)
= t = A(s), then X(t) = 0 and, for intervals (A(sn ), A(sn)), s E D converging to t as n - oo, max
to (A(sn-), A(sn))
I X(t) I - 0
as
n -- oo
by the above remark. * cf.[234]for construction of Brownian motions in more general cases than that treated here.
SOME APPLICATIONS OF STOCHASTIC CALCULUS
127
To identify X(t) with a Brownian motion, we appeal to Theorem II-6.1. We may assume that p is given as an (.9')-Poisson point process with respect to some reference family Then A(t) is (. r} adapted and hence fi(t) is an (.9i)-stopping time for each t. Let -9; (r) and .Y_ _ JV0 (r)_ be the usual o-fields; in particular fir)_ is the o-field generated by sets of the form An (s < cS(t)}, A E -W;, s ei [0, oo). For each fixed t > 0, set F (s, w, co) = w(t - A(s-)) IuaA<s_)>.
It is (.9`-s)-predictable and belongs to F,, n FD. Indeed
f :f
I w(t - A(s))I&ZAts)) IR,(dsdw) T"
= f:ds fr+ o
+ f dsf r- [-w(t - A(s))I,,,,wln-(dw) 0
2f =2
f
K+(t - A(s), x)xdx
dsI,2ZA(,)) f
o
u
I12AWIdS 0
where *,(dsdw) is the compensator of the point process p. Also,
OM f
r I w(t - A(S))I f 4 ,)) 121d,(dsdw)
= 2 f ds I rZA(,p f o
K+(t - A(s), x)x2dx ro,m)
8 f" (t - A(s))1"2 I
IrZA (*)
ds.
o
Set N,(dsdw) = N,(dsdw) - 1G,(dsdw). Then, clearly the X(t) defined above is expressed, for each fixed t, as cr) +
f
X(t) = f o = f"')'f o
Fr(s, w, )N,(dsdw)
r FF(s, w, )N,(dsdw)
since
f¢Q)+ f o
r FF(s, w, )1Y,(dsdw) = 0.
128
STOCHASTIC CALCULUS
A°, = .l_V cr[p(0(t))(u - A(O(t)-)); u < t] e .*' We will show that X(t) is an ( °)-martingale such that <X)(t) = t. Set H(m) = Let
H)(co)H2(co) where Hi(m) is bounded and ,9_-measurable and H2(m) G(p(¢(s))SA(O )_)). Here G(w) is a bounded .gc( '*)-measurable function on '* *2 and for w E V'* and s > 0, ws E V* is defined by w,-(t) = w(t A s). It is sufficient to prove that (4.13)
E(X(t)H(m)) = E(X(s)H(m))
and
E([X(t)2 - t]H(w)) = E([X(s)2 - s]H(m)).
(4.14)
(4.13) is proved as follows. E(X(t)H(m))
W+
= E[ f o = E[
fr FF(u, w, o)N,(dudw)H(m)]
f
o
r FF(u, w, m)N,(dudw)H(m)]
E
= E[
FF(z, p(r), co)H(m)]
=E[ T<0 s),TED,F,(z, p(z), w)H(m)] + E[F:(O(s), p(O(s)), m)H(m)] = 1 + 12.
Then Il = 0 because FF(z,p(r),m) = 0 if r < g(s) < $(t). It is known (cf. [15]) that there exists a bounded (J"-,)-predictable process H.(')(m) such that Hi(m) = Then 12 = E(FF(O(s), p(a(s)), (D)H1(w)H2(w))
= E(
EDp
= E(fo =
_
E{ fo E { f¢ c:) o
(s-A(T-)
I`-Aw)
r
duHL')(m)[ f I(,-A(u)
p(s) e 7/'', s e D is extended by setting 0 ifs ir; D,. #2 70* = 7/U {0j, where 0 is the function 0(t) = 0.
SOME APPLICATIONS OF STOCHASTIC CALCULUS
129
Here we used (4.15) below. The following are the fundamental properties
of the measure n: if t > s > 0 and g(w) is bounded and , ,(V)-measurable* then
f
(4.15)
w(t)g(w)n(dw) =f w(s)g(w)n(dw)
and
fr
(4.16)
[w(t)2-tAQ(w)]g(w)n(dw)= f [w(s)2_sAc(w)]g(w)n(dw)
Now the above argument can be reversed to see that
E{f
0(d)
f rw, )G(w=-,A())n(dw)]}
o
= E(X(s)H(co)). Equation (4.14) is proved in a similar way by taking FF(s, w, co)
A(s-))z - [(t -
and by using
(4.16).
= (w(t In this
case FF(z, p(r), )
0 for r < ¢(s) but it is easy to see that F,(r, p(r), ) _ F,(r, p(r), ) if r < ¢(s). Consequently X(t) is an ( °t)-Brownian motion by Theorem 11-6.1. Next we prove that 0(t) is the local time at the origin of X(t). It is clear
that 0ct>+
X(t) I= f o
fr I w(t - A(s-))I u>ac:-f N,(dsdw).
Noting that f ,,, w(t) n(dw) = 2 f7 xK+(t,x)dx = 2 for every t > 0, we see that 0ct>+
X(t) I
=
f fr o
w(t - A(s-))rit>a(,-)i I 9,(dsdw) + 2¢(t). l
By the same argument as above, we can prove that
I w(t - A(s-))I a>act-), I N,(dsdw) is an (X)-martingale, and so by Theorem 4.2 we can conclude that * A(2r) is the a-field on 7V generated by Borel cylinder sets up to time s.
130
STOCHASTIC CALCULUS
2c(t) = lim 2e f 810
Iro.a)(I X(s) I )ds.
o
The inverse A(t) of 0(t) has expression A(t) = f t o+ f o(w)N,(dsdw). 9r
We know that it is a one-sided stable process with exponent 1/2. Thus we have established the following formula describing a Brownian sample path X(t) in terms of a Poisson point process of Brownian excursions p:
f f 3r w(t - A(s-))N,(dsdw) A lt) f f 6(w)NDld d w) (t+)
IX(t) = (4.17)
o
s
`+ 0
9
land i(t) is the inverse of t
A(t).
This expression may be regarded as a formula for the decomposition of a Brownian path into its excursions. Using it, many results on the local time 0(t) and the zero set X of X(t) can be obtained. Before we proceed with some examples, we first introduce the following maps (4.18)
TI:
-- (0, oo) defined by T1w = v(w)
and
(4.19)
T2: V -- (0, oo) defined by T2w = max I w(t) I . 0
T1 oo) and T2 (0, by induce stationary Poisson point processes T1(p) and T2(p) on DT, (P) = DD
and
T1(P)(s) = TT(P(s)),
s e Dr1(n),
i = 1, 2.
The characteristic measure n1 of T1(p) is given by (4.20)
n1([x, oo)) = 2n+({w; 6(w) > x}) =
2f-
dt
x 127ct3
and the characteristic measure n2 of T2(p) is given by
=2
2 ViLx
SOME APPLICATIONS OF STOCHASTIC CALCULUS n2([x, oo)) = 2n+({w; max
osrsolw>
131
w(t) > x))
= lira 2n+({w; a (w) > e, max w(t) > x} ) eSr
C LO
(4.21)
= Jim 2 fW K+(e, y)Pr(6= < cio)dy 810
= 1e m 2 10
0
/2 JO0
(_jX7) Y' Y Ax 2E
7Le3 y eXp
x
dY
_ 2x,
where P. is the Wiener measure starting at x and v, is the first hitting time to a.*' For a Brownian path X(s) and s > 0, let rte(t) = the number of excursion intervals in (0, t) whose lengths are not less than a,
(4.22) and
de(t) = the number of down-crossings of X(s) from a to 0 and up-crossings of X(s) from -e to 0 before time t.
(4.23)
It is immediate from (4.17) that r/5(t) = Nr, (,)((0, 0(t)) X [e, -)) and
de(t) = Nr2(v)((0, fi(t)) X [e, oo))
It follows from the strong law of large numbers that
P(lim 2 NTI w((01 a) x [e, no)) = 2a
for all a > 0) = 1
and
P(lim eN,2(,)((0, a) X [a, no)) = 2a for all a > 0) = 1. 510
Consequently, we have proved the following results of Levy *2: *1 The formula P,(v, < 6b) = (b - a)l(b - c), b < a < c, is well-known and is easily derived from the fact that w(t AQ, Aub) - a is a P; martingale. *2 Levy only conjectured (4.25); for different proofs, cf. e.g. Ito-McKean [73], ChungDurret [11] and Williams [179].
STOCHASTIC CALCULUS
132
(4.24)
P(lim 610
J 2 71r(t) = 2¢(t)
for all
t > 0) = 1
and
(4.25)
P(lim sd5(t) = 20(t) for all t > 0) = 1. 810
Let p+ and p be the restriction of p on V'+ and W'- respectively. Then p+ and p are stationary Poisson point processes on 7+ and Vwith the characteristic measures n+ and n- respectively; moreover, they are mutually independent. We set, respectively, t
(4.26)
A*(t)
= J o f a(w)N,,(dsdw) "
and
X`-(t) = p:'*(s)[t - A*(s)]
where A*(s--) S t:5; A±(s) under the convention that p±(s) - 0 if s E DDt. The unique s such that A±(s-) < t< A±(s) are denoted by O±(t), respectively, so that 0-t(t) are the inverse of t - A`-(t). Also we set 0+(t)
=
o
I,
.
, (X(s))ds
and
0-(t) =
f: I(`m o3(X(s))ds
where X is the Brownian motion given by (4.17). It is immediately seen
that X+(t) = X(r+(t ))
and
X-(t) = X(i (t ))
where r+(t) and T -Q) are the inverse of t --- - 0+(t) and t -w 0-(t) respectively. Thus X+ and -X' are mutually independent reflecting Brownian motions as being functionals of p+ and p respectively and therefore, we recover what was explained at the end of the previous section. If we set B+(t)
(4.27)
=
p+(O+(t )[t
- A+(c+(t))] - O+(t)
B-(t) = -p-(O-(t)) It - A-(O-(t))] -
and (t),
then, by the same proof as for (4.17), we can show that B+ and B- are Brownian motions and that X+(t) = B+(t) + 6+(t) and -X-(t) =
B-(t) + ¢-(t) are Skorohod equations for X+ and -X- respectively. In particular,
SOME APPLICATIONS OF STOCHASTIC CALCULUS
¢+(t)
lim 8 10
c (t)
2a -
lam 2E
133
f 1..,(X-(s))& and
f
o
1(-a.o](X (s))ds.
If A(t) is defined by (4.17), it is immediately seen that At(t) = 9±(A(t)). Let a < 0 and set as = inf{t; X(t) = a). Then the excursion p(e) E W,
of X such that A(e-) < a, S A(e) must coincide with p -(e) E V where
A-(e'-) < as = infit; X-(t) = a) S A-(e'). Hence e = e' = -(aa) and e = inf{s e D,; M_[p(s)] < a} where
M_(w) = inf OSt
w(t),
w E W-
(w)
Then d(e)
9+(aa) = f0
I[o. m)(X(s))ds = A+(e) = A+(O-(aa ))
e is exponentially distributed with mean -a since
P[e > u] = P[N,((0, u] x {w; M_(w) < a)) = 0] = exp[-u n({w; M_(w) -
X+ coincides with [X+(t); 0 < t < A+(e) = inf{u; +(u) > e}] where e is an exponential holding time independent of the process X+ with mean
-a. The process defined in the second way is known as the elastic barrier Brownian motion with parameter y = -a/(1 - a), (cf. Example IV-5.5) and the above shows that it can also be constructed from a Brownian motion X(t) by a time change as in the first way. Let us give one more application of the above consideration. Let t > 0 be fixed and set u = T+(t). Then with probability one, X(u) = X+(t) > 0 and X(u) is contained in the excursion p[6(u)] = p+[O+(t)] E
STOCHASTIC CALCULUS
134
V". Hence 0(u) = 0+(t). Now it is obvious that
t - A+(¢+(t) -) = u - A(O(u)-) = u - A(O+(t) -) and hence
u = t - A+(0+(t) -) + = t + A-(O+(t) -)
A(c+(t) -)
= I + A-(O+(t)).
Noting the independence of X+ and X- and the fact that A-(t) is a one-sided stable process with E[exp(-1%A-(t))] = exp[-tyr())] where Yi(A) =J 0(1 - e z")n-({w; a(w) E du})
we obtain the following formula due to Williams [ 235 ], (cf. McKean [114]):
E[exp (-,IT+(t)) I X+(s); 0 < s < oo]
= exp[-2t -
2.1+(t)].
This formula can be used to prove the aresine law:
P[B+(T) < t] = -!sin-' J T,
(4.28)
0
T.
Indeed
E[exp(-A-c+(t))] = E[exp(-fit
-
and, noting 0+(t) and max X(s) are equivalent in law (Corollary of os,sgr Theorem 4.2), this is equal to 2e zt ('o e' 2
= 2e za(2nt)-112
I2exprof
oe
2t
]dx
z"(2nu3)-1i2x exp(_ -,j.-)du }
x expHTF)dx
=
f {t(u - t)3}-1J2(7cu)-'ezut(u - t)du r
from which (4.28) is easily obtained.
SOME APPLICATIONS OF STOCHASTIC CALCULUS
135
In the remainder of this section, we shall present, within our framework of Brownian excursions, several results of Williams on decomposi-
tions of Brownian paths, especially on a beautiful description of the Brownian excursion law n or n+ = n 1 .-+ . For w E Wl+, we define w E=- 7/"+ by
W(t) - I w(a(w) - t) l
0
for 0 < t < a(w) fort ? ai(w).
v
Clearly a(w) = a(w).
Lemma 4.3. n is invariant under the mapping: w - w. + n {w; a(w) = Tj is invariant under the Proof. P' on mapping: w --- w (cf. Example IV-8.4). Since n+(B) = f oPT(B n {a(w) = T}) (2xxT3) _fdT,
B E B(V+),
the assertion is obvious.
For a continuous path w E C([0, oo) - R) and t > 0, define wt and w, in C([0, oo) -- R) by w, (u) = w(t + u) and w, (u) = w(t A u).
Define aa(w), a e R, by aa(w) = inf{t; w(t) = a). In the following, a Brownian motion X(t) with X(0) = a is denoted by BM" and its probability law on C([0, cc) -- R) is denoted by Pa. A Bessel diffusion process Y(t) with index 3 (i.e., the radial process of a three-dimensional Brownian motion, cf. Example IV-8.3) such that Y(0) = a (a ? 0) is
denoted by BES°(3) and its probability law on C([0, cc) - R.,) is denoted by Q. where R+ = [0, cc). BES°(3), a ? 0, is a diffusion process on [0, oo) with transition probability q(t, x, y)dy where (4.29)
1 p°(t, x, y)y
for t, x, y > 0
K+(t, y)y
for t, y > 0, x = 0
q(t, x, y) = x
with p°(t, x, y) and K+(t; y) given as above.
Lemma 4.4. (i) If a > 0, (4.30)
n+({w; w;Q(w) E B1, w,. (.) E B2} 1aa < cc) = Qo[wQ° E B)}P4[w;o E B2}
STOCHASTIC CALCULUS
136
for any Borel subsets B1, B2 in C([0, oo) --- R).
(ii) If 0 < b < a, (4.31)
B2)
Pb[{w; W--. (w) E B19 w a(w)
IQa < a'o]
= Qb[w;a E B1]Po[w E B2]
for any Borel subsets B1i B2 in C([0, oo) -- R). Proof. (i) We know n+({w; aa(w) < oo }) = 1/a. Also by (4.12), n+
has the following property: for any Borel subsets B1, B2 in CQo, co) - R+) n+({wr E B1, o(w) > t, wj E B2))
= E'+[E w(r)[wvo(W.) E B2]: wt E B1, a(w) > t] where E""' (EFa) denotes the integration by n+ (resp. Pa). Hence if f1i fz, . , fm and g1, g2, ... , ga are bounded Borel functions on R+,
O
. .
. fm(w(tm))Iftm
x g1(w(Qa + sl))g2(w(oa + s2)) . E'+[f1(w(t1))f2(w(t2)) .
x
EPw1m)[g1(w(oa
.
.
gn(w(o'+
. fm(w(tm)) IIOnaa>rm)
.
+ s1))gz(w(a + s2)) .. .
gn(w(Qa +
E'+ (f1(w(t1))f2(w(t2))
x
.
.
. fm(W (tm))I (aAa >rm,
Epw'(tm)[[E°a[g,(w(s1))g2(w(s2)) /... ga(w(Sa))Iua
= E'+U 1(w(t1))f2(w(t2)) .
.
. fm/(W((/tm))I (oAQ >tm) Pw (tm) [Qa < 60]]
x E1a[g1(w(sl))g2(w(s2)) .
E" (fl(w(t1))f2(w(t2))
.
.
x EPa[gl(w(s1))g2(w(s2)) .
.
.
. f,a(w(tm))I(oa>tm,w(tm)] .
. ga(w(sn))]I,ta
and by using (4.32) below, this is equal to E4o[i(w(t1))f2(w(t2)) .. . fm(w(tm))I(aa>rm)] x E1a[g1(w(s1))g2(w(s2)) ...
a.
Then the proof of (i) will be completed. So we prove the following: for any Borel subset B in C([0, oo) --- R+), (4.32)
E'+[w(t)I(wl Eg)] = Qo [w, E B].
SOME APPLICATIONS OF STOCHASTIC CALCULUS
For, if 0 <s, <s2 < ... <sn E"+[g1(w(s1))g2(w(s2))
f0mf0r
.
t,
... gn(w(sn))w(t)]
.. f K+(sl, x1)P°(s2 - s1, x1, x2) 0 o
x P°(s - sn-1, xn-1) xn) x gl(xl)g2(x2) . . . gn-1(x,,_1)gn(xn)xndxldx2 .
=
ff 0
137
.
0
f
o
q(sl, 0) xl)q(s2-s1, x1, x2) .
x q(sn - sn-1, xn-1, x,,) .7 .7 x g1(x1)g2(x2) . . . g,(xn)dx1dx2. EQ0[gl(w(s1))g2(w(s2)) .
.
.
.
.
.
dxn
.
,V
. dxn
. gn(w(s,))]
and this proves (4.32). The assertions (ii) can be proved in the same way as (i).
Lemma 4.5. Let a > 0 and set, for w E C([0, oo) --- R), (4.33)
1,°(w) = sup{t; t < Q°(w), w(t) = a)
(4.34)
1,,(w) = sup It; w(t) = a) .
Then (4.35)
P,,[w7o(w) E B] = Q,[wla(w) E B]
for any Borel set B in C([0, oo) -- R+).
Proof. It is sufficient to prove that for bounded Borel functions . . , fn on (0, oo) and 0 < tl < t2 < .. . < tn,
fi, f2,
EP°[ II.f (w(tJ))J(,o>,d] = EQa[ II f
(4.36)
J-1
We have
the left hand side of (4.36) n
=e[ II f (w(tj))I(Q,(wtn)<,°(wn), EP°[ JII f (w(tJ))g(w(tn))Iltn
J-1
STOCHASTIC CALCULUS
138
=
o
. .
fo
. J °fII{P°(1J - ti-1, xJ-1, xJ)f (xJ)}g(xn)dx1dx2
dxn
a and
where to
g(x) = P= [6° < Q°] = x
a
a
x > 0.
Also
the right hand side of (4.36) = EQa[ III f (w(tJ))1 a(w)<_,] n
= EQa[ lI f (w(t,))h(w(tn))] J=1
J of 0
f JI{q(tt - tJ-1, xJ-I, xx)J(x1)}h(xn)dxldx2 . o
.
.
dxn
where to = 0, x° = a and h(x)=Qx[6°<00]=
x
a
a,
x>O.
The equality (4.36) is now easily verified by noting q(t, x, y) P°(t, X, Y)Y/x.
Corollary 1 (Williams [179]). For given a > 0, let {X(t)} be BM° and { Y(t)} be BES°(3). Then {X(cr - t), 0 < t S 010x) is equivalent in law to {Y(t), 0:9 t S 1; } where
Qo = inf{t; X(t) = 0} and to = sup{t; Y(t) = a}.
Proof. From (4.30), we see that {w(t); 0:s: t < o(w)} is decomposed, under n''(
I ca < oo), into mutually independent parts { w(t),
0 < t < Qa} and {w(t + va), 0 < t < a - va}, the former being equivalent in law to { Y(t), 0 < t < o- } where as = inf { t; Y(t) = a} and the latter to {X(t), 0 < t < a§ j. Hence, the part {w(t), 0 < t < la(w)} (1°(w) is defined by (4.34)) is decomposed, under n*( I o , < oo), into mutually independent parts {w(t), 0 < t < o,} and {w(t + 6a), 0 < t < la - Qa}, and latter being equivalent in law to {Z(t); 0 < t < 1.01 where Z(t) is BES°(3) by virtue of (4.35). Hence we can conclude that the part { w(t); 0 < t < /.I is, under n( - I a. < oo), equivalent to { Y(t); 0 < t< Ill. Since {Qa(w) < oo} ={oa(W) < co}, it follows from
SOME APPLICATIONS OF STOCHASTIC CALCULUS
139
Lemma 4.3 that {w(t), 0 < t < 1,()7v)} is, under "n( I a, < oo), also equivalent to { Y(t), 0 < t < 11}. But it is obvious that 1,(W) = a(w) -
a,(w), w(t) = w(a(w) - t) and we know that {w(t + a,), 0 < t < a-a,} is, under n( - Ia, < oo), equivalent to {X(t), 0 < t < ao}. This clearly implies the assertion. Let p+ = p I,,+ be the above point process on 7+ and define X+(t) and A+(t) by (4.26). If we define B+(t) by (4.27) then we know
that B+(t) is BM° and hence X(t) = -B+(t) is also BM°; that is, the process X defined by (4.37)
X(t) = 0+(t) - [p+(O+(t))7(t - A+(O+(t) - ))
_ 0+(t) - X+(t)
is BM° and X(s). c+(t) = max °5r
We define a continuous process Y(t) by (4.38)
Y(t) _ c+(t) + [p+(O+(t))] (t - A+(O+(t) - ))
= 2 max X(s) - X(t). °5s5t
We fix a > 0 and define a point process p on V+ by
(0, a); a-$
DD = {s e
D,+} U {se (a,oo); seD,+} and
( p+(a - s) p(s) = jl p+(s)
for s e DD n (0, a) for s E D. n (a, oo).
By Lemma 4.3, it is immediately seen that p has the same law as p+. If we define k(t) from p in the same way as Y(t) is defined from p+ by
(4.38), then it is immediately seen that A+(a) = A(a) : = =E a[p(s)l,
A+(a) = inf{t; X(t) = a},
A(a) = sup{t; Y(t) =a I and
k(t) = a - X+(A+(a) - t)
for 0 < t < A+(a).
Therefore, Corollary 1 implies that { Y(t), 0 < t < 1; } is equivalent to
{Z(t), 0 < t < If) where Z is BES°(3). Letting a T oo, we see that Y(t) defined by (4.38) is actually BES°(3). Thus we have the following:
STOCHASTIC CALCULUS
140
Corollary 2 (Pitman [141)). If X(t) is BM°, X(t) is BES°(3).
Y(t) = 2 max X(s) 053-.S;`
Here, we present a useful result for an (,F)-stationary Poisson point process p on a general state space (X, (X)) with characteristic measure
n(dx). Let A E . ([O, oo)) x .q(X) and assume that f IA(s, x)n(dx) = C,,(s) < and
for all t > 0
f 'o C(s)ds < oo but f OC,,(s)ds = oo.
Define (4.39)
0 = inf{s E D,; (s, p(s)) E Al.
Then 0 is an (.)-stopping time and P[0 < oo] = 1 because
P[0> t]=P[NN({(s,x)EA; s
f ' C,,(s)ds] -- 0
as
0
t - oo.
Lemma 4.6. (i) The point process p* on X defined by D
{t;t +
0 E Dp} and p* (t) = p(t + 0), t E Dp*, is independent of Y. (ii) ._ and p(O) are conditionally independent given 0: To be more (X)-
precise, if H(w) is bounded-measurable and G(x) is bounded measurable, then we have, for I > 0, E[HG[P(0)]e ze]
x)n(dx)}ds = f mex,rE[I,,,,Ha] f G(x)I4(s, x 0
where H, is a bounded predictable process such that H = Ho (cf. [15]).
From this formula, we can conclude that (4.40)
P[HI 0= s] =
E[I,OZ,,H,]eS°°Acu>a
SOME APPLICATIONS OF STOCHASTIC CALCULUS
E[G[P(0)119 = s] = f
=f x
(4.41)
141
x)n(dx) { C4(s) } -' XG(x)I4(s, x)n(dx){ f
a(s, x)n(dx)}-1
X
.
Proof. (i) This is essentially established in Theorem 11-6.5, because
(Y )-stationary Poisson point process p is always independent of .9. (ii) We have E[H G[P(0)]e x9
= E[ .ED,, E H,G[P(s)]ea'IA(s,P(s))] s$B r
= EI If B+ f H,G(x)ex°I4(s, x)N, (dsdx)] x
J
ox
L
= E[f'e A,H, f G(x)I,, (s, x)n(dx)ds] [ E[I,B?, H,f G(x)I4(s, x)n(dx)}ds = fWe1 o x
and hence the result. Example 4.1. Let p be the above Poisson point process of Brownian
excursions given as an (Y )-stationary Poisson point process on and construct BM° X by (4.17). Let a > 0 and define A E .6((0, co)) x
.6(') by
A = (0, cc) x { w; M(w) ? a) where M(w) = max w(s). If 9 is defined by (4.39), then 0 = inf {s
D,; M[p(s)] > a)
and hence, A(0--) < QQ < A(O) and
A(9-) =
sup { t;
t < u , X(t) = 0) : =
l'.
If G(w) = O(wa), where 0 is a bounded Borel function on C([0, cc) -.R+), we have E[G(p (0)) 10 = s] = E[G(P(0))]
f =f
=
n(dw){n({M(w) > a})} 0(wva)n(dwI
0 < cc)
STOCHASTIC CALCULUS
142
=
L
CO, -) -> a+)
iP(w'°)Qo(dw)
by (4.30). Thus we obtained the following: (4.42) If X is BM° and Y is BES°(3), then {X(10' + t), 0:!9 t< 6a 1o x} is equivalent in law to { Y(t), 0 < t < u }. Furthermore, {X(10 X
+ t), 0 < t < a - 10X} is independent of {X(t), 0 < t S l0X}. The second assertion follows from the fact that {X(t), 0 < t < l0X} is 9_-measurable and E[G[p(9)] 10 = s] is independent of s. Now we discuss the first decomposition result of Brownian paths Take the (.9,)-point process p+ = p I ,,.+ and, for a > 0, define BMa X by
due to Williams [ 179].
X(t) = a + O+(t) - p+[O+(t)](t - A+(9i+(t)-)) Then
a+0+(t)=maxX(u). Osust
Define A E R((0, oo)) x .°.dl(''+) by
A= {(s, w); a+s-M[w]<0}. If 0 is defined by (4.39), it is easy to see that A+(0-) < cr < A+(B),
a + 0 = max X(t) 0:!9t--50X 0
A'(0)
a
r=A"(B--)
0
a
Fig. 1
a+0
SOME APPLICATIONS OF STOCHASTIC CALCULUS
143
and y = A+(0-) is the unique time point in [0, u] which attains this maximum. Then
p(0)(t) = a + 0 - X(y + t) and by (4.41) E[G[p(0)] 10= s]
=fr+ G(w)!,
, a°+,,n+(dw) {
n+({ M(w) > a + s})}-'
= fr+ G(w)n+(dw I a°+r < oo). By (4.30) we can conclude that, given 0, {X(y + t), 0 < t < oo - y}
is equivalent in law to {a + 0 - Y(t), 0< t< q.11+0} where Y is BES°(3). But, if 0 is given, {a + 0 - Y(t), 0 < t < o+e} is equivalent to { Y(Q+e - t), 0 < t < a+9} as is easily verified by Corollary 1 above and (4.42).
Next, we consider the part (X(t), 0 < t < y}. This part is_ -measurable and, for a bounded Borel function 0 on C([0, co) R+), we set H = 0(XY) where XT is defined by Xy (t) = X(y A t), t > 0.
Then H = He where H. is a predictable process defined by Hs = 0 (X;+(2_)). By (4.40), we have E[O(Xy) I O= s]
= E[!(
(X;+(_))]{Efreas,]}-
,)
= E[O(X;x +) I oa+ <
]
since A+(s-)=A +(s) = al, for each fixed s > 0. By (4.31) we can conclude that, given 0, {X(t), 0 < t < y} is equivalent to { Y(t), 0 < t < o+e } where Y is BES°(s). Note that, given 0, p(0) and 9 _ are independ-
ent; in particular p(O) and {X(t), 0 < t < y} are independent. Finally, we have P [O E ds] _ { exp [- f oCA(u)du] } C4(s)ds =
a ds (a+s)2
because
C,,(s)=n+({M(w)> a+s})= a+ s. Now we can summarize the above to obtain the following result of Williams: On a probability space, we set up the following three independent BES°(3) Y', BES°(3) Yz and a random variable 0 E random elements:
STOCHASTIC CALCULUS
144
(0, oo) distributed by
P(@ E ds) _
(a+ s)2d
Define
_ Y'(t)
At) -
,
Y2 Q+t), 111 ' (Qae + co 2
Y1
0
Then {Z(t), 0 < t < a; 'e + o B} is equivalent in law to {X(t), 0 < t < o- ox) where X is BMa. This result, combined with (4.30), yields immediately the following
description of Brownian excursion law n+ due to Williams: Let Y' and Y2 be mutually independent BES°(3) and, for a > 0, set
Y'(t) X°(t) = Y2(o'a' + o
for 0
- t)
for as" < t < Qa1 + aa2
for t>Qa'+az
0
Let R. be the law on 9Y+ of Xa. Then n+ = f W (Ra / a2)da. 0
1
a
0
Fig. 2
SOME APPLICATIONS OF STOCHASTIC CALCULUS
145
Next, we discuss the second decomposition result of Brownian paths due to Williams [179]. We start with BES°(3) Y defined by (4.38)
from the (,9 )-point process p+ on
V+.
Let a > 0 and define A E
.,d((0, oo))x.,a9(1Y+)by A = {(s, w); s + M(w) > a}. If 0 is defined by
(4.39), it is easy to see that A+(9-) < as < A+(B+), 0 = min Y(t) dQ StGla
and y = A+(0) is the unique time point in (a;, la') which attains this minimum. Set X(t) = a - Y(1; - t), 0 < t < 1,,Y. Then la = aa, 1; ca' = to X: = sup{t; t < a., X(t) = 0) and, by Corollary 1 above X is a part of a BM° up to its hitting to a. The process Y* defined by Y*(t) = Y(t + y) - 9 is independent of -9-9 because Y* can be constructed from the point process p* of Lemma 4.6 (i) in the same way as Y from
p+. Then, under the conditional probability P( IJ), { Y(t + y), 0 < t
< 1; - y) can be expressed as {9 +Z(t), 0 < t < eel where Z is BES°(3) independent of 0. Since Y(t + y) = a - X(aa - t - y), 0 < t < of - y, we see by Corollary 1 again that {X(t), 0 < t < as - y}, under P( - I9), can be expressed as {W(t), 0 < t < o e} where W is BM° independent of 0. Also we know by (4.42) that {X(1g"X + t), 0 < t< as is independent of {X(t), 0 < t < 10a,'} and is equivalent in law to {Z(t), 0 < t :!!g as } where Z is BES°(3).
Finally we study the part {X(aa - y + t), 0 < t < lg X - as + y}.
Noting that lg" - as+y=y - aa and X(aa-y+t)=Y(y-t), 0
CA(s)=n+({s+M(w)> a})= a
1
s,
0<s
and hence
P[0 E ds]
exp(- f :CA(u)du) }C4(s) = s
and
E[G(P+(O))10 = s]
= f,.+G(w)1(M(w).za-s)n+(dw){n+({s + M(w) > a}))-' = E"+[G(w)laa_,, < oo]. Hence, by (4.30),
We can
146
STOCHASTIC CALCULUS E[(P([p+(O)l 0'a_8[ ,+(e)3) 10
= s]
= En+[0(w a_) I o'a-.s < co] = Era-s[P(wo°))
for a bounded Borel function 0 on C([0, oo) -- R+). This implies that
{ Y(a + t), 0 < t < y - o'a } is equivalent in law to {a - W(t), 0 <
t < 7 B} where W is BM° and 0 E (0, a) is independent of W and uniformly distributed on (0, a). This, combined with Corollary 1 above,
implies that {X(Q' - y + t), 0 < t < lo'r - of + y} is equivalent in law to {a - 0 - Z(t), 0 < t < l; a} where Z is BES°(3) independent of 0. We can now summarize the above to obtain the following result of Williams: On a probability space, we set up four independent random elements: BM° X, BES°(3) Y' and Y2, a random variable S uniformly distributed on (0, a). Define
Z(t) Then
X(t)
for 0 < t , ,
6- Y'(t -aa) Y2(t - QX la)
for Of
for 0, + la' < t < aX + la' + cr
{Z(t), 0 < t < aX + is + Qa2} is equivalent in law to
o
.
{ X(t),
4.4. Some limit theorems for occupation times of Brownian motion. Let X = (X(t)) be a one-dimensional Brownian motion such that X(0) = 0. Letf(x) be a continuous function with compact support. We are interested in the problem of finding the limit process of
f oe f(X(s))ds
as A -- 00
where u(A) is a some normalizing function. The situation will be quite different accordingly as f = fRlf(x)dx * 0
or I= 0.
Theorem 4.4.* (i) If f
0, the family of continuous stochastic pro-
cesses f AtJ (X(s))ds,
A > 0,
* Cf. Papanicolaou-Stroock-Varadhan [137]. The corresponding problem for a twodimensional Brownian motion was discussed by Kasahara and Kotani [82].
SOME APPLICATIONS OF STOCHASTIC CALCULUS
147
converges in the sense of law on the space of continuous functions to the continuous process t - 2f0(t, 0) as A --- co where 0(t, 0) is the local time of X(t) at zero.
(ii) If f = 0 but f is not identically zero, the family of continuous stochastic processes
A>0,
t I- T1 /-4 f 0 .f(X(s))dg,
converges in the sense of law on the space of continuous functions to the
continuous process t - 2,1B(O(t, 0)) as 2 -- oo where 0(t, 0) is the local time of X(t) at zero and B(t) is another Brownian motion independent of X(t) with B(0) = 0 and _ --- fR,f R1 I x - y I f(x) f(y)dxdy.
Proof. The proof of (i) is easy. Indeed, by the scaling property of the Brownian motion, {X(t)} q {,% 112X(1t)} for each A > 0. From this it is
easy to conclude that {¢(t, a)} s {¢(1t,
Jo
.f(X(s))ds =
2J
Rl
for each A > 0. Then
LlO(At, a).f(a)da
¢(t, a/.1/Tf(a)da.
But 2f R,c$(t, a/I-A )f(a)da converges, as A - oo, to 20(t, O) f Rl f(a)da uniformly in t on each bounded interval a.s. Thus the assertion of (i) is proved. To prove (ii), set
F(x) = f x f(y)dy and
G(x) = f x F(y)dy. 0
Sincef(x) is of compact support and null charged, i.e., f = 0, F(x) is of compact support and G(x) is bounded. By Ito's formula,
G(X(t)) = f' F(X(s))dX(s) + 0
Therefore,
2
f f(X(s))ds.
STOCHASTIC CALCULUS
148
f .f(X(s))ds = '4 G(X(2t)) - i f F(X(s))dX(s) 4
o
11 + I2.
Clearly II - 0 uniformly in t as 2 - oo a.s. Set
- ,2 f F(X(s))dX(s).
MA(t)
4
For each 2 > 0, MA(t) is a continuous martingale such that <Mz)(t)
= A4z fof F(X(s))2ds,
and by the result (i), this family of processes converges in law to the pro-
cess 8F2O(t,0) as 2 -- oo. Define a family of three-dimensional continuous processes ZA(t), 2 > 0, by
ZI(t) _ (Ma(t), <Ma)(t)s T1iz X(2t))
First we show that the family of laws of Z. is tight. For this it is clearly sufficient to show that the family of laws of Ma(t) is tight. By Theorem 3.1,
we have for s < t, that E([MM(t) - Ma(s)]6)
< const. E([<M >(t) - <Mx)(s)]3) < const. A-112 f
as AS du f as u
dv asf ' dw Rf dx f RI
exp(_ x2) exp(_ (Y - x)2) exp( x
2n(v(v w))
2irww
< const. A-312
< const.
(t -
at
Es as
du
u
v
a,
AS
f dv f
dw
dy f dz RI
(z - Y)z)
2ir(u u v)v) F(x)zF(Y)z F(z)z %/2irw 27r(v
1
- w) 127r(u - v)
s)312.
Hence, we can conclude from Theorem 1-4.3 that the laws of Ma(t) constitute a tight family. Next we show that the set of limit points of laws of Za consists of a single point. Let Za, -- Z in law for some subsequence
SOME APPLICATIONS OF STOCHASTIC CALCULUS
149
and Z = (Z1(t), Z2(t), Z3(t)). Then Z3(t) is a Brownian motion and Z2(t) = 8F20(t, 0) where ¢(t, 0) is the local time of Z,(t) at zero. By Theorem
1-4.2, we may assume that Zxt(t) -- Z(t) uniformly on each compact interval a.s. Then it is easy to prove that (Z1, Z3) is a system of continuous martingales such that , = Z2(t) = 8F 0(t, 0), 1 = t and
WA,
Z3>
A1/2
zim (-
2 a J o` F(X(s))ds)
=0
since
p-17f of
F(X(s))ds --- 2F0(t, 0) = 0 in law.
By Theorem 11-7.3, we can conclude that Z1(t) = where B(t) is a Brownian motion with B(0) = 0 which is independent of Z,(t). This shows that the law of Z(t) is uniquely determined. In particular it has been shown that Mt(t) converges in law to the process Z,(t). The assertion of (ii) now follows if we notice that = 2F2 but this is immediate since
- f xj xl I x - y I f(x)f(y)dxdy = -2 2f f
xY (J
y
J J x>Y
(x - y)f(x)f(y)dxdy
dz)f(x)f(y)dxdy = -2 f f f
-W
= 2f dz f -f(x)dx f _f(y)dy = 2f
x>=>yf(x)f(y)dxdydz
(fs
f(y)dy)2dz
= 2f'- F(z)2dz. 5. Exponential martingales
Let (Q, 9; P) and be given as in Section 1 and consider a quasimartingale X(t) such that X(0) = 0. We denote by Mx the martingale part of X(t). Then the exponential quasimartingale defined by (5.1)
M(t) = exp{Xt - 2 <Mx>t}
is the unique solution of the stochastic differential equation
STOCHASTIC CALCULUS
150
IdM = M0 = 1, as is easily seen by Ito's formula and Theorem 2.1. Define the Hermite polynomials L-1)n
exp exp ( 2t) a n
Hjt, x]
(-
2t )'
n > 0,
that is, (5.3)
E ynH [t,x] = exp(yx - y2)
for every y e R.
In particular, H0[t, x] = 1,
HI[t,x]=x,
x3
- xt
6
T
HZ[t, x} H3[t, x] =
2 ,
For each . E R set z
MA(t) = exp{{Xt - 2 <Mx>t} = exp L't -
(5.4)
<Max>t
Thus M2(t) is the unique solution of
dM2(t) = 2Mx(t) dX(t) {Mx(0)
= I.
By (5.3),
MA(t) = E 2nHn[<Mx),, X,] n-0
E n° nlt),
where we set Xn(t) = Hn[<Mx>,, X,]. Let N be a positive integer. Letting Ax be the bounded variation part of X(t), we set rt(t) = max{X(t), <Mx>(t), 1Ax1(t)}, aN = inf{t; i1(t) > N}
0 < t < oo,
EXPONENTIAL MARTINGALES
151
where 1A,1(t) is the variation of the mapping: s - As(s) on the interval [0, t]. We note that HH[t, x], n = 0, 1,2,
.
. can be expressed as
HH[t, x] =k-0 EE an(k, J)t jxk j-0
Then we can easily verify by induction on n that n
n-k
E Ej-0 I an(k, j) I<< 1. k-0
This implies that I Hn[t, x] I < Nn, n = 0,1,2, and I x I < N. Hence
... ,
provided t < N
for 0 < t < QN.
IXn(t)I < Nn
Then it is easy to see by (5.5) that for every A satisfying 0 < A < MN(t A QN) = 1+ A
frnoN 0
Mx(s)dX(s)
= 1 + A f ""v(E AnZn(s))dX(s) n-0
0
1+Af
oN(E n-0
0 = 1 + E An+1 =1
AnX (s))dMx(s) + A f 'AVN(
o^°NXn(S)dMX(S) I
0
-0 An
n(s))dAx(s)
+ E An+l fo °NZn(s)dAx(s)
+E An+1f t/.uNXn(S)dX(S) n-0
0
E AnZn(t A 0 N).
X-0
Then we have
1 and Xn(t A 7N) = f
X0(t A OWN)
f o r n = 0, 1,2,
tAON o
°` n-1(S)dX(S)
.. ..Since
lim O'N = 00,
we deduce from this that X0(t) =-I and Z .(t) = $ Za-1(s)dX(s) for n = 1, 2, .... Hence
STOCHASTIC CALCULUS
152
X°(t)
= f ' dX(t,) J
"42"42) ... f 'n-' dX(t
)
Thus we have proven the following result ([64] and [113)).
Theorem 5.1. (5.6)
Jo
dX(t,)J, dX(t2) ...
ro -' dX(tn) J
=
X j,
n = 1,2, .. .
Now we will investigate some properties of the exponential quasimartingale exp {X, - <Mx>,121.
Theorem 5.2. If X E ..or, then the exponential quasimartingale M. is a continuous local (,,` )-martingale. Furthermore, M, is a supermartingale, and it is a martingale if and only if
E[M,] = I
for every t Z 0.
Proof. Since M, is the unique solution of (5.2)
Mr-1= fM3dX(s)E.4'. Thus, it is easy to see by Fatou's lemma that M, is a supermartingale.
Theorem 5.3. (Novikov [132]). Let X E .4' and set
M,=exp{X,-
2<X>,}.
If (5.7)
E[e""2] < co
for every t Z 0,
then M t Z 0, is a continuous (Y )-martingale, i.e., (5.8)
E[M,j = 1
for every t Z 0.
Proof. By Theorem II-7.2', on an extension (5,0' ) of 0 with a reference family (J' ), there exists an (.')-Brownian motion B = (B(t)) with B(0) = 0 such that X(t) = B(<X>(t)) and <X>(t) is an (.`,)-stopping
EXPONENTIAL MARTINGALES
153
time for each t > 0. Set
a. = inf {t; B(t) < t - a). Then, if a > 0, we have for Z > 0 that (5.9)
E[e-zaaJl
Indeed, setting
u(t, x) = e zre-c+zz-ux we have by Ito's formula that
u(t,B,-t)-u(0,0)= foa(s,B,-s)dB, +f =f Thus t
o
( 'at
- ax + i ax
2)(s, B, -
s)ds
(s, B, - s)dB,. o
a
u(t A 6a, BA,, - t A 6a) is a martingale if a > 0 and hence
E[u(t Aa'°, B,A,, - t AQ°)] = u(0, 0) = 1. Letting t t oo, we have by the bounded convergence theorem that
/ E[u(Q°, B, - a.)] = 1. This proves (5.9). From this we can conclude that E [exp (2 6°) ] = e < oo. Therefore,
E [exp(B(o ) - 2 O'a)] = e °E[exp(2 6°)] = 1. Combining this result with Theorem 5.2 and setting
154
STOCHASTIC CALCULUS
Y(t) = exp(B(aaAt)
- 2 a.At),
we can show that Y(t), t E [0, oo], is a uniformly integrable (,5;',)-martingale. Hence for any (.)-stopping time a, E[exp(B(aa A a) -
as A a)] =
1.
2
in particular, we have 1 = E[exp(B(aa A <X>) - 2 a. A <X>:
exp(-a + -T a-)] +
=
E[Itca>(A-)d
exp(X(t)
-
<Xx)] a
Since E[ItQas<x),,
exp
(-a + 2aa)] < e
aE[exp(2 <X> )]
we have
1 = lim E[Ii,a><x>r, exp (X(t) -
<X>,)]
a-
= E[exp(X(t) -- 2 <X>,)]
2
and this completes the proof.
Remark 5.1. By a slight modification of the proof, Kazamaki [83] proved the following stronger result: if instead of (5.7) we assume that E[exp(X(t)/2)] < co, then the same conclusion (5.8) holds. The following result is an immediate consequence of Theorem 5.3.
Corollary. Let X e .///. If <X>, is locally bounded, i.e., for every t > 0 there exists a positive constant C(t) such that (5.10)
<X), < C(t)
a.s.,
then M, is a continuous (.F)-martingale.
CONFORMAL MARTINGALES
155
6. Conformal martingales Let (Q, J V, P) and
be given as usual.
Definition 6.1. By an n-dimensional conformal martingale (local conformal martingale) with respect to (?,), we mean a C°-valued continuous (. )-adapted process Z, _ (Zl, Z, ... , Z,) such that, writing
Za=X'+
Xa, Y'
(6.1)
Ya, a=1,2,...,n,
(resp. Xa, Y1,1 E dL=
z ba), a = 1, 2,
,n
and their stochastic differentials satisfy
dX" dXa = dYa dY,'
(6.2)
= 1, 2, ... , n.
a,
-dX? dYa
(6.2) implies, in particular,
a=1,2,...,n.
(6.3)
A one-dimensional conformal martingale (local conformal martingale) is simply called a conformal martingale (resp. local conformal martingale). We complexify the spaces .4', .ci 62, d. d -V, d6 ' in an obvious way and define, for Z, = X, + Y, E ..
dZ,=dX,+../=1dY,and dZ,=dX,-VTdY,. It is immediately seen that (6.2) can be equivalently written in the complex form as (6.2)'
dZa dZs = 0,
a, fi = 1, 2, ... , n.
In other words, Z, _ (Z', Z,, .
.
.
,
Z,) is an n-dimensional local
conformal martingale if and only if all Z', Z, `Z,,O, a, ,6 = 1, 2, . are (-F",)-local martingales.
.
.
, n,
Example 6.1. Let B, = (B', B?, ... B,", BI) be 2n-dimensional (.F )-Brownian motion and set Za = B,a-I + B2, a = 1, 2, . . . , n. Then Z, = (Z, Z,2, ... , Z,") is an n-dimensional conformal martingale. Z, is called an n-dimensional complex Brownian motion. Example 6.2. Let (B', Br, B,, B4) be a four-dimensional Brownian motion and let [f be the proper refence family of (Bt, B,2) (i.e.,
STOCHASTIC CALCULUS
156
the filtration generated by the process (B,, B,1)). Let A, be a continuous (,rB'.BZ)-adapted increasing process such that Ao = 0 a.s. Define (Z,, /-1B,4,t
by
Z, =Br +
B,
Z?=B,,+
and
If {,F-,) is the proper refence family of (Zr, Z,) (i.e., the filtration generated by the process (Z;, Z,2)), then it is not difficult to see that Z, is a two-dimensional local conformal martingale with respect to (.7;).
The following two propositions are easily obtained by Doob's optional sampling theorem (Theorem 1-6.11) and Knight's theorem (Theorem 11-7.3').
Proposition 6.1. Let A = (A,) be a continuous (9 )-adapted process
such that Ao = 0 and t - A, is strictly increasing. Furthermore we assume that lim Ar = oo, a. s. r' .91-, «, . Then, for every n-dimenthe time change sional local conformal martingale Z, with respect to (. process 2, = Z, (r) is an n-dimensional local conformal martingale with
Set C(t) = inf { u; A. > t) and
respect to (F,). Proposition 6.2. Let Zr = (Z;, Z,2, . . . , Zf) be an n-dimensional local conformal martingale such that dZ" dZs = 0, a, 6 = 1, 2, ... , n, a #,6. Then there exists an n-dimensional complex Brownian motion C(t) = (C'(t), C2(t), . . . , C"(t)) such that Za
= Ca(,),
a = 1, 2, .
.
.
, n,
where , is defined to be the common processes <Xa>, = (Ya>,.
Let AX 1, x2, ... , x", y', y2, ... , y") be a CZ-function on RZ^. By setting za = xa + ya, it can be regarded as a C2-function f(z', z2, . . , z") on C. We introduce the differential operator in the usual way : aa
2 (axa
- aya
and
aaa = 2 (-w +
V
aya ).
Then the Ito formula for 2n-dimensional continuous semimartingale
C, = (Xl, X?, ...
, X,-, Y; Y,2
df(C,) =aE(
.
.
.
, Y,-) :
cr)dXa+ a-y Ct)dY')
CONFORMAL MARTINGALES (6.4)
+
-xaaX «.± (a a
p
2
dY°
(C,)dXr
+2 a -f a
157
dYfl+
Yf ((r)dYa dY?)
a
can be rewritten, in the complex form, as follows. Setting
Z:=(Z,,Z?,
. .
Zf=X, iY«,
.
df(Z,) = E (az Z:)dZg + a (Z,)d2") (6.4)'
+
,(a as
2 «.
Zr)dZa
dZF + 2 aaaa Zt)dZ' d2!
Z
+ az aZr)dZ,
d2F
.
If Z, = (Z,, Z2, . .. , Zr) is an n-dimensional local conformal martingale, then dZ« dZ8 = 0 and hence d2,« dZf = dZ d = 0. Therefore (6.4)' becomes a simple form: df(Z,) = (6.5)
(Zr)dZa +
\ aZ«
+ «E aaaa (Zt)dz«
f (Z,)-d2 r)
a
d2f.
An important consequence of (6.5) is the following: If f(z) is holomorphic, i.e.,
of = 0,
a=1,2,...,n,
we have (6.6)
df(Z,) = «E a (Zr)dz .
In particular, f(Z,) is a local martingale. Furthermore, it is a local conformal martingale, because f(z)2 being also holomorphic, f(Z,)2 is
also a local martingale. More generally, let f = (f', f2, ... , fm): C° ---- Cm, be holomorphic, i.e.,
=0, of' az« Then
i=1,2,...,m
and
a=1,2,...,n.
STOCHASTIC CALCULUS
158
df `(Z,) = QE aZ (Zr)dZa ,
i = 1, 2, ... , m
and
d((f`'f)(Z,)) = } a a 'f') (Z,)dZ",
i,1 = 1, 2a-1
.
.
.
m
are local martingales and hence f(Z,) = (f'(Z), f2(Z,), ... , f°(Z,)) is an m-dimensional local conformal martingale. Thus we obtain
Theorem 6.3. Let Z, _ (Z', Z,2, ... , Zl) be an n-dimensional local conformal martingale and f = (f', f 2, ... , f): C° - Cm be holomorphic. Then
f(Z,) =
(f'(Z,), f2(Z=),
... , f'"(Z,))
is an m-dimensional local conformal martingale.
Corollary. Let Z, = (Z', Z?, .
. . ,
Zf) be a Cn-valued continuous
(,P' )-adapted process. Z, is an n-dimensional local conformal martingale
with respect to (,9) if and only if for every holomorphic function f.Cn --- C, f(Z,) is a local (.Y)-martingale.
Indeed, "if" part follows by taking f = z" and f = z"zp, a, f _ 1, 2, .
. .
, n. "only if" part follows at once from Theorem 6.3.
Remark 6.1. It Z, = (Z', Z,2,
. . . ,
ZZ) satisfies that Z, E D for
all t > 0, a.s., where D is a domain in Cn, then the above results
remain valid for every J" D --- C'" which is holomorphic in D.
CHAPTER IV
Stochastic Differential Equations
1. Definition of solutions Let Rd be the d-dimensional Euclidean space and let Wd = C([O, oo) Rd) be the space of all continuous functions w defined on [O,oo) with values in Rd. For w,, w2 E Wd, let p(w., w2) = E 2-k( max I m(t) - wz(t) I A 1), O5t
where I denotes the Euclidean metric in Rd (see Chapter I, Section 4). Wd is a complete separable metric space under this metric p. Let (Wd) be the topological a-field on Wd and . ,(Wd) be the sub-a-field of , (Wd)
generated by w(s), 0 < s< t. In other words, .,dl,(Wd) is the inverse afield pp'[.(Wd)] of.'(Wd) under the mapping p,: Wd - Wd defined by (ptw)(s) = w(tAs).
Definition 1.1. We shall denote by Vd.r the set of all functions a(t, w) : [0, co) x Wd - Rd®Rr such that (i) it is .'([O,oo)) x ;q( Wd)/ q(Rd©Rr)-measurable, and (ii) for each t E [0, co), Wd D w a(t, w) E Rd®Rr is .4,(Wd)/ ll(Rd(DRr)-measurable.
Here we denote by Rd®R' the totality of real d x r matrices ; ,9(Rd(& R') is the topological v-field on Rd®Rr obtained by identifying Rd x®R( with dr-dimensional Euclidean space.
We shall denote the (4 j)-entry of the matrix a(t,w) by ajl(t,w),
i= 1, 2, ... ,d,j= 1, 2, ... , r. Suppose we are given a E.sid-r and 6
Consider the fol-
lowing stochastic differential equation for a d-dimensional continuous process X=(X(t)),Z0: 159
STOCHASTIC DIFFERENTIAL EQUATIONS
160
i = 1, 2,
dXi _ E a4(t,X)dB'(t) -I- ,8'(t,X)dt
(1.1)
/®1
,d
or sometimes simply written as (1.1)'
dX, = a(t,X)dB(t) + fl(t,X)dt.
A precise formulation is as follows.
Definition 1.2. Let a = (aj(t,w)) and fl = (flt(t,w)) be given. By a solution of the equation (1,1), we mean a d-dimensional continuous stochastic process X=(X(t)),z0 defined on a probability space (Q,-r,P) with a reference family such that (i) there exists an r-dimensional (.F)-Brownian motion*' B = (B(t)) with B(O) = 0 a.s.;
(ii) X=(X(t)) is a d-dimensional continuous process adapted to (_971,0, i.e., X is a mapping: wEQ
X(co) E Wd such that, for each t E
[0, oo), it is ,/.9,(Wd)-measurable; (iii) the family of adapted processes 05(t, co) and v'(t, co) defined by IfiXt, co) = ai(t,X(w)) and
vv`(t, co) = ,l`(t,X(w))
belong to the spaces 2' *2 and Y l- respectively, where 2' °° is the set of all measurable (_0";)-adapted processes P such that for every t > 0, S
l W(s,co)Ids < c a.s.*' (iv) with probability one, X(t)=(X'(t),X2(t),
=(B'(t), B2(t), (1.2)
. . .
... ,Xd(t)) and B(t)
,B'(t)) satisfy '
X1(t) - X1(0) = E f' ai(s,X)dB'(s) + f fl`(s,X)ds, -1
0
0
i=1,2,...,d, where the integral by dB'(s) is Ito's stochastic integral as defined in Chapter II, Section 1.
In equation (1.2), the first term on the right-hand side is called the martingale term and the second term is called the drift term. To emphasize the particular role of the (Y;)-Brownian motion B = Cf. Chapter I, Section 7. *2 Cf. Chapter IT, Definition 1.6. *3 Y,
are identified if f o I''(s, w)-'!1'(s, w) I ds = 0 for every t Z 0 a.s.
DEFINITION OF SOLUTIONS
161
(B(t)) in Definition 1.2, we call X =(X(t)) a solution of (1.1) with the Brownian motion B = (B(t)), or sometimes we call the pair (X,B) itself a solution of (1.1).
Remark 1.1. The condition (iii) of Definition 1.2 is satisfied if a and fiare bounded* or more generally, if sup{Ila(t, w)II+IIf(t, w)II; t E [0, TI,
IIwIIT <
M} < o0
for every T and M > 0, where IIwIIT = max 11w(t)II 05t5T
and Ilall = fE E Iai12
for a E Rd® R'.
The stochastic differential equations which are most important and which are mainly studied in this book are of the following type. Definition 1.3. Let a(t,x) _ (aj'(t,x)) be a Borel measurable function (t,x) E [0,00) x Rd -- Rd®Rr and b(t,x) = (b°(t,x)) be a Borel measurable
function (t,x) E [0, oo) x Rd - Rd. Then a(t,w) and $(t,w) defined by a(t,w) = a(t,w(t)) and fl(t,w) = b(t,w(t)) clearly satisfy a E Vd.l In such a case, the stochastic differential equation (1,1) is said to be of the Markovian type. The equation then has the following form: (1.3)
dX(t) = a(t,X(t))dB(t) + b(t,X(t))dt
or, in terms of its components, (1.3)'
dX'(t) _ E ak(t,X(t))dBk(t) + bt(t,X(t))dt, i = 1, 2, ... , d.
Furthermore, if a and b do not depend on t and are functions of x E Rd alone, then the equation (1.1) is said to be of the time-independent (or time homogeneous) Markovian type.
Note that an equation of Markovian type reduces to a system of ordinary differential equations (a dynamical system) X, = b(t,XX) when a =_ 0. Thus a stochastic differential equation generalizes the notion of an ordinary differential equation by adding the effect of random fluctuation. Now we will present several definitions concerning the uniqueness of
solutions. Given a E.r&' ' and fl E said 1, we consider the stochastic differential equation (1.1). We suppose that at least one solution of (1.1) exists. * i.e., all components of a and j6 are bounded.
STOCHASTIC DIFFERENTIAL EQUATIONS
162
Definition 1.4. We say that the uniqueness of solutions for (1.1) holds
if whenever X and X' are two solutions*' whose initial laws*z on Rd coincide, then the laws of the processes X and X' on the space Wd coincide.
Remark 1.2. The above definition is equivalent to the following: the uniqueness of solutions for (1.1) holds if whenever X and X' are two
solutions of (1.1) such that X(O) = x a.s. and X'(0) = x a.s. for some x E Rd, then the laws on the space Wd of the processes X and X' coincide. The equivalence is easily seen if we notice the following fact: if X is a solu-
tion of (1.1) on the space (9Q,,9-,P) with ( )too, then, setting P°' _ P( I.9),*3 we have, for almost all fixed co, that X is a solution of (1.1) on the space (Q,. Pw) with (Y lary of Theorem 1-3.2).
such that X(0) = X(0, co) (cf. the corol-
The uniqueness defined in Definition 1.4 is sometimes called "the uniqueness in the sense of probability law". On the other hand if we consider
stochastic differential equations as a tool for defining sample paths of a random process as functionals of Brownian paths, then the following definition might be more natural. Definition 1.5. (pathwise uniqueness). We say that the pathwise unique-
ness of solutions for (1.1) holds if whenever X and X' are any two solu-
tions defined on the same probability space (5 ,, ,P) with the same reference family (,) and the same r-dimensional (.)-Brownian motion such that X(0) = X'(0) a.s., then X(t) = X(t) for all t > 0 a.s. Remark 1.3. We may also consider the following more strict definition of pathwise uniqueness. We say that the pathwise uniqueness holds (in the strict sense) if when-
ever X and X' are two solutions such that X(0) = X'(0) a.s. which are defined on the same probability space (Q,.F P) with the reference families () and (() respectively, and with the same Brownian motion B(t) which
is both an and (X,)-Brownian motion, then X(t) = X'(t) for all t > 0 a.s. Since it is not necessarily true that B(t) is (9v 2°)-Brownian motion, the equivalence of this strict definition and Definition 1.5 is not trivial. However it can be proved as an easy consequence of the following Theorem 1.1. *' They may be defined on different probability spaces. *x The law of X(0) of a solution X of (1.1) is called the initial law or initial distribution of the solution. *3 P( 1-9-.) stands for the regular conditional probability. We can always represent any solution (X,B) on a standard measurable space (b2, .9) without changing the law of (X, B).
DEFINITION OF SOLUTIONS
163
Remark 1.4. Just as in Remark 1.2, we need only consider nonrandom initial values; i.e., X(0)=X'(0) =x a.s., for some fixed xERd. To understand some of the implications of pathwise uniqueness, it is convenient to introduce the following notion. In the following, a function O(x, w): Rd X Wo --- Wd *1 is called P(Rd x W(,)-measurable if, for any Borel probability measure Ic on Rd, there exists a function &,1(x, w) : Rd x W,- Wd which is O (Rd X Wo0K>
"/
-9(Wd)-measurable and for almost all x (u) it holds 0(x, w) = c,,(x, w), a.a. w(PW). *Z For such a function O(x, w) and for an Rd-valued random variable and an r-dimensional Brownian motion B = (B(t)) which are B) where u is the law of mutually independent, we set B): _ . By this it is a well-defined WI-valued random variable. Definition 1.6. (strong solution). A solution X = (X(t)) of (1.1) with a Brownian motion B=(B(t)) is called a strong solution if there exists a Wd *' which is P(Rd x Wo)-measurable function F(x,w): Rd x Wo F(x, w) is ;,aft( Wo)pw/8' (Wd)-measurable for and, for each x z Rd, w every t > 0 and it holds (1.4)
X = F(X(0),B) a.s.
We shall say that the equation (1.1) has a unique strong solution if there
Wd with the same properties as exists a function F(x, w): Rd x Wo above such that the following is true: (i) for any r-dimensional (,fit)-Brownian motion B=(B(t)) (B(0) _ 0) on a probability space with a reference family (Y) and any Rd-valued -measurable, the continuous process X= random variable which is F(c,B) is a solution of (1.1) on this space with X(0) = a.s.; (ii) for any solution (X, B) of (1. 1), X = F(X(0),B) holds a.s. Thus, a strong solution may be regarded as the function F(x, w) which produces a solution X of (1.1) if we substitute an initial value X(0) and a Brownian motion B.
Theorem 1.1. Given a E.sal'd,r and /3 EJ fd.l, the equation (1.1) has a unique strong solution if and only if for any Borel probability measure It on Rd, a solution X of (1.1) exists such that the law of initial value X(0) coincides with p and the pathwise uniqueness of solutions holds.
Proof. If a unique strong solution of (1.1) exists, this means by defi*1 R o = {w E C((O, °p) -.- R`); w(0) = 01.
t2 Pw is the (r-dimensional) Wiener measure on Wo (i.e., the probability law of B).
STOCHASTIC DIFFERENTIAL. EQUATIONS
164
nition that a function F(x, w) : Rd x Wo - Wd exists such that (i) and (ii) above hold. So for a given Borel probability measure p on Rd, let B = (B(t)) be an r-dimensional (.)-Brownian motion and be an .measurable Rd-valued random variable which is distributed as p defined on some suitable probability space with a reference family (. ). Then if we define a continuous process X by X =F(g,B), X is a solution of (1.1) such that X(O) = a.s. Also, if two solutions (X,B) and (X',B') exist on the same probability
space such that B(t) =B'(t) and X(0) = X'(0) a.s., then X = F(X'(0),B) = F(X'(0),B') = Y. This implies that the pathwise uniqueness of solutions holds.*' Thus what we have to prove is that the existence of a solution for each given initial distribution and the pathwise uniqueness imply the existence of a unique strong solution. So let us assume that for any initial distribution a solution of (1.1) exists and the pathwise uniqueness of solutions holds. Let x Rd be fixed and let (X,B) and (X',B') be any solutions of
(1.1)*2 such that X(O)=x and X'(0)=x, a.s. Let P,x and P;' be the probability distributions of (X,B) and (X',B') on the space Wd X Wo w2 G Wo is the projection, then both marginal distributions rr(P.,) and rr(P,) coincide with Pw, the Wiener measure on W. Let Qw2(dw) and Q'w2(dw,) be the regular conditional distributions of w, given w2; that is, (i) for a fixed w2 E Wo, Qw2(dw,) is a probability measure on (Wd, respectively. If rr: Wd x Wo ED (w,, w2)
_q (Wd))2
(ii) for a fixed A e_q(W' ), w2 . Qw2(A) is (iii) for every A,E..d(W') and PX(A, X A2) =
J AZ
(Wr)P"'-measurable,
Qw2(A,)PW(dw2).
Q'w2 is defined similarly from P. We define, on the space 0 = Wd X Wdx Wo, a Borel probability measure Q by Q(dw1dw2dw3) = Qw3(dw,)Q'w3(dw2)Pw(dw3).
Let -9- be the completion of the topological a-field .(Q) by Q, and
.
=
n (-q,+a V _r), where Rf = -9,(Wd) x .gr(Wd) x -9,(Wo) and _41- is a the set of all Q-null sets. Then clearly (w,,w3) and (X,B) have the same distribution and as does (w2, w3) and (X',B'). In fact, it implies the pathwise uniqueness in the strict sense of Remark 1.3. #2 They may be defined on different probability spaces.
DEFINITION OF SOLUTIONS
165
In order to complete our argument, we first need to prove two subsidiary lemmas.
Lemma 1.1. For A
E Wo ' -- Qw(A)
or Q'w(A) is
9t(WW)P"'-measurable.
Proof For fixed t > 0 and A E ;, t(W' ), there exists a conditional probability Qtl(A) such that w c=- Wo - Qw(A) is Rt(WW)P '-measurable and P,X(Ax C) = Jc Q,w(A)Pw(dw) for every CER,(W,,). If we can show
that this equality holds for all CE, f(Wo'), then this implies that Qw(A) = Qw(A) a.a. w(PR') and the assertion of the lemma holds. We may assume that C is of the form C = {w E Wa; p,w E = -
9w E = -
A,, AZ ER(Wo),
where Ot is defined by (Ow)(s) = w(t + s) - w(t). Then since Otw and . t(WW) are independent with respect to P w, we have f Qw (A)P w(dw) A2) tp,WEA,)
= P,(AX {p,w E A,})Pw(0,w E A2) Px(A X { p,w e A, })PX(Wd X {B,w E A2} )
= P(X (-=A, p,(B) E A,)P(Ot(B) E A2) = P(X E=-A, pt(B) E A,, Ot(B) E A2)
=P(XEA,BEC) = P,(A x C) since {X E A, p,(B) (--- A,} E . This proves the lemma.
and 9,(B) and .
Lemma 1.2. w3 = (w3(t)) is an r-dimensional (. on (Q, Q).
are independent. )-Brownian motion
Proof. It is only necessary to prove the independence of w3(t) w3(s) and . L
for every t > s. For this, it is sufficient to prove that
IA,XAZXA;1l *
* EQ stands for expectation with respect to the probability Q.
STOCHASTIC DIFFERENTIAL EQUATIONS
166
= exp [-(I
12/2)(1 - s)]Q(A1 X A2 X A3)
for E R',
A,, A2 E=-a5(Wd) and A3 E.,r95(W0.
But using Lemma 1.1, we have that the left hand side
f
A3
e(c°.w3U)-w3(')) Qw3(A1)Q'w3(A2)PR'(dw3) \
/
(
= eXp [-(I S 12/2)(t - s)] J
= exp [-(I
A3
Qw3(A1)Q'w3(A2)PW(dw3)
12/2)(t - s)]Q(A1 x A2 x A3).
Now returning to the proof of Theorem 1.1, we conclude from Lemma
1.2 that (w,,w3) and (w2,w3) are solutions on the same space (0,_v;Q) with the same reference family Hence the pathwise uniqueness implies that w, = w2, Q-a.s. This implies that Qw x Q'w(w, = w2) = 1 PR'-a.s. Now it is easy to see that there exists a function w E Wo Wd such that Qw = Q'w = Fx(w) Pwa.s. By Lemma 1.1, this function FF(w) is .q,(W,' )`/.W,(Wd)-measurable. Clearly Fx(w) is uniquely
determined up to Pw measure 0. Next, let ,u be any given Borel measure on Rd and let (X,B) be any solution of (1.1) such that X(O) is distributed as tr. Then (X,B) is also a solution on (Q,Y;P( /. )) with respect to (. ) and hence P(F,(,)(B) _ X{ ) = 1. From this it is easy to conclude that F,,(w) is ip(Rd x Wo)measurable and X = Fx(o)(B) a.s. Thus the existence of a unique strong solution is now proved. Corollary. The pathwise uniqueness of solutions implies the uniqueness of solutions (Definition 1.4). Indeed, in the above proof, we showed that P. = PX which means that the laws of (X,B) and (X',B') coincide. Then, of course, the laws of X and X' coincide. This implies the uniqueness in the sense of Definition 1.4 (cf. Remark 1.2). Finally, we shall give an example of a stochastic differential equation for which the uniqueness of solutions holds but the pathwise uniqueness does not hold. This example is due to H. Tanaka.
Example 1.1. Consider the following one-dimensional stochastic differential equation of the time-homogeneous Markovian type:
EXISTENCE THEOREM
167
dX(t) = a(X(t))dB(t),
(1.5)
where a(x)=1 for x > 0 and a(x) = -1 for x < 0. For any Borel probability u on R' there exists a solution X(t), unique in the law sense, such that the law of X(0) coincides with u. Indeed, let B = (B(t)) be an (.Y)Brownian motion and let be an X;-measurable random variable having the distribution u defined on some suitable probability space with a ref-
erence family (.'). Set X(t) = c+B(t). Then B(t) = foa(X(s))dB(s) is )-Brownian motion by Theorem 11-6.1 and X(t) = + f o a(X(s)) dB(s), that is, (X(t), B(t)) is a solution with the initial value X(0)=t; distributed as p. The uniqueness in the sense of law is clear since for any soluan (.
tion (X(t), B(t)), f oj(X(s))dB(s) is a Brownian motion which is independent of X(0). However the pathwise uniqueness of solutions does not hold for (1.5).
For example, if (X(t), B(t)) is a solution such that X(0) = 0, then (-X(t), B(t)) is also a solution. In this case we can prove that a[B(s); s < t] _ a[ I X(s) I
; s < t]; indeed, as we saw in the proof of Theorem III-4.2,
1X(t) I
= J o cr(X(s))dX(s) + 0(t) = B(t) + 0(t),
where 0(t) = li a 10
2e
f
I.,)(1 X(s) I )ds. Thus a[B(s); s < t] ca[ I X(s) I
;
o
s < t]. Also it follows from the proof of the same theorem that I X(t)
B(t) - minB(s). This proves the converse inclusion. This relation of o<,,
a-fields implies immediately that no strong solution exists for the equation (1.5).
Another example will be given in Example IV-4.1.
2. Existence theorem*' Consider the stochastic differential equation (1.1)
dX(t) = a(t,X)dB(t) + 13(t,X)dt,
where aE.Vd' and flE_Vd,'. For feC2(Rd),*2 let *1 An existence theorem for stochastic differential equations was first obtained by Skorohod [150].
*2 Cb(R') = the set of real m-times continuously differentiable functions which are bounded together with their derivatives un to the m-th order.
STOCHASTIC DIFFERENTIAL EQUATIONS
168
(2_1)
i
(Af)(t,w) =
d
rEl a"(t,w) a
s (w(t )) + E f`(t,w) ax (w(t)), f
t E [0, oo), w E Wd,
where (2.2)
r
a,,(t, w) _ Z ak(t, w)ar(t, w). k-1
If (X(t),B(t)) is a solution of (1.1) on a probability space (Q,.3P) with a reference family
(2.3)
then by Ito's formula we have
f(X(t)) -f(X(0)) - J o (Af)(s,X)ds
=E Ef
(X(S)) dB k (S) 0
and hence
(2.4)
f(X(t)) -f(X(0)) - f 0 (Af)(s, X)ds E.,ft2 rx for every f E C2 (Rd).
Conversely, if a d-dimensional continuous adpated process X=(X(t)) defined on a probability space (Q,, ,P) with a reference family (-W-,) satisfies (2.4), then on an extension (Sl,. P) and (.
) of (Q, -9-, P) and
(.9-,) we can find an r-dimensional (fl')-Brownian motion B = (B(t)) such that (X, B) is a solution of (1.1). Indeed, let B, = {x E Rd; I x
f :nor 81(s,X)ds E 0
i,
i=1,2,...,d. Thus,
M,(t) = X°(t) - X°(0) - f ' f'(s,X)ds E..1f21oc, i = 1, 2, ... , d. 0
By choosing f E Cb(Rd) such that f(x) = x'xJ, x that
B,, we see similarly
EXISTENCE THEOREM (2.5)
<M1,Mj>(t) = f
169
a'J(s,X)ds. o
)-Brownian motion B = (B(t)) on an extension (52,.{P) and (S,) such that By Theorem 11-7.1', we can find an r-dimensional (.
M,(t) _ k-1 E f'0 a,(s,X)dBI(s), i = 1, 2, ... , d. Hence (X, B) is a solution of (1.1). If X satifies (2.4), then its probability law Px on (W",.'(Wd)) satisfies (2.6)
f(w(t)) -.f(w(0)) - f o (Af)(s,w)ds E.0Z l°c *1
for every f E Cb(,Rd). Clearly, X(t,w) = w(t) is a stochastic process on (Wd, q(Wd),Px) with (6Wr+(Wd)) satisfying (2.4). Thus we have the following result.
Proposition 2.1. The existence of a solution of (1.1) is equivalent to the existence of a d-dimensional continuous process X satisfying (2.4), and this is also equivalent to the existence of a probability P on (Wd,_4(W')) satisfying (2.6).
Theorem 2.2. Suppose that a E.sld' and f
are bounded
and continuous.*Z Then, for any given probability u on (Rd,.g1}- (Rd)) with
compact support, there exists a solution (X, B) of the equation (1.1) such that the law of X(O) coincides with p, i.e., P {X(0) E A} = u(A) for any A E.4 f(Rd).
Proof. By Proposition 2.1, it is sufficient to construct a process X with
the property (2.4) and P {X(0) E A) =p(A) for every AE.q(Rd). For each I = 1, 2, ... , let O,(t) be defined by 01(t) = k/2'
for k/2' < t < (k + 1)/2' (k = 0, 1, 2,
... ),
and set a,(t, w) = a(O,(t), w) and fl,(t, w) = f(O1(t), w). Clearly, a, E '_Q1d., and /3, On a probability space (52,.P1;P) with a reference family
we prepare an r-dimensional (.Y)-Brownian motion B = (B(t)) *1 is defined with respect to the reference family .°.i{,+(Wd). *Z i.e., the function (0, co) x Wd a (t,w)' a(t,w) E Rd®R' or fi(t,w)ERd is bounded and continuous.
STOCHASTIC DIFFERENTIAL, EQUATIONS
170
and a d-dimensional .-measurable random variable c such that P( A) = u(A)*' for every A E.i(Rd). Define a d-dimensional continuous process X, (1= 1, 2, ... ) inductively as follows: X,(0) _ , and if X,(t) is defined for t < k/2', then we define X,(t), for k/2' < t < (k + 1)/21, by Xr(t) = X,(k/2') + a(k/2`,Xr,k)(B(t) - B(k/2r)) + fl(k/2r,X,,k)(t - k/2') where
X,k(t) =
X10),
{X,(k/2'),
t < k/2', t > k/2'.
Clearly X, = (X,(t)) is the unique solution of the equation dX(t) = a,(t,X)dB(t) + fl,(t,X)dt
X(O) = Since IIa,112 + IIf,II < M for some constant M> 0, we can apply Theorem 111-3.1 to conclude that for every T > 0 and m = 1, 2, ... , (2.8)
sup sup E f I X1(t) 12ml < C. I
0:5t-5T
and (2.9)
supEfIX,(t)X,(s)12m)SCmIt-sIm,
t,sE[0,TJ,
where Cm (m = 1, 2, ...) is a constant. Indeed, for (2.9) we have
X,(t) - X,(s) = f t a,(u,X)dB(u) + f 81(u,X)du and hence*2
tar(u,X)dB(u)II2mj
E[I XI(t) - Xr(s)I2m} < C,("E[il f + C,,,2>E[II s: flr(u,X)dull2m]
Note that s is bounded since y has a compact support. *2 In the following, C.111, Cm', ... are positive constants depending on m, T and M.
EXISTENCE THEOREM a.MO E[(
171
rs Il a,(u,XNI2du)m] + Cm)Fl(f ` Ils Nllu,X 11 du)m]
< C. It - sum.
Applying Theorem 1-4.2 and Theorem 1-4.3 for a = 4 and f = 2, we and d-dimensionX,, and ,',,(t) converges to ?(t) uniformly on each compact interval of [0, oo) as obtain a subsequence {11} , a probability space
al continuous processes ±, .',,, i = 1, 2, ... , such that X,,
i ---- oo a.s. If s < t and if F is a real bounded continuous and 0s(Wd)measurable function on Wd, then for everyf E C;(Rd),
E{(f(X(t)) -M(S)) (2.10)
- f 1(Af)(u,)du)F(X)} s
lim E {(f(t11(t)) -f(±1,(s)) - f` (A(lr'f)(u, t1)du)F(2,,)} s = 0,
where the operator A%) is defined in a similar fashion as A from a,, and fl,,. Equation (2.10) implies that
f(,f(t)) -f(±(0)) -
f
(Af)(u,I)du
o
is an (f )-martingale, where
= e>o n a[,t (u); u
Remark 2.1. The condition that u has compact support is technical and may be removed. Indeed, by what we have shown, for each xE-=Rd
there exists a solution X" such that p = 8x. Let P. = Px(x). If we can choose P. in such a way that x .---- PX is .9(Rd)/.(_7( Wd))-measurable* or g'(Rd)/.h9(,117(Wd))-measurable then P(.) = fRd is a probability on (Wd,_9(Wd)) which satisfies (2.6). Such a selection is always possible because of a general selection theorem ([160], p. 289). The boundedness assumption on a and fl can be weakened, but some kind of restriction on the growth order of a and fl is necessary in order to * .c(Wd) is the set of all probabilities on (W4, °.'(Wd)) with the topology of the weak convergence (cf. Chapter I, Section 2).
STOCHASTIC DIFFERENTIAL EQUATIONS
172
guarantee the existence of a global solution (i.e., a solution defined for all t E [0, oo)). We will not, however, discuss this kind of problem in general (see, e.g. [102] and [188]); we shall only discuss it in the case of equations of the Markovian type.
Let v(x) = (ak(x)): Rd - Rd Qx R' and b(x) _ (b'(x)): Rd -- Rd be continuous. Consider the following stochastic differential equation (2.11)
dX(t) = a(X(t))dB(t) + b(X(t))dt,
or, in terms of its components,
dX'(t) _
k-1
ok(X(t))dBk(t) + b'(X(t))dt, i = 1, 2, ... , d.
If x i--- a (x) and x ---- b(x) are bounded, then we know by Theorem 2.2 that a solution of (2.11) exists for every given initial distribution with compact support. If we remove this condition of boundedness, then a solution does exist locally but, in general, blows up (or explodes) in finite time. Therefore it is more convenient to modify the notion of a solution given in Section 1 so as to include solutions admitting explosions. Let Rd = Rd U (Al be the one-point compactification of Rd and Wd = {w; [0, co) t h- w(t)ERd is continuous and such that if
w(t) ==J, then w(t') = d for all t' > t). Let we set (2.12)
be the o-field generated by Borel cylinder sets. For w E Wd,
e(w) = inf{t;w(t) = d}
and call e(w) the explosion time of the trajectory w.
Definition 2.1. By a solution X = (X(t)) of the equation (2.11) we mean a (W d, °.,d (J'd))-valued random variable defined on a probability space (Q,YP) with a reference family (. )rZ0 such that (i) there exists an r-dimensional (.W;)-Brownian motion B = (B(t)) with B(0) = 0,
(ii) X = (X(t)) is adapted to (.3r), i.e., for each t, w F-- X(t, (0) E Rd is .-measurable and (iii) if e(w) = e(X(w)) is the explosion time of X(w) E Wd, then for almost all w,
EXISTENCE THEOREM
X°(t)
ft
- Xt(0)
cik(X(s))dB'(s) +
x
f
o
173
bt(X(s))ds,
i=1,2,...d,
for all t E [0, e(o,)). Remark 2.2. The stochastic integral
f t ak(X(s))dBk(s)
t E (0, e(co))
0
is well-defined, for if a,(co) = inf {t; I X(t) I > n} , then is bounded in (s,co) and hence dBk(s) =
f t0
is defined for t E [0, oo). Thus t E [0, a,) ' -- f ock(X(s))dBk(s) is defined for every n = 1, 2, ... , and hence it is defined on [0, e(co)) since e(co) = lira a,(cw).
e(co) is called the explosion time of the solution.
Now the uniqueness of solutions, the pathwise uniqueness of solutions etc. are defined in the same way as in Section 1: just replace Wd by Wa.
Theorem 2.3. Given continuous a(x) = (ak(x)) and b(x) _ (b'(x)), consider the equation (2.11). Then for any probability p on (Rd,.}(R')) with compact support, there exists a solution X = (X(t)) of (2.11) such that the law of X(0) coincides with p.
Proof. As in Proposition 2.1, it suffices to show the existence of a Wd-valued random variable X = (X(t)) on a probability space (Q,Y,P) with a reference family (.f) such that X is (Y)-adapted, P(X(0) E dx) = p(dx), and for every f E CI(Rd) and n = 1, 2, ...
.f(X(tAa»)) -f(X(0)) - f
tho,
(Af)(X(s))ds
0
is an (.
)-martingale. Here a, = inf {t; I X(t) I > n} and d
(Af)(x) = 2
a'J(x)
tZJ-
1
a`'(x) ax a f (x) +
ak(x)o'k(x). k-l
bt(x) a (x),
STOCHASTIC DIFFERENTIAL EQUATIONS
174
Let p(x) be a continuous function defined on Rd such that 0 < p(x) < I for every x e Rd and all p(x)a`i(x) and p(x)b;(x) are bounded. Clearly we can choose such a function. Let (Af)(x; = p(x)(Af)(x). By Theorem 2.2,
there exists a d-dimensional continuous process I = (X(t)) (i.e. a Wdvalued random variable) on a space (0,51"',P) with a reference family (fl') such that P(1(0) e dx) = p(dx) and for every f e Cb(Rd),
f(X(t)) -f(X(0)) - J o (Af)(X(s))d is an (_9' )-martingale. (2.13)
Set
A(t) = f o p(X(s))ds
and
(2.14)
e=
f
p(X(s))ds. u
Since p < 1, A(t) < co for every t and 0 < e < oo. The inverse function
a(t) of t - A(t) is defined for t E (0, e) and lim 6(t) = oo. Since A(t) trc is .l-adapted, it is easy to see that c (t) * is an (.f)-stopping time for each t. Set (2.15)
3, = 'R:(*
and
(2.16)
X(t) =
1(17(t)), I'd
< e,
t z e.
By the next lemma, we see that X= (X(t)) is an (. )-adapted Wd-valued random variable. Lemma 2.1. If e(co) < oo then lim X(t)=d in Rd for a.a. w. trc
Proof. It is equivalent to show that, with probability one, if f o p(X(s))ds < oo then bin X(t) = d in Rd. Take 0 < a < b such that
1X(0)1 < b a.s. and define * We set a(t) = oo if t Z e.
EXISTENCE THEOREM
175
6, = 0
z, = inf It > 6, ; I X(t) I > b} ,
62=inf{t> f1; I1(t)I
f,=inf{t>a,; 11(t) I >b},
We show that, with probability one, if f o p(X(s))ds < oo, then there exists an integer n such that zn < 00 and 6n+, = oo. It is sufficient to show that
fo p(X(s))ds = oo a.s. on the set {3n such that 6n < oo and Zn = co} U {6n < oo for every n). First, if there exists an integer n such that 6, < 0o and T. = oo, then I X(t) I < b for all t > 6n and hence f u p(X(s))ds = 00
since min p(x) > 0. Next, we show that if 6n < oo for every n, then Izlsb
57,
('zn - 6n) = 00. If we can show this, then n
f
0
p(X(s))ds ? E fan p(X(s))ds > min p(x) E (zn - fin) = 00.
It is clearly equivalent to show that
<..,} expf-En (Tn - an)} = 11(Ildn<_, expf-(Zn - 6.)D {II1 n n
= 0, a.s., i.e., (2.17)
E(IlI{,. <,j exp n
[-(zn -
6n)]) = 0.
We have m+1
E( II Ildn<w n-1
)
exp [-(f. - 6n)} I _R'
11llan<°D7 exp f-(Zn -
x E(exp [-(t.+, - &.+,)] I
d,n+,)
In the following we assume for simplicity that d= 1; a necessary modification of the proof in the general case is left to the reader. Now X(t) is of the form .
%(t) =.9(0) + M(t) + f
c(s)ds, 0
STOCHASTIC DIFFERENTIAL EQUATIONS
176
where M(t) is a continuous (. )-martingale such that <M>(t) = f'd(s)ds and I c(s) I + I d(s) I < c (c > 0 is a constant). We may assume (Q, Y)
is a standard measurable space (e.g we may always take Q = Wd and _V-= and let P( - I "m+l) be a regular conditional probability given ,gym}1. By Doob's optional sampling theorem, N, = M(t+6m+)M(v 1) is a martingale on [a.+,<-) with respect to the probability P( I m+1) and the reference family JP, = . +dm}1 with , _ f a +i 1d(s)ds. By Theorem 11-7.2', there exists a Brownian motion b(t) (b(0) = 0) which is independent of f =m+1 such that N, = b((N),). Then { I1(t + Qm+i) - X(Qm+1) I > a} C {I M(t + rrm+1) - M(Qm+i) I
= I b(,) I> a/2} U{ f
I c(s) I ds > a/2} .
Consequently, a
inf{t; I x(t+am+1) - X(6.+1) I > a} >
2c
'
6a/z c
where 6a,z = inf {t; I b(t) I > a/21. Hence I,dm+,<_, E(exp
{-(Tm+1 -
P
<E(exp
jm+1)} I
1b-a
(n
2c
A
ocb-a,1z
_C1 k<1.
J
Thus M
E(II n2+1Ild,<-, exp [-(ta - )}) < kE( nil n-1
exp [-(f. - 6n)D
and (2.17) is now obvious. Therefore if f a p(X(s))ds < oo then there exists an n such that fa < and I X(t) I > a for all t > za. Since a was arbitrary, we have lim 2(t) = d itm
in Al". The proof of the lemma is now complete. Now we return to the proof of Theorem 2.3. We have only to show
that
EXISTENCE THEOREM
177
f0 tntr
f(X(tAon)) -.f(X(0)) -
(Af)(X(s))ds
J
is a martingale. Since
f(X(t)) -f(X(0)) - f o (pAf)(X(s))ds is an (.
)-martingale, we have by the optional stopping theorem that ftne
f(X(t A 60) -1(1(0)) - J o (pAf)(X(s))ds is an (. )-martingale for n = 1, 2, ... , where J. = inf {t; I X(t) I > n} . Again by the optional sampling theorem
f(X(c(t) A &n)) -f(X(0)) -
o Wno
J
(pAf)(X(s))ds
and hence is an (.Y)-martingale. It is easy to see that c(t) A Qn = o(t A X(Q(t) A an) = X(t A a.). Also, we have t = f u 1/p(X(s))dA(s) and hence o(t) = f opt' 1/p(X(s))dA(s) = f o 1/p(X(s))ds. Consequently,
Jo
(pAf)(X(s))ds =
Jo
(pAf)(X(s))di(s) = f o
(Af)(X(s))ds.
The proof of the theorem is now complete.
Theorem 2.4. If o(x) _ (ot(x)) and b(x) = (br(x)) are continuous and satisfy the condition (2.18)
IIo(x)112 + lIb(x)112 < K(1 + 1x12)
for some positive constant K, then for any solution of (2.11) such that E(I X(0) 12) < co, we have E(I X(t)12) < oo for all t > 0. Thus e = oo a.s.
Proof. Let inf {t; I X(t) I > n} and f E Cb(Rd) be chosen so that f ( x ) =!x12 if J X J < n. Then since
f(X(tAa.)) -f(X(0)) - J o
Qn(Af)(1(s))ds
STOCHASTIC DIFFERENTIAL EQUATIONS
178
is a martingale, on
E(I X (t A (7 ) 12) = E(I X(0)1 2) + E[
Jo
(E a"(X(s))
d
+ 2 E X°(s)b°(X(s)))ds]. i-x
By (2.18) we have for some constant c > 0 that
E(IX(0)12) + c
f
o
{1 + E(I
From this we can conclude that E(I X(t A a.) 12) < { l ± E(I X(0) 12)} e°' - 1.
Letting n - oo, we have E(I X(t) 1 2) < { 1 + E(I X(0)12)} e°' - 1,
which completes the proof.
Thus the condition (2.18) is a sufficient condition for non-explosion of solutions. A more general criterion for explosion or non-explosion will be given in Chapter VI, Section 4.
3. Uniqueness theorem In this section we only consider stochastic differential equations of the time homogeneous Markovian type. So suppose we are given a(x) _ (ok(x)) :
Rd -- Rd (x R' and b(x) = (b'(x)): Rd - Rd which are as-
sumed to be continuous unless otherwise stated. We consider the following stochastic differential equation (3.1)
dX(t) = o(X(t))dB(t) + b(X(t))dt
or in terms of its components,
dX'(t) _ E vk(X(t))dBk(t)+bt(X(t))dt, i = 1, 2, ... , d. k-1
Theorem 3.1. Suppose a(x) and b(x) are locally Lipschitz continuous, i.e., for every N > 0 there exists a constant KN > 0 such that
UNIQUENESS THEOREM (3.2)
179 y12
Ilo(x) - o(y)112 + II b(x) - b(y)I12 < KNI x for every x, y E BN.*
Then the pathwise uniqueness of solutions holds for the equation (3.1)
and hence it has a unique strong solution. Proof. Let (X, B) and (X', B') be any two solutions of equation (3.1) defined on some same probability space (0,-9-,P) with some same reference
family such that X(0) = X'(0) = x and B(t) - B'(t). It is sufficient to show that if oN = inf {t; I X(t) I > N} and QN' = inf {t; I X'(t) I 2:N} then oN = oN and X(t) = X7(t) for all t < o,, (N = 1, 2, ... ). But ^ ^ X(tAoNAoN)-X'(t/\6N/\6N) = Jo
o(X'(s))JdB(s) + f
AaNAON
- b(X'(s))Jds
o
and hence, if t
[0, T],
E{I X(tno'NA(Q - X'(tAoNAo )12) t0 AQNAON
- o(X'(s))JdB(s)12}
< 2E { I f 'o oNA N
+ 2E { I f
[b(X(s)) -
0
2E I
J
'AuNAoN
II6(X'(S)) - a(X'(S))II2dS J
o
+ 2TE {
b(X'(s))]ds 12}
f
oN^oN
t
< 2E{J,
I b(X(s)) - b(X'(s))12ds} I1k(X(SAuNAa2) -a(X'(SAo'NAoN))Il2ds}
+ 2TE{ f 0 17(X(5AoNAok)) - b(X'(SAoNAo)) I2dd}
< 2KN(l + T) f E{IX(sAoNA ) - X'(SAuNn6N)12) ds. 0
It is easy to conclude from this inequality that E {I X(t A6N^/\o'N) - X'(t AUNA o'N) 12} = 0
* By = {x; I xI S NI .
for all t E [0, TJ
STOCHASTIC DIFFERENTIAL EQUATIONS
180
and hence, letting T t o0o,we /have
X(tAoNAu ) = X'(tAoN/\QN)
for all t
a.s.
0.
Since X and X' are continuous in t a.s., we can conclude that X(t) = X'(t) for all t c- [0, ON/ \ oN) a.s. This clearly implies that oN = 6N a.s. and the pathwise uniqueness of solutions of (3.1) is proven. From Theorem 1.1 and Theorem 2.3 we now conclude that the unique strong solution* exists for the equation (3.1).
The existence of a strong solution may be proved more directly by using the method of successive approximations as follows. For simplicity
we assume that the Lipschitz condition (3.1) holds globally: i.e., there exists a constant K > 0 such that (3.3)
11a(x)
- a(y)11' + 11b(x) - b(y)Il2 < KIx - yI2
for all x,yeRd. Then, by changing the constant K if necessary, we may assume that (3.4)
for all x E Rd.
11a(x) IIZ + Ilb(x)HH2 < K(IxI2 + 1)
In the following we essentially repeat the same proof as that of Theorem 111-2. 1. Let x c- Rd be fixed. For a given r-dimensional Brownian
motion B = (B(t)), we define a sequence X. =
(n = 0, 1, 2, ...)
of d-dimensional continuous processes by
(3.5)
f'
x + f'0
b(X.-1(S))ds,
0
n=1,2,.... Set
XX(t) = f
'
f'
[b(XX(s)) - b(Xn-1(s))I^.'
= 11(t) + 12(t)
* The space 1Wd is now replaced by 011.
say.
UNIQUENESS THEOREM
181
By Doob-Kolmogorov's inequality, E(sup I II(s) I2) < 4E(I II(t) 12)
= 4E( E
II a(X»(s)) - v(Xn-1(s))II2ds)
o
< 4K f
E(I XX(s) - Xn-,(s) 12)ds. 0
Also, if t E [0, T], E(osupt I IZ(s) 12) < E({ f II b(X,,(s)) - b(Xn-1(s))II ds} 2) 0
< TE(f o
11''(Xn(s)) - b(X"-I(s))II2d )
< TK f 0 E(I X (s) - Xn-1(s) 12)ds. Hence E( uppt I Xn+1(s) - X.(s) 12)
< 2K(4 + T) f E(I XX(s) - Xn-1(s) 12)ds 0
and therefore,
E(sup I X (S) 0<s
< {2K(4 +
-
X .(S) 12)
T)} ° f0t dt1 f 'o dt2 .
.
f,04-, dt,E(I X1(t>,) -
Since
E(I X1(t) - X0(t)12) < 2E(116(x)B(t)I12 + I1b(x)112t2) < 2(IIu(x)II2t + 11b(x)112t2)
< 2K(1 + T)T(1 + Ix12), we have
E( sup I 05t5T
Consequently
Xn(t)12) < const. {2K(4 + T)} "T"/n!.
STOCHASTIC DIFFERENTIAL EQUATIONS
182
P1 sup IX.+I(t) - X"(t) J >112n) < const. {8K(4 + T)} "T"/n! 05r57'
By Borel-Cantelli's lemma, we see with probability one that X"(t) converges uniformly on [0, T], and, since T was arbitrary, limX"(t) = X(t) W--
determines a d-dimensional continuous process which is clearly a solution of (3.1). This solution is of the form F(x, B) for some function F(x, w) on
Rd Qx W' since each X. is so. Thus X is a strong solution of (3.1); the uniqueness is clear from Theorem 3.1.
The Lipschitz condition (3.2) for the pathwise uniqueness of solutions
of the equation (3.1) can be weakened considerably in the case d = 1. For simplicity we state the theorem in the global form assuming that o(x) and b(x) are bounded but it may be localized as Theorem 3.1 in an obvious way.
Theorem 3.2. Let d = r = 1*1 and suppose that o(x) and b(x) are bounded. Assume further that the following conditions are satisfied: (i) there exists a strictly increasing function p(u) on [0, cc) such that p(0) = 0, f o+p '(u)du = co and I Q(x) - a(y) 1 < p(I x - y I) for all x, y ER'*2;
(ii) there exists an increasing and concave function K(u) on [0, 00)
such that x(0) = 0, fox-I(u)du = oo and Ib(x) - b(y)e < lc(Ix - y1) for all x,yER1. Then the pathwise uniqueness of solutions holds for the equation (3.1) and hence it has the unique strong solution.
Corollary. If o is Holder continuous with exponent 1/2 and b is Lipschitz continuous, then the pathwise uniqueness of solutions holds for the equation (3.1) in the case d 1.
Proof of Theorem 3.2. Let 1 > a, > a2 > ... > a" >
> O be
defined by
f'
p-z(u)du = 1,
ai
f al
a
p Z(u)du = 2, .... ,
f
a"
' p-2(u)du = n,
... .
a"
Clearly a" --- 0 as n -. oo. Let v(u), n = 1, 2, ... , be a continuous function such that its support is contained in (a", a"_,), *1 r may be arbitrary. We assume r = 1 only for simplicity. *2 This nice condition for a was found by T. Yamada ([1831) improving an idea of H. Tanaka in [1621.
UNIQUENESS THEOREM
0 S iin(u) < 2p-2(u)/n and
183
1.
-1 fan
Such a function obviously exists. Set ]XI dY Pn(x) = f 0 f Y v.(u)du, 0 j
R'.
x
9,(x) t I x I as n - oo. Suppose that we are given two solutions (X1(t),BI(t)) and (X2(t), B2(t)) on the same probability space with the same reference family such that X1(0) = X2(0) = x and B1(t) = B2(t) (: = B(t)) a.s. Then we have It is easy to see that 97n E C2 (RI), I ipn(x) I < 1 and
X1(t) - X2(t) = f'0 [6(X&)) -
+f
17(X2(s))]dB(s)
b(X2(s))]ds
0
and by Ito's formula,
Vn(XI(t) - X2(0) = J o V.(Xj(S) - Xz(s))[Q(Xj(S)) - 6(X2(s))]dB(s)
+f
o
,pn(X1(s)
+2f
o
- X2(s))[b(X1(s)) - b(X2(s))]ds
q''(1 (s) - X2(S))[Q(X1(S)) - Q(X2(s))]2ds.
Since the expectation of the first term in the right-hand side is zero, we have E[con(X1(t) - X2(t))] = E[ f' 09'(X,(s) - X2(s)) {b(X1(s)) - b(X2(s))} ds]
+ E[ f
o
(Pn (X1(5) - X2(S)) {6(X1(5)) - 6(X2(5))} 2d5]
= I1 + I2. Now I I1 I
f E[ I b(XI(s)) - b(Xz(5)) I Ids f E[k(i X 1(s) - X2(5) J )Ids o
0
STOCHASTIC DIFFERENTIAL EQUATIONS
184
'
0
K[E(I X1(S) - X2(s) I )]dS
by Jensen's inequality, and i r2 i
<_ z f E n < tin 0
p-2( l X. (s)
- X2(s) I)p2(I X. (S) - X2(S)
as n
1)] ds
oo.
Consequently by letting n -- oo, :
E(I X1(t) - X2(t)j) < f 0 ic[E(I X1(s) - Xz(s)1)]ds. Since f
o+x-1(u)du = +oo, the above inequality implies that E(I X1(t) -
X2(t) 1) = 0 and hence X,(t) = X2(t) a.s. This proves the pathwise uniqueness for (3.1).
The continuity condition for the function o in the theorem is, in a sense, the best possible. Indeed, suppose for simplicity that b(x) = 0 and o(x) is such that o(x0) = 0 and fxo±a a' 2(y)dy < co and o(x) Z 1 for I x - x0 I > c. Then the equation dX(t) = o(X(t))dB(t) (3
I X(0)
= xo
has infinitely many solutions. Clearly X(t) = x,, is a solution. Also, for a one-dimensional Brownian motion b(t) with b(O) = 0, we set for each p > 0, fi(t) = xo + b(t), Ap(t) = 2 f -_O(t,y)o 2(y)dy + pc(t,xo) fo Q x0) and X1(t), (Ap1(t)), where Apt is the inverse function of t'--- Ap(t) and ¢(t,y) is the local time of fi(t).*1 Then Xp(t) is a solution of (3.6), because M, = Xp(t) - xo is a continuous martingale with
(M), = AP 1(t)=
$4
1W
p
o
o2(Xp(s))ds
(cf. Proposition 2.1). It is easy to see that the probability law of Xp is differ-
ent for different p. So far we have only presented conditions for pathwise uniqueness.*2 Chapter III, Section 4. *z Except the cases which can be covered by Theorems 3.1 and 3.2, little is known about the pathwise uniqueness and the existence of the strong solutions. Cf. [125] and [189].
UNIQUENESS THEOREM
185
There are also several important results for the uniqueness of solutions in the sense of probability laws due mainly to Stroock-Varadhan [157] and Krylov [90]. In particular, Stroock-Varadhan showed that the uniqueness of solutions holds for the equation (1.3) if the matrix a(t,x) _ u(t,x)o (t,x)* (in component form, a'f(t, x) _
6k(t,x)Qk(t,x)) is con-
tinuous, bounded and uniformly positive definite, and if b(t,x) (b'(t,x)) is bounded and Borel measurable. Here we shall content ourselves with presenting only a particular case of this beautiful and important result.
Theorem 3.3. Consider the equation of the time homogeneous Markovian case (3.1). If a(x) = a(x)o.(x)* is uniformly positive definite, bounded and continuous and b(x) is bounded and Borel measurable, then the uniqueness of solutions holds. Proof. We assume b(x) = 0; the general case is obtained by a transformation of drift which will be discussed in the next section. Set Af (x)
2 r.E
aif (x) a.7a
(x), f E C2(Rd).
By Proposition 2.1, it is sufficient to prove that if P. is a probability on (Wd,.6(Wd)) such that
(i) P. {w; w(0) = x} = 1 and (ii) f(w(t)) - f(w(0)) - f' (Af)(w(s))ds is a (PX,:9,( Wd))-martingale for f e Cb(Rd),
then P. is uniquely determined. As we shall see in the corollary of Theorem 5.1, it suffices to prove that Ex[ f o e-'x'f(w(t))dt] is uniquely determined for every f E Cb(Rd); that is, for any two probabilities P. and Px on (Wd,.g(W')) satisfying (i) and (ii), (3.7)
EE[ f o e z`.f(w(t))dt] = Ex[ so e_1'f(w(t))dt]
for everyf E Cb(Rd). We shall obtain this result by a perturbation argument on Wiener measure. First we shall show that there exists a positive constants such that if a(:x) = (a'f(x)) satisfies (3.8)
1 a'f (x) - Sif 1 < e
for all x e Rd and i,j = 1, 2, ... , d,
then for any x and for any two probabilities P. and Px satisfying (i) and (ii), (3.7) holds.
STOCHASTIC DIFFERENTIAL EQUATIONS
186
We now list some analytical properties of operators related to Brownian motion which we shall need in the subsequent discussion. Set
x2
t>0, x E= R°,
g:(x) = (27rt)-d'2 exp(- 12t ),
A>0'
vx(x) = f e-A' &(x)dt,
x E Rd
0 o
and
(V2f)(x) = f ,d vz(x - Y)f(Y)dy. Then the following facts hold. Let A > 0 be fixed. (1) Vx is a bounded operator on 2,(Rd) into itself such that II Vill, S Ilvill, = 1/2. This is a consequence of the well-known Hausdorff-Young's inequality. (2) If p > d V 1, then there exists a constant A. depending on p
and d only such that for every f E 2,(Rd) and x E Rd.
I Vxf(x) I <_ Apll f III
Indeed, by Holder's inequality, I VVf(x)l < II1'.IIQIIfIIp
1/p + 1/q = 1,
where
and
Ilvxllq S foe-x` IISrllgdt < const.
f
e
x°t- (1-Q) dt < -
o
if
d(1 -q)= p< 1. a2
(3) For each i and j, -V1 f for f c---- C;(Rd) * can be extended to a ax,ax,
bounded operator on 9',(Rd) into itself for every p > 1; that is, there exists a constant C, depending on p and d only such that
avfll,
* CX(R4) is the space of C'-functions on Rd with compact supports.
UNIQUENESS THEOREM
187
The proof in the case of p = 2 is easily furnished by the Fourier transform, but in the general case we have to appeal to the deep Y theory for singular integrals (cf. [152]).
Suppose a(x) = (ati(x)) satisfies (3.8) and let P. be a probability on (Wd,.I(Wd)) satisfying the above conditions (i) and (ii). Then
for f e Cb(Rd)
EX[f(w(t))] =f(x) + f' EE[(Af)(w(s))]ds 0
and hence for fixed 2 > 0 and x e Rd,
2EX[ 0f e 'f(w(t))dtl =f(x) + Ex[ f. e-xt(Af)(w(t))dt]. 0 Consequently, denoting EX[Jo e-Atg(w(t))dt] by,uA(g), we have that 1uA(2f
-
df) =f(x) + 2 A(t,E
a
as
where c"(y) = atU(y) - Sty. Letting f = VAh for h e C;(Rd), IuA(h) = VAh(x) + 2
p (r,E ct!(.)axia (.)
Using the above properties (2) and (3) for p > d/2 V 1, IpA(h)I <_ A,Ilhll, + 2 d2C, sup IPA(f)I 11h11'2
if sup IpA(f)I = IIPAIiq < co, then we can conclude that Ifl,51
II/1All, < AD/(1
- 2 d2C,) for all e > 0 such that 1- Ed 2C, > 0. This
can be verified as follows. Set
Yr(t,w) = w (2 Am), t e [k/2m,(k + 1)/2m), k = 0, 1, .. . and
.Ym(t,w) = x + f' Q(Ym(s))dB(s), 0
STOCHASTIC DIFFERENTIAL EQUATIONS
188
motion such
where B = (B(t,w)) is a d-dimensional that
w(t) = x + f
c,(w(s))dB(s) o
(cf. Theorem 11-7.1). Then it is easy to see that the probability law P" of the process (Xm(t)) converges weakly to Px and, in particular,
fE Ca(Rd).
Jul'(.f): = Ex[ foe 1f(Xm(t))dtJ
Since {w(t), t E [k/2m, (k + 1)/2m)} is, with respect to the regular conditional probability Pm(. I Ak,am(Wd)), a linear transformation of the ddimensional Brownian motion by the constant matrix v(w(k/2m)), we see from (2) that I EX[ f0 .C 1f(Xm(t))dt]l < oo. IIPz h = sup pflp5l
Then by the same argument as above we have A,I(1
- 2 d2C,
Finally, by letting m IIp
Iq <_ lim II/4'Iiq < A,/
(1
-
dzC,). 2
Let Pz be another probability on (Wd,..°g(Wd)) satisfying the above conditions (i) and (ii). Then for each fixed x and f e CK(Rd), Vx f(x)-1 1.La(Kx f)
and p. (f) = VJ(x)+pz(Kzf)
where Kx f(Y) =
2 Q- c`J('') axvaz (Y)
and px is defined in the same way as Aux from P. Therefore
UNIQUENESS THEOREM
189
(pz - u)(f) = (uz - kz)(Kzf). But we have IIKzf 11, <
22
X,11 f 11,. Consequently
z sup I (uz - u) (f) I < d EC, sup I (uz - #D(f) 1 2 1rEp51
UI,S1
2
and hence, if we choose e > 0 such that
EC, < 1, then pz(f) _ ,uz(f ).
Thus we have proved the uniqueness of the probabilities {PX} XERd 2
provided ari(x) satisfies the condition (3.8) for s > 0 such that 2 sC, < 1.
Clearly (5,,,) may be replaced by any positive definite constant matrix
C = (C"), ands > 0 can be chosen independent of C if A < A(C) < A(C) < B for some positive constants A, B, where ..(C) and (C) are the least and the largest eigenvalues of C respectively. Now we remove the restriction (3.8) by the following standard localization argument. Set T(w) = inf {t; max I a"(w(t)) - ati(w(0)) I > e} . i51.j5d
Then the result we obtained implies that for any {P.,'} satisfying (i) and (ii), (3.9)
P. {w.- E Al = P;' {w.; E Al
for every x and AE,9I(Wd)
where w. E Wd is defined by w (u) = w(T A u). We denote P. {w; E Al, x e Rd, A e O(Wd), by p {x, A). By Doob's optional sampling theorem, it is easy to see that if 0 is a (.4,(Wd))-stopping time such that P,(0 < oo)
= 1, then for P, -almost all w, P,'(A) = P,,(we E AI-VB(Wd)), A .q(Wd) satisfies the above condition (ii) and Pw(w'; w'(0) = w(0(w))) = 1.*
Here wg E Wd is defined by (wg)(u) = w(0 + u). Using this fact, it is easy to conclude the following: letting To(w) = 0 Tj(w) = T(w)
T2(w) = Tl(w) + T(w,) * Cf. the corollary of Theorem 1-3.2.
STOCHASTIC DIFFERENTIAL EQUATIONS
190
Zn+1(w) = %(w) + Z(wrn)
and
for n=0,1, ... ,
(w zn
faop
f
{x, dwo} J Al p X
{wo(r(wo)), dwl} f ...
f
=PX{w;00wEA0,0,wEA,f...,45nwEA.). Since z (w) - oo for every w, this implies P., = PP.
4. Solution by transformation of drift and by time change Certain stochastic differential equations can be solved (i.e., we can show the existence and the uniqueness of solutions) by some probabilistic methods. These methods sometimes apply even for equations which are not covered by the theorems obtained so far. 4.1. The transformation of drift. Let (f2,,' 7;P) be a probability space with a reference family (. ). In the following, we assume that (Q,,Y`;P)
and (.),>o possess the below property: Suppose, for every t > 0, that ,u, is an absolutely continuous probability measure on (52,.Y) with respect to P such that ,u, restricted (4.1)
on .Y coincides with lt, for any t > s > 0. Then there exists a probability measure u on (52,.x) such that p restricted on .W' coincides with lc, for every t > 0.
That is, we assume that any consistent family of absolutely continuous probability measures with respect to P can be extended to a probability measure on For example, if (52,.x") is a standard measurable space, then the above condition (4.1) is satisfied.
For XE % 1OC we set (4.2)
M(t) = exp[X(t)-<X)(t)/2].
We assume that M is a martingale. This is true, for example, by Theorem 111-5.3 if E(exp ('<X>,)) < oo for every t > 0. Then if we define P, for
each t > 0 by
SOLUTION BY TRANSFORMATION OF DRIFT
P,(A) = E[M(t): A],
(4.3)
AE.
A E -9,-,
for t > s > 0. In fact, if
and
fl, is a measure on (Sl,
191
,
P,(A) = E[M(t): A] = E[E(M(t)A] = E[M(s): A]. By assumption (4.1), there exists a probability P on
such that
P is called the probability measure which has density M with respect to P. We denote P as P = M.P. In this way we obtain a new system (52,.,P) and The spaces etc. of martingales with respect to this system are denoted by The following theorem was established by Girsanov [40]* in the case when X(t) is a Brownian motion.
Theorem 4.1. (i) Let Y E -WZ'°°. If we define Y by (4.4)
P(t) = Y(t) - (t),
then Y E -4?2c 1".
(ii) Let Y1, Y2 E -Oz I- and define Y, and Y2 by (4.4). Then (4.5)
= <-'i,Is>-
f
Remark 4.1. Since P -, and ,mot are mutually absolutely continuous for each fixed t > 0, there is no ambiguity in the statement of (ii): (4,5) holds P-a.s. and P-a.s. ,
Proof. Assume first that Y(t) is bounded in the sense that, for each t > 0, Y(t) E .°'(Q, -F-). By Ito's formula, d(M(t)Y(t))
= d{M(t)(Y(t) - (t))} = Y(t)dM(t) + M(t)dY(t) - M(t)d
M(t)d<X,Y>(t). This
STOCHASTIC DIFFERENTIAL EQUATIONS
192
-0[Y(t) I -7;] = E[M(t)Y(t) I. -,}M(s)-' = IF(s)-
In general, we choose a sequence of (.Y)-stopping times such is bounded in the above sense. Since Y(t A that, for each n, t b-- At A r° is defined by <X, Y>(t A Y°n(t), where Y E Y(t A by the above proof and hence Y Y°»(t) = Y(t A a.), Y°° E
E .°.1°° z The proof of (ii) can be given in a similar way.
Corollary. Let YE-111c. e, 0 EY°°(Y) and Z(t) =$ )(s)dY(s). 2 Then 45 (=-Y'2-(?) and (4.6)
22(t) = Z(t) -
f
0(s)d?(s). o
Proof. By the definition of stochastic integrals, it suffices to prove (4.6) for 0 (=- -70; but in this case the assertion is clear.
Theorem 4.1 implies that if we transform the probability measure P into t = M. P, every continuous local martingale Y with respect to P is transformed under the probability P to Y = a continuous local martingale+ < Y, X>. That is, the transformation of the probability measures P - P = M. P induces a drift for every local martingale Y. For this reason, the transformation of measures P -16 is called transformation of drift. It is also often called the Girsanov transformation. The method of transformation of drift can be applied to solve a class
E
of stochastic differential equations. Suppose a and let us consider the stochastic differential equation (1.1)
dX(t) = a(t,X)dB(t) + fl(t,X)dt.
Suppose that a solution (X, B) is given on a probability space (Q,..91;P) with a reference family (,). Without loss of generality, we may assume that this system satisfies condition (4.1). Choose y e ,sad'-' such that it is bounded or, more generally, satisfies
E[exp (2 f II y(s,X)I jzds) < o a
for every
t > 0.
Then
(4.7)
M(t) = exp { f ' y(s,X)dB(s) -
f ' IIy(s,X)IIZds}
2
SOLUTION BY TRANSFORMATION OF DRIFT
193
)-martingale. Let P = M.P. By Theorem 4.1,
is an (,
B(t) = B(t)
(4.8)
-f
y(s,X)ds 0
is an r-dimensional (,9';)-Brownian motion on the probability space (0, Jw;P) with the reference family (J '-,),z0. Indeed, since X(t) in (4.2) is now
f
oy(s,X)dB(s) (=
rE
f
oy`(s,X)dB`(s)) := Y(t),
we have B`(t)
= Bt (t) -(t) = BI(t)
- f y°(s,X)ds e o
and <,91,k>(t) = (t) = 6, jt, implying that B is an r-dimensional (.9 -t)-Brownian motion. By the corollary of Theorem 4.1, we have (4.9)
dX(t) = a(t,X)dB(t) + [fl(t,X) + a(t,X)y(t,X)]dt.
This implies that (X(t), B(t)) is a solution of the stochastic differential equation (4.9) on the probability space with the reference family ()rao Thus we get a solution of (4.9) by applying the transformation of drift to a solution of (1.1). Furthermore, if the uniqueness of solutions (cf. Definition 1.4) holds for the equation (1.1), then it also holds for the equation (4.9). To show this, we may assume without loss of generality that y E _Vr" I is of the form y(t,w) = a*(t,w)ri(t,w) with some ri E .sat'd,' (a*(t, w) a jjr.d is the transposed of a(t,w); as usual, we regard it as a linear mapping Rd --- R'.) Indeed, let R' = a*(t,w)(Rd) Q a*(t,w)(Rd)1
be the orthogonal decomposition and y(t,w) = yl(t,w) + y2(t,w) under this decomposition. Then y, E a7',' y, is clearly bounded if y is bounded
and a(t,w)y(t,w) = a(t,w)yl(t,w) since a(t,w)y2(t,w) = 0 as we see by (a(t,w)y2(t,w),y) = (y2(t,w), a*(t,w)y) = 0 for any y E Rd. Finally choose
7(t,w) from the affine space 1y; a*(t,w)y = yl(t,w)} in Rd such that
rl e VV-1; for example, let rl(t,w) be the unique element in the affine space which attains the minimal distance from the origin. By replacing y by yl we can conclude the above assertion. Then M(t) given by (4.7) is a well determined functional of X:
M(t) = exp }f o
y(s,X)dB(s) - 2 f0
II Y(s,X)II Zds}
STOCHASTIC DIFFERENTIAL EQUATIONS
194
exp{ f ' i(s,X)dMX(s)
- 2 $ ' ll
a*(s,X)rl(s,X)Il2dsJ
where
'
MX(t) = X(t) - X(0) - $' fl(s,X)ds = f a(s,X)dB(s). 0
0
Thus the uniqueness of solutions for the equation (1.1) implies that the law of the joint process (X(t),M(t)) is uniquely determined by the initial law of X. Now we shall show that the uniqueness of solutions for the equation (1.1) implies that for the equation (4.9). In fact, starting with any solution (i(t), fi(t)) of the equation (4.9) on a probability space with a reference family (.' )rz0 satisfying the condition (4.1), we define
M(t) - exp [-f
y(s,
t)d.(s)
o
2 f' Ily(s, 1)II2d ] , 0
B(t) = B(t) + f' y(s,t)ds, 0
and
Then (X(t), B(t)) is a solution of equation (1.1) with respect to the probability P and it is easy to see that if we apply the above transformation of drift to the solution (,?(t), B(t)), then we come back to (t(t), h(t)). Thus any solution of (4.9) is obtained by the transformation of drift from some solution of (1.1), and hence combined with the above remark, the uniqueness of the solutions of (1.1) implies that of (4.9). Summarizing, we have the following result.
Theorem 4.2. If the equation (1.1) has a unique solution, then the equation (4.9) also has a unique solution; moreover a solution of (4.9) is obtained from a solution of (l.1) by the transformation of drift. Corollary. Suppose f E said' is bounded. Then the stochastic differential equation (4.10)
dX(t) = dB(t) + f(t,X)dt
(i.e., the stochastic differential equation (1.1) with a = I, the identity
SOLUTION BY TRANSFORMATION OF DRIFT
195
matrix) has a unique solution and is constructed as follows. We choose on some probability space (Q,.cP) with a reference family (9 )tzo satisfying
the condition (4.1) a d-dimensional (Y)-Brownian motion B(t) with B(0) = 0 and a d-dimensional, ,`,-measurable random variable X(0) with given distribution g on Rd. We set
X(t) = X(0) + B(t),
M(t) = exp [fo fi(s,X)dB(s) - 2
f
'o
Jjf(s,X)j 2ds],
f Then (X(t), B(t)) is a solution of (4.10) on the probability space (52,,f-,P) with the reference family In this way, the stochastic differential equation (4.10) is always solved
uniquely by the method of the transformation of drift, (cf. Maruyama (ii0)). But the solution thus obtained is not necessarily a strong solution in general. In fact, Cirel'son [12] gave an example of a stochastic differential equation of the form (4.10) for which the solution is not strong.
Example 4.1. (Cirel'son [12]). Let d = 1 and /3(t,w) E .sa 1.1 be defined as follows: let 9(x) = x (mod 1) = x - [x] E [0, 1), x e R1, and let,
{tk; k = 0, -1, -2, ... } be a sequence such that 0 < tk_1 < tk and lim tk = 0. For w e W1, set 9(w(tk) (4.11)
4
fl(t,w) = 0
- w(tk_1)) lft E [tk, tk+l),k= -1, -2, tk_3 ,
...,
if t=0 or t>t0.
It is clear that /t(t,w) E .sz"1.1 and is bounded. However, the one-dimensional stochastic differential equation (4.10) does not have a strong solution. Proof.* Suppose on the contrary, this stochastic differential equation has a strong solution (X(t), B(t)). We may assume that X(0) = x a.s. for * We learned the following simple proof from Shiryaev by private communication.
STOCHASTIC DIFFERENTIAL EQUATIONS
196
some constant x E R'. Then, by the definition of strong solution, we have
.PX=Q[X(s);s
Jilt tk
= B(tk+l) - B(tk) + 0(X(tk) tk - tk_1
X(tk-1))
(tk+1 - tk)
,fork=-1,-2,..., and hence if we set
Ik =
B(tk-1) - X(tk-1) , Sk = B(tk)tk -- tk_1
X(tk)
tk - tk-1
xx
k=-1,-2,...,
then
(4.12)
11k+1 = Sk+1 + 0(rlk)
Consequently e2atnk+1 = e2atfk+l e2nltlk,
and therefore, for every. 1 = 0, 1, 2,
k ...
e2n' k+1 = e2.ilk+1 e2
(4.13)
,
2atgk-t.
)-adapted, we have that G+1 is independent of a[t;k, bk-I, Sk-2, ... , ,Ik, Ilk-1, >Ik-2, ] and so from (4.13) Since B(t) is an (Y)-Brownian motion andxx(X(tYY), B(t)) is (.
E[e2n1.7k+1]
= n-0 II E[e2n`ek_n+1]E[e27tt,lk-1). I
Since JE[e2"`nk-t] j < 1 and E[e2nZn] = exp [-27r2/(tn - tn_1)], we have E[e2n lk+1 ]
t
exp [ - 2ic2 E (tk_n+1 - tk_n)-']. n-0
Letting 1---- oo, we deduce that (4.14)
E[e2it1nk+1] = 0,
k = -1, -2, ... .
SOLUTION BY TRANSFORMATION OF DRIFT
Set
u< tk_,]
197
a[B(t) - B(s); tk_t < s < t < tk+1]. Then a[X(u), B(u), is independent of -,g,kl+1 and hence, by (4.13) and
(4.14), E[e2ntak+] I gk+l]= e27iiZk+t e21 k ... e2n'4k-t+'E[e2ntnk-I] = 0.
Since v 'q1+1 = , ft ®o
k +19
k+1
mink+1 = E[e2" tnk+1 I.
'
it follows by letting I t co that
I-
] = lira E[e2"'nk+1 I.4r1,+1] = 0,
(cf. Theorem 1-6.6). But this is clearly a contradiction.
4.2. Time change. Another probabilistic method which is sometimes useful in solving stochastic differential equations is method of random time change. The general theory of time changes in the martingale theory is well known; we have already discussed it somewhat in Chapter II and in
Chapter III, Section 1. In order to avoid undesirable complications, we restrict the class of time changes and formulate it as follows. Let I be the class of functions
0: t E [0, oo) - 0, E [0, co) satisfying
(i) Oo = 0, (ii) 0 is continuous and strictly increasing and (iii) 0, t oo as t t oo. Clearly, if 0-1 is the inverse function of 0 E I, then 0'1 E I. I is a subset of W1 and the Borel fields induced by . '(W'), .9 ,(W') are denoted by .,II(I) and .t(I) respectively. Each 0 E I defines a transformation TO of Wd into itself by (4.15)
TO: w E Wd _ (TOw) E Wd
where (T#w)(t) = w(o, 1), t e [0, oo). TO is called the time change defined I. by 0
Let a probability space (Q,-9;P) with a reference family (.
)t?o be
given. Consider a mapping 0:0 D o) '---- O(co) E I which is f/. r(I)measurable for each t. Such a 0 = (0,(co)) is called a process of the time change. Clearly 0 = (0,((o)) is an (. )-adapted increasing process, and hence if '(cv) is the inverse function of t i---- 0,(w) then 071 is an
198
STOCHASTIC DIFFERENTIAL EQUATIONS
(,Y;)-stopping time for each fixed t E [0, co). If X = (X(t)) is a continuous (.)-adapted process, then TOX = ((TOX)(t)) defined by (TOX)(t) = X(¢71) is a continuous (9 1)-adapted process. The transformation X i- T#X defined above is called the time change of X by the process of time change 0. Given a process of time change 0, we define a new reference family
(fl) by
t E [0, oo). The class -,I lc I°` with respect to ., is denoted by _4?2' . It is an important consequence of Doob's optional sampling theorem that if XE. f 1°`, then TOXE. cyl-, and if X, Y E ./21oc, then (4.16)
= TO<X,Y>.
Now we apply the method of time change to solve stochastic differential equations. In the following we consider the case of d = 1, r and,B = 0. Thus we consider for given a(t,w) (4.17)
the equation
dX(t) = a(t,X)dB(t).
We assume for simplicity that there exist positive constants C, and C2 such that (4.18)
C1 < a(t,w) < C2.
As we saw in Theorem 4.2, if we can solve the equation (4.17), then we can
solve the general equation having drift term 6(t,w)dt by the method of transformation of drift. Theorem 4.3. (i) Let b = (b(t)) be a one-dimensional (.)-Brownian motion with b(0) = 0 given on a probability space (Q,,P) with a reference family (_9';),,,0 and let X(0) be an .
-measurable random variable.
Define a continuous process g = (c(t)) by fi(t) = X(0)+b(t). Let
_
(O) be a process of time change such that (4.19)
0_ = f 0 a(0,,,T00-Zds
holds a.s. Then if we set X =T0 (i.e., X(t) = (O ') = X(0) + b(071) ) and . = g-1, there exists an (. )-Brownian motion B = (B(t)) such that (X(t), B(t)) is a solution of (4.17) on the probability space (0,57-,P) with the reference family ()rzo.
SOLUTION BY TRANSFORMATION OF DRIFT
199
(ii) Conversely, if (X(t), B(t)) is a solution of (4.17) on a probability space (0,9-,P) with a reference family then there exist a reference family (f an (.F)-Brownian motion b = (b(t)) with b(O) = 0, and a process of time change = (c$) with respect to the family (A) such that, if we set fi(t) =X(0)+b(t), then (4.19) holds a.s. and X =TOE. That is, any solution of (4.17) is given as in (i).
Corollary. Suppose we are given a one-dimensional (Y )-Brownian motion b = (b(t)) and an -measurable random variable X(0). Define = (fi(t)) by g(t) =X(O)+b(t). If there exists a process of time change 0 such that (4.19) holds and if such a 0 is unique, (i.e., if' is another process of time change satisfying (4.19), then 0(t) = V(t) a.s.), then the solution of (4.17) with the initial value X(0) exists and is unique. Moreover, the solution is given by X = TOE'.
Proof. (i) If b = (b(t)), X(0) and 0 = (O,) are given as in (i) of the theorem, then M = TOb E.A la and <M>(t) If X = TOE, then by (4.19), t = f' a(O,,X)2dO, and hence
<M)(0 = 07 1 = f o` a(O,,X)2dO, =f'0 a(s,X)2ds.
We set B(t) = f' (a(s,X))-'dM(s). Then B E .
i ra and
(t) _ f o a(s, X)-2d<M>(s) = t. This implies that B is an (F)-Brownian motion. Since M(t) = X(t) - X(0) = f' a(s, X)dB(s), (X, B) is a solution of (4.17). (ii) Let (X(t), B(t)) be a solution of (4.17) on a probability space (Q, ,9'-,P) with a reference family (.J' Then M(t) = X(t) - X(0) E
and <M>(t)=f' a(s,X)2ds. Set y/(t) = <M>(t), Or = y-' and .. _ ,7v1 1. Clearly 0 = (c,) is a process of time change with respect to )-Brownian motion (, ) and the process b = (b(t)) = (M(0 )) is an (. (Theorem 11-7.2). If we now set fi(t) = X(0) + b(t), then TOE = X. Furthermore, since t = f' a(s, X)-2dyr it follows that r = f Or a(s,X)-2dyi3 =
f
0
a(0L,X)-2du
= f a(Oy, TOO -2dU O
and consequently now complete.
satisfies (4.19). The proof of the theorem is
200
STOCHASTIC DIFFERENTIAL, EQUATIONS
Example 4.2. Let a(x) be a bounded Borel measurable function on R'
such that a(x) > C for some positive constant C. Set a(t,w) = a(w(t)). Thus we are considering a stochastic differential equation of time-independent Markovian type. If X(0) and b = b(t) are given, then equation (4.19) may be written as (4.20)
0, = fo a[T
f
o
where fi(t) = X(0) + b(t). Consequently c5, is uniquely determined from fi(t) and hence the stochastic differential equation dX(t) = a(X(t))dB(t)
is solved as X(t) = Example 4.3. Let a(t,x) be a bounded Borel measurable function on
[0,co) x R' such that a(t,x) Z C for some positive constant C. Set a(t,w) = a(t,w(t)). In this case, equation (4.19) is given as (4.21)
0, = f '
f o a[os, c(s)]-2ds.
This is equivalent to the following differential equation for 0, along each fixed sample path of g(t): (4.21)' 100 = 0.
One simple sufficient condition that (4.21)' has the unique solution is that a(t, x) be Lipschitz continuous in t; in this case the stochastic differential equation
dX(t) = a(t,X(t))dB(t) is solved uniquely as X(t) = (c$ 1) (cf. Yershov [187]). On the other hand,
Stroock-Varadhan [159] proved that the above stochastic differential equation always has a unique solution, and this fact, in turn, can be used 0, =
dt0,
SOLUTION BY TRANSFORMATION OF DRIFT
201
to show that (4.21)' has a unique solution 0, along each fixed sample path of fi(t). (cf. Watanabe [167]).
Example 4.4. (Nisio [130]). Letf(x) be a locally bounded Borel measurable function on R', a(x) be a bounded Borel measurable function on
R1 such that a(x) > C for some positive constant C, and y e R'. Set a(t,w) = a[y+ f o f(w(s))ds], differential equation is (4.22)
w E W. The
corresponding
stochastic
dX(t) = a[y+ f of(X(s))ds]dB(s)
and the equation (4.19) is given as (4.23)
' ' 0, = f 0 a(y+ f 0 f [(T #X (u)]du)-zds.
Now f 0Sf[(T OO(u)]du = f "'f
= f of
1)]du
fo
udu.
Thus (4.23) is equivalent to
, = l/a(y+ f
(4.24)
100
= 0.
and this can be solved uniquely for each given fi(t) as follows. Set
Z(t) = f 0 Then
Z(t) and hence
STOCHASTIC DIFFERENTIAL EQUATIONS
202
f a(y+Z(s))22(s)ds = f f(c(s))ds. o
Consequently, if we define A(x) by
for x > 0,
f x a(y+z)2dz, 0
A(x)
- f a(y+z)2dz,
for x < 0,
x
then A(Z(t)) = f o f(c(s))ds and therefore Z(t) = A-' [f where A''(x) is the inverse function of x - A(x). Thus 0, is solved uniquely as
0, = f a(y + Z(s))-gds = f a(y+A-'[ f o
o
and so the equation (4.22) is solved uniquely as X(t) _( '). Note that in the special case off(x) = x, equation (4.22) is equivalent to the following equation of motion with random acceleration:
(4.25)
dY(t) = X(t)dt dX(t) = a(Y(t))dB(t) Y(0) = Y.
5. Diffusion processes Diffusion processes constitute a class of stochastic processes which are characterized by two properties: the Markovian property and the continuity of trajectories. Since it is beyond the scope of this book to discuss diffusion processes in full generality,*' we restrict our attention to the class of diffusion processes which can be described by stochastic differential equations. This class of diffusions is known to be sufficiently wide both in theory and applications*2; furthermore, the stochastic calculus provides us with a very powerful tool for studying such diffusions. First, we give a formal definition of diffusion processes. Let S be a topological space. We sometimes find it convenient to attach an extra point *1 Only in the one dimensional case is a satisfactory theory known, cf. Ito-McKean [73] and Dynkin [22].
* 2 There are, however, many recent results on multi-dimensional diffusion processes which can not be covered by the method of stochastic differential equations; see, e.g: Fukushima [30], Ikeda-Watanabe [53], Motoo [124] and Orey [135].
DIFFUSION PROCESSES
203
A to S, either as an isolated point or as point at infinity if S is locally com-
pact. Thus we set S' = S U {Al. A is called the terminal point. Either S or S' is called the state space. Let W(S) be the set of all functions w:
t -- w(t) e S' such that there exists 0 < C(w) < oo with the
[0, co)
following properties: (i) w(t) E S for all t e [0,C(w)) and the mapping t e [0, C(w)) w(t) is continuous; (ii) w(t) = A for all t > C(w). C(w) is called the life time of the trajectory w. For convenience, we set w(oo) = A for every w E W(S). A Borel cylinder set in W(S) is defined for
some integer n, a sequence 0 < t, < t2 < . . < t and a Borel subset
AinS'"=S'xS'x . xS'as (
'
)
t" : W(S) --- /S'" is given (/by
where zctl ty //
7rtl.t2....,t"(w) = (x'(tl), ti' 02), ... , w\t"))
Here, of course, a Bore] subset in a topological space is any set in the smallest Q-field containing all open sets. Let .'(W(S)) be the a-field in W(S) generated by all Borel cylinder sets and let Rt(W(S)) be that generated by all cylinder sets up to time t, i.e., sets expressed in the form 7r It ,,,,r"
(A) where t" < t. A family of probabilities {P,,,x eS'} on
(W(S),.'(W(S))) is called Markovian*' if (i) PX {w; w(0) = x} = 1 for every x E S', (ii) x e S -- P,t(A) is Borel measurable (or more generally, universally measurable)*2 for each A e .'(W(S)), and (iii) for every t > s > 0, A E R,(W(S)) and a Borel subset 1' in S', (5.1)
P. c,) [w;
Px(A (1 {w; w(t) E T}) = f
w(t - s) E T]P,,(dw')
A
for every x e S'. A Markovian system {Px, x e S'} is called conservative if (5.2)
P. {w; C(w) = co} = I
for every
x e S.
In this case we need not consider the point A since almost all sample paths *1 We only consider the time homogeneous case. *2 Cf. Chapter I, §I.
204
STOCHASTIC DIFFERENTIAL EQUATIONS
lie in S. For a Markovian system {P,,,, x c= S'}, t e [0, oo), x E= S' and a Borel subset r of S', we set
P(t,x,r') = Px {w; w(t) E T} .
(5.3)
The family {P(t,x,rl} is called the transition probability of a Markovian system. By the successive application of (5.1), it is easy to see that Px[w(t1)E A1, w(t2)E=- A2,
=
(5.4)
JAP(t1, x, dx1) ,
J
.
,
w(tn)E An]
P(t2 - t1, x1, dx2) Ay
X An
J
A3
..
P(t - to-1, xn-1, dx )
for0 0, we set ,F (W(S))
(1 t+e(W (S))Px*1 and ."-..(FV(S)) = V _F7(W(S)). A mapping _ (1 t>O e>0 xcS' w E W(S) ---- u(w) E_ [0, oo) is called a stopping time if it is an (.
(W(S)))
-stopping time, i.e., if for every t > 0, {w; a(w) < t} E-=.(W(S)).
For a stopping time a, we set fl (W(S))={A E 9 (W(S)); A(1 {w; a(w) < t) E ,. (iW(S)) for every t > 01. The Markovian system (P.') is called a strongly Markovian system if for every t > 0, stopping time a, A E .a ;(W(S)) and a Borel subset r of S', we have that Px(A n {w; w(t + 6(w)) E r) ) (5.5)
= J
A
Pw'(o(w')) [w; w(t) E r']Px(dw')
for every x E: S'. *1
Definition 5.1. A family of probabilities {PP} xes, on (W(S), -9(W(S))) is called a system of di, fJusion measures, or simply a diffusion if it is a strong-
ly Markovian system.
Definition 5.2. A stochastic process X = {X(t)} on S' defined on a It follows from this definition that .9;(W(S)) is right continuous, i.e., .Y 40(W(S))
f (W(S)) for every t Z 0. *2 We need only assume that (5.5) is valid for bounded stopping time; otherwise, replace a by aAn, A by An [a S nl and then let n j oo.
DIFFUSION PROCESSES
205
probability space (Q,J ,P) is called a diffusion process on S if there exists a system of diffusion measures {P} xeS, such that, for almost all co. [t i-X(t)] E W(S) and the probability law (i.e. the image measure) on W(S) of f3, [t - X(t)] coincides with where p is the Borel measure on S' defined by p(dx) =P {co;X(0, w) E dx} and is called the initial distribution of X. If X = (X(t)) is a diffusion process and if we set C(cv) = inf {t; X(t) _
A), then it is clear that, with probability one, [0, 0 ED t - X(t) E S is continuous and X(t) = A for all t > C. C is called the life time of the diffusion process X. X is called conservative if C(w) = oo a.s.
Now, let C(S') be the Banach space of all real or complex valued bounded continuous functions defined on S' and (A,fIf(A)) be a linear operator on C(S') into itself with the domain of definitions le(A). Let {Px, xES'} be a system of probability measures on (W(S),.%` (W(S))) such that x i -- PX(A) is Borel measurable (or universally measurable) for AE W(S). The following definition is based on an idea of Stroock and Varadhan [160]. Definition 5.3. {P,,,} is called a diffusion measure generated (or determined) by the operator A (or simply as an A-diffusion) if it is a strongly Markovian system satisfying
(i)
Px {w; w(0) = x} =1
for every x,
(ii)
f(w(t)) f(w(0)) -f o (Af)(w(s))ds
is a (Px, 0,(W(S)))-martingale for every f e ff(A) and every x. Theorem 5.1. Suppose that {Px,x E S'} is a system of probability measures on (W(S),%(W(S))) satisfying conditions (i) and (ii) of Definition 5.3. Suppose further that {P.} is unique; i.e., (iii) if {P,'} is any other system of probability measures on (W(S),
.( W(S))) satisfying the conditions (i) and (ii) of Definition 5.3, then P;' = Px for every x. Then {Px} is a system of diffusion measures generated by the operator A.
Proof. It is only necessary to show that {Px} is a strongly Markovian system, i.e., it satisfies (5.5). Since
Xf(t) =.f(w(t)) -MOD- f 0 (Af)(w(s))ds
STOCHASTIC DIFFERENTIAL EQUATIONS
206
is a (Px,P1fr(W(S)))-martingale for every x E S' and t F--- Xf(t) is right
continuous, it is clear that it is also a (P.,,.F(W(S)))-martingale. If a is a bounded stopping time, then by Doob's optional sampling theorem t - Xf(t+o) is a (Px,.9+,(W(S)))-martingale. In particular, for every t > s, A E. +,(W(S)) and CE .B(P(S)), we have EX(Xf(t+o) - Xf(s+Q): A fl c) = 0. This implies that
E=(Xf(t+o) - Xf(s+o"): AI. `;(W(S))) = 0, a.a. w(P.). Therefore, if Pw(A) = P,(B;'(A)T. (W(S))), AE.6(W(S)) is the regular conditional probability given .(W(S)) where B,: W(S)--- W(S) is defined by (O,w)(t) = w(a(w)+t), then Pw(w'; w'(0) = w(v(w))) = 1 for a.a. w(PP) and Xf(t) is a (Px',.$$(W(S)))-martingale. By assumption (iii), we have P* = P,(,(,)) which clearly implies (5.5).
Corollary. Let {P., x E S'} be a system of probability measures on (W(S),. (W(S))) which satisfies the conditions (i) and (ii) of Definition 5.3. The uniqueness condition (iii) of Theorem 5.1 then follows from the weaker condition (iv): (iv) if {P,'} is any other system of probability measures on (W(S), .(W(S))) satisfying (i) and (ii), then (5.6) J W(S)
f(w(t))P.(dW)
= f W(s)f(w(t))PX(dw)
for every t > 0, x E S' and f E=-_5--, where 9 is some total family in C(S').* The equality (5.6) can also be replaced by (5.7)
f f Svgs,
o
e
z'f(w(t))dtP.(dw) = f i^"
x° f(w(t))dtPP(dw) Jo e
for every A > 0, x E S' and f G,597, where 9 is some total family in C(S'). Proof. Just as in the above proof, we have * .3 C C(S') is called total if for any two Borel probability measures u and v on S',
f,,f(x)µ(dx) = f,,f(x)v(dx) for any f E.9 implies that p = v.
DIFFUSION PROCESSES
f 'f (w'(t))P-(dw') = $
207
f(w'(t))PW( (W»(dw')
for every bounded stopping time a and hence {P} is a strongly Markovian system. Since .T is total, our assumption (5.6) implies that the transition
probability P(t, x, r) = P,(w(t)Ef) is uniquely determined. Consequently, by (5.4), PS is uniquely determined as a measure on O(W(S)), i.e. (iii) is satisfied. The equivalence of (5.6) and (5.7) is obvious.
The following theorem is an easy consequence of Theorem 5.1.
Theorem 5.2. Let (A,.(A)) be a linear operator on C(S') and let
(52,.P) and (. )rzo be given as usual. Suppose that for each x E S' there exists an S'-valued, (-F-,)-adapted stochastic process X. = (X(t)) such that (i) with probability one, X(O) = x, [0, C(w) = C(XX)) D t - X(t) E S is continuous and X(t) = d for t > C and (ii) for every f c- .V(A),
Xf(t) =f(X(t)) -f(X(0))
- f (Af)(X(s))ds
is an (.)-martingale. Let P. be the probability law of the process Xx on (W(S), .°(tP(S))) and suppose that x P. is universally measurable and, furthermore, that P. is uniquely determined for every x E S. Then {P.} XEs, is the system of diffusion measures determined by the operator A and the stochastic process X. is a diffusion process satisfying X(0) = x. We call Xx the A-diffusion process starting at x E S'.
Example 5.1. Let S = Rd and S' = Rd U {A} , where A is attached to
Rd as an isolated point. Let J(A) _ { f E C(S'); f!Rd EC;(Rd)} and define A on 11(A) by
(5.8)
Af(x) =
1 A.f(x),
X E Rd,
2 0,
x=A.
Then the operator A generates a unique diffusion {P,} ; This is just ddimensional Brownian motion i.e., P. is the Wiener measure on Wd starting
at xERd. To prove this, we first show that P., x ERd, is conservative. Indeed, J '(x) defined byf(x) I Rd - 0 and f(A)=1 belongs to O(A) and Af(x) - 0. Thus I4(w(t)) is a martingale with respect to Px for every x, and hence if
STOCHASTIC DIFFERENTIAL EQUATIONS
208
x ERd, I4(w(t)) = 0, a.a. w(P.), that is, P,(C(w) = oo) = 1. Consequently for every f e Cb(Rd) and x E Rd, f(w(t)) f(w(0))-J0' (df)(w(s))ds is a PX martingale and we can apply the same proof as that of Theorem II6.1 to show that Px is the Wiener measure starting at x. Thus the A-diffusion is just d-dimensional Brownian motion.
Example 5.2. Let S' and _T(A) be as in Example 5.1. Define A on -q(A) by
(5.9)
Af(x) =
J--4f(x)
+-
c, a
0,
(x),
,
x e Rd,
x ='d'
where c = (c,) E Rd is a constant. Then the operator A generates a unique diffusion {Ps} ; Ps is the probability law of the process X(t) = x+B(t)+ ct where B(t) is a d-dimensional Brownian motion with B(O) = 0. This diffusion is called the d-dimensional Brownian motion with drift c. This can be proved in the same way as in Example 5.1.
Example 5.3. Let S' = Rd U {d} be as in the previous examples. Let A'(A) = If (= C(S'); A IRd E C2(Rd) and f(d) = 0} and define A on 11(A) by Af(x)
(5.10)
=
2
(d f)(x) - cf(x),
x E Rd.
x =J,
0,
where c > 0 is a constant. Then, the operator A generates a unique diffusion {P} on Rd; Px is the probability law of the process X. defined as follows. Let (B(t)) (B(0) = 0) be a d-dimensional Brownian motion and e be an independent exponentially distributed random variable with mean 1/c. We define
x+B(t), Xz(t)
- A,
if
t<e,
if
t > e.
This diffusion is called the d-dimensional Brownian motion with the random absorption rate c. To prove this assertion, we first note that the system {Px} clearly satisfies the conditions (i) and (ii) of Definition 5.3. Secondly the function ff(x), E Rd, defined by
DIFFUSION PROCESSES
fi(x)=
209
e Rd,
I
x=A,
0,
is in g(A), and hence if {P,'} satisfies (i) and (ii) of Definition 5.3, then
f (w(t))- A(w(0))-J' (Afe)(w(s))ds is a Px martingale. Thus, if x E Rd, EX[er< wct»:
and so
2
C > t] = e'< x>
EX[h(w(t))]=E.'[e'q wu'>:
+c
Ex[el< .wcs >: C > s]ds*
+c>t C > t] = e'<>-'2"2.
Since { ff;
E Rd} is total, we conclude that {P;'} coincides with {PX} by the corollary of Theorem 5.1.
Example 5.4. Let S = R+: = {x = (x',x2, ... xd) E Rd; xd > 0}, S' = R+ U {A} where A is attached to R+ as an isolated point. Let &(A) _ { f E C(S') ; f IRd+ E Cb(R+) and
of
ax as = 0} where aS = {x E R+; xd = 0}
and define A on P21(A) by
(5.11)
Af(x) = 2 0,
Af(x),
x E R+, x = A.
Then the operator A generates a unique diffusion {Px} on R+; P. is the probability law of the process XX(t) defined as follows. Let B(t) = (B'(t), B2(t), ... ,B"(t)) be a d-dimensional Brownian motion with B(0) = 0 and
XX(t) = (x'+B'(t), x2+B2(t), ... , xd-'+Bd-'(t), I xd+Bd(t) I). This diffusion is called the reflecting barrier Brownian motion on R. To prove this, we first note that by Chapter III, Section 4.2 XX(t) = xk+ Bk(t)+bkdo(t) for k = 1, 2, ... , d where B(t) is a d-dimensional * Ex stands for expectation with respect to F.
STOCHASTIC DIFFERENTIAL EQUATIONS
210
Wiener martingale* and $(t) is the local time of XX(t) at 0: 0(t) Jim L
I(o,.) (X d(s))ds. Hence, by M's formula, o
J (X (t)) -f(X) (x) -
J02
df(X.(s))ds
= E J Oa
(XX(s))dBk(s)
=
(X=(s))dBk(s)
J o axk
if (Xx(s))ras(Xx(s))do(s) + f ' axd o
if f E ff(A).
Thus {Px} satisfies the conditions (i) and (ii) of Definition 5.3. To prove E Rd, the uniqueness of any such system {P,'} we set, f o r =
A(x)
11
= k-1
elf's' cos dxd,
0,
x e R+9
x=d.
Then ffE.J(A) and hence
ff(w(t)) -}f(x) - J o (A.fe)(w(s))ds 2
is a P,'-martingale. Noting that Aff = - 2I ff, we have E'f.ff(w(t))] =.ff(x) - 12L 22 J o
EX[.ff(w(s))]ds.
2
This equation implies that EX[ ff(w(t))] = exp (-
2
t) fi(x), and so since
{ ff, E Rd} is total, we can apply the corollary of Theorem 5.1 to conclude
that PX = Px. Example 5.5 ([73]). For simplicity, we consider the one-dimensional
case only. Let S=[0, oo) and S'=[0, co) U {A), where A is attached to S as an isolated point. For a given parameter y (0 < y < 1), let .f(A) = If E C(S'); f I [0,o,, E Cb([0,oo)) and (1 - y)f(0) = yf'(0)} and define A on .J1(A) by * Indeed, B'(t) = B'(t) for i 4 d, and &d(t) = f,, sgn(xd+Bd(s))dBd(s).
DIFFUSION PROCESSES
211
2
(5.12)
Af(x) = 2 Jdx2.f(x),
x E [0,00),
The operator A generates a unique diffusion {Ps} on [0, oo) which is called the elastic barrier Brownian motion with parameter y. Px is the probability law of the following process XX(t). Let x(t) be the reflecting Brownian mo-
tion starting at x and 0(t) be its local time at 0: 0(t) = lim 2s 5 o'O,C) 840
Let e be a random variable which is exponentially distributed with mean y/(1 - y) and is independent of fix. Set x(t), if t < C: = inf {t; 0(t) > e} ,
X.(t) _ I'd,
if t > C.
The proof follows by a similar argument as in Examples 5.3 and 5.4: we take for E R fF(x) =
yc cos x + (1 - y) sin fix,
x E [0, 00),
x=d.
I 0,
Example 5.6. Let S be a bounded smooth domain in Rd and S' = S U (A), where d is attached to S as a point at infinity.*' Let ff(A) = Cg(S)
twice continuously differentiable in S and tends to 0 at A}) and define A on P(A) by (5.13)
Af(x) =
2 d f(x),
x E S,
0,
x=d.
Then the operator A generates a unique diffusion {Px} on S' which is called the absorbing barrier Brownian motion on S or the minimal Brownian motion on S. P. is the probability law of a Brownian motion starting at x E S stopped at the first instant when it hits the boundary of S (which is identified with d). Again it is clear that {P.} satisfies the conditions (i) and (ii)
of Definition 5.3. Suppose {P,'} also satisfies these conditions. Let Ga(x,y), a > 0, be the Green function of S for the operator La = a - d/2 with Dirichlet boundary conditions: i.e., if f E C;(S) *2 then Gaf(x) f SGa(x, y)f(y)dy is the unique solution of *' i.e., S' is the one point compactification of S. *2 CX(S) is the set of all C'-functions f with the support S(f) c S.
STOCHASTIC DIFFERENTIAL, EQUATIONS
212
au - Zdu=f, u!as = 0.
Let f e C;(S) and set
u = Ga f.
Then u E ff1(A) and Au = au -f.
Hence
u(w(t)) - u(w(0)) - 2 f o du(w(s))ds
t > 0,
is a Px-martingale and, therefore for every
E,[u(w(t))] - u(x) = a f EX[u(w(s))]dsf E=Cf(w(s))]ds. 0 0 Consequently
f
o
--E' [u(w(t ))]dt Jo
u(x)
[ f o E,'[u(w(s))]ds]d(e a') EX'[f(w(s))]ds]d(e-a')
+ a f o[ fo
= foe a'E'[u(w(t))]dt - a f
o
e
atE' f(w(t))]dt
and hence
u(x)= f 0o e-asEX[f(w(s))]ds =Gaf(x) Now apply the corollary of Theorem 5.1. 6. Diffusion processes generated by differential operators and
stochastic differential equations Suppose that we are given a second order differential operator A on Rd:
(6.1)
Af(x) = 2
,, a"(x) as (x) + E b'(x) az (x), d
f
where a`f(x) and b'(x) are real continuous functions on Rd and (atf(x)) is d
symmetric and non-negative definite; i.e., a°f(x) = af"(x) and E a'f(x) i.f-t
A-DIFFUSION PROCESSES
213
0 for all _ (ct)ERd and all xERd. We take, as the domain of the definition of A, the space CK(R') consisting of all twice continuously differentiable functions having compact support. The notion of the diffusion process generated by the operator A (A-diffusion) was defined in the
previous section. To be precise, we formulate it again as follows. Let All =Rd U {d} be the one-point compactification of V. Every function f on Rd is regarded as a function on R' by extending it as f(d)= 0. Let Wd be defined as in Section 2 and e(w) be defined by (2.12).
Definition 6.1. By a diffusion measure generated by the operator A(or simply an A-diffusion), we mean a system {P., xERd} * of probabilities on (Wd,R(Wd)) which is strongly Markovian and satisfies (i)
for every xERd,
Px {w; w(0) = x} = 1 (ii) t
f(w(t)) - f(w(0))- f (Af)(w(s))ds 0
is a (P.,.gr(J ))-martingale for every f E C1(Rd) and every xERd. Remark 6.1. By Theorem 5.1, we know that any system {Px, x E Rd} of probabilities on (Wd,3 (Wd)) satisfying (i) and (ii) and that x Px is universally measurable is strongly Markovian and hence is an A-diffusion if it satisfies further the following uniqueness condition: (iii) if {P.,'} is another system of probabilities on (Wd, (Wd)) satisfying the above conditions (i) and (ii), then Pz = PX for all x. Also, by the corollary of Theorem 5.1, (iii) may be replaced by the following weaker condition: (iii)' if {P, } is another system of probabilities on (Wd, M(Wd)) satisfying the above conditions (i) and (ii), then
f
6.df(w(t))r'.x(dw) = f R,d.f(w(t))PP(dw)
for every t > O, x E Rd and f in a total family of functions on Rd.
Definition 6.2. A stochastic process X=(X(t)) on All is called a diffusion process generated by the operator A or simply A-diffusion process,
if almost all samples [t i -- X(t)] belong to Wd and the probability law of X coincides with
f Rd
where {Px} is a diffusion measure
* To be precise, Pa should be included in the system but, since Pa is the trivial measure a.., where wa(t) = A, we usually omit it.
214
STOCHASTIC DIFFERENTIAL EQUATIONS
generated by A and ,u is the probability law of X(0). Our problem now is to consider the existence and uniqueness of A. diffusions. Let o _ (ok(x)) E Rd©Rr be such that (6.2)
x
v(x) is continuous and a"(x) = E (Tk(x)Q,jXx) k-1
fori j=1,2,...,d. Clearly such a 6 exists for some r. We choose one such o and fix it. We now consider the following stochastic differential equation r
(6.3)
dX'(t) = E vk(X(t))dBk(t) + b'(X(t))dt, k-1
i = 1, 2, ... , d.
According to Theorem 2.3, we know that for every x E Rd there exists a solution X(t) of (6.3) such that X(O) = x. By Ito's formula (Theorem II5.1),
f(X(t)) -J (X(0)) _ E
d
Ef
0
ax,
f o (Af)(X(s))ds for every f e CK(Rd). From this, it is clear that the law P. on Wd of the process X satisfies the conditions (i) and (ii) of Definition 6.1. We shall now show that the uniqueness of solutions for the stochastic differential equation (6.3) is equivalent to the uniqueness condition (iii) in Remark 6.1. Indeed, it is obvious that (iii) implies the uniqueness of solutions of (6.3). On the other hand, if {Pz} is the system of probabilities on Wd satisfying the conditions (i) and (ii) of Definition 6.1, then we can conclude by the same proof as in Section 2 that there exists some extension
P) with a reference family (,) of the probability space (Wd,.q(Wd),Pp) with the reference family (.gt(Wd)),* and an (,7)-Brownian motion B = ((B(t)) such that setting X(t) = w(t) and e = e(w), we have for t Ei [0, e),
X'(t) = X' + k-1 f o'k(X(s))dBk(s) + f 0b'(X(s))ds, i = 1, 2, ... , d. 0 This implies that (X(t), B(t)) is a solution of (6.3) such that X(0) = x. Since * 4j(Wd) is defined in the same way as .q,(Wd).
A-DIFFUSION PROCESSES
215
the probability law of the process X(t) is clearly Px, the uniqueness of solutions of (6.3) implies the uniqueness condition (iii). Thus we have the following result.
Theorem 6.1. Let the differential operator (6.1) be given as above and choose any o- _ (a'(x)) such that (6.2) holds. Then the A-diffusion {P,,xERd} exists uniquely if and only if the uniqueness of solutions holds for the stochastic differential equation (6.3). In this case, P. is the probability law on (it'd, of a solution X= (X(t)) of (6.3) such that
X(0)=x.*1 Stroock-Varadhan's result (Theorem 3.3) implies that if (a`i(x)) is bounded, continuous and uniformly positive definite and (b°(x)) is bounded, then the A-diffusion exists uniquely; moreover, it is a conservative diffusion. In the general case when (a' (x)) may degenerate, we have by Theorem 3.1 that if we can choose the above o(x) = (o (x)) such that 6(x) and b(x) = (b'(x)) are locally Lipschitz continuous, then the A-diffusion exists uniquely. Furthermore, this A-diffusion is conservative (i.e.,
Px(e = oo) = 1, for every x E Rd) if a(x) and b(x) satisfy the growth condition jjo(x)II+IIb(x)jI < K(l+I x1) for some positive constant K. In the one-dimensional case, the Lipschitz condition on o(x) may be weakened
as in Theorem 3.2. In particular, if we can choose a(x) such that it is locally Holder continuous of exponent 1/2 and if b(x) is locally Lipschitz continuous, then the A-diffusion exists uniquely. An important question now is to determine when we can choose a sufficiently smooth o such that (6.2) holds for a given matrix a. For this we have the following result.
Proposition 6.2. (i) Let Sr be the set of all r x r symmetric, nonnegative definite matrices. If a(x): Rd --- Cam' is in the class CI(Rd),*2 then
the square root a (x) (i.e., c (x): Rd - S' satisfying the property v(x)Q(x)* a(x)) is uniformly Lipschitz continuous on Rd. (ii) If a(x): Rd -- Cam' is twice continuously differentiable, then the square root 6(x) is locally Lipschitz continuous. Proof. Clearly it is only necessary to prove (i). We shall prove this in the case d = 1; the general case follows from the fact that a function is uniformly Lipschitz continuous on Rd if it is uniformly Lipschitz con-
tinuous in each variable x` E R' for fixed * = (x',x2,
...
,
x°-',xr+l,
xd) with the Lipschitz constant independent of .9. Fix x0E R'. We
*' The universal measurability of x'--- P. is seen as in the proof of Theorem 1.1. *2 We say that a(x) is in C2(Rd) if every component of a(x) is in C2(Rd).
216
STOCHASTIC DIFFERENTIAL EQUATIONS
can choose an orthogonal matrix P such that Pa(xo)P* is a diagonal matrix. Set a(x) = Pa(x)P*. For a positive constant e > 0, let ae(x) = a(x) + al and ae(x) = Pae(x)P* = d(x) + e1.
The square roots of ae(x) _ (aeJ(x)) and a8(x) = (a J(x)) are denoted by c,,(x) = (a,"(x)) and Oe(x) = (5'(x)) respectively. Then a ,(x) and 0'e(x) Po.8(x)P* are clearly in Cb(R1). By differentiating both sides of a8' (x) _ E O'8k(x)vek(x) k-1
(6.4)
at x = x,,, we have
a,",(xo) = (a (x0) + d 1(xo))Qa1(XO).*
Let K =1Si,JSr,xcR sup
a`f(x) I =
sup
I o,'(x) I
15t.J5r,xeR1
and set f(x) =
for 2 e R. Then sincef(x) > 0 for all x E R', we have
0 < f(x + h) = f(x) + f (x)h +
f(x) + f(x)h +
f (x+Oh)h2
2
(E I A, I )2Kh2
2 and hence
f(X)2 < 2f(x) (E;_1I2,I )2K.
Letting A be 6, _ (Stk)k=1, 8f = (a k)k:, and 8, + 8, respectively, we obtain that a6 `(x)2 <
a8'(x)2 < 2Ka8J(x)
and
(ae`(x) + 2881(x) + 8,'(X))2 C 8K(ae`(x) + 2ae'(x) + Qe'(x)).
Consequently there exists a constant c(K) independent of s > 0 such that I %1 (X) I < C(K)(aet(X)112 + a8j(X)1 /2)
for all x E R'. If we set x = xO, then * For every f e C6(R'), we setl(x) = dxf(x) andr(x) = d f(x).
STOCHASTIC DIFFERENTIAL EQUATIONS
217
a-'(xo) I <- c(K)(Q'(xo) + 6E'(xo))
and combining this with (6.4), we have I iT;'(x0) I < c(K). Since oE'(xo) _ (P*Q&(xo)P),, it is easy to see that I o'e'(xo) I < rc(K). But xo is arbitrary and hence,
I d '(x) I< rc(K)
for all x E R'.
Thus I oE'(x) - Qtl(y) I < rc(K) I x - y 1. Letting s 10, we conclude that I o'f(x) - a" (y) I < rc(K) I x - y 1. Corollary. If the coefficients of the differential operator (6.1) satisfy (i) a'f(x) is twice continuously differentiable, i, j = 1, 2, ... , d, and (ii) b'(x) is continuously differentiable, i = 1, 2, ... , d, then the A-diffusion exists uniquely.
7. Stochastic differential equations with boundary conditions In the previous section we discussed a class of diffusion processes described by second order differential operators. If we consider the case of a domain with boundary, a diffusion is usually described by a second order differential operator plus a boundary condition. A general class of boundary conditions was found by Wentzell [175]. Here we will discuss the construc-
tion of such diffusions by means of stochastic differential equations.*' For simplicity we only consider diffusion processes on the upper half space
R+, d> 2. So let D = R+ = {x = (x', x2, ... , xd); xd > 0}, aD = {x E D; xd = 0} be the boundary of D and D = {x E D; xd > 0} be the interior of D. Suppose we are given a second order differential operator on D acting on C, (D):*'
where a'f(x) and b'(x) are bounded continuous functions on D and (a'f(x)) is symmetric and non-negative definite. Suppose also we are given a boundary operator of the Wentzell type; i.e., a mapping from C' '(D) into the space of continuous functions on aD given as follows: *1 [49], [166] and [169].
R2 CX(D) = the set of all twice continuously differentiable functions with compact support.
STOCHASTIC DIFFERENTIAL EQUATIONS
218
a'f(x)
Lf(x) = 2
d- I
asj (x) + E f`(x) az (x)
a
(7.2)
xE
+ 5(x) of (x) - p(x)Af(x),
aD,
where a`f(x), fi'(x), $(x) and p(x) are bounded continuous functions on 8D
such that (a'f(x))t and p(x) > 0.
',
is symmetric and non-negative definite, b(x) > 0
Definition 7.1. By a diffusion measure generated by the pair (A,L) of
operators given above, or simply (A,L)-diffusion, we mean a system {Pz, x E D} of probabilities on (W(D),.' (W(D))) *1 which is strongly Markovian (cf. Section 5) and satisfies the following two conditions: (i) Pz {w;w(0) = x} =1 for every x ED; (ii) there exists a function ¢(t,w) defined on [0, oo) x W(D) such that
a) for a.a. w(P.), 0(0,w) = 0, t
¢(t,w) is continuous and non-
decreasing, and
f ` IaD(w(s))do(s,w) = O(t,w),
for all t > 0,
0
b) for each t > 0, w - q' (t,w) is
*2
f(w(t)) - f(w(0)) -J ' (Af) (w(s)) ds --f o (L.f)(w(s))d0(s,w)
(7.3)
is a (PI( W(D)))-martingale for every f ECK(D) and
f IaD(w(s))ds = f ' p(w(s))dO(s,w)
(7.4)
a.s. (Pr).
o
Remark 7.1. Suppose (A,L) and (A',L') are two pairs of operators as above such that Af(x) =A'f(x) and Lf(x) = c(x)L'f(x) for ail fE CK(D), where c(x) is a positive continuous function on aD. Then an (A',L')diffusion is also an (A,L)-diffusion since
f (LJ)(w(s))d0'(s,w) = 5' o
W(D) = C((O, oo) - D) = the space of all continuous functions w: [0, oo) E) t F--w(t) e D with the topology of uniform convergence on bounded intervals. #2 .q,(W(D)) is the a-field on W(D) generated by cylinder sets up to time t.
STOCHASTIC DIFFERENTIAL EQUATIONS
219
where ¢(t, w) = f c(w(s))d¢'(s, w) and ¢ satisfies the conditions a) and b) of Definition 7.1. Consequently, there is one degree of freedom in defining the operator L. Remark 7.2. If we define another boundary operator L' by
L'f(x) = Lf(x) + p(x)Af(x) (7.5)
1
=
a-1
a'J(x) .
i
a-i
ay f
axfax (x) +
(x) + a(x) of (x)
fl`(x) aX,
then the expression (7.3) is given as follows:
.f(w(t))--f(w(0)) -J o (A.f)(w(s))ds- f a (L.f)(w(s))d0(s,w)
=.f(w(t))-f(w(0)) -f o I.d(w(s))(Af)(w(s))ds -$ ' IaD(w(s))(Af)(w(s))ds
-f' (Lf)(w(s))d0(s,w)
= f(w(t)) f(w(0)) - f o I,n(w(s))(Af)(w(s))ds -5' p(w(s))(Af)(w(s))d0(s,w) f ' (Lf)(w(s))d0(s,w) (by (7.4)),
=f(w(t))-f(w(0)) -5' Ib(w(s))(Af)(w(s))ds
- f'
(L'f)(w(s))dO(s,w).
Thus (7.3) is equivalent (under (7.4)) to the statement that
f(w(t)) - f(w(0)) - 5' Ib(w(s))(Af)(w(s))ds (7.3)'
- f (L'f)(w(s))d0(s,w) o
is a (Ps, A(W(D)))-martingale for every f E CX(D).
Definition 7.2. A continuous stochastic process X=(X(t)) on D is called a diffusion process generated by the pair of operators (A,L), or simply
(A,L)-diffusion process, if the probability law of X on (W(D),_q(W(D)))
STOCHASTIC DIFFERENTIAL EQUATIONS
220
coincides with where {Pr} is a diffusion measure JD generated by (A,L) andp is the probability law of X(0).
Remark 7.3. By Theorem 5.1, we know that any system {Pz, xED} of probabilities on (W(D),.T(W(D))) satisfying (i) and (ii) of Definition 7.1 and that x i -- P,, is universally measurable is strongly Markovian and hence is an (A,L)-diffusion if it satisfies further the following uniqueness condition: (iii) if {P,'} is another system of probabilities on (W(D),.°.(W(D))) satisfying the above conditions (i) and (ii), then P,, = P. for every xE D.
Now we will discuss the existence and uniqueness of (A, L)-diffusions by the method of stochastic differential equations. First we will formulate a stochastic differential equation which describes an (A, L)-diffusion pro-
cess. For this, we choose a(x) = (ak(x)): D -- Rd®Rr and a(x) _ (z',(x)): 8D -- Rd-1®R' which are continuous and (7.6)
a"(x) = E ak(x)ak(x), i, j = 1, 2, ... , d, k=1
and (7.7)
a`J(x) 1=1
i, j = 1, 2, ... , d - 1.
Consider the following stochastic differential equation r
dX'(t) _ E a'(X(t))Ib(X(t))dBk(t) + b'(X(t))ID(X(t))dt k=1
+ E r(X(t))I8D(X(ti=
))dO(t),
1,2, --- d- 1,
(7.8)
dX"(t)
r
k=I
ak(X(t))ID(X(t))dBk(t) +b d(X(t))Ib(X(t)dt
+ a(X(t))dO(t), I0D(X(t))dt = p(X(t))dO(t).
An intuitive meaning of this equation is as follows. 0(t) is an increasing process which increases only when X(t) is on the boundary aD and is called
the local time of X(t) on aD. dO(t) acts only when X(t) E aD and causes
STOCHASTIC DIFFERENTIAL EQUATIONS
221
the reflection at aD. {Bk(t), M,(t)} is a mutually orthogonal*' system of
martingales such that d(t) = dt, k = 1, 2, ... , r, and d<M'>(t) = d6(t), I = 1, 2, ... , s, i.e., B is an r-dimensional Brownian motion in the ordinary time, and M is an s-dimensional Brownian motion if the time is measured by the local time fi(t). The function p(x) represents the rate of sojourn of X(t) on the boundary: note that p(x) =_ 0 if and only if Jo IaD
(X(s))ds = 0 for every t > 0 a.s. In this case, we say that the boundary aD is non-sticky; otherwise it is called sticky. A precise formulation of (7.8) is as follows.
Definition 7.3. By a solution of equation (7.8)*2 we mean a system of stochastic processes 3E = [X(t) _ (X'(t), X2(x), ... , Xd(t)), B(t) = (B'(t), B2(t), . . . , Br(t)), M(t) = (M'(t),M2(t), ... , MS(t)), fi(t)] defined on a probability space (0,7-,P) with a reference family (. )rzo such that (i) X(t) is a D-valued continuous (.F)-adapted process, (ii) 0(t) is a continuous (. )-adapted increasing process such that 0(0) = 0 and f 'o IaD(X(s))d¢(s) = 0(t),
t > 0,
a.s.,
(iii) {B(t), M(t)} is a system of elements in .4'° ° such that (t)
bkjt, = 0 and <M', Mm>(t) = 8,m¢(t), and (iv) with probability one,
X`(t) = X`(0) +k-l E f'0 ck(X(S))Ib(X(S))dBk(s) + f 0b`(X(s))Ib(X(s))ds
+ 1-1 f 0 za(X(S))IaD(X(S))dM'(s) + f
fi'(X(s))IaD(X(s))dO(s)
i = 1, 2, ... , d - 1,
(7.8')
Xd(t) = X d(o)
o
+ F f ' c(X(s))II (X(s))dBk(s) kel
0
+ f o bd(X(s))Ib(X(s))ds +fo a(X(s))d¢(s),
f Ia.(X(S))ds 0
= f' p(X(s))do(s) 0
Definition 7.4. We say that the uniqueness of solutions for (7.8) holds With respect to the random inner product < , > of Definition 11-2.1. * Also we call it a solution corresponding to the coefficients [a, b, T, 9, E, p].
STOCHASTIC DIFFERENTIAL EQUATIONS
222
if whenever T and 3' are any two solutions of (7.8) whose initial laws coin-
cide, then the probability laws of X=(X(t)) and X'=(X'(t)) on (W(D), 91(W(D))) coincide.*
The following theorem can be proved in almost the same way as Theorem 6.1. Theorem 7.1. Let the differential operator A and the boundary operator L be given as above and choose continuous a and r satisfying (7.6) and (7.7). Then the (A,L)-diffusion { P,,, x e D } exists uniquely if and only if, for every probability p on (D, ,91(D)) there exists a solution of
(7.8) such that the probability law of X(0) coincides with p and the uniqueness of solutions holds for (7.8). Px is the probability law on (W(D), .91(W(D))) of a solution X(t) of (7.8) such that X(0) = x. Theorem 7.2. We assume for the stochastic differential equation (7.8)
that or, b, T, 6, b, p satisfy the following: a and b are bounded and Lipschitz continuous on D, z, /3 and S are bounded and Lipschitz continuous on aD and p is bounded and continuous on aD. Furthermore, we assume that a satisfies add(x) _ E ad(x)ad(x) > c, k-1
(7.9)
x E 8D,
and
8(x) > c, x e aD for some positive constant c. Then for any probability p on (D, .9(D)) (7.10)
there exists a solution X(t) such that the probability law of X(0) coincides with p. Furthermore, the uniqueness of solutions holds for the equation (7.8).
Corollary. For a given pair of operators (A,L) satisfying (7.9) and
(7.10), suppose that we can choose or and T for some r and s such that (7.6) and (7.7) hold and a, b, T, /i, 8, p satisfy the assumption of Theorem 7.2. Then (A,L)-diffusion exists uniquely. Proof of Theorem 7.2. Let c(x) be a continuous function on aD such
that c, < c(x) S c2, x E aD, for some positive constants c1 and c2. Then it is easy to see that X = [X(t), B(t), M(t), 0(t)] is a solution corresponding to [a, b, T, /B, 6, p] if and only if _ [X(t), B(t), M(t), ¢(t)] with X(t) = X(t), B(t) __ B(t),
M(t) = fo {
c(X(s))}-1dM(s),
¢(t) =f {c(X(s))}-'ds
* We sometimes call the process X = (X(t)) itself a solution of (7.8).
STOCHASTIC DIFFERENTIAL EQUATIONS
223
is a solution corresponding to [a, b, ,/ca, c/i, 6, cpJ. Under the assumption (7.10), therefore, we may always assume that b(x) is normalized to be 8(x) __ 1. We will prove the theorem in the following three steps. (1°) The case of non-sticky boundary; i.e., p(x) __ 0 and
,(x) 1,
ak(x) = 0, k=2,3,
, r, and bd(x) = 0. (2°) The case of non-sticky boundary i.e.; p(x) = 0. (3°) The general case. (1°) The case of p(x) = 0, ad(x) 1, add(x) = 0, k = 2, 3, ... , r,
and bd(x) - 0. First we show the existence of solutions. Letp be a given Borel probability on D. On a probability space we construct the following three objects such that they are mutually independent. (i) x(0) = (x'(0),x2(0), ... ,x'(0)), a D-valued random variable with the distribution p, (ii) B(t) _ (B'(t),B2(t), ... ,B,(t)), an r-dimensional Brownian motion with B(0) = 0 and (iii) B(t) _ (B1(t),B2(t), ... ,A'(t)), an s-dimensional Brownian motion with B(0) = 0. Define ¢(t) and X"(t) by (7.11)
t < o0: = min {t; B'(t)+xd(0)=0} , 0, fi(t) _ {-min(B'(s)+xd(0)), t > go a0gs
and (7.12)
Xd(t) = xd(0) + B'(t) + 0(t).
As we saw in Chapter III, Section 4.2, Xd(t) is a reflecting Brownian mo-
tion on [0,oo) and ¢(t) is the local time of X11(t) at 0: c(t) = lim2s1 J0 810 Iio,,)(Xd(s))ds. Next define M(t) = (M'(t),M2(t), ... ,Ms(t)) by M(t) _ where -'F;' is the a-field generated by x(0) .9(0(t)). Set .9 = (1. and {B(u),M(u)}.«. It is then clear that {B(t),M(t)} is a system of elements in 412'°` satisfying -the conditions in (iii) of Definition 7.3. Consider the following stochastic differential equation for X(t) = (X'(t), X2(t), ... ,Xd-l(t)):
dX'(t) =
ak(X(t), Xd(t))dBk(t) + b'(X(t), Xd(t))dt km]
224
STOCHASTIC DIFFERENTIAL EQUATIONS
(7.13)
+ E zt(X(t), 0)dM'(t) + /3'(X(t),0)d0(t), 1-1
i = 1, 2, ... , d - 1.
X'(0) = x'(0),
By Theorem 111-2. 1, the solution 1(t) exists uniquely. X(t) _ (X(t), Xd(t)) is a continuous D-valued process satisfying f' IaD(X(s))ds = f 1, (Xd(s))ds = 0 for every t > 0 a.s. and f oIaD(X(s))do(s) = f ol,a,(Xd(s))d¢(s) = ¢(t) for every t > 0 a.s. In particular,
I,y(X(t))dBk(t) = dBk(t),
k = 1, 2, ... , r,
and I$(X(t))dt = dt. Consequently 3E_ [X(t),B(t),M(t),5(t)] is a solution of (7.8). Next we show the uniqueness of solutions. The equation (7.8) implies
that dXd(t) = dBk(t) + d¢(t)
and by Theorem 111-4.2, Xd(t) and 0(t) are uniquely determined from Xd(O) and BI(t) as (7.12) and (7.11). By Theorem 11-7.3, {B(t), B(t) = M(O-'(t))} is an (r+s)-dimensional Brownian motion which is independent of X(0). Therefore, the probability law of [X(0), (B(t)), (M(t))] is uniquely
determined from the law u of X(0). Since the solution X(t) of (7.13) is unique and is constructed as in Theorem 111-2.1, it is clear that the law of X=[X(t) = (X(t), Xd(t)), B(t), M(t), 0(t)] is uniquely determined from the law p. (2°) The general non-sticky case: p(x) - 0.
First we discuss some transformations of solutions. (a) Transformation of Brownian motion. Let 3E=[X(t),B(t),M(t),¢(t)] be a solution on a space (Q,,7-,P) with (3;) corresponding to the coefficients [a,b,r,fl,0]. Let p(x) = (pf(x)):
D - 0(r) be a continuous function defined on D with values in the r-dimensional orthogonal group 0(r). Set
Bk(t) = Z f J-1
' o
p,-(X(u))dBJ(u),
k = 1, 2, ... , r.
Then B(t) = (Bk(t)) is an r-dimensional (Y )-Brownian motion (Example II-6.1) and 3r = [X(t), B(t), M(t), 0(t)] is a solution on (Q,, 7-,P) with (-57;) corresponding to the coefficients [8, b, z, fl, 0], where S = up-'. The transformation 3r -- t is called a transformation of Brownian motion deter-
STOCHASTIC DIFFERENTIAL EQUATIONS
225
-D-
t. Clearly X is also obtained from t by transformation of the same type determined by p ' : t mined by p and is denoted by X
(b) Time change.
Let X = [X(t), B(t), M(t), fi(t)] be a solution on a space (Q,.7-,P) with (.
) corresponding to the coefficients [a, b, r, f, 0]. Let c(x) be a con-
tinuous function on D such that c, < c(x) < c2 for some positive constants c,, c2. Set A(t) = f o c(X(u))du and denote by A-1(u) the inverse of A(t). Let X(t) = X(A-'(t)), B(t) _ (Bk(t)) whereB"(t) = f o t
dBk(A-'(u)), M(t) = M(A-'(t)) and ¢(t) = ¢(A-'(t)). Also set .9 _ Then we see at once (cf. Chapter III, Section 1 or Section 4.2) that
_ [X(t),B(t),M(t),¢(t)J is a solution on (52,.9 ',P) with (. ) corresponding to the coefficients [c-12a,c-'b,r,fi,0]. The transformation X - i is called a transformation of time change determined by c and is denoted by X°) i. Clearly X is also obtained from by transformation of the same type determined by c-1:
.
C-1
(c) Transformation of drift.
Let X = [X(t), B(t), M(t), ¢(t)] be a solution on (Q,,9',P) with (.)* corresponding to the coefficients [a, b, z, fl, 0). Let d(x) = (d'(x), d2(x), ... , d'(x)) be a bounded R'-valued continuous function defined on D and set p(t) = exp [
f
dk(X(s))dBk(s) o
2
f dk(X(s))2ds]. o
Then p(t) is a positive (.9i)-martingale and P = u P is defined by Defini-
tion 4.1. Set E(t) = [X(t), B(t), M(t), 0(t)], where . ' (t) = Bk(t) f o dk(X(s))ds, k = 1, 2,
-
... , r. It is easy to see from Theorem 4.1 that i(t)
with (.9i) corresponding to the coefficients is a solution on [a, b = b-kid, z, A, 0]. The transformation X -- fE is called a transformation of drift determined by d and is denoted by `d . Clearly X is also obtained from i by transformation of the same type determined by
-d;
`='d
X.
With these preparations completed, we will now show the existence and uniqueness of solutions in the case p - 0. Let a, b, z and fl satisfy the assumptions in Theorem 7.2. Then there exists p(x): D - 0(r) such that each component of p(x) is Lipschitz continuous and * We may assume without loss of generality that (&2,,r) is a standard probability space.
STOCHASTIC DIFFERENTIAL EQUATIONS
226
*
cr(x)P(x)-' = *
where ai(x) = (6k(x))ke, is the i-th row of a(x) and
e(x) = JkE
1(71, W1 2,
i = 1, 2, ... , d.
Indeed, p, (x) = Qd(x)/ I Qd(x) I : D --- Sr1 = {xE= R'; I X I = 1 } is Lipschitz
continuous. We choose pk(x): D ---- S'-1, k = 2, 3, ... , r such that pk(x) is Lipschitz continuous and the system [p,(x), p2(x), .. , p,(x)] is orthonormal in R' for every x E D. Such a selection of pk(x) is always
possible. Then p(x): D -- 0(r) whose k-th row is pk(x), k = 1, 2, r, is what we want. Next set c(x) = Qd(x) 12 and define d(x) = (d'(x),d2(x),
... ,
... , d'(x))
by
d'(x) = -bd(x)/c(x) and d'(x) - 0, i = 2, 3, ... , r. Let X be a solution corresponding to the coefficients [o, b, z, /3, 0], If we operate on 3r by the successive transformations
(4) 3e, p
(b) c
' 3E2
(c) ' 3E3, d
then 3r3 is a solution corresponding to the coefficients [8, b, i, if, 0], where
8=(Qp-')/,/c,b=c-1 b+Sd,i=randf=/3. Clearly 81"(x) - 1, ad(x) = 0, k = 2, 3, ... , r, and P(x) - 0. By the result of case (1°), the law of 3r3 is uniquely determined from the law P of X(0). Since 3r3
(cd_
3.-2
(b) - 3r, (a)
3E,
P-1
the law of 3r is uniquely determined from p. This completes the proof of the uniqueness. The existence of solutions is also clear. We know the existence of a solution 3r3 corresponding to [8, b, f, if, 0]. Consequently 3r
is obtained by the above transformation. (3°) The general case. Let [Q, b, r, /3, p] satisfy the conditions in Theorem 7.2. Construct a solution 3r = [X(t), B(t), M(t), 0(t)] on a space (52, with (-5F;) cor-
STOCHASTIC DIFFERENTIAL EQUATIONS
227
responding to [oi, b, -c, fi, 0]. Taking an extension of Q if necessary, we may
assume that there exists an r-dimensional Brownian motion B* = (B*(t)) on 0 which is independent of t. Let A(t) = t + f' p(X(s))do(s) and A-1(t)
be the inverse of t i A(t). Set X(t) = X(A-'(t)), M(t) = M(A-1(t)), j(t) = ¢(A-'(t)) and .l = ,,a;l_,(,) V a {B*(s); s S t). Also set B(t) = B(A-'(t)) + f IaD(X(s))dB*(s) Then = [X(t),B(t),M(t),¢(t)] is a solution corresponding to [a,b,z,fl,p]. This can be proved easily if we note the following relations '0
A-1(t) = f I,y(X(s))ds and
f
o
IaD(X(s))ds =
fa p(X(s))dj(s).
These are the consequences of f I,y(X(s))dAs 0
= f Ij,(X(s))ds = t 0
and f r IaD(X(s))dA: 0
= f p(X(s))do. 0
Next we show the uniqueness of solutions. Let t = [X(t), B(t), M(t), ¢(t)] be any solution corresponding to [a, b,,r, f3, p]. Set A(t)= f oI$(X(s))
ds. Then t i---- A(t) is strictly increasing a.s. Indeed if this is not true, there exist 0 < r, < r2 such that if we set S2r,,r, = {co;A(r1)=A(r2)} , then P(S2rl,r,) > 0. But «Q.r1,
r2 c
{r2 - r, = f
a.s.
c{
Ms.
r2
r,
r2
r2
IaD(X(s))ds = f ri P(X(s))dj(s)} rl
dc(s) > 0}
and SGr1,r2 c {Ib(X(s)) = 0 for all s e [r1, r2]} a. s.
c { ke1 f r2Qk(X(s))Ib(X(s))dBk(s) =0 rl
a.s.
and f r2 ba(X(s))1,y(X(s))ds = 0). r,
Therefore,
228
STOCHASTIC DIFFERENTIAL EQUATIONS
Sl, ,z C {1' (rz) =Xd(ri)-f (rz)-S(ri) > xd(ri)} MS.
C {X(rz) ED}. But this is clearly a contradiction.
Thus the inverse A-'(t) oft A(t) is continuous. Set ae = [X(t) _ (A_'(t))] X(A-'(t)), Is(X(s))dB(s), M(t)=-M(A-'(t)), 0(t) _ Then it is easy to see that X is a solution corresponding to [a, b, r, fl, 0]. Also,
t = f ' Ib(X (s))ds + ft IOD(X (s))ds = A(t) + 0
0
f' p(X(s))d¢(s) 0
and hence
A-'(t)
= t + f 0 p(X(s))dc(s)
This implies that t is obtained from X as above. Since the law of X is unique, this implies that the law of t is also unique. Thus, we have constructed a general class of (A,L)-diffusion processes by means of stochastic differential equations. But we assumed that d(x) >
0 everywhere on aD and normalized it so that 8(x) = 1. From a probabilistic point of view, this assumption is too restrictive and should be
weakened to the condition that 8(x) + p(x) > 0 everywhere on 0D. Roughly speaking,a(x) > 0 implies that there is reflection at x and p(x) > 0 implies that there is sojourn at x. Therefore it is intuitively clear that 8(x) = p(x) = 0 is impossible but 8(x) = 0 and p(x) > 0 may be allowed. We can give another method of constructing (A,L)-diffusion processes which covers the general case of 8(x) + p(x) > 0. This method, which is similar to the one given in Chapter III, Section 4.3, consists in piecing together excursions from the boundary to the boundary. As we shall see, the probabilistic structure of the diffusion is clearly revealed by this type of construction ([170], [174] and [229]). For simplicity, we consider the case of A = A/2 leaving the general
case to [174] and [229]. So let D = R+, A = A/2 (i.e., a°"(x) = 8j, b'(x) = 0) and let L be given by (7.2). We assume that (7.14)
inf[p(x) + a(x)] > 0. XEaD
Assume furthermore that there exists a(x) = (z`,(x)): aD - Rd-' Qx R. such that
229
STOCHASTIC DIFFERENTIAL EQUATIONS J
a`J(x) =
(7.15)
1-1
r (x)Tj(x),
x E aD,
i, j = 1, 2, ... , d - 1,
and assume that all the functions z;(x), fi'(x),,u(x) are Lipschitz continuous on aD. Let 7i (D) be the totality of all continuous functions w: [0, oo) --- D
with w(0) = 0 such that there exists a(w) > 0 having the property that if 0 < t < a(w), then w(t) E D and if t > a(w), then w(t) = w(a(w)) E aD. Let R(7//^o (D)) be the a-field generated by Borel cylinder sets and let n
be the a-finite measure on ((D),R(V'o(D))) defined as follows. In Chapter III, Section 4.3, we defined the path space V-1. and the a-finite measure n+ on ( +, 6B(dY*)). Let P0 be the Wiener measure on Wo-1 starting at 0, and define n as the image measure of P0 x n+ under the map Wo-1
x ?y+
where co;
(co, w) - ((Do(w), w) E7P(D)
is defined by co;(w)(t) = d-1
K +(tex) = II
Aa(w)). If we set
(xl12
1
2nt
exp
2
n0 xd exp -
2t
(x'1)2
2t
for t> 0, x= (x1, x2, ... , xd)e D, and d-1
P° (t,x, y) _ 1
1
27rt exp (
1 (x` - yi\ ( ( l 2t 1) /27ct leXp 1
- exp (- (xd
22))
then n is the unique measure on (2Y(D), . n({w;1rwAi (t1) E A1, w(t2) E A2,
=J
for
(x4 - d121 l
2t } 1
I
t > 0, x, y E D,
(7.1/'o(D))) such that
... , w(t°) E A.))
K+(t1, x1)dxl $AyP°(t2 - t1, x1, x2)dx2 $.!g . X
P (tn - 4-1 xn-1f xn)dxn
for 0 < t1
< t and Al E .q(D). Let //'(D) be the totality of continuous paths w: [0, co) -- D such that w(0)EaD and w(t) = w(t A a(w)), where a(w) = inf { t > 0; w(t) = aD }. Let 0 denote the constant path 0(t) = 0 E aD. Let us define a mapping TT : V'0(D) - Wl o(D) U ( 0 } for every c > 0 by
STOCHASTIC DIFFERENTIAL EQUATIONS
230
(7.16)
c > 0, c=0
cw(t/c2),
(Tw)(t) = 10,
and let us also define a map 0: aD X W;(D) EB (x, w) i -- O(x, W) E 2Y(D) by (7.17)
t > 0.
-P(x,w)(t) = x + (TS(x) w)(t),
Clearly P(x,w)(0) = x and c[i(x,w)] = $(x)2ar(w). Let us define 0: aD x (D) 2 E) (x, w) F--- 0(x,w) E aD by (7.18)
O(x,w) = 0(x,w)(o[-P(x,w)]) - x = 8(x)w(6(w))
It is easy to see that for every x, y E aD
f
I O(x,w) - c(y,w) 12 n(dw)
9rp(D) f1 to(w)
= 1 a (x) - J(Y)12 (7.19)
=(d-1)V
f
I w(o(w)) 12n(dw)
? 0(D) f) (o(w)Sll
1 o(x)-8(y)12*
(i) 9?, C.> o; an increasing family of sub o-fields of .91 and a d-dimensional (fir)-Brownian motion B(t) _ (B'(t)) with B(0) = 0, (ii) an s-dimensional (.'7;)-Brownian motion B*(t) _ (B°(t)*) and (iii) an (J'9-;)-stationary Poisson point process p on (D))) with the characteristic measure n. We shall now construct a path function of an (A,L)-diffusion process. Let x e D be given as the initial point. Firstly, we set * Note that n(iw; a(w) a dt, w(o(w)) (=- dxj)
=
(2nt')-1/zdt(2tct)-(0-1)'2
exp
(- I2t
z
)dx,
t > 0, x e 8D.
STOCHASTIC DIFFERENTIAL EQUATIONS
Xx(t)
(7.20)
= x + B(t)
231
for t < Qo,
where &0 = inf It > 0; x + B(t) E 0). Set o = Xx(oo). Then co is an (.J)-measurable dD-valued random variable. Secondly, we solve the following stochastic differential equation of jump type for the process fi(t) _ (fi(t)) on aD:
V(t)=0, S`(t) = SO +
+f
r+
(7.21)
+
0
f' fl
f r0
1-1
0
f
p0(D)
(s))ds
o`(c (s-), w)1,(w)s11 ND(dsdw)
t+
0
0(D)
O'(c(s-), w)Ito(w)>ll ND (dsdw),
i = 1, 2, . .. , d - 1. (7.21) is a stochastic differential equation of the jump type which will be discussed in Section 9. Noting the Lipschitz continuity of z and ft, (7.19) and that n({w; Q(w) > 1}) = j 7 (2at 3)-1l2dt < cc, we can apply Theorem 9.1 to conclude that fi(t) is determined uniquely as an (-O';)-adapted rightcontinuous process on aD with left-hand limits. Thirdly, set
A(t) = co + ,) 0
(7.22)
f
0(D) er[
NN(dsdw) + f o
Qo +s$r, E sE6D,(t(s-))2Q[P(S)] +
f
r 0
It is easy to show using (7.14) that A(t) is an (.7)-adapted right-continuous process such that t -- A(t) is strictly increasing and lim A(t) = 00 rtm
a.s. For every t > 0, there exists a unique s > 0 such that A(s-) < t < A(s). If s = 0, i.e., 0 < t < co, X1(t) was already defined by (7.20). If s > 0 and A(s-) < A(s), then this implies that s e D, and we set Xx(t) =
(7.23)
0
(7.24)
and A(s-) = A(s), then t(s) = c(s-) and we set
Xx(t) = (s).
In this way we have defined a stochastic process Xx(t); it is obvious
STOCHASTIC DIFFERENTIAL EQUATIONS
232
by the way of construction that t p- Xx(t) is continuous a.s. The remaining problem is to show that Xx(t) is an (A,L)-diffusion process and it is unique. We can show that this Xx(t) satisfies (7.8) by using the results
in [234] or, we may argue as follows: We can show the existence of solution X(t) to (7.8) by a similar tightness argument as in Section 2. We can show, by decomposing any such solution X(t) into excursions { X(t), t c ea } where eQ is one of intervals (A(s-), A(s)) with A(t) = 6-'(t), that X(t) is obtained as explained above from a Poisson point process of Brownian excursions p and auxiliary Brownian motions B(t) and B*(t). From this we can conclude the uniqueness of solutions to (7.8) and, at the same time, that the above constructed process X11(t) actually satisfies (7.8). For the details, we refer to [229].
8. Examples Example 8.1. (Linear or Gaussian diffusions). Let or = (ok) be a constant
d X r-matrix and
E fW, x =(x',x2, ... k-1
be a constant d x d-matrix. Set Y(x) _ xd) E Rd. Consider the following stochastic
differential equation (8.1)
dXr, = E vkdBB + b'(XX)dt, k-1
i = 1, 2, ... , d,
or in matrix notation, (8.1)
dX, = odBr + flX,dt.
We know by the general theory (Theorem 3.1) that the solution exists uniquely; it is given explicitly as follows. Let Ik
k
k-o k!
Then the solution X(t) of (8.1) is given as r0
(8.2)
X(t) = ear(X(O) + f e`.8° odB(s)),
or in component form,
EXAMPLES
XI(t)
(eflt)i {X'(0)
+ E kE J r
233
(e-P')1 o dBk(s)} . 0
The proof is easily seen from the relation
d(e ftX(t)) = e P(dX(t) -,9X(t)dt) = e-fl, adB(t).
In particular, if the initial value X(O) is Gaussian distributed, then X(t) is a Gaussian process.* For example, if d = 1 and
dX(t) = dB(t) - yX(t)dt
(8.3)
(y > 0),
X(t) is solved as (8.4)
X(t) = e-vt(X(0) + f evsdB(s)).
(see also Example 2.1 of Chapter III, Section 2). Suppose that X(0) is Gaussian distributed with mean 0 and variance a2. Then the covariance of X(t) is given as E(X(t)X(s)) = e-V (t+s) a2 + f e-r " e-) "u) du o
Qz _ 1 a-v (t+s) + 1 e-r U-e> 2y) 2y '
if t > s.
In particular, if 62 = 1/2y, X(t) is a stationary Gaussian process [18]. The equation (8.3) is known as Langevin's equation and the solution X(t) in (8.4) is known as Ornstein-Uhlenbeck's Brownian motion.
A slight general equation (8.5)
dX(t) = (aX(t) + b)dB(t) + (cX(t) + d)dt
can be solved in a similar way where a, b, c and d are real constants. First, we note that (8.5) is equivalent to
dX(t) = aX(t)odB(t) -
a(aX(t) + b)dt + bdB(t)
2 + (cX(t) + d)dt
(8.6)
= aX(t)edB(t) + (c
-
a2) X(t)dt + bdB(t) 2
* By the definition of solutions, X(O) and B(t) are always independent.
STOCHASTIC DIFFERENTIAL EQUATIONS
234
+ (d- Zab)dt. If M(t) = exp J_aB(t) - (c - 2 a2t then dM(t) = -aM(t)odB(t) - (c -
ab)M(t)dt
and consequently
M(t)-'odM(t)
[adB(1) + (c - -,La 2 )dt] .
Now (8.6) is equivalent to
dX(t) = -X(t)M(t)-'odM(t) + bdB(t) + (d -
ab)dt
2 d(M(t)X(t)) = W(t)-dB(t) + (d - 2 ab)M(t)dt. Therefore X(t) is solved uniquely as (8.7)
X(t) = M(t)-'[X(0) + b f M(s)odB(s) + (d - 2 ab) f M(s)ds] a
u
where M(t) =exp {-aB(t) - (c - 2 a2)t}. Similarly if we consider a multi-dimensional stochastic differential equation d
dX'(t) =
(,E LD,X/(t)+co)odBP(t) +(E L`o/X/(t)+co)dt P=1
X`(0) = x`,
/=1
/_I
i = 1, 2, ... , d,
where Lp/, i, j = 1, 2, ... , d, p = 0, 1, ... , r, are constants such that L;/ = 0 for i > j, then the solution is given by
EXAMPLES
Xd(t) =
235
Md(t)-1(xd + fo Md(s)odrtd(s))
with rjd(t) _
d
=I
c,,BD(t) + cgt and
X'(t) =
M`(t)-1(x
+ J o M'(s)odii(s))
with rt`(t)
!E L
X,(s)°dbj(s) +DZ c;B"(t) + cot,
i=d-l,d-2,...,1.
Here
LjBD(t) + L',t
S!(t) = D=1
and
M'(t) = exp
i, j = 1, 2,
In this case, the Lie algebra 3(Lo,LI, d
L,=E
d
E(L,Jx!+c')axt, a
...
... , d. ,L,.) generated by vector fields
P=0, 1, ...
d,
is solvable. A general result for the representation of solutions in such a case was obtained by H. Kunita [96].
Example 8.2. Let a, c, d be real constants such that a > 0. Consider the following one-dimensional stochastic differential equation: (8.8)
dX(t) = (2aX(t)VO)112dB(t) + (cX(t)+d)dt.
Since the coefficients or(x) = (tax V 0)' /2 and b(x) = cx + d satisfy the condition of Theorem 3.2 and also the growth condition (2.18), a global strong solution X(t) exists uniquely for every given initial value X(0). If d > 0 and X(0) > 0 a.s., then X(t) > 0 for all t > 0 a.s. Indeed, in the case d = 0, it is obvious that X(t) = 0 a.s. if X(0) =_ 0 a.s. by the uniqueness of
solutions. Setting a = inf{t;X(t) = 01 we see that X(t) = X(t+u) is a
STOCHASTIC DIFFERENTIAL EQUATIONS
236
solution of (8.8) with X(0) = 0 on the space 02 = {w; a(w) < oo} , .,9` =
_91a, 13 = P( 1a)) and hence X(t) = 0 a.s. on 6. This implies that X(t) - X(tAc) a.s. and consequently, X(t) > 0 a.s. if X(0) > 0 a.s. In the case d > 0, set a_, = inf {t; X(t) = -e} where e > 0 is such that -ce +d > 0. Assume that P(o_, < oo) > 0. Then, with probability one, if we take any r < a_, such that X(t) < 0 if t EE (r, a_8), we have dX(t) _ (cX(t) + d)dt on the interval (r, a_8) and hence t - X(t) is increasing on this interval., This is clearly impossible.
Thus the solution of (8.8) defines a conservative diffusion process {P.} on [O,oo) in the case d > 0. It is the L-diffusion process where L is the operator a
Lf(x) = ax j-xf(x) +
(8.9)
(ex
+ d) d_J(x)
acting on Ck([0, co)).
We shall now prove the following formula: (8.10)
Ex(e zw(t)) = c'l (e" [ L
- 1) +
1]-d1a
exp 1
I
a (e«'eCex - 1) + " 1
C
(If c = 0, we understand that L(ec, - 1) = t.) Indeed by Ito's formula, with respect to Px we have that
u(t,w(t)) - u(0,x) = a martingale + f [at + Lu](s,w(s))ds o for every u(t,x) E C .2([0, oo) X [0, oo)). * Noting that the function v(t, x)
in the right-hand side of (8.10) satisfies that at = Lv and v(0+, x) = e'AX, we set u(t,x) = v(to - t,x) for fixed to and apply Ito's formula for u(t, x). Then v(to- t,w(t)) - v(to,x) is a Px martingale and hence E=[v(to-t,w(t))] = v(to,x).
Letting
t = to, we
have
E=(e-''w(0)
v(t0, x). u(t, x) implies that all the derivatives of u up to the first order in t and up to the second order in x are continuous and bounded. * Cb Z([0, oo) x [0, oo))
EXAMPLES
237
Let x > 0 and set 6O = inf {t; w(t) = 01. Then (8.11)
if 0
P.(ao0 =1
and (8.12)
if d > a.
Px(aO = oo) = 1
For the proof of (8.11) and (8.12), set s(x)
- J eXp{- J
Y
Czaz d
}dy = e0lo
f
1
eXp
r_ a y] ya1adY
and
x(x)
r= J
eXp L -J
ycz+d az
1
=crl+d
y
j
dz {
eXp L J
arl
dz az} dy.
It is easy to see that s(0+) = -oo if and only if d > a and s(oo) = oo if and only if c < 0 or c = 0 and d< a. Note also that ic(0+) < oo if d < a. Now the assertions follows from Theorem VI-3.1 and Theorem VI-3.2. We also remark that the boundary x = 0 is regular* and reflecting if 0 < d < a, and exit and absorbing if d = 0. If d >_ a the boundary is entrance, and it is easy to conclude that (8.13)
PO(w(t) > 0 for all
t > 0) = 1.
Indeed, letting ) t oo in (8.10), we have PO(w(t) > 0) = 1 for every t > 0. Combining this with (8.12), we can conclude that PO(w(t + s) > 0 for all s > 0) = 1 for every t > 0. Since t is arbitrary, this implies (8.13).
Example 8.3. (Bessel diffusions). For a > 0, let La be the differential operator on [0, oo) defined by z
(8.14)
Laf(x) = 2 dx' f(x) + a x 1 dxf(x)1
* Ito-McKean [73].
STOCHASTIC DIFFERENTIAL EQUATIONS
238
with the domain IT (L«) = {f E C61([0,co)); for some constants 0 < a, < a2 and
f(x) = cx2
if x E [0, a,] and f(x) = 0 if x E [a2ioo)} .
(8.15)
For fE i(La), we set Laf(0)=c(a-1) so that LafECb((0,00)). There exists a unique conservative diffusion process generated by the operator L. which is called the Bessel diffusion process with index a. Bessel diffusion processes are essentially a particular case of the diffusions discussed in the previous example. Let
L«f(x) = 2xd 2f(x) + adxf(x) and c (8.16)
{f(x)=.f(/);.fE
(La)}
.
Then an L,,-diffusion {Px} xeCO. W, exists uniquely. Indeed, the diffusion of Example 8.2 in the case a = 2, c = 0 and d = a is clearly an L«-diffusion.
Conversely, by the same proof as in Theorem 6.1, we can prove that if is La diffusion then {X(t, w) = w(t)} is a solution of equation (8.8) for a = 2, c = 0 and d = a with X(0) = x. By the uniqueness of solutions of (8.8), we can conclude that the La diffusion is unique. It is easy to see that (8.17)
(La.f)(x2) _ (L«f)(x),
f E -2r(L«)
where f(x) = f(,/ T). Now we can conclude that the La diffusion {PX«'} xa(o,,, is unique and Pxc«> is the image measure on W([O,oo)) of the
measure Pxi) under the mapping W([0, cc)) D w
w E W([0,oc)),
w (t) = That is, the Bessel diffusion Xx,(t) of index a starting at x is obtained as XQ(t) = /Yx where the path I-w is defined by
where YQ2 is the unique solution of (dY(t) = 2(Y(t) V 0) 1 12dB(t) + adt
Y(0)=x2. By (8.10), we see that
239
EXAMPLES xw(r)2) = (2.1t + 1)- a12 EX (a) (e
(8.19)
exp
(- 22tLz+ 1).
Inverting the Laplace transform, we see that
E( )(.f(w(t))) = J
'p
x, y)f(y)dy,
where
p(a)(t, x, y) = exp
(8.20)
[O _ (y)
+ y2)/2tl3,a_<
x*
_I( T
tY)
x zn
x° (2
and
2
°°
nlr(v + n + 1)
is the modified Bessel function.
We can provenan interesting property of family of Bessel diffusions by using equation (8.18). Let B, and B2 be two independent Brownian motions and a, and az be positive constants. Consider the equations
dY,(t) = 2(Y,(t)VO)"2dB1(t) + aldt jl
Y1(0) = Y. (-= [0,oo) and
dY2(t) = 2(Y2(t)V 0)1"zdB2(t) + a2dt
Yz(0) = Y2 E [0,o)
Set Y3(t) = Y1(t) + Yz(t) and B3(t)
Y1(s)
41(s) + Y2(s)
dB, (s) +
Y2(s)
t JO
dBz(s) x
Y1(s) + Y2(s)
Then B3(t) is a Brownian motion by Theorem II-6.1 and dY3(t) = 2(Y3(t )V 0)1"zdB3(t) + (a1 + az)dt Y3(0) = Y1 + Y2-
Thus the law of Y3 is P 0) = 1 for every t > 0.
240
STOCHASTIC DIFFERENTIAL EQUATIONS
independent Bessel diffusions of index a and fl respectively, I X'(t) 1 2 1 Xf(t) 2 is a Bessel diffusion of index a + P. In particular, if a = d, d = 1, 2, ... , X11(t) can be identified with the radial process of d-dimensional Brownian motion. See [73], [147], [168] for further information on Besse] diffusions.
Example 8.4. (Brownian excursions).*' Let T > 0 be fixed. Consider the following stochastic differential equation
(8.21)
dX(t) = 2(X(t) V 0)' "2dB(t) + (3 - 7, (t t)dt X(0) = 0.
This is an equation similar to (8.18). Hence it can be shown that there exists a unique solution X(t) for t e [0, T) and that (8.22)
P(X(t) > 0 for all t c- (0, T)) = 1.*2
Furthermore, X(t) defines a time-dependent Markov process. To be precise, for 0 < s < T and x e [0, oo), let W,,, be the totality of all continuw(t)E [0, oo) such that w(s) = x and w(t) > ous paths w: [s, T) B t
0 for all t E (s,T), , (W,,x) be the a-field on W. generated by Borel cylinder sets and A(W,,x), s < t < T, be the sub a-field generated by Borel cylinder sets depending only on the interval [s, t]. Let Po,o be the probability law on (W,.o, . (Wo,o)) of the solution X(t) of (8.21) and more generally P,,,, be the probability law on (W,,,, (W,, ,)) of the unique solution {X(t)}:Et,.T) of
(8.23)
dX(t) = 2(X(t) V 0)'"2 dB(t) + (3 - 2P) )dt X(s) = X.
The Markovian property of Po,o is now formulated as follows;
for 0< s < t and f e B([0, o0)),*3 (8.24)
Eo.o[f(N'(t))
a.a. w(Po.o)
More generally, for 0 < u < s < t, x E [0, oo) and f E B([0, oo)), *1 [731.
*2 To prove (8.22) rigorously, apply the comparison theorem (Theorem VI-1.1) to equa-
tion (8.8) with a = 2, c = 0, 2 < d < 3 and X(0) = 0 and equation (8.21). *3 B([0, oo)) is the totality of all bounded Borel measurable functions on [0, co).
241
EXAMPLES
(8.25)
))l
E,.W c=1
a.a. w (P,,,x)
The proof of the above can be given as in Section 5 using the uniqueness of solutions of (8.23) for every s and x. Let P,,s be the image measure on (W, x, ° (W, X)) of the measureP,,xz
under the mapping W,.,2 w - / w E W, W. where the path is defined, of course, by 1W(t) = w(t). Then it is obvious that the Markovian property (8.25) also holds for {P3,x}. Set
(8 . 26)
p° (t, x, y) =
2nt
(x
(exp (
- y)) - exp - x + (
2t
y)z
))
t>0, x,yE[0,co), (8.27)
K(t, x) _ ,Jt3 x exp (-
2 ),
t > 0, x EE [0, 00),
and
KT- t,y)po(t-s,x,y), K(T - s, x) (8.28)
p(s, x; t, y) =
if0<s
if 0 <s0. We shall show that for every x > 0 and s > 0, P,,. 1w; w(t1) E dx1, w(t2) E dx2, ... (8.29)
,
E
= p(s, x; t1, x1)p(t1, x1; t2, x2) ...
x dxldx2 ... dx,,, for every s < t1 < t2 < . (8.30)
E:,,[f(w(t))) =
< to < T. It is sufficient to prove that
f
CO, ")
p(s, x; t, y)f(y)dy,
0<s
STOCHASTIC DIFFERENTIAL EQUATIONS
242
u(S, x; t) = fo AS, x; t, y)f(t, y)dy, where f(t,y) is a bounded smooth function. Then we can verify by direct calculation that u(s,x; t) satisfies 2
a (s,x;t)-{2 ax'
(8.31)
+
Tx sa}u(s,x;t), s E (0, t), x E (0, oo),
lim u(s, x; t) = f(t, y).
$It.x--.Y
If we set
u(s, x; t) = f p(S, .1-X; t, y)f(t, y2)dy, u(s, x; t) satisfies z
(8.32)
I-as
(s, x;1)_{2%!+3
T-s)a }u(s,x,t)
lim u(s, x; t) = f(t, y).
s1t,x"Y
By (8.32) and Ito's formula, we see for each r < T that [s, r) E) t u(t, X(t); r) is a martingale if X(t) is the solution of (8.23). Hence
E(u(t, X(t); z)) = u(s, x; z) for every t e [s, z). Letting t t z Eff(r, X(T))] =
E.,x[f(T, w(r))] = u(s, x; z).
,Consequently
E=.x[f(r, w(r))] = E5.5Z[f(z, ./w(T))] = f o
p(s, x; r, y)f(r, y)dy
and (8.30) is proved. It is immediately seen from (8.29) that P0,0 {w; w(t1) E dxj, w(t2) E dx2,
. . . ,
w(t.) E dx}
243
EXAMPLES
= Po,o {w; w(T - ti) E dx,, w(T - t2)
dx2, ... , w(T -
-for every 0 < t, < t2 < . . . < t < T. This shows that Po,o is invariant -under the time inversion w - w, where w is defined by w(t) = w(T - t). From this, we can conclude that (8.33)
Po,o {w; lim w(t) = 01 = I. ttT
Hence P°,o may be regarded as a probability on the space Wo.o = {w; [0, T] E) t w(t) is coninuous, w(0) = w(T) = 0 and w(t) > 0 for
tE(0,T)}.
Example 8.5. (Pinned Brownian motion). Let X(t) be a one-dimensional Brownian motion such that X(0) = 0. For fixed to > 0 and x, y c= R', define the process Xz°'Y = (XX'Y(t))oststo by (8.34)
XtX'Y(t) = x + X(t) + to (-X(to) + (y - x))
=x+ tot (y- x)+Xo,o(t). It is easy to verify that the probability law of Xx', coincides with P.( I w(to) = y), where P. is the Wiener measure starting at x. The process X.' 0,-' is called a pinned Brownian motion. Consider the following stochastic differential equation
(8.35)
J
t =t- -t
dX()
dB()
y
X t) d
t - to
1X(0) = X.
Clearly the solution X(t) exists uniquely for t e [0, to). By (8.35) we have
(t - to)d (X(t) - y) = dB(t), t - to and hence X(t) is solved as * The process Xo'0is sometimes called the Brownian bridge.
STOCHASTIC DIFFERENTIAL EQUATIONS
244
X(t)=x+
(8.36)
o(y-x)+(t-to) fodB_t
t
It is now easy to identify the process X(t) with X °"(t). Both XP'00) and
(t - t0) f ` dB(s) are centered Gaussian processes with the covariance oS - to
r(s, t) = t /\ s - t . 0
Thus the equation (8.35) is the stochastic differential equation determining the pinned Brownian motion X', ''.
9. Stochastic differential equations with respect to Poisson point processes So far we have only considered stochastic differential equations with respect to Brownian motions. For such equations, the solutions are always continuous processes. We can also consider more general stochastic differential equations* which include Poisson point processes as well as Brownian motions; in this case, however, the solutions are usually discontinuous processes. For simplicity, we consider such general equations in the case of the time-homogeneous Markovian type. Let (U, .qu} be a measurable space and n(du) be a a-finite measure on it. Let U0 be a set in .9 u such that n(U\ U0) < oo. Let a(x) = (uLL(x)) be
a Borel measurable function Rd -- Rd Qx R', b(x) = (b'(x)) be a Borel measurable function Rd -- Rd, and f(x, u) _ (f'(x, u)) be a -q(Rd) x function Rdx U -- Rd such that for some positive constant K, (9.1)
JJa(x)112 + 1!b(x)l12 + f
U°
II.f(x, u)112n(du)< K(1 + 1x12), x E Rd.
Consider the following stochastic differential equation
XI(t) = X`(0) + j] f' Qa(X(s))dB`(s) + f' b`(X(s))ds k=I
(9.2)
+ f o+f + f o+f
0
0
uf'(X(s-), u)1uo(u)ND(dsdu)
uf'(X(s-), u)Iu.u0 (u)N5(dsdu),
i = 1, 2,
* They are also called stochastic differential equations of the jump type.
... , d,
POISSON POINT PROCESSES
245
where B = (Bk(t)) is an r-dimensional Brownian motion, p is a stationary Poisson point process on U with characteristic measure n and Np and ND are defined in Chapter II, Section 3. A precise formulation is as follows. By
a solution of the equation (9.2), we mean a right continuous process X = (X(t)) with left hand limits on Rd defined on a probability space (Q, -9-,P) with a reference family (Y-,) such that Xis (J0";)-adapted and there
exist an r-dimensional ()-Brownian motion B = (Bk(t)) and an (.)stationary Poisson point process p on U with characteristic measure n such that the equation (9.2) holds a.s.
Theorem 9.1. If a(x), b(x) and f(x, u) satisfy in addition to (9.1) the Lipschitz condition (9.3)
II a(x) - a(y)11' + IIb(x) - b(y)II" + f UO IIf(x, u) -fly, u)II2n(du)
x,yER°,
then for any given r-dimensional (.)-Brownian motion B=(Bk(t)), any (.)-stationary Poisson point process p with characteristic measure n and any R°-valued F-0-measurable random variable c defined on a probability space with a reference family (.Y), there exists a unique d-dimensional
(.)-adapted right-continuous process X(t) with left-hand limits which satisfies equation (9.2) and such that X(0) =
a.s.
Proof. Suppose B = (Bk(t)), p and are given as above. Let D = Is E Dp; p(s) E U\U0} . Since n(U\U0) < co, D is a discrete set in (0, co) a.s.
Let a, < a2 < .
.
be the enumeration of all elements in
D. It is easy to see that an is an (,F;)-stopping time for each n and lim an = n1
0o a.s.* First we shall show the existence and uniqueness of solutions in the time interval [0, a,]. For this, consider the following equation
Y'(t) k=1
f' ak(Y(s))dB'(s) + f 0
+ f "f of
g(Y(s-),
b`(Y(s))ds 0
1(Y(s-), u)IU0(u)Sr (dsdu),
Noting the following general formula
E[ { f "f,,
t
2]
* We disregard the trivial case of n(U\U0) = 0.
i = 1, 2, ... , d.
STOCHASTIC DIFFERENTIAL EQUATIONS
246
= f ods f
voE[g2(Y(s),u)ln(du)
and the assumption (9.3), we can show by the same argument as in the proof of Theorem 3.1 that the solution Y(t) of (9.4) exists uniquely and is constructed as follows: if = y, a constant point in Rd, the solution is constructed by the successive approximation as in the proof of Theorem 3.1. The solution is a measurable function of y, B and p in the obvious sense. The solution for a general initial value is obtained by replacing the variable y of this function with . Set XI(t) )
Y(t),
0
Y(o'1-) +f(Y(U1-), P(al)),
t = a,.
The process {X,(t)}tE[o,,,j is clearly the unique solution of (9.2) in the time interval [0, or,]. Next, set 5 = X,(o,), B = (Bk(t)) where Bk(t) = B'(t-f-6,) - Bk(cl,), and p = (p(t)) where D; = {s; s+Q, ED,,) and p(s) = p(s+v,). We can determine the process X2(t) on [0, 61] with respect to 5, B and p in the same way as X,(t). Clearly 61, defined with respect top, coincides with c2 - Q,. Define {X(t)} 10% 123 by
_
X(t) -
X,(t),
t E [0, 011],
X2(t - 0'1),
t E [o r, 62}
It is easy to see that {X(t)} tE10,,2] is the unique solution of (9.2) in the time interval [0, 62]. Continuing this process successively, X(t) is determined uniquely in the time interval [0, for every n and hence X(t) is determined globally.
We have actually proved, under the assumption (9.3), the unique existence of the strong solution of (9.2). The uniqueness in law is obvious from this stronger result.
CHAPTER V
Diffusion Processes on Manifolds
1. Stochastic differential equations on manifolds Let M be a d-dimensional C°°-manifold i.e., M is a Hausdorff topological space with an open covering {U,} aEi of M, each Ua provided with a homeomorphism Oa with an open subset 0a(Ua) of Rd such that, if Ua fl Uf o the function Oeoo ' from ¢a(Uafl Ua) into Op(Ua fl UU) is a Cmfunction. Ua is called a coordinate neighborhood and for x E Ua, Oa(x) _ (x', x2, ... , xd) E Rd is called a local coordinate of x. In this book we always assume that M is connected and a-compact. It is well-known then that M is paracompact and has a countable open base.*
A function f(x) defined on an open subset D of M is called C' (or smooth) if it is C°° as a function of the local coordinate, i.e., f6¢;' is C'° on ¢a(Ua fl D) for every a. Let F(M) be the totality of all real valued C"functions on M and F0(M) be the subclass of F(M) consisting of all functions in F(M) with compact support. F(M) and F0(M) are algebras over
the field of real numbers R with the usual rules of f+g, fg and Af (f, gEF(M) or Fo(M), 2ER). Let x E M. By a tangent vector at x we mean a linear mapping V of F(M) into R such that
V(fg) = V(f)g(x) +f(x)V(g) The set of all tangent vectors at x forms a linear space T'(M), called the tangent space at x, with the rules
(V+V')(.f) = V(f)+V'(f)
and
* Cf. [1121. 247
(A VV) = .%V(f ).
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248
Let (x',x2, ... xd) be a local coordinate in a coordinate neighborhood U of x. Every f E F(M) is expressed on Uas a C°°-function f(x',x2, ... ,xd).
Then f ----- ()(x) is a tangent vector at x for every i = 1, 2, ... , d. This is denoted by (a) . It is easy to see that {(a )
forms a }
base for TT(M).
By a vector field we mean a mapping V: x EM' -- V(x)ETT(M). V is called a C°°-vector field if for every feF(M), (Vf)(x): = V(x)f is a C°°-function. Thus V is a C°°-vector field if and only if V is a linear mapping
of F(M) into F(M) (or F0(M) into F(,(M)) such that V(fg)=V(f)g+ fV(g). In this book we only consider C°°-vector fields unless otherwise stated. The totality of C°°-vector fields is denoted by 3E(M). Let A0,A,, ... A, E X(M). We consider the following stochastic differential equation given in an intuitive form
dX(t) = Aa(X(t))odBa(t) + A0(X(t))dt.*'
(1.1)
A precise formulation is as follows. Let M =M or M U {A} ( = the onepoint compactification of M) accordingly as M is compact or non-compact. Let IP(M) be the path space defined by Tl?'(M) = {w; w is a continuous mapping [0, oo) --- Al' such that w(0)E M and if w(t)=A then w(t') =A for all t' > t} and let . (JP(M)) be the or-field generated by the Borel cylinder sets. The explosion time e(w) is defined by e(w) = inf{t; w(t) = d}.
Definition 1.1. A solution X=(X(t)) of (1.1) is any (.F)-adapted J'(M)-valued random variable (i.e., a continuous process on M with A as a trap) defined on a probability space with a reference family (,) and an r-dimensional (,.9)-Brownian motion B = (B(t)) with B(0) = 0 such that the following is satisfied: for every fEFo(M),*Z (1.2)
f(X(t)) -- f(X(0)) =
J
0(Aa.f)(X(s))odB9(s) +
J'(Ao.f)(X(s))d ,
*1 According to the usual convention, the summation sign is abbreviated for repeated indices appearing once at the top and once at the bottom. *2 We define f(d)=0 for every feFo(M).
STOCHASTIC DIFFERENTIAL EQUATIONS ON MANIFOLDS
249
where the first term on the right-hand side is understood in the sense of the Fisk-Stratonovich integral defined in Chapter Ill, Section 1.
The results of Chapter IV applied to each coordinate neighborhood enable us to obtain a unique strong solution of (1.1). Namely we have the following result.
Theorem 1.1. There exists a function F: Mx W,'- W(M) which is fR(M)x.°Bt(Wa)""Pwlq,(W(M))-measurable`' for every t> 0 such P
that (i) for every solution X = (X(t)) with respect to the Brownian motion B = (B(t)), it holds that X = F(X(0),B)
a.s.,
and (ii) for every r-dimensional (.,57;)-Brownian motion B=(B(t)) with B(O) = 0 defined on a probability space with a reference family (,F-,) and an M-valued (,7)-measurable random variable , is a solua.s. tion of (1.1) with X(0) =
Proof. Take a coordinate neighborhood *2 U and express A,,= v.,(x) axi , a=0, 1, ... , r, under the local coordinates (x',xz, ... xd) in U. Extend the functions u ,,(x) to bounded smooth functions on Rd and then consider the following stochastic differential equation (1.3)
(dX1, = Q.1(XX)°dB1(t) + oo(X,)dt
Xo=x`,
i=1,2,...,d.
Note that (1.3) is equivalent to (1.3)'
dXi = Q«(XX)dBa(t) + 601(X,)dt X,0
X"
i= 1, 2, ...,d,
where *1 Here /u runs over all probabilities on (M,,9J'(M)). ,,r, P", ,9,,(Wo') have the same meaning as in Chapter IV: Wa is the space of continuous paths in R' starting at 0, Pa" is the Wiener measure on Wo and R,(Wo') is the cr-field generated by the Borel cylinder sets up to time t.A(W(M)) is defined similarly. *z Here we choose a relatively compact coordinate neighbourhood. Such a remark will sometimes be necessary in the future but usually we do not mention it.
DIFFUSION PROCESSES ON MANIFOLDS
250
(1.4)
Q (x) = a (x) + 2
(axx rJ(x))o (x)
It follows from the results of Chapter IV that the unique strong solution of (1.3) exists; i.e., there exists a mapping F: Rd x Wa W" with the properties as in Theorem 1.1 such that any solution X of (1.3) is given as
X=F(x,B) where x=(x',x, ... ,xd). F(x,w) - (X(t,x,w)) itself is the solution of (1.3) with respect to the canonical realization w=(w(t)) of Brownian motion on { W,, P''} with the reference family {. °} defined by -9;1 = . 0'(W°r), t > 0. Take x = (x', x2, ... , xd) E U and set inf {t; X(t,x,w) U). Define X = (X (t,x,w)) by Xi,(t,x,w) = X(t A
For each xE M and coordinate neighborhood U containing x we construct the local solution XU as above. It is easy to see that if U and U are two coordinate neighborhoods and x E Ufl U, then Xu(t,x,w) = Xc(t,x,w)
for all t <
A r0(w). Indeed, if A.=
ordinate z = (z',ic2, (1.5)
...
a
,
under the local co-
jcd) in U, then we have
6.(9(x)) = Q.k(x) ak
and the equation for X0(t,x,w) is of the form (1.6)
dX; = 6ra,(Xr)°dwa(t) + 6o(Xr)dt.
On the other hand, it follows from the chain rule (Theorem 111-1.3) that the
process Xu under the local coordinate z in U, i.e., z(X (t,x,w)) = (9,9, satisfies
d2 = axk (X(t))°dXk(t) axk
(X(t))oQ(X(t))°dwa(t )+ 5 (X(t))oo(X(t))dt
= Qa(xr)°dwa(t) + Q'(xr)dt.
satisfies the same equation (1.6) as I, = Thus R, = X0(t,x,w), and so by the uniqueness of solutions, we conclude that XU(t,x,w) = Xo(t,x,w) for all t < r (w) A ra(w). Now we will patch together the local solutions into a global solution.
STOCHASTIC DIFFERENTIAL EQUATIONS ON MANIFOLDS
251
We first choose two systems of coordinate neighborhoods ( U,') and { Va } which form locally finite coverings of M such that U. C Va and and Pa is also contained in another coordinate neighborhood W,,. Let x E M and U1, U2, , U1 be the totality of coordinate neighborhoods in the system { Ua } containing x. Then the process l (t,x,w) _ Xv1(t,x,w) is well-defined for t E [0, f,(w)] where f,(w) = max
16151
{'r, (w)}. Here we set ,t(oo,x,w) = d, for covenience. Define T1(w) = and X(t) = X(t) for t E [0, r1]. Inductively, if and X(t) = (X(t,x,w)) are defined for t E_ [0, r (w)], then on the set { w; T,,(w) < we define
X. = X(T;),
*
W. = 0rnw,
Tn+1 = rn + Txn(wn)
and X(t) = A3(t for t c- [,r., ri+1]. In this way, X(t) is defined for t e [0, r-) where r = lim T,,. We now show that (1.7)
lim X(t) = d
on the set {w; r.(w) < oo
ttzm
We can choose an increasing sequence { M. } of finite sums of { Va } such
that
U M. = M and M C
for every
n-1
Furthermore we may assume that if Vk (1 8Mn For each n we define
n ? 1.
c, then Vk c M.}1.
of =0
v1 = inf{ t > oi; X(t) E Mn+1 }
oZ = inf{t> &i; X(t) E
Uz = inf{t> ozi X(t) E Mn+1}
o,=inf{t>62; X(t)E=M,}
6,=inf{t> o,; X(t)E=Mn+1}
It is sufficient for (1.7) to show that for every n, on the set {w; rm(w) < oo }
(1.8)
, there exists an integer k such that 6k(W) < oo
and
Qk+1(w) _ 00
To show this, it suffices to prove that r,,(w) = no on the set * 0,: Wo -- Wo is defined as in Chapter IV: (B,w)(s) = w(t+s)-w(t).
DIFFUSION PROCESSES ON MANIFOLDS
252
{ w; 3 k such that ok(w) < oo and Qk(W) = 00 } U { w; ok(w) < oo for every k }.
First if ok(w) < 00 for every k, then we can show E {dk(W) - 0k(W) } = 00
(1.9)
k=1
by the same argument as in the proof of Lemma IV-2.1. This clearly implies that a (w) = oo. Next, consider the case when there exists k such that ok(w) < oo and dk(W) = oo. Then we have X(t) E Mi+1
(1.10)
for all
t > ok.
We can now conclude by the same argument as in the proof of (1.9) that r-(w) = oo. We set X(t)=d for t > r- on the set { w; zm < e0 }. Thus we have defined X(t) = (X(t,x,w)) as a mapping Mx Wo F) (x, w) i--- X = (X(t,x,w)) E W(M).
It is easy to see that it is a solution of (1.1). Indeed, it is obvious that for every f E Fo(M), nT,
fto
f(X(tAi )) - f(x) =
f
o
aX, (X(s))od(X(S))ds
1T,(Aof)(X(s))ds.
,AT,(Aa.f)(X(S))°dW"(s)+ 0
0
Similarly, on the set {w;
oo},
f(X(t A zn+1)) - f(X(t A Zn))
-
(wn)
(flf)( a
0
f = tAT.
(S,x,,,Wn))
odWn(s)
(t-:A 0
(A0f )(X(s,xn,wn))ds
f fAz,
(Aof)(X(s))ds.
STOCHASTIC DIFFERENTIAL EQUATIONS ON MANIFOLDS
253
Summing up, we have
f(X(t)) - f(x) = f(X(t A t.,)) - f(x)
= rrnt 0
(A"f)(X(S))"dw"(s) +
= f (Aaf)(X(s))odw"(s) + f 0
f nTW 0
(Aof)(X(s))d
' 0
(Aof)(X(s))ds.
The uniqueness of solutions is also easily proved.
Remark 1.1. We can also construct the solution of (1.1) more directly by appealing to Whitney's imbedding theorem ([176]). M is imbedded into RId+' as a closed submanifold of Rad+' and the vector fields Aa(x) are restrictions on M of smooth vector fields Aa(x) on R'a+'. The stochastic differential equation corresponding toAa(x) is defined globally in the Euclidean coordinate system and the solution is constructed as in Chapter IV. If the initial value is on M, then it is easy to see that the solution remains on M. Thus this solution actually defines the solution of (1.1). A construction of the Brownian motion on a sphere given in Chapter III, Section 2 is a typical example of the method of imbedding.
Theorem 1.2. Let P. be the probability law on J'(M) of the solution X=(X(t)) of (1.1) with the initial value X(O)=x. Then {Px} zEM is a diffusion generated by the second order differential operator (7.11)
Af
r
E Aa(Aaf) + Aof,
f EFo(M).
Proof. Using the uniqueness of solutions we can show that {PX} has the strong Markov property. Actually, we can prove the following stronger
result: for any (_57;1)-stopping time a(w), we have X(t+v(w),x,w) = X(t,X(a(w),x,w), O,w) for all t > 0 and almost all w such that o(w) < oo. Since for f c= F0(M),
df(X(t)) = (A"f)(X(t))odw"(t) + (A0 f)(X(t))dt = (Aaf)(X(t))dw"(t) + (Aof)(X(t))dt `I- 2 and
d(Ap f)(X(t)) = Aa(Ap f)(X(t))"dw"(t) + (AOAp f)(X(t))dt,
DIFFUSION PROCESSES ON MANIFOLDS
254
we have
d(A"f)(X(t)) dw"(t) = E A"(A"f)(X(t))dt. Consequently,
df(X(t)) = (A"f)(X(t))dw"(t) + (Af)(X(t))dt,
where (Af) is defined by(7.11). This proves that X=(X(t)) is an Adiffusion.
Remark 1.2. By a similar argument as in Chapter IV, we can deduce the uniqueness of the A-diffusion {P.}.,,-, on FP(M) from the uniqueness of solutions of (1.1).
2. Flow of diffeomorphisms
Given vector fields A"E BE(M), a= 0, 1, ... , r, we constructed in Section 1 a mapping X = (X(t,x,w)): Mx Wo E?(x,w)
X(.,x,w)
E JP(M). This may also be regarded as a mapping: [0, oo)xMx Wo = (t, x, w) -- X(t,x,w)EM The main purpose of this section is to show X(t,x,w)EA is a local diffeomorphism of M that the mapping xEM for each fixed t > 0 and for almost all w such that X(t,x,w) E M. First we discuss the case of M = Rd. Let o (x) = (o'k(x)) E Rd( R, and b(x) = (b'(x)) E Rd be given such that they are smooth functions (i.e.,
C°°-functions) on Rd, 11o(x)tj+jb(x)j
(2.1)
{ dX = va(XI)dw"(t) + b'(X1)dt
X0=x,
i=1,2,...,d,
defined on the space (W(,, PW) with the reference family (7°). As we saw in Chapter IV, the solution X=(X(t,x,w)) exists uniquely and E{ X(t)12) < oo for all t > 0.* This result will be strengthened below as E { J X(t) P}
< oo for all p > 1. In particular, e = oo a.s. First of all we shall prove a lemma on an approximation of solutions * E stands for the expectation on the space (No, P").
FLOW OF DJFFEOMORPHISMS
255
by polygonal paths (cf. [111]). Let (2.2)
O"(s) = k/2"
if s E [k/2", (k+1)/2"),
k = 0, 1, ... .
Lemma 2.1. Let A(x) = (A'(x)) E Rm®x Rr and fi(x) = (,Bt(x)) F=R' be given and satisfy the following conditions; (i) there exists a positive constant K such that
IIA(x)II + If(x)I < K(1+IxI)
for every x E R"`,
(ii) for every N > 0, there exists a positive constant KN such that
II A(x) - A(y)II + I fi(x) - P(y) I <- KNI x - y j
for every x,yE R'° such that I x I 5 N and I y I < N. Let a(t) and a"(t), n = 1, 2, ... , be continuous R°'-valued (,F°)-adapted *' processes such
that for some p > 2 (2.3)
supE{sup Ia"(t)Ip+'}
OSr T
,.
as n -- oo.*Z Let Y(t) and Y"(t), n = 1, 2, ... , be continuous R"'valued (-9;0)-adapted processes such that (2.4)
Y`(t) = a(t) + J ° Al(Y(s))dwa(s) +
pr(Y(s))d
*s
J°
and (2.4)'
Y"(t) = a(t) + f ° Aa(Y"(O"(s)))dwa(s) + J ° fl`(Y"(¢"(s)))ds
fori=1,2,...,m, n=1,2,.... Then, for every T > 0, (2.5)
E { sup I Y"(t) - Y(t)I') -0
as n --- oo.
0G_t5T
.
*1 As in Section 1, we consider the Wiener space (W',,°.i! (W'),Pw) and .;° _ Pw(WW), t > 0.
*Z E stands for the expectation on the Wiener space. T> 0 is any fixed constant. *3 w(t) = (wa(t)) is the canonical realization of r-dimensional Brownian motion on
the space (Wa, P w).
DIFFUSION PROCESSES ON MANIFOLDS
256
Proof. Let T>O be arbitrary but fixed. First we remark that (2.3) implies (2.6)
E { sup i a(t) I °*'} < 00 0st5T
and
(2.7)
E j sup I an(t)
0
OSt
as
n - oo.
(2.6) follows easily from Fatou's lemma and (2.7) follows from E { ssup I an(t)-a.(Or,(t )) I °}
< K,[E { sup I
I °} + E { supTI an(t)-a(t) I °} ],
the right-hand side tending to zero as n - oo by the dominated convergence theorem and (2.3). In the following we assume for simplicity of notation that m=1 and r = 1. We shall show that (2.8)
sup E I sup I Y,, (t) I °+'} < 00.
From (2.4)' we have E { sup I Yn ( s ) I °+'} < KZ[E { sup I a (s) I F+'} 0<sGr
05S5r
+ E{ sup I f s A(YY(0 (u)))dw(u) 1 °+'} o<s
0
0,.,,, I + E{ sup f so
I °+'} ]
for t c- [0, T]. By Theorem 111-3.1 and Holder's inequality, E{ sup I f A( Y (O (u)))dw(u) I °+'} osssr 0 I (p+') /2}
< K3E { I f o
* In the following K K2, ... are positive constants independent of n (which may depend on T).
FLOW OF DIFFEOMORPHISMS
< K. f t E { I A(Yn(0 (s)))
257
I'+1} ds
0
< KS f' [1 + E { I 0
Y.(O (s)) I o+l}J
and
E{ o
If
I°}1}
S K6 f t E{ I fi(Y.(0.(s))) I p+l} ds 0
< K, f
t
+ E{ I Y.(O.(s)) 11+'} }ds.
{1 0
Consequently (2.9)
E J's
s K8(1 + fo E {I YY(O,r(s))I°+'} ds).
P I
Then obviously 1,P+ 1) < K3(l + f'
E{I
E
{ I Y.(O.(s)) 1,P+ I I ds)
and we can deduce from this (using here a similar truncation argument as in the proof of Theorem 111-3.1 or Theorem IV-2.4) that
E{I
Ksexp{K8t}
Substituting this inequality into (2.9), we obtain (2.8). Similarly we can prove (2.10)
E{ sup I Y(t)ID+'} < 00. OntnT
Next we set
Q = inf {t; I Y (t) I > N} and
vN = inf {t; I Y(t) I > N}
258
DIFFUSION PROCESSES ON MANIFOLDS
for every N > 0. Then, for tE [0, T], E{
s
np.N
AN I Y(s)
D}
K9[E { sup I a (s) - a(s) I D} OSsst
+E +E{
I f0 0S
sup
I
f"
A(Y(u))} dw(u) I D}
{/t(YY(O (u)))
0
- fl(Y(u))} du III l
and by estimating similarly as above, this is dominated by * Klo[E { sup I a (s) - a(s) I P)
+E{f
aknsN
I A(YY(O.(s))) - A(Y(s)) I'ds}
o
+ E{ fo
o'
n°N
I fl(YY(6.(s))) - fi(Y(s)) I 'ds} J
tA0N /WN
S o(1) + LNE { f o
I YY(Y$n(s)) - Y(s) I'ds}
t
0(1) + LN f0 E{I
A0N)) - Y(sAcr Ao"')I'}ds.
By writing s' = s A an A o N for simplicity we have E { I YY(O5(s'))
- Y(s') I'}
K, j(E { I YY( ¢ (s')) --
I D} + E { I
Y(s) I '} )
and E{I
YY(s') I'}
< K12[E { I a.(O.(s'))
+ E { I A(
-
I D}
w(0 .(s'))) I P1
+ E{ I fl(Yt,(6 (s')))(s' - 0.(s')) I'} l = 0(1)
by (2.7). Consequently * LN and LN are positive constants which may depend on N. o(1) denotes quantities which tend to zero as n - oo uniformly in t E [0, Tl.
FLOW OF DIFFEOMORPHISMS E{
259
supI Ys) - Y(s) I °}
OSsSt/bha'V
/`Qn ^naN)-Y(snUNnOIN ID}ds.
0(1)+LN f 0t E{I
Yn(S^^
We can conclude from this that (2.11)
E{
sup
05s5TAa AUN
I Y (s) - Y(s) I D} - 0 as n -- co
for every N > 0. Finally E { sup I Y (s) - Y(s) I D} 05s5T
<E{
sup
O5s5Tno4' naN
Y (s) - Y(s) I
+ E sup (I Y(s)
QN < T}
o5s5T
+ E { sup (I Yr(s) + I Y(s) I )P: QN < T} O5s5T
<E{
sup A",, I Y(s)
D}
OSsSTA ,,
+ 2E { supT(I Y (s) I + I Y(s) I)P: sup (I YY(s) I + I Y(s) I) Z N}
< E {o5rsnpPF AaN
I US) - Y(s) I D}
+ N E{ 5 pT(I YY(s)I + I Y(s)I)°+I}.
Now we can easily deduce from (2.8), (2.10) and (2.11) that
E { sup I Y (s) - Y(s) I D} - 0 o5s5T
as
oo.
n
q.e.d.
For given a(x) and b(x) as above, let XX(t) _ be the solution, on the Wiener space (Wo, PR') with respect to the canonical realization w(t) = (wa(t)) of Brownian motion, of the equation (2.12)
X (t) = x + f 0 Qa(Xn(0n(s))) dwa(s) + f
b(XX(cn(s)))ds,
0
in component form, (2.12)'
Xn(t) = xt +
ft 0
a, X.(O.(s)))dwa(s) +
f
t 0
b`(Xa(On(s)))ds,
i=1,2,...,d.
DIFFUSION PROCESSES ON MANIFOLDS
260
X"(t) is uniquely determined. Indeed, X"(0) = x and if X"(t) is determined
for tE [0,(k-1)/2"], then X"(t) = X"((k-1)/211) + a"(Xt,((k-1)/2"))(w"(t)-w"((k-l)/2"))
+b(X"((k-1)/2"))(t-(k-1)/2") for t e [(k-1)/2", k/2"]. It is also clear from this that X"(t) is expressed as X"(t) = F(x,w(1/2"),w(2/2"), ... ,w([2"t]/2"),w(t))
for some C"-function F: Rd x (R')[2'13+1 - Rd.
In particular, x.
X"(t,x,w) is C" for every t > 0 and we W. Let D" =
alai
axial ... pad
for a = (al, a2, ... ,ad), I a I =a) +az+ - - - +ad and set D"X;(t,x,w).
Proposition 2.1. For each xe Rd, 1 < i < d, and each a, there exists a unique process YQ(t) = (YY(t,x,w)) such that (2.13)
YQ(t) I D} - 0
E { Sup I Y.
as
n -- oo
for all T > 0 and p > 1. Furthermore, the convergence (2.13) is uniform in
x on each compact set in Rd. Proof. First consider the case of I a I = 1. If we set Y'. () (t,x,w) = TxjX,t,(t,x,w), Y> (t) _ (Yj,(,,)(t, x, w)) is determined by the equation (in
matrix notation) Y(") (t) = I + f 'o o (X"(O"(s))) Y c.) (c"(s))dw"(s) (2.14)
+f where o (x)i = axe
b'(X"(O.(s)))Y(,)(O"(s))ds,
a"(x), b'(x)j =
axj bt(x)
and I = (b5.
Applying
Lemma 2.1 to the system of equations (2.12) and (2.14) combined together, we have that (2.15)
E { sup I X"(t) - X(t) I -0} + E { sup I I Y(") (t) - Y(t)Ij °} tE[0,T]
te[o,T)
as n- co
0
FLOW OF DIFFEOMORPHISMS
for all T > 0 and p
261
1, where X(t) is the solution of (2.1) and Y(t)
(YY(t, x, w)) is the solution of (2.16)
Y(t) = I + fo a'(X(s))Y(s)dwa(s) + f
b'(X(s))Y(s)ds. o
As is immediately seen in the proof of Lemma 2.1, the convergence (2.15) is uniform in x on every compact set. Next, set a2
Y$.k.(n)(t) =
Xn(t,x,w)-
axfaxk
Then Y`, (n)(t) = (Yj
is determined by the equation
(n)(t) = f t0 + f t b'(Xn(3 (s)))k Yi,,j2, (n)(j Y ,,(S))ds 0
(2.17)
+ f' o.(Xn(On(s)))k. Yi,. (-)(O- (S)) YY2, (n)(On(s))dwa(s) 0
+ f' b"(Xn(On(s)))k.,YJ,,
(,)(On(S)) Y52.
where a2
a(x)k., = axkaX, or(x)
2
,
b"(x)k./ = ax axI b°(x)
If we denote by aj(,J2,(n)(t) the sum of last two terms on the right hand side of (2.17), then noting Theorem 111-3.1 and (2.15) it is easy to conclude
that (2.18)
E ( sup I af,,i2, (n)(t) - a51,12(t) 1 °} - 0 CE[O, T]
for all T > 0 and p > 1. Here (2.19)
af1ij2(t) = f 0 or,.,(X(s))`j.1Yi,(S)Y;2(s)dwa(s)
as n --- oo
DIFFUSION PROCESSES ON MANIFOLDS
262
+ f` 0
and this convergence is uniform in x on each compact set. Now we can apply Lemma 2.1 * to conclude that E { sup I Y5,.12, (n)(t) - Yi,,12(t) D} -- 0 reCO,T)
(2.20)
as n --- 00
for every T > 0 and p> 1, where Yjl,,12(t) is determined by the equation
Y(t) = f 6a(X(s))%Y1,,j2(s)dwa(s) 0
(2.21)
+ f t b'(X(s)), Y1jL ,.12(s)ds + ai,,>2(t).
Furthermore, the convergence is uniform in x on each compact set. By continuing this process step by step for higher order derivatives we complete the proof of the proposition.
on
Now we shall introduce the following seminorms for smooth functions Rd. For a bounded domain QcRd, p > 1 and m = 1, 2, ... , we set
HOIIp,,m = E { I&l m
f a I Dao(x) I °dx} "',
IIcI!E,m =
E sup I Da0(x) I
Ia&Sm x
For each 0 and m, we can find S2'DQ, m' > m, p > 1 and a constant K > 0 such that the following inequality holds for all smooth functions 0 on Rd: (2.22)
110112,m < KIIcII° mt.
This is an obvious consequence of the well-known inequalities of Sobolev [151].
Now (2.13) clearly implies that
E { sup
05t$T
I Ya,
Y«, (m)(t,x,w)1 P} -' 0
as n, m -- oo for all T > 0 and p > 1 uniformly on each compact set in x. Consequently, * We apply this lemma to the system of equations (2.12) and (2.17) combined together.
FLOW OF DIFFEOMORPHISMS
f
E { Sup J OStST
<
J
263
Yom, (n)(t,x,w) - Yom, (m)(t,x,w) I D Al A
dx E { sup I Y',
Yom, (m)(t,x,w) I °}
05t5T
-0
as
n, m - oo
for every bounded domain 0. This implies that E { sup I I 05,5T
w) - Xm(t,
,
w) I Ip ,} --- 0
as n, m - oo for all T > 0, p > 1, 1= 1, 2, ... and bounded domain Q. By a standard argument we can extract a subsequence of X,,, denoted by
X. again, such that for almost all w (PI'), u+'SpTIIX (t, ,w) - Xm(t, ,w)IIa,--- 0
as n, m --- oo f o r all bounded domain 0, p > 1, 1= 1, 2, ... and T > 0. The inequality (2.22) then implies that, for almost all w, (2.23)
sup II X (t, ,w) - Xm(t, ,w)II2., --- 0
o It5T
as n, m -- oo for all bounded domain 0, 1 = 1, 2, ... and T > 0. Consequently, for almost all w, llm
i(t,x,w)
exists uniformly in (t, x) on each compact set in [0, oo) x Rd; furthermore
(t, x) '--- -k(t,x,w)ERd is continuous and for each tE=-[0, oo), x, t(t,x,w) is a C°°-mapping from Rd into Rd. We also have by (2.15) that
Pw[w; ,t(t,x,w) = X(t,x,w) for all
t > 0] = 1,
for all x, i.e., (I(t,x,w)) is a modification of the family of solutions (X(t,x,w)) of the stochastic differential equation (2.1). Thus we have obtained the following result.
Proposition 2.2. Let X=(X(t,x,w)) be the solution of (2.1). Then a modification of X(t,x,w), denoted by X(t,x,w) again, can be chosen so that the mapping x e Rd -X(t, x, w) E Rd is C" for each fixed t, a.s.
Next we shall show that the mapping x X(t,x,w) is a diffeomorphism of Rd. In doing this, it is more convenient to rewrite the equa-
DIFFUSION PROCESSES ON MANIFOLDS
264
tion using Fisk-Stratonovich differentials. So instead of the equation (2.1), we consider the following equation dXl = Qr(XX)odw'(t) + b'(XX)dt (2.24)
i=1,2,...,d.
X0=x,
Note that (2.24) may be transformed into an equation of the form (2.1) with b`(x) replaced by b1(x) where (2.25)
5'(x) = b`(x)
+
2
(
axk 6"(x)) 6,a(x)
Clearly c,(x) and b'(x) satisfy the same assumptions as Q,(x) and bt(x) and
therefore, it is just a matter of convenience as to whether we write the equation in the form (2.1) or in the form (2.24). We shall now show that the C°'-mapping x X(t,x,w) defined by the solutions of (2.24) is a diffeomorphism. By (2.16), the Jacobian matrix (YJ(t)) = (--
xX'(t, x, w))
satisfies (in matrix notation) (2.26)
Y(t) = I + f' o,' (X(s))Y(s)dw'(s) + f' b'(X(s))Y(s)ds
where b is given by (2.25). It is easy to see that (2.26) is equivalent to (2.26)'
Y(t) = I + fo Q,(X(s))Y(s) odw°(s) + f ' b'(X(s))Y(s)ds.
Now let Z(t)=(ZJ(t )) be the solution of (2.27)
Z(t) = I - f o Z(s) i, (X(s))odwa(s)- fa Z(s)b'(X(s))ds.
Then
d(Z(t)Y(t)) = Z(t)odY(t) + dZ(t)oY(t) = 0 and therefore Z(t) Y(t) - I. This proves that Y(t) is invertible and Y(t)-' = Z(t). Consequently, the C--mapping is a local diffeomorphism into R". By using the next Lemma it is easy to see that it is a bijection. * To be precise we consider the system of equations (2.24) and (2.27) combined together. It is a stochastic differential equation on R° X RdZ whose coefficients satisfy the same kind of regularity and growth conditions.
FLOW OF DIFFEOMORPHISMS
265
Indeed, x =X(t, .(t,x,w),w) and x = ?(t,X(t,x,w),w) for every x a.s. Thus the mapping is a diffeomorphism of R.
Lemma 2.2.* Let ?(t,x,w) be constructed as above from the equation dXl = oQ(XX)odwa(t) - b'(XX)dt
(2.28))
Xo=x.
Then, for every fixed T > 0, we have
X(T - t, x, w) = t(t, X(T, x, w), iv)
(2.29)
for every 0 < t < T and x, a.s. (PA'). Here w is another Wiener process defined by (2.30)
w(t) = w(T-t)-w(T),
0:!g t < T.
Proof. Let Bn(t, W)
Y.
n)
- tn) znf
wa(tn) + ! n
waQ.),
a = 1, 2, ... , r,
where i` = (k 2 1 T and t,, = 2 T ifT < t < (k ... , 2"-1, n = 1, 2, .... It is clear that Bn(T-t, w) - wa(T) = Bn(t,w),
1)T
2
,
k = 1, 2,
0 < t < T.
Consider the following two systems of ordinary differential equation for
every n = 1, 2, ...
dd- (t) = : o (X.(t)) dd ° (t,w) + X,(0) = x and
* Malliavin [106].
i = 1, 2, ... , d,
DIFFUSION PROCESSES ON MANIFOLDS
266
d "(t) _' a.1 ii (X (t))--" d(t,w) - b`(e'n(t)) t "
X" (0) = X.
i= 1,2,...,d,
Solutions will be denoted by X"(t,x,w) and t (t,x,w) respectively. It is immediately seen by the uniqueness of solutions that
X"(T-t,x,w) = t
n = 1, 2, ... .
We can apply the corollary of Theorem VI-7.3 to obtain (2.29) by taking the limits. Summarizing we state the following result.
Theorem 2.3.*1 Let X(t,x,w) be the solution of (2.24) (or (2.1)) on the Wiener space (Wo, PH'). Then a modification of X(t,x,w) can be chosen
so that the mapping x
X(t,x,w) is a diffeomorphism of Rd a.s., for
each t e [0, oo). The Jacobian matrix (YJ(t,x,w)) = (-2- X'(t,X,W)) is determined by the equation (2.26)' (resp. (2.16)).
Thus we have a one-parameter family of diffeomorphisms X,(w): x'--- X(t,x,w) for tE [0,oo). Clearly Xo(w)=the identity and X,(6,w)o X,(w) = X,+,(w) for almost all w. X(t,x,w) Furthermore it actually holds a stronger statement: a.s., x is a diffeomorphism of Rd for all t > 0. For details, see Kunita [209].
Now we return to the case of a compact manifold M. The solution (X(t,x,w)) of (1.1) may be considered as a family of mappings X,: x X(t,x,w) from M into M. Theorem 2.4. Assume that M is a compact manifold. X(t,x,w) has a modification,*-' which is denoted by X(t,x,w) again, such that the mapping X,(w): x X(t,x,w) is C°° in the sense that x i--- f(X(t,x,w)) is
C" for every f E F(M) and all fixed tE[0, oo), a.s. Furthermore, for each x e M and t E [0, co), the differential X(t,x,w)* of the mapping x
X(t,x,w); X(t,x,w)*: T,,(M) i--- Tx(,.=.W)(M)
Cf. Funaki [31], Malliavin [106] and Elworthy [25] *z By a modification of X(t,x,w) we mean a process X(t,x,w) such that P w[ X(t,x,w)=
X(t,x,w) for all t? 0} = I for all x.
FLOW OF DIFFEOMORPHISMS
267
is an isomorphism a.s. on the set {w; X(t,x,w)E M}.
Proof. Let x0 E M and t c- [0, co) be fixed. Then for almost all w such that X(t,x0,w) E M, there exist an integer n > 0 and a sequence of coordinate neighborhoods U,, U2, - - , U. such that -
{ X(s,xo,w); s e [(k - 1)t/n, kt/n] } c U.,, k = 1, 2, - - , n. -
By Theorem 2.3, we can easily conclude that if U is a coordinate neighborhood and { X(s,y0,w); s E [0, to] } C U, then y X(to,y,w) is a diffeomorphism in a neighborhood of yo. Consequently, since X(t,x0,w) = [Xt/n(e(n-1)t/nw)o - -
oX,/n(et/nw)oXt/n](x0)
the assertions of the theorem follow at once. In case of non-compact manifolds, the solution (X(t,x,w)) of (1.1) may be considered as a family of mappings X,: x -4 X(t,x,w) from M into M = MU { d } and local results can be deduced from the compact case. However to obtain global results about the family of mappings X, we must assume some additional conditions. Elworthy [197], Chapter VIII can be consulted for detailed information about these. Now let us introduce the bundle of linear frames GL(M) on M.* By
a frame e=[el,e2, ... ed] at x we mean a linearly independent system of vectors e, ETT(M), i = 1, 2, ... , d, i.e., a basis of TX(M). GL(M) is defined as the collection of all frames at all points x E M:
GL(M)= {r = (x, e) ; x e M and e is a frame at x).
GL(M) can be given a structure of a C"-manifold as follows. Let {U,,, lo.} be a coordinate system of M. Set
UQ = {r = (x, e) EGL(M); x E U. and e is a frame at x} and define the mapping
from U, onto O,(UQ) x GL(d, R) C Rd X Rd2
by
c,,(r) = (ba(x) = (x`, x2, ... , xd), e;, i, j = 1, 2, ... , d) where
of =
a e/
" [7) and [1311.
DIFFUSION PROCESSES ON MANIFOLDS
268
Clearly (U*, ¢Q) defines a coordinate system of GL(M) and so GL(M) has the structure of Cm-manifold of dimension d + d2. An element a of the
group GL(d, R) acts on GL(M) from the right by (2.31)
r a = (x,ea),
where ea = [(ea)1, (ea)2, ( e a ) i = aie1,
r = (x,e) , (ea)d] is a frame at x defined by
j = 1, 2, ... , d.
Thus GL(M) is a principal fibre bundle with the structural group GL(d, R). The projection 7C: GL(M) -- M is defined, as usual, by it(x, e)=x.
Every vector field LE3w(M) induces a vector field L on GL(M) as follows. Let fEF(GL(M)). Then Lf is given by (2.32)
(L'f)(r) = d f((exp tL)x, (exp tL)*e)I,_o
where r=(x, e) and (exptL)*e = [(exptL)*e1i (exptL)*e2, ... , (exptL)*ed]. Here, of course, exptL is the local diffeomorphism x r--- x(t,x) defined by the differential equation dt (t,x) = a'(x(t,x)),
(L = a(x)
),
x(O,x) = x
and (exptL)* is its differential which is an isomorphism TT(M) -for each xEM. Let A0, A1, ... , Ar EX(M) and X, = (X(t,x,w)) be the flow of diffeomorphisms on M constructed above. Then A0, A,, ... , A. E3;(GL(M)) defines a flow of diffeomorphisms r, = (r(t, r, w)) on GL(M), and it is easy to see from the definition that r(t,r,w) = (X(t,x,w), e(t,r,w))
where r = (x, e) and e(t,r,w) = X(t,x,w)*e.* The expressions under a local coordinate are as follows: X(t,x,w) is determined by the equation (2.24) where A,,(x) = a ,',(x) ax, (2.33)
, a = 1, 2, ... , d, Ao(x) = b'(x) ax' and
eXt,x,w) = Y,(t,x,w)ejk
* X(t, x, w)* is the differential of x .--- X(t,x,w) and, of course, X(t, x, w)*e =
[X(t,x, w)*e,, X(t,x,w)*e2, ... , X(t,x,w)*edl.
HEAT EQUATION ON A MANIFOLD
269
where Yk(t,x,w) is determined by the equation (2.26'). E F(GL(M)) for each, Let L E 3r(M). We shall define a function
i=1,2,...,d by
(2.34)
fi(r) = (e-')kak(x)
in a local coordinate (xt,e= (e5)) of GL(M), where L = at(x) a and e-' is the inverse matrix of e. It is easy to see that (2.34) is independent of a choice of local coordinates and thus defines a global function on GL(M). It is also easy to prove that for LI,L2EX(M), (2.35)
(L'ftz2)(r) =ffel = 1, 2, ...
, d.
Here L, is defined by (2.32) for L, and [L,, L2]=L,L2-L2L, is the usual Poisson bracket. Therefore, we have that
f'L(r(t,r,w)) - fi(r) (2.36)
=
f o3'La.L1(r(s,r,w))°dwa(s) + Jf'fi40.L](r(s, r, w))dr
for every Le 3r(M) and i = 1, 2,
... , d.
3. Heat equation on a manifold
Let M be a C°°-manifold and A0,A,, ... , ArEX(M). Let X, = (X(t,x,w)) be the flow of diffeomorphisms on M constructed in the preceding section. Then x --- f(X(t,x,w)) is C°° for any fE F0(M). Throughout this section, we shall assume that the vector fields Ak, k = 0, 1, ... ,r, have the property that E[ sup sup I Da { f(X(t,x,w))} I] < co te[0.T] xeU
for all f E F0(M), every coordinate neighbourhood U such that U is compact, every T > 0 and every multi-index a. This condition is satisfied if M=R d and if the coefficients (b'(x)) and
and Aa(x) = ox(x) ax , a = 1, 2, ... , r, a satisfy the condition of the preceding section. It is also satisfied if M is (o,',(x))
in A0(x)=b"(x)
imbedded into R"' (cf. Remark 1.1) such that Ak, k = 0, 1, ... , r are restrictions of vector fields Ak on R'" which themselves satisfy the condi-
270
DIFFUSION PROCESSES ON MANIFOLDS
tion. In particular, if M is compact the above condition is always satisfied. Define the second order differential operator A acting on F(M) by (3.1)
4f(x) = 2 aE Aa(Aaf)(x) + (Aof)(x)
We shall show that the function u(t,x) defined by (3.2)
f E Fo(M)
u(t,x) = E[ f(X(t,x,w))],
is a smooth solution of the following heat equation (3.3)
J(tx) = (Av)(t,x) at lim v(t,y) =f(x).
elo.r-X
First, we shall prove that the function u(t,x) defined by (3.2) belongs to C°°([0, oo) x M).* It is clear that u(t, x) is a C°°-function of x E M since x f(X(t,x,w)) is C°° and the differentiation under the expectation sign is justified by the above condition. By (1.2), we have
f(X(t,x,w)) - f(x) = a martingale +
fu (A f)(X(s,x,w))ds
for each xE M, and hence (3.4)
u(t,x) =f(x) + f"0 E [(Af )(X(s,x, w))]ds.
Since A"f EF0(M), n = 1, 2, ... , we have
u(t,x) = f(x) + t(Af)(x) + fr dt, f" E[(A2f)(X(t2,x,w))]dt2 0 0
=f(x) + t(Af)(x) +
+
f 0 dt1
f"0
(A2f)(x)
dt2 f0 E[(A3f)(X(t3,x,w))]dt3
=f(x) + t(Af)(x) +
+ ... +f
2
0
2
(A2f)(x)
dt1 f 0
dt2 ...
f
-' E[(A"f)(X(t,,,x,w))]dtn 0
* C-([0, oo)xM) is the class of C'-functions on (0, oo)xM.
HEAT EQUATION ON A MANIFOLD
271
Now it is clear that u(t,x) C°°([0, oo) x M). Next we shall show that (Au,)(x) = E[(Af)(X(t,x,w))],
(3.5)
where for each fixed t > 0, we set u,(x) = u(t,x). Applying Tto's formula to the smooth function u,(x), we have (writing X,=X(s,x,w)) ut(X,) - ur(x) = f so
j"
a
Let U be a relatively compact neighborhood of x and let or = inf {t; X, U1. Then we have
E[ut(Xno)] - u,(x) = E[ If
o o
(Au,)(X,,)du],
and hence
(Aut)(x) -
E[u,(X,no)] - u:(x)
E[sAc]
E[ f o {(Au,)(XX) - (Au5)(Xo)} du] E[s A 6]
I
E[ fo du f o (Au) (XE)dg]
E[ f o
E[s A 6]
where Y. = f: (AaAu,)(XX)dwa(g). Clearly,
E[
f,Au du o
s: (A'u,)(XE)df] E[s A Q]
Also E,[ f,nQ
Y du]
E[s A Q]
I
< s maa (AZUr)(x)
DIFFUSION PROCESSES ON MANIFOLDS
272
E[(sAa) sup
Osu5snc
(E[(sAa)2D112(E[ sup 1Y.1-D"
I Yull
Osussno
E[s f\ al
E[s A a]
A D1i2, 5 K(E[(sAa)2D1'2(E[ E[s A a]
(by Theorem 111-3.1),
(E[(sAa)2]) "2(E[sAa])112 (max E
((AaAur)(x))2)112
xep a=1
E[sAa]
< Ks1'2(max
((AaAur)(x))2)112,
YeU a=1
where K is a positive constant. Consequently, (3.6)
lim E[ur(Xsno)] - ur(x) ` (Aur)(x)-
E[sAa]
sio
On the other hand, by the strong Markov property of Brownian motion, E[ur(Xsna)l - ur(x)
= E[f(X(t, X A,,, esnaw))] - E[f(X(t,x,w))] = E[ f(X(t-f-s A a,x,w))] - E[f(X(t,x,w))]
= E[ = E[
+sn(Af)(XX)du]
f
o
Q(Af)(Xu+r)du].
Therefore,
(3.7)
lim E[ur(Xsn,)] - ur(x) s10
E[sAa]
f lim E[ 810
SAV o
(Af)(Xy+r)dul
E[sAa]
It is immediately seen that E[s A a] = s + o(s) since s-E(s A a) < sP(a < s) = O(S). So (Af)(XX+r)du] + o(s) E[ f a (Af)(XX+r)dul = E[ fu
since Af is bounded. Therefore we can conclude that the limit in (3.7) is equal to
HEAT EQUATION ON A MANIFOLD
lim
s E[
f
0
273
(Af)(X>.+,)du] = E[(Af)(XX)]
This completes the proof of (3.5). From (3.4) and (3.5), it follows that
u(t,x) = f(x) + f (Au)(s,x)ds 0
and therefore we have
a[ (t,x) = Au(t,x), i.e., u(t,x) satisfies the heat equation (3.3). Conversely, let v(t,x)E C',z([O, no) x M) * be a bounded solution of (3.3). We shall further assume that v(t,x) satisfies the following condition: (3.8)
lim E[v(t-a,,,X(c,,,x,w)): 6n < t] = 0 nt^"
x, w)(D.} and D.
for every t > 0 and xEM, where
is an increasing sequence of relatively compact open sets in M such that
UDn=M. Clearly (3.8) is satisfied for any bounded v if, for instance, (X(t,x,w)) oo) = 1 for every xEM. This is is conservative, i.e., true since lim 6n. n1.
Now by Ito's formula, we have for each to > 0 and 0 < t < to, E[v(t0 -tAon, X(t A a.,x,w))] - v(to,x)
- E[ f
tAon
{(Av)(to_s, X(s, x, w))
at
(t0-s, X(t,x,w))}ds]
= 0. Letting n t no and noting (3.8), we obtain * C'.'((0, oo) x M) is the totality of all functions.f(t, x) on [0, o) x M which are continuously differentiable in t and twice continuously differentiable in x.
274
DIFFUSION PROCESSES ON MANIFOLDS
t} = v(t0,x).
E{v(t0 - t, X(t,x,w)):
Now letting t t to, we see by bounded convergence theorem that the left hand side tends to E[f(X(t0,x,w))] = u(t0ix). Therefore v(t,x) must coincide with u(t,x). The above results are summarized in the following theorem.
Theorem 3.1. For f c-F0(M), define u(t,x) by (3.2). Then u(t,x)E C°°([0, oo) x M) and satisfies the heat equation (3.3). Conversely, if v(t,x) E C" 2([O,oo) x M) is bounded and satisfies the heat equation (3.3) and the condition (3.8), then v(t,x) must coincide with u(t,x).
Remark 3.1. Generally a bounded solution of (3.3) is not unique and a kind of conditions like (3.8) is necessary in order to assure the uniqueness. For example, if M=(O,oo) and the operator Au = u"/2,
i.e., A0 = 0 and A,=
x,
then, for ffF0(M), both
v`(t,x) = J o /2nt (exp l
(x 2t y)Z
)- exp
(- (x 2t Y)Z))f(Y)dd
and
V,(t,x) =
Jo
1271t
(exp
(
(x 2t Y)z )
+ exp
(x
Y)Z
2t
))f(Y) dY
are solutions of the heat equation. v,(t,x) is the solution which satisfies the condition (3.8). Now let c(x) E=-F(M) such that it is bounded from above: sup c(x) < XEM
co and
f
x' --- E[exp { t c(X(s,x,w))ds} f(X(t,x,w))] 0
is C°° for any f E F0(M) and t > 0. This condition is always satisfied if c E F0(M).
Theorem 3.2. The function (Feynman-Kac formula) u(t,x) defined by (3.9)
u(t,x) = E[exp { f ! c(X(s,x,w))ds} f(X(t,x,w))], f eF0(M) 0
NON-DEGENERATE DIFFUSIONS AND HORIZONTAL LIFTS
275
is a solution in C°°([0, oo) x M) of the heat equation (3.10)
(t,x) = (Av)(t,x) + c(x)v(t,x)
at
Jim v(t,Y) = f(x)
t I0,A--x
which is bounded on [0, T] x M for each T > 0. Conversely if v(t,x) is a solution in C"2([0, 00) X M) of the equation (3.10) such that it is bounded on [0, T] x M for each T > 0 and (3.11)
lim E[exp { an c(X(s,x, w))ds} v(t-o J
nim
,
X(Q,,,x,w)): Q < t] = 0
0
then v(t,x) coincides with u(t,x) given by (3.9).
The proof is given in the same way as in Theorem 3.1. This time, however, we use Ito's formula as follows:
d[exp If' c(X,)ds} f(X)] = exp If' c(X,)ds} (Aa f)(Xt)dwa(t)
+ exp If'
c(X,)ds} (A f(XX)+c(X,) f(XX))dt
and
d[exp If' c(X,)ds} v(t0-t, X,)] 0
= exp {
f t c(X,)ds} (Aav)(to-t, X,)dwa(t) 0
+ exp { f c(X,)ds} (- at (t0-t, Xt) + (Av)(t0-t, X,) 0
+ c(Xt)v(t0--t, X,))dt.
4. Non-degenerate diffusions on a manifold and their. horizontal lifts Consider a two dimensional Brownian motion on a plane and suppose that the trajectory of a Brownian particle is traced in ink. We roll a sphere on the plane along the Brownian curve without slipping. The resulting path which is thus transferred defines a random curve on the sphere, and, indeed, it defines a Brownian motion on the sphere. This idea of constructing spherical Brownian motion was first proposed by Bochner. Such a method
DIFFUSION PROCESSES ON MANIFOLDS
276
can also be carried over to a general Riemannian manifold. By making use of the connection in the sense of Levi-Civita, we can "roll" the manifold along a smooth curve in Euclidean space and, although a Brownian curve in Euclidean space is not smooth, stochastic calculus enables us to "roll" the manifold along such a curve. A Brownian motion on a Riemannian manifold can be obtained in this way. By generalizing the connection of Levi-Civita to a class of affine connections, we can obtain more general diffusions on a manifold; indeed, the most general non-degenerate smooth diffusions can be obtained in this manner. Such a process will be carried
out below by constructing a flow of diffeomorphisms on the bundle of orthonormal frames over the manifold. This method is due to Eells and Elworthy [23].*' As we shall see in the next section, it is closely related to stochastic parallel displacement which was first introduced by Ito [68]. Before going into details, let us quickly recall several fundamental notions in differential geometry. Let M be a C"'-manifold. A tensor of type
(p,q) at x is an element in the tensor product Tx(M). Here Tx(W, = ®... CO TT(M) Ox TT(M)* Ox TT(M)* Ox ... Qx Tx(M)*
TT(M) Ox TT(M) P
4
is the linear space formed of all multi-linear mappings u: TT(M)* X TT(M)* X . . . X TT(M)* X T.(M) X TT(M) X
x TT(M) -- R
4
P
with the usual rules of addition and scalar multiplication.*2 Choosing
a local coordinate (x',x2, ... xd) introduces a basis
(ad ) x
(a) ax'
) x
axe (a
in TT(M); its dual basis is denoted by (dx')x, (&2).,,,
We denote by
9 ... Q (a;) 0 (dx'')x xQ ax"( x z
Qx
,
...
, (a [Q)
,l ... u fpP 31f11 ... aj4
for every k,, k2i (4.1)
.
0
, k, 1,, 12, ... ,1,. Clearly the system ax2
& ... (&
8x1V z
&(
')x 0 (
*' Cf. Also their works given in the references of [23] and Malliavin [104]. *2 is the dual space of TT(M).
2)x
,
(dx°) x.
(dx'o)x
element u e Tx(M)Q such that u((dxk')x, ... ,(dXkP)x, (axtl)x'
... x
the
277
NON-DEGENERATE DIFFUSIONS AND HORIZONTAL LIFTS
O .. ((dxlq)x}
i1, i2,
,Jq = 1, 2,
,ip,Ji,J2,
,
d
forms a basis of TT(M)Q. A C°°-tensor field of type (p, q) *' is a mapping
u(x)eTT(M)q
u: M_=3 x
whose components uiI;2.. q(x) with respect to the basis (4.1) are C°° in every coordinate neighborhood. The uj""2:.q obey the usual rule under a change of coordinates:
(4.2)
ax'' x )= J,l2...lq( axk1 1112...lp
a912
ax'o ax'i ax'2
axk2
axkp 571 5T2
..
ax'g _.k,k2... kp W, rrlll,t2.. tQ x ()
Conversely, if a system of C°°-functions {u,'1'2 'y(x)} is defined in every coordinate neighborhood and satisfies (4.2), then there exists a unique (p, q)-tensor field whose components coincide with it. A (1,0)-tensor field is just a vector field. A (0,1)-tensor field is called a differential 1 form. Generally, a (differential) p -form is a (0,p)-tensor which is alternate, i.e., its components satisfy uo(1,)0li2)...(,P)(x) = sgn(o)u,,12...lp(x)
for every permutation sgn(a) dx°('O) Qx dx°('2) Qx
If we set
dx" A dx'2 A ... A dx'p =
Q dx'"d, then a p-form u(x) is
expressed as u(x) = ui, l2...19(x) dx" A dx'2 A ... A dx'o 1,(x)dx" A dx'2 A ... A dx'°. =P! F. u,112... fi<12<. . . <'p
The exterier product a A fi of a p-form a and q-form ft is a (p+q)form defined by (4.3)
(a AY)(x) = ak,k2...kp(x)akp.+,kp+2...kp+q(x)dxkl Adxk2 A ... A dxkp+q.
The exterior derivative da of p-form a is a (p+1)-form defined by *' Also we call it simply (p, q)-tensor field. *2 Such a property is clearly independent of the choice of coordinates. The notion of a symmetric tensor can be defined similarly.
DIFFUSION PROCESSES ON MANIFOLDS
278
(4.4)
a`12.-.iD(x) dxr A dxti A ... A dxiz.
(da)(x) = a
By an affine connection P we mean a rule which associates to every X E X (M) a linear mapping 17x : 3E(M) - X (M) having the following properties: (i) PxY is bilinear in X and Y; (4.5) (ii) 17rx+a r = fVx + gVr (iii)
Vy(fY) =fVxY+ (Xf)Y.*
The operator vx is called covariant differentiation with respect to X. A system of functions {rjk(x)} is defined in a coordinate neighborhood by
Pa/aJ = F(x) axk
(at
=
The r (x) are called the components of the connection F. In a local coordinate system, PxY may be expressed as (4.6)
PxY = [X1(x) YJ(x)ra(x) + X1(x) ax-, Yk(x)] -L axk a.
where X = X'(x) a, and Y = Y'(x)
a
.The components of the connec-
tion obey the following transformation rule under a change of coordinates f;k (4.7)
_ axD axq
azk
az1 avv axe
I, +
a2x'
axk
az`agi ax'
Dq
Conversely, any system of smooth functions Tu(x) defined in each coordinate neighborhood and satisfying the rule (4.7) determines an affine connection by (4.6). Given a tensor field u(x) = (uj'i'2- q(x)) of type (p, q), a tensor field (Vu)(x) = (UJlJ2': jo.k(x)) of type (p,q+1) is defined by tD (x)(: = 2... q(x)) 1
(4.8)
tpp (x) + _ axk utli2... 11J2--q
,n
a=1
rk!(x)u111 .J ] 2 .q
=D (x)
fii2... tD
rk.Je(x)uJjJ2...m...Jq(x)
* For f eF(M) and X el(M), fX el(M) is defined by (fX)g = fX(g) g eF(M).
for every
NON-DEGENERATE DIFFUSIONS AND HORIZONTAL LIFTS
279
where the indices 1 and m in u take the place of is and j# respectively. By the transformation rule (4.7), it is easy to see that 17u(x) is a tensor field.
Pu(x) is called the covariant derivative of u(x). For X = X`
EX(M),
the (p,q)-tensor Fxu defined by (VX 1112..J, = U)[112... ry
XkUW2...1, J,J2..Jo.k
is called the covariant derivaitive of u(x) in the direction X. Note that if u =YEX(M), the above coincides with the original FxY. Let c: I E) t F-- c(t) E M * be a (piecewise) smooth curve in M and
u(t) = (ujI 2 : tya(t)) be a tensor field along c; that is, u(t)ET,(,)(M)q for t E I and t,--- u(t) is (piecewise) smooth. u(t) is said to be parallel along d dt (4.9)
c (with respect to the connection P) if ui112. 12
(t) + a=
rk-1(c(t))u;'j...
.a(t)dck(t
t 2... ,
o
- E ` kJm(c(t ))u B
B=1
a
dt
1J2.. m..-J,(t
)
dck(t) = 0, dt
t E I.
In particular, a tensor field u(x) is parallel along c if and only if (176(ou) (c(t )) (= = ui;J2...`.%,k (c(t ))
dcdlt))
= 0, t E I. For to, t, E I, to
tt,
u(t,) is uniquely determined from u(to) as a solution of (4.9) and we say that u(t,) is obtained from u(to) by parallel displacement along the curve c(t).
Consider a manifold M with an affine connection F= {I'fk(x)}. Let GL(M) be the bundle of linear frames. For each r E GL(M), (4.10)
H. _ IX
=
at
(a `` x
-
Tk,(x)eeak ae}
(a') C R-1
is a linear subspace of T,(GL(M)) which is clearly independent of the choice of local coordinates (x', e55. A tangent vector X in H, is called horizontal. An affine connection may also be defined as a rule which assigns a linear
subspace H, of T,(GL(M)) at each r E GL(M) (cf. Nomizu [131)). H, is called the horizontal subspace. Let 5 E TT(M). 5 E T,(GL(M)) is called a horizontal lift of c if 5 is horizontal, ir(r) = x and (dir),5 = . 5 is unique if r such that qt(r)=x is given. Given X E 3E(M), there exists a unique ffGX(GL(M)) such that X is the horizontal lift of X,,(,) for every r e GL(M). X is called the horizontal lift of X. In a local coordinate, * I denotes an interval of R'.
DIFFUSION PROCESSES ON MANIFOLDS
280
(4.11)
X = x`(x) ax` - r9'(x)x`(x)e' ae° D
if X = X'(x) axi . Given a smooth curve c: I D t ,- c(t) E M, a smooth curve c: I ED t,--- c(t)EGL(M) is called a horizontal lift of cif (i) W-(t) is
horizontal and (ii) lr(c"(t)) = c(t) for t E I. Clearly,
if r =
(x,e = [e,, e2, ... , ed]) is given where x is the starting point of c, then a horizontal lift c starting at r exists and is unique. Indeed, c(t) = (c(t), e(t) = [e,(t), e2(t), ... , ed(t)]), where e,(t)E TT(1)(M) is obtained from
e, by parallel displacement along the curve c. For each j = 1, 2, ... , d, there exists a unique vector field L1E3E(GL(M)) such that (L1),. is the horizontal lift of e. ETX(M) for every r = (x, e =[e,, e2i ... , ed]). In a local coordinate (xi, eD, L, may be expressed as (4.12)
a
a
L, = ej' 8x` - rZ,e e`Dae;.
{L,,L2, ... ,Ld} is called the system of canonical horizontal vector fields or basic vector fields ([7] and [131]). Let u(x) = (u,'j22. ;Q(x)) a (p,q)-tensor field. Define a system of
smooth functions (4.13)
(r)} on GL(M) by
u(x) = Fyi;i2:.1,(r)ei, O ei2 x0 ...
e,,®e Ox e*i
... Ox e*Q
for r = (x, e =[e,, e2, ... , ed]) and e* =[el, a*, ... , ed*] is the dual base of e. In a local coordinate (xi, ef), it may be expressed as (4.14)
i, (r) = u(x) 112Z..
elz ... a fk f Z ... f tp
where (f j) is the inverse matrix of (ef).
is called the scalarization of the tensor field u(x) or the system of components of the
tensor field u(x) read in the frame e. F(r) is GL(d,R)-equivariant in the sense that (4.15)
F
F.k ZZigD(r. a)aik,dx2 ... ail
b j12 ... bfq
for every a = (aj) E GL(d, R) where r- a is the action of a on r defined by (2.31); (b,) is the inverse of (aj). Conversely, every GL(d, R)-equivari-
ant system F(r)
Fi,'2;: fi(r) }
of smooth functions on GL(M) is
given as F = F. for some uniquely determined tensor field u.
NON-DEGENERATE DIFFUSIONS AND HORIZONTAL LIFTS
281
Proposition 4.1. (4.16)
Lm(Fyj;jz':,)(r) _ (Fvu)i;i2: >,:m(r)
for every it, i2, ... , i j,, j2, ... ,. and m = 1, 2, ... , d, where {Lm) is the system of canonical horizontal vector fields. The proof is left to the reader.
For an affine connection P= {rf}, Tu = ru - rat are the components of a tensor T of type (1,2). T is called the torsion tensor. An intrinsic definition of the torsion tensor T is given by
T(X,Y) = PXY - PYX - [X, Y],
X, Y EX(M).
An affine connection r= {TI} is called torsion free or symmetric if the
torsion tensor is zero, i.e., r; = r
.
A C°°-manifold M is called a Riemannian manifold if a tensor field g = (gt,) of type (0, 2) is given on M such that (i) g is symmetric, i.e., gtj(x) = g,,(x); (ii) g is positive definite, i.e., 0 for all x and 0E Rd.* g is called fundamental tensor field or Riemannian metric (tensor field). It defines an inner product on each tangent space TT(M) by (4.17)
<, ) = gt,(x)tl',
= t ()
x
and q = rlt (axt)r.
An affine connection V= {T } is called compatible with the Riemannian metric g if the inner product is preserved during a parallel displacement
of tangent vectors. That is, for every smooth curve c(t) and tangent vectors '(t) axt and rlt(t) axt at c(t) dc'
dt + T;k(c(t))
dc'(t) dt
k
dnt
dcJ(t) k t (t) = 0 and dt + Tik(c(t)) (t) = 0 dt
imply that dt (gt,(c(t))`(t)rl'(t)) = 0. From this it is easy to conclude that V is compatible with g if and only if * Properties (i) and (ii) are clearly independent of the choice of coordinates.
282
(4.18)
DIFFUSION PROCESSES ON MANIFOLDS
as g, = gljrk! + g11Tk, for all i, j, k = 1, 2, ... , d.
An affine connection compatible with g is not unique (see Proposition 4.3
below), but if we assume further that it is symmetric then it is unique. Indeed, (4.18) together with Tk = T implies that (4.19)
T U = j,1 j)
a
=
a
axamglJ) g km 2 (axt gmj + ax j gim -
This connection is called the Riemannian connection or the connection of Levi-Civita. The {!k j) are called the Christoffel symbols. Let O(M) be a submanifold of GL(M) defined by (4.20)
O(M) = {r = (x,e) EGL(M); e is an orthonormal base of TTM} .
In a local coordinate (x1,ej!) of GL(M), r c= O(M) if and only if (4.21)
gkleeJ1 = aij,
or equivalently, a
(4.22)
E emeJm = g1j M=1
where (g"J) is the inverse matrix of (g1 j). The equivalence of (4.21) and (4.22)
is easily verified as follows. Set e = (ei) and G = (g,j). Then (4.21)
e*Ge = I
G = (e*)-1 a-1
G-1 = ee* (4.22).
Now the orthogonal group 0(d) acts on O(M), and O(M) is a principal fibre bundle over M with the structural group 0(d). O(M) is called the bundle of orthonormal frames on M. Let P be an affine connection com-
patible with g and c: [a, b] -- M be a smooth curve in M. If r = (c(0), e) E 0(M), then the horizontal lift c(t) = (c(t), e(t)) of c(t) lies in O(M) since e(t) is an orthonormal frame at c(t). Similarly, a horizontal vector field X of X eX(M), if restricted to O(M), is a vector field on O(M), and the canonical horizontal vector fields Lm, m = 1, 2, ... , d, are vector fields on O(M).
NON-DEGENERATE DIFFUSIONS AND HORIZONTAL LIFTS
283
Let M be a Riemannian manifold and V= {rI} be an affine connection compatible with the Riemannian metric g. The connection P enables us to "roll" M along a curve y(t) in Rd to obtain a curve c(t) in M as the trace of y(t). Intutitively the infinitesimal motion of c(t) is that of y(t) in the tangent space which can be identified with Rd by choosing an orthonormal frame and the infinitesimal motion of the frame is given by the connection i.e., parallel displacement along the curve c(t). To be precise, let y: [0, co) ED t F---- y(t)ERd be a smooth curve in Rd. Let r (x, e) EO(M) and define a curve 6(t) = (c(t), e(t)) in O(M) by a
dt (t) = dt (t)ea(t ) (4.23)
7tc,>e(t) = 0
J
c(0) = x L
e(0) = e.
In local coordinates, d
i= 1, 2, ... , d
dtt (t) =ea(t) dt (t), (4.24)
dcm
dt (t) _ -
71
c`(0) =
(t)
i,a= 1,2, ...,d.
e(0) = era
The equation (4.23) may be written as a
(4.25)
dC (t) = La(C(t )) dt (t)
c(0) =
where 1f,, Z2, ... , Ld} is the system of canonical horizontal vector fields. The curve c(t) _ 7r(c5(t)) in M depends on the choice of the initial frame e at x; we shall denote it as c(t) = c(t,r,y), r = (x, e). It follows at once that (4.26)
c(t, r a, y) = c(t, r, ay),
t E [0, oo), a E 0(d),
where r- a is defined by (2.31) and the curve ay in Rd is defined by
DIFFUSION PROCESSES ON MANIFOLDS
284
(4.27)
(ay)(t) = ay(t),
i.e.
(ay)` (t) = a,' y'(t).
Now let w(t) = (wa(t)) be the canonical realization of a d-dimensional Wiener process. Stochastic calculus enables us to define a random curve X(t) in M in the same way. Let r(t) = (r(t, r, w)) be the solution of the stochastic differential equation
dr(t) = La(r(t))odwa(t)
(4.28)
r(O) = r.
r(t,r,w) is the flow of diffeomorphisms on O(M) corresponding to the canonical horizontal vector fields L,, L2, ... , Ld and the drift vector field * Los 0. In local coordinates, (4.28) is equivalent to dX°(t) =
(4.29) I
dea(t) _ - 1,'.(X(t))ea(t)adXm(t),
i = 1, 2, ... , d i, a = 1, 2, ... , d,
where r(t) = (X'(t), ea(t)). That the solution r(t) = (X°(t), ea(t)) lies on O(M) if r(0) E O(M) is clear since La is a vector field on O(M). Of course, one can also verify directly that d(gij(X(t))eQ(t)ee(t)) = 0 by using (4.18).
Now a stochastic curve X(t) = (XI(t)) on M is defined by X(t) = n[r(t)]. By (4.26) we have (writing X(t) = (X(t,r,w))) (4.30)
X(t,
w) = X(t,r,aw) for t > 0, a e 0(d) and w r= W.
But aw = (aw(t)) is another d-dimensional Wiener process and hence the
probability law of r.a, w) is independent of a e 0(d). In other words, the probability law of X(., r, w) depends only on x = 2r(r). We denote it by P. It is now easy to deduce the strong Markov property of the system { Px } from that of r(., r, w). Remark 4.1. Of course r(t, r, w) can be defined as a flow of diffeomorphisms on GL(M) for any affine connection but its projection to M is not strong Markov in general because it is not usually true that the law of r, w)] depends only on x = rc(r). This is the reason why we restrict
ourselves to an affine connection compatible with g and to a flow of diffeomorphisms on O(M). Thus we have a diffusion {P,,} on M. We shall now show that it is A-diffusion process where the differential operator A is given by * In the stochastic differential equation (1.1), the vector field Ao is called the drift vector field.
NON-DEGENERATE DIFFUSIONS AND HORIZONTAL LIFTS
285
A= 2 dM+b.
(4.31)
Here J. is the Laplace-Beltrami operator on M given by z
dMf = g1 V i7 f * = 9 as'
(4.32)
- g`' {`k!} axk'
and b = b°(x) axt is the vector field given by
b' =
(4.33)
gmk( {m'k} - TT,k). 2
Indeed, considering f(r)-f(x) for r = (x, e), we have
f(X(t)) - f(X(0)) = f(r(t)) -f(r(o))
f =
u
(Lg,f)(r(s))°dw (s)
f T. (r(s))dwa(s) + f o
2
La(La f)(r(s))ds. o
Therefore it is sufficient to show that d
aE La(Laf) = Af.
Note that A given by (4.31) may also be written as
A 2 g"Vivj
2 I g`j a axe -
9`'T fag}.
By Proposition 4.1,
La(.I f) =
(Fvvr)aa= (V,VJf)e`aea
.
Hence d
d
La(LaJ) = Ea=1(V V, )ea,4 = g'iV1V1f a=1 * pR =
is the Riemannian connection.
286
DIFFUSION PROCESSES ON MANIFOLDS
by (4.22). The above results are summarized below.
Theorem 4.2. Let M be a Riemannian manifold with an affine connection V which is compatible with the Riemannian metric g, and let fl, L2, ... , Ld be the system of canonical horizontal vector fields. Consider the stochastic differential equation (4.28) on O(M). The solution defines a
flow of diffeomorphisms r(t) = (r(t,r,w)) on O(M) and its projection X(t) = 7r[r(t)] defines a diffusion process on M corresponding to the differential operator A given by (4.31).
Definition 4.1. The process r(t) in Theorem 4.2 is called the horizontal lift of the A-diffusion X(t). Definition 4.2. In the case A = AM/2, the A-diffusion X(t) is called the Brownian motion on M.
Thus the horizontal lift r(t) of a Brownian motion X(t) on M is constructed by means of the Riemannian connection. Now we shall prove the following Proposition 4.3. (i) For every vector field b = bt(x)
8
on a Rieman-
nian manifold M, there exists an affine connection P = {T,} on M compatible with the Riemannian metric g such that (4.33) holds. (ii) Two affine connections V= {Tu} and P'= {T' } compatible with the Riemannian metric g satisfy (4.34)
gJkTjk = gJkf jk
for all i = 1, 2, ... , d
if and only if (4.35)
T7,=T'g, i=1,2,...,d,
where {Tu} and {TM 'k are the torsion tensors of P and P' respectively.
Proof. (i) Define f5k
2
= {/k} + d - 1(55bk - gJkb`)
where b, = gt1bJ. Then, since (5}bk - gfkb' are the components of a (1, 2)tensor, V= {Tik} satisfies (4.7) and hence defines an affine connection (j185]). It satisfies (4.18) since
NON-DEGENERATE DIFFUSIONS AND HORIZONTAL LIFTS
a
b a g" - 6pmq`
T
p D - 9,1.
287
!q
_ j gpq - gmq llmp} - gpm llmq}
,,
r
2 1
(gmgVi by - gmgglPbm + gDmV! bq -
$Dmglgbm)
2
d-_ 1 (g,gbp - bgglp + gDlbq - glgbp) 0.
Also,
d
2 glk({llk} .- rlk)
1
1 glk(albk - g1kbl)
d-1 (glkbk - b°d) = bl. (ii). First we note the following identity for any affine connection compatible with g: if {Tjk} is the torsion tensor and Sfk =
{T k}
a-
g1mgJ. Tmk, then
rlk = {j1k} + 2 TJk + 2 (SJk + S%J)
(4.36)
Indeed, by (4.18), a
gsj - gmj! ks - gsmrkJ - 0, 0, and
axj gsk - gmkrls - gsmr'lk = 0 a
T+
+ gmkf' + gjm k = 0.
axagjk
Hence by (4.19), 1
1,
2g
{J k}
2
a a.Xk
a
gsJ
axj gsk
glsgmj(rks - rsk) +
2
a
)
ax,,, gJk
glsgmk(JJs
- r;) +
2
(T.1, + rjk)
DIFFUSION PROCESSES ON MANIFOLDS
288
2
(SJk + SO +
(rkJ +
-T (SJk + SW + FJk
rk)
%, TJk
This proves (4.36). Since gikTik = 0, (4.34) holds if and only if for all
giksJk = $Jks'Jk
i=1,2,...,d,
that is gJkgimg1 (Tn,k
- T'n,k) = g'-(T;. - T'mn) = 0, i = 1, 2, ... , d.
Finally this is equivalent to gkrg'm(T mn - T 1 1= )=(_ 1
(T k -
0,
k = 1, 2, ... , d.
q.e.d.
From this we easily see that the correspondence between P= {T jk} and b defined by (4.33) is a bijection if d = 2, while it is a many to one surjection if d > 2. Let M be a differentiable manifold and A be a second order differential operator on M which is expressed in local coordinates as (4.37)
Af(x) =
2
a'J(x)
ax ax j (x)
+ b'(x) af- (x), f E=F(M)
where (a'J(x)) is symmetric and non-negative definite.*' If (a`J(x)) is strictly positive definite, i.e., (4.38)
0
for all
x and
= (c,) E Rd\ {0},*2
we say that the operator A is non-degenerate. The corresponding A-diffusion is also called non-degenerate. Now any non-degenerate diffusion on M can be constructed by Theorem 4.2. Indeed, let A be a non-degenerate differential operator. In a change of local coordinates, (a`J(x)) and (b'(x)) transform accordingly as (4.39)
4"'J(x) = ak'(x)
a;V aJCJ
axk ax'
*',*7 These properties of (a'J(x)) are independent of the choice of local coordinates.
NON-DEGENERATE DIFFUSIONS AND HORIZONTAL LIFTS
289
and
(4.40)
ail
b'(x) = bk(x)
a
+ i akl(x) aXkax'
(4.39) implies that (a'J(x)) is a tensor field of type (2,0) and hence its inverse matrix (gjj(x)) defines a tensor field of type (0,2) which is also symmetric and positive definite. Thus, it defines a Riemannian metric g on M and so M is a Riemannian manifold. Then we have (4.41)
A=
dm+b 2
where b = b'(x) (4.42)
ax'
is given by
El(x) = b'(x) +
g" {r'J}
.
2
Obviously b is a vector field on M. Now choose an affine connection P= { flk} on M compatible with g such that b'(x) = 2
gmk( {m'k}
j
\#
This is always possible by Proposition 4.3. The A-diffusion is now constructed as in Theorem 4.2. Remark 4.2. In this construction of an A-diffusion we have made use of an affine connection. An A-diffusion can also be constructed using only the Riemannian connection yR. In this case, we first construct a flow of diffeomorphisms r(t) = (r(t, r, w)) on O(M) as the solution of the stochastic differential equation (4.43)
dr(t) = La(r(t))odwa(t) + L,(r(t))dt,
where (r1, LZ, ... , E) is the system of canonical horizontal vector fields corresponding to PR, and the drift vec t )r field to is the horizontal lift (with respect to PR) of the vector field E. The A-diffusion X(t) is then obtained by the projection: X(t) = 7c[r(t)]. Finally, we shall study some problems related to the invariant measure * g'-' = a-' by definition.
290
DIFFUSION PROCESSES ON MANIFOLDS
of a non-degenerate A-diffusion. For simplicity, we shall assume that M is compact and orientable. As we saw above, we may assume without loss of generality that M is a Riemannian manifold and A is of the form (4.44)
A
Am + b,
where 4M is the Laplace-Beltrami operator and bE=-X(M). Let JP.) be the system of diffusion measures determined by A (i.e., the A-diffusion). P. is a probability measure on fk(M).*' The transition semigroup T, of the A-diffusion is defined by (4.45)
(Tt f)(x) = f
f(w(t))P,(dw),
f c= C(M).
W(M)
Let 0 be a domain (i.e., a connected open set) in M and define paw E #1(92), W E=- W(M) by
if t < %(w) if t > -Ca(w)
w(t)
(paw)(t) = 1A
where zs,(w) = inf {t; w(t) Q). The image measure of P. (x E Q) under the mapping pa is denoted by P. It is a probability measure on R92). As is easily seen, JP- Q) XE 2 defines a diffusion on 0 and it is called the minimal
A-diffusion on Q. Its transition semigroup is defined by
(T?f)(x) = f
f(w(t))Po(dw) W(Q)
(4.46)
f
w(M)f(w(t))I{Tn(W)>) Px(dw),
fECb(Q).*2
Definition 4.3. A Bore] measure ,u(dx) on M is called an invariant measure of the A-diffusion {PX} if (4.47)
f T f(x),u(dx) = f f(x),u(dx) M
M
for all f E C(M).
Definition 4.4. (i). An A-diffusion (P} is said to be symmetrizable if there exists a Bore] measure v(dx) on M such that Since M is compact, *(M) = W(M): the set of all continuous paths in M. #z We set f(d) = 0 for f e Cb(Q).
NON-DEGENERATE DIFFUSIONS AND HORIZONTAL LIFTS (4.48)
f Tf(x)g(x)v(dx) M
=
f
291
f(x)(Trg)(x)v(dx) M
for all .f, g E C(M).
(ii). An A-diffusion {Px} is said to be locally symmetrizable if {PQ} is symmetrizable for every simply connected domain Q cM, i.e., if there exists a Bore] measure va(dx) on 0 such that (4.49)
f Tff(x)g(x)va(dx)
n
=f
f(x)TQg(x)va(dx)
n
for all f g E Cb(Q). It is clear that if {PX} is symmetrizable, the measure v in (4.48) is an invariant measure. A differential 1-form cob is defined from the vector field b by (4.50)
cvb = br(x)dxr,
where b = b'
ax,
and b, = gr,b' in local coordinates. By a well-known
theorem of de Rham-Kodaira ([145]), wb has the following orthogonal decomposition:* (4.51)
wb=dF+Sfl+a
where Fe F(M), fi is a 2-form and a is a harmonic 1-form. Here we briefly
recall some of necessary notions. An inner product is defined on the totality A,(M) of all p-forms on M by (4.52)
(a, 9), = fm dx,
where a
Adx'P, f r.r.....i z a gr>J,grzlz ... g dx" A dx'2 A ... A dx °, Q
,I<12<...<rp
dx = /det(gv(x)) dx'dx2 ... dx°. The operator 6: A,(M) - A,.,(M) is defined by * With respect to the inner product defined by (4.52) below.
292
DIFFUSION PROCESSES ON MANIFOLDS
(4.53)
a EA,_1(M), f EA,(M).
(da, )a = (a,
: A,(M)
De Rham-Kodaira's Laplacian
A,(M) is defined by
= -(dd + 8d).
(4.54)
a E A,(M) is called harmonic if a = 0. The totality of all harmonic p-forms is denoted by H,(M). It is known that a = 0 if and only if da = 0 and 8a = 0. For f e F(M), grad f E X(M) is defined by
grad f -J axj ax' ,
(4.55)
and for X E X (M), divX E F(M) is defined by
div X = -bco, (4.56)
_
det G ax,
(X`
det G).*
Then the Laplace-Beltrami operator (4.32) is also given as
div(gradf) = -8df
(4.57)
for f EF(M).
It is easy to see that y is an invariant measure of the A-diffusion {Px} if and only if
f Af(x),u(dx) = 0
(4.58)
for all f e F(M).
Indeed, we know by Theorem 3.1 that u(t,x) = Tf(x), f E=- F(M), is the unique solution of 8u = Au(t,x)
at
lim u(t,Y) = f(x). r 10.Y-'x
Also, we saw in the proof of Theorem 3.1 that Au(t,x) = TT(Af)(x). Hence if (4.47) holds, we have by differentiating with respect to t that
* G=(gr,) and X=X'a
.
NON-DEGENERATE DIFFUSIONS AND HORIZONTAL LIFTS
293
for all f EF(M).
fm Tt(Af)(x),u(dx) = 0
We obtain (4.58) by letting t 10. Conversely, if (4.58) is satisfied, then
0= J M (Au)(t,x)p(dx) = dt
f
m
T, f(x)p(dx).
Consequently (4.47) holds for f E=-F(M) and hence for f EC(M). Proposition 4.4.
(Af, h)o = (f, A*h)o,*'
f, h EF(M),
where (4.59)
A*h = - 2 8dh + 6(hwb) = 2
div(hb).*2
The proof is immediate from the definition. Proposition 4.5. An invariant measure p(dx) of the A-diffusion exists and is unique up to a multiplicative constant; moreover, p(dx) is given as v(x)dx where veiF(M) is a solution of
A*v = 0.
(4.60)
Proof. The equation (4.58) is equivalent to A*,u = 0 in the sense of Schwartz distributions on M. Since A* is elliptic, any solution must be of the form p =vdx, v EF(M), by Weyl's lemma ([1]). Furthermore, 'I = 0
is the largest eigenvalue of the eigenvalue problem (A - 2)¢ = 0, it is simple and {coo; c ER} is the eigenspace where 00(x) - 1. Therefore, 2 = 0 is also the largest eigenvalue of the eigenvalue problem (A* - .1)O = 0, it is simple and its associated eigenspace is given as {co*o ; c E R} where we can choose 0o such that 0 *0(x) > 0 for all x E M.*' Hence all invariant measures of the A-diffusion are given as p = c
f
f(x)g(x)dx, dx = /detC, dx'dx2 ... dx°. *2 For f EF(M) and a e A,(M), a = a,,,,2.....1,dx" Adx'2A ... Adx'p, fa is defined by fa = (fa,,.r2...... )dx" Adx'2A ... Adx'D.
This fact is well known. It is also a consequence of the theory of positive operators
[89].
DIFFUSION PROCESSES ON MANIFOLDS
294
We choose U(x) EF(M) so that A*(e-U) = 0, i.e., a-Udx is an invariant measure. Since A*(e-U) = - 2 6d(e-U)+6(e °c)b) = 0, a-Uwb
- 2 d(e-U) = Sfi,+a, for some /3,EA2(M) and a,EH1(M), i.e., (4.61)
a-°wb = 2 d(e-U) + aflx + a,,
where fl, E A2(M) and a, c- H, (M). Conversely, if U satisfies (4.61) with
some fl, and a, then A*(e-U) = 6 e-uw,, - 2 d(e-U)) = 5(afl + a,) = 0 and hence is an invariant measure. (4.61) gives the de RhamKodaira decomposition of the 1-form a Uwb. Theorem 4.6. (i). The A-diffusion is symmetrizable if and only if (4.62)
6/3 = a = 0
in (4.51);*
and this is equivalent to (4.63)
bf, = a, = 0
in
(4.61).
The condition (4.62) or (4.63) is equivalent to the condition that b be given as (4.64)
b = grad F,
F E F(M),
and in this case the invariant measures are of the form constant x e2p`x' dx. (ii). The A-diffusion is locally symmetrizable if and only if (4.65)
8fl = 0
in
(4.51);
this condition is equivalent to (4.66)
dcob = 0.
* [88] and [127].
NON-DEGENERATE DIFFUSIONS AND HORIZONTAL LIFTS
295
(iii). The A-diffusion has a measure cdx (c > 0: constant) as its invariant measure if and only if dF = 0 in (4.51) and this condition is equiv-
alent to
acob = -div b = 0.
(4.67)
Corollary. The A-diffusion is symmetric with respect to the Riemannian volume dx (i.e., it is symmetrizable and the measure v in (4.48) is dx) if and only if it is the Brownian motion on M. Proof. Let U°(x) E= F(M) be determined by
J
M
e u°(=)dx = 1
and
A*(e-u°)
= 0.
Then the measure a uo(X)dx is the unique invariant probability measure of
the A-diffusion. If we introduce another inner product on F(M) by (4.68)
fm
u(x)v(x) eu°' dx,
then it is easy to see that
Av = 2 4MV - (b + grad U°)v.
Suppose the A-diffusion is symmetrizable. As we remarked before, the measure v in (4.48) is an invariant measure, and hence if v is normalized so that it is a probability measure, we must have v(dx) = e u0(")dx. Thus = = =
b = -(b + grad U°).
DIFFUSION PROCESSES ON MANIFOLDS
296
Hence
b = - 2 grad U0. Conversely, if b = grad F, F E F(M), then cob = dF and so ezrmb
= I d(e'"). Therefore U = -2F satisfies the equation (4.61) with bfl, = a,= 0. Hence e-Odx is an invariant measure and U = Uo+c for some constant c. Consequently we have b = - 2 grad Uo and this implies that Av = Av, v E F(M). But generally, the A-diffusion {PS} and A-diffusion
(P.) are connected to each other, through their transition semigroups T, and L, respectively, by the relation
u, v E F(M).
= , Indeed,
ji = <-AT,_,u, t v> +
= - + = 0, and hence
- = - f,
ds
ds
=0. Therefore A = A implies that = and consequently the A-diffusion is symmetrizable. Now the proof of (i) is completed. The proof
of (ii) can be given similarly. Finally, the measure dx is an invariant measure if and only if the function U in (4.61) is a constant, that is, cob = 8fl + a, for some fl EAZ(M) and a EH,(M). Remark 4.3. It might be interesting to give an probabilistic interpretation for conditions like "a = 0 in (4.51)", "a, = 0 in (4.61)" or "fl, = 0 in (4.61)". In this connection, Manabe [109] obtained the following result. He first defined "the stochastic Kronecker index" I(X[0, t], c) between a d-1 chain c in M and an orbit X[0, t] _ {X(s); s E [0, t]} of A-diffusion
STOCHASTIC PARALLEL DISPLACEMENT
297
X(t) as a stochastic process. He then showed that a, = 0 in (4.61) if and only if for every d-1 cycle c lim I(X[0, t], c)/t = 0 itm
a.s.
5. Stochastic parallel displacement and heat equation for tensor fields
Let M be a Riemannian manifold and A be the Laplace-Beltrami operator. Then, as we saw above, the solution of the heat equation au (5.1)
1
at = 2 d u uIt_o =f
can be solved uniquely as (5.2)
u(t,x) = E[.f(X(t,x))]
where X(t,x) is a Brownian motion on M starting at x. In order to generalize this fact to the case of heat equation for tensor fields, Ito ([68], [72])
introduced the notion of stochastic parallel displacement. His idea is as follows. Consider the equation (5.1) where u and f are now tensor fields on M and Au = g°"171V u, 17 being the covariant differentiation with respect to the Riemannian connection. Then the solution u is given by (5.3)
u(t,x) = Ef f(X(t,x))*],
where f(X(t,x))* is a tensor at x obtained from the tensor f(X(t, x)) at X(t, x)
by parallel displacement along the time reversed Brownian curve. The difficulty in this procedure is in obtaining f(X(t,x))* as the parallel translate off(X(t,x)) along the time reversed Brownian curve. Ito defined it as a limit of the parallel translate along a piece wise geodesic curve from X(t,x) to x which approximates the time reversed Brownian curve. But as we saw in the previous section, we can use stochastic calculus to perform a
parallel displacement along the Brownian curve from x to X(t,x) in the usual sense of time. Moreover, we can actually realize Ito's idea using only
such parallel displacements: instead of translating a tensor at X(t,x) to a tensor at x, we translate an orthonormal frame e at x to an orthonormal frame e(t) at X(t,x) along the Brownian curve and read the tensor at X(t, x) using this frame e(t). This approach is due to Malliavin [104]. It is now
DIFFUSION PROCESSES ON MANIFOLDS
298
clear that the process r(t) = (X(t,x), e(t)) is the horizontal lift on O(M) of the Brownian curve X(t,x) constructed in the previous section. The heat equation (5.1) for tensor fields is solved in this manner. By modifying the expectation with a Feynman-Kac type weight we can also solve the heat equation for differential forms as at
(5.4)
1
Da
al,-. =f where
is the Laplacian of de Rham-Kodaira (4.54).
Let M be a compact Riemannian manifold and O(M) be the bundle of orthonormal frames over M. Let {L,, LZ, ... , Ld} be the system of canonical horizontal vector fields on O(M) with respect to the Riemannian connection {J'k} . As in the previous section, the flow of diffeomorphisms
r(t) = (r(t,r,w)) on O(M) is defined by the solution of the stochastic differential equation
(5.5) jl
dr(t) = L«(r(t))-dw«(t) r(O) = r.
r(t) defines a diffusion process on O(M) which corresponds to the differ-
ential operator 2 do(M), where (5.6)
docmr> _ E L«(L«).
r = (r(t)) is called the horizontal Brownian motion on O(M) and douM> is called the horizontal Laplacian of Bochner. As we saw in the previous section, the projection X(t) = lr[r(t)] is the Brownian motion on M. We write r(t, r, w) = (X(t, r, w), e(t, r, w)). We know that (5.7)
r(t, r a, w) = r(t, r, aw) a
(5.8)
X(t,
w) = X(t, r, aw) and e(t, w) = e(t, r, aw)a for every a E O(d).*
* See (2.31) for the definition of the action
of a on r.
STOCHASTIC PARALLEL DISPLACEMENT
299 (FTJiJ2
Let T(x) _ (x)) be a (p, q)-tensor field and F,.(r) = ' (r)) be its scalarization. It is a system of smooth functions on O(M). Let d be the Laplace-Beltrami operator acting on tensor fields defined by
()
ll 12...1p
5.9
1J
1112...!,
(4T)JiJ2::; = g`J(P(PT))JJ2...P, _ g Ti,J2...Jo.t;J
where PT is the covariant derivative of T with respect to the Riemannian connection. By Proposition 4.1, we have (5.10)
(d O(M)
F y r2... rpp (r = T1,J2... Jq)( )
1112...1, FeTJ,J2...Jq( T)
for every il, i2, ... , ip jl,j2, .. , j.Indeed
0(M)(FTJj12.. !q
aLa(T`TJJJ2..lq) 1112...1,
_
Fv7TJ1J2..Jeaa
a
--
11 erelaeJ a t e/12
...
a
=
e322
= (F
...
1g
eJ f tl, k f r2k2..
eil fk11fk2 ...
f 1P112... iqt!
./ L,(dT)1l122.:.
,(FJ7T)'; k2-..k,
P
112
J2 .. Jq
since Z e,',eJ, = g"J. a
For a given (p, q)-tensor field f = (f(x)), we define a system of functions Ui",f2;; q(t, r) on [0, co) x O(M) by (5.11)
UJ;i2:: q(t, r) = E[(Fr)j;i2.. !q(r(t, r, w))]
for every il, i2, ... , i jl, j2i ... , jq = 1, 2, ... , d. By Theorem 3.1 U 1'2* .1iq, is the unique solution of the heat equation
av
at =
(5.12) 1
I
2
doCM,V
V1,_0 = (F!)j f2.: Iq 1
300
DIFFUSION PROCESSES ON MANIFOLDS
Since Ff = {(Ff) j1'2.: `y) is 0(d)-equivariant, we can easily prove by (5.7)
and (5.8) that U(t) = {U;;2..Q(t, )} is 0(d)-equivariant for every t > 0. Therefore there exists a unique (p,q)-tensor field u(t, ) _ {uj,',22..fq (t, .)} such that UJilp.'JDQ(t,r) _
,q(r)
By (5.10), it is clear that u(t, -)is the unique solution of (5.1). In this way
the heat equation (5.1) for tensor fields can be solved uniquely. Next we consider the equation (5.4) for differential forms. Let
a(x) _ E t,
ar1 r2...r (x)dx`1 A dx2 A ... A dx`D D
be a p-form. We agree to define a1112...1D(x) for all system of indices by
the alternation property, so that ar1r2...tp(x) = 0 unless the indices i1, i2, - - , p are all distinct and ar1i2...iD(x) = for any permutation o. Then.
a(x) = p1
!
ar1r2...rp(x)dxrl A dxr2 A ... A dxrD.
Following [145] or [128], we identify a(x) (in a coordinate neighborhood) with the system of functions (ar1r2...rD(x)) and write a(x) = (ar1r2...1D(x)). Note that these components (ar1r2...rp(x)) of p -form a are p! times of those when a is regarded as an alternate (0, p) tensor.
Since the Riemannian connection is torsion free, we have (5.13)
D+1
(da)t1t2.:.1,+1
where the circumflex denotes omission. Combining this with (4.53) it is easy to see that (5.14)
(Sa)1,1,...r,,
By (5.13) and (5.14),
v=1
and
= - grkVkar 11 ... x,_1
STOCHASTIC PARALLEL DISPLACEMENT
(bda)t1i2...1 = - V`Vlat1t2 .1. -
301
(-1)°i7 Vlyar r,...ty..1
Then it is clear that (5.15)
(-1)v(VwV` -
VtVrayt2..1y -
Notice Ricci's identity D
(VIVk - VkVt)ayt2...i, =
t...rD
where (
R'/kt = a.
{It/}
l) XJ {k'!} + ( J ltafl lk'a} -
J
(kal) {Ita} )
is the component of the curvature tensor and
[145]. Applying this identity to (5.15) we see that (5.15) can be rewritten as
(da)t,i2...1, - L (5.16)
iv ah 1,...Yu.-p
-2 E (-1)µ+vRhtv , a t'k
v
The relation (5.16) is called Weitzenbock's formula ([128], [145]). For the 1-form a = a!(x)dx`, (5.17)
(fla)t = (da)t + Rtaj
where RJ, = R'.k ;. In exactly the same way as above, we see that the equa-
tion (5.4) is equivalent to the following equation for alternate and 0(d)equivariant systems V(t, .) = {V,1t2...,p(t, r)} of functions on O(M)
a
Vi,i2...1(t, r)
= (5.18)
1
1
2
h
JO(M)Vt,lZ...rp(t, r) - 2
- E (-
IV
,(t. r)
D(t, r)
DIFFUSION PROCESSES ON MANIFOLDS
302
=2
V),1 2...ty(t, r),
Vttt2...t,(0, r) = (Fr)t,t2...1',(r),
where { J/t(r) } is the scalarization of R/t = RJkki and { J`/kt(r) } is the scalarization of f R f }k1 }. Just note that o (M, F° = FE],, for a CAD(M) by (5.16). By using the tensor { Rik, } we can form various mixed tensors by raising and lowering indices via Ri k i = $/°Rt°kt
Rt/kt = $t°R°jkt,
etc.
Then a straightforward calculation in local coordinates shows that
Rt/kt = Rktt/ _ - Rutk The Ricci tensor Rt/ is defined by
Rt/ = Rk Then we have R/ = R/,. We note that J/t(r) = J/t(r) and Jt/kt(r) = Jl/kt(r) for all i, j, k, 1 where (J/t(r)) and (Jt/kt(r)) are scalarizations of (R/t(x)) and (Ri/k,(x)), respectively: It is a general fact that raising and lowering indices of components are irrelevant for scalarizations of tensors. The equation (5.18) is apparently complicated and so following [194] we now rewrite it into more compact and manageable form by introducing several algebraic notions, especially exterior and interior mul-
tiplications (creation and annihilation operators) over exterior (Grassman) algebra. 6d Let Rd be the d-dimensional Euclidean space and 8', 82, be the canonical basis, i.e.,
,
6'=(0,...,
Let ARd = E
(@
0,
t-th 1
,
0,..., 0).
ARd be the exterior (= Grassman) algebra over Rd:
m0
Euclidean space of dimension (p) with basis ARd is the '0_0 6t,A61eA...A6,
1
and hence, ARd is the Euclidean space of dimension 2d. An element
STOCHASTIC PARALLEL DISPLACEMENT CO of
303
ARd is uniquely expressed as
CO =
ll<12<
<(p
0)1112...[p6" A a12 A ... A 6'p
and we agree to define w1112...'p for all system of indices by the alternalp = 0 unless il, i2, , i, are all distinct for any permutation a. Then we and 0)11'2.. 1p = sgn((7)w, ul) c (12) have
tion property so that
w=-iw1112...16t1A$12A ... A51. We denote co = (w1112._'p) and call w11'2...'p components of co.
If V1, V2 are vector spaces, the space of all linear mappings from Vl and V2 is denoted by Hom (VI, V2). For a vector space V, Hom(V, V)
is denoted simply by End(V). Thus End(V) is the algebra of all linear mappings on V. For each i = 1, 2, d, let a! E End (ARd) be defined by
(5.19)
a,* (w) =Straw,
ARd.
co
Thus a* is a linear mapping on ARd sending i1Rd into it Rd for each p (creation operator). The dual of a* is denoted by a1. Hence a' E p
p+l
End(ARd) which sends ARd into A Rd (annihilation operator). easy to see that
{a,,a1} ={a*,aj*}=0
It is
and
(5.20)
[a,*, a, } = E1JI
where
{a, b} = ab + ba
for a, b c- End(ARd).
In terms of components, these linear mappings satisfy the following: If co = ((o1112...tp), i.e., CO =
pw'1t2 i ,6'1A 5'z ... As'p
304
DIFFUSION PROCESSES ON MANIFOLDS
then, for i = 1,2, ... , d, p+1
(5.21)
[a*(w)]l,ty...lp+1 = - Z (- 1)° (.O1112 ...7v'..lp_1vl,1y
(5.22)
[at(w)]t,1y...lp_i = w!ll...lp_i.
For a - (alj) E Rd & Rd, define D1[a] E End(ARd) by d
(5.23)
D1[a] = ± alja*aj. bJ-1
D1[a] sends ARd into ARd for every p and we see by (5.21) and (5.22) that (5.24)
(D1[a]((0))l1,2..t, =
- L1 Z (-
1)v
wjll...pv _lpalvJ
For p = (P Jk) E Rd ® Rd O Rd QX Rd, define D2LY] if co = E End (ARd) by (5.25)
/
D2L8]
a
.E1
/
Ptjkla*ak aJat.
GJ
D2[#] also sends ARd into ARd for every p. Furthermore, if fl satisfies (5.26)
Pijkl = Jlklij
we see easily that (D2[8](0-)))111y...lp d
(5.27)
= 2 1Sv
if co = (w1112...1p). Hence, if fi satisfies furthermore (5.28)
Prjkl --YJtkl = -Yljlk
then we have
STOCHASTIC PARALLEL DISPLACEMENT (D2[fl](w))s1 r2.. lr
(5.29)
=2
305
d
E (-l)y+,e COlkl1
ISv<,,Sp 1,k-I
lpflklj,liy
Noting that J,f(r) = J1,(r) and that Jlfkl(r) satisfies both (5.26) and (5.28), we can now rewrite (5.18) in the form
at
V= 2
OCM1 V
= 2 Jo(m) V + 2 D1[Jij(r)] V - 2 Dz[Jjk,(r)] V
(5.18)'
V(0,r) = Ff(r). If we further define, for f _ (fffk,) E Rd ®x Rd ®x Rd ®x Rd, DZ[f] E End(ARd) by d
(5.30)
Dz[fi] =i,f,k,l-l E $ifkla*afak are
then it is immediately seen from (5.20) that D2[Q] + Di[a]
bz[fl]
where d
a = (aif)
Rd Qx Rd,
a,j
kL...il f lkkj.
Hence (5.18) can finally be rewritten in the following simple form ([194]):
aV
at (5.18)
1
=2
oiM1v
= 2 Ao(M) V + 2 Dz[Jifkl(r)] V
V(0,r) = Ff(r). Thus, to find a solution a(t,x) of (5.4) is equivalent to find a ARd-valued, 0(d)-equivariant function U(t,r) = (U1112.,p(t,r)) defined on (0, co) x O(M) which satisfies (5.18)".
Now we are going to show that a ARd-valued smooth function U(t,r) on [0, oo) x O(M) which satisfies (5.18)" exists and is unique; furthermore, this U(t,r) is 0(d)-equivariant. In order to construct this
DIFFUSION PROCESSES ON MANIFOLDS
306
U(t,r), we modify the expectation in (5.11) with Feynman-Kac type weight. So let r(t) = r(t,r,w) be the solution of (5.5) defined on the ddimensional Wiener space (W(d, PK') and, for each w E Wo and r E O(M), consider the following ordinary differential equation for End (AR')-valued function M(t): (5.31)
dM (t) = 1 M(t) D2[Jl>k,(r(t,r,w))] dt M(O) = I (: = the identity in End(ARd)).
Clearly the unique solution M(t) E End(ARd) of (5.31) exists. We denote M(t) = M(t,r,w) to clarify the dependence of M(t) on r = r(O) and w.
For a given p-form
I(x) =
p
l,iV
1,(x)dx" A dx12 A ... A dx`p,
let Ff(r) = {(Ff),liz...,p(r)} be its scalarization and define a
APRd-valued
function U(t,r) on [0, oo) X O(M) by (5.32)
U(t,r) = E[M(t,r,w) Ff(r(t,r,w))].
We can conclude as in Section 3 that U(t,r) is smooth on [0, oo) x O(M). Note that the following Ito formula holds for any smooth
valued smooth function V(t,r) on [0, oo)xO(M): M(t)V(t,r(t)) - V(0,r)
f (5.33)
+f
r 0
0
M(s) (LaV)(s,r(s))dwa(s)
M(s) {(aV (s,r(s)) +
do(M) V(s, r(s))
+ 2 DZ[Jitkl(r(s))] V(s,r(s))}ds
= a martingale +
f M(s)( aV + o(M) V) (s, r(s))ds. o
Then, by the same argument as in Theorem 3.1, we can show that U(t,r)
defined by (5.32) is the unique solution of (5.18)". Finally, we show that U(t,r) is 0(d)-equivariant. For this, we intro-
STOCHASTIC PARALLEL DISPLACEMENT
307
duce some notations. Let g = (gJ1) E 0(d) and define A(g) E End(ARd) by (5.34)
A(g) [451' A 4512 A ... na=p] = (5 g) A (b12g) A ... A (&5'g)
where 61g = g16afl, i = 1,2,
,
d. In other words, A(g) E End(ARd)
which sends ARd into ARd for every p and satisfies (5.34)'
[A(g) (w)]'1'2...1p
= g,',,g22 ... g,'.
fip
if co = (co1112...1) E APRs. Note that A(gh) = A(h)2(g) for g,h E- 0(d) and hence, A defines an action of 0(d) on ARd from the right. The fact
that Ff(r) is 0(d)-equivariant can now be stated as A(g)F(r)
(5.35)
where the action
for every g E 0(d) and r E O(M),
of gEO(d) is defined by (2.31). Also, for g =
(g) E 0(d), define an Rd Q Rd-valued function [r(g)J]1J(r) and an Rd Qx Rd ®x Rd Qx Rd-valued function [T(g)J]1jk1(r), both defined on O(M), by (5.36)
[z(g)J]1(r) = g?gfJap(r)
and (5.37)
[z(g)J],Jk1(r) = g?gfgkgfJapra(r)
The fact that { JJ(r) } and { JJJk1(r) } are 0(d)-equivariant can now be stated as (5.38)
.It, (r'g) = [z(g)J]1J(r)
and (5.39)
JJJkl(r'g) = [z(g)J]1Jk1(r)
for every i,j,k,1, g E 0(d) and r E O(M). It follows from (5.24) and (5.29) that D1[[r(g)J]1J(r)] = A(g)D1[J,J(r)] and
A(g)-,
DIFFUSION PROCESSES ON MANIFOLDS
308
/
D2[[T(g)J]ljkl(r)] = A(g)D2[J11k1(r)] 2(g)-'.
Hence, combining these with (5.38) and (5.39), we have (5.40)
2(g) D1[J1,(r)]
2(g)-1
and (5.41)
D2[Jlfki(r'g)] = 2(g)D2[Jljkl(r)] A(g)-'
We can conclude by (5.40), (5.41) and (5.7) that 2(g)M(t,r,gw) 2(g)-',
(5.42)
g e 0(d).
From this and (5.35), we see immediately that U(t,r) satisfies U(t,rg) = 2(g)E[M(t,r,gw)Ff(r(t,r,gw))] = 2(g)E[M(t,r,w)Ff(r(t,r,w))]
= 2(g)U(t,r).
This means that U(t,r) is 0(d)-equivariant. Malliavin [104] used the above to obtain an interesting generalization of a vanishing theorem of Bochner ([185]) for harmonic 1-forms.
6. The case with boundary conditions We shall now discuss a similar probabilistic construction of the solution of the heat equation (5.4) for differential forms in the case of a manifold with boundary.*
Let M be a Riemannian manifold of dimension d with smooth boundary. The interior and boundary of M are denoted by )I( and aM respectively. Near the boundary, we can choose a coordinate neighborhood U and a local coordinate x = (x', x2, . . . , xd) in U such that xd > 0 for all x E U and x E U fl aM if and only if xd = 0. The tangent vector
n(x) = nl(x) (6.1)
a a
at x EaM given by
nl(x) = gld(x)I
gdd(x),
i= 1, 2, ... , d
is called the inward pointing unit normal vector at x. For a smooth function f defined in U, The material in this section is adapted from [55] and [172].
THE CASE WITH BOUNDARY CONDITIONS
(x) = nr(x) of (x),
(6.2)
309
x e am,
an
is called the normal derivative off at x. Let
7:
al112... tp
and -"'2(6.3)
Id
be the skew symmetric (0, d) tensor field defined by
Ertr2...td(x) =
det(g11(x)) x1112. atd .
For a p-form a, its adjoint *a is a (d - p)-form defined as follows. If a is expressed as
a(x) =
(6.4)
'1<'2<...<tp
at, r,... r (x) dxl, A dx'2 A ... A gyp,
then
*a(x) =
(6.5)
E
a*J1J2...Jd ,(x)dxf, Adx'2A ... Adx i-p,
where (6.6)
-
a,*, J2..Jd_p(x)
"
r1t2...tD/1J2...Id_p(
t ! xa12...lpx
and
a't12...lp(X) = gtti1(x)g',J2(x) ... g'pip(x)a1112..Jp(x).
Let 0 be a p-form. We denote by 9taa the restriction of 0 to aM and call it the tangent component of 9. We define the normal component of 0 by Onorm = 0 - 0tanLet r7(x) be a differential 1-form such that (6.7)
r71 a,,,(x) = r7t(x)dx',
x E am
where rjt(x) = gt1(x)nJ(x) and nt(x) is defined by (6.1). Then it is easy to
see that the restriction
(-1)p(d-p)+p-1 [*(*0Ar7)]ArilaM of thep-form
(-1)p(d-p)+p-1 [*(*6Ai7)] A'7 is uniquely determined from 0 and coincides with 9norm-
DIFFUSION PROCESSES ON MANIFOLDS
310
Definition 6.1. A differential form 0 is said to satisfy the absolute boundary conditions if Onor,, = 0 and (dO)norm = 0 ([144]).
We want to solve the heat equation (5.4) with the absolute boundary conditions, namely, as F'-
1
2E]a
(6.8)
anorm '- O,
(da)norm = 0
where f EA (M) is given. In order to avoid undesirable complications, we shall restrict ourselves to the case of 1-forms. Near the boundary, we can choose a coordinate neighborhood U and a local coordinate x = (x',x2, ... xd) in U such that the following holds:* (i) xd > 0 for all x E U;
(ii) x E Un am if and only if xd = 0; (iii) the metric tensor g(x) = (g,,(x)) satisfies g,d(x) = 0 for
.. ,d-1.
i = 1, 2,
In this local coordinate, it is easy to see that OEA,(M) satisfies the absolute boundary conditions if and only if (6.9)
Od(x) = 0 and
a
0,(x) = 0,
i= 1, 2, ...,d-1, xEUfaM. Indeed, we have anorm
= Od( x)dxd and ( d0 )norm
= d-1 (LO,
i-1 axd
aO
dxd
A dx',
and so (6.9) follows at once.
Let O(M) be the bundle of orthonormal frames over M. If F9(r) = ([FB],(r)) is the scalarization of 0, i.e., [Felt (r) = O1(x)ei,
r = (x', e!),
then (6.9) holds if and only if (6.10)
f[F9]1(r) = 0 and fJ, 5T [FO]j(r) = 0, i = 1, 2, ... , d - 1,
* Cf. [13], [144].
THE CASE WITH BOUNDARY CONDITIONS
311
where (f;) is the inverse of (es). Thus, the initial value problem (6.8) for differential 1-forms is equivalent to the following initial value problem for Rd-valued functions (Ut(t,r)) on O(M) which are 0(d)-equivariant:
a `(t, r) = 2 [Jo(m, UU(t, r) + Ji(r)U1(t, r)} (6.11)
UU(0, r) = (Ff),(r) a U1(t, r) I aocmn = 0, f, axd a
i= 1, 2, ... , d- 1,
f'dUj(t, r)Iaocnn = 0
where { J,(r) } is the scalarization of the tensor { R,(x) } and aO(M) _
{r = (x, e) EO(M); xe aM ). We note that by using same notation as in Section 5, (6.11) can also be rewritten in the following form:
at U =
2 o (M) U
=
[do(M) + D1[Jj]]U 2
U(0, r) = Ff(r)
fi axd
UP, r) I ao (M) = 0,
i = 1, 2, ... , d - 1
A d U j(t, r) I ao (M) = 0.
We will now solve this initial value problem using the horizontal Brownian motion (r(t)) on O(M). (r(t)) can be obtained by solving stochastic differential equations with boundary conditions (Chapter IV, Section 7). Since the construction of solutions can be localized, we
do not hesitate to put the following assumptions. Assumption (A). M is the upper half space of Rd:
M = {x; x =(x',x2,
...
xd) (=-Rd, xd > 0}
and
aM = {x E EM; xd = 0).
Assumption (B). The Riemannian metric tensor (gj(x)) under the coordinate x=(x', x2, ... , xd) consists of C°°-functions which are
312
DIFFUSION PROCESSES ON MANIFOLDS
bounded together with all of their partial derivatives. Furthermore, (g,1(x))
is uniformly positive definite and satisfies gtd(x) - 0 for i = 1, 2,
... ,
d- 1. We consider the following stochastic differential equation for the process (X(t), e(t)) on R . X Rd : dX'1 = ek(t)odBk(t) + btddO(t) (6.12)
de(t)
{I'k} ((X(t))ea(t)odX'(t) {z'k}(X(t))ea(t)em(t)odBm(t) - {h} (X(t))e (t)dO(t)
i,a= 1,2, ... ,d. Here {1'k} (x) are the Christoffel symbols and B(t) = (B'(t)) is a d-dimensional Brownian motion. 0(t) is a continuous non-decreasing process which increases only when X(t) EBM. (6.12) is a particular case of the stochastic differential equations discussed in Chapter IV, Section 7, and so by Theorem IV-7.2, we know that for any Borel probability measure p on R + X Rdz, the solution (X(t), e(t)) of (6.12) with the initial law It exists uniquely. In the same way as in Section 4, we see that if gk1(X(0))e, (0)ei(0) = arj
then for all t > 0 gk1(X(t))ei(t)eJ'(t) = at!
holds almost surely; i.e., if (X(0),e(0)) E O(M), then (X(t), e(t))E O(M) for all t > 0 a.s. Thus we have a diffusion process r(t) = (X(t), e(t)) on O(M). It is called the horizontal Brownian motion on the bundle of orthonormal frames O(M) with reflecting boundary. Let fl, L2, ... , Ld be the canonical horizontal vector fields: (6.13)
(LmF')(r) = emax' (r)
-
{k'l} (x)emei ae (r),
(r = (x, e = (ei)),
m=1,2,...,d, and define the horizontal Laplacian of Bochner by d
(6.14) Set
(T
do(M) = E Lm(Lm)
THE CASE WITH BOUNDARY CONDITIONS (6.15)
add(r) = gdd(x),
313
amt(r) = -em {a°k} (x)gdd(x).
Theorem 6.1. Let r(t) =(X(t),e(t) = (e,,(t))) be the horizontal Brownian motion with reflecting boundary constructed above from the solution of (6.12). (i) For any smooth function F(t, r) on [0, oo) x O(M),
dF(t,r(t)) = (LmF)(t,r(t))dBm(t) + I 2 (Jo(M,F)(t,r(t))
+ OF (t, r(t ))} dt + (XdF)(t, r(t))dT (t), where Id is the horizontal lift of the vector field Xd = as which is given explicitly as (6.16)
(XdF)(t,r) = a-xd(t,r) + add(r)
aem`(t'r).
add(r(t))dt
(6.17)
m, i = 1, 2, ... , d.
admt(r(t))dt,
Proof. (i) is immediately obtained from Ito's formula. (ii) is easily proved once we notice e`' (t)el.(t) = g`'(X (t))
Theorem 6.1 implies that the process (r(t)) on O(M) is determined by
the differential operator 2 do(,) with the boundary condition XdF = 0 on aO(M). (6.17) implies that the process (r(t)) is a normally reflecting diffusion process in the sense of Definition 6.2 given below.
In order to obtain a solution of (6.11), we consider the canonical realization { r(t, w),w E W(O(M)),P, } of the horizontal Brownian mo-
tion on O(M) with reflecting boundary: W(O(M)) = C([0, oo) -O(M)), P, is the probability law on W(O(M)) of the solution r(t) of (6.12) with r(O) = r, and r(t, w) = w(t) for w e W(O(M)). For a Borel probability measure p on O(M), P. is defined by P,,(B) = Jo (M) P.(B)
x p(dr), B E .q(W(O(M))).* Let
n 0(W(0(M)))P'1 and . , _ K
* .4 (W(O(M))) and .°J(W(O(M))) are defined as usual.
314
DIFFUSION PROCESSES ON MANIFOLDS
I
{A E Y ; for any p there exists BA such that B, E 9 (W(O(M))) and P,, (A p 0 }. We now fix p and give the following discussions on the probability space (W(O(M)), Writing r(t, w) _ (X(t) X(t, w), e(t) = e(t, w)), we set
fi(t) = lim 81o 2e
1r
o
I0 -$)(Xd(s))gdd(X(s))ds
and
B'(t) = f r [e(s)-']ko[dXk(s) - &dO(s)] 0
Then {B'(t)} is a d-dimensional (,.W-,)-Brownian motion and {r(t) _ r(t,w), O(t)} satisfies the equation (6.12).
Lemma 6.1. (P, } is invariant under the action T. of a, a E 0(d),
from the right; that is, if w W(O(M)) by (w a)(t) = w(t) a and if Ta(Pr) is the image measure of P, under the mapping w i- w a, then (6.18)
T,(P,) = Pr.,
.
Proof. Let r(t) be a solution of (6.12) with B(t) and ¢(t) such that is a solution of (6.12) with B(t) = a-'B(t) and fi(t) = 0(t) such that F(0) = B(t) is another ddimensional Brownian motion and hence (6.18) holds by the uniqueness of solutions. By this lemma we see that X(t) defines a diffusion process on M and it is easily seen as in Section 4 that X(t) is determined by dM/2 with
r(0) = r. Then for a E 0(d), i(t) =
boundary condition an = 0 on M. This diffusion is called the Brownian motion on M with reflecting boundary or simply, reflecting Brownian motion
on M. Let Rd ®Rd be the algebra of all d x d real matrices a = with the norm IIalI2 =
d
l.J='
endowed
I aj 12. It is convenient for our purpose to define
the multiplication in Rd®x Rd by the following rule:* for a = (aj) and b = (bj), ab = (cj) where (6.19)
c; = a7 b'k.
* This convention is used only in this section.
THE CASE WITH BOUNDARY CONDITIONS
315
Let P = (pig where
if i=d and j=d
I
P5 = I
otherwise
0
and Q=I -P. In the following we fix a Borel probability measure p on O(M) and restrict our attention to the probability space (W(O(M)),,.O;P,.). Then r(t, w) = (X(t, w), e(t, w) = (e5(t, w))) (also denoted simply by r(t) _ (X(t), e(t) = (ej(t)))) is the solution of (6.12) with B(t) and 0(t) defined as above. Following H.Airault [2], we consider the following stochastic differential
equation for an Rd®Rd-valued process K(t)=(Kj(t,w)) as described below: (6.20) (6.20)a
(i)
for any t> 0
such that X(t) E M,
dK'(t): = dK(t)P = K(t) {e(t)-'de(t) + 2 R(X(t))dt}P
where R(x) = (R5(x)) is the tensor defines by R1 = R>.k;, (cf. Section 5)
(ii) for any t > 0, dK2(t): = dK(t)Q (6.20)b
= K(t) {e(t)-'de(t) + 2 R(X(t))dt}I,,,(X(t))Q; (iii) with probability one, t'---K'(t)=K(t)P is right-continuous with left-hand limits: furthermore
K'(t) = 0
if X(t) EaM,
and the initial valuedis given by (6.21)
K'(0) =
K2(0) = e(0)Q.
Remark 6.1. (a) A precise formulation of (6.20). is as follows: if a continuous process Y(t) is defined by a semimartingale integral Y(t)
= f 0 K(s) {e(sylde(s) + 2 R(X(s))dsjP,
DIFFUSION PROCESSES ON MAINFOLDS
316
then, with probability one,
K'(t) - K'(s) = Y(t) - Y(s) for all s < t such that X(u) (-=A? for every u c- [s, t].
(b) (6.20), (ii) automatically implies that t i--- Kz(t) is continuous with probability one. The above stochastic differential equation may also be expressed in the equivalent form of a stochastic integral equation as the next lemma shows.
Lemma 6.2. An Rd®x R°-valued process K(t) adapted to (5) is a solution of the above stochastic differential equation (6.20) with the initial condition (6.21) if and only if
K'(t): = K(t)P
= I.,,(e(0) + f ' K(u)[e(u)-'de(u) + (6.22)
-f- 1($
7)
ft
K(u)[e(u)-' de(u) +
2 R(X(u))du])P R(X(u))du]P
T (1)
I
2
KZ(t): = K(t)Q = e(0)Q + f' K(u)[e(u)-'de(u)
+I where (6.23)
Q=
inf {s; X(s) E aM} (cc
if
{
} _ 0,
is the first hitting time of X(t) to aM and (6.24)
z(t)=
sup {s; s < t, X(s) E aM} if
j 0
I=
is the last exit time before t from 3M. Remark 6.2. f ITa> is understood, of course, as Y(t) - Y(t(t)) where Y(t) is the continuous process given by Y(t) = f o . The proof is easy and omitted.
THE CASE WITH BOUNDARY CONDITIONS
317
Let 8 be the
totality of all R°Qx Rd-valued processes g(t) = fi(t) defined on (W(O(M)),3",P) adapted to (.3) such that t
is right continuous with left-hand limits a.s. and satisfies (6.25)
'
for all T > 0.
oo
7
Define a mapping fi: E -- S by
0'((t): = O((t)P (6.26)a
= I1o>A(e(0) +
+ Ib«i f ,p2(
(6.26)b
f
(u)[e(u)-' de(u) + 2 R(X(u))du])P o
(u)[e(u)-' de(u) + 2 R(X(u))du]P
'
(t): = 45(0(t)Q
= eQ +
f
(u)[e(u)-' de(u) + 2 R(X(u))du]I,;s(X(u))Q. o
Let A(t) be the right-continuous inverse oft F- -- ¢(t) and set (6.27)
D = Is > 0; A(s-) < A(s)}.
If t > 0 is fixed, then z(t) = A(O(t)-) a.s. By Theorem 6.6 given below,
we see that if g(t) is an (.)-well measurable process such that t E [g(t)2] is locally bounded then E,,[ { f
(6.28)
' g(s)dB' (s)} 2] Tcn
< E,j
AWN
a;
g(s)dB'(s)} 2] = E,,[ f 0 { f d(a' )At
g(s)2ds].
It is easy to show from this that for every T > 0 there is a constant K = K(T) > 0 such that (6.29)
K(1 + f o E,,[ll
(u)II2]du)
for all t c- [0, T].
This proves that 0( EH if E Again using (6.28), we have that, for ',1J E3,
DIFFUSION PROCESSES ON MANIFOLDS
318
(6.30)
E'[II,p( (t) - 0(ii)(t)II2] <_ Kf'
E [0, T]. Theorem 6.2. The stochastic differential equation (6.20) with the initial condition (6.21) has one and only one solution K(t)ES. Proof. Let
E3, n = 0, 1, ... be defined by o m 0 and
=
n = 1, 2, .... Using (6.30) we can show that there exists E.?, such that
E'j u IISn(t) - b(t)II2J
0.
Then clearly is a solution of (6.22). The uniqueness also follows from (6.30). The arguments are standard and similar to those given in Chapter III or Chapter IV and so we omit the details. Let K(t) = (KJ(t, w)) be the solution of (6.20) with the initial condition (6.21) and define M(t) = (MM (t, w)) by (6.31)
M(t,w) = K(t,w)e(t,w)-',
t > 0.
Theorem 6.3. M = {M(t,w)} is an Rd ®Rd-valued MOF of the horizontal Brownian motion on O(M) with reflecting boundary;* i.e., (i) M(t,w) is (Y )-adapted,
(ii) for every t,s > 0, M(t+s,w) = M(s,w)M(t,O,w) a.s., where the shift operator 0,: W(O(M)) - W(O(M)) is defined by (O,w)(t) = w(t+s). Proof (i) is obvious. To prove (ii), we fix s and set k(t) = K(t+s,w), X(t) = X(t+s,w) and e(t) = e(t+s,w). Then it is clear that K(t) satisfies the above stochastic differential equation (6.20) with respect to (X(t), e(t)). On the other hand, by applying the shift operator 0, to K(t), we see that f(t) = K(t, O,w) satisfies the same equation with respect to (X(t, O,w), e(t, Ow)) = (X(t), e(t)). If we set
K'(t) = K(s,w)e(s,w)-'K(t), then k'(0) = K(s,w)e(s,w)-'(I,;f(X(s,w))e(s,w)P + e(s,w)Q) * [139).
THE CASE WITH BOUNDARY CONDITIONS
319
= K(s,w)(I,;s(X(s,w))P + Q)
= K(s,w) by (6.20) (iii). Hence K(t) and K'(t) satisfy the same equation and the same initial condition. Consequently K(t) - K'(t) by the uniqueness of solutions. That is, K(t+s,w) = K(s,w)e(s,w)-1K(t,9sw).
Multiplying by
e(t+s,w)-1
= e(t, 0,w)-1 from the right yields (ii).
The following lemmas delineate some properties of MOF M= {M(t,w)}
.
Lemma 6.3. If X(0) E aM, then
Pe(0)-'M(t) = 0
for all
t > 0.
Proof. It is enough to prove that Pe(0)-'K(t) = 0 for all t Z 0. If X(0) E 3M, then
Pe(0)-'K(0) = Pe(0)-'(Is(X(0))e(0)P + e(0)Q) = PQ = 0. Since K(t)=Pe(0)-1K(t) satisfies (6.20), K(t)=0 by the uniqueness of solutions.
Lemma 6.4 M(t, w- a) = aM(t, w)a-',
t > 0, a E 0(d).
Proof. Since X(t, w a) = X(t,w) and e(t, w a) = ae(t,w),* we see at once that K(t, w a) = aK(t,w) by the uniqueness of solutions of (6.20). Thus M(t, aK(t, w)[ae(t,w)]-' = aK(t,w)e(t,w)-1a'' = aM(t,w)a-1, which completes the proof. Now we can solve the equation (6.11). First of all, however, we shall
adopt the following convention in addition to the multiplication rule (6.19) : for a d-dimensional b = (b) and a= (af) E Rd( Rd, ab = c is the d-dimensional vector c = (c1) defined by (6.32)
c, = a,b,.
* This was denoted by e(t, w)a in (2.31). Here we are adopting the multiplication rule (6.19) and hence it should be written as ae(t, w).
DIFFUSION PROCESSES ON MANIFOLDS
320
Under this convention, (6.11) is rewritten as aU= 2 {do(M)U+JU} (6.11)
UI,-o = Ff
=0
(Pe-' U + Qe-' a U-
xa)
if r = (x, e) E aO(M).
More generally, we consider the following initial value problem for the heat equation of Rd-valued functions U(t, r) = (U,(t, r)) on O(M): au
T= (6.33)
I
2
{do(er) U -I- JU}
Ul,co = F aU
Pe-' U + Qe-' I
ax
-
aU aem
{dik} (x)em + e1d(x)e ' U]
=0
if r = (x, e) E aO(M), where Td(x) E Rd(2 Rd is defined by Td(x) = ({d°,r} (x))
If U(t,r) is 0(d)-equivariant, this implies that U(t,r) = e(Ax), r = (x,e), where U(x) is a smooth Rd-valued function on M, and hence
aem
{dtk} (x) em = eTd(x)e-' U.
Thus the initial value problem (6.33) is reduced to the initial value problem (6.11) in the case of 0(d)-equivariant functions. We now construct a semigroup corresponding to the initial value problem (6.33) by using the MOF M. Let C0(O(M) --- Rd) be the set of all bounded continuous functions F(r) on O(M) taking values in Rd such that (6.34)
Pe-'F(r) = 0
if r = (x, e) EaO(M).
For F E C0(O(M) - Rd) and t > 0, set (6.35)
(HF)(r) = E,[M(t,w)F(r(t,w))].
THE CASE WITH BOUNDARY CONDITIONS
321
Theorem 6.4. (i) {H,} defines a one-parameter semigroup of operators
on C0(O(M) -- Rd). (ii) If F is 0(d)-equivariant, then so is HF for all t > 0. Proof. If r e 8O(M), then by Lemma 6.3 Pe-'(HF)(r) = 0. The continuity in r of the functions HF(r), t Z 0 follows from the continuity of
r , P, E.9'(W(O(M))),* which in turn is a consequence of the uniqueness of solutions of the stochastic differential equations. The semigroup property of {H,} is obvious since M is an MOF. Finally, (ii) follows from Lemmas 6.1 and 6.4. Theorem 6.5. Let F(t, r) _ (FF(t, r)) be a smooth function on [0, oo) X
O(M) taking values in Rd such that for each t > 0, r'--- F(t, r) is a function in C0(O(M) - Rd). Then, with probability one,
M(t)F(t, r(t)) - M(0)F(0, r(0))
f ' M(u)(L.F)(u, r(u))dB'(u) 0
+f
o
M(u)
([M (u, r(u)) +
{d o (M) F(u, r(u))
2
+ J(r(u))F(u, r(u))} J) du
(6.36)
+f
o
M(u)e(u) Qe(u)-' [x7, (u, r(u))
aem
(u, r(u))
x {d'k} (X(u))em(u) + e(u)1'd(X(u))e(u)-'F(u, r(u))1 dq(u).
Proof. As we remarked above, the diffusion (r(t)) is a normally reflecting diffusion on O(M) in the sense of Definition 6.2 given below. As we shall see, a characteristic feature of such a diffusion is that if f(t, r) is a smooth function on [0, oo) x O(M), and if g(t) is an (,9;)-adapted process such that s g(s) is right continuous with left-hand limits and s -E,(g(s)Z) is locally bounded, then the following identity holds: (6.37)
E* f A(s-)/fir g(s)df(s, r(s)) = f
s5¢(t) SED
' 0
g(s)df(s, r(s)),
*.Y(W(O(M))) is the totality of all probabilities on W(O(M)) with the topology of weak convergence.
DIFFUSION PROCESSES ON MANIFOLDS
322
where the integral is understood in the sense of stochastic integral by the semimartingale s F---f(s, r(s)), f'(")-' is defined similarly as in Remark 6.2 and the sum is understood as the limit in probability of the finite s 10. A(s)s-A(sL )>s
First we prove the following lemma. t such that X, E
Lemma 6.5. For any (6.38)
dM(t) = 2 M(t)J(r(t))dt,
i.e., if u ED, then for every s < t such that [s, t] c(A(u-), A(u)) we have (6.39)
M(t) - M(s) =
f J M(u)J(r(u))du. 2
Proof. First we note that J = eRe ' by the convention (6.19). Then by (6.20) and Ito's formula, if X(t)EM, dM(t) = K(t)e(t)-'{de(t)e(t)-' + e(t)d(e(t)-') +
2 K(t)e(t)-'J(r(t))dt K(t)e(t)-'d(e(t)e(t)-') + 2 K(t)e(t)-'J(r(t))dt
= 2 M(t)J(r(t))dt. Now we return to the proof of (6.36). By Lemma 6.5 and Ito's formula, {M(t A A(s))F(t /\ A(s), r(t A A(s))) ss1(Dr)
sE
- M(A(s-))F(A(s-), r(A(s-))} A(ti (6.40)
,5 (t) A(s-)
M(u)dF(u,r(u))
sc-D
+
1
2 E(*
f
A(s)tiAr
M(u)J(r(u))F(u,r(u))du.
A(s
seD
Using (6.37) and Theorem 6.1 (i), (6.40) is equal to
THE CASE WITH BOUNDARY CONDITIONS
f
0
323
M(u)dF(u,r(u)) + Z fo M(u)J(r(u))F(u,r(u))du
= f M(u)(LmF)(u,r(u))dBm(u) 0
(6.41)
+ f M(u)
(u,r(u)) + 2 {Ao (M) F(u, r(u)) L
+ J(r(u))F(u, r(u))} J du
+f
aF 0
M(u)
Caxd
(u, r(u))
8F
aem(u, r(u)) {d'k} (X(u))em(u)]do(u)
On the otherhand, for every s < t,
M(t)F(t, r(t)) - M(s)F(s, r(s))
= K'(t)Pe(t)-'F(t, r(t)) - K'(s)Pe(s)-'F(s, r(s)) + K2(t)Qe(t)-'F(t, r(t)) - K2(s)Qe(s)-'F(s, r(s)). Noting that Pe(t)-'F(t, r) = 0 if r EaO(M) and that r(A(u)) EaO(M) if u c- D, and r(A(u-)) E aO(M) if u e D and u > 0, we see that the first line of (6.40) is equal to
[K'(t)Pe(t)-'F(t, r(t)) - K'(0)Pe(0)-1F(0, r(0))] (6.42)
+ E* {K2(t A A(s))Qe(t A A(s))-' F(t A A(s), r(t A A(s)))
ss VED
- K2(A(s-))Qe(A(s-))-'F(A(s-), r(A(s-)))}. By (6.20) and the fact that
f IaM(X(u))du = 0,
dK2(u) = K(u) {e(u)-'de(u) +
R(X(u))du} Q
- K(u)e(u)-'de(u)IaM(X(u))Q. Hence it is clear that d {K2(u)Qe(u)-'F(u, r(u))} has the form g,(u)de(u) + g2(u)du + g3(u)d(e(u)-1) + g4(u)dF(u, r(u))
-IaM(X(u))K(u)e(u)-lde(u)Qe(u)-'F(u, r(u))
g,(u), i = 1, 2, 3, 4, are all right continuous (.9)-adapted where u processes with left-hand limits. Using again the general formula (6.37),
DIFFUSION PROCESSES ON MANIFOLDS
324
the second term of (6.42) is equal to A(t) N
AS-)
3s (t) SED
d {K2(u)Qe(u)-'F(u, r(u))}
= f og1(u)de(u) + f o ga(u)du + fo g3(u)d(e(u)-1)
+f
u
g4(u)dF(u, r(u))
= KZ(t)Qe(t)-'F(t, r(t)) - KZ(0)Qe(0)-'F(0, r(0))
+f
la,,,(X(u))K(u)e(u)-'de(u)Qe(u)-'F(u, r(u)). o
Because of (6.12) and since IaM(X(u))Pe(u)-'F(u, r(u)) = 0, f IaM(X(u))K(u)e(u)-'de(u)Qe(u)-'F(u, r(u)) _$t
0
_ -f
K(u)e(u)-'e(u)Ta(X(u))e(u)-'F(u, r(u))do(u) K(u)Ta(X(u))e(u)-'F(u, r(u))dd(u).
u
Thus we have
M(t)F(t, r(t)) - M(0)F(0, r(0))
- f M(u)e(u)T4(X(u))e(u)-'F(u, r(u))d¢(u) 0
= f '0M(u)(E,,F)(u, r(u))dBm(u)
+f (6.43)
o
[
(u, r(u)) + 2 {do(M)F(u, r(u)) M(u) Ti
+ J(r(u))F(u, r(u))} I du
+f
o
M(u) [8.xa (u, r(u))
ae (u, r(u)) (A) (X(u))em(u)] do(u).
Finally we remark that if g(u) is (,91)-well measurable process, then
f
0
M(u)g(u)d0(u)
THE CASE WITH BOUNDARY CONDITIONS
325
f'
= (6.44)
K'(u)e(u)-'g(u)dc(u) + f` K2(u)e(u)-'g(u)dg5(u) 0
0
= fr K2(u)e(u)-'g(u)d$(u) 0
= f' 0M(u)e(u)Qe(u)-'g(u)d0(u) since IaM(X(u))K'(u) = 0. Now (6.36) follows from (6.43).
q.e.d.
Theorem 6.5 may be regarded as a martingale version of the statement that u(t, r) = HF(r) solves (6.33). In the remainder of this section, we shall elaborate on the above mentioned notion of normally reflecting diffusions and especially on the formula (6.37). Let D be the upper half space of Rd and Q(x) = (ok(x)),
b(x) = (b`(x)), r(x)=(r,(x)), i(x)=(lt(x)) be given as in Chapter IV, Section 7. Consider the non-sticky stochastic differential equation (7.8) in Chapter IV, Section 7 corresponding to [a,b,r,f,1,0]. Let a'J(x) and r`j(x) be defined by (7.6) and (7.7) in Chapter IV, Section 7. Let 3r = (X(t), B(t), M(t), $(t)) be a solution. We know that X(t) is a diffusion process on D determined by the differential operator d
(6.45)
Af(x) = 2
c. aj(x)aaj (x) + tE b`(x) ax< (x)
with the boundary condition
Lf(x) = 2
(6.46)
d-1 15
1
a
d-1
a'i(x) a aj (x) + E 6'(x) of (x) + of (x) = 0
on aD. Set
Z= Is > 0; X(s) E aD} .
(6.47)
Since SOIaD(X(s))ds = 0 a.s., X has Lebesgue measure 0 a.s. and (0, co) U ea, where ea=(la,ra) are mutually disjoint open intervals. Each a
ea is called an excursion interval and the part (X(t), t Eea) is called an excursion of X(t). Let A(t) be the right-continuous inverse function of
t i- q(t). Set D = Is a [0, oo); A(s)-A(s-) > 01.* Then it is easy to see that the totality of excursion intervals coincides with the set of intervals {(A(u-),A(u)), u E D). Let .° be the completion of v[X(u), * A(0-) = 0.
DIFFUSION PROCESSES ON MANIFOLDS
326
B(u), u < t] and let g(s) be an .7°-well measurable process such that s E[g(s)2] is bounded on each finite interval. Then Y11(t) = !o g(s)dBc(s) is a continuous Y;°-martingale. For each excursion interval ea _ (Ia, ra),
f
sa n CO.e
g(S)dBk(s) = Y"(r a A t) - Yk(1. A 1)
by definition. It is also denoted by
A(u-)At
if e,, = (A(u-), A(u)).
g(s)dBk(s)
Theorem 6.6. (i).
E(4E
(6.48)
[
J
g(s)dBI(s)]2) = E[ J 0 g(s)2d ],
A(u)
k=1,2,...,r, t>_0. (ii). Assume further that s s--- g(s) is right-continuous with left-hand limits and that a'J(x) is C' on D. Then At
57,*
g(s)dB"(s)
J A(-)
(6.49)
t
+f
0
= f g(S)dBk(s) o
x(s)) g(s)aas(X(s)) d¢(s),
k = 1, 2, ... , r,
t>0.
Here ('A(u)Ar
u6D
A(u-)/fit
is defined as the limit in probability of finite sum
E D
A(u)U6 .4("-)>a
A(u)At
f A(u-)At
if and only if the limit exists.
Proof. We shall first prove the special case of the reflecting Brownian motion and reduce the general case to this special case.
THE CASE WITH BOUNDARY CONDITIONS
327
(a) The case of the reflecting Brownian motion: i.e., the case 6((x) _
bk with r = d, b(x) - 0, r(x) __ 0, fi(x) - 0 and S(x) - 1. In this case, the system 3;(t) = (X(t), B(t), ¢(t)) is determined by the equation (6.50)
i= 1,2, ...,d- I
Xt(t)=X(0)+BI(t), I. yd(t) = Xd(0) + Bd(t) + c(t)
Let the path spaces W(D),
2Y(D), the corresponding o-fields
.g'(W(D)), .5F(V(D)), and the a-finite measure n on -9(W-o(D))) be defined as in Chapter IV, Section 7. If we set D, _ {u E (0, oo); A(u) - A(u-) > 0} = D\ {0} and
X(t + A(u-)) - X(A(u-)), 0 < t < A(u)-A(u-): = o[p(u)] t Z A(u)-A(u-)
p(u) - X(A(u)) - X(A(u)),
for
u ED
then we know that p : D, E) u i--- p(u) E 7i (D) is an (,l)-stationary Poisson point process on (01,(D), -9(711,'(D))) with characteristic measure
n, where ,5 _ JA (f). Let nc, ED, be the image measure on V(D) of n under the mapping w --- + w.* We shall now introduce the following notations (6.51)
p,: W(D) --- W(D) defined by (p,w)(s) = w(tAs) (stopped path)
(6.52)
6,: W(D) - W(D) defined by (O,w)(s) = w(t+s); (shifted path)
(6.53)
paD : W(D) - W(D) defined by (paDw)(s) = w(s A a(w))
where (6.54)
u(w) = inf {t > 0; w(t) E 0}
(stopped path on reaching the boundary).
Let, by X, be denoted the path t - X(t). Clearly this is a W(D)-valued random variable.
* ((+w)(t) _ + w(t).
DIFFUSION PROCESSES ON MANIFOLDS
328
Excursion formula I. Let Z(s) be an (.9 )-predictable non-negative process and f(s,w,w') be a non-negative Borel function on (0, oo) X W(D) x V(D). Then E{ s5r.seD, E Z(s)f(A(s-), p,(,-)X, PaD[ea(r-)X])} (6 55)
=E{
f Z(s)[ f? r
0
f(A(s), p,(,) X, w)nxca(1)>(dw)]ds} . (D)
This follows immediately from the fact that the sum under the expectation on the left is just r+ 0
9ro(D)
Z(s)f(A(s-), Pa(s-)X, X(A(s-)) + w)N,(dsdw)
(cf. Chapter II, Section 3). By a random time change t
¢(t) in (6.55) we have
Excursion formula H. Let Z(s) be an (..)-well measurable non-negative process andf(s, w, w') be as above. Then
El E (6.56)
= E{
(rZ(A(s-))f(A(s-),Pa(r-)XPaD[ea(,-)X])}
f Z(s)[ f r(D) f (s, P,X,w)nXcv(dw)]d0(s)} . 0
Let a(t) be defined by (6.24). By setting f(s, W, W') = At - S, s, w, W')I(a(W,)>r-dl
in (6.56),*1 we have
Last exit formula. Let Z(s) be an (9)-well measurable non-negative process and g(s,s',w,w') be a non-negative Borel function on (0, oo) x (0, oo) X W(D) X W,'(D). Then E{Z(r(t))g(t - r(t), a(t), PT(r)X, PaD[OT(r)X])IIT(f)>o,} (6.57)
= E { f r0 Z(s)[ f 2 (D)
For each i = 1, 2, continuous
... , r
g(t - s, s, PsX, W)I acw)>r-.,, nx(s) (dw)]d¢(s)}.
and to > 0, w'(t+to) - wr(to) is a with respect to n{( I o(w) > to).*2
*1 This idea is due to Maisonneuve [103]. *2 t eaD is fixed. a(w) is defined by (6.54).
THE CASE WITH BOUNDARY CONDITIONS
'(D))-well measurable process cP(s, w) such that
Hence, for any .%t( a(w)nt
f r(n) [ f
329
10(s,
w) 12ds]n4(dw) < oo
for each t > 0,
0
we can define the stochastic integral eno(w)
i
45(s,w)dw'(s). tpna(w)
It is easy to see that
lim f to 10
tna(w) tono(w)
0(s,w)dw'(s) = f
tno(w)
45(s,w)dw'(s)
o
exists in 2'2(?1'(D), ne), tna(w)
(6.58)
f r(n) [(fo
= f 9 (D) [ f
ds(s,w)dw'(s))2]W(dw) tna(w)
P(s,w)2ds]n1(dw) 0
and for every , 0(2Y(D))-measurable H(w) in 2'2(V(D), n{),
(6.59)
f r(D) (f t 0(s, w)dw'(s)]H(w)n{(dw) 0
= f ,. (n) [ f
0(s,w)dw'(s)]H(w)n1(dw),
t > to.
0
First we shall prove (6.48). Without loss of generality we may assume
that X(u) is given in canonical form, i.e., X(u,w) = w(u), w E W(D). Now let g(s) = g(s,w) be (.° }t(W(D)))-well measurable process such that
s - E(g(s)2) is bounded on each finite interval. For given s > 0, w E W(D) and w' E=(6.60)
'(D), set
s(u, w, w') =
g(s + u, [w, w't) 10 1
w'(0),
otherwise
where (w, w'], is defined by (6.61)
s, [w, w']t(u)
w
(u'- s ,
u0
> s. <
DIFFUSION PROCESSES ON MANIFOLDS
330
Let t > 0 be given and fixed. Let (' (r-t)Ao(w') J (6.62)
fi(s,w,w') =
[
t>s>0
P8(u,w,w')dw"(u)}2,
o
t<s
0,
in the sense explained above. By (6.58),
f
(D)
[ fsAt Au, [w, w'],)2du]n' (,) (dw)
f; (s, w, w')n' (') (dw') =
if w(s) = w'(0),
9 (D)
otherwise.
It is clear from the definition that f, (A(s-), PA(,-)`Y, PaD[eA(,-)x])
(6.63)
= (f AG,)
g(u)dx`(u))2 =
(f A(a t
g(u)dB1(u))2.
By the excursion formula and (6.62), A(,)nr
E (Z p ( f
g(u)dB°(u))2}
A(,-)nt
= E {(f o
g(u)dB((u))2} + E{
E
Ed(t(f A() t g(u)dB`(u))2}
t
vnt
= E{ f g(u)2du} + E{ f do(s) f o
(o(w')+:)nt
g(u,[w,w'],)2du]n"(')(dw')}
cD) [$ sAt A(e) 't
= E { f opt g(u)2du} + E {ED 57, F(t)
f
g(u)2du}
t
= Elf 0 g(u)2du]. Now we shall prove (ii). In the case (a), (6.49) is given as (6.64) s
and
* E f A((s-
r
g(u)dB'(u)
= fo
g(u)dB'(u),
i = 1, 2, ... , d - 1
THE CASE WITH BOUNDARY CONDITIONS
331
First we prove (6.65). In the following, we shall call g(s) a step process
if there exists a sequence of (( W(D)))-stopping times ao = 0 < a,
< a2 < ... < an < ... - oc and
{.
,(W(D))} -measurable random
variable g, such that g(s) = g, if s e [a,, a,+,) for i = 0, 1,
... .
Lemma 6.6. Let g(s) be a step process. Then (6.65) holds.
Proof. If, for example, g(s) = 1, then (6.65) is trivially true: we have A(s)Ar
f A0,dBd(u) =
(s)At fA1A
)At
r
dX d(t) =
= Xd(t), Xd(a A t) - Xd(O),
Xd(A(s) A t) - X d(A(s ) AA t)
s e D., A(s) < t or A(s-) > t s EDD, A(s-) < t < A(s) s=0
and hence the left-hand side of (6.65) is equal to Xd(t) - Xd(O) = Bd(t) + 0(t). A similar argument applies if g(s) is a step process.
Lemma 6.7. Let g(s) be a .hl,(W(D))-adapted process such that s g(s) is right-continuous with left-hand limits. Then for every e > 0, there exists a step process ge(s) such that (6.66)
I ge(s) - g(s) I < e
for every s.
Proof. Let {a,} be defined by ao = 0 and
an = inf It > an-, ; I g(t) - g(an-1) I > e} An. has the desired properties.
Then ant co, and ge(s) = Lemma 6.8. For c E aD and t > 0, let (6.67)
u{,r(B) = nl(B I a > t) =
W(B: a> t)
W(a>t)
for
B E.8(W (D)).
Then jO-' is a Markovian measure on 7P(D) concentrated on {wE
w(0) = , a(w) > t} such that
'(D);
332
DIFFUSION PROCESSES ON MANIFOLDS
#1.t {w;w(t,)EE,, W(t2)EE2i ... , W(tn) EEn} (6.68)
=
n-1
dxl r
dx2 ... f, dxnK{(ti, x1) n P(tt, x,; J E2 J n
E,
xr+i)
for 0
- NO - s, x) KI(s, x) = K+(s, x K(t)
S>0' x ED
h(t - u, y) p(s,x;u,y)= h(t - s, x)p°(u-s,x,y), 0<s
h(s, x) = f Dp°(s, x, y)dy =
J0
exp {- 2s} l
and
K(t) = f K+(t, x)dx = 2t , where K+(t,x) and p°(t, x, y) are given as in Chapter IV, Section 7. This lemma is easily proved from the properties of the measure n. Corollary. The process
(0:!9 s < t)
ba(s) = wd(s) - f o A(t, u, w(u))du
is a one-dimensional Brownian motion with respect to the probability measure ,A` on (D), where
(6.69)
A(t, u, x) =
lh(t - u, x)) ax
-(Xd)2
exp
=
h(t - u, x)
Xa
f X0 exp {
2(tu
}
n2
2(t - u)}
Lemma 6.9. Let g(s) be a bounded (A(W(D))} -well measurable process. Then for fixed s, w e W(D) and for any s >- e' > 0,
f I
(6.70)
f2r(D)
0
0g(u, w, w')dw'a(u) I
<_ (J+ 1) Ilsll
THE CASE WITH BOUNDARY CONDITIONS
333
where IIghI W =sup I g(u,w) I and 0,1 is defined by (6.60). u, w
Proof. By the corollary to Lemma 6.8,
f
J T (D)
nw (s) (dw')
f0" 08(u, w, w')dw'd(u) I 1
s J 9 (D) (I f
`PP(u, w, w')dbd(u) I },uw(s).t(dw')nw(s)(o.(
o
) > E)
8
-}- f
{J 9 (D)
I sbg(u,w,w')A(s,u,w'(u)) I du} pw(,).a(dw')nw(s)(a(w) > E)
0
< 11g11. e'uz VI
<
, + Ilgllm
1) I le...
Here we used the following facts:
1
W(v(w) > s) = f D K+(e, x)dx = and
f
3r (D)
[ f 0 A(e, u,
e)
[$8
A(e, u, w(u))du]p 8(dw)n1(Q(w) > e)
< f ST(D)
0
wd(s)n1(dw) (D)
= f xdK+(e, x - jdx = 1. D
Now let g(s) be a {°.,i,(W(D))}-well measurable process such that s'. -. E(g(s)2) is bounded on each bounded interval. We introduce the following notation: (6.71)
Sa(t) =
(6.72)
YB(t) =
A(Ai s)
AU-) g(s)dBd(s),
E s$6(t),A(s)-A(s-)>a f J
do(s)[ f r(D) { f
O(w')A(t-s) o
w')dw'4(u)1(d(w-)>.)}
`k'(u, w, x nw(s)(dw)],
DIFFUSION PROCESSES ON MANIFOLDS
334
M8(t) = Se(t) - Y8(t).
(6.73)
d(S) Lemma 6.10. Me(t) -- f 'g(s)dB 0
in 2'2(P) as e j 0.
Proof. Set ('u-s)no(,1)
(6.74)
P2,(s, w, w') =
0s(u,
0
for 0< s S t,
w, w(s)+w')dw'd(u)
w E W(D) and W E
(D).
Then clearly 011Vg(u)dBd(u)I,o>a
Ma(t) = f 0
+f
(t) o
fro(o)'f z(A(s-), PA(.,-)X,
NP(dsdw')
in the sense of stochastic integrals (Chapter II, Section 3). Consequently
for each fixed t, Ma(t) -- M(t) in 2'2(P) as c -- 0, where o(t)
trAr
M(t) = f
g(u)dBd(u) + f
$ ,.o(D) fi(A(s ),PA(t-)X, w')Np(dsdw')
and
E(M(t)2) = E[
o
= E[f o
g(u)2du] + E[ f dO(s) f
7'0(D)
{fi(s,P3X,w')} 2n(dw')]
g(u)2du1.
Next we assume that g(s) is bounded and prove that M(t) is an {..t(W(D))}-martingale. It is sufficient to show that for any bounded Borel measurable functions F,(w), F2(w) on W(D) and 0 < tl < t2, (6.75)
E(M(t2)H) = E(M(t,)H)
where
H(w) = We prove the following estimate
THE CASE WITH BOUNDARY CONDITIONS
E(M8(t2)H) = E(Me(t))H) + o(l)
(6.76)
335
(c 10)
from which (6.75) follows. First, it is clear that E(J o 2 S(u)dBd(u)I,o>e, H)
= E($0 " S(u)dB°(u)I (>e, H)+ o(1).
By the martingale property of the stochastic integral with respect to N we have
E[J
oo
f 2 (A(s-), Pacs-) X, w')I
J
0(D)
= E[f (r') 0
N,(dsdw')H)
'2(A(-) s PA(S-) X w')I {o(we)>E) 1VD (dsdw')H].
fyr0 (D) J 2
Now (A(s-), Pa(,_,X, w')I,o(w,)>a,NP(dsdw') J r cD)f 0
Jo
E
,ED,s-<4(r0,Q(0A(,-)X)>e
-
1,
J
o
f2 (A(s-), PA(,-)X, PaD[6A(,-)X))
d(s)5yw0(D)f2(s,PsX, w')I(rr(w,)>e)n(dw')
.=Ill -Iz, where PaD[eA(s-)X] = paD[BAU-)X] - X(A(s-)). If s < t2 - t, and s S t], rp(D)
}'2(s, Ax, w')I(a(wr)>e)n(dw')
f (D) [ f
(r2-s)/W
s
9r (D)
[
':(u, p,X, w')dw'd(u)]I(u(w,)>a,nx(s)(dw')
0
0
-P8(u, P,X, W')dW'd(u)]I b(w,)>a, nx(s) (dw')
and hence
Ii = f
d0(s) f ,(D)f2(s, P,X, w')I(,(w,)>,,nx(s)(dw')
0
+f" f ('
J
do(s) { J rcD)
("s) 0
[ f o P;(u, P,X,
45s(u, P,X, w')dw'd(u)]I
w')dw'd(u) (Q(w,)>r)nx(s)(dw')}
.
336
DIFFUSION PROCESSES ON MANIFOLDS
If we denote the second term by 6(8), then by Lemma 6.9, E(8(e)) = 0(1). Next, aEn
cr)
f'2(A(sL
+ fZ
),
Ps(r,)X, PaD[BT(r,)X])
.=1611+11612
where r(t,) = A(O(t,)-). By the last exit formula, E(I i2H)
E[ f o F1(P,X)d6(s) { ,) 7-(D) [ J
oW)n
fc,)
X
= E[ J o Fi(P,X)dc(s) { J q cn) [ J o X
s(u, p,X, w')dw'd(u)]
l
" -,) Os(u,
p,k. w')dw'd(u)]
jfe(wr)>av(r,-,)iF2(Pr,-,wr)nX(,)(dwr)} ]
fy(D) [ "-' 0,,(U, p,X, wF)dw'd(u)]
= E[ f"0
J
0
X 11,,(w)>,v(t,-41 F2(Pr,-,w')nXG)(dw')} ]
f rc.[ J o 4i;(u, p,X, w )dw"(u)
+ E[ f=,-a
- f"-5 01,(u,
P,X, w')dw'd(u)]l ((wr)>,l F2(Pr,-,w')n11) (dw')} l
:= al(e) + a2(8).
Then clearly E(I811H) + a)(8) = E({
E
.f?(A(s-),PA(,-)X,PaD[e,1(,-)X])) M1)
By Lemma 6.9, a2(8) = o(1). Now the proof of (6.76) is complete. Let g(s) be a bounded step process. By Lemma 6.6,
S,(t) - f
r 0
g(s)dBd(s) + f g(s)do(s) a.s. as 0
a-0
and hence
Y,(t) = SAO - MAO --- f g(s)dBd(s) + f0g(s)d0(s) - M(t) 0
THE CASE WITH BOUNDARY CONDITIONS
337
in probability as a --- 0. By Lemma 6.9, we can easily conclude that this for some bounded adapted process limit is of the form $a h(s). Therefore, r
f r h(s)dd(s) a
= f 0 g(s)dBd(s) + f 0t g(s)db(s) - M(t).
We conclude from this that
f
o
h(s)d¢(s) = f g(s)ds(s) o
and
f g(s)dBd(s) = M(t). 0
Let g(s) be general. Then we take a sequence {gk(s)} of bounded step processes such that
E[ f
I gk(s) - g(s) I zds]
0
(k ---- oo).
It is easy to see that Mk(t) corresponding to gk(s) converges to M(t) and hence M(t) = f ' g(s)dBd(s). 0
Lemma 6.11. Let g(s) be a {Rj(W(D))}-adapted process such that s g(s) is right-continuous with left limits and s --- E[g(s)2] is locally bounded. Let Ye(t) be defined by (6.72). Then (6.77)
Y8(t) --- f g(u)do(u) in probability as e -- 0. o
Proof. First assume that g(s) is a step process. Then by Lemma 6.6
S8(t) -. f o g(s)dBd(u) + f o g(u)do(u) and by Lemma 6.10,
a.s.,
DIFFUSION PROCESSES ON MANIFOLDS
338
M8(t) -- f
r
in
g(u)dBd(u) 0
_.(P)
Thus (6.77) holds. Now let g(s) be general. By Lemma 6.7, we can choose a sequence {gk(s)} of step processes such that
gk(s) - k < g(s) < 9k(S) + I Since Y8(t) may also be expressed as
Y8(t) =
rn[ f
ft
dc(s) 0
0
og(u,w,w')A(e,u,w(u))du x I (wr)>Jnx() (dw'),
we have
Y8(t) -
p(t) < Ye(t) < YB(t) + LO(t)
k where YE corresponds to gk. The desired conclusion then follows by first
letting e -k 0 and then k - oo. Now we are ready to conclude the proof of (6.65). By Lemma 6.10 and Lemma 6.11 Se(t) =Me(t) + Ye(t) converges in probability to
f
0
g(s)dBd(s) + f
'g(s)do(s) 0
and this proves (6.65). The proof of (6.64) is immediate from the following reasoning. Suppose that a family of disjoint open non-random intervals {ea} in [0, oo) is given such that [0, oo)\ U ea has zero measure. Then it is obvious that a
E*f a
[o,rlneag(u)dB'(u) = f
0g(u)dB'(u)
Indeed, if E is the union of all ea such that I ea I > e,
THE CASE WITH BOUNDARY CONDITIONS
Ef lCal>8
eag(u)dB'(u) ro
t3 n
339
_ f' 1 (u)g(u)dB'(u)
since the e,, are non-random intervals; moreover, it is clear that f' IE(u)g(u)dB'(u)
-f
g(u)dB'(u)
in
-5-712(P)
as E --- 0.
o
Since {A(u)} is defined only through {Ba(t)}, it is independent of {B'(t),
i = 1, 2, ... , d-1} . By Fubini's theorem, the intervals (A(s-), A(s)) can be treated as non random intervals and hence (6.64) follows. (b) General case. The proof of (i) is similar to the proof in case (a). As for (ii), we first
remark that we may assume d= r. Indeed, if r < d,we set Q4(x)__0
for r
and then adjoin d-r independent Wiener processes B'+'(t), Br}2(t), Ba(t). If r > d, we consider the r-dimensional process (Y'(t),Y2(t), ... , Yr-d(t),X'(t), ... ,Xd(t)) by setting e.g., Y'(t)=B'(t), Y2(t)=B2(t), ... , Y1'd(t)=Br-d(t). First, we consider the case ok(s) = blk' and bd(x) _ 0. Then [B'(t), B2(t),
... ,Bd-'(t)'Xd(t) = Xd(0)+Bd(t)+¢(t)] is a refelcting Brownian
motion and the proof in case (i) applies. Secondly, we consider the case ok(x) = ak. Then by a change of drift (Chapter IV, Section 7), it is reduced to the first case. Thirdly, we consider the general case. It is reduced to the second case by the following change of coordinates and transformation of Brownian motion. Since the a'J(x) are C3 by assumption, we can find a
C2-function f(x) on D such that f(x) > 0, f(x) = 0 if and only if x(-= aD and
a'i(x)
of a __ 1 ax' ax
on
D.
For X=(X(t),B(t),M(t),¢(t)), set i=(.t(t),B(t),M(t) 0(t)) where t'(t) = X'(t), i = 1, 2, ... , d-1 and fd(t) =J(X(t)). X corresponds to [d4i,fl,1,0] with add(x) = 1. By a transformation of Brownian motion from B(t) to h(t) (Chapter IV, Section 7), we may assume that dk(x) _ q d,F.
V
The proof of Theorem 6.6 is now complete. Let 3E=(X(t),B(t),M(t),O(t)) be given as above and f(t,x) be a
DIFFUSION PROCESSES ON MANIFOLDS
340
smooth function on [0, oo) x D. Then f(t, X(t)) is a continuous (.97°)semimartingale.
Theorem 6.7. Let g(t) be an (. °)-adapted process such that t'--g(t) is right-continuous with left-hand limits and t - E[g(t)2] is locally bounded. Then we have f
A(s)A
E* J
A(s->nrg(u)df(u,X(u))
=f
0
g(u)df(u,X(u)) - E 1= t
jd
+
g(u) of (u, X(u))v (X(u))dM,(u)
add(X(u))) a
1-1
o
0
adl(X(u)) of
1
- f g(u) 57
(6.78)
f
(u,X(u))
I id-1
C'(X(u)) axiai (u,X(u))} do(u)
Proof. Let A and L be defined by (6.45) and (6.46). By Ito's formula, g(u)df(u,X(u)) = g(u) at (u,X(u))du
+, d-1
g(u) axt (u,X(u))6`(X(u))dBk(u) s
g(u) ax, (u,X(u))r (X(u))dM`(u)
+
+ g(u)(Axf)(u,X(u))du + g(u)(Lzf )(u,X(u))d0(u).
By Theorem 6.6, the left-hand side of (6.78) is equal to
f
o
g(u) at (u,X(u))du -f -' r
d
[
d k
i=1
af
-1
+ t=1 k=1 f 0g(u) axl (u,X(u))
f
° g(u) TX, (u,X(u))u'
o_ik(X(u))Qk(X(u))
(X(u))dB'(u)
do(u)
add(X(u))
+ f g(u)(Axf)(u,X(u))du 0
= f ° g(u)df(u,X(u)) - E 21 f° g(u) az r
(u, X(u))zi(X (u))dM'(u)
d ad1(X(u)) of
f° g(u)[L.f(u,X(u)) - E add(X(u)) axt Since
KAHLER DIFFUSIONS
341
Lll (u,x) - E,.1add x) axr of (u,x) a (x)
=
tf x)
add(X)I
ax' (u'x) + 2
"`'(x)ax
(u,x),
a
we have obtained the conclusion. Corollary. The identity A(s
A(s)1V-
(6.79)
sE* J
Ig(u)df(u,X(u)) =
J og(u)df(u,X(u))
holds for every smooth f(u, x) on [0, oo) x D if and only if (6.80)
a°"(x) = 0 and
8'(x) = a
identically on aD,
i,j=1,2,...,d-1. Definition 6.2. We say that X is a normally reflecting diffusion process if (6.80) is satisfied.
7. Kiihler diffusions Let be a mapping from an open set D of C^ into Cm. Then we can where q` is a complex-valued function write (z) _ (O'(z), defined on D. As in the Section 111-6 if each 0' is holomorphic on D, then 0 is called a holomorphic mapping from D into Cm. A Hausdorff topological space M is called a d-dimensional complex manifold if M has an open covering { Ua }QE such that for each Ua there is a homeomorphism Oa from Ua onto an open set Da of Cd satisfying the following
property: if Ua fl Uf # o the mapping 0, o 0a' from c6a(Ua fl Up) into OF(Ua fl U.) is a holomorphic mapping. { (Ua, Oa) }«Ee is called a system of holomorphic coordinate neighborhoods of M. Let U be an open set of M provided with a homeomorphism 0 from U onto an open set D of Cd. (U, 0) is called a holomorphic coordinate neighborhood of M if the follow-
ing property holds: If Ufl Ua *0, a E A, then the mapping 0a a 0-' from own Ua) to ¢a(Ufl Ua) and the mapping 0 o o;' from Oa(Ufl Ua) to O(U fl Ua) are both holomorphic. For a holomorphic coordinate neighborhood (U, 0), we set 0(z) = (Z'(z), zz(z),..., zd(z)), z E U
DIFFUSION PROCESSES ON MANIFOLDS
342
and we call (z', z2, . . . , zd) the system of complex local coordinates on (U, 0). Since we can identify Cd with RZd, a d-dimensional complex mani-
fold M can be considered as a 2d-dimensional real manifold. Hence a holomorphic coordinate neighborhood (U, 0) of M is a C" coordinate neighborhood. For a system of complex local coordinates (z', z2,..., zd) on (U, 0), we denote by xk and yk the real and imaginary parts of xd, yd) is a system of local zk respectively. Then (x', y', xZ, coordinates of M. Hence (ay')'...'(a
\ax,),'
L' \ayd/:1'
is a basis of T,(M) and { (dx'):,
(dy'):,..., (dxd):, (dyd). }
is a basis of T, (M). As in the Section 111-6, we set
(aak/ and also set (dzk): = (dxk)= +
(dyk)=
(dzk)r = (dxk), -(dyk),. Then we see immediately that
{():
(aZl
:,...I
():' (aad )_ }
is a basis of the complexification T°(M) of T,(M) and { (dz'):,
(dz'):,.... (dzd):, (dzd), }
is a basis of the complexification T=`(M) of Ti (M). Define a linear transformation J= of T,(M) by
J'(axk/:
- (ay,)' j'(-ayk):
-(axkl='
k
d.
KAHLER DIFFUSIONS
343
It is easy to see that J= can be defined independently of the choice of the
system of complex local coordinates (z', z --- J= is called the almost complex structure attached to M. A Riemannian metric g on a complex manifold M is called a Hermitian metric
on M if g satisfies (7.1)
for X, Y E T=(M)
&(J=X, J=Y) = g=(X, Y)
at each point z of M. We extend the value of g at z E M to a symmetric bilinear form on TT(M) by defining g=(X +
Y)
Y, X' +
= (g=(X, X') - g=(Y, Y')) +(g=(X, Y') + g=(I', X')) for X + N/=1 Y, X' + ../=1 Y' E Tj (M). We set gap(Z) = g=(( 11a )z' (ZS )J'
gaP(Z) =
gaa(z) = g=(( aaa )_,
gap(z) = g((aa
(-7))' lo
aaa
/ _' \ a / _)
a
'
(-))
, a, fl = 1,2,..., d,
and call gap, gap, gap, ga, g the components of g with respect to(z',zz,".,
zd). It is easy to see that gap = gpa, gas = gpa, gap = gpa,
gap = gap,
gap = gap.
In terms of components of g the condition (7.1) is expressed as (7.2)
gap = gap = 0,
a, ,B = 1, 2,..., d.
For a Hermitian metric g, we set (7.3)
co,(X, Y) = g=(J=X, Y)
for
X, Y E T=(M).
The mapping w: z -- co, defines a real differential 2-form on M. Co is called the differential form on M of degree 2 attached to the Hermitian metric g. co is expressed as
w = -1 E gagdza A dzfl. a.fim1
If Co is a closed form, i.e.
DIFFUSION PROCESSES ON MANIFOLDS
344
dco = 0,
(7.4)
then the Hermitian metric g is called a Kahler metric. A complex manifold endowed with a Kahler metric is called a Kahler manifold. We note
that in terms of components of g the condition (7.4) is expressed as ag
(7.5)
g,
a, .8, y = 1, 2,..., d.
fi
We now note that if (M, g) is a Kahler manifold, the Laplace-Bel-
trami operator d on M may be written as ga#(z)aaaz d =E1 a
(7.6)
a
l
g#a(z)azaav
where (g aP(z)) and (g 0a(z)) are given by (7.7)
Eg,9(z)g?fl(z) = af,
Eg11(z)gr# (z) = ae
a,fl=1,2,...,d, respectively. Indeed a straightforward calculation in local coordinates shows that if g is a Hermitian metric, then the principal part of A is equal to the right hand of (7.6). Combining (7.5) with V-(4.32) we can show that the coefficients of the lower order terms of A identically vanish. These imply (7.6).
Definition 7.1. The Brownian motion on a Kahler manifold M, i.e. the diffusion process on M generated by the differential operator given by (7.6) is called the Kahler diffusion on M. We note that the Riemannian connection V defined by IV-(4.19) is extended by complex linearity to act on complex vector fields, i.e., for complex vector fields Yk
k=1,2
we set Vzl Z2 = Vx1 Xy - V rl Y2 +
(17x, Y2 + V rl X2)-
Define the components of the connection j7 with respect to aza and a azp by
KAHLER DIFFUSIONS V
a a
azw
aza
a
Y(r azY +
V
aaY
az"
P
r
a2
a
345
a) aiY /
+ r9g azY /
r; azy + lie aZr
aZa
Va
aae
= FY-\r8# a Y + T"e
azY /
aZa
If (M, g) is a Kahler manifold, we obtain
(7.8)
Tay=r.J11O rPy = T;fi,
rfly = rPy.
Furthermore, for a Kahler manifold the coefficients of p are determined by
57,g,3 r e = az
,
zg,3 rah = as
.
6
In the rest of this section, unless otherwise stated we assume that (M, g) is a Kahler manifold. We introduce the bundle of unitary frames over M: By a unitary frame e = [e,, ed] at z we mean a system of complex tangent vectors ea E T.. (M) such that gs(ea, ed= Sap
a, P = 1, 2,..., d
and each ea is of holomorphic type, i.e., it has no component of azp ,
fi = 1, d. U(M) is defined as the collection of all unitary frames at all points z E M:
U(M) = [ r = (z, e); z E M and e is a unitary frame at z }. Then U(M) is a principal fibre bundle with the structure group U(d). This bundle is called the bundle of unitary frames over M. For details,
DIFFUSION PROCESSES ON MANIFOLDS
346
see [207]. Since M is Kahlerian, U(M) is invariant under the parallel displacement with respect to the connection p. As in the Section 4, the Kahler diffusion on M is constructed by the
solution of the stochastic differential equation corresponding to the canonical horizontal vector fields. In this case, it is described more conveniently by using the complex structure. Let (Wo d, P R') be the standard 2d-dimensional Wiener space and define a system of complexvalued martingales C(t) _ (ra(t))a_1 by
la(t)
a = 1, 2,..., d.
2 (w2a-1(t) +
As stated in the Example 111-6. 1, C(t) is called a d-dimensional complex Brownian motion and it is a typical example of d-dimensional conformal martingale. We consider the following stochastic differential equation (7.9)
dr(t) = La(r(t)) o dC'(t)
Ld } is the system of the canonical horizontal vector field on U(M), i.e. La(r) E T; (U(M)) is defined to be the horizontal lift of ea E T. (M), a = 1, d where r = (z, e = [e1, ed]) E U(M). The meaning of (7.9) is as follows: we say that r(t) is a solution of (7.9) if r(t) is a continuous process on U(M) such that, for every F E F(U(M)), F(r(t)) is a semimartingale and satisfies where {L,,
F(r(t)) - F(r(0)) =
J0
L,,F(r(s)) o dC11(s) + $ (L,,F) (r(s)) ° dCa(s). 0
In a complex local coordinate, it is given as follows :
(7.9)'
dZ1(t) = ea(t) o dla(t)
i = 1, 2,.-., d
de,(t) = -
o dZl(t ),
i, a = 1, 2,---, d.
Since dsa - dCfl = 0 and dZa - dZZ = 0, (7.9)' is equivalent to dZ°(t) = eta(t)dC'(t)
i = 1, 2,..., d
(7.9)"
den(t) = - Tyfl(Z(t))ea(t)dZ''(t),
i, a = 1, 2,---, d.
If r(0) E U(M), then the solution r(t) = (Z°(t), ea(t)) lies on U(M) and
KAHLER DIFFUSIONS
347
Zd(t)) is a d-dimensional local conformal the Z(t) = (Z'(t), margingale. Combining these results with III-(6.5), we obtain the following: The solution r(t) to (7.9) with r(0) E U(M) lies on U(M) and its projection Z(t) = x(r(t)) defines the Kahler diffusion on M where n: U(M) -- M is the natural projection given by 7c(r) = z for r = (z, e) E U(M).
Example 7.1. Let D = { z e C; I z I < 1 } be the unit disc in the complex plane C endowed with the Riemannian metric (7.10)
ds2 = IdzI2/(l - IzI2)2
z E D.
Then D is a Kahler manifold and the metric g given by (7.10) is called the Poincare metric. Since (D, g) is a realization of the Lobachevskii plane, the Kahler diffusion on (D, g) is called the Lobachevskii Brownian motion in the unit disc. This diffusion is obtained from the one-dimensional complex Brownian motion by a transformation of time change
determined by the function c(z) = (1 - I z
12)-2. Let C(t) be the one-
dimensional complex Brownian motion defined above. For z EE D, we set
NO = z + C(t) and let
o = inf{t; I NO I =1 }. We also set A, = f' c(Cz(s))ds,
t < Q.
0
Then it holds that lim,,,A, = oo a.s. and hence its inverse C, = inf { u; A. > t } can be defined for t E [0, oo). To see this, we note that r(t) I C'(t) I is BESIz1(2): the Bessel diffusion process with index 2 (Example IV-8.3) and r(Q is a one-dimensional diffusion on [0, 1) starting at I z I with generator (1 - r2)2 2
d2
I d
( dr2 + r T,-).
By Theorem VI-3.2 we can see that if I z I * 0, r(Cr) can not hit 0 nor 1 in a finite time almost surely. Noting that lim+1,A,: = e < oo implies
DIFFUSION PROCESSES ON MANIFOLDS
348
limt, ,r(C,) = 1, we can conclude that lim+, aA, = oo a.s. Then Zf(t) = l z(C,) is a local conformal martingale by Proposition 111-6.1 and it possesses the properties that Zz(t) E D, t > 0 a.s. and
limt1_Zf(t): = Z. exists and Z. E 8D = {z; I z I = 1, z E C} a.s. It is clear that { Zz(t), t > 0 } defines the Lobachevskii Brownian motion
starting at z in the unit disc. Remark 7.1. Let p(t, z,, z2), t > 0, z1, z2 E D be the transition den-
sity function with respect to the Riemannian volume dxdyl
m(dz)
z=x+
1 - z 22 )
yED
on D of the Lobachevskii Brownian motion. Then it holds that
t > 0, z1, z2 E D
p(t, z1, z2) = p(t, p(z1, z2)),
where p(z1, z2 ), z1, z2 E D is the Poincar6 distance between z, and z2, i.e.
log 1 - r ,
p(zi, z2) =
r=
z1z2
1 - 21z2
2
and
p(t, p) =
_t
e$ (2itt)3/2 J p
r2
re 2t dr
chr-chp
Finally we give one of simplest examples in which conformal mar-
tingales can be applied to problems in complex analysis.
Example 7.2. Let D be a polydisc in C2 given by
D={z=(z',z2)EC2; Iztl <1,i=1,2} and let zo = (zo, z(2,) E D. Let Z(t) = (Z'(t), Z2(t)) where Z'(t) and Z2(t) are mutually independent Lobachevskii Brownian motions in the
unit disc in C as given in the Example 7.1 such that Z'(0) = zo, i = 1, 2. Then Z(t) is a two-dimensional conformal martingale such that Z(0) = za and with probability one, Z(t) E D for all t z 0, Z. exists and 1-1
ZmE6D={z=(z',z2); Iz'I =1, i=1,2}.
MALLIAVIN'S STOCHASTIC CALCULUS
349
Note that the usual topological boundary aD of D is given by
aD = (z = (z', zz); I z' I < 1 and I z2 I = 1 or 1 z' I =1 and Izz1 :511 and aD is much smaller than 6D. aD is called the distingushed boundary
of D. Now if 0: D -- C is holomorphic, q(Z(t)) is a local conformal martingale. As a simple application of this fact, we show the following: Let c: D = D U aD - C be continuous and c, restricted on D, be holomorphic. Then maxD 10(z) I is always attained on aD: (7.11)
max I c(z) I = max I (z) I
Indeed ¢(Z(t)) is a bounded martingale and limt1, O(Z(t)) = 0(Z.). Hence c(zo) = E[6(Z(0))] = E[O(Z,,)]
and therefore, I O(zo) I < E[ 10(Z.) I ]
We know that Z,,, E aD a.s. and hence I O(zo) I -< max I c(z)1 zeaD
Since zo can be choosen as any point in D, this clearly implies the assertion (7.11) For these topics, we refer the reader to Debiard-Gaveau [195] and Kaneko-Taniguchi [205]. For further applications of conformal diffusions to complex analysis, see Gaveau [200], Malliavin [108], Durrett [196], Fukushima-Okada [198] etc.
8. Malliavin's stochastic calculus of variation for Wiener functionals As we have seen above, strong solutions of stochastic differential equations are functions of Brownian motions. Such functions are often called Wiener functionals or Brownian functionals and they have been studied by many people with a variety of motivations (cf. e.g. [10], [64],
350
DIFFUSION PROCESSES ON MANIFOLDS
[79], and [178]). Recently P. Malliavin ([106], [107]) gave a new approach to the analysis of Wiener functionals, especially to the analysis of strong solutions of stochastic differential equations. We follow the assmptions
and notations of Section 2 and let X = (X(t, x, w)) be the solution of (2.1) realized on the r-dimensional Wiener space (W0', P R'). Recall that
Wa is the totality of continuous functions w: [0, oo) _ R' such that w(0) = 0 with the topology of uniformly convergence on every bounded interval, i.e. the topology induced by a countable system of norms Ilwlln = maxIw(t)1, OStSn
n = 1,
w E Wo.
If t (> 0) and x are fixed, the mapping: w --- X(t, x, w) is a d-dimensional Wiener functional, i.e. a P 0' measurable function of w, but it is not in a
class of functionals to which the classical calculus of variations or Frechet differential calculus on a countably normed space Wa can be applied. Generally, it is not even continuous in w. It is an important discovery of Malliavin that this functional, however, can be differentiated in w as many times as we want if the differentiation is understood properly. Moreover he showed that these derivatives can actually be used to produce fruitful results. Examples of such successful applications initiated by Malliavin, then followed by Kusuoka-Stroock, Bismut, Watanabe, L6andre and so on, are among others, in the problems of. regularity, estimates and asymptotics of heat kernels. As we saw in Sections 3 and 5, the initial value problems of heat equations can be solved by probabilistic method of representing solutions as expectations of certain Wiener functionals. With a help of the Malliavin calculus, we
can proceed one step further and represent the heat kernel, i.e. the fundamental solution of a heat equation, as a generalized expectation (in the same sense as in the Schwartz distribution theory) of a certain generalized Wiener functional. This method, as we shall see in the subsequent sections, is quite useful in the above mentioned problems of heat kernels. We start with the r-dimensional Wiener space (Wo, PI). We write
simply Wa = Wand Pw = P when there is no confusion. As usual, Pmeasurable functions defined on the Wiener space (W, P) are called Wiener functionals and two Wiener functionals with the same range space are identified whenever they coincide P-almost everywhere. Needless to say that the most important function spaces of Wiener functionals are L, spaces. In the following, we denote by E a real separable Hilbert
space. As usual, we denote by L,(P; E) (1 < p < oo) or simply by L,(E) the real L, -space formed of all E-valued Wiener functionals F such that I F( w) I E is p-th integrable and endowed with the norm
MALLIAVIN'S STOCHASTIC CALCULUS
351
EP(dw))'I'
I]FIIo =
< e, e > E 2 is the norm of e E E and < , >E is the inner product of E. In the case E = R, L,(E) is denoted simply by L,. In the where I e
Malliavin calculus, we introduce, besides L, -spaces of Winer functionals, a system of Sobolev spaces of Wiener functionals so that we can develop
a differential calculus of Wiener functionals. Malliavin defined the notion of derivatives and Sobolev spaces in terms of Ornstein-Uhlenbeck processes over Wiener spaces. These notions have been studied further
by Shigekawa [148]; Meyer[218], Kusuoka-Stroock [211] and Sugita Sugita [226], in particular, showed the [225], [226], among others. equivalence of apparently different approaches of these authors. Here we develop the theory along the line of [204] and [232]. Let H be the Hilbert space formed of all h E W such that each component of h(t) = (h'(t), hr(t)) is absolutely continuous in t and has square-integrable derivatives. Endow H with the Hilbertian norm
hEH
IhIH= f0lh(t)I2dt,
Ar(t)) and
where h(t) = (1'(t),
1'(t) =
d
h'(t),
i = 1,
r.
This H is often called the Cameron-Martin subspace of W. For h c- H, the stochastic integral (often called Wiener integral) (8.1)
[h](w) _
f -h`(t)dw°(t) 11 0
is well defined and [h] E L2 as a function of w. Indeed, L2-norm of [h] coincides with I h I H i.e. the linear mapping: H E) h - [h] E L2 is an isometry. Definition 8.1. (i) A function F: W -- R is called a polynomial func-
tional if there exist an n E N, hl, h E Hand a real polynomial p(xl, of n-variables such that (8.2)
F(w) = p([h1](w),
[h2l(w),..., [hh](w))
The totality of all polynomial functionals is denoted by P.
DIFFUSION PROCESSES ON MANIFOLDS
352
(ii) A functional F: W -- R is called a smooth functional if there exist
an n E N, h,, h E H and a tempered C--function f(x,, on R" such that (8.3)
F(w) =.f([hd(w), [h2](w),...,
Here, f is called a tempered C°°-function if it is C°° and all derivatives of
f are of polynomial growth order: i.e. for each multi-index a = (a,, of E Z*, positive constants Ka and N,, exist such that for all
Da f(x) I < Ka(1 + I x 12)Na
x c=- R°,
where D. is the differential operator defined by
D.=(
a
a
-57Y1 .
ax2) ... (
a
(previously denoted by D° in Section 2). The totality of smooth functionals is denoted by S (iii) An E-valued functional F: W- E is called an E-valued polynomial functional (smooth functional) if there exist an m E N, e,, e2, , em E E and F,, Fm E P (resp. S) such that F(w) = FI(w)e, + F2(w)e2 +...+ Fm(w)em.
The totality of E-valued polynomial functionals and that of E-valued smooth functionals are denoted by P(E) and S(E) respectively.
Remark 8.1. In (8.2) and (8.3), h,, h2i..., h E H can be chosen to satisfy the orthonormality condition: (h,, h1) = S,f, if = 1, n, if n and p or f are suitably modified. In this case, the degree of polynomial p is uniquely determined from F and it is called the degree of the polyno-
mial functional F. Note also that the joint law of ([h,](w), [h,](w)) is the n-dimensional standard Gaussian distribution N,,, i.e. (2ic)
exp(- 2 rxt
The totality of F E P of degree at most n is denoted by P,,. In the following, we often state definitions, properties, etc. in the case E = R for simplicity: All the statements are valid in the case of general E with obvious modifications.
353
MALLIAVIN'S STOCHASTIC CALCULUS
Remark 8.2. P c S c L, for every 1 < p < oo and P is dense in L. Gnerally, P(E) c S(E) c L,(E) and P(E) is dense in L,(E). Proof. We fix an ONB { h, } in H. Note that, for every c > 0, n
E[exp{ cE I Ak[hk](w) I }] < 00 k-1
(8.4)
for every n = 1, 2,
and 2 = (,11,
An) a Rn
where E denotes the expectation and .'(W) = a[[h1](w), i = 1, with respect to the Wiener measure P. It immediately follows that
PcSc
(1 1:9P
L,
and
M(W) = V
where .q"(W) = a[[h,](w), i = 1,
n
n]. We prove that P is dense
in L 1 S p < 00. Suppose the contrary. Then there exists a non-zero X E LQ, (1/p + l/q = 1) such that E[X F] = 0
(8.5)
for every
F e P.
Noting (8.4) and that 1 < q S oo, we have E[ I X I exp{ E I ,k[hk](w)1 }] < 00 k-1
and, combining this with (8.5), we can conclude that E[X exp{ ,,/1 E1k[hk](w) }] k-1
(8.6)
M-0
E[X(EAk[hk](w))'"]
M!
for every n = 1, 2,... and A = (', If X" = E[XI.Tn(W)], then there exists f(xl, such that X,,(w) = f([h1](w), E[XX exp{ k-1
Ak[hk](w) 111
= E[X exp( /-1E)k[hk](w) }] k-1
.V) E R".
E Yl(R", Nn) [h"](w)) and (8.6) implies that
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354
= 0,
0
for every A = (Al' A2'...' An) E R.
By the uniqueness of Fourier transform, we can conclude that f(x)NN(dx) = 0 i.e. f(x) = 0 a.e. x(NN), implying that X. = 0 a.s. By Theorem 1-6.6, X = lim X. a.s. and hence X = 0, leading us therefore to a contradiction.
L2 is a Hilbert space and is decomposed into the direct sum of mutually orthogonal subspaces known as Wiener's homogeneous chaos:
L2=Co(D C1( ...ED C.O... where Co = [constants) and
Cn=Pn n [co(+C,(D ...E) c.-,]J..
Here - and 1 denote the closure and the orthogonal complement i L2, respectively. The projection operator L2 -.. CC is denoted by J. Let HH(x), n = 0,
be Hermite polynomials defined by 2
(8.7)
Hn(x) = ( n!On exp(2 ) dx" exp(- 2 ),
n = 0, 1,
(previously denoted by Hn[1, x] in Section 111-5). Let
A = {a = (a,,
a, E Z+, a, = 0 except for a finite number
of i's)
and, for a E A, set
a!=rIa,!
and
1-1
I a I =Eat. 1-1
Let { ht )t:, be an orthonormal base (ONB) of H and define
H,(w)EP,aEA,by
MALLIAVIN'S STOCHASTIC CALCULUS
(8.8)
355
H,(w) = 1T H. ([h, ](w)). t-I
t
Since H0(x) = 1 and a, = 0 except for finite number of i's, this product is actually finite and defines a polynomial functional. We note that
H,EP.
if
aI
Proposition 8.1. (i)
{
n.
a !H,(w); a E A } is an ONB in L2.
{N/a H,(w); a E A, I a I = n) is an ONB in C.
(ii)
Proof. First recall that { I; Ti H,(x)) } is an ONB in 22(R, N,), n-0 where N, is the one-dimensional standard Gaussian measure, i.e. N,(dx)
_ 2n expj -
2I
Z2 }dx.
Since { [h,](w) } are independent identically distributed random variables
(i.i.d.) with the one-dimensional standard Gaussian distribution, it is immediatly seen that
f
wH.(w)H,(w)P(dw) = - i ao. a,
a, b E A.
Since P is dense in L2, the assertion of the proposition is almost obvious.
Corollary. If F E P, then JnF E P and F =
is a finite sum.
Indeed, if F is represented as (8.2) with orthonormal (h} extend {h,} to an ONB of H and apply the proposition. Since p(x,, xn) can be expressed as a linear combination of llHaj(x,), the assertion is obvious.
Hence, for a given real sequence
= (On)
, we can define an
n-0
operator To on P by (8.9)
TTF = Z0nJ,F,
F E P.
n-0
T can be extended to a self adjoint operator on L2 (denoted again by
DIFFUSION PROCESSES ON MANIFOLDS
356
TO) by setting
the domain ff(T#) _ {F E L2; and
TTF=
F E -V(T#).
B-0
If 0. = - n, To is denoted by L and called the Ornstein-Uhlenbeck operator or number operator. Also (I - L)s, for s E R, is defined to be TT with 0. = (1 -{- n)$, n = 0, If 0, = exp{ - nt }, t >_ 0, n = 0, 1,
, we denote To by Tr. Namely, we define (8.10)
TF = EeJ"'J F, B-0
F E P, t > 0.
It is clear that { T, } defines a one-parameter semigroup of operators on P. If (8.10) is extended to F E L2, it define a contraction operators on L2, i.e., II T,F112 < 1I F112,
F E L2.
Hence { T } defines a one parameter semigroup of symmetric and contraction operators on L2 and is called the Ornstein-Uhlenbeck semigroup. Proposition 8.2. The Ornstein-Uhlenbeck semigroup T, on P is also given by
T,F(w) = f F(e'tw + /Y--- e Ztv)p(dv) (8.11)
, FE P,
= f T,(w, dv)F(v) where T,(w, dv), w E W, is the image measure on W of P under the mapping: WED v -. a-tw + .,/1 - e Ztv E W. Proof. We denote the operator defined by the right-hand side of (8.11) by 1',. Let h E H and
MALLIAVIN'S STOCHASTIC CALCULUS
357
F(w) = exp{../-1 [h](w) + I h 1 .2/2). Then noting that [h](w) is Gaussian distributed with mean 0 and variance I h 1 H, we have
1',F(w) =
exp{ N/-1e t[h](w) + Jw
1 - e-2t[h](v)
+
I h 1 H}P(dv)
2
= exp{
(8.12)
e`[h](w) +
x =exp{
I h 12}
2
f exp{,/1 - ezt[h](v) }P(dv)
e t[h](w) + Z e-2t I h i H}.
Let A = (%', A2 ..... .In), n E N, and h =,,'hl + )2h2 { h` } E H is orthonormal. Setting F(w) = tfJ exp{ ./-1.1`[hr](w) - 2 (
,`)2 }
we have, by II-(5.3) and (8.7) n
F(w) =
E ml, m2, -' , mnmo J-1
IIH,J([hj](w))
J-1
Applying 1t to both sides, we obtain 2 e2t I hjH}
=1 n
ml, M2, ..., mn-0 J-1
n
t-1
Since the left-hand side is IIexp{ n
2'et[hJ](w) -
= IIJ-1{ k-0 E(% -_jJet)kHk([hJ](w)) }
)2}
).nh. where
DIFFUSION PROCESSES ON MANIFOLDS
358
n
a (ml+my+...+mn)r7f
ll(v J-1
ml, m2,..., mn..o
tnm
'_ 1lJ)m]HH.,([hjl (w)), J-1
we can conclude that e-(m1+m2+...+mn)tfHHJ([h1}(w)),
1t(11Hmj([hj]())(w) = J-1
J-1
that is (T'tHH)(w) = e-'-11H,(w)
for any a E A.
This, combined with Proposition 8.1, implies
't = Tt
which completes the proof. Thus, the Ornstein-Uhlenbeck semigroup is a symmetric Markovian semigroup, i.e., a semigroup given by a probability kernel TT(w, dv) as (8.11) which is symmetric in the sense of (4.48) with respect to its invariant measure v = P. It can be extended to F in S by (8.10) and it is easy to see that TF E S, t > 0. Furthermore, it can be extended to any Borel function F(w) on W with growth order I F(w) I < C1 exp{ C2IIwll7I }
for some positive constants C1, C2 and n. As we have noticed, { Tt } is an L2-contraction symmetric semigroup on L2. Furthermore it is a strongly
continuous contraction semigroup on L I < p < co, i.e., for t > 0 (i)
IIT;FII, <_ IIFII,
(8.13)
for any F E L,.
(ii) limllT;F - Fll, = 0 t1o In fact, it is easy to see that for any F E L, satisfying IF Jo E L2,
f w I T,F I D(w)P(dw)
f
If
T:(w, dv) F(v) I' P(dw)
f {f Tt(w, dv) I F(v)I'}P(dw)
MALLIAVIN'S STOCHASTIC CALCULUS
359
= f Tt(I F I °)(w)P(dw) =f
I F D(w)(Tl)(w)P(dw)
=f
I F J D(w)P(dw).
(by the symmetry of T1 in L2)
In the general case, for any F E L,, we choose F. E L, such that IF" I P E L2 and F.-Fin L,. (ii) of (8.13) is easily concluded first for F E P and then by a standard limiting argument.
We have defined the Omstein-Uhlenbeck operator L on P by
LF =
(8.14)
n20
F E P.
Clearly
dt T,F = TT(LF) = L(T,F),
(8.15)
F E P.
We now determine the expression of L. Let F E P be given by (8.2) with 4(w) = ('(w), 2(w),"., 4"(w)) = (h1 } orthonormal. Then, writing, ([h1](w), [h2](w),..., [hn](w)) E R" and a1p = ap/axt, i = 1,2,..., n for simplicity,
d
(TTF)(w) = dt f a"p(e!` (w) +
1-
e-Zryi)r
7,)-e- 11112 dye
)ne
f gn ±e1. 1` `(w)(a1p)(e-` (w) +
+f,
n
f
1
/1
d?l
e-2tyi)( 21c
t -2t
"
ia2Z
J Ig12
1
2
eP)(e-`(w) + 1 - e
dj
2
gn
Ee `4'(w)(a,p)(e-`4(w)
f gn1-1 Ee-`((a?p)(e-%w) +,%/I
+
- e-stii))( j-_) e
by an integration by parts. Hence
e
2j
dri
&I
DIFFUSION PROCESSES ON MANIFOLDS
360
LF(w) = (8.16)
d
(T,F) 1,-,(w)
n
= E((a2P)((w)) -`(w)(arP)((w)) i-1 Similarly, if F E S is given by (8.3), LF(w) =
d (TF) 1,_0(w) is also
given by (8.16) with p replaced by f. For F E P (or S) and h E H, we define the derivative of F in the direction of h by DhF(w) = lim
F(w + sh) - F(w) a
alo
It is clear that there exists unique DF E P(H) (resp. DF E S(H)) satisfying
h s H.
DhF(w) = 3
Indeed, if F E P is given by (8.2) (8.17)
DF(w) = E(8,P)(4(w))h,, i-1
and similarly in the case of F E S. More generally, if F E P(E) then DF E P(H(& E), HQ E being the tensor product of H and E. Remember that the tensor product El Qx E2 of two separable Hilbert spaces El and E2 is a Hilbert space formed of all linear operators A : El - E2 of Hilbert-Schmidt type endowed with the Hilbert-Schmidt norm E2 } "2
IIA IIas = { 1,
_1
for some (= any) ONB's {e, } in El and { eJ } in E2. Indeed, if F(w) = EPk([hl](w), k-1
ek E E
[h2](w),...,
then
DF(w) = E E(atPk)([h1](w), k-I t_1
[h2](w),...,
where h1 Qx ek E H © E is defined by
[h, Q ek](h) = a ek
h E H.
ek
MALLIAVIN'S STOCHASTIC CALCULUS
361
If F E P, then DZF E P(H (3 H) and D2F, as a linear operator on H, is of trace class for almost all w e W. Indeed, if F E P is given by (8.2) with { h, } orthonormal, then extending { h, } to an ONB in H,
trace DZF(w) = EH 1-1
(8.18)
"
_ roe From (8.16), (8.17) and (8.18), we have (8.19)
LF(w) = trace DZF(w) - [DF](w).
Here, for G E P(H) generally, [G] E P is defined as follows: If G is represented by G = EG,(w)h, with G, E P and h, e H, tm1
(8.20)
[G](w) = ZG,(w)[hr](w)
This definition of [G] is well-defined independently of a way of repre-
senting G. If we introduce a linear operator D*: P(H) - P by (8.21)
D*F(w)
trace DF(w) + [F](w),
F E P(H),
then we have (8.22)
L = - D*D.
All the above definitions and formulas extend to the case of S and S(H). In the same way as we obtained (8.16), we can prove (8.23)
f w< DF(w), G(w)>HP(dw) = f F(w)D*G(w)P(dw) w for all F E S and G E S(H).
Also it is easy to prove the following chain rules: If F, E S, i = 1, 2,...,
n and g(x,, x2,..., x") is a tempered C"-function on R", then (8.24)
D(g(FI, F2,..., F")) _ Eax,- (Fj, F2,..., F")DF,
DIFFUSION PROCESSES ON MANIFOLDS
362
L(g(F,
Fn)) n
(8.25)
a29
i.
l aX axi (Fl,
+
ax (FI,
F2,..., Fn) H F2,.... F,)LF,.
In particular, if F, G E S,
D(FG) = FDG + GDF
(8.26)
and
$ =
(8.27)
2
{ L(FG) - (LF)G - FLG }.
If, F,G,J E S, then
z,, DJ>, = g®a + H H.
If F E S and G E S(H), then FG E S(H) and (8.29)
D*(FG) = - s + FD*G.
Hence, if F, G E S (8.30)
D*(FDG)
$ - FLG.
As before, let E be a separable real Hilbert space. Definition 8.2. For 1 < p < oo and s e R, we define a norm I on P(E) by (8.31)
lI, ,
IIFIID,d = II(I - L)°'2FII,-
Note that (I - L)''2F E P(E) is defined by the Wiener chaos decomposition as
(I - L)"2F = I(1 + n)'"2J7F. -0
Proposition 8.3. These norms possess the following basic properties :
(i) (Monotonicity) if s < s' and I < p':5; p < oo, then,
MALLIAVIN'S STOCHASTIC CALCULUS
363
F E P(E).
IIFII,., < IIFIID,, ,,,
(ii) (Compatibility) if s,s' E R and pp' E (1, oo) and if F. E P(E) satisfy IIFFII,,, - 0 as n - oo and J IF,, - Fmil,,, ,, -- 0 as n,m --- 00,
then IIFnI-.0 as n -- oo.
(iii) (Duality) if s r=- R, p E (1, oo) and G E P(E), then IIGIIQ.
-, =
sup( I f W EP(dw) I ; F E=- P(E), IIFII,.,
1}
where i/p + l/q = 1. Proof. (i) Since T, is a contraction semigroup on L,(E), i < p < oo, the operator
(I - L)-s = r(s)
f
o
e °ts-Trdt
is also a contraction on L,(E) if s > 0: II(I - L)-SFII, < IIFII,,
F E=- P(E).
From this, (i) is easily concluded. 0 as n, (ii) Let G. = (I - L)s'"2Fn E P(E). Then IIFF - Fm11,, m - oo means that IIGn - GmII,, -. 0 as n, m -- oo and hence, there exists G E L,(E), such that IIG,, - GII,, -. 0 as n -. oo. But IIFFII,,, --
0 as n
oo implies that L)(,-1,)'2G'nII,
II(I -
-0
as n -- oo.
Let J E P(E). Then ,,-s
(I - L) 2 J E P(E) c LQ(E) for every I < q < oo. Hence
f
W
EP(dw) = Jim f
= lim f n-
<(I - L)s Z G,(w), (I - L)s W
=0 and this implies that G = 0. Then
J(w)>EP(dw) Zs
J>EP(dw)
DIFFUSION PROCESSES ON MANIFOLDS
364
as n -- oo.
IIFFIIp,, e = IIGnlI,, -- 0
(iii) This can be easily obtained by the well-known duality between
L,(E) and LQ(E) if 1/p + 1/q = 1. Definition 8.3. Let 1 < p < oo and s e R. Define D,,t(E) to be the completion of P(E) by the norm II Ilp,tIt is easy to see that D,,,(E) coincides with the completion of S(E) IIp, t Now it is clear that
by the norm 11 (8.32)
DO (E) = L,(E)
and by (i) and (ii) of Proposition 8.3, we have (8.33)
for s S s' and 1 < p' S p < oo
D,,, ,(E) C D,, ,(E)
where c denotes the continuous inclusion. By (iii) of Proposition 8.3, we have
t(E) = D, -,(E),
(8.34)
s E R,
p E (1, oo)
under the obvious identification where 1/p + 1/q = 1. Set (8.35)
D.(E) =
n,
10
D,,,(E)
and
(8.36)
D__(E) =
U
10
D,._t(E).
Then D (E) is a complete countably normed space (Frichet space) and is its dual. The spaces DD,t(E), D (E), are denoted
by DD,t, D_, D_..... if E = R. Since D,, 3(E) c D,, 0(E) = L,(E) elements in
D,,t(E) U 1o
ifs > 0,
are Wiener functionals in the usual sense,
but elements in D,,t(E) for s < 0 are usually not so. We have a clear analogy with the Schwartz distribution theory and it may be natural to call such elements as generalized Wiener functionals. The coupling D-oa(E)(0, F)D.(E),
45 E D__(E), F E D.(E).
MALLIAVIN'S STOCHASTIC CALCULUS
365
is also denoted by E(<'1, F>E) and it is called a generalized expectation.
In particular, 1 (= the functional identically equal to 1) E D. and, for 0 a D_,,, the coupling D__(cI, 1)D is called the generalized expectation of 0 and is denoted by the usual notation E(,P) because it is compatible when
45 ei
Ornstein-Uhlenbeck operator L: P(E) - P(E) can be extended uniquely to L: D_..(E) and it is continuous as a operator D,,,+2 (E) -- D,, (E) for every p e (1, oo) and s E R. The following important result is due to Meyer [218]. Unfortunatly, we can not have enough space here to give its complete proof. For the proof, see [225] and [232]. Theorem 8.4. (Meyer [218]). For p c- (1, oo) and k E Z+, there exist positive constants c,,k and C,,k such that k
(8.37)
c,,klIDkFlln c IIFIID.k <_ C,.kEIID'Fllp r-o
for every F E P(E). By using this we obtain the following theorem. Theorem 8.5. (Meyer [218], Sugita [225], P. Kree-M. Kree [208]). The operator D : P(E) -- P(H (D E) (or S(E) -- S(H (& E)) can be D_m(H (& E) so that its extended uniquely to an operator D: restriction D: D,,,+,(E) -- D,,,(H(D E) is continuous for every p e
(1,00) and sER. By the duality, we immediately have the following
Corollary. The operator D*: P(H Qx E) --- P(E) (or S(H (D E) -S(E)) can be extended uniquely to an operator D*: D_ ..(H (DE) D_.(E) so that its restriction D*: D,.,+,(H (& E) -- D,,,(E) is continuous for every p e (1, co) and s E R.
It is now clear that L = - D*D in this extended sense. These results imply, in particular, that operators D, D* and L, first defined on the space of polynomial functionals or smooth functionals, are closable and therefore can be extended to Sobolev space D,,3(E). It may be in-
DIFFUSION PROCESSES ON MANIFOLDS
366
teresting to note that the following operator on P; P 2) F- trace D2F E=- P
is not closable. Indeed, let r = 1 and Fk E P, k = 1,
be defined
by 2k
Fk(w) _
It is easy to show that Fk -- 0 in L, for every 1 < p < oo but trace DZFk = 2 for every k = 1, To prove Theorem 8.5, we prepare the following lemmas and propositions.
Lemma 8.1. (Commutation relations involving D). For a real sequence 0 = (0.,).'=O,
DTI = TT+D
P(E)
on
where
0+(n) = ¢(n + 1),
n = 0, 1,...,
Proof. For simplicity, we only show the lemma in By Proposition 8.1, it suffices to prove DTOH,(w) = TT+DHQ(w)
for every
case of E = R.
a E A,
where Ha(w) = jfl HH,([h,](w))
and [h,1,'!, is an ONB of H. Since TTH,(w) = 0(I a I)H4(w)
aEA,
we have (8.38)
DTTHQ(w) = cS(I a I)DHH(w).
Furthermore, noting that H;(x) = H,-,(x), we easily DH,(w) = aj>0 Z Haj-j([hi](w)) IIi*jH.,([hj](w))hi
see
MALLIAVIN'S STOCHASTIC CALCULUS
367
and hence
DHR(W) E C. This implies TO+DH,(w) _
n-1
O(n +
o(I a I)DH,(w).
Combining this with (8.38) yields the conclusion.
Proposition 8.6. (Hypercontractivity of the Ornstein-Uhlenbeck semigroup, Nelson [220]). Letp E (1, oo), t > 0 and q(t) = (p - 1)e2` + 1. Then for every G E
IIT,Gllgco c IIG1Ip
L.
Proof. The following proof is due to Neveu [221]. Let { B$'>, t > 0 }, i = 1, 2, be two n-dimensional Brownian motions with B0('> = 0 defined on a probability space {92, . ; Q }. We assume that { B;'', t > 0), i = 1,2 are independent. For given 2 E (0, 1), we set
q=q(1)=(p-l)1-2+1 and define q' = q'(2,) by
1 f-q,=1. It is easy to see that { (p - 1) (q' - 1) }1J2 = A. We define ]3> = Br(')
and
i ;2) = ABP) +
12B,(2) ,
and ._f"('), i = 1, 2, the proper reference families of o.[tt1>, 0 <
0
i.e..i7 _
s < t], i = 1, 2. Let f and g be C`-functions defined on R"
such that
a
forsome0
Define martingale { M,, 0 S t S 11 and { N1, 0 < t < 1 } by M, = E[f(Jiu)a' 1.
-t11)], N, = E[g(4112))D 1 _11',(2)],
0 < 1 < 1.
Then these can be written in the following forms (Theorem 11-6.6):
DIFFUSION PROCESSES ON MANIFOLDS
368
N, = No+fov/,dks2>.
M,=Mo+foO3d4,111,
By Ito's formula, we have EQ[M, /q' Nl.1/P] - EQ[Mo /ra'N (N/P]
2
EQ(J
IM -2 NP -2 L P
P) M,' V,1 - 2 pq, M`N`O,V, 11
1
Q
2
E
1_
l2
1
l
2 N`
LJ
-
1_2 (-, P1 +-L (1
1 N,0`/ Idt]
( by -,1(p
-1)(q'-1)=2 )
where EQ denotes the expectation with respect to Q. This implies that EQ[f( i1))g( i2))] = EQ[M'I N1i I lP
_
< E12[M101q, N'01']
Hence we obtain r
e
1
2
d1/q' {
J
Rnf(4)q,(N/1
27l
1
K
)"e-
Rng(S)P('V
2
n_ 1812
2e
_ If12
1
g(24 + /1 J Rn J Rn r
2 d dtt
n2 IfI2
27t
X2
Taking 2 = e ', t > 0, we have
{ f Rng(2 C
If
i - eryl)( /n_ "2
K
Rn g(S)P('V
27G
e
)ne_
jj fr 1/P
d4 2
1
812drl
z
IFI2
1
/1
d1/P
n_ I
2e
IZ
2d
j""
where 4(t)' = q(e °)'. By combining this with Remark 8.2 and Proposition 8.2, we easily obtain
f TG(w)F(w)P(dw)
s IIGIIPIIFIIQ(=),
for every G E LP and F E L4(,)I, which completes the proof.
MALLIAVIN'S STOCHASTIC CALCULUS
Proposition 8.7. J,,: L2 - C. is a bounded operator on L,,
369
PE
(1, cc).
Proof. Let p > 2 and t be the positive constant such that
e2.=p-1. Then by Proposition 8.6 we have
F E L, .
IITTFIIp <- IIFII2
In particular IIJVF112 s IIFII2 < IIFIlp
Since e-11114FII,,
this implies that (8.39)
for F E L,
c e''lIF11
For 1 < p < 2, consider the adjoint J,* of J. Then (8.39), applied to p'
such that 1/p + l/p' = 1, yields that IIJ.*FI1, < e-"'IIF11,
for F E P
But, for F E P, J,*F = 4F. By the denseness of P, the result follows. Lemma 8.2. (L, -multiplier theorem).* Let 0 positive sequence such that
0(n) = E a, (n--L ) k, for some no E N,
EIaxI(
for
0(n));.,, be a real
n > no
1 > a > 0 and o)k
Then for every I < p < oo there exists a positive constant C, such that First obtained by Meyer [218]. Proof given here is due to Shigekawa (Private communication).
DIFFUSION PROCESSES ON MANIFOLDS
370
F E P.
for every
IIT#FII, S C,IIFII,
Proof. First we consider the case a = 1. We set ng --1
To = n-0 L c(n)J + E 0(n)Jn
: = TO(1I + TO2)
X no
By Proposition 8.7, T0(1) is L; bounded, i.e., there exists a positive constant CD, (8.40)
IITT"FII, S C,IIFII,
for
F E L,.
We now show that there exists C > 0 such that (8.41)
IIT0(I - Jo - J, -...- Jn°-1)FII, < Ce-n°`IIFII for all t > 0
and
F E L,.
First we consider the case p = 2. For F (=- L2, we have
II Tt(I - J° - J, -- Jn°-1)FII22 (8.42)
= IIE ek,JkFl12 = Ek°n0 IIe kSJkFII2 k®n0 e20° k-n° E IIJkFII22 S e "ofIIFIIi
Next we consider the case p > 2. Take to such that p = e20 + 1. By Proposition 8.6, we obtain
II T°+# - J° - J, -...- Jn0-1)FII2 Jno-1)FII2 < II Tt(I - Jo - J,
< IIE e (8.43)
n-n0
e2n°n_'0 E IIJJFIIi e-2n0`IlFl12
< e2n°tIIFII2
=
e2n°t°e2n°a+t°,IIFII2
Furthermore for t :!!g to we obtain, by (8.13), 11 T, (I - J° -- J1 - ... - Jn°-1)FIIP
MALLIAVIN'S STOCHASTIC CALCULUS (8.44)
< 11(I - Jo - J,
< III - Jo - J, where III -
Jo - J, -
371
- J.0-1)FII, Jn0-111DIIFII,
- J.0_111, is the operator norm of (I - Jo
- J, - ... - JJ0_1) in L.
Combining (8.42), (8.43) and (8.44) we obtain (8.41) with p > 2. For 1 < p < 2, the (8.41) follows by duality. We now set
R"0 = fo
T,(I - J. - J, - ... -
Then, from (8.41) we obtain IIRnoFII, < Cno IIFII,.
Since
R"0F =
J of 0
T:+S(I - Jo - J, -
-
dtds
we obtain IIRn20FIIo < Cno IIFIID
and repeating this we have (8.45)
IIRn0FII, s C LIIFIIP,
k = 1, 2, ... .
Note that RnoF = nkJnF,
k = 1,
Hence we obtain
T1Z)F =n-n0k-0 E E akRnoJTF =k-0 E By (8.45),
if F E Cn, n
no.
DIFFUSION PROCESSES ON MANIFOLDS
372
IITT2'FIID < C(EIakl o)k)IIFIID
Combining this with (8.40), we can conclude that there exists C, > 0, such that
F E P.
IIT#FII < CPIIFIIP
For the general case i.e., 0 < a < 1, define Qia' = E e natJJF =
Ja
(T:F) a)(ds)
where p;°) is the probability measure defined by
f
asp160(ds) = 0
e
exp (-
A61t)
for every A > 0.
As in the case a = 1, write To = TOW + TO(2)
By the same reason, TT1' is L, bounded. From (8.41), we have II
Qia)(I -
< c$
J0 - J1
Jn0-1)FIID
IIFIIe-1'0°p,('1'(ds) = C exp (- not)IIFII,. o
Define R.0
f0
Qt-)(I
-J0-J1
and proceed as in the case a = 1. Then we obtain that TT2) is LD bounded. This concludes the proof.
Proof of Theorem 8.5. In case of E = R, we show the theorem. By Lemma 8.1, we obtain
I (I -
L)112DF
I H = I DR(I -- L)sl2F I H
where R is the operator given by n R=E(n+11
J,,.
FEP
MALLIAVIN'S STOCHASTIC CALCULUS
373
Note that R = To with 10,
1(n)
s/2
s/2 n (n+1) -(1+ n) ' 1
n=0 n> 1
and
h(x)=(1+x 1
is analytic near x = 0. By Theorem 8.4 and Lemma 8.2, there exist posi-
tive constants C, and C; such that II(I - L)1'2DFIIP = II DR(I - L)"ZFll, s
,
< C,II(I - L)TR(I - L)'TFII, = C,II R(I -
CII(I -
L)<s+ll'2Flip
Therefore IIDFII,,s < C;IIFII,,s+,
FEP
from which the result follows by a limiting argument.
Proposition 8.8. Let E, and E, be real separable Hilbert spaces. For any p, q c= (1, oo) and k = 0, 1, 2, such that
there exists a constant C,, q, k> 0 such that (8.46)
IIF0 GIIr,k C Cp.q.klIFIIp.kIIGIIq,k
for every F E P(E) and G E P(EZ). Proof. Indeed,
DIFFUSION PROCESSES ON MANIFOLDS
374
D(FQG)=DF®G+FQDG and hence ID(F(& G)IHeEIeEZ < IDFIS®E,IGIE,+ IFIE,IDGIR®Ey
from which (8.46) in the case of k = 1 is easily obtained by noting the equivalence of norms IIFII, + IIDFII, and IIFII,,1 (which is a consequence of Meyer's theorem). We can obtain (8.46) by the same argument successively.
From (8.46) and the duality (Proposition 8.3, (iii)), it is easy to obtain the following.
Corollary. For every p,q e (1, oo) such that 1
1 + q =
r
< 1
and k = 0, 1, , there exists a constant C,q,k > 0 such that (8.47)
IIF® GIIr.-k < CD q,kjIFII,,kIIGIiq.-k
for every F E P(E,) and G E P(E2). By (8.46) and (8.47), we see that F Q G E D,, k(E, (g EZ) is welldefined for every F E DD, k(E,) and G E Dq, k(EZ) and, F Q G E D,, _k(E,
(D E2) is well-defined for every F E D,, k(EI) and G E Dq, -k(EZ) provided
+ q = r < 1. In particular, D. is an algebra, i.e., if F,G
D_ then FG E D.. Furthermore, D (E) and D__(E) are D,.-modules, i.e.,
if F E D_ and G E D_(E) (D_..(E)), then FG E D_(E) (resp.
D--(E)). Note that if F E Dm and G E D__, the generalized expectation F>D,,. E(FG) of FG E D__ coincides with We note that chain rules (8.24)-(8.30) can be easily extended to Wiener functionals belonging to Sobolev spaces in an obvious way: In particular these formulas hold if P and P(E) are replaced by D.. and D_(E) respectively. Also the integration by parts formula (8.23) can be extended to the case F E=- D,,, and G E Dq,,(H) such that
I + q =1.
PULL-BACK OF SCHWARTZ DISTRIBUTIONS
375
Of course, (8.23) holds in the case FED,,,+, and G E DQ, _,(H),
P + 9 =1,
sER,
if the both sides of (8.23) are understood as the natural coupling between D,, 3(H) and DQ, _,(H) (= (D,,,(H))') and D,, and Dq respectively.
9. Pull-back of Schwartz distributions under Wiener mappings and the regularity of induced measures (probability laws) We now apply the results of the previous section to the study of re-
gularity and asymptotics of probability density of finite dimensional Wiener functionals. For this, we regard a d-dimensional Wiener functional as a d-dimensional Wiener mapping F: W- Rd and discuss the pull-back of Schwartz tempered distributions on Rd under this mapping. This pull-back can be defined as an element in D_-, i.e., as a generalized Wiener functionals, if the Wiener mapping satisfies certain conditions on regularity and non-degeneracy usually referred to as Malliavin's conditions. If the Dirac delta function 8X at x E Rd is pulled back by a Wiener
mapping F: W- Rd, then the generalized expectation E[Sx(F)] of this pull-back 8z(F) coincides with the density p,(x) of the probability law of functional F with respect to the Lebesgue measure on Rd. We can discuss the continuous or differentiable dependence on parameters of the pull-back thereby deduce the regularity of the probability density. So let (W, P) be the r-dimensional Wiener space and let F(w) _ (F'(w), FZ(w), , Fd(w)) be a mapping W- Rd. We assume that it is smooth in the following sense of Malliavin: (A.1)
FED_(Rd),i.e.F'ED.,
i=
Then we can define (9.1)
or'i(w) = x,
i,.l = 1, 2, ... , d
and we know that a'J E D-. The matrix a(w) = (o'"(w)) is non-negative definite and hence det v(w) > 0 for almost all w(P). 6(w) is called the Malliavin convariance of F. Definition 9.1. We say that F is non-degenerate in the sense of Malliavin if it satisfies
DIFFUSION PROCESSES ON MANIFOLDS
376
(det v(w))-1 E f1 L,.
(A.2)
1
Here it is understood that 1/0 = oo. Assume that F: W-- Rd satisfies Malliavin's conditions (A.1) and (A.2). For every e > 0, define v8 = (oa) by aa1=au+6(/11
we have where t5'1 is Kronecker's delta. Then, setting (yt1) = y,8, E D_, i, j = 1, 2, ..., d because there exist tempered C"-functions f1(x), i, j = 1, 2, , d on R1 such that i, j = 1, 2, ... , d.
Y91= jt,(a8),
It is easy to prove by the chain rule that d
Dy,, = --
(9.2)
k,1-1
Y'rkYi1Do
Furthermore it is easy to deduce from (A.2) that in L,(H®k)
DkYi1- Dky,1
for every k=0,1,-.., 1
H®k = H Qx H Q k
i, j = 1, 2,
,
d
where
(Q`7)-1 This shows that
i,j= 1,2,...,d.
Yr1 e D_,
Also by (9.2), we have d
Dy1
(9.3)
= - k,E1-1
YrkYllDdkl
Let 0: Rd - R be a tempered C --function. Then, by (8.24), d
D(¢ -F)
(a,0 o F)DFr. t=1
Hence d
,y = E (a,OoF)H 1-1
PULL-BACK OF SCHWARTZ DISTRIBUTIONS d
=
a,0°F. Qu [n1
and therefore, d
(9.4)
X. y,JE.
3r 5 °F
Jm1
Let g E D... Then, we obtain by (8.23) that fw ai¢ ° F(w) g(w) P(dw) d
_ 57, f
.P(dw)
J-1
w
_ J-1 E
w
0° F(w) D*(yjJ(w)g(w)DFJ(w)]P(dw)
= f w 0 ° F(w)
01(w; g)P(dw)
where d
01(w; g) _ EJ-1D*[yu(w)g(w)DFJ(w)l d
Z (g(w) H
(9.5)
J-1
+ 7,j(w) H + y,J(w)g(w)LFJ(w) )
by (8.26) and (8.30). Hence
f
w
(atlat2
... ask 0) °F(w) . g(w)P(dw)
= $o F(w) where 0'3, ,2, Oft, 12,
.
0t1, t2, ... r Ik (w; g)P(dw)
,k(w; g) E D. is determined successively by iv(w; g) = O1(w; 'Pt1r 12' ... '
-
; g))
We can conclude from (9.5) that 0,1, 1 .....:k(w; g) has a form
377
DIFFUSION PROCESSES ON MANIFOLDS
378
011. 12..... 'k(w; g) (9.7)
_ W0(w)g(w) + < .'1(w), Dg(w)>H + .. . + < Ylk(w), Dkg(w)> Hek
where
v=0,1,2,.
Tp EDW(H9%
which are obtained as polynomials in y,j, F= and their derivatives. a Similarly we have for d = E (a/ax`)2, i-3
J w111 + 1x12 - 2) ko)o F(w) . g(w)P(dw)
= f 0 - F(w)
?lzk(w; g)P(dw)
where 772k(w; g) E D. has a similar expression as (9.7). Indeed, 772k(w; g)
is obtained in the same way as 0,i, =z. . ,k (w; g) with some more polynomials of F multiplied in each step of the integration by parts. In particular, for every q > 1, (9.9)
sup{ II 2k(
; g)IIj; g E= Dq.2k, IlgIIq.2k s 1 } < 00
II, denotes the L,-norm. We introduce a system of Sobolev spaces on Rd as follows: Let .So(Rd) be the real Schwartz space of rapidly decreasing C"-functions on Rd. For 0 E .S'(Rd) and k = 0, + 1, + 2, , introduce the norm where II
(9.10)
110112k = II(1 + 1x12 -A/2)'011.
where II
II- is the sup-norm on Rd. Let S°2k, k = 0, ±1,±2,
be the completion of 9o(R2d) by the norm II
II2k. Then we have
c .9'2 c Y,, = : { f(x); continuous on Rd and lim I f(x) I 1X1--
= 0}cs°_2c n _9"2k = .7(Rd)
k>0
and
U -9"-2k = Y'(Rd).
k>O
Theorem 9.1. Let F: W--- Rd be a d-dimensional Wiener mapping satisfying Malliavin's conditions of (A.1) and (A.2). Then, for every p > 1 and k = 0, 1, 2, , there exists a positive constant C =
PULL-BACK OF SCHWARTZ DISTRIBUTIONS
379
C(p, k; F) such that the following estimate holds: For every 0 E Y(Rd), (9.11)
IIsoFIIp.-2k -<- C11011-2k-
(Note that ¢ o F E D_).
Proof. Define q by 1/p + 1/q = 1. Then by the duality I I0e FI Ip, -zk = sup{
f w O o F(w) IIgllq.2k C
g(w)P(dw); g E Dq, zk, 1}
and, by (9.8) and (9.9),
i f 0 o F(w)
=I
f
g(w)P(dw) I
((I + 1x12 - d/2)k(l + I x 12 - d/2)-ko } o F(w)
x g(w)P(dw)
=I
f
{ (1 + 1x12 - A/2)-'o } o F(w)
< 11(1 + I x 12 - e/2)-k¢II W IIi72k(
<
172k(w; g)P(dw) I
, 8)111
II0II-2kll 2k( ; g)111-
Now (9.11) is established by setting C = C(p, k; F) to be the supremum in (9.9)
Corollary. The mapping Y(Rd) E3 - 0 o F E D. can be extended uniquely to a continuous linear mapping 5'-2k E) T- ToF E Dp, _2k
for every I < p < oo and k = 0, 1, 2, - - . In particular, for every T E 5"(R,), ToF E_ D__ is well-defined and -
(9.12)
ToFEU
(1
k-1 1
Dp, -2k-
This ToF, denoted also by T(F), is called the pull-back of Schwartz distribution T E S'(Rd) on Rd by the Wiener mapping F: W_ Rd, or the composite of distribution T on Rd and d-dimensional Wiener functional F.
DIFFUSION PROCESSES ON MANIFOLDS
380
The fact (9.12) is important because ToF can be coupled with test functionals in a class much larger than D_: By introducing the notations (9.13)
D_ = n
(9.14)
D__ = U
D,,
U
k-I I <,
fl
k-1 1<,
and
DD. -k,
if G E D_, then G. ToF E D_" is well-defined and hence the generalized expectation
D_ 1,,,
p_a, <1, is well-defined.
Example 9.1. Let r = I and so W = W,'. For w E W, set t
G(w) = exp{ a f w(s)2ds }, 0
a E R.
Then G(w) E D. if and only if a < rc2/8 and if f(x) is a C"-function on R with compact support, G(w)f(w(l)) E D" if a < n2/2. The proof of these facts can be easily provided by the formulas (6.9) and (6.10) of
Section VI-6. Note that G(w) E D_ if and only if a S 0. The above result can be applied to show the existence of smooth density of the probability law F,k(P) of F, i.e. the image measure of P under F. For this we first note the following fact.
Lemma 9.1. Let Sy E .9'(Rd) be the Dirac S-function at y e Rd and m be a positive integer. Then
Sy E Y-2. and D,Jy E -9"-'.-2k where a = (a,,, a2i -
+
,
if m> d and I a I S 2k,
ad), a, E Z+, is a multi-index, I a I = a1 + a2
+ ad and D. is the differential operator: Da =
as)a, 1
ax2
1a2
...
axd )ad
PULL-SACK OF SCHWARTZ DISTRIBUTIONS
381
Furthermore if m > d/2, then the mapping: Rd E) y -- Sy E .- 2m-2k
is C2k, i.e. 2k-times continuously differentiable.
Proof. First we note that the operator A = : (1 + I x 12 - d/2)-' is defined by a kernel A(x, y) :
Af(x) =
J Rd
A(x, Y)f(Y)dY
and A(x, y) is given by
A(x, y) = f e 'p,(t, x, y)dt where p1(t, x, y) is defined in (6.12) of Section VI-6 (in the case d=2, but similarly in general). From this, we see that A(x, y) > 0, A(x, y) is C°° in (x, y) E (Rd x Rd)\{ (x, y); x = Y1, and
const. x I x- y I- cd-2)
A(x, y)
as Ix - y j
d 0.
Also it is immediately seen that x, y e Rd
A(x, y) < A(x, y),
where A(x, y) is the kernel of the operator A = :(1 - A/2)-'. Hence ((1 + I x l 2- d/2) mo)(x)
- fRd
Rd
f...
fRd A(x,
f f ... f Rd
Rd
Rd
Yz) ... A(Ym-a, Y)dyidy2 ... dym-,
A(x) Y1)A(Y1, Y2) ... "(Ym-t, Y)dY1dY2 ... dym-I
I )d
2: f Rd 1+ I
12 m 2
)
and the right-hand side is in Soo as a function of x if m > d . From this
we can easily deduce that
DIFFUSION PROCESSES ON MANIFOLDS
382
(1 + I x z- d/2)-mb,, E .9%
for each y
and the mapping Rd EE) y --- (1 + I x 12 - d/2)-mby c- .9'0 is continuous
provided m > d/2. This proves the first assertion and the second assertion can be proved similarly. The last assertion is obvious if we notice that Day = (_ 1)"'(Dy)aby
where (Dy)a is the differentiation with respect to the parameter y.
Theorem 9.2. Let F: W- Rd satisfy the conditions (A.1) and (A.2) and let Fk(P) be the probability law of F. Then F*(P) has the smooth density pp(y) with respect to the Lebesgue measure dy on Rd and pp(y) can be given by (9.15)
PF(Y) = E[by(F)]
as generalized expectation of by(F). More generally, for every multiindex a, (9.16)
(Dapp)(Y) = E[((Dy)aby)(F)] = E[(-1)1a'(Daby)(F)].
Proof. Let k be a non-negative integer. By the lemma above and the continuity of the mapping: S 2m T -- T-F E Dp, _2m, we can deduce at once that the mapping: Rd E) y --- by(F) E D, -2m0-2k
is C2k where mo = [d/2] + 1 and hence, for every g E Dq,2mo+zk, I/P +
1/q = 1, the mapping: Rd ii) y -- E[gby(F)] is C21. In particular, the y - E[gby(F)] E R is C- if g E D,,. Thus the mapping: Rd m y -- E[by(F)] E R is C-. (9.15) can be verified by noting that for mapping: Rd
every c E So(Rd)
fRd
c(Y)E [by(F)]dy = E [J
xd
O(Y)by(F)dy]
= E[(f ad 0(Y)by( )dy) - F] = E[0 - F]. Now (9.16) is obvious. In the above proof, it was shown that if g E U Dp, 2m0+2k where mo p>1
PULL-BACK OF SCHWARTZ DISTRIBUTIONS
383
[d/2] + 1 and k is a non-negative integer, then the mapping: Rd D y- E[g6y(F)] is C2k. It is easy to see that (9.17)
E[gby(F)] = pF(y)E[g I F = y]
for almost all y such that pF(y) > 0. Thus we have obtained a regularity result for conditional expectation:
Corollary. If g E U DD.2m0+Zk where mo = [d/2] + 1 and k is a y>1
non-negative integer, the conditional expectation E[g I F = y] has a C21 -version on the set {y; pF(y) > 0 ). In particular, it has a C°°-version
on the same set provided g E E. Example 9.2. Let r = d and F: W(= Wod) - Rd be defined, for fixed t > 0 and x E Rd, by F(w) = x + w(t). Then F E D_(Rd) (actually E P(Rd)) and Qtj(w) = out, i, j = 1, 2, , d. Hence yj = Stj/t where 81j is Kronecker's delta and hence (A. 1) and (A.2) are clearly satisfied. Let k E C2-(Rd) which is tempered in the sense D`xk, I a I :!S; 2m, are all polynomial growth order and
HEk(x)/Ix12=a
If a is sufficiently small (actually it is sufficient to assume a < 1/2t), we
can easily see that g(w) = exp{ f u k(x + w(s))ds) E pU1 D,.,,.. Hence
p(t, x, y): = pF(y) = E[(exp{ f' k(x + w(s))ds })O7(x + w(t))]
is C2k in y if k = m - mo > 0. By the Feynman-Kac formula (Theorem 3.2), it is easy to identify p(t, x, y) with the fundamental solution of heat equation au
I
at = 2 4u + ku.
The generalized Wiener functional O,(x + w(t)) is known as Donsker's 0-function (cf. [210]).
DIFFUSION PROCESSES ON MANIFOLDS
384
Now we consider a family { Fa(w) }ael of d-dimensional Wiener
functionals depending on a parameter a, the parameter set being assumed to be an m-dimensional interval I e Rm, for simplicity. We assume that Fa E for all a E I and also that, denoting by o(a) = (&'(a)), oif(a) = the Malliavin covariance of Fa, suplidet o(a)-111, < oo aEl
for 1 < p < oo.
Then it is clear that the constant Ca = C(p, k; Fa) in Theorem 9.1 can be chosen such that sup C. < CO. aal
Hence, for T E 2k, if we choose 0. E .99(R4) such that 0. --- Tin S°_2k as n -- oo, then for every I < p < oo, 0.- F,, - T°Fu in D,, _2k as n -- oo uniformly in a E I. From this we can easily conclude the following: If the mapping: I a -- Fa E D. is continuous (2m-times differentiable) then, for T E S? 2k, the mapping: I Z) a - T°Fa E D,, -2k is continuous for every 1 < p < oo (resp. the mapping: I E) a --T°Fa (z D, _2k-2m is 2m-times differentiable for every 1 < p < cc). We can deduce from these facts the continuity or differentiability in a of the generalized expectation if the mapping: I iD a-. ga E DQ,2k is continuous or differentiable where 1lp + 1/q = 1. Note that, in the differentiable case, we have a
d aFQ 8T
as T ° Fa) _ E aa (3x"
°
Fa,
F. = (F1, F«, ... , Fad)
which is easily verified by a limiting argument as 0. - T, On e .9'(Rd). Finally, as an example of dependence on parameters, we discuss on
the asymptotic expansion of Wiener functionals. Consider a family { F(e, w) } of E-valued Wiener functional depending on a parameter e E
(0, 1] where E is a separable Hilbert space. We assume
F(e, w) E D (E)
for every e.
We define F(e, w) = O(em) in D.(E) as e 10 if F(e, w) = O(sm) in D,, k(E) for' every 1 < p < oo and k E N as a 0, i.e. lim e-mI!F(e, w)ll,.k < oo. #to
PULL-BACK OF SCHWARTZ DISTRIBUTIONS
E
Let fo, f,, (9.17)
385
We define
fo + ef, + s2f2 + ..
ass J, 0 in if, for every n = 0, 1, 2, - - - ,
F(s, w)
F(e, w) - (1 0 + ef, + ... + e"fn) = O(e^+1) in D_(E) as a J, 0. Also, if { 0(e, w) }
is a family of elements in D_ depending one
(0, 1] and 1Vo, yr1,
-
- e D_..(E), we define in D__(E)
(9.18)
if, for every n = 0, 1, 2, that ds(c, w), s e (0, 1]
- -
, we can find
as s 10
kENand 1
and yro, yrli - - - , V/. are all in D,,-,(E)
and
0(e, w) - (+ilo + e'1 + ... + snpn) = O(en+1) in D, -k(E) as e 10. We define D_(E) and b_(E) for general E in the same way as (9.13) and (9.14), i.e.
D.(E) = k-1 fl 1
D,. k(E)
and
b_(E) =u fl k-1 1
Dp.-k(E)
If { G(e, w)) is a family of elements in D&(E) depending on e E (0.1] and go, g1, - - - E D_(E), we define (9.19)
G(e, w) --- go + eg, + e2g2 + -
if, for every n = 0, 1, that
in b_(E) as s y 0
and k e N, we can find 1 < p < oo such
G(e, w), e E (0, 1] and go, g1, -
-
, gn are all in D, k(E)
and
G(e, w) - (go + sg1 + ... + e"gn) = 0(s- 1) in Dp. k(E) ass 10.
DIFFUSION PROCESSES ON MANIFOLDS
386
If ( !P(e, w)) is a family of elements in D_..(E) depending on e E (0,1] and if yro, yrli E D_..(E), we define (9.20)
9'(e, w) -- yro + eyrl +
as e.L 0
, we can find k c- N such that
if, for every n = 0, 1, 2, Vl(e, w), a E (0,1]
in D_m(E)
and
, V. are all in
yro, yr1,
(1 D,,_k(E)
1
and, for every 1 < p < oo, TI(s, w) - (yro + eyrl + - - - + e"+v) = O(e°+1)
in D,, _k(E) as a J, 0.
Noting the continuity of multiplications in Sobolev spaces as given
by inequalities (8.46) and (8.47), it is easy to prove the following:
Proposition 9.3. (i) If F(e, w) E D_(E) and G(e, w) E D. (D_), e e (0,1] such that (9.21)
F(e, w) -fo + Ef1 +
in D_(E) as e J 0
with f, E D_(E), i = 0, 1, - - and (9.22)
G(e, w) - go + eg1 +
with g, E D. (resp.
i = 0, 1,
in D (resp. ,
as e.0
then H(e, w) = G(e, w)F(e, w)
satisfies (9.23)
H(e, w) - ho + eh1 + -
and ho, h1, tiplication : (9.24)
in
(resp.
as a 10
E D.(E) (resp. D-(E)) are obtained by the formal mul-
ho = gofo, h1 = gof1 + g1fo, h2 = go.fz + g1fi + gzfo, .. .
(ii) If G(e, w) E
and O(e, w) E
e E (0, 1] such
that (9.25)
G(s, w) --- go + sg1 +
with g1 a D. (resp. D_), i = 0, 1,
in D. (resp. D_) as e.0 and
387
PULL-BACK OF SCHWARTZ DISTRIBUTIONS
(9.26)
00 + eO, + -
fi(s, w)
with 0, E D__(E), i = 0, 1,
in D_..(E)
as e
0
, then W(e, w) = G(e, w) fi(e, w) satis-
-
fies
(9.27)
V`(s, w) - V'o + eVr, + ...
in D_..(E) (resp. D__(E))
as 810 and Vro, Vr,,
.
are obtained by the formal
- E D__(E) (resp.
multiplication : (9.28)
Wo = 9000, w, = g001 + g160, Wz = 9002 + 8101 + g7-00,
.
--
(iii) If G(s, w) E D_ and fi(e, w) E D_(E), c E (0, 1] such that (9.29)
with g1ED_, i=0, 1, (9.30)
in D. ass 0
G(e, w) - go + eg, + and
in D_ . (E) as s
fi(a, w) --- Oo + e6, +
0
with 0, E i = 0, 1, , then (9.27) holds in D_..(E) with given by (9.28). Wr E D_..(E), i = 0, 1, -
Theorem 9.4. Let { F(e, w), a E (0,1] ) be a family of elements in such that it has the asymptotic expansion: (9.31)
F(e,w)-..fo+ef,+ i
(9.32)
= 0, 1,
and satisfies
limll(det
oo
for all
1
00
810
where a(s) = (o-"(s)) is the Malliavin covariance of F(e, w): 6'J(e) = H. Let T E .9'(Rd). Then fi(e, w) = T o F(e, w) has the asymptotic expansion in b-. (and a fortiori in D_..): (9.33)
45(e, w)
00 +eO,+
and 01 E D_,,, i = 0, 1,
- -
in D__
asa10
are determined by the formal Taylor
DIFFUSION PROCESSES ON MANIFOLDS
388 expansion:
P(E, W) = T(fo + [Efi + E2f2 + ... l)
= Ea t1(D.T) o fo[efi + -2f,. + ...
(9.34)
]
= o +Eq1 + where (i) the summation is taken over all multi-indices and (ii) for every , ad) E Rd, we multi-index a = (a,, a2, , ad) and a = (a,, a2, set as usual a! = a1!a2! ... ad! and a' = a, Ia22 ... aadd.
In particular, denoting at = a/axt, i= 1, 2,
, d,
d
Oo = T(fo),
01 = i-1 Ef(a1T)(fo)
02 = tEff(a,T)(fo) + 3, (9.35)
03 = tn()(f0) + 2! Eff3(ataaT)(fo)
+ 3!t,J,k-1 E fff'fk(ata akn(fo) and etc. Here
fk = (fk,fk,
fkd) (E D..(Rd)),
k = 0, 1, 2, -
-
Remark. fo E D (Rd) satisfies the condition (A.2) by virtue of (9.32).
Proof. We set F(0, w) = fo, ai'(0) = ,, y(E) = (Ytu(E)) = v 1(E), E E [0,1]. Then it is easy to see from (9.32) that y,,(8) has the asymptotic expansion: Yrj(e) -
YtJ(0) + eYj' + El(2) + .. .
k=1,2,
with ¢1,
First, we show that, for every k e N, we can find s E R and o, ,vk-1 E n D,,, such that 1
T o F(e, w) E
fl
1
for all c E [0, 1]
PULL-BACK OF SCHWARTZ DISTRIBUTIONS
389
and (9.36)
ToF(e, w) = 00 + go, + ... + ek-10k-1 + O(ek) in Do., as e,I,0
for all 1 < p < oo. To prove this, we take m E N so large that 0 _ (1 + Ix 12 - d/2)-m T is a bounded function on Rd, k-times continuously
differentiable with bounded derivatives up to k-th order. Then, for every J E D", we have, by the same integration by parts as (9.8), w). J] = E[oo F(e, w)-1,(J)]
E[T o F(e,
where 18(J) E D. has a similar expression as (9.7): 2m
1e(J) = E R®: 1-o
where P1(e, w) E D" (H®r), i = 0, 1, , 2m, which are polynomials in F(e, w), its derivatives and y(e). By the assumption on 0, we have OoF(e' w)
IalE-l -a!(Da0) ° A [F(e, w) - fola + Vk(e, w) zgk
where for some constant M > 0, I Vk(e, w) I< M Ial E -kI [F(e, w) - fo]a 1
Hence, for every p' E (1, oo), we can find C, > 0 such that 11 Vk(e, W)11,, < C,ek
for all e e [0, 1].
Let q' E (1, oo) such that l/p' + l/q' = 1 and choose q such that q > q'. Then a positive constant C2 exists such that 111e(J)11q1 < C211J11q,2m
for all e E [0, 1]
and J E D".
Hence, for all e E [0, 1] and J E D", I E[Vk(e, w)1e(J)]I C 11Vk(e, w)IID,111e(J)IIq,
C,C211Jll,,,2me'.
Also lalE-1-
(D60) °fo
[F(e, w) -folale(J)
(9.37)
<- (DaO) ° fo E I«E [Rk-i a!
[F(e, w) - fo]aP1(e, w), D`J>a®,
DIFFUSION PROCESSES ON MANIFOLDS
390
and since [F(e, w) - fo]aP1(e, w) has the asymptotic expansion elafea.r.o(w) + siai+lea,1.1(w) + .. .
[F(e, w) _ f0]aP,(e, w)
in D-(H®`)
as a 10,
we have
(9.37) = Zo(w) + aZ1(w) + ... + ek-1Zk-1(w) + Uk(e, w) where
Z,(w) = 1=0 E iai+v-l
< 11(Da0) °.fo - ea.t.v. D'J>R®i. a.
We can easily find C3 > 0 such that I E[Uk(e, w)] l S C3ekIIJIIq.2m
for all s E [0, 1] and J E D. Set
-f. 01 =/=0 E lal+v1 E_ (D*)a (1(Da0) a.
ea r v),
I = 0, 1, ... , k - 1.
It is easy to see that
01 E
II
1
Dp.-2m
and this, combined with the above, yields k-1
I E[T°F(e, w)
J] - E etE[O,J] I < (C1C2 + C3)ekIIJIIq.2m 1-0
for all a E [0,1] and J E D=. Hence we have k-1
II ToF(c, w) -
-"01lip. -2m
C (Cl C2 + C3)ek
where I/p + 1/q = 1. It is clear in the above argument thatp E (1, oo) can be chosen arbitrarily and thus (9.36) is obtained. It remains only to show that 1 can be determined through the expression (9.34). Since
APPLICATIONS TO HEAT KERNELS
07 = E imiE='
(D*)i(- (Da((1
391
+ Ix1z - d/2)--T}) o f. ea,j."
we see that the mapping ,7'(Rd) E) T- 01 a D_ is continuous in the sense that for every n c- N, there exist s > 0, p E (1, oo) and C such that
<
CIIT1I-2n.
On the other hand, it is clear that 01 is uniquely determined from T and must be calculated from the expression (9.34) in the case T E S°(Rd). Hence, it must also be given by (9.34) for general T E 9"'(Rd). This completes the proof.
10. The case of stochastic differential equations: Applications to beat kernels Let X = (X(t, x, w)) be the solution of a stochastic differential equation on Rd given in the same way as in Section 2:
dX(t) = a.(X(t))dw"(t) + b(X(t))dt
(2.1)
IX(O)
=x
which is realized on the r-dimensional Wiener space (W, P). We assume,
as before, that all coefficients a0(x) = (c,(x)) and b(x) = (b'(x)) are C" with bounded derivatives of all orders >_ 1. We know from the results of Section 2 that, for almost all w EE W (P), (i) (t, x) -> X(t, x, w) is continuous, (ii) x -> X(t, x, w) is a diffeomorphism of Rd. If Y(t) = (Yf(t)) is defined by YXI)
ax711(t, x, w),
then Y(t) satisfies (2.16)
dY(t) = a'(X(t))Y(t)dw°(t) + b'(X(t)) Y(t)dt
IY(0)=I
where
Q.(x)i = axe' a,' (x)
and
b'(x)i
=
xb'(x)
DIFFUSION PROCESSES ON MANIFOLDS
392
and I = (o) is the d x d identity matrix. The equations (2.1) and (2.16), put together, define a stochastic differential equation for the process r(t) = (X(t), Y(t)) and r(t) takes values in Rd x GL(d, R) which can be identified with the bundle of linear frames GL(R") over Rd. If we introduce a system of vector fields Va, a = 0, 1, 2, , r by Va(x) = a,(x)
a
a = It 2, ... , r
t,
and
V0(x) = [b'(x) - 2 aE aa(x)5o (x)] ax`
then (2.1) and (2.16) can be rewritten as dX(t) = E Va(X(t)) C dwa(t) + V0(X(t))dt a-1
(10.1)
X(O) = x and
dY(t) = E aVa(X(t))Y(t) o dwa(t) + a Vo(X(t)) Y(t)dt a-1
(10.2) I
Y(0) = I
where (aVa(x))i = axi Valx)
Proposition 10.1. Let t > 0 and x E Rd be fixed. Then (i) X(t, x, w) E D (Rd), i.e., XI(t, x, w) E Dm, i = 1, 2, (ii) the Malliavin covariance a(t) = (o`"(t)), a`f(t) _
,
d,
a, is given by (10.3)
a'J(t) _
[Y(t)Y(s)_1Va(X(s))]1[Y(t)Y(s)-1Va(X(s))]'ds. a-1 J
0
Proof. For X(t) = X(t, x, w), we define X("'(t) = X1n'(t, x, w) as
If A = (A',) is a d x d matrix and x = (x') E R°, we denote (Ax)' = A "X"
i=1,2, ...,d.
APPLICATIONS TO HEAT KERNELS
393
in Section 2: X (n) (t) = x +
(10.4)
(O.(s))dW"(s) +
Qa(X (1> 0
' b(X (n) 0
(On(s)))ds
where
if s e [k/2n, (k + 1)/2"), k = 0, 1,
0"(s) = k/2"
.
We know that X (n) (t) E S and I I X (") (t) - X(t)I!, --- 0 as n -* oo for every 1 < p < oo. It is easy to see that D,,X (") (t) _ (DhX (") "t(t)), , defined by DhX(n)'i(t)
= H,
heH
satisfies DhX(n)(t)
+ Jf
=f
(
a .(X (n)(On(s)))DhX (n)(O \ (s))dw'(s)
r
0
b'(X (n) (0.(s)))DhX (n) (0.(s)) ds + J 0 Oa(X (n) (0.(s))) ha(s)ds
and hence, if we denote :
DhX (") (t) = f i t ,)(v)hfi(v)dv,
h = (hfl(t)) E H,
0
thenr (v), t
v, ft = 1, 2, f` J wn(v)nt
(10.5)
+ft
, r, satisfies
o (X (n> (cn(s))) ` (s),a(v)dwa(s) bt (X (n)
stnW M
n(s),Q(v)ds +
(n) (Y (v)),
where yin(v) = k/2"
if v E ((k - 1)/2", k/2"], k = 0, 1, .. .
By applying a slight modification of Lemma 2.1 to (10.4) and (10.5), we
have, for every I < p < oo, (10.6) and
E[ sup I X(")(s) - X(s) l'] - 0 se [0,:]
DIFFUSION PROCESSES ON MANIFOLDS
394
sup E[ sup
(10.7)
Je [y, (]
us,St
I
S (v) - :. (v) I'] -- 0
as n - co where ,, f(v), t > v > 0, satisfies fit. ,6(v) = f t o'a(X
,(v)dwa(s)
(10.8)
+f Note that (10.9)
b'(X(s))cs. (v)ds + a (X(v)). v
(v) is uniquely determined by (10.8) and is given by c,,,(v) = Y(t)Y(v)-'6.,(X(v)),
fl = 1, 2,
, r.
Indeed, it is immediately seen that ,, (v) given by (10.9) is a solution of (10.8) and the uniqueness is obvious. This proves
X(t) E (1 D, ,(Rd)
for each t > 0
1
and
(10.10)
= f o
1 jv)1 fi(v)dv,
hEH
where g,,,(v) From this, (10.3) can be concluded. Repeating this argument as in the proof of Proposition 2.1, we see easily X(t) E D (Rd). Next we shall investigate the assumption (A.2), namely the condition (10.11)
(det tr(t))-' E
(1 I
L,
where o(t) = (v"f(t)) is the Malliavin covariance of X(t, x, w). It is easy
to see that a(t) can be written as (10.12)
E f"0 Q`'(t) = a-1
where fea(r), r e GL(Rd), is defined by (2.34). Hence (10.13)
6(t)= Y(t)6(t)Y(t)*
where *
Y(t)* is the transpose of Y(t).
APPLICATIONS TO HEAT KERNELS
395
3'1(t) = , f r fva(r(s))fva(r(s))ds. a=1 0
(10.14)
We have seen in Section 2 that
for all 1 < p < co
E[II Y(t)I Ip + II Y(t)-' 11p] < oo
and hence (10.11) is equivalent to (det d(t))-' E fl Lp.
(10.15)
1
Note that f,,(r(t)), V E X(Rd), is a d-dimensional Ito process such that fv(r(t )) - fv(r(0)) = E f r fc0va, v](r(s)) - dwa(s) a.1
(10.16)
+ f' fc vo, I(r(s))ds
as we know from (2.36) of Section 2. Now we obtain a general result on Ito processes by which we can discuss the condition (10.15). First we state several simple lemmas. In are positive constants the following, c0, c1i c2, and ao, a,, a2, independent of n = 1, 2, and w E W. Lemma 10.1. Let q: W--- R be a real Wiener functional. Suppose
that c i = 0, 1, 2, 3 exist such that P[ I t7 l < n-°o] S c1 exp[-c2n`3], n = 1, 2,
(10.17)
Then
.
for all p > 1.
E[ I >7 -p] < oo
The proof is obvious and is ommitted. Let 5Fd be the totality of dx d symmetric, non-negative definite matrices. For g = (g11) E $#d and 1= (1') E Sd-1 = { x E Rd; I x j _ 1 }, we set
= MIgII
d
d
{i,J-1 Z(g`')2}I"2
and
g(1) _ Eg`11`11. i,/-1
We write g1 > g2, g1, g2 E Yd if g, - g2 is non-negative definite. Lemma 10.2. Let >7 and 4 be .Sod-valued Wiener functionals such that >7 ? rl a.s. Suppose that c, , i = 0, 1, 2, 3, 4 exist such that Ir/II < co a.a.w. and
DIFFUSION PROCESSES ON MANIFOLDS
396
sup P[il(1) < n-01] < c2 exp[- c3nc4], for n = 1, 2,
(10.18)
lcSd-1
Then
for all
E[(det r/)-p] < oo
Proof. Let W
1 < p < oo.
w; q > it and 11111 < c, 1. Then P(W) = 1. Set
W.(1) = { w e W; #(I) > n-I }
for I E Sd-' and n = 1, 2,
.
Then (10.18) implies that P[ W (1)c] < c2 exp[- c3n`4]. If w E W,
Ij(1) -i1(1')I c 2coI1-1'I
for all
Clearly there exist It E Sd-1, i = 1, 2,
1, 1' E Sd-1
, m such that
an
U {I E Sd-1; I I - lk I < 11(4con°1) } =
Sd-1
k-1
m
where m < can cd-11 °1 for some cs. TL l lien, w e (1
implies that
k-1
inf #(I) > 1/(2n`1) 111-1
and hence det
1/(2n°1)d/2. Since d
det r1 > det r/
if w E W, we have
an
P[det ri < l/(2n°1)T] 5 P[{ (1 Wf(Ik)}°] k-1
< csn
By Lemma 10.1, we have E[(det rl)-D] < oo
for all 1
which completes the proof. In the following, we consider the case when = (11J) E .7d is given by f1 Er
`' =
0 a-1
fa(s)ff(s)ds
and s - fa(s) (=- Rd, a = 1, 2, . , r, are {Rt }-adapted continuous processes where { .fit } is the proper reference family on W for the rdimensional Wiener process w(t) = (w"(t)) canonically realized on
APPLICATIONS TO HEAT KERNELS
397
(W, P). By Lemma 10.2, we can easily conclude the following: Lemma 10.3. Suppose that there exist constants c1, i = 0, 1, 2, 3, 4
and {0, }-stopping times 0 < a1 < a2 < I such that I fa(s) I < co for s E (a1, a21, a = 1, 2, , r a.s. and
sup P[z
(10.19)
f12
a-1 al
IeSd-1
[1
fa(S)]2dS < n-cl] s c2 exp[- c3n`4],
for every n = 1, 2,
,
where d
i -rats) = E 1fa(S). f=1
Then
for all 1 < p < co.
E[(det ri)-P] < oo
Proof is immediate if we notice 11(1)
= a=1 E
f1
0
[1
fa(s)]2ds
and choose
`' = E fa2fa(s)I' (s)ds, a=1 al
i,j=1,2,...,d,
in Lemma 10.2.
As usual, we call a continuous (-q,)-adapted process 4(t): [0, oo) . . , r, if
R an Ito process with the characteristic ta(t), a = 0, 1,
m) = S(0) +a=1 E f'0 Sa(S)th x(s) + f' y0(s)ds 0
where ha(s), a = 0, 1, such that
, r, are
(
.}-adapted measurable processes
r
f o I Us) I Zds +
Lo(s) I ds < oo
for every t > 0, a.s.
fo
We are now in position to state a key result for Ito processes which plays a fundamental role in obtaining sufficient conditions for (10.15). This result is first obtained in a weaker form by Malliavin [107] and then improved by ideas of Kusuoka and Stroock (cf. [211] and [212]).
DIFFUSION PROCESSES ON MANIFOLDS
398
Lemma 10.4. (Key lemma). Let i(t) be an Ito process with char, r. Assume furthermore, that ko(t) is acteristics fa(t), a = 0, 1, also an Ito process with characteristics to, a(t), a = 0, 1, - , r. Suppose that there exist c1, i = 0, 1, 2, 3 and {.q, }-stopping times 0 < a1 < az
< 1 such that (i) for almost all w (P) r I
r
(S) I +E I a(S) 12 + 14o(S) I + E 1 4o. Q(S) I -< Co
for all s e [al, az] and (ii) for every n = 1, 2, a1 <
]
< cl exp [- c2n°3].
n
Then for every given c4, there exist c1, i = 5, 6, 7, 8 (which depend only on c0, c1, c2, c3 and c4) such that r
o2A(ol+I/n)
(10.20)
P[f o1
o2A(o1+1/n)
14(s) I gds S l/n°5, E fof
14a(s) I gds
1/na4] S c6 exp [- c,n°$]
for every n = 1, 2, Here we present a proof of this lemma along the lines of [203] and [232]. It should be remarked that a much simplified different proof was given recently by Norris [222]. First we give a series of probabilistic lemmas.
Lemma 10.5. Let K > 0 and X(t) be a one-dimensional continuous semimartingale
X(t) = X(0) + m(t) + A(t) such that <m>(t) = f o a(s)ds, A(t) = f /3(s)ds and I a(s) I < K, I a(s) o
< K. Then
P(aa < A) < 4 exp [-
a2/(8K2)]
for all
a> 0
and
A E (0, a/2K]
APPLICATIONS TO HEAT KERNELS
399
where o, = inf { t; I X(t) - X(0) I > a). Proof.* By Theorem 11-7.2', there exists a one-dimensional Brow-
nian motion b(t) (b(0) = 0) such that X(t) - X(O) = b(<m>(t)) + A(t). Since { I X(t) - X(0) I > a } c { I b(<m>(t)) I > a12) U { I A(t) I > a/2 }, we have Qa > (a/2K) A (aa12/K) where 0,12 = inf { t ; I b(t) I > a12). Hence, if 0 < d < a12K
P(oa<1)
<
e- - 2 / 1 , -dx < J a/2
<
(K). exp[- a2/(8K).)})
exp[- a2/(MA)]
This completes the proof. Before stating the next lemma, we shall prepare some notation. For a finite interval I and a square-integrable function a(s) defined on I, we set
o,(a) = I j f r [a(s)2 - [22]ds I
where
a=
r f a(s)ds. I
It is easy to see that o1(a1 + a2) < ol(al) + o,(az)
Lemma 10.6. Let b = (b(t)) be a one-dimensional Brownian motion.
Then for every a > 0 and E > 0, (10.21)
P(oro a,(b) < 8)
exp - 2,82 a ].
Proof. Without loss of generality, we may assume that b(0) = 0. We use the following well known expansion formula of b(t) in Fourier series: * The proof was essentially given in Theoren IV-2.1. We shall repeat it however because of its importance.
DIFFUSION PROCESSES ON MANIFOLDS
400
= to +
[b(t)
At
(cos 27rkt - 1) 2rrk
+ ilk
sin 2irkt 1
2nk
J
where { Sk, r/k } are independent random variables with common distri-
bution N(0, 1) ([75], [136]).* Then
b(t) -j b(s)ds = (t - 1/2) 0 + .// T t [4k cos 2i kt + Ilk sin22iv t 27rk ] and since { cos22rkt } and { (t - 1/2), sin 27rkt } are mutually orthogonal in X2([0, 1]), we have 62
= Qio, ii(b)
k-i kl(2nk)2.
Hence for z > 0, E(e 12o2) < E(exp{ - 2z2 E 2 k/(2nk)2 }) k-1
J1 J E(exp{
-
Z142
k/2 r2k2 }) = 11 (1
+z2/iv2k2)-112
= v z/sinh z< y/2e-S 12. Consequently
P(cr < e) < exp[2z282] E(exp[- 2z2Q2]) < /-f exp[2z282 - z/4]
and by setting z = 1/1682 we have (10.22)
P(or < s) < /T
exp[_
L
21 2 a2
Since { )_b(at)}. { b(t) }, we have
13(b)
°3(b). Then(10.21)
follows from (10.22)
Lemma 10.7. Let b(t) be a one-dimensional Brownian motion on [0, A] where A is a positive constant. Then, for every 0 < y < 1/2, there exist positive constants a1i a2, a3 such that (10.23)
P[ sup L,SE(O. A] 1$S
I b(t)
- b()
I t - sly
I
> n] < alexp[- a2n°3]
* N(0, 1) stands for the normal distribution with mean 0 and variance 1.
APPLICATIONS TO HEAT KERNELS
for every n = 1, 2,
401
-
- -
.
Proof. First we note (10.24)
P[ sup
t,se[O,A] t*s
I b(t) - b(s) I < oo] = 1 It - SIY
for every 0 < y < 1/2.
Indeed it is proved in the proof of Corollary to Theorem 1-4.3 that
P[ sup IX(t)-X(s)I <00]=1 r,se[O,A]
t*s
It-SI
for any continuous stochastic process X(t), t e [0, A], satisfying
E[IX(t)-X(s)Ia)<MIt-sI'+B,
t,sE[0,A],
where a, /3, M are positive constants and a is any constant such that 0 < a < fl/a. Since
E[Ib(t)-b(s)I2m]=(2m- 1)(2m-3) ... 3 x 1 x It-slm,
m=1,2,-..7 (10.24) is now easily concluded. For one-dimensional Wiener space (Wo , P) with time interval restricted to [0, A], set
Wor={wE W0;IIwIIY
05sSA
I w(s)I +05s
Then Wo r is a Banach space which is however not separable and, if 0 < y < 1/2, (10.24) implies that P(Wo r) = 1. If WY is the closure of the Cameron-Martin Hilbert space H with respect to 11 I I ,-norm, then WY is a separable Banach space with respect to II IlY norm and it is
easy to see that Woi r, C WY
if 0 < y < y' < 1/2.
In particular, P(WY) = 1 for every 0 < y < 1/2. Then applying Fernique's theorem to the separable Banach space B = W. for 0 < y < 1/2,
DIFFUSION PROCESSES ON MANIFOLDS
402
we have
for some e > 0.
K.: = E[exp{sllwllf }] < -_ Hence
- b(s) I > n] P[ sup Ib(t) It - SI' tSA 0!9$
S P[Ilwllr > n] < Keexp[- en2],
n = 1, 2, -
-
For completeness, we give a proof of Fernique's theorem following Kuo [210).
Fernique's theorem: If B is a separable real Banach space with norm
II and u is a mean zero Gaussian measure (i.e. { 1(x), 1 E B' } c L2(B,,u) and it is a mean zero Gaussian system with respect to u). II
Then s > 0 exists such that Ja
exp[ellxll2]k(dx) < 00.
Proof. Consider the product probability space (B x
x+Y
x-y
and
/ 2
(D ,u). Then
for (x,y)EBxB
are B-valued random variables over this probability space such that they are mutually independent and both of them are p-distributed.
Hence, for t < s < 0, p(IIxII <_ s),u(IIxII > t)
O,u(Ilx-YII <s)pO,u(IIx+YII >t)
u
2
>t) =a® (Ilx - YII <sand IIx+YII ,./2 /
,./2t)
S
> %7
P(IIxII
= u(IIxII >
t
2s
)Z
and Ilyll > t
s1
APPLICATIONS TO HEAT KERNELS
For given e > 0, define t,,, n = 0, 1, ,/-f tn. Then ( / .L )n+1 - 1
to = 11/ 2 - 1
-
403
by to = s and ti}1 = s +
s = ((- 7)"' - 1) (-,I2 + 1)s.
Set
an = p(Ilxll > tn)lp(I xll S S). Then an+1 = p(Ilxll > s + %/ 2 tn)/u(Ilxll < s)
= iz(Ilxll > S + ./ 2 tn)AI xI i < s)/(u(llxll < S))2
and applying the above obtained inequality to the numerator, we obtain an+1 C u(llxll > tn)2/(.u(Ilxll C s))2 = an.
Hence
an S am = exp[2n log ao], that is n+1
4a(IIxii > (2
2
- l)(21 + 1)s) < iz(llxll < s)exp[2° log ao], I
n=1,2, .
Choose s so large that p(Ilxll < s) > 0 and ao =,u(llxll > s)/,u(llxii < s) < 1. Then the estimate obtained above can be written in the form p(llxll > u) < b exp[- au2]
for all u > c
with some positive constants a, b, c and the Fernique theorem follows
at once from this. Let ra(t), t E [0, 1], be an Ito process with bounded characteristic 'ia(s). By Theorem 11-7.2', we can find a Brownian motion b(t) such that rt(t) = ;,7(0) + b(a(t)) + f o ?7o(s)ds
404
DIFFUSION PROCESSES ON MANIFOLDS
where
a(t) _ a-1
f
r 0
17a(s)2ds.
Hence Lemma 10.7 implies immediately the following: Lemma 10.8. Let 17(t), t E [0, 1], be an Ito process with bounded characteristics.
Then c,, i = 0, 1, 2 exist such that (10.25)
I17(t) - rll 3)
P[ sup
os:
I
t-sI
> n]
co exp[- C1n`2]
forn=1,2, Finally we need the following real variable lemma due to Kusuoka and Stroock.
Lemma 10.9. Let f(t) be a continuous function on [a, b] and set g(t) = g(a) + f ' f(s)ds,
t c= [a, b].
O
If a
sup Sb
I f(t) -f(s)I It-SI1/3
n
and
f.' I f(s) gds > s2 for some s > 0 such that s3 < 22K3(b - a)512, then
(10.26)
(b - a)a2Ia.b](g)
2-133-IK-9611(b - a)-11/2.
Proof is easily provided if we note that, first to E [a, b] exists such
that
If(t0)I
>s(b-a)-1/2
and then an interval I, I e [a, b], exists with length s3K-32-3(b - a)`3/2 on which f is of constant sign and I f I > s2-1(b - a)-1/2. We now note
that there exists t1 E I such that
APPLICATIONS TO HEAT KERNELS
g(t)) _ (gl,), (Sl r) = 11-1
405
g(t)dt. J
Then
(b - a)a2ta.b)(g) ? f (g(s) - g)2ds
f, { f
r'
f (s)ds } 2dt
z 82 4-1(b - a)-'f (t - t,)2dt > (48)-'s2(b - a)-)III 3 which completes the proof. Proof of the key lemma. In proving (10.20), we may clearly assume are positive constants independent of n and w. We know(Theorem 11-7.2') that a one-
n ? 2 and co > 1. In the following a, and d1, i = 1, 2,
dimensional Brownian motion B(t) with B(O) = 0 exists such that
F,(t) = 4(o) + B(A(t)) + g(t), a, < t < 1 where
= A(t)
5f a-1
d
I ba(s) I gds
and
g(t) = f'al
4o(s)ds.
Set zn(o,+t/n)
W,
a=0
I 4a(t)12dt ;-> 1/nc4]
f',,'
and
w2 = [ f
I fi(t) 12dt < 1/n°5].
Here c, is given and c5 will be determinded later. Set o2A(a,+1/n)
Wi,
[f
o,
I o(t)12dt > 1/(2n`4), A(a2 A (a, + 1/n)) < 1/n-1]
and
W1.z = [A(az A (o + 1/n)) > 1/nal].
DIFFUSION PROCESSES ON MANIFOLDS
406
Then W1 c W1.1 U W1, 2
(10.27)
if a,
c, + 1
since
l/nal < 1/n`4+1 < 1/(2n°4)
for n ? 2.
Set
W3 = [as - a, ? l 1n], W4 = [
SUP
°15
10(t) _ 40(3)1 < n] I t- S I 1/3
and
W5 = [ sup I B(t )1 < 1 /n°2]. 0SK1/na1
By Lemma 10.9, we can find a3 such that W3 n W4 n [f I 0(t) 12dt >_ 1/(2W4)] In
(10.28)
c W3 n [a n(g) > 1/n°3]
where In = [a,, a, + 1/n]. Hence W3 n [0,2 (g) < 1/n°3, f I do(t) I Zdt >_ 1/(2n`4)] !n
(10.29)
c w3 n w4. We have w1,1 n w5 n W3 c [sup I B(A(t)) I < 1/n°2] n w3 tEln
C [an(B(A( ))) < 1/n2a2] n w3 c [a;,(B(A( ))) < 1/(4n°3)] n W3
if a2 > a3/2 + 1. Also
APPLICATIONS TO HEAT KERNELS
407
W2 n w3 c [fin 14(t) I gds/ I In I < l/n°s-'] n W3 C [ai,(k) < 1/n`5-1] n W3 C [o n W3
if c5 > a3+3.
Since
a,n(g) :!5: a,,,(B(A(
))) +
if a2 > a3/2+1 and c5 ? a3 + 3, w), n w2 n w3 n w3 (10.30)
C [Q n(g) <
1/na3] n W,,, n W3
CW; by (10.29). Hence, if a, > c, + 1, a2 > a3/2 + 1 and c3 > a3 + 3,
W,.,nW2CWWUW UWs. By Lemma 10.5, (10.31)
P[WS] < d,exp[- denal-2a2]
if a, > 2a2.
Hence we can conclude, by the assumption (ii) in Lemma 10.4, Lemma
10.8 and (10.31), that if c5 > a3 + 3 and a, > (a3 + 2) V (c4 + 1), (10.32)
P[W11 , n W2] < d3exp[- d,n,s].
Next we estimate P(W12 2 n W,). The idea is to use the fact a,(B(A( ))) is comparatively larger than a,(g) if the interval I is sufficiently small. If w E W3i I. C [6,, v2]. We divide I. into nm equal intervals
In.k = [Q) + k/n'+m, a, + (k + 1)/n' +m], k = 0, 1, ... , n^'- 1
where we choose m e N such that (10.33)
m > a,.
Then if w e W3 $In k I k(t) 12dt = $ Ink 14(al) + B(A(t)) + g(t) 12dt
> f I k(o) + J A
B(s) + g(A-)(s)) 12dA-)(s)
DIFFUSION PROCESSES ON MANIFOLDS
408
>
1
14(c'1) + B(s) + g(A-'(s)) I Zds.
A (Jn.k)
CO
Here we note that E I ha(s) I
co
a-1
by the assumption (i) in Lemma 10.4 and we set A(II, k) = { A(t); t E In. k }. Set
Jk = [A(6, + k/n'+m), A(o1 + k/n'+m) + 1/na,+m],
Then, setting g(s) = g(A-'(s)), W3 fl
{
I A(I,,, k) I ? 1/na'+m }
= W3 f l { A(IJ. k)
c W3 n {
J
Ink
Jk }
I fi(t) I Zdt >_
c W3 fl if
if co
I t(t) I Zdt
I
I fi(t) I Zdt >
I
Jn.k
I R(ay) + B(s) + g(s) I Zds }
g(' )) }
Co
Jn,k
c W3 n { f
Jk
I
co
[7'1(B) - oJk(g)]Z }
Since
g(s) = J r 40(u)du
and
140(u) I
<_ co on [a,, Qz],
we have
I g(t) - g(s) I < co I A-' (t) - A-' (s) I .
Therefore, setting t0 = A(o, + k/n'+m),
(7 40)
I Jk
S coZ
I
IJkI
f
f
J
I g(t) - g(t0) 12dt
k
Jk
(A-1(t) - A-1(t0))2dt
APPLICATIONS TO HEAT KERNELS
co[A-'(A(a, + (k +
409
1)/nl+m))- A-'(A(6, + k/nl+m))]2
C2/n2+2m 0
W3 n { I A(In, k) I > 1Ina1+m } and hence Jk C A(In, k). Therefore
if w
W3 n [O',/k(B) > 2Cp/nl+m, I A(I4, k) I > 1 /nal+m }
C W3 n [f
14(S) 12dS s
2
J
I CO
nC+m
I
n2+3m+a1
'n. k
Set
W6 =nm-, n0
{
6,k (B) > 2cp/n]+m }.
Since nm-1
A(C2 A (Q1 + 1/n)) = A(vl + 1/n) _k-0 Z IA(I,,.k)I > 1/na' on W3 n W1.2,
there exists k such that
I A(I,,, k) I
>_ 1/nal+m Hence
w1, 2 n w3 n W6 n,a-1
CU k-0
{
I A(In, k) I
1/na'+m' alk(B) > 2Cp/nl+m } n W3
nm-1
Cp/Y12+3m+a' } n W3
U .k I b(S) 12dS C k0 ln ('o2h >i +l ln)
C{
J
I 4(S) I
2
dS > 1/n 2+3m+4 1 }
a1
since we assumed co >_ I. Therefore if cs ? 2 + 3m + a,,
w1.2 n w2 c w3 u W6 and by Lemma 10.6 (note that A(u, + k/21+m) is a stopping time for B(t)), nn+-1
P[W6] S E P[v,k(B) < 2Cp/n'+ml k-0
nm-1
S E d6exp[- d, I Jk I k-0
d8exp[- d9nm+2-al]
1
(1/n1+m)-2]
DIFFUSION PROCESSES ON MANIFOLDS
410
d8exp[- d,n2]
because of (10.33). Noting the assumption (ii) in Lemma 10.4, we have thus obtained (10.34)
P[W12 2 n W2)
- d10exp[-
Therefore, if c5 is chosen sufficiently large, (10.32) and (10.34) imply the
estimate (10.20) because W, n w2 c (W,,, n W2) U (W,, 2 n w2). This completes the proof of Lemma 10.4.
Remark 10.1. In the key lemma, the variable n E N can be replaced obviously by a continuous variable T E [1, oo).
Remark 10.2. By examining the above proof carefully, we can deduce the following: Let 0 E (0, 1] and we assume in Lemma 10.4 that 4(t), ?;a(t), u, and a2 may depend on the parameter 0 but they satisfy the same assumptions (i) and (ii) where the constants CO, c1, c2, c3 are in-
dependent of 0. Then, for any given c, there exist c,, i = 5, 6, 7, 8, 9, 10 (which depend only on c0, c1, c2i C3, c4 and hence are independent of 0,
in particular) such that P[ d'2A(71+1I
J
a,
r
('o2h(a,i 1/n)
/, E J
(s) I2ds S o n°5
S c6nc7exp[- c80`9n`10]
a-O o,
2
I Za(s) I ds
for all n = 1 , 2,
01n1.]
and 0
(0, 1].
Now we return to the stochastic differential equation (10.1). We inX(M),
troduce the following notation: for V
(Va,V)=[Va,V]
a = 1, 2,
,r
(10.35)
(Vo, V) = [V0, V] + 2
[Va, [Va, V]].
Then, by (10.16), we have 1 - fv(r(t)) - I -.f,,(r(0)) r
(10.36)
= a=1 E J r0 1
- f[ va, 17(r(s)) o dwa(s) +
= E J o I- A',', v)(r(s))dwa(s) + f
f 0
r o
I - .f v0. 3(r(s))ds
1-.fcvo, )(r(s))ds
APPLICATIONS TO HEAT KERNELS
411
for every I E Sd-1 and V E 3; (Rd). For each n = 0, 1, set E. e X (Rd) successively by
,
define a
E,o={V1, V2, ...,V,} (10.37)
n={(Va, V);
,r} for n
1, 2,
and set
±n = Eo U E1 U ... U F,.,
(10.38)
for n = 0, 1,
.
Now we introduce the following hypoellipticity condition of Hormander
type for the vector fields Va, a = 0, 1,
, r.
, r satisfy the assumpAd E £M Ad(x) are linearly independent.
Definition 10.1. We say that Va, a = 0, 1,
-
tion (H) at x e Rd if there exists M E N and A1, A2, such that A1(x), A2(x), -
- -
,
Cleary (H) is satisfied at x if and only if there exists M E N such
that
inf E [1 - A(x)]2 > 0.
(10.39)
leSd-1 AeZM
If (10.39) is satisfied, then we can find S > 0 and bounded neighborhood U(x) in Rd and U(I) in GL(d, R) of x and I = (8;) respectively such that
inf E [1
(10.40)
f4(x)]2 >- 8
if r E U(x) X U(I).
leSd-1 Ae±M
Set a = inf{s; r(s) U(x)x U(I)} where r(s) is the solution of (10.1) and (10.2) put together. Set a, __ 0 and a2 = a A 1. Then a, and a2 satisfy the condition (ii) of Lemma 10.4 by Lemma 10.5. Cleary inf
1aSd-1
r02+ (a 11-1 /n)
J a1
{ Z [I - f4(r(s))12 Ids > 8/n AaZM
on the set W. = [a2 - 91 >- 1 /n] and
alexp[- a2n], n = 1, 2,
- -
. Let N
Then for
DIFFUSION PROCESSES ON MANIFOLDS
412
every I
Sd-1, we can find ao, a1, 1
,
ak, 0 S k < M such that
l n (0'2+1/-)
(10.41)
J al
[! f Vao.
ai ... , ak(r(s))]zds ? 8/(nN)
on W.
where
al. .... ak = (Vak, (Vak-1, (... (Val, V0)) ... ).
Va0.
Noting (10.36) and applying the key lemma successively, we can easily
deduce the following: For each j = 0, 1, , k, there exist positive constant cl, j, i = 1, 2, 3, 4, independent of n = 1, 2, and I E Sd-1
such that o2n(a1+1/n)
P[fJ o
[! ' f'ao, al, ..., aj (r(s))]2ds < n-0,.1] 1
<- cz.1exp[- C3,jna4.>]
for all n = 1, 2,
.
In particular, we can conclude that positive constants c,, i = 1, 2, 3, 4 exist independent of n such that
(10.42)
sup P[r az [I. f,(r(S))]2ds < n--l] a-1 J al
Ie Sd-1
S czexp[- cina4]
for every n = 1, 2,
By Lemma 10.3 and (10.14), we obtain 11(det 6(l))-11I, < oo
(10.43)
for all 1
Now introduce a parameter 1 > e > 0 and consider the following stochastic differential equations r
dX(t) = EE Va(X(t)) o dwa(t) + e2VO(X(t))dt a-i X(O) = x
(10.1)6
and
dY(t) = 81,EaVa(X(t))Y(t) e dwa(t) + ezaV0(X(t))dt (10.2)6
a-1
{(
Y(0) = I.
Denote the solution by r6(t) = (X8(t), Y6(t)). It is immediately seen that
APPLICATIONS TO HEAT KERNELS
413
( r(e2t), t ? 0 ) is equivalent in law to (re(t), t ? 0 }. In particular, r(e2) is equivalent in law to re(1) and 3e(1) defined similarly for re(t) is equivalent in law to 6(82). Suppose that ( Va, a = 0, 1, , r) satisfy the condition (H) at x. We can give the same arguments as above for the vector field Vq, e E_ (0, 1], where Va = SVa, a = 1, 2, - , r and Vg = &V0. -
In this case, however, b > 0 above is replaced by 8ek for some k e N and 6 > 0 which is independent of e e (0, 1] and we apply the key lemma in a generalized form as stated in Remark 10.2. Finally we obtain, instead of (10.42), su P [E a=i
tnd-1
Z [1 f vs (re(s))]2ds < ekn-ai]
for n = 1, 2,
and e E (0,1] where c1, i = 1, 2, 3, 4, 5, 6 are independent of n and e. Here all and a2 are also defined in the same way for
the process r8(t). From this we can deduce, as above, the following estimate:
P[det 38(1) S
T]
< a,Tazexp[- a3ea4Ta5]
for all T E [1, oo) and e E (0, 1] where a1, i = 1, 2, 3, 4, 5 are positive constants independent of e and T. We can now conclude that (10.44)
II(det de(1))-hII = II(det 3(e2))-'Il, <
K1(p)e-K2
for all pE(1,oo)and eE=(0,1] where K1(p) > 0 may depend on p but not on a and K2 > 0 does not depend on p and e. Thus we obtain the following result due to Kusuoka and Stroock. Theorem 10.2. Let V0, V,, , V. satisfy the assumption (H) at x Er Rd. Then, for every t > 0, X(t, x, w) E= D_(Rd) and it satisfies (A. 2): More precisely, there exists a positive constant K2 and for every
I < p < oo, there exists a positive constant K1(p) such that (10.45)
II(det a(t))-'IIp <
KI(p)t-K2
for all t E_ (0, 1] and 1 < p < oo. Remark 10.3. If (H) is satisfied everywhere in a domain D c Rd, the
DIFFUSION PROCESSES ON MANIFOLDS
414
estimate (10.44) holds uniformly in x E D for every ID clear from the above proof.
D. This is
Thus if (H) is satisfied at x E Rd, then for any Schwartz distribution
T E 7 '(Rd), T(X(t, x, w)) E D_ is defined for every t > 0. In particular, 8y(X(t, x, w)) is defined for every y E Rd and p(t, x, y) = E[8y(X(t, x, w))]
which is C" in y, coincides with the fundamental solution of 8u = Au
at where
r
A= -E Va,+V0 2 a_1
u(t, x)
= fRd p(t, x, y)f(y)dy
solves the initial value problem 8u = Au,
at
ur,_0 =f
uniquely among tempered (i.e. of polynomial growth) solutions for a tempered function f on Rd. More generally, for g(w) E b.,, E[g(w)by(X(t, x, w))]
is well-defined. In particular, if C(x) is a tempered C°-function such that
C(x) < K for some constant K > 0, then g(w) = exp[ f ' C(X(s, x, w))ds] E D_ and
E[g(w)oy(X(t, x, w))] = p°(t, x, y) * The vector field V. is regarded as a differential operator.
APPLICATIONS TO HEAT KERNELS
415
is the fundamental solution of au = (A + C)u.
at
Now assume that V., a = 0, 1, -
-
, r, are bounded on Rd and
satisfy (H) at x E Rd. Let O(f;) be a C--function on Rd such that O(t;) = 1
if 141 < 1/3 and 0(4) = 0 if I I > 2/3. Let y E Rd and define
u(z)_
10(
(_z-y 1X-Y1 ) 1
Then
if
x*y
if
x = y.
Way = ay and hence
p(t, x, y) = E[ay(X(t, x, w))] = E[V(X(t, x, w))by(X(t, x, w))].
By the integration by parts as discussed in Section 9, this is easily seen
to be a finite sum E[a,(X(t, x, w))F,(w)]
where a,(x) is of the form of a finite sum bi(x)DO#V(x)
with bounded continuous functions b,(x) and F,(w) E D. is a polynomial in the components of X(t, x, w), their derivatives and [det a(t)1-1. By Lemma 10.5, we can deduce that P[ I X(t, x, w) - x I
>-
I x - y I /3] < axeXp[- a2 I x - Y 12/t ]
where a, and a2 are positive constants independent of x, y E Rd and t E [0, T]. Combining this with (10.44), we obtain the following estimate:
Theorem 10:3. Suppose that Va, a = 0, 1,
, r, are bounded
and satisfy (H) everywhere in a domain D of Rd. Then, v > 0 exists and
for every compact set K C D and T > 0, positive constans c, and c2 exist such that (10.46)
p(t, x, y) S t° exp[- c2I x - y 12/t] for all
t E (0, T], x E K and y c- Rd.
DIFFUSION PROCESSES ON MANIFOLDS
416
Next, we shall study the short time asymptotics of the fundamental
solution p(t, x, y). For this we introduce, as before, a parameter e E (0, 1] and consider the equations (10.1)` and (10.2)e. We denote the solutions by Xe(t, x, w) and Ye(t, x, w), sometimes more simply by XX(t)
and P(t), respectively. Then we have (10.46)
x, y e Rd.
p(e2, x, y) = E[by(Xe(1, x, w))],
We introduce the following notation: For a = (a1, a2, {0, 1,2, ,r}m,m= 1, 2, ... , we set
am) E
hail=m+#{v; Also, let Sa(t, w) = f0 o dwa1(t1) (`' ° dwa2(t2)
frm-1 0
0
o
dwrm(tm)
be a multiple stochastic integral in the Stratonovich sense for a = (a,, a2,
,
am) E {0, 1, 2,
, r j- where we set w°(t) = t.
Theorem 10.4. Let x E Rd be fixed. Then X8(1, x, w) has the asymptotic expansion (10.47)
(10.48)
d as a J. 0 in D-(R)
X8(1, x, w) tifo + efi + e2f2 +
and f E D-(Rd), n = 0, 1,
D. (Rd)
, are given by
f0=x
and (10.49)
f = E Vam . Vam-1 ... Va2(Va)(x)Sa(1, w), a;Iag-
n=1,2, .
Here the vector field Va, regarded as a differential operator, is denoted by Ya, i.e.
Va(f)(x) = VV(x)ax, 2f (x),
.f (Z C-(Rd -* R)
and
VV(y) = (V«(y)) E C-(Rd -> Rd),
a = 0, 1, ... , r.
APPLICATIONS TO HEAT KERNELS
417
In particular, fl(w) _ E Va(x)wa(1).
(10.50)
ac,
The expression in (10.47) is uniform in x e K for any bounded set K in
Rd.
Proof. The proof is easily provided by successive applications of the Ito formula:
X8(1, x, w) - x
=8
a-i fo
Va(X8(S)) o dw"
0
aE Va(x)wa(1) + E
+ s'
o
J0
[Va(Xo(s)) - Va(x)] ° dWa
Vo(X8(s))ds
Ef + 82[i &1-
+f
+ elVo(X8(s))ds fl
a
Va2(Vai)(XE(u)) o dWa2(u)} o dwai(S)
J 0{ J 0
Vo(X 8(s))ds + E
f o{ f o
o r
Vo(Va)(X 8(u))du } o dwa(s)]
r
= efi + e2[Vo(x) + a, =i a2-1 E + 8 [ aE1 a2E f 't
f'
Va2(Va,)(x)Sca,.a2)(1)]
E(u))
Va2(Va,)(x)) ° dwa2(u) }
o dwa'(s) + f [Vo(X8(s)) - Vo(x)]ds o
+ ei f j f
V0(V.) (X8(u))du } o dwa(s)] 0
and so on. Continuing this process and applying Theorem 111-3.1, it is easy to see that X8(1, x, w) =fo + ef1 + ... + ,nfn + O(En+i)
in L,(R") for every p E (1, oo). Since DkX8(t, x, w) [h,, h2i
, hk]
are determined successively by stochastic differential equations as we saw in the proof of Proposition 10.1, it is easy to conclude that DkX8(1, x, w) = f0) + fl(k) + ... En ,(k) + 0(e+1)
DIFFUSION PROCESSES ON MANIFOLDS
418
in L,(H®k (3 Rd) as a J, 0. This completes the proof. X8(1, x, w) is not uniformly non-degenerate in the sense of (9.32) because f0 = x which is completely degenerate. However, if we consider
F(E,w)=
X8(l,x,w)-x,
(0, 1].
e
then we have the following: Theorem 10.5. The family F(e, w), e E (0, 1] defined above, satisfies (9.32) if and only if (Atf(x)) defined by A"(x) = E V.(x)V., (x) amt
is non-degenerate, i.e. det(A"(x)) > 0. Proof. Since in
F(e, w) - f1 + e. f2 + -2f3 + ...
where ft are given in Theorem 10.4 and the Malliavin covariance off, coincides with A(x) = (Atf(x)), A(x) must be non-degenerate in order that F(e, w) satisfies (9.32). Conversely, suppose that det(Atf(x)) > 0. Denoting by a(a) = (a"(e)) the Malliavin covariance of F(e, w), we have by Proposition 10.1 that a(e) = f 10 Set
r = inf{s; (Y8(s))-1A(XS(s))((YS(s))-1)* < A(x)/2}.
Applying Lemma 10.5, it is easy to see that P[a < 1/n] < clexp[- c2n`3],
n = 1, 2,
,
where c, i = 1, 2, 3, are positive constants independent of e E (0, 1] and n. Note also sup jldet(Y8(1))-111, < oo
86(U,1)
for all
p E (1, oo)
APPLICATIONS TO HEAT KERNELS
419
because (Y8(t))-l is the solution of the equation (2.27) in which a,', and b' are replaced by eca and e2b' respectively. Then det a(e)
(det Ye(1))2det f
IAT
(Ye(s))-1A(Xe(s))((Ye(s))-')*ds 0
2 (det Ye(1))Zdet(A(x))(1 A i)d
and we can now easily conclude that sup Ildet(a(E))-'llp < co
for all p E (1, oo),
se (o.1)
which complete the proof.
Suppose that det(A(x)) > 0. Then by Theorem 9.4, T(F(e, x, w)), T E .5(Rd), has the asymptotic expansion in f)--. Since 6.(Xe(l, x, w)) = 8X(x + eF(e, w)) = e'8o(F(E, w)), we have the following:
Theorem 10.6. Suppose that det(A(x)) > 0. Then 8,(X8(1, x, w)) has the following asymptotic expansion in b-. as a 10: (10.51)
8,(X8(1, x, w)) ^' a-d(0o + eS1s1 + 82452 + ... )
and 1 k E D__ can be obtained explicitly by Theorem 9.4: (10.52)
0o = ao(fi)
and
(10.53)
tkk(w) _
I
a
I
1
n2 ... fn'
fork= 1,2, where f. = (f9 is given by (10.49), f1 by (10.50) in particular. In (10.53), , the summation extends over all a = (a,, a2, , a,) E { 1, 2, such that n1 + n2 d}' and n = (n1i nz, , n,), n1 ? 2, 1 = 1 , 2,
+ ... +n,-1=k. Also D,,= aal aa2 ... as , = 1 if a = (a,, az,
.
ak =
xk
and a
, a).
From this we see that E[Ok(w)] = 0 if k is odd since, then, Pk(-w)
DIFFUSION PROCESSES ON MANIFOLDS
420
= - 0k(w) and the mapping: W D w -a - w E W preserves the measure P. Corollary. Suppose that det A(x) > 0 where A(x) = (Atf(x)). Then At, x, x), t > 0, x E Rd, has the asymptotic expansion (10.55)
p(t, x, x) ^' t-d12(Co(x) + c1(x)t +
and ct(x) is given by (10.56)
i = 0, 1,
ct(x) = E[O21],
where 0t is given by (10.52) and (10.53). In particular CO = { (27C)d det A(x) 1-1/2.
Remark 10.4. We can discuss by a similar method the asymptotic expansion of p(t, x, x) in the degenerate case det A(x) = 0 under some hypoellipticity condition and also the expansion of p(t, x, y), x * y, as t ,j 0, cf. [192], [212], [214], [215] and [228].
We now consider an application of our results to heat kernels on manifolds. Let M be a compact Riemannian manifold and b e X(M). Let e(t, x, y) be the fundamental solution of the heat equation at = 2 Amu + b(u)
uit-o =f where Am is the Laplce-Beltrami operator for the Riemannian metric
g on M i.e. u(t, x) = fm e(t, x, y)f(y)m(dy)
(m(dy): the Riemannian volume)
solves the above initial value problem. Let r(t, r, w) = (X(t, r, w), e(t, r, w)) be the stochastic moving frame realized on the d-dimensional Wiener space (W, P) by the solution of stochastic differential equation (4.43) with the initial value ro = r E O(M). Let S, be the Dirac 6-func-
tion at y E M, i.e. the Schwartz distribution on M such that 6Y(O) = 0(y)
for every 0 E F(M).
APPLICATIONS TO HEAT KERNELS
421
Then we can define S,,(X(t, r, w)) E D_ and e(t, x, y) can be expressed as
e(t, x, y) = E[6,.(X(t, r, w))]
where ir(r) = x. Actually, we can extend our results obtained so far to the case of manifolds: We can develop a general theory of Wiener mappings with values in manifolds and pull-back of Schwartz distributions on M by these mappings. We do not go into details on these topics, however, only referring interested readers to [230] and [190]. In order to apply our results above to the asymptotic expansion of e(t, x, x), we take two bounded coordinate neighborhoods U, and U2 of x such that U, c U2, view U2 as a part in Rd and extend the Riemannian metric (g,j(y)) and the vector field b = (b'(y)) outside UZ to a metric g' = (g,j(x)) and a vector field b' = (b"(x)) on Rd respectively, so that
g,j(y) = 8,f and b"(y) = 0 near oo. Let e'(t, x, y) be the fundamental solution of the heat equation
at = 2
d'u + Y(u)
on Rd
where d' is the Laplace-Beltrami operator for g'. Then we have (10.57)
sup I e(t, z, y) - e'(t, z, y) S c, exp[- c2/t]
y,zeU,
as t.0
for some positive constants c, and c2. For the proof, let v r=- C"(Rd) with
support contained in U, and set u(t, z) = fa, [e(t, z, y) - e'(t, z, y)]v(y)%/det g(y)dy,
z E U2.
Then u is a solution of
at = 2 Amu + b(u)
on U2
and
lim u(t, z) =0 do
on U2.
By Ito's formula applied to u(t-s, Xs) where X, = X(s, r, w) with 7L(r) _
z, we deduce that
DIFFUSION PROCESSES ON MANIFOLDS
422
u(t,z)=E[u(t-a,Xo);Q<_ t],
z E U2
where
a=inf{t;XXE8U2}. Hence
u(t, z) I <
(10.58)
sup
z E U2
I u(s, 1;) I ,
sE [o, r7, f e aU2
Noting that e(t, z, y) and e'(t, z, y) have estimates similar to (10.46), in particular, sup
Ceaa2,yG U1
I e(s, , y) - e'(s, , y) I < cl exp[- c2/s],
s E [0, 1]
we obtain from (10.58) that u(t, z) I < c1(exp[- c2t])IIvIIL,, z e U1, t E (0,1] and (10.57) follows from this. Hence, in order to obtain the asymptotic expansion of e(t, x, x) in powers of small t, we can instead take e'(t, x, x). This e' can be given by a generalized Wiener functional expectation as e'(e2, x, x) = E[bx(Xe(l, x, w)]
where X`(t, x, w) is the solution of the following stochastic differential equation on Rd: 1
dX`(t) = kE Qk(X(t))dwk(t) - e2[ Z L J,'gik(X(t))rk(X(t)) + b'(X(t))]dt X(0) = X.
Here, we denote the components of the above extended g' and b' by g,j and b`; rk is the Christoffel symbol for g,1. Also, (g`1) is the inverse of (g,a) and
is the square-root of (g`i), so that g`i _
vkvk. Then,
in the same way as above, we can obtain e(t,
x, x) = t-dJ2(co(x) + tc,(x) +
)
as
t.0
and each c,(x) can be expressed explicitly by a generalized Wiener func-
APPLICATIONS TO HEAT KERNELS
423
tional expectation. Usually, c,(x) is a very complicated polynomial in components of curvature tensor, their covariant derivatives and the vector fields b. By choosing our coordinates to be normal for the metric g, we can compute these expectations to obtain c0(x) = (27r)-l" and
cl(x) = (27r)-d/2( ZR(x)
-
-
div(b)(x) 2
IIbii2(x)) 2
where R(x) = R''(x) is the scalar curvature. However the details are left to the reader. For related topics, cf. [217] and [231].
Finally we apply our method to prove the Gauss-Bonnet-Chern theorem. Let M be a compact oriented manifold of even dimension d
= 21. As Section 4, let A,(M), p = 0, 1,
,
d be the spaces of
differential p-forms and
A(M) _
O+ A,(M). P-0
Consider the heat equation on A(M)
(5.4)
uit-o = a where
= - (d + 8)(d + S) = - (d8 + 8d) is the Laplacian of de
Rham-Kodaira. In Section 5, we solved this initial value problem by a probabilistic way in the form (5.32) where U(t, r) is the scalarization of u(t, x). For given r = (x, e) E O(M), there is a canonical isomorphism r`: i1Rd - ATx (M)
for each
p = 0, 1,
and hence
r: ARd = E ARd -),AT*(M) = P-0
E
AT,*(M),
P-0
defined by
r(Sal A da2 A ... A
aap) =P l Alan A ... fay
,
d,
DIFFUSION PROCESSES ON MANIFOLDS
424
where (fl, f"
, f d } is the ONB in T,*(M) which is dual to the
in TT(M) and ba is defined by ba = (0, 0, 1:'0' , 0), (cf. Section 5). Using this isomorphism, the scalarization F. E C°(O(M) -+ AR d) of a E A(M) can be written as ONB e = {e,, e2,
, ed }
a
Fa(r) = F -' (a(x)),
r = (x, e) E O(M).
Then by (5.32), the solution u(t, x) of (5.4) is expressed as (10.59)
u(t, r) = FE[M(t, r, w) F(t, r, w)-'a(X(t, r, w))]
where r(t, r, w) _ (X(t, r, w), e(t, r, w)) is the stochastic moving frame, i.e. the solution of (5.5). Note that the right-hand side of (10.59) is independent of a particular choice of a frame r = (x, e) E O(M) over x: This just corresponds to the fact that U(t, r) = F-'u(t, x) is 0(d)-equivariant, (cf. Section 5). Hence
u(t, x) = fM e(t, x, y)a(y)m(dy) and the fundamental solution e(t, x, y) E Hom(AT*(M), AT,*(M))
can be expressed by a generalized Wiener functional expectation as (10.60)
e(t, x, y) = E[FM(t, r, w)F(t, r, w)-'b,,(X(t, r, w))].
We now introduce the decomposition A(M) = A+(M) OO A_(M) where
A+(M) =p:even E Q Ap(M) and A_(M) _
QQ A,(M). p:odd
Define a linear operator (- 1)F on A(M) by
(- 1)Fw = 1-0)w
if w E A+(M) if co E A_(M).
Then the operator Q = (d +b) satisfies
Q(- 1)F = - (- 1)FQ,
APPLICATIONS TO HEAT KERNELS
i.e.,
425
{ (- 1)", Q) = (- 1)FQ + Q(- 1)F = 0. In particular, Q sends
A+(M) into A., (M). For .l >- 0, let H,i = {co E A(M); w + )co = 0)
and Hx = { w E A±(M); w + tw = 0 }. Then H,. = Hx Q Hi- and 0< dim Hx < co (cf. [145]). We have (10.61)
dim Ht = dim Hi"
if A > 0,
because the mapping Q: Hx a Hx is an isomorphism. Indeed, if w E HZ, then
Qw = - Q'w =
-.LQw
showing that Qw E Hi-. If co e HZ and Qw = 0, then co = (Q2w)/) = 0 showing that the mapping Q : Hx - Hr is one to one. Finally if w E Hj , then co = Q[(Qco)/.1] and (Qw)/.1 E Hx showing that the mapping Q: Ht -* Hi- is onto. In the same way, we define the decomposition ATx (M) = A+TX (M) Q+ A-T,*(M),
X E M,
and
AR" = A, R" (D A-Rd.
Then (- I)F E End(AT,*(M)) and (- 1)F E End(ARd) are defined similary as above. For A E End(AT,*(M)) or A E End(ARd), define the supertrace Str(A) of A by Str(A) = Tr((- 1)FA). If A E End(AT, (M)) leaves A,T,*(M) invariant, then clearly Str(A) = Tr(A 14+(T"(M)) - Tr(A I .t_(T. (M) )-
Now we claim that (10.62)
fm Str[e(t, x, x)]m(dx) = X(M),
t>0
X(M) being the Euler characteristic of M, (cf. [145]). Indeed, by the eigenfunction expansion of e(t, x, y), e(t, x, y) = E e-Ant n-0
E
c(x) 0 9S(y)
c5:ON8 in HA n
DIFFUSION PROCESSES ON MANIFOLDS
426
for some 0 S 20 < .1, < .. and hence Str[e(t, x, x)]
(10.63)
= -0 E e xnt where II
E #:ONB 3n HZn
II0(x)II2 --0 E
e_xn,
E
110(x)1 2
s:ONB in Hxn
is the Riemannian norm of AT*(M). Hence
II
f Str[e(t, x, x)]m(dx) = E e-xn(dim Hx; - dim Hxn)
-0
M
and, noting (10.61), this is equal to a
dim Ho - dim Ho = E (- 1)pdim HH(M) P-0
co E Ap(M); fjco = 0) is the space of harmonic p-
where HH(M) forms. Since
dim HH(M) = p-th Betti number of M by the de Rham theorem, (cf. [145]) (10.62) is proved. Hence if we can show that (10.64)
Str[e(t, x, x)] = C(x) + o(1)
as t 10 uniformly on M, then X (M) =
f
M
C(x)m(dx).
Now we show that (10.64) actually holds and we identity the limit C(x)
with an explicit polynomial in terms of curvature tensor as given by (10.89) below. This result was first obtained by Patodi [223]: In the same way as above, e(t, x, x) End(AT,*M) has the asymptotic expansion e(t, x, x)
t-d(co(x) + ci(x)t + ... + ce(x)t` + cr+1(x)tr+i + ... )
as t10, where I = d/2 and c,(x) E End(AT,*M), i = 0, 1, , c,(x) being a very complicated polynomial in components of curvature tensor and its covariant derivatives. However, Patodi showed that a remarkable cancellation takes place for the supertrace, namely,
APPLICATIONS TO HEAT KERNELS
427
i=0,1, ,1- 1
Str[c,(x)] = 0,
and furthermore, Str [c,(x)] is a polynomial of curvature tensor only, the terms involving covariant derivatives being completely cancelled. After Patodi, many simpler proofs were invented (cf. Gilkey [202], Getzler [201] etc.). A probabilistic method was first given by Bismut [193]. Our method, like Bismut, consists in obtaining cancellation at the level of functionals before taking expectation which simplifies much the proof of cancellation. In the study of short time asymptotics of e(t, x, x), we may assume as before, by choosing a coordinate neighborhood and extending the components of the metric to whole Euclidean space, that the metric tensor gjj(y) are defined globally on R' and furthermore, we may choose a normal coordinate around x so that x is the origin of Rd and near the origin (10.65)
g1(y) au +
Rtmk1(0)Ymyk + O(1 y 13)
and (10.66)
T7k(Y)
Rrikm(0)Ym -
3
Rik1m(0)Ym + O(I Y I Z)
Let (g'1(y)) and (vj(y)) be the inverse of (g,1(y)) and the square root of (g'1(x)) respectively. Consider the following stochastic differential equation on Rd x GL(d, R) with a parameter a (0, 1): dX'(t) = .6k'k(X(t))dW''(t) - 2 g1k(X(t))rJ (X(t))dt (10.67)
def(t) = - F.VX(t))ei(t) o dXm(t)
i,j=1,2,...,d. (X(0), e(0)) = (0, I)
Denote this solution by re(t) = (X8(t), ee(t)) and their components by X*)' and e8(t)j' respectively. We know that re(t) E O(M) _ {(y, e) (=R" X GL(d, R); g1(y)e' e'p = aQ } for all t 0 a.s. and that { r8(t) } is equivalent in law to { r(ezt, r, w) } where r(t, r, w) is the solution of (5.5). Let H8(t)E=- End(ARd) be defined by (10.68)
178(t): 6' A 3" A ... A 8r° -+ e8(t)" A e(t)'= A ... A ee(t)',
428
DIFFUSION PROCESSES ON MANIFOLDS
where e8(t)' _ (eg(t)', e`(t)i, , ee(t)`a) E Rd, i = 1, 2, fine Mg(t) E End(ARd) by the unique solution of (10.69)
dM dt t)
-
,
d. De-
= .ZM(t)DZ[ 2 J,,km(re(t))]
M(0) = I, (cf.(5.30)) of Section 5 for notations). Then, by (10.60), we can conclude
that e(e2, x, x) = e(e2, 0, 0) (10.70)
= E[M8(l)n8(l)So(X8(l))],
e E (0, 1].
By (10.65), (10.66) and (10.67), we can easily deduce as in Theorem 10.4
that (10.71)
X8(1) = ew(1) + O(e2)
in D-(Rd)
as e j 0
and, by Theorem 10.5, we see that X8(1)/e is uniformly non-degenerate in the sense of (9.32). Hence by Theorem 9.4, we have (10.72)
8o(X8(1)) = e-dSo(w(1)) -I- O(c d+1)
in
D--
as
a J, 0.
Similarly by (10.65), (10.66) and (10.67), we can deduce e8(1)5 = S] + (10.73)
3
[R[m,k(0) + Rr mk(0)] fo wk(s) o dwm(s)
+ O(s3)
in b.
as a J, 0, i, j = 1, 2,
, d.
From (10.67) and (10.68), we see that 178(t) is the unique solution of (10.74)
178(t) = I + f 178(s) o d08(s) o
where ®8(t) is an End(ARd)-valued semimartingale defined by d
(10.75)
08(t) = D1(08(t)) _ E 6,8,(t)a*aj
(cf. (5.23) for notations) where 08(t) is Rd x Rd-valued semimartingale given by
APPLICATIONS TO HEAT KERNELS (10.76)
O(t) = - f
429
Tm,(X8(s)) o dXr(s)m. O
By (10.66) we can deduce (10.77)
in D.
O1,(t) = e2Cj(t) + O(e3)
ass 10,
where
cu(t) =
[Rimjk(0) + R1fmk(0)3f wk(S) 0
3
dwm(s),
i,j=1,2,...,d.
By (10.74), we have
178(1) = I + f
(10.78)
378(s) o de2(s) o
= I +98(1) +
f
f" 0
n8(t2) c d08(2) c dO8(t,)
0
= i +98(1) + f 08(t,) c d98(t,) 0 o
+f
0
f o' f0
178(13) ° d98(t3) o d98(tz) o d98(t,)
and so on. Hence, setting (10.79)
A. = f f ... 0
0
f`m o d98(tm) - d08(tm_,) a ... -deco), 0
it is easy to conclude 178(1) = I + A, +
(10.80)
+ A, + 0(s21+2)
in
D_(End(ARd))
as a 10.
Furthermore, by (10.77), we can deduce for each m = 1, 2,
A. (10.81)
-', dD,[C(tm)l = Elm f f 0' ... f'--'-
that
0 dD,[C(tm-,)l
o
o dD,[C(tl)] + 0(e2m+,)
in
D_(End(ARd))
as a J0.
On the other hand, denoting by P(t) the Rd Qx Rd 3Rd ®x Rd-valued process J,Jkm(r8(t))/2, we see from (10.69) that
DIFFUSION PROCESSES ON MANIFOLDS
430
M8(1) = I + E2f l Me(t)D2[J8(t)]dt 0
= I `f'. elf l Dz[J8(t)]dt 0
+ e4f 'f"0 Me(t2)D2[J8(t2)]D2[`18(tl)]dt2dtl 7
O
and so on. Hence, setting (10.82)
B. =
e2n f
oJ
of ...
f
Dz[Je(tt)]
o
dtndtn_l ... dtl,
it is easy to conclude that (10.83)
M8(1) = I + Bl +
+ Bt + O(e21+2)
D.(End(ARd))
as a 10.
Furthermore, it is easy to deduce of each n = 1, 2,
, that
in
(10.84)
B. = n (D2[J])n + O(e2n+l) i
in
D..(End(ARd))
as a ,1. 0
where J = J8(0) = (Rljkm(0)/2)c- Rd 0 Rd (2) Rd ® Rd.
We now need the following algebraic lemma:
Lemma 10.10. Assume that d = 21 is even. (i) Let al, a2, , am E Rd 0 Rd and bl, b2i
, bn E Rd Q Rd Qx Rd ® Rd. Let A C- End(AR4) be a product of D1[al], Dl[a2], , Dj[an] and D2[bl], . 2[b2], , D2[bn] in some order. Then
(10.85)
Str(A) = 0
(ii) Let b = (10.86)
if m + 2n < d = 21. E Rd 0 Rd 0 Rd 0 Rd and suppose it satisfies
bukm = - bnkm = - brjmk = bkmij bifkm + btkm/ + bimik = 0.
Then
and
APPLICATIONS TO HEAT KERNELS
431
Str((D2([b])1)
(10.87)
=
E sgn(v)sgn(u)bv (1) v (2) p (J) p (2)
1
per'(2l)
21
by (21-1)P(21)p(21-1)p(21)
x by (3)y(4) 4(3)p(4)
where S(21) is the permutation group of order 21. Proof will be given at the end of this section. Remark 10.5. The curvature tensor R1jkm satisfies the relation (10.86).
By (i) of Lemma 10.10, we have
if 2n + m < 21 = d.
If 2n + m = 21 and m > 0 or if 2n + m > 21, then by (10.81) and (10.84), B.A. = O(e2m+2n) = O(e21+1)
as a 10.
in D_(End(ARd))
Hence we obtain by (10.80) and (10.83) that Str[M8(l)178(1)] = e2Str[B1] + 0(e"')
in D.
as a 10.
Therefore, it follows from (10.84), (10.87) and Remark 10.5 that 8
Str [M8(1)178(1)] _
Str[(D2[R1fkm(0)/2])1] +0(S21+1)
e2t
(10.88)
= 2211) v, E (zn
sgn(v)sgn(u)Rv (1) v (2) p (1) p (2) (0)
x Rv(3)v(4)µ(3)µ(4)(0)
+ O(E21+1)
. Rv(21-1)v(21)1U(21-1)p(2l)(0)
in D.
as a 10.
Combining this with (10.70), (10.71) and (10.72) and noting E[ao(w(l))] =
(23)-d/2 =
(27c)-1
we finally obtained the following: Str[e(e2, 0, 0)] = E[Str[M8(l).I78(l)]bo(X8(1))] (10.89)
1
= 231 111
v ,,
zq
sgn.(v)sgn(u)Rv (1) v (2) p (1) /I (2) (0)
432
DIFFUSION PROCESSES ON MANIFOLDS X Rv(3)v(4)IL (3)k(4)(0) ... Rv(21-I)v(21)k(21-1)A(21)(0)
+0(6) as e 10. It is clear that 0(e) in (10.89) can be estimated uniformly with respect to
x E M which we take to be the origin of the normal coordinate. Generally, denoting by J,fkm(r) the scalarization of the curvature tensor Rljkm(x) as before, define a function C(r) on O(M) by C(r) = 231ir111 k (2n sgn(v)sgn(p)Jv (I) v (2) P (I) ,u (2) (r) (10.90)
X Jy(3)v(4)p(3),(4)(r) ... Jv(2l-1)v(21)µ(21-1)µ(21)(r), r E O(M).
It is easy to see that C(ra) = C(r)
for every a e 0(d)
and hence C(r) depends only on 7t(r) = x E O(M). Thus we may write C(r) by C(n(r)). This function C(x) on M is known as the Chern polynomial. We have now established (10.64) and therefore X (Al) = J
M
C(x)m(dx).
This formula is known as the Gauss-Bonnet-Chern theorem. In the case
d = 2, C(x) = K(x)/2n where K(x) __
RI232(x)
det(glj(x))
is the Gauss total curvature and hence
1 f K(x)m(dx) = X (M) = 2(1 - g) 27c m
where g is the genus of the surface M. This is the classical Gauss-Bonnet theorem.
Proof of Lemma 10.10. A direct proof may be found in Patodi [223]. Following [194], we now give a proof based on a unifying idea of supersymmetry. Let r1Rd = E Q ARd be the exterior algebra over R4 and a*, al E End(AR4) P-0
APPLICATIONS TO HEAT KERNELS
433
be defined by (5.19) and its dual. Let ACRd be the complexification of ARd and let y, E End(A,Rd), i = 1, 2, , 2d, be defined by Yzt-i = at + ar
i= 1,2,
d.
Yzt = 1=11 (a* - at) Then from (5.20) we have { y,, yy } = 2S,f I and y* = yt. For every subset K = {u1 ,,t2i < luk of { 1, 2, , 2d} let ; Ik}, A, <,Uz < )k(k-1)12Yt<1Yuz ... yIIk
Ys=
and
Then y = I and y,*K = yr. Also, it is easy to see that Tr(y4) =
0
if A * 0
2d
ifA=0.
Indeed if A = 0, let u, = min{ i I i e A } and let A = A\{,u, }. Then,
if #A = k = even, Tr(Ys) _
)k-'Tr(Y,1,
_-
YA)
1)k-'Tr(YA
= - -1)k'Tr(Yk,
Yr,,) VA)
Tr(ye)
and hence, Tr(y4) = 0. If #A = odd, choose u E4 A and write Tr(ya) = Tr(Yr4 ' Yt4Y.4) = - Tr(Y,.
= - Tr(Y,. ' Y.t
Y" - Y"')
Y;') = - Tr(YA)
Hence Tr(y,,) = 0. It is easy to see that the system { y}, where K is a subset of { 1, 2, 2d }, forms a basis of End(A,Rd) : Independence of this system is easily seen from
Tr(y1y ,)
-
10 2d
if K* K' if K= K'.
Also, dim End(ACRd) _ (2d)2 = 2 zd = the number of the system {y1}
DIFFUSION PROCESSES ON MANIFOLDS
434
and the assertion follows. Thus every A E EndA(Rd) is expressed uniquely as (10.91)
A = E CS(A)ya
C1(A) E C
K
and (10.92)
Tr(A) = 2dCO(A).
(10.92) is known as the Berezin formula. Next we claim that (10.93)
(- 1)F = (- l)dyn.2.
,2dt
Indeed, if we denote the right-hand side by a, then { y,,, a) = 0 and hence
, d. From this, it is easy to conclude a = [at*, a) = 0, i = 1, 2, (- 1)F if we can show that a w = co for w e Ac(Rd) = C. But 1(akak - akak)
72k-I72k = %
and hence if co E C.
72k-I72k(t)
Thus acv = (- 1)d(%/
1)2d(2d-1) /2(
1)dco = (- 1)d(%/
)2dZco
= co.
Hence, combining this with (10.92), we have (10.94)
Str(A) = Tr((- l)FA) = (- 1)d2dC,1. 2, ...,2d)(A).
Now the proof of (i) is easy: If we express A in the form (10.91), then, in each term, y's appear at most 2m + 4n times. Hence if m + 2n < d, C41, 2...., 2,n (A) = 0.
For the proof of (ii), we first note that d
1 2N = 3L 2 brjkmY2t72JY2k-1Y2m-i (10.95)
+ a polynomial in y's of degree 2
APPLICATIONS TO HEAT KERNELS
435
if (btJkm) satisfies (10.86). Indeed, omitting the summation sign for repeated indices, we have 132[b] = biJkma$aJak am
= 24 Y,
1)`btlkmYt5JYkYm
where Yt = yet or Y21-, and the exponent a depends only on the way of this choice. Noting the well-known property of (btUkm) satisfying (10.86) that the alternation over any three indices vanishes, it is easy to deduce that b,jkma,*ajak am _ 4 { btJkmY2tY2JY2k-IY2m-1 + b1kmY21-1YZJ-1Y2kY2m bt)kmY21YZJ-1Y2k-1Y2. + bifkmY21-1Y2JY2kY2m-1
- btJkmYztY2J-1Y2kY2m-1 - bjjkmY21-1Y2JY2k-1Y2.)
+ a polynomial in y's of degree 2. It is easy to deduce from (10.86) that the first two terms are equal and the remaining four terms cancel. Hence the proof of (10.95) is finished. From this and (10.94), we can easily conclude (10.87).
Remark 10.5. If we take another supertrace Str 12) (A) = Tr[y (2, 4,
, u, A }
and do the same calculation as above, we can obtain the Hirzebruch signature theorem. Main difference in this case is that A. in (10.83) does not disappear in the process of cancellation, that is, an effect of the paral-
lel translation operator 17e(l) remains in the final form of signature theorem. For details, cf. [204]. Indeed, we can give a similar probabilistic proof of Atiyah-Singer index theorem for every classical complex, cf. [193] and [224].
CHAPTER VI
Theorems on Comparison and Approximation and their Applications
1. A comparison theorem for one-dimensional Ito processes Suppose now that we are given the following: (i) a strictly increasing function defined on (0, co) such that p(O) = 0 and (1.1) J
a+P(S)-Z a' = no;
(ii) a real continuous function 6(t, x) defined on (0, oo) x R' such that (1.2)
1 cr(t,x) - c(t,y) I < P(I x - y I),
x, y e k, t > 0;
(iii) two real continuous functions b1(t, x) and b2(t, x) defined on [0, oo) x R' such that (1.3)
b1(t, x) < b2(t, x)
Let (Q,.Y,P)
t > 0, x E R'.
be a probability space with a reference family (. )tz0.
Theorem 1.1. * Suppose that we are given the following stochastic processes: (1) two real (Y )-adapted continuous processes x1(t,c)) and x2(t,c));
(2) a one-dimensional (,)-Brownian motion
B(t,cv)
such that
B(0) = 0, a.s.;
* This theorem covers the previous results obtained by e.g. Skorohod [150], Yamada [182] and Malliavin [108]. 437
COMPARISON AND APPROXIMATION THEOREMS
438
(3) two real (-O';)-adapted well-measurable processes /f,(t,w) and ,62(40We assume that they satisfy the following conditions with probability one: (1.4)
x,(t)- xx(0) =
(1.5)
x,(0) < x2(0),
(1.6)
fi,(t) < b,(t,x,(t))
for every t
(1.7)
fl2(t) > b2(t,x2(t ))
for every
Ju
Q(s,x,(s))dB(s) + J'
b`(s)ds,
1= 1, 2,
0,
t > 0.
Then, with probability one, we have
xl(t) < x2(t)
(1.8)
for every t > 0.
Furthermore, if the pathwise uniqueness of solutions holds for at least one of the following stochastic differential equations
dX(t) = Q(t,X(t))dB(t) + b,(t,X(t))dt,
(1.9)
i = 1, 2,
then the same conclusion (1.8) holds under the weakened condition (1.3)'
b,(t,x) < b2(t,x)
for t > 0, x E R'.
Proof. * By a usual localization argument we may assume that o(t,x) and b,(t,x) are bounded. Step 1. We assume that b,(t,x) is Lipschitz continuous, i.e., there exists a constant K > 0 such that
I bi(t,x) - bi(t,y) I< K I x - y 1,
x, y E R'.
Choose a sequence V. (u), n = 1, 2, ... of continuous functions as in the proof of Theorem IV-3.2 and set
7n(x) =
0, fx
x<0, fY
f. dy J o
* The following proof is due to T. Shiga [146].
x > 0.
COMPARISON THEOREM FOR ITO PROCESSES
439
It is easy to see that 0. E C2(R'), 0.(x) = 0 for x < 0, 0 < 0'(X) < 1 and 0,(x) t x+ as n - co. An application of Ito's formula yields 0n(x1(t)-x2(t)) = 11(n) + 12(n) + 13(n),
where r1(n) = f 0n(x1(s)-x2(s)) {o(S,x1(S)) - Q(S,x2(s))}dB(s), 0
12(n)
=f
0
0n(xl(S)-x2(S)) {MS) - kS)}ds
and
= 2 f", 0, (x1(S)-x2(s)) {a(S,x1(s)) - 0,(s,x2(S))} 2ds.
13(n)
It is clear that E(11(n)) = 0 and E(I3(n)) < 2 E(f o cn'(x1(s)-x2(s))p(I x1(S)-x2(S)1)2ds) 5 n . Also, 12(H)
f 0'(x1(s)-xz(s)) {b1(s,x1(s)) -b2(s,x2(s))} ds o
= f' 0n(x1(s)-x2(s)) {b1(s,x1(s)) - b1(s,x2(s))} ds + f' O (xl(s)-x2(s)) {bl(s,x2(s)) - b2(s,x2(s))} ds :!9f
{b1(S,x1(S))-b1(s,x2(s))} ds o
01[.1cs»Y2c,»
Ix1(s) - x2(s)Ids
< K f o (x1(s) - x2(s))+ds.
Hence, by letting n -- oo, we have E[(x1(t) - x2(t ))+] < KE[ f 0 (x1(s) - x2(S))+ds]
= K f r0 E((x1(s) - x2(s))+)ds.
440
COMPARISON AND APPROXIMATION THEOREMS
We can deduce from this that
E[(x,(t) - x2(t))+] = 0
for all t > 0,
that is P(x,(t) < x2(t)) = 1
for all t > 0,
and so by the continuity of paths we conclude that (1.8) holds.
If instead we assume that b2(t,x) is Lipschitz continuous, then a similar argument leads to the same conclusion (1.8).
Step 2. In the general case we choose b(t,x) such that
b,(t,x) < b(t,x) < b2(t,x) and b(t,x) is Lipschitz continuous. Let X(t) be the unique solution of the following stochastic differential equation dX(t)
o(t,X(t))dB(t) + b(t,X(t))dt
X(0) = x2(0)
Then the result from step 1 gives that X(t) S x2(t) and x,(t) S X(t) for all t > 0 a.s. Consequently we can conclude that (1.8) holds. Step 3. We now turn to the proof of the second assertion. We assume that the pathwise uniqueness of solutions holds for one of the stochastic differential equations (1.9), say for i = 1. Let X(t) be the solution of the equation (1.10)
dX(t) = c(t,X(t))dB(t) + b,(t,X(t))dt
X(0) = x0)
For s > 0, let X(tg)(t) be the solutions of dX(t) = a(t,X(t))dB(t) + (b,(t,X(t)) ± s)dt
X(0) = x0) respectively. By what we have already proven,
V-4)(t) < X(t) < X(+B'(t)
for every t > 0,
APPLICATION TO OPTIMAL CONTROL PROBLEM
441
and if 0 < e2 < e1i then X(-ej)(t) < X(-e2)(t) and X`+e2)(t) < X`+aj1(t)
for every
t > 0.
Hence by the continuity of bl(t,x) and the pathwise uniqueness of solutions for (1.10), we have
lim X (-i)(t) =e1Olim X(t) = X(t) CIO
for every
t > 0.
Again applying what we have already proven to x1(t) and X'+E1(t), we have (1.11)
x,(t) S X (-')(t)
since fl1(t) < b1(t,x1(t)) ting s 10 in (1.11), (1.12)
x,(t) < X(t),
for every
a.s. and bi(t,x) < b1(t,x) + e. Hence, by letfor every t > 0.
2(t) > b2(t,x2(t)) a.s. and X'-e1(t) < x2(t); letting e j 0 gives Since
X(t) < x2(t)
t>0
for every
b2(t,x) > b1(t,x) - s, we have
t > 0.
Combining this with (1.12) we obtain the inequality
x1(t) < X(t) < x2(t),
for every t > 0,
which completes the proof of the second assertion.
2. An application to an optimal control problem As an example of an application of the comparison theorem in the previous section we shall consider the following stochastic optimization problem.
Let k(r) be a non-decreasing and non-negative function defined on [0, co). Let (Be, u) be a system of stochastic processes defined on a probability space (Q,- 9-,P) with a reference family (,') such that (i) B1 (Bo = 0) is a d-dimensional (.57;)-Brownian motion and (ii) u, is a d-dimensional (9)-well measurable process such that d u, I< 1 for all t> 0 a.s.
442
COMPARISON AND APPROXIMATION THEOREMS
Such a system (B,, u,) is called an admissible system or an admissible control. Let XE Rd be given and fixed. For a given admissible system, the response XX is defined by (2.1)
Xf = x + B, + E u,ds. o
The optimization problem then is to minimize the expectation E(k(I X,1)) among all possible admissible systems. The solution is given as follows. Let U(y) be defined by (2.2)
= o,Y11Y1,
y E 0 d\ E {01
U(Y)
a
Consider the following stochastic differential equation (2.3)
dX, = dB, + U(X,)dt Xo = X.
By the corollary to Theorem IV-4.2 we know that a solution (X,,, B,) exists uniquely. Set (2.4)
of = U(X,).
Then the admissible system (BOgo) gives an optimal control; that is, for any admissible system (B,,u,), we have (2.5)
E(k(I X, I )) <_ E(k(1 1 i I ))
We shall now prove this fact as a simple consequence of Theorem I.I. It was originally obtained by Benes [3] by different techniques.
Theorem 2.1. Let (B,,u,) be any given admissible system and, for a fixed x E Rd, let the response {Xf} be defined by (2.1). Then on an appropriate probability space we can construct Rd-valued processes 1±3 and 1±01 such that (i)
{XIM}
{'j},
(ii) {X°} and
(iii) 1 xf 1 < 1 'f 1
for every t > 0, a.s.
APPLICATION TO OPTIMAL CONTROL PROBLEM
443
Corollary. Let Wd = C((0, co) - Rd) and F(w) be a non-negative Borel function on Wd with the following property: if w,,w2 E Wd and
(2
F(wl) <
w2(t) I
I w1(t)
for every t Z 0, then
F(w2).
Then for any admissible system (Be, u,) we have
E(F(X`)).
(2.7)
That is, the solution IX,*) of (2.3) is optimal in the sense of minimizing the expectation of F(X').
Note that the particular case of F(w) = k(I w(l) 1) clearly satisfies (2.6).
In order to prove the theorem we first state the following lemma. Lemma 2.1. Let (X1, Bt) be a pair of d-dimensional continuous (-g;)adapted processes defined on a probability space (0,.Y1P) with a reference family (.J9'-,' such that {B,} is a d-dimensional Brownian motion with
B0 = 0. Let (Y,,B') be a similar pair defined on another space (S2',,`r'P') with (. '). Then we can construct a probability space (52,F P) with a reference family (,,) and a triple of d-dimensional (JP,')adapted processes such that (i) (X, B:) e (ii) (Y:, B'D
and
(iii) (B,) is a d-dimensional (.,9)-Brownian motion. The proof follows in exactly the same way as in Theorem IV-1.1.
Proof of the theorem. Let be a given admissible system and X, = x+B,+ f o u,ds be the response. Let (XO,B,) be a solution of (2.3). Choose an 0(d)-valued Borel function (p,j(x)) such that (2.8)
xJ/IxJ
x= (x',x2, ... ,x') k- 0
45,J
x=0.
p,j(x) =
Set Bt = f' p(X,)dB, and ff = f' p(XS)dB:. Then we have :
(2.9)
:
Xt" = x + f 0 P-'(Xs)dB, + f 0 u,ds
COMPARISON AND APPROXIMATION THEOREMS
444
and t
(2.10)
X: = x+ f
p-'(X s)dB s+ f' U(X o,)ds.
0
0
Now we apply Lemma 2.1 to (X",Br) and (X0,.§,0). We then have a triple
(t- Xo, Bt) of (.)-adapted processes on a probability space with a reference family (..) such that {Bt} is a d-dimensional (. )-Brownian motion and (Xu,. ) i' (X',B') Clearly there exists an (..)-well measurable d-dimensional process {12r} such that 19, I < 1 for
every t > 0 and =x+ft0p-'(.ftdBs-{-J0t tds. Applying Ito's formula to x1(t) = I to 12 and x2(t) = I X! 2, we have
dx2(t) = 2±j p-'(X jdBt + 21'. a dt + ddt = 211 7 I dB,' + [2 . U', + d]dt =2,/x,-(t)
dB t + [2t" ut + d] dt
and
dxl(t) = 2I .p-'(X°)dBt + 2A3o U(Xa,)dt + d dt
= 2I k I d d f + [-2I t I + d]dt dd.' + [-2 xl(t) + d]dt,
= 2/)
where st = (Bt, Bj, ... , Ba). (Note that Set a(t,x) = 2 x,p11(x) = ate 1 x I).
E -2 ,lx- O+d, fl1(t) = -2 xl(t)+d and
'(x)]1= x V 0,
xj(p-'(x));t =
b1(t,x) = b2(t,x) =
/32(t) = 2 ' tt+d. Then clearly fl1(t) = b1(t, x1(t)) and fi2(t) > -2 1 Xi I +d = b2(t, x2(t)). As we shall see in the next lemma, the pathwise uniqueness of solutions holds for the stochastic differential equation (2.11)
dX(t) = a(t, X(t))dB(t) + b1(t, X(t))dt = 2(X(t) V 0)112dB(t) + [-2(X(t) V 0)''2 + d]dt.
Therefore we can apply the second assertion of Theorem 2.1 and obtain the result that x1(t) < x2(t) for all t > 0, a.s., that is I Ito I < I X'j I for
all t > 0, a.s.
APPLICATION TO OPTIMAL CONTROL PROBLEM
445
Lemma 2.2. The pathwise uniqueness of solutions holds for the equation (2.11).
Proof. First we remark that the maximal and minimal solutions of (2.11) exist; that is, there exist strong solutions X,=F,(X1(0),B) and X2=F2(X2(0), B) of (2.11) such that if (X(t), B(t)) is any solution of (2.11)
and X1(0)=X(0)=X2(0), then X1(t) < X(t) < X2(t) for every t > 0, a.s. such that Indeed, choose smooth functions b( ,"(x) and
< ba'(x) < bi2)(x)
and
lim b;,"(x) = lim b,1,2'(x) = b(x).
Here b(x) = b,(t. x) = -2(x V 0)' 12+d. Then the solutions XX(t) of dX(t) = a(t,X(t))dB(t) + b,',"(X(t))dt { X(O) = x
satisfy that X (t) < n = 1, 2, ... , by Theorem 1.1 and we set F1(x, B) = lim X.(.). F2(x, B) is similarly defined. Clearly they possess the above property. Set
s(x) = f
1
exp I - f2z' dil dy, ,
x > 0.
Then s(0+) > - oo if d = 1 and s(0+)
oo
if d > 2. First we
consider the case d = 1. By Ito's formula s(X(t)) - s(X(0)) =
ft
s'(X(s))a(s,X(s))dB(s)
for any solution X(t) of (2.11) with X(0) > 0. Hence we have E(s(X(t))) = E(s(X(0))).
From this we can conclude that X1(t) = X2(t) a.s. for all t > 0 if X1(0) = X2(0) a.s. This implies the pathwise uniqueness of solutions for (2.11). If d > 2, we see by Theorem 3.1 given below that inf X(t) > 0 a.s. for every 05cST
COMPARISON AND APPROXIMATION THEOREMS
446
T > 0 if X(0) > 0 a.s. The uniqueness of solutions for (2.11) can be easily deduced from this; we leave the details to the reader.
The optimization problem discussed above is a special example of stochastic control problems. For the recent development of the stochastic control theory, we refer the reader to Krylov [91].*1 3. Some results on one-dimensional diffusion processes *2
Let I = (1, r) be an open interval in R1 (-oo < 1 < r < oo). Let 6(x) and b(x) be sufficiently smooth real valued functions*3 on I such that 62(x) > 0 for all x e I. For x E I, the stochastic differential equation
dX(t) = a(X(t))dB(t) + b(X(t))dt { X(0) = x
(3.1)
has a unique solution X'(t) up to the explosion time e= lim Tn, where nt""
Tn = inf {t; Xx(t)
[an, bn]} (an and b (n = 1, 2, ...) are chosen such that
I < an < bn < r and an j 1 and bn t r). By the same proof as in Lemma IV-2.1, we can show that limXx(t) exists and is equal to I or r a.s. on tie
the set {e < oo}. We define Xx(t) to be this limit for t > e on the set [I, r] {e < oo}. Let JP, be the set of all continuous paths w: [0, oo) such that w(0) E I and w(t) = w(e(w)) for all t > e(w): = inf {t; w(t) = I or w(t) = r} . e(w) is called the explosion time of the path w. Then XX = {X"(t)} defines a W,-valued random variable with e[Xi] = e. Let PX be the probability law on 41, of P. As we saw in Chapter IV, {Px} defines a diffusion process on I. It is called the minimal L-diffusion, where the differential operator L is defined by 2
(3.2)
L =
+ b(x).
62(x)
2
dxd 2
Let c e I be fixed and set (3.3)
s(x) = f c exp {- fy
y 62() dz} dy.
In particular, this book contains important results by Krylov on the estimates for the
distributions of stochastic integrals, an extension of Ito's formula and stochastic differential equations with measurable coefficients. *Z Cf. [731 for more complete information. #3 In the following discussion it is sufficient to assume that v and b are of class C'.
RESULTS ON ONE-DIMENSIONAL DIFFUSION PROCESSES
447
Then s(x) is a strictly increasing smooth function on I satisfying
Ls(x) = 0
(3.4)
on
1.
oo and s(r-) = co, then
Theorem 3.1. (1) If s(1+)
Px(e = oo) = Px(lim X(t) = r) = Px(lim X(t) = 1) = I
(3.5)
ft-
7T__
for every x.* In particular, the process is recurrent, i.e. Px(a,, < oo) = 1 for every x, y E I where ay = inf {t; X(t) = y} . (2) If s(1+) > -oo and s(r-) = oo, then lim X(t) exists a.s. (P.) and tte
Px(1im X(t) = 1) = Px(sup X(t) < r) = 1
(3.6)
tte
t<e
for every x. A similar assertion holds if the roles of 1 and r are interchanged.
(3) If s(1+) > - oo and s(r-) < oo, then lim X(t) exists a.s. and tle
(3.7)
Px(1im X(t) = 1) = 1 - Px(lim X(t) = r) = tte
tte
s(r-) -s(x) s(r-)
Thus the process is not recurrent in cases (2) and (3).
Proof: Let 1 < a < x < b < r and z = inf{t; Xt Er; [a, b]}. By Ito's formula and (3.4), s(Xx(tA r))
tAT
- s(x) = f0o
st (XX(s))a(Xx(s))dB(s)
and hence EE[s(X(t AT))] = s(x).
By letting t t oo, we have
EE[s(X(T))] = s(a)Px(X(r) = a) + s(b)Px(X(T) = b) = s(x). Combining this with
1 = Px(X(T) = a) + P,(X(T) = b) * X(t) = X(t, w) = w(t), w e a',. Also e = e(w).
COMPARISON AND APPROXIMATION THEOREMS
448
we have
PS(X(z) = a) =
(3.8)
s(b) - s(x) s(b) - s(a)'
PP(X(T) = b) = s(x) - s(a)
s(b) - s(a)
Suppose s(r-) = -s(1+) = oo. Then lim PX(X(z) = b) = 1. all
Hence
PP(sup X(t) > b) > lim P,,(X(a) = b) = I all
r<e
for every b < r and consequently P.,(sup X(t) = r) = 1. r<e
Similarly
P=(inf X(t) = 1) = 1. Now it is easy to conclude the assertion of (1). Next, t<e
assume that s(l+) > -oo and s(r-) = oo. We can deduce as above that (3.9)
PP(inf X(t) = 1) = 1. Ke
Now Y,-' = s(X(t A Sr)) - s(1+) is a non-negative martingale and, letting
a j 1 and b f r, we can easily see that Y, = s(X(tAe))-s(1+) is a nonnegative supermartingale. Consequently lim Y, = lim s(X,) - s(1+) exists :T"
tle
as a finite limit a.s. (Theorem 1-6.4). Therefore lim X, exists a.s. and in rte
view of (3.9) we must have (3.6). The proof of (3) is similar and thus omitted.
Let 1 < c < r be fixed and set (3.10)
lc(x) = 2
exp c
f 2b(z) dz IT C exp [fcn 2 b(z) dz1 d J dc, Q(z) 16(11) c0 2(Zj tfc xE=_ I.
Lemma 3.1. Let u(x) be the unique solution of
Lu(x) = u(x) (3.11)
u(c) = 1 u'(c) = 0.
RESULTS ON ONE-DIMENSIONAL DIFFUSION PROCESSES
449
Then (3.12)
x E 1.
1 + K(x) < u(x) < exp {K(x)},
Proof. u(x) is the solution of (3.11) if and only if it satisfies u(x) = 1+ f X ds(y) f Y u(z)dm(z), C
0
where
ds(y) = exp [_ f y 2"zj dzl dy
.z
I
l
and
&n(z) = 2 exp [f c Q() d i]
az) dz.
Hence u(x) = E uk(x) k.-0
u (x) > 0
with u0(x) - I and un(x) = f x ds(y) Jy and u,(x) = K(x). Consequently
u(x) > I + u,(x) = I + K(x). On the other hand, if we suppose that K(z)n/n!, then
f : ds(y) f u,,(z)&n(z) Y
n l s: ds(y) f K(z)°dm(z)
n f ' ds(y)K(y)" f: dm(z) = f : K(y)nd[K(Y)) t
t
K(x)n+1 (n + 1)!,
Consequently
450
COMPARISON AND APPROXIMATION THEOREMS x()n u(x) _ n=0
E=0
n!
= exp{x(x)}.
Remark 3.1. It is obvious that (i) x(r--) < oo s(r-) < o0 and
(ii) x(1+) < 00 = s(I+) > - 00. Theorem 3.2. (1) If x(r-) = x(1+) = oo, then (3.13) (2)
(3.14)
Px(e = oo) = 1
for all x E I.
If rc(r-) < oo or x(l+) < oo, then Px(e < oo) > 0
for all x E I.
Px(e
for all xEI
(3)
if and only if one of the following cases occurs (i) x(r-) < oo and x(1+) < 00. (ii) x(r-) < oo and s(1+) = -oo. (iii) x(1+) < oo and s(r-) = oo.
Proof. Let u(x) be defined by (3.11). Let 1 < a < x < b < r and z = inf {t; X(t)
(a, b)}. By Ito's formula
de tu(X(t)) = e-tu'(X(t))a(X(t))dB(t) + e-t(-u(X(t)) + (Lu)(X(t)))dt = e-tu'(X(t))c (X(t))dB(t) and hence a t^tu(X(t A z)) is a martingale. Letting a I I and b t r we immediately see that exp (-t A e)u(X(t A e)) is a non-negative supermartingale. If x(r-) = x(1-}-) = oo, then by Lemma 3.1, lim u(x) _ xII
lim u(x) = oo. Consequently Px(e < oo) > 0 is impossible because t xtr
exp(-(t A e))u(X(t A e)) is bounded a.s. as a non-negative supermartingale. This proves (1).
Next suppose that x(r-) < oo. Without loss of generality, we may assume that c < x. Let a' = inf {t; X(t) = c). By Lemma 3.1, u(r) < oo and hence (3.15)
exp (-tAr')u(X(tAr')) is a bounded Px martingale.
RESULTS ON ONE-DIMENSIONAL DIFFUSION PROCESSES
451
Consequently (3.16)
u(x) = EE[exp (- t A r')u(X(t A T'))].
Letting t t oo in (3.16), we have
u(x) = EE[exp (-e)u(r-); lim X(t) = r] rieA,'
4- EE[exp (-T')u(c); lim X(t) = c]. t t eAr'
If EE[exp(-e); lim X(t) = r] = 0 then we have t i ehr
lim X(t) = c] < u(c).
u(x) = u(c)EE[exp
r t ens'
This is clearly a contradiction. Therefore E.,[exp(-e); lim X(t) = r] > 0 t i cATI
which implies that Px(e < co) > 0. Finally we shall prove (3). If K(r-) < oo and if none of (i), (ii) or (iii) hold then we must have s(1+) > - oo and x(1+) = co. Then Px(lim X(t) _ tie
1) > 0. Since exp(-t A e)u(X(t A e)) is a non-negative supermartingale and lim u(x) = oo, we must have e = oo a.s. on the set {lim X(t) = 11. xlr
tie
Consequently Px(e = oo) > 0. Therefore if Px(e < oo) = 1, then at least one of the conditions (i), (ii) and (iii) must hold. It is now sufficient to show that each of conditions (i), (ii) and (iii) implies that Px(e < oo) = 1 for all x. First we shall assume (i). Define G(x, y), x, y E I, by
G(x, y) =
(s(x) - s(1-))(s(r-) - s(y)) s(r-) - s(1+) (s(y) - s(1-))(s(r-) - s(x))
s(r-) - s(1+)
x
y<x.
If f(x) is a bounded continuous function on I, then the condition (i) implies
that u(x) = f G(x, y)f(y)m(dy) is a bounded function. In particular
ul(x) = f G(x, y)m(dy)
COMPARISON AND APPROXIMATION THEOREMS
452
is a bounded function. It is easy to prove that u E C2(1), u(1+) = u(r-)
0 and Lu = -f. In particular, Lu, _ -1. Hence by Ito's formula, u,(X (t A e)) - u, (x) + t A e = f
ui(X(s))dB(s). o
Consequently
-E,(u,(X(tAe))) + u,(x) = E,(t/\e).
Letting t t oo yields u,(x) = E,(e) < oo. This proves that P,(e < 00) _
1. Next we shall assume
(ii).
For each n = 1, 2, ... , set 8n =
inf {t; X(t) = 1-x- 1/n} and a, = inf {t;X(t) = r). Then lim U. A o, = e. n,m
By the result of case (i), P,(5, A a, < 00) = 1 for all x. By Theorem 3.1,
1im P,(6 > a,) = 1 since s(r-) < eo and s(1+) = -co. Clearly {a, < n,m
oo} = U {a, < 8n} and hence P,(6, < oo) = 1. Consequently n
P,(e
4. Comparison theorem for one-dimensional projection of diffusion processes
Let a _ (aL(x)) be a sufficiently smooth function: Rd E) x i --- a(x) ERd(x Rd and b = (bt(x)) be a sufficiently smooth function: Rd E) x _ b(x)ERd. We consider a diffusion process X = (X(t)) on Rd defined by the solutions of the following stochastic differential equation (4.1)
dXf =
d
k-,
uu(X,)dBk + b'(X,)dt,
i= 1, 2, ... , d.
The diffusion X is defined up to the explosion time e (cf. Chapter IV, Section 2). As explained in Section 6 of Chapter IV, this is the diffusion process generated by the differential operator a2
(4.2)
L=
2 i.E,
al(x)
aaxe +
b`(x) ax`
COMPARISON THEOREM FOR DIFFUSION PROCESSES
453
where atf
d
(x) = E Qk(x)Qklx). k-l
Let p(x) be a smooth real function on Rd and let I = { = p(x); x e R,11. Then I is an interval in R'. Let S be the set (possibly empty) of all x ERd such that p(x) is an end point of I. Let I° be the maximal open interval contained in I and I be the minimal closed interval in [-cc, cc] which contains I. We assume that 12
PP(x) I = (t,±i a"(x) a (x) ax (x))
>0
for all x E Rd\S.
Set
(4.3)
a(x) = rE) a`1(x) a (x) ax (x),
(4.4)
b(x) = (Lp)(x)/a(x),
x E Rd\S
and
a+(c) = sup a(x), xe D(G;D)
(4.5)
b+(0 = sup b(x), xeD(I;D)
a-() = xED inf((;y)a(x) b-(c) = inf b(x), xED(4OD)
where D(g;p) = {x; p(x) = c} for E I°. We assume that at(e) and bt() are locally Lipschitz continuous functions on I° and at(e) > 0. On the interval I° we consider the following four minimal diffusion
processes tt =
which are generated by the operators L±± respec-
tively; 1 d2
L++ = a+() (2 dr2 + b+() d )' 2
L+- =
(2 de2 +
b-(0
(4.6)
L-+ = a-() (2 de2 + b+(0 2
L-- =
(2 de2 +
b-(0
d
d
454
COMPARISON AND APPROXIMATION THEOREMS
Each of the diffusions is given as in Section 3; thus, the sample paths **(t) are defined for all t > 0 and are !-valued continuous paths with
1\I° as traps. Let X, be a sample path of the above diffusion (4.1) starting at x0 e
Rd\S and set C = inf {t; t < e, Xr (z S). We assume that if C < oo then lim p(XS) exists in I. We set p(XX) = Sit
lim p(XS) for t > C. Thus p(X,) is defined for all t >_ 0 as an 1-valued Six
continuous path.
Theorem 4.1 ([54]). Let x° E Rd\S be fixed and S0 = p(x0) E 10. Let X = (Xr) be the above diffusion starting at x0. Then we can construct 1valued continuous stochastic processes rjr, rtf +, i7t+-, rtf +, ilt - on the same
probability space such that the following properties (i), (ii) and (iii) hold. (i) (,7r) has the same law as (p(XX)).
(ii) (r7t*) has the same law as** _ (**(t)) starting at 0 for each of four combinations of ± ±. (iii) If we set l r = max i1,
ii, -tir = min 0;!9j-'V
q±± =maxr;*
r7* =minr7;*, -t SsSr
ossst
and
o5s$r
then with probability one we have
It-<1,
(4.7)
for all t>0
and for all
(4.8)
t>0.
Proof. For simplicity we assume that C = oo a.s. and that are all conservative diffusion processes on I°; the general case can be proved with a slight modification. Set
+(t)
=
Jo
[a(XS)/a+(p(X:)))ds
and
0-(t) = $o [a(X)Ia-(p(X))]ds.
COMPARISON THEOREM FOR DIFFUSION PROCESSES
455
Then clearly q+(t) < t < O-(t)
for all
t > 0.
Let +(t) and e (t) be the inverse functions of t - 0+(t) and t 1---- 0-(t) respectively and set Xr+) = X()V+(t))
and
X;-) = X(Vl (t)).
As in Chapter IV, Section 4, we see that Xt+) = (X=+)',X(+'z , ... , X1+)°') and XI-) = (Xt-)1,X;-)z, ... X,(-)d) satisfy the following stochastic differential equations with appropriate d-dimensional Brownian motions B,+) _ (Bt+)',B,+)z, B,+)d) with Bo+) = 0 and B,-) _ (B,(-) 1,B,-)Z, ... ,BB-)d) with Bo) = 0 respectively: a
dX,+)r =
(a+(P(X,1+)))la(X,(+)))'1z E ak(X,+))dB,+)k k-1
(4.9)
+ Xo+) = xo, d
dXj-)'
= (a-(P(X,(-)))la(XI-))))rz E ak(X,-))dB(-)k k-1
(4.10)
+
i = 1,2, ... , d
Xo-) = xo.
By Ito's formula, dP(Xr+)) =
(a+(P(X,(+)))la(X,(+)))1iz
(4.11)
[a+(P(X,'+)))la(X,+))] (LP)(X,(+))dt
P(X o+)) = o and
ai(X,-))axt(x,(-))dBt-)'
dp(X,(-)) = (4.12)
+ [a-(P(X,(-)))la(X,(-))](LP)(X,(-))dt
P(Xi-)) = o Hence, if we set
456
COMPARISON AND APPROXIMATION THEOREMS
B° = t.Z l J 0
(l/a(X=t)))',2Qf(X(±))
LP
(X=t))aBst);
a then (B, ) and (BT) are 1-dimensional Brownian motions and we have (4.13)
(a+(p(X,(+')))';zdB
dp(XX+') =
+ a+(p(X (+)))b(X'+')dt
p(Xoc+> _
and
(4.14)
dp(X,-)) = (a-(p(X,(-')))'/2dB, + a-(p(Xc'))b(X( ))dt p(X o )) = o-
Let ,, = p(Xf+') and 7, = p(X,(-)). Then by (4.13) and (4.14) we have (4.15)
d?7t =zz(a+(t1,+))'nd.§ + a+(rl+b(X()lt
no =bo and
(4.16)
d,7 = (a-(17-))'12d.§;7 + a-(i7-)b(X=`))dt no = 0.
Consider the following stochastic differential equation (4.17)
drl, + =
(a+(j7t'+))'12dB+ + a+(r!s +)b+(ij, +)dt,
hoso
In Theorem 1.1, take
X'lt) = ili, a(t, ) =
X2(t) = 17r++2 (a+())''a,
fi'(t) = b(X,+')a+(ij ), R:(t) = b+(tl,++)a+(tl++)
and
b'(t, 4) = b2(t, c) = b+( a+(D are locally Lipschitz continuous, Since we assumed that a+() and the pathwise uniqueness of solutions holds for the equation (4.17) and hence, by Theorem 1.1, we have
COMPARISON THEOREM FOR DIFFUSION PROCESSES
(4.18)
for all t > 0,
r/+ < r/++
a.s.
Similarly, if r/+- is the solution of the stochastic differential equation d,lr+(4.19)
= (a+(,l+ -))1izd.§ + a+(?I, -)b-(rj; -)dt
lho = ,o,
then we have
(4.20)
r/+- < r/+
for all t >_ 0,
a.s.
Also, if 1f + is the solution of the stochastic differential equation (4.21)
,l-+ = (a-( +)) izdB + a-(r/,-+)b+(r7T +)dt 11/0 -+ = o,
then we have
(4.22)
r/< < r/ +
for all
t > 0,
a.s.
and if ?7,-- is the solution of
- _ (a (, -))11zdB + a-(ii -)b-(q -)dt (4.23)
then
(4.24)
?17 -
for all
t > 0,
a.s.
Finally, set tj, = p(XX). Then, since
max rI; = max rl o5s5r
OSsS
(c)
min r/+ = min 7,
OssSr
OSs ,+(t)
and t < yr+(t), we have (4.25) max rI+ > max r/, and min 7s < min rl,. 05.t OSsSt 0:915r 05s5t Similarly, using the fact that t > V -Q), we have (4.26)
max r/s < max ri, and min r/, > min r/,.
oar
0_<<5:5r
ossst
of ., t
457
458
COMPARISON AND APPROXIMATION THEOREMS
Combining (4.18), (4.24), (4.25) and (4.26), we have
max,s <maxna <maxris=rig 0<s5r 0<s5t
Osssr
and
rtj=maxrj,<maxris <maxna+=;t++, O5z:
osssr
ossst
Similarly we can obtain ni+<17,
This proves the theorem. Remark 4.1. If a(x) in Theorem 4.1 depends only onp(x), i.e., if there exists a function defined on I such that a(x) = a"(p(x)), then a-(0 = a() and therefore we may assume that ?1,+,+ = n7 + and n; - = n_ In this case we have
nj - < nr < j+,+
for all
t > 0,
a.s.
As an application of Theorem 4.1 we shall now investigate the possibility of explosions for non-singular diffusions. Let X=(X(t)) be the ddimensional diffusion process determined by the solution of (4.1). We assume d > 2 and det (a'J(x)) > 0 for all x E Rd. Our problem then is to find a criterion which determines whether or not explosion happens in finite time, i.e., whether e is finite or infinite. It is easy to see that X(t) leaves any bounded set containing X(0) in a finite time a.s., and hence we may assume for this problem that a'J(x) = 8tf and b'(x) = 0 for I x < 1
and i, j = 1, 2, ...,
d.
a
Let p(x) = j x j 2/2 = E (x')2/2, xE Ra. Then t-1
I= [0, oo) and S = {0} . In this case, a
a(x) = E a'°(x)x'xJ '.1-I
and
b(x) =a(x) 1{
t-t
a"(x)/2
Following (4.5), we set
+ E b'(x)x'} t-1
for x e Rd\ 101.
COMPARISON THEOREM FOR DIFFUSION PROCESSES
a+(r) = max a(x),
a-(r) = min a(x)
b+(r) = max b(x),
b-(r) = min b(x)
459
IXI -Zr
IXI-
and
XI-
I X1-,/V
for r E(0, oo). It is easy to see that these functions are locally Lipsc-
hitz continuous and at(r) > 0 for r E(0, oo). Let r+ = (r,) and r- _ (rr) be the minimal diffusion processes generated by the operators + dl r(1 d2 `(r) l 2 dr2 + b (r) dr)
and
1 d2 dl a-(r) (2 dr2 + - b-(r) drl respectively. Set
c+(r) = exp f 2b+(u)du,
c-(r) = exp f r 2b-(u)du
and
s+(r)
f i c+(u)
du,
s -(r)
=
f t c (u)
du
for r e (0, oo). By the above assumption,
s+(r) = s-(r) =
-(2 - log
- 1) (l/r"_' -1)
I r
d>2
d=2
for r e(0,1). Hence s+(0 -) = s-(0 -) = - oo. By Theorem 3.2, if e+ and e- are the explosion times for (r+(t)) and (r-(t)) respectively, then
P; {e+ = oo} = 1 or P, {e+ < oo} = 1 accordingly as
for r e (0, oo)
COMPARISON AND APPROXIMATION THEOREMS
460
JI
[l /c+(r)] fr [c+(u)/a+(u)]dudr = 00
or < oo;
also
P, {e- = oo} = 1 or P,{e- < 00} = 1
for r E(0, oo)
accordingly as
S1 [1/c-(r)] f 1 [c-(u)/a-(u)] dudr = o0
or
Here P, and P,- are the probability laws of r+ = (r+(t)) with r+(0)=r and r- = (r-(t)) with r-(0)=r respectively. By Theorem 4.1, we may assume that max r-(s) < max I X(s)12/2 < max r+(s), a.s. OSsSt
OSssr
O5s5i
and hence
e+ < e < e-,
a.s.
Consequently we have the following result. Theorem 4.2. (Hashiminsky [46]). Let P. be the probability law of the
solution X = (X(t)) with X(0) = x of (4.1). (i)
If
f 1/c+(r) f' [c+(u)/a+(u)]dudr = oo,
then PP(e = oo) = 1 for all (ii)
x E=- R4.
If
f 1/c-(r) f' [c-(u)/a-(u)]dudr < then Px(e < oo) = 1 for all x ERd. 5. Applications to diffusions on Riemannian manifolds First we shall introduce some necessary notions in differential geometry.
APPLICATIONS TO DIFFUSIONS ON MANIFOLDS
461
Let M be a d-dimensional complete Riemannian manifold and V be the Riemannian connection (cf. Chapter V, Section 4). To every pair X, Y E 3r(M) we associate a mapping Rx,.: 3r(M) - 3r(M), called the curvature transform, by (5.1)
RXY = Fcx.r) - (V VY - V4Px).
In local coordinates (x', x2, (5.2)
... , xd),
RxrZ = X'YJZkRalajak
where X = X'a1, Y = Y/a1 and Z = Z1a, (5.3)
where R1
= R1Jkl *
(al
=
x1) ; furthermore,
and R,Jk, = gfhRhJk,
was defined in Chapter V, Section 5 by
R'Jk1
Al = ak{11J} - al{k1J} + ({1a J} {k'a} - {kal} {lla})-
A plane section at x e M is a 2-dimensional subspace of TI(M). Let be a plane section at x and let X,YeTT(M) be an orthonormal of is defined by basis for . The sectional curvature
K(c) = .
It can be shown that depends on alone and is independent of the particular choice of X and Y. In local coordinates, K(b) = X'YJ YkXIR1Jk1
if X = X1a1i Y = Yla1 and {X, Y} are orthonormal. For X E=- X(M) such that 11XII = <X, X> =1, the Ricci curvature p(X) of the direction X is defined by d
p(X)=ZK({{X, Y1}}), 1-2
where {X,Y2i ... , Yd} is an orthonormal basis at TI(M) for each x and { {X, Y1} } is the plane section generated by X and Y, p(X) is independent of the choice of {YZ,Y3i ... ,Yd} . In local coordinates, *
is the inner product in each tangent space.
462
COMPARISON AND APPROXIMATION THEOREMS
p(X) = X'XJR1 j and
Rt j = Rkrjk
where X = X'at with IIXII = 1.
Definition 5.1. M is called a space of constant curvature if K(c) is independent of both the choice of the plane section at each x E M and of the point x E =-M.*'
Definition 5.2. Let y: I -- M be a smooth curve. y is called a geodesic if the tangent vectors y* = dt are parallel along y; in local coor-
dinates, y(t) = (y'(t),y2(t), ... ,yd(t)) is a geodesic if and only if (5.4)
dt2- (t) + {`kJ} dti dtj =
0,
t e I.
The following are a consequence of the existence and uniqueness theorems for the differential equation (5.4) :*2 for every x EMandX E TT(M) there exists a unique open interval J(X) in R' which contains 0 such that
(i) there exists a geodesic y: J(X) - M such that y(O) = x and y*(0) = X;
(ii) if a < 0 < b and c: (a,b) ----M is geodesic such that c(0) = x and c*(0) = X, then (a,b) C J(X) and c(t) = y(t) for t E (a,b). Definition 5.3. The above geodesic y is denoted by yx. We set
T,(M) = (X (E TX(M); 1 E J(X)) and define the exponential mapping exp: T=(M) --- M by (5.5)
exp X = yx(1).
It is also denoted by exp,X. Clearly
exp tX = y1(t).
Set
ro = rro(M) = max {r; there exists a ball in T.. (M) of radius r about 0 on which expxa is a diffeomorphism}. *t By Schur's theorem, the first property implies the second property if d Z 3 ([7]). *2 e.g. [47].
APPLICATIONS TO DIFFUSIONS ON MANIFOLDS
463
If M is a simply connected space of constant curvature c then r, is given as (5.6)
c>0
ro(c) )
c S 0.
100,
We also state the following fact known as Meyer's theorem. If the lower bound of the Ricci curvature is not less than (d-1)a2 > 0 * then M is compact and the diameter of M is not greater than 7r/a. Furthermore, the fundamental group of M is finite ([7] and [122]). Let = (01,02, ... , 9d-1) be the spherical polar coordinate on the sphere in Txo(M). (r = d(xo, x),0',92, ... ,9d-1) induces a local coordinate in a neighbourhood U of x0, called the geodesic polar coordinate, by the exponential map (r, ) 1-- expxo
Under this local coordinate, the Riemannian metric has the form 1
(gi,) =
0
ll
0(g,)I'
where (gt,(r, 01,02, ... , Bd-1)) is a Riemannian metric on the geodesic sphere S(xo;r)= {x; d(xo,x) = r, x E M} . The Laplace-Beltrami operator has the following expression L2 1
are
(a)a ,,/det
air
d-1 a + ,,/jet G i.,-1 (e(J/at---) a9! ae where G = (g11) ([47]).
Let h be the Jacobian determinant of expxo. Then we have (5.8)
ar log h
d
r 1- ar log
et G.
Also, if M is a space of constant curvature c, then * We assume a > 0.
464
COMPARISON AND APPROXIMATION THEOREMS
(5.9)
a a log A, ar log /det G = ar
(
where
1
- sin '/ a r)
0 < r < ro(c),
d-'
c>0
,
c=0
A=A(r,c)= rd-', 1
(
- sinh /=c r)
d'1
c < 0.
The following lemma plays an important role in our further discussions.
Lemma 5.1. Let the Ricci curvature p(X) satisfy that
(d - 1)b < p(X),
X E TT(M),
and the sectional curvature section . Then we have
ar
a for every plane
satisfy that
log h < ar log A(r,b) - d r
(i)
JlXii = 1,
log h > ar log A(r,a) -
1
,
d-1
r c- (0, r E (0, ro(a))
and
(ii)
h(r) < A(r, b)/rd-', h(r) > A(r, a)/rd-',
r E (0, rro(M)) r E (0, ro(a))
For a proof, see [7] (pp. 253 -r 256) and [42].
Generally, a coordinate mapping 0: U -- Rd, Uc M is called a normal coordinate mapping at x = ¢-'(0) if the inverse images of rays through OeRd are geodesics. A normal coordinate neighborhood N, the domain of a normal coordinate mapping 0, has the property that every y E N can be joined to 0-'(0) by a unique geodesic in N. As a corollary to Lemma 5.1 we have the following.
Corollary. If the Ricci curvature p satisfies p(X) > (d - 1)c, X E TT(M), JJXJJ = 1, x E M, then the volume of a normal coordinate ball
APPLICATIONS TO DIFFUSIONS ON MANIFOLDS
465
B(xo;r) is smaller than or equal to the volume of a normal coordinate ball of the same size in the simple space form of constant curvature c. By the simple space form we mean the sphere if c > 0, the Euclidean space if c = 0 and the hyperbolic space if c < 0, (cf. [7], p 256).
Now we shall apply the comparison theorem to the Brownian motions on M. Let x0 E M be fixed and let B(xo,r)= {x E M; d(xo,x) < r). For a fixed r* < r,,o(M), let X = (X,) be the minimal Brownian motion on B(xo,r*). As we saw in Chapter V, Section 4 and Section 8 the transition probability density p(t,x,y) of X exists with respect to the Riemannian volume m(dx): PX(X, E dy) = p(t, x, y)m(dy);
moreover p(t, x, y) is C" in (0, oo) x B(xo,r*) X B(xa,r*) and p(t, x, y) = p(t, y, x). Let r* < ro(c) and let ° = be the minimal L(c)-diffusion on (0, r*), where the operator L(C) is given by
Lcc> = I (d' 2 `dr
log A(r, c)) W'-) = + (d 1dr dr
1
1
d
2 A(r,c) dr (
A(r, c)
r
Let us assume that d > 2. Then
s () r
Y A (z, c) A(ro,c) = Jr,o' exp (-Jrro A(z, c) dz dy - Jf.o A(y,c) dy
*I
satisfies s(0 +) = -co. Hence the results of Section 3 imply that 0 cannot be reached in a finite time for °. Also, the transition probability density p (Q' (t, , r7)
exists with respect to the measure A(ri, c)dii, is C" in (0, oo) x
(0, r*)x(0, r*) and satisfies p(°'(t, g, r7) = p(°'(t, r7, ). Noting now (5.7), (5.8) (5.9) and Lemma 5.1, the comparison theorem immediately implies the following results. Theorem 5.1. Let the Ricci curvature p(X) satisfy that (d - 1)b < p(X)
for all X ETT(M), JJXIj = I and x EM. (Let the sectional curvature K(g) satisfy that
a for all plane sections
at every point x.) Let us
fix r* such that 0 < r* < rro(M) (respectively 0 < r* < ro(a)).*2 Then we can construct, on a suitable probability space, the above diffusions (X,) and (Ib)) ( respectively (X,) and such that o°) = d(xo,Xo)
*T 0
COMPARISON AND APPROXIMATION THEOREMS
466
(respectively o > = d(xo,Xo)) and the following is satisfied : if r, = d(xo,X,), then
fit') > r,
(5.10)
(respectively
t°' < r)
for t > 0, a.s. Corollary 1. (A.Debiard-B.Gaveau-E.Mazet [14]). (i)
limpca>(t,r,r1)
r110
(respectively lim p<°'(t, r, r) > p(t, y, x(,)). r,1o
(ii) If C is the explosion time (i.e. the hitting time to r* for (]") (resand the hitting time to 8B(x(,,r*) for (X,)), then E;')(C)
pectively
EY(O, y = (r, 0), 0 < r < r* (respectively E(o)(O > E,,())) where E() and E. stand for the expectations with respect to the probability measures with o°) = r and (X,) with Xo = x respectively. of
Remark 5.1. 0 is an entrance boundary for
and
in the sense
of Ito-McKean [73]. It is well known then that limp`°'(t, r, r1) and r,lo
lim p`"(t, r, rl) exist. rIl0
The proof is immediate from the above theorem and the corollary to Lemma 5.1.
Corollary 2. Let M be a connected, complete d-dimensional Riemannian manifold (d > 2) with non-positive sectional curvature for all plane sections. Furthermore, we assume that the Ricci curvature p satisfies the condition
0 > p(X) > (d - 1)c > -co, X E TX(M), IIXII = 1, x E M, then (X,) is conservative i.e.,
P(t,x,M)= 1
for all xE M.
Proof. Without loss of generality, we may assume that M is simply connected because the Brownian motion on M is obtained as the projection of the Brownian motion on the universal covering space M of M. Then r* = oo by the Cartan-Hadamard theorem ([7]) and the assertion follows immediately from the theorem.
467
STOCHASTIC LINE INTEGRALS
In this result, the condition on sectional curvature can be dropped. For details, cf. [186] and [206]. Similarly, we see that if M is a complete simply connected Riemannian manifold and satisfies the condition
0 > a > K(J > -oo
for all plane sections
then (X,) is non-recurrent. Further applications of comparison theorems, especially, applications to the smallest eigenvalue of the Laplacian, can be found in Azencott [191] Malliavin [108], A. Debiard-B. Gaveau-E. Mazet [14] and M. Pinsky [140].
6. Stochastic line integrals along the paths of diffusion processes Let M be a manifold and y be a smooth bounded curve in M. Then the line integral f r a is defined for any differential 1-form a. Now let
X = (X(t)) be an M-valued quasimartingale, i.e., t - f(X(t)) is a quasimartingale in the sense of Chapter III, Section 1, for every f E F0(M). We can then also define the "stochastic" line integral fxto tj a along the curve {X(s), s E [0,t]} for any differential 1-form a by making use of stochastic calculus; moreover the resulting process t f X10.03 a is a
quasimartingale. One standard way of defining this is as follows. We choose a locally finite covering (W.1 of M consisting of coordinate neighborhoods and choose coverings {UA} and of M such that
Define sequences {Q,F "} and {Tk )} of stopping times by
Q0, Tk> = inf {t ; t > ak711, XX E U.)
UP) = inf It ; t > Tk ), X,
Let a = a,(x)dxt and X(t) _ (X'(t)), rk) S t < o ), under the local coordinate in W,,. We define fxto,q a by setting f J x[o.r]
a
=
f
k
J k>nr
(X(s))odXi(s)
where { v,,} is a partition of unity subordinate to {
It is easy to see that
COMPARISON AND APPROXIMATION THEOREMS
468
f1ra.0 a is well defined and is independent of the particular choice of coordinate neighbourhoods (cf. [50] and [51]). In the following, we shall restrict ourselves to the case when X(t) is a non-singular diffusion on M. As we shall see, fx
dr(t) = Lk(r(t))odwk(t) + Eo(r(t))dt
(6.1)
{r(0)=r.*
Here {L1, L2, ... , L,,} is the system of canonical vector fields corresponding to the Riemannian connection and f, is the horizontal lift of a vector field b on M. The projection X(t) = n[r(t)] of r(t) onto M is the diffusion process on M generated by the operator
A+b. For simplicity,
we shall assume that X(t) is conservative. Let a EA1(M) be a differential
1-form and (a1(r), a2(r), ... , ad(r)) be its scalarization. Recall that ai(r) = aj(x)e;, i = 1, 2, ... , d, if a(x) = a f(x)dx> and r = (x', eD in local coordinates. Let a(b) E F(M) be defined by a(b)(x) = a,(x)b'(x) if a = a,(x)dx' and b = b'(x) ax' in local coordinates. Definition 6.1. We set (6.2)
J xro,:j a = ,1 o
ak(r(s))odwk(s) +
Jo
a(b)(X(s))&
fx[o,,I a is called the stochastic line integral of a E A1(M) along the curve {X(s), s E [0, t]}. This definition coincides with the above mentioned one. Indeed, if U is a coordinate neighbourhood and a < r are any stopping times such that X(t) E U for all t E [Q, r], then
ft o
ak(r(s))odwk(s) + f' a(b)(X(s))ds v
= f' a,(X(s))[ek(s)odwk(s) + bJ(X(s))ds] * As in Chapter V, Section 4, (w' (t)) is the canonical realization of the d-dimensional Wiener process.
STOCHASTIC LINE INTEGRALS
= fo a,(X(s))edXX(s)
469
for any t
It is obvious that t -- ixto.r7 a is a quasimartingale. Its canonical decomposition is given in the following theorem. Theorem 6.1. (6.3)
LOX a =
f
o
ak(r(s))dwk(s) + f o
(a(b)
-
2 aa) (X(s))ds
where b: d,(M) -- F(M) is defined as in Chapter V, Section 4. Proof. We have
f
J X[o, r)
a = f' ak(r(s))°dwk(s) + f' a(b)(X(s))ds 0
0
= fo ak(r(s))dwk(s) +
+f
2
f
o
d[ak(r(s))]
desk(s)
a(b)(X(s))ds. o
Since r(t) is the solution of (6.1),
d[ak(r(s))] = (Ljak)(r(s))°dw''(s) + (Loak)(r(s))ds
and hence d [ak(r(s))] . desk(s) _ (LJak)(r(s))dw'(s) dwk(s)
= Ek-l(Lkak)(r(s))ds. By Proposition V-4.1, we have
Lkak = (Pa)kk where {(pa)ts} is the scalarization of the (0,2)-tensor Pa. Consequently Lkak = (Pa),jerkee,
and by (4.22) and (5.14) of Chapter V,
COMPARISON AND APPROXIMATION THEOREMS
470
k-I
Lkak = E k-i(Va)!Jekek = gtJ(Pa)fJ = -8a.
Example 6.1. Let M = R2 and X = (X,(t), X2(t)) be the two dimensional Brownian motion. If
a=
(xidx2 _ x2dx'),
x = (x', x2),
2 then
2{ j X.(s)dX2(s) - f
LO ,r a
o
X2(s)d1l(s)}.
o
This stochastic integral in the case X(0) = 0 was introduced by P. Levy ([100] and [1011) as the stochastic area enclosed by a Brownian curve and its chord. We set
S(t) = f x10,=] a= 1 { f o X,(s)dX2(s)
(6.4)
-f
X2(s)dX,(s)} o
and
r(t) = ..IX,(t)2 + X2(t)2. Then by Ito's formula, r(2)2
(6.5)
=f
o
r(s)dB,(s) + t
where
B,(t)
'X2 (s) dX,(s) + f r(3) dX2(s). r = f -(s)) o
o
Clearly the system of martingales (S(t),B,(t)) satisfies the relations , = t, , = 0 and <S,S>, =
where ct is the inverse function of
t I--
4
f
r(s)Zds. 0
I f' r(s)2ds. Let B2(t) =S(Or)
4
o
STOCHASTIC LINE INTEGRALS
471
By Theorem 11-7.3, the Brownian motions B1(t) and B2(t) are mutually independent. We suppose from now on that X(O) = x is a fixed point in R2. Then we know that r(t) is the pathwise unique solution of (6.5) with r(O) = I x I (Chapter IV, Example 8.3). In particular, this implies that a[r(s), s < t] c a[B1(s), s S t], and consequently the processes {B2(t)} and {r(t)} are mutually independent. Thus we have obtained the following result. The stochastic area S(t) has the expression (6.6)
S(t) = B2(4 J'
r(s)2
)
where B2(t) is a one-dimensional Brownian motion independent of the pro-
cess {r(t)}. Suppose that X(0)=x 0. Then X(t)* 0 for all t > 0 and hence we can introduce the polar coordinate representation: X1(t) = r(t)cos0(t) and X2(t) = r(t)sinO(t). An application of Ito's formula yields
dS(t) = 2 {r(t)cos0(t)sin0(t)dr(t) + r(t)2(cos0(t))2d0(t) - r(t)cos0(t)sin0(t)dr(t) + r(t)2(sin0(t))2d0(t)}
= 2 r(t)2d0(t), and hence S(t) = 2 f o r(s)2d0(s).
Then
0(t) : = 0(t) - 0(0) =
f, r(s)2 dS(r),
and consequently d,
= r( t)2
d, = 0 and
d<0, 0>, = r4 d<S, S>, = r(t)2 dt.
Again by Theorem 11-7.3, there exists a Brownian motion
{B3(t)}
(B3(0) = 0) independent of {B1(t)} (and hence independent of the process
{r(t)}) such that
COMPARISON AND APPROXIMATION THEOREMS
472
r
B(t) = B3
J r (1s) 2 is). o
The formula
(6.7)
[Xi(t) = r(t) cos [9(0) + B3 (f o r(s)2 ds)] X2(t) = r(t) sin [0(0) + B3 (f
o r(s)2
ds),
is known as the skew product representation of two dimensional Brownian motion. As an application of (6.6) we can obtain the following formula in the
case X(0) = 0: (6.8)
E(et1stt)) = (cosh Zt)-
for
E R.
Indeed, noting the independence of {B2(t)} and {r(t)}, we have E(eicsct)) = E(exp [icr B2
(4 f
r(s)2ds)]) o
= E (exp [- g2 fo r(s)2ds])
1)2 = = IE(exp[ - gf o,i(S)2dr])
where ti(t) is a one-dimensional Brownian motion with r/(0) = 0. Since I
c
ri(ct)}
(q(t)) for every c > 0,
E(er{sa>) = JE (exp [- 82 t-
f o'i(s)2ds])}
2.
Therefore it is sufficient to prove the formula (Cameron-Martin [10] and Kac [80]) (6.9)
E(exp [- A f o 17(s)2ds]) = (cosh
/2--A)-112
(I > 0).
Let K(t, s) = t As and consider the eigenvalue problem in X2([0,1]):
STOCHASTIC LINE INTEGRALS 2
J
0
K(t,s)g$(s)ds
473
_ (t)
The eigenvalues and eigenfunctions are given by
(n+2)7r
2n
2,
n=0, 1,
= 2 sin (n + 1 7rx,
...
Now it is easy to see that
q(t) = n
where the n are independent random variables with common distribution N(0,1). Then 2
f Jor72(s)ds=EAR, and hence
E(exp[-A
Jo
rt2(s)ds]) = n E (exp [0
A.
ll
z J)
82
= R (1 + (2n + 1)2n2)
= n-0 II
1+2 A.
-1/2
_ (cosh
2A
We can also obtain a formula which is more detailed than (6.8) (Gaveau [33]). Let X(t) be the two dimensional Brownian motion with X(0) = 0. Then for every x EE R2, t > 0 and E R, (6.10)
scr>X(t) = x) = E(e'I
r t exp I (1 L 2sinh Zt
- 2t coth Lt) x2t
2
2J
(6.8) is an easy consequence of (6.10) and hence the following proof also provides us with another proof of (6.8) (or (6.9)). * * Still another proof of (6.9) may be found in [114].
474
COMPARISON AND APPROXIMATION THEOREMS
Proof. By the rotation invariance of the Brownian motion it is easy to see that E(e"sa) I X(t) = x) = E(etls
I r(t) = r)
where r = I x 1. By (6.6), we have E(etcsct) I X(t)
= x) = E (exp [- 4
f
r(s)2ds I
I r(t) = r).
0
But we know (Chapter V, Section 3) that
u(t, x) = E((exp [-a f' I X + X(s) 12ds])f(x+X(t))), t > 0, x E R2, 0
where a > 0 is a given constant, is the solution of the initial value problem
at=zGu-aIxL2u
(6.11)
luIt_0 =.f Let {HH(x); n = 0, 1, ... } be the Hermite polynomials and set n m(x) = (exp C- I x 12])Hn(xl)Hm(x2) for x = (x),x2) and n, m = 0, 1, ...
Since n m
satisfies
(A - I x 12)5b,,,m + 2(n + m + 1)0n,m = 0,
we can solve (6.11) by the method of eigenfunction expansion:
u(t, x) = f RZPa(t, x, y)f(y)dy where
Pa(t, x, y) _ E
n,m-0
en.m(x)en,m( )
and
en m(x) =
A formula on Hermite polynomials yields
STOCHASTIC LINE INTEGRALS
475
H,((2a)i'4xr)Hn((2a)'14Y)
Pa(t,x,Y)=II E e `' r-, -0 (6.12) 2
exp
)` e eTa(z?+r2)12
("
[_, 2a coth
_
2"n!(2a)-,l4
/7G
2at(x? - 2xysech tat +y J]
27z sinh /Tat (2a)-'12 x = (x,, x2),
Y = (Y,, Y2)
Therefore,
E(e"() I X(t) = x) =
0, x) (27tt exp [- 121 2) _3 t
tt exp (1 - 2 cosh2)
12t
2
2 sinh Zt
Finally we consider the three dimensional process
X(t) = (X1(t), X2(t), X3(0)+S(t)),
where (X1(t), X2(t)) is a two-dimensional Brownian motion and S(t) is defined by (6.4). It is a diffusion process on R3 with generator
A=
(L21 + Li), 2
where
It is a degenerate diffusion, but since L dim a {L1, L2} x
L
a
=3 for every x. Thus the smooth transition probability
density p(t, x, y) exists by Theorem V-8.1. It is easy to obtain from (6.10) that m
z
P(t, 0, x) _ u;;)
J
sinh /2 eXp
x + X2 2t
{t x3
1/2
tanh
/2}
dd,
x = (x,, x2, x3)
476
COMPARISON AND APPROXIMATION THEOREMS
(cf. Gaveau [33]). Also, by Example 8.1 given below, we see that the topological support 5S'(P.) of the law P,, of {X(t)} with X(0) = x is the whole space W, = {w; C([0, oo) -- R3), w(O) = x). For a further detailed study of this diffusion, we refer the reader to Gaveau [33].
Remarks 6.1. The formulas (6.8) and (6.10) can be obtained more directly by the following Fourier series expansion, (cf. [33], [216]): Let X(t) = (X'(t), X2(t)) be a two-dimensional Brownian motion such that
X(0) = 0 and set, for i = I and 2,
v/2 f oin 2nkt dX() and
k = -./ 2
f1o
os 2irkt W(t),
k= 1, 2,
Note that [ X'(1), X2(1), k'), 412), Yfj1), rlk2), k = 1, 2, ... )is a system of independent Gaussian random variables with mean 0 and variance 1. Let S(1) be the stochastic area: .
S(l) = I f
0{XI(s)dX2(s) - X2(s)dX'(s)}.
Then it admits the following expansion in the sense of .2 (P) and almost surely :
M S(1)
27Ik
2 X2(1))Sk1 }.
`(i 1) - ., 2 X'(1)JSk)
This can be deduced, at least heuristically, from the Fourier series expansion of X'(t): X1(t) = tX°(1) + / T E (k° f 'sin 2xcks ds + rik° f 'cos 2xcks ds) k-t
0
i = 1,2. It can be proved rigorously as follows: For F(s, t)E-9 ([0, 1]2), the multiple stochastic integral I (F)
= J of
(s, t
o
)dX' (s)dX 2(t )
is defined to be the iterated stochastic integral
STOCHASTIC LINE INTEGRALS
477
f o{ f F(s, t)dX'(s)}dX2(s). 0
This is well-defined because, setting .
=v[X'(u),XZ(v); 0
for 0
45(t). = f (s, t)dX'(s) is (.9)-adapted measurable process and X2(t) is an (.9;)-Brownian motion. It is easy to see that I(F) is also given by f o{ f F(s, t)dX2(t) }dX'(s) 0
and that, if F, G E 9Z([0, 1 ]2), then E[{ I(F) - I(G) }2] =
f 1J [F(s, 0
t) - G(s, t)]2 dsdt.
0
In particular, if F,,, FE=-2'([0, T]2) and F. -* F in .2([O, 1]z), then 2'2(P). Also it is clear that, if F(s, t) = f(s)g(t), f,gE
I(F)
Y 2([O, 1]), then
f g(s)dX2(s).
I(F) = f 'f(s)dX'(s)
0
Now define F(s, t)E P2([0, 1]Z) by
0<s
1/2, F(s,t)-{-1/2,0
Then I(F) = S(l). Also F(s, t) can be expanded into Fourier series in .9' ([0, 1]Z):
F(s, t) =
°° ( (cos 27cks - .. I -T) sin 27rkt
2nk
_
sin 2ncks(cos 27rkt - ../_T) 2nk
Now, (*) can be easily calculated in 9'Z(P)-sense and, since this is a sum
of independent random variables, a.s. covergence also holds. Then, if I a ( < 2n,
478
COMPARISON AND APPROXIMATION THEOREMS
I: = E[eas(l) IX'(l) = x1, X2(l) = x2}
-
= E[exp[Ek-1°k2n{ (77k1)
=kfI E[exp[ 2nk (ylkl) L
2x2) T ]]]
2 x')S 2) - (77k(') 2
a (n1k) x')kk2)]]E[eXp[- 2nk
-
for (x1, x2) E R2. We see easily that, if a e R, E[exp[ 2nk (rly) -
= E exp [-
=
(2n)-1J m
(1 -
2 nk J
2 a)tL2)]]
(nk2) - - 2 a)tk' )
mexp[ 2nk 2
(x -
)exP[(1 -1/2
2 a)y - (x2 + y2)/2]dxdy
- (2nk)) (_2_nk) a ]. o
2
o
1
2 2
Hence
_
-( I - k?1(1
7
2nk)2)_.
= 2 sin a/2 exp[(1
1
exp[k-1(1 -(
- 2cot 2) I x
Q
2
2nk))
2nk)21
1
12
x
12]
by using well-known formulas sin x x
x2 ) = II( 1 - n22 n ,
1
cot x = x + 2x
1
x2-n2n2
By an analytic continuation and the scaling property of the Wiener process, we obtain (6.10). The proof of (6.8) is similar. For another interesting proof of (6.10), Cf. Yor [236]. Example 6.2. Let M be a Riemannian manifold. The horizontal Brownian motion r = {r(t)} is a diffusion process on O(M) determined by the equation (6.1) with La = 0: (6.13)
dr(t) = Lk(r(t))°dwk(t) I r(0) = r.
STOCHASTIC LINE INTEGRALS
479
Let b be a vector field on M. The differential 1-form co(b) is defined as
usual by co(b) = b,(x)dx' where bo(x) = g11(x)b'(x) and b = b'(x) ax' It is easy to see that the scalarizations of b and w(b) coincide:
cv(b):(r) = b,(x)e{ = b'(x)fi = b'(r),
i = 1, 2, ... , d.
Suppose that b'(r) = a(b)t(r) is bounded on O(M) for i = 1, 2,
... , d.
Then (6.14)
M(t) =exp { 5 f b'(r(s))dw'(s) - 2
Jo
[b'(r(s))]2ds}
o
is an exponential martingale. By the transformation of the drift determined by M(t) (cf. Chapter IV, Section 4.1), we obtain a d-dimensional Wiener
process w(t) = (W(t)) where V(t) = W(t) - f' b'(r(s))ds. The equation (6.1) now becomes
dr(t) = Lk(r(t))°dwk(t) + Lo(r(t))dt
(6.15)
r(0) = r
I
where Lo is a vector field on O(M) given by Lo(r) = bk(r)Lk(r). It is immediately seen that Lo coincides with the horizontal lift of b. Thus the solution of (6.1) is obtained from the solution of (6.13) by the transformation of drift determined by M(t). In particular, the diffusion generated by the
operator
d+b is obtained from the Brownian motion by the same
2 transformation. Note that M(t) can also be expressed in the form
M(t) = exp L1 xro,tl
.(b) +
a[W(b)](X(s))ds
Jo
2
2 J o Jkw(b)ll2(X(s))ds]
=
exp [J
X1010
w(b)
2Jo
div(b)(X(s))ds 2 J o I!b11Z(X(s))dr,.
This follows immediately from (6.3) and
COMPARISON AND APPROXIMATION THEOREMS
480
E Y(r)2 =
d
e{e;bj(x)b,(x) = gfl(x)bb(x)b,(x)
= g,,(x)b'(x)bt(x) = Ilbll2(x)
7. Approximation theorems for stochastic integrals and stochastic differential equations As we have seen in this book, stochastic integrals and solutions of stochastic differential equations are most important and typical examples of Wiener functionals. If, in these definitions of integrals or equations, a Brownian path is replaced by a smooth path, then they are defined by using ordinary calculus and we are thus provided with functionals defined on smooth paths. A question naturally arises: if we substitute in these functionals defined on smooth paths an approximation of Wiener process (i.e., a process consisting of smooth sample paths which converge to Brownian paths uniformly a.s.), do they converge to original Wiener functionals? Since these functionals are usually not continuous in the uniform topology of the path space, the answer is negative in general and the limiting Wiener
functionals, if they exist, need to be modified according to ways of approximations. For such familiar approximations as piecewise linear approximations or regularizations by convolution (mollifiers), however, the answer is affirmative if we adopt the symmetric multiplication in the definition of stochastic integrals or stochastic differential equations. Let (WW,PW) be the r-dimensional Wiener space * with the usual reference family {.fi}. PW is simply denoted by P below. The shift operator Wo is defined by (0,w)(s) = w(t+s)-w(t). We shall 0, (t > 0): Wo consider the following class of approximations for a Wiener process. Definition 7.1. By an approximation of a Wiener process, we mean a family {Bo(t,w)=(B(t,w),B2(t,w), ... ,Bj(t,w))}6>o of r-dimensional continuous processes defined over the Wiener space ( W0', P) such that (i) for every w, t -. B5(t, w) is piecewise continuously differentiable, (ii) Ba(0,w) is .9a-measurable,
(iii) Ba(t+k(5,w) = B5(t, 6kaw)+w(k6) for every k = 1, 2, ... , t > 0 and w, (iv)
E[B6'(0,w)] = 0
for
i=1,2, ... , r,
* Wo = 1w a C([0, oo) --- R'); w(0) = Of and Pw is the Wiener measure on W.
APPROXIMATION THEOREMS
481
(V)
E[ I Ba(O,w)16] < c63
for i = 1, 2, ... , r, *
E[(fa
I .$Xs,w) I d$)6] < C63
for
i = 1, 2, ... r,
o
where
=
Ea(s,w)
d
Ba(s,w)
If {Bo(t,w)} a,o satisfies the above conditions, then for every T > 0
we have that (7.1)
E{ max I w(t) - Ba(t,w)12} - 0 05r5T
as o 10.
Thus {Ba(t,w)} a>o actually approximates the Wiener process {w(t)} . (7.1) is an obvious consequence of Theorem 7.1 below. A simple application of Holder's inequality yields that
E[(f
a 0
I a (s,w) I ds)D'(
f6
I A6 (s,w) I ds)D2 ... (fa I Ea (s,w) I ds)pm] 0
0
if p,> 1 and p,+Pz+
<
By (iii), this estimate also holds if f Holder's inequality, it follows that o
E[(f "a I E`a (s,w) I ds)D'(f 0
(7.2)
"2a
(*+1)6. Again by is replaced by fk4
I Ea (s,w) I ds)P2 ... ( f "mal Ea (sw) I ds)Dm] 0
g0
<
+p.::96.
n"m(/'f(D1+D2+...+Dm>
if 1 < P,, p, +p2+
+Pm < 6 and n, E Z+.
Let us introduce the following notations:
(7.3)
fr
s,j(t, S): =
E 11
t
[Ba(s,w)l&a(s,w) - &(s,w).a(s,w)]ds} o
2
* c, c', c,, c2, ... are positive constants independent of 6.
COMPARISON AND APPROXIMATION THEOREMS
482
and
c,,(t, S): = l-E { f E (s,w)[Be(t,w) - BB(s,w)]ds}
(7.4)
u
for i, j = 1, 2, ... , r and t > 0. Clearly (s1,(t,S)) is a skew-symmetric r x r-matrix for each t and S. We shall make the following assumption.
Assumption 7.1. There exists a skew-symmetric rXr-matrix (st,) such that
s,,(S, S) - s,
(7.5)
510.*
as
Set
(7.6)
ct, = si, + 2 Sty,
i,j=1,2,...,r.
Lemma 7.1. Let k(b): (0, 1 ] -. Z+ such that k(S) t oo as S 10. Then (7.7)
lam ctj(k(6)6, S) = c,,.
Proof. We set S* = k(6)6. Then for every n = 1, 2, ... , 2ncsrl(na, S)
= k-
a+l) S
= k 0 E[
{Bgs, w).
(s, w) - B'(s, w)-66'(s, w)} ds]
E[f o {(BB(s, ekdw) + w°(kiS)),&6(s, ekow) - (B/s, ekaw) + w'(k6))E (s, ekaw)} ds]
=
0) +
n-
E[w`(kO)(BB(S, ekaw) - BB(0, 0 w))
r
- w'(k0)(Bb(a, ekdw) - Ba(0, Okaw))]
= W s,(a, S) +
`
n-1
k
w) - Ba(0, w)]
- E[wJ(kS)]E[Ba(S, w) - Ba(0, w)]} * We may consider the case that s,1(Sn, s,, for some sequence 6. 10. All results below also hold in this case if we replace {SI by the sequence {an} .
APPROXIMATION THEOREMS
483
= 2n5s,,(8, 6).
Hence a*st,(8, a)
= a*s,,(ar*, a) 2
E[
d r ds (B(s, w)B'(s, w))ds] - E[ J o* B (s, w)E(s, w)ds] J o*
= 2 E[B6((5*, w)B6'((5*,w)) - 2 E[Ba(0, w)&(0, w)]
+ E[
fob*
E (s, w) {BB(S*, w) - BB(s, w)} ds]
- E[BB(S*, w) {Ba(8*, w) - B6(0, w)} ] = 2 E[(Ba(0, Oo*w) + w°((5*))(B,1(0, 05*w) + wJ(8*))] 2 E[Ba(0, w)Bb(0, w)] + 6*ci,((5*, a)
-E[(B'a(0, 05*w) + wf(6*))(Ba(0, Oj*w) + w`(S*) - B6(0, w))]
= 2 E[Ba(0, w)Ba(0, w)] + 2 E[wi(8*)w'(6*)]
- 2 E[Ba(0, w)Ba(0, w)] + a*c,,(a*, S) - E[BE(0, w)BB(0, w)] - E[wt(8*)wf(S*)] + E[wJ(a*)B3'(0, w)]
= a*c,,(a*, d) -
E[wl(6*)wJ(S*)] - E[Ba(0, w)Bb(0, w)]
+ E[w1(&*)BX0,2 w)].
The assertion is now clear since E[w°(8*)wf(6*)] = 6,,a*, I E[Ba(0, w)Ba(0, w)] I < (E[Ba(0, w)'])' "(E[BgO, WAY c25 = o(o*) and I E[w'(8*)B6'(0,<_w)] I
(E[w'(S*)2])l'2(E[BXO, w)2])1 /2
s C,(a* /251/2= OW).
l
/Z
COMPARISON AND APPROXIMATION THEOREMS
484
Before stating our main results, we shall first give several examples. Let 0 be the space of continuously differentiable functions 0 on [0, 1] such
that ¢(0) = 0, 0(1) = 1. Let 0 = dt 0 and dkw' = w'(k8+b) - w'(kS). Example 7.1. Choose 0' E sls, i = 1, 2, ... , r, and set BB(t, w) = w'(k8) + 0'((t - k8)/8)dkw',
(7.8)
if kSo satisfies all conditions (i) (vi) of Definition 7.1 and hence it is an approximation of a Wiener process. For example, E[{ f' Iha(s,w)Ids}6]=E[1dow'16](f I
0
Also, it is clear that sj(5, 6) = 0 and hence Assumption 7.1 is satisfied with
sty = 0. (Thus c11 = 2 S,J in this example.) If, in particular, ¢'(t) = t for i = 1, 2, ... , r, then {Bo(t,w)} a>o is the familiar piecewise linear approximation. Example 7.2. (McShane [115]). Let r = 2 and choose 16'E rh, i = 1, 2. Set
(7.9)
w'(k8) + o'((t - kS)/8)dkw',
Bs(t, w)
w`(ka) + 03-`((t - k8)/8)Akw',
dkw'dkw2 Z 0, dkw'dkw2 < 0,
if k6o is an approximation of a Wiener process. In this case, however,
2f
o
_
{.B (s,
w)E (s, w) - Ba(s, w)aa(s, w)} ds
J dow d0w2 1
2
(1-2 f
APPROXIMATION THEOREMS
485
and hence Assumption 7.1 is satisfied with
(1 - 2 f 01(S)o2(S)dS).
S12 =
Example 7.3 (mollifiers). Let p be a non-negative C°°-function whose support is contained in [0, 1] and f' p(s)ds = 1. Set P,($) =
for 8> 0
P (S)
S
and
(7.10)
Bgt, w) = f W w'(s)pb(s - t)ds = 0
f w'(s + t)pa(s)ds 0 o
for
i = 1, 2, ... , r.
It is easy to verify that {Ba(t, w)} 610 is an approximation of a Wiener process. Indeed, (i) . (iv) in Definition 7.1 are obvious and (v) and (vi) are verified as follows. We have I Ba(0, w)
12m
= ( fo
wt(aS)P(s)dS)2mam
and hence
E[ I BB(O,w) 12m] = (SmE[( f ' w'(s)P(s)ds)2m].
Also,
f w'(s+BJp'()dd
961(s, w)
and hence
E[( f 0
ds)2m]
I A;(s,w) I
= E[(f
I .a( s, w) Ids) 2162m 0
= E[(f0 I f
w'(8(s + ))P'()dd 1 ,*)2m] 0
486
COMPARISON AND APPROXIMATION THEOREMS
= E[(f 11 0 f 01 W"'(" +
S))
= E[(f 1 j f 1 wt(s +
p'( dd
N/
0
0
pi(b)dS I ds)Zm]am
ds)sm]8'".
Clearly s j(6,E)= 0 and consequently Assumption 7.1 is satisfied with
stj=0. First, we shall consider the approximation of stochastic integrals. Let a be a differential 1-form on P given by
a = E at(x)dxt. t-1
We shall assume that all partial derivatives of at(x), i = 1, 2, ... , r, are bounded. In particular, we have
I a,(x)1 < K(1 + xI),
i =1, 2, ... , r,
for some positive constant K. Set (7.11)
A(t, a; w)
= Lo,t] aE f
at(w(s))adwt(s) 0
and
(7.12)
A(t, a; B5) = f
a = t-1 Ef ablo,t]
t
at(Bo(s))ha(s)ds. 0
Theorem 7.1. Let {B5(t, w)} ago be an approximation of a Wiener process satisfying Assumption 7.1. Then
lim E[ sup I A(t, a; B5) - A(t, a; w) (7.13)
J10
OStST
-ftE
0 ij-1 stj ataj(w(s))ds
1 Z] = 0
for every T > 0. Here 8a j = a a j. Proof. * It is sufficient to show that * We are indebted to S. Nakao for the main idea of the proof.
APPROXIMATION THEOREMS
487
lim E[ sup I f' u(Ba(s, w))h6(s, w)ds osrsr
j10
(7.14)
0
- f u(w(s))
° dw'(s)
0
-f
0
(Ei-1s1 u,(w(s)))ds I Z] = 0
for every T > 0 and j = 1, 2, ... , r. Here u: Rr --- R is a C2-function such that all partial derivatives are bounded and u1(x) =
8
u(x).
First we shall introduce the following notation. Choose n(8): (0,1] --Z+ such that
n(6)4S 10 and n(8) j oo as 6 j 0.
(7.15)
We shall then write a = n(8)&,
(7.16)
[s]+(a) _ (k + 1)3
(7.17)
if kb < s < (k + 1)3
[s]-(a) = kb
and (7.18)
m(t) = m(t)(6) = [t]-(a)/3
Integration by parts yields ck+1)a
f
u(Ba(s, w))dBB(s, w) ka
= -f
ck+1)a
u(B6(s, w))
ka
TS
[BB((k + 1)b, w) - Ba (s, w)]ds
= u(Ba(k3, w))[Bb(k3 + 3, w) - Ba(kb, w)] (7.19)
+if -1
(k+1)a kJ
u,(B5(s,
w) - BB(s, w)]ds
_ (u(Ba(k3, w)) - u(w(k3)))[Ba((k + 1)a, w) - Ba(k3, w)]
+ u(w(k3))[B4((k + 1)a, w) - w'((k + 1)b)] -u(w(k5))[Ba(k3, w) - wl(k3)] + u(w(kS))[w1((k + 1)b) - wf(kS)] (k+1)a
+ i-1 E f ka
u1(Ba(s, w))h (s, w)[B'((k + 1)S, w) - BB(s, w)]ds.
COMPARISON AND APPROXIMATION THEOREMS
488
Also,
u(BB(s, w))dBBs, w)
r
5
= -u(Ba(t, w))[BB([t]+((5), w)
- B (t, w)]
+ u(Ba([t]-(a), w))[BJ([t]+(a), w) - Bi[t]-(b), w)] ur(Ba(s, w))fa(s, w)[Ba([t]+(b), w) - BB(s, w)]ds
+riJ
(7.20)
= -[u(B4(t, w)) - u(B5([t]-(a), w))][Ba([t]+(a), w) - Bit, w)] [u(Ba([t]-(a), w)) - u(w([t]-(a)))][8 ([t]+(a), w) - B'$t, w)]
-
- u(w([t]-(a)))[B'([t]+(a), w) - Bit, w)] + u(Ba([t]-(a), w))[B'a([t]+(a), w) - Ba([t]-(b), w)] t
+E
w
ut(Ba(s, w))B6(s, w)[B[t ]+(a), w) -
w)]ds.
By (7.19) and (7.20) we have
f u(Ba(s, w))dB s, w) 0
-f
u(w(s)). dw'(s)
0
- f t Es,,u,(w(s))ds 0 r-1
fo u(Ba(s, w))dBgs, w) - f o u(w(s))dw'(s) - f
t
01-1
c, ju,(w(s))ds
= - [u(Ba(t, w)) - u(Bo([t]-(a), w))][BB[t]+(a), w) - B'(t, w)] - [u(Ba([t]-(a), w)) - u(w((t]-(a)))][B'([t]+(a), w) - BJ(t, w)] - u(w([t]-(a)))[B'a([t]+(a), w) - Ba(t, w)] + [u(Bo([t]-(b), w)) - u(w([t ]-(a)))][B'([t]+(a), w) - B ([t]-(a), w)] + u(w([t]-(a)))[B6([t]+(a), w) - BB([t]-(a), w)] (7.21)
+'f' ,_
Cr7-rte
ur(Bo(s, w))E (s, w)[B [t]+(a), w)
-
B'a(s, w)]ds
- u(w([t ]-(a)))[w'(t) - w'([t ]-(a))] [u(w([t]-(a)))(w>(t)
- w'((t]-(a))) -
f` u]
u(w(s))dw'(s)]
N cr1ut(w(s))ds [t]-m r r
+
m1 (t -[u(B5(ka, k-0
w)) - u(w(ka))][B1((k + 1)a, w) - B'(ka, w)]
APPROXIMATION THEOREMS
489
M(r) -I
+ E u(w(kS))[BB((k + 1)S, w) - w!((k + 1)S)]
-
k-0
m(t
1
u(w(ka))[Ba(ka, w) - w'(kb)]
k-o
1u(w(k6))(w'((k + 1)g) - w'(kb)) - f o]
+[
r m(t -1
+E t-1
0
k-0
u(w(s))dw'(s)1
(k+1)a
fka
[ut(B,(s, w)) - ut(w(ka))].a(s, w)[Ba((k + 1)S, w)
- B(s, w)]ds
+
t -1 r m(E
ut(w(kb)) f
(k+1)a
[$a (s, w)(BB ((k + 1)S, w)
- B(s, w)) - cj(S, 8)]ds r m(t -1
+ t-1 E k-
r m(t -1
-I-N E
k-0
ut(w(k3))[cj(3, S) - ctj]a
f
(k+1)a
[u(w(kb)) - ut(w(s))]ds c,
ka
r
= I1(t) + I2(t) + 13(t) + 14(t) + I5(t) + E Its(t) + I7(t) t-1 + 18(t) + 19(t) + 110(t) + 111(t) + I12(t) + I13(t) r r ' r r + E Ii4(t) + E Iis(t) + E Ii6(t) + E I17(t) t-1
1-1
t-1
1-1
It is clear that (7.22)
E[ sup I I9(t)12] -- 0
as 610
and, by a martingale inequality,
(7.23)
E[ sTj 113(t) + I8(t) 12] S c4E[f {u(w([t]-(a))) - u(w(t ))} 2dt] o
-0
as
By Lemma 7.1, it is clear that (7.24)
E[ sup 1116(t )1 2] o5t5r
We have E[ sup I I,7(t)12] 0:9 5r
0
as
&10.
510.
COMPARISON AND APPROXIMATION THEOREMS
490
< c, Et-1E[(f T 0I wt([s]-(J)) - wt(s) I ds)9 (7.25)
< c,T E f a E[ I w`((sl-(b)) - %I(s) 12lds 1-1
< c,T2rS - 0
b 10.
as
Also, by (7.2),
E[ sup
I11(t)12]
05t5 r
< C6 E E[ sup (Bgt, w) - B`a([t]-(b), w))2(B'd([t]+(3), w) - B'B(t, w))ZJ i-1
05t5 T
< c6 E E[ sup i-1
r (7.26)
C6
ftj+(
I
hAs,
w) I
05i5T J (]W (f(k+ua E[ max
t-1 r
(
05k5m(T)
(k+1)d
m((Tr)
E[(f
< c6(m(T) + 1)
+1)a
.a(s, I
I $'a(s, w) I ds)21
[t7- (d) ds)2 (f"41
I Ea(s, w) I i
ka
C6 E A E[(f kd
flt)Tcd)
ds)2(
w) I
ds)2(f
(k+1)3 kd
Z
s w) I ds) l a(,
I
Ef(s, I
w) I
ds)2J
I ,§b(s, w) I ds)2(f I E (s, w) Ids)-] o
< c7(m(T) + 1)n((5)4b2 < csn(b)36 _ 0
as
b j 0.
As for I2(t), E[ sup I
I2(t)121
05f- 5T
< c, E E[ sup (BB([tl-(b), w) - wt([tl-(3)))2 1-1
05t5 T
x (BB([t]+(b), w) - B(t, w))21 < c9 E E[ sup (BB([tl -(S), w) - wi([tl -(V)))2( i-1
0
5t5 r
f
(k+1)a
= C9i-1 E E[0$k<m max(T)(Bj(k(5, w) - wt(k5))2( (7.27)
< c9E
m(T)
i-1 k-0
E[(Ba(k3, w) - wt(kb'))2(f
ka
(k+1)a I kd
ca)
I 1%(S' w)
Ct]-(a)
s, w) I ds)2J $Ja(f
his,
w) I
ds)2l
d
< c9(m(T) + 1) E E[(BB(0, w))2( f 0 B$(s, w) I ds)2] i-1
I Ba(s, W) I d3)4J} 1/2
< c9(m(T) + 1) E {E[(B6(0, w))4JE[(f 1-1
0
I ds)2J
APPROXIMATION THEOREMS
491
1)(a2n(a)4a2)1/2
S c1o(m(T) +
< c118n(8) - 0
as
&10.
In the case of I,(t), we have E[ sup I I3(t)12] 0-5f: 5T
'+ (a)
S c17,i-1E E[OstsD sup (1 + I wr([t]-(a))12)(f[r]-(a) I.$j(s, w) I ds)9 ((k+1)a
< c17, E {E[ sup (1 + I wt(t)12)2]E[ max t-1
(7.28)
OStST
< c13(E[ mE k-0
OSkSn(T)
kJ
I
ha(s, w) I ds)4]} 112
(Irk+ua
I E (s, w) I ds)4])liz
ka
< c14((m(T) + 1)n(8)462)112 as
< c15(n(8)'d)"2 --- 0
&10.
Similarly, as in (7.27) and (7.28), we can prove that (7.29)
E[ sup 1 I4(t)1 2] -- 0
as 610
E[ sup I IS(t)12] J --- 0
as
OStS T
and
(7.30)
OSrST
610.
For I6(t), E[ sup I I5'(t )12] OSrST
< c16E[ sup (
OStS T J [r]
(T)
(a)
I ha(s, w) I ds
I h j(S, fI"") w) I ds} 2] 4-M
(k+1)a
(7.31) < C17 Eo E[ { f ka
(k+1)3
I B6(s, w) Ids f
I Ej(s, W) I ds} 2] ka
< c1,(m(T) + 1)E[ { f o I Es(s, w) I ds f 01 .j(s, w) I ds} 2] < c18 n(8)8 n(t5)462
= c18n(8)38 -- 0
As for I,(t), E[ sup II,(t)12] OSt T
as 6 j 0.
COMPARISON AND APPROXIMATION THEOREMS
492
c19E[ sup 0+ I w([t]-(b)) 12) sup I w'(t) - w'([t]-(b)) 12] 0515 T
O5t5 T
sup I w'(t + kb) - wJ(kS) 14]}' 12
< c20 {E[ max
OSksm(T) 0
(7.32)
mE
< c20 {
k-0
r 112 E[ sup I wJ(t + kJ) - wJ(kZ) 14]} 05t5a
c21
1
n(8)8
(n(8)8)1
= c21(n(6)6)112 --- 0
as 8 10.
We estimate 110(t) by E[ sup I I10(t) 12] 0:9t-19T
m(t)-1
< E[ sup
[u(Ba(kS, w)) - u(w(kS))]2
OSt$T k-0
m( t
-1
k-0
[Ba((k + 1)9, w)
- B'(kS, w)]2]
< E[
m(T)-i
m(T)- 1
[u(B5(kS, w)) - u(w(kS))]z E [BJB((k+ 1)S, w) k-0
k-0
(7.33)
S {E[m(T)
m(T
-
Ba(k3, w)]2]
1
[u(B5(kS, w)) - u(w(kS))]4) m(T)-1
X E[m(T) E [BB((k + 1)S, w) - Ba(kS,w)]4} 112 {m(E
< c22m(T) i-1
k-0
1 E[ I B6'(kS, w) - w'(ko) 14] m(7)-1
x E E[ I BB((k + 1)g, w) - BB(kS, w) 14} 1 iz < c23m(T)28 {2E[ I BJa(0, w) 14] + E[ I w(8) 14]} 111,
c24m(T)28S < c25n(8)-' --- 0
as 810.
In the case of 111(t), we first note that nn(w)
=
u(w(k3))(BB((k + 1)8, w) - wJ((k + 1)8)) n
=k
u(w(k3))Ba(0, B(k+llaw)
APPROXIMATION THEOREMS
493
Hence, by a martingale
is an {.} -martingale where inequality, E[ sup 1111(t)12] 0<s5 T
= E[ max
1 f]n(w)12] OSnSm(T)-1
5 C26E[J?lm(T)-l(W)12] (7.34)
i l E[u(w(k3))2(B'a(0, 0(,+l)aw))2]
= C26
= C26 m(E l E[u(w(kg))2]E[(B'a(0, w))2] k-0
S c27m(T)&
as 610.
= c28n(5-1- 0 We can show in the same way that E[ sup 1112(t)12] -- 0
(7.35)
as 8 10.
O5r T
As for 114(t), E[ sup I I'14(t) 1 21 051ST
m(1 -1
(k+1)a
f k-0 kJ
< E[ sup { 0$t5T
[ I u1(Ba(s, w)) - u1(w(kb)) I I Ea(s w) I X
W)
(`(k+1)1
, m(T)_1
(k+l)a
k-0 fk8
I d ]ds} 2]
6\
u B S w)) X r(k+ua
h1 S w)
u W kb
w)Idflds}2]
Jka
(7.36)
m(r)_1 ('(ktl)a [ I B6(0,0kaw)
C29 E E[ {
k-0 . ka
1-1
I
X II h AS' w) ,
< c29m(T)
+f S
m(T)-1
E[ k-0 E{f 1-1
+f'
f(k+l)a I g6( ,w) I dc]ds} 2]
(k+1)1 ka
[(I Ba(0, Okaw) ka+1)3
ka 0
ka E,(, , ekaw)dti
I N17, 0,,,w) I d1l) I -aa(s, w) I c29mm(T)2
r-1
J
w) I dal ds} 2]
E[ { I Bfa(0, w) I f a I Ea(s, w) I ds f a0 I Ea(s, w) I ds 0
COMPARISON AND APPROXIMATION THEOREMS
494
+ J'
I Ba(r1, w) I d j Jo I Ba(s, w) I ds f o
w) I d } zl
I
a
a
< c3om(fl2 E {E[ I Bl(0, w)12(f 0
1-1
+ E[(f
o
I BX77,
I Ea(s, W) I ds)2( f
W) I d3)2l
I
0
W) I dn)2(f 0 B'{s, w) 12ds)2( f o I Ba(s, w) 12ds)2]} I
S C30m(i )z !E {E[ I Ba(O, w) 1
671 ! 3E[( f
0
B6ls,
W)
I ds)3 (foIBSS'w)Id)3]z/3
x + E[(oI Ba'(,1, W) I dn)2(f 0 Ea(s, W) I ds)2(f o C31 n(6)262 (S(n(S)663)213
0
< C32n(8)48
as
I Ba(s,
W) I ds)2]}
+ n(6) 6&3)
S}0.
Finally we shall prove that (7.37)
E[ sup 11113(t) 12] -- 0 OSrgz
as S 10.
For this, we first note that (7.38)
3c,,(3, S) = 6cu(3, S) + (a - b)c,,(J - S, S)
where c*,(3, b) = E[
f6 E(s, w)[Ba3,w) - BB(s,w)]ds]/S. o
Hence
Iis(t) =
\
mc')-1
ck+1){B(s, w)(B'6((k + 1)a, w) - Ba(s, w)) u.(w(ka)) fka+a
-c,(S-5,5))ds
+ E m
(7.39)
k-0
4+6
1
ui(w(kS)) f kJ
{B`a(s, w)(BB((k + 1)6, w) - Ba(s, w)) - c *j(5, S)) ds
= ' J1(t) + J2(t).
APPROXIMATION THEOREMS
495
It is easy to see that '7.(W) = kFJO ut(w(kS))
J ka+6
a Ins, w)(B ((k + 1)b, w)
- Ba(s, w)) - c,(3 - 6, 8)} ds 3-6
_ k-0 u,(w(k3)) f0
-
{Ea(s, Bka+6w)(B6(3 - 8, eka+6w)
Ba(s, eka+Sw)) - c, (3 - (S, b)} ds
is an {..} -martingale where Y"' =
Consequently
I J)(t) 12] E[OStSsup
= E[0
-
c33E[I lm(T)-1IZJ
m(
c33
k-0 1 E[u,(w(k3))2(
-6 {h61s, eka+6w)(B(3
- (S, 04+0)
fa0
- B(s, Bka+6w)) - c11(8 - 6, 8)} ds)2] a-6
< c34m(T)E[(f
{E (s, w)(B (3 - 6, w) - Ba(s, w))
0
- c, (8 - 8, 8)} ds)2]
(7.40) 30-6
{Ba(s, w)(B'6(3 - 8, w) - Ba(s, w))} ds)2]
< c34m(T)E[(J a
c34m(T)E[(f0 I B]{s, w) I ds)2(f 0 I fts, w) I ds)2] < C35
-i- n((5)462
= c33n(8)38 --- 0
as
6 j 0.
Also E[ sup 14(t) 12] 0515 T
7.41)
m(t) - 1
< c36E[ sup ( O5t5T
k-0
k3+6
I f ka {Ba(s, w)(B'6((k + 1)8, w) - B(s, w))} ds 1)7, + m(T)2S2c(b, 8)2].
By (7.38) and Lemma 7.1, we have
COMPARISON AND APPROXIMATION THEOREMS
496
as 810
Scu(8,'5) --0 and consequently (7.42)
m(T)x8xc (b, 8)2
- c (a cu(b, 8i1
x
-- o
as
810.
On the other hand, m(:)-1
ka+6
E[05tST supk-0(E f ka S E[ sup m(t) < m(T)
m
Ea(s, w)(Ba(ka + b, w) - Ba(s, w))ds I)']
m(')-1
ka+a
E (f kJ
E 1 E[(J
-96'(s, w)(Ba(k8 + 8, w) - B](s, w))ds)x]
ka+a
h s, w)(Ba(k3 + 8, w) - BB(s, w))ds)x]
(7.43)
= m(T)2E[(f
0
B`a(s, w)(Ba(8, w) - Ba(s, w))ds)2]
< c38m(T)x {E[( f
a.b 0
(s, w)(Ba(8, w) - Ba(s,w))ds)x]
+ E[(Be(o, w) - B61(0, < c38m(T)x {E[( f o I
+ E[( f I Ee(s,
Ba`(s,
w))2(BB(3,
w) - B6J(8, w))2]}
W) I ds)2(f I Ba(s, W) I ds)x] o
w) I ds)2 (Ba(0, 0aw)S-
Ba(0, Oaw)
+ (wi(S) - w'(8))} 2]} < e39
1
n(6)xd2
(82 + 8xn(8)) -- 0
as 81 0.
This completes the proof.
Next we obtain an approximation theorem for solutions of stochastic differential equations. Such theorems have been discussed by several authors: McShane [115], Wong-Zakai [181], Stroock-Varadhan [158], Kunita [93], Nakao-Yamato [126], Malliavin [106] and Ikeda-NakaoYamato [52]. Our result covers most of these results in a unified way. Main idea of the proof is due to S. Nakao. Let Ao
b`(x)
8x`
and A. =
,
o (x)
X, ,
n = 1, 2, ... , r
APPROXIMATION THEOREMS
497
be vector fields on Rd such that 6n E CI(Rd) and b' Es CI(Rd) {Bo(t,w)} a>o be an approximation of Wiener process defined on the Wiener space (Wo, P). We assume that {B5(t,w)} a>o satisfies the Assumption 7.1. Consider the following equations *1 r
(7.44)
dXX(t,w) = Z A,(Xa(t, w))dB6n(t,w) + A0(X8(t,w))dt n-1
Xa(0,w)=x and
dX(t,w) _
A,,(X(t, w))odwn(t) + A0(X(t, w))dt
(7.45)
+ E snm[An,AmJ(X(t, w))dt lsn5m5r
X(0 , w) = X.
These are equivalent to the respective integral equations (7.44)'
I (t, w) - x' =
n
f
, (Xa(s,
o
w))B$(s, w)ds +
Jo
bt(Xo(S,
w))ds
and
X'(t, w) - x' (7.45)'
o (X(s, w))dwn(s) + n-1
0
J
r b'(X(s, w))ds 0
+ n.m-l ac_, E Cnm $ Qn (Va:Qm)(X(s, w))ds i = 1, 2, .. .
,
d.
0
For any given x E Rd, the solutions of (7.44) and (7.45) are unique; we shall denote them by Xa = (Xo(t,x,w)) and X = (X(t,x,w)). In the following discussion, the common initial value x is arbitrary but fixed and so we shall often suppress it.
Theorem 7.2. For every T > 0 we have (7.46)
lira E[ sup X(t, w) - X5(t, w) 1 Zj = 0. 05t r 810
Proof. *2 First we note by (7.44)' that *1 We use the same notations as in Chapter V. *ZIn the following, K, K KZ, ... are positive constants. S has the same meaning as in the proof of Theorem 7.1.
COMPARISON AND APPROXIMATION THEOREMS
498
(7.47)
w) -
1
w) I < K( nF f s
.
(u, w)1 du + (t - s))
for every 0 < s < t. We have for each i = 1, 2, ... , d that (7.48)
X(t, w) - X`(t, w) : = H1(t) + H2(t) + H3 + H4(t)
where
H1(t) =
n-
-
(7.49)
a (Xo(s, w))Bgs, w)ds
fr Cr]-(a)
n-1
ft
c,`,(X(s, w))dwn(s)
CAS-(a)
t
- En.m(-1 E Cnm f Ct]-(a) (6naaO'm)(X(s, w))ds, a-1 -m H2(t)
r
=
{Ja]
- fu]-'
(7.50)
°n(Xa(s, w))Ena(s, w)d
a.'(X(s, w))dw'(s)
a
E c,, -E m-1 a-1
f -E
H3 =
om06aan(X(s, a a
a
an(Xs(s, w))Bna(s, w)ds -
r
n-1
(7.51)
w))d },
16-(a)
n-1
0
f 6n(X(s, w))40(s) 0
d
E Cnm J a0 QnaaCm(X(S, w))ds
n.m-1 a-1
and
(7.52)
H4(t) = f
'00
b'(XX(s, w))ds - f 'o b1(X(s, w))ds.
It is clear that (7.53)
E[sp H4(t)12] < K1 f o E[ I Xo(s, w) - X(s, w)12lds
for every 0
H1(t) = H11(t) - En-1H"12(t) - H13(t)
APPROXIMATION THEOREMS
499
Then it is obvious that (7.54)
E[ sup I H13(t)1 2] <- K2b2. 0
By a similar estimate as for (7.28), we have (7.55)
BE uprI H11(t )12] <
K3(n(6)s6)112.
Also
H-2(t) = o (X([t]-(S), w))(w"(t) - w"([tl-(S))) t
+f
(cn(X(s, w)) - Q"(X([t ] (S), w)))dw"(s)
= Hi21(t) + H122(0By a similar estimate as for (7.32), we have (7.56)
BE sup I Hi21(t)1 2] < K4(n(3)5)1 i2 G-e
As for Hi22(t), E[ sup I H4122(t)1 2]
< E[ sup
sup { f
OSksm(T) OStsa
< kE)
E'[E[ sup
ka+'
(on(X(s,
w)) - Q (X([s]-(b), w)))dw"(s)} 2]
ka
{ f 0 (u.(X(s, y, w)) - 6n(y))dw"(s)l 2]
X> E'[E[ f0 (o"(X(s, y, w)) - Q"(y))2ds] I Y-X(ka.wl)l (7.57) < K, ffl(T) kN
< K6
/
E'[E[ f
O
X'(S, y, w) - Y' 1 gds] 1 y-X(ka.w')]
< K,(m(T) + 1) f o [s + s2]ds < K88.
Here we used the estimate
I X'(s, y, w) - y' 12 < K9[(' f" "-1
*
0
E' stands for the integration with respect to W.
y, w))dw"(b))2
1Y-X(ka.w')l
500
COMPARISON AND APPROXIMATION THEOREMS
+ (f
y, w))dd) 2],
o
and hence
E[ I XJ(s, y, w) - Y' 2] < K1o(s + s2).
Putting these estimate together, we conclude that as 310.
E[ sup I H1(t) 12] = o(1)
(7.58)
ggr
It is obvious that I H3 I S sup I H1(t) I and hence 0:r5 T
E[IH3I2] = o(l)
(7.59)
6 10.
as
Finally we shall estimate H2(t). We have w)ds
fka
_ on(X6(k3, w))[Ba((k+1)b, w) - Bg(k3, w)]
+ Ed
rck+lfa
$=1 J ka
0vn(X5(s, w))[E ofl(Xo(s, w))EJa(s, w) + bfl(Xo(s, w))] J®1
X [Ba((k+1)3, w) - Ba(s, w)]ds d
= J1(k) + E Jjx(k) fe1 Also
J1(k) = Qn(X6(ka - 6, w))(w"((k+1)3) - w"(k3)) + [o';(X6(k3,w)) - on(Xa(k3 - 6, w))](Ba((k+1)S, w) - B,-5(k3, w))
+ 6n(Xe(k3 - 6, w))(Ba((k+1)b, w) - w"((k+1)3)) + Q'(X5(kJ - b, w))(w"(kb) - Ba(k8, w)) = J11(k) + J12(k) + J13(k) + J14(k)-
Hence, writing H2(t) = E H-2(t), n-1
Hi(t) = I1(t) + IZ(t) + 13(t) + 14(t) + I5(t), {51} is the coefficient of the vector field A0 + E ;_[AA, AA,]. ALAI
APPROXIMATION THEOREMS
501
where
f
[cr(a)
m&)-1
I1(t) =
(7.60)
op(X(s, w))dw"(s)
a
m(r)-1
(7.61)
II(t) = E
(7.62)
15(t) = E { k-1 E Jj(k) --
M(r)-1
d fl-1
f [fl-WEr
w))ds}.
1-1
a
First we write 11(t) as
= I1(t)
[an(X5([sl'(a) - (5, w)) - an(X(s, w))]dw"(s)
and then use a martingale inequality to obtain E[ sup I I1(t) 12] < K1 1E[ I I1(t1) 12]
S K12 (7.63)
f
uiJ-a)
E[ I X5([s]-(3) - 8, w) - X(s, w) 12]ds
a
K13 { f 'o E[ I X6(s, w) - X(s, w)
+f S K1a{fo
w-(a) E[ I
12]d'
yX3([s]-(a) - 8, w) - X3(s, w) 12lds}
a
E[IXo(s,w)-X(s,w)I2]ds
[s]t(J)
+ f T E[1-IE (f is]-W-6
Al6(t, W) I d 2 + (b + 3)2]ds}
a
K1s { f E[ I Xj(s, w) - X(s, w) 12]ds + n((5)Za}
for every
0 S t1 S T.
In the case of I2(t), E[ sup I I2(t) 12] 0sC r
C
m(77-1
1
E[ E (an(Xa(kJ, w)) - Qn1(Xa(k8 Y - 6, w)))2 k-1
502
COMPARISON AND APPROXIMATION THEOREMS
x (7.64)
m(E
1)3, w)
k-1 m(T
< K16 {m(T)
-
Ba(k3, w))2]
1
- E[ I Xa(k3, w) - X6(k3 - 8, w) 14]
k-1
m(79-1
x m(T) E E[ I Ba((k+ 1)3, w) - Ba(k3, w) 14]] 1 J2 k-1
S K17(m(T)2U2m(T)23211)2 1
C K1S(n(a))-1 _.. 0
as
610.
Similarly as for (7.34), we can show that (7.65)
as 610
E[ sup 113(t) 12] < K19(n(!S))-' --- 0 05tST
and (7.66)
E[ stpTI I4(t) 12] <- K20(n(6))-'
as a j 0.
0
Finally we shall consider I5(t). Write I5(t) as
15(t) =g-1 E "'(0, where m(0-1
m(t)-1
E f k-1
S(t) - LIk-1f 12(k) r-1 m(t)-1
E3I + m(t)-1 r
(k+l)a
(k+1)a r
i
ka
E J-1
c1"(QSaQQn)(X(s, w))ds
[(a9a8Qn1(Xa(s, w))
w))]Ea(s, w)[B"l((k+1)b, w) - Bl(s, w)]ds
f
(k+1)a
(s, w))ds
ka
m"-' r(k+1)d
+EE
(asaso")(Xa(ka, w))[B(s, w)(Ba((k+1)8, w)
J ka
- Ba(s, w)) - c1"(b, S)]ds r m(:)-1
(k+l)a
+ E kr) Jfka r
[(apon)(xa(kb, w)) - (69afO"`)(X(s, w))]ds c1"
m(r) -I
+EJ-1E "(Qjfasa")(Xo(ka, w))(cj.(a, 6) - c1") k-1
APPROXIMATION THEOREMS r
r
503 r
r
= E 151(t) + 152(t) + F. I'53(t) + E I54(t) + J-1
155(t)
1-1
J-1
It is obvious that (7.67)
E[ sup 11-1.55(t)12]
As for 54(t), E[ sup I I54(t)1 2l 0SrSr,
< E[( o I (6iae6 (X'a([sl -(b), w))
S TE[ (7.68)
Ja
-
w)) I ds)Zki
w)) -
I
(6jfla,66n)(X(S w)) 12d$ jc12
<_ K22 f E[ I X(s, w) -- Xo([s]-(b), w) 12lds 0
< K23[ f"0E[ I X(s, w) - Xa(s, w)121ds + f 0 E[ I X5(s, w) - Xo([s]-(3), w) 12]ds]
< K24[
O
E[ I X(s, w) - XX(s, w) 12lds + n(l )26]
for 0
m(r)-I
(k+1)d
E[OSrST sup (k-]E f kd (7.69)
< K25E[ {
(k}1)d LL
w)) I ds f
I
m(T)-1 _ (k+i)a
f
k-1
ka
m(T)-1
K26b2m(T)
ES
I Ea(s, w) I ds} 2] (k+1)3
E[(f ka
I B$(s, w) I ds)2]
< K26V2m(T)2n(V)2S
< K2,n(8)2S -- 0 We estimate I51(t) by
as 610.
kd
I E (s, w) I ds)2]
504
COMPARISON AND APPROXIMATION THEOREMS
E[ sup J r5,(t) 12] 0
d
m(T)-1
l-i
k-1
< K,,, E E[ {
(k+1)a
f
I Xa(s, w) - Xa(k6, w) I I Bas, )+') I ds
ka
r(k+i)a
x m(T)-i
J
(k+l)S
(k+1)d
d
Kz9E[ k-1 { E (E f ka 1-1
I B`a(s, w) I ds + 6)f
f
x
(7.70)
< K30
1-1
m(T)
m(T
k-1
1
{E[(f
B"a(s, W) I ds} 23
(k+i)a
(k+1)a
I Ba(s, w)
(f
I Bs(s, w) I ds
ka
xa+1)a I
)2(fka
ka
x
+ 32E[(f
I Ba(s, w) I cuss} 2]
kJ
I Ba(s, W) I dy)2
(k+ua I B$(3, W) I ds)2]
kJ
+1)a
ka+ua
I B'a (s,
w)
Ids)'( f ka
IBa(s, w) I ds)2]}
< K,im(T)z(n(b)6E3 + n(8)684)
< K,,n(b)46 -. 0
as
61 0.
By repeating the same proof as for (7.37) we see that z
(7.71)
+ n(5)-1) --- 0
E[ upTI I53(t)121 S K33 (n(6)36 +
as 610.
Now combining (7.67), (7.68), (7.69), (7.70) and (7.71), we obtain the estimate (7.72)
01
E[ sup i I J5(t) 12] < K34 f E[ I X(s, w) - X5(s, w) 12]ds + 0(1)
as
610,
where o(1) is uniform in ti E [0, T]. By (7.63), (7.64), (7.65), (7.66) and (7.72), we have (7.73)
E[ sup I H2(t)12] <_ K35 f o E[ I X(s, W) - Xa(s, w) 12]ds + 0(1),
where again o(1) is uniform in t1E[0, T]. Finally, by (7.53), (7.58), (7.59) and (7.73), we have
APPROXIMATION THEOREMS
505
E[ sup I XX(t, w) - X(t, w) I'] 0st
(7.74)
1
S K36 f E[ I XS(s, w) - X(s, w) I Z]ds + o(1).
As before o(1) is uniform in t, E [0, T]. (7.46) then follows from (7.74) by a standard argument. This completes the proof.
Remark 7.1. If the vector fields A,, A2, ... , A, are commutative, i.e., [A., Am] = 0 for n, m = 1, 2, ... , r, then the limiting process of (X5(t, w)) does not depend on the choice of the approximation {BB(t, w)} as Theorem 7.2 shows. But this is seen more directly in this case if we notice that X(., w) is a continuous functional on Wo by the result of Doss (Example 2.2 of Chapter III, Section 2). If A,, AZ, ... , A, are not commutative, however, X(., w) is not continuous on Wo and the limiting processes of w) are controlled by {sj} . For piecewise linear approximations or approximations by mollifiers, we have sty = 0 and so the limiting process
is the solution of the stochastic differential equation corresponding to vector fields A,, A2i ... , A, and to A0 as was defined in Chapter V, Section 1. Remark 7.2. In the proof of Theorem 7.2, we have actually shown that (7.75)
lim sup E[ sup I Xa(t, x, w) - X(t, x, w)1 2] = 0. &10 xeRd
05t5T
In the following, we shall restrict ourselves to a class of piecewise linear approximations. It is an open question whether our results below can also be obtained for more general class of approximations. We shall assume that the coefficients an and b' of vector fields A0, A,, ... , A, are C°° and bounded together with derivatives of all orders. In the following discussion, the terms involving the coefficients b` do not cause any trouble and so we shall assume, just for simplicity of notations, that A0 = 0. Thus we consider the stochastic differential equation (7.76)
dX(t) = a(X(t))odw(t). *
Also, setting for each n = 1, 2,
... ,
wn(t)-nr(lnt)w(n)+(t- n)w(7n 1)1 * We shall use the matrix notations below; in particular a = (aa.) and w = (wt).
COMPARISON AND APPROXIMATION THEOREMS
506
jn < t < j+1 n
if
we consider the following ordinary differential equation
d "(t) =
(7.77)
d ° (t).
The solutions of (7.76) and (7.77) with the initial value xERd are denoted by X(t,x,w) and X"(t,x,w), respectively.
Lemma 7.2. Let T and N be arbitrary given positive constants. Then we have
sup sup E[ sup II D"X"(s, x, w)II'] < oo
(7.78)
IxISN
,.
05sST
for every p > 2 and multi-index a.
Proof. First we consider the case a = (0, 0, ... , 0), i.e., DOX. = X. We denote
dkw=w(kn1)-w(nk )' Then, if
k
X"(t, x, w) - X. (_t_ , x, w)
k/n
a(X"(s, x, w))dkw ds n
=Q(X"(n , x, w))dkwn(t
- n)
+ Jki" [ais, x, w)) - a (X°( and hence
X"(t, x, w) -- x
=%' x' w))dkw k-0 a(`Y"(n '
n , x, w))]ds dkw n
APPROXIMATION THEOREMS
+
(k+1)/n
(nr]-1
k-0 Ski,, k1.
507
[a(X(s,x'w))-o(X,1(n'x,W))]dsdkwn
+ a(X,11[nt)' x, wd(nt]wt - [n] )n r
J'nt]/n[a(Xr(S. x, w)) - o (X.1 ni' x, + =Il(t)+I2(t)+I3(t)+I4(t)
w))] ds AE t]w n
By Theorem III-3.1, E[ sup III1(t)II'] 05t5T
KIE1(Ent]
<
w))II2)''2]n
I
Ila(Xn(n f x,
<
D/z *1
K2nD/ln-y/2
=K2
E[ sup I I2(t) I') 05t5T
<
T°_'ln°-'E[05t5supT 1nt1 -1 k.E0
f(k/n k+l)/n
IIJ
-or < T °-lnP-1E
[t
[ci(Xn(s, x> w)) (X,(n,x,w))]dsdkwIl']n'*2
1 IIJ /`(k+l)/n
k-0
k/n
xf w))
L
-Q(Xn(n'x,w))]dsdkwll°]n' (nT]-1 f(k+1)/n
< T°-'n°-'n'n-(°-')k-0 E
J
k/n
- a (X,1(n < K3n' k-31J
k/n
E II Q(X,(s, x, w))
' x, w))] dkwlV]ds
1)/n E[II Xn(sf xf w) - X. (n , x, w)II° IldkwIl] dr
*1 K,, K2, ... are positive constants independent of n and x. k k *2 Generally, if A,,> A 2, ... , Ak are matrices of the same type IIE A,II 5 (E IIA,IIY
'
k
5 ka-1 E IIArII'. ,-1
508
COMPARISON AND APPROXIMATION THEOREMS [nn-1 (k+1)/n
Ef
K3n2p
s
E[II f k/n o(Xn(u, x, w))dkwdulI'II dkwII ']ds
k/n
[n77-1 f(k+1)/n
< K3n2p E
J k/n
[nTf)t-1
< K4n2p
ft
E[(J k/n
Lf
II d kwII du)1II dkwIl p]ds p
(k+1)/n I
k/n
S
n) ds EQldkwll2')
< KSn2pn n- (p+1) n-p
=K5
<E[ sup
Ilo'(Xn([n],
x, w))d[nt)wllp]
< K6E[ sup IId[nt)wlip] o
< K6E[ max Ildkwllp] 0
[nfl
< K6 E E[II dkwllp] k.-0
< K,([nT] + 1)n-p/2
< E[ sup
t
0
I I (6(Xn(S, x, W))
- o(X ([n t], x, < E[ sup
W)))d[nt)wlIpdS]npn-(p-1)
([nt)+1)ln
0<,
llo'(Xn(S, x, w))
6 (XX([nti, x, W))IIPII4[nt)wllpds] n
<
nfl
K9n[E
k-0 E
Kton1-p
(k+1)/n
[f k1n
II Xn(S, x, w) - xn(n , x, w) II°IldkwIIpdS]
APPROXIMATION THEOREMS
509
K11(1 + IxV').
E[ sup II Xx(t, x, w)II'] 05t5T
Thus (7.78) is proved for a = (0, 0, ... , 0). Next, we consider the case of the first order derivatives. Setting Y.01 x, w) = (8-J X.*' x, w))
and Da = (axJ 011.) '
we have
Yn(t, x, w) = I +
Do (X (s, x, w))YY(s, x, w)wn(s)ds
Hence
Yn(t,x,w)-I [n
I
k-00
Do(X"(nw)) (k+t)/n
Cnt]-1
+ A J k/n
n
w)dkw
IDc(X,(s, x, w))
-Dv(Xn(n,x,w))]dsYn(n,x,wAkwn (k+1)/n
Cat]-1
+
k-0
+ DQ(Xn ( ( J
Dcr(X,(s,
$ k/n [nt]
[cJis+
w))(Y,(s, x, w)
x,
, x, w)) Yn(
[nt]
, x, w)A[ntlw n(t
w))dsdkwn
)
, x, w))
[at]/n
- DQ(Xn ([nt]n +
- Yn(n ' x, - [nt]
x, w))] ds y n
([nt] x, w)Akw n n
ft
Cnt]/n
Dc(X,(s, x, w))(Y,(s, x, w) - Y"( nt]' n x,
w))ds Akw n
= J1(t) + J2(t) + J3(t) + J4(t) + Js(t) + J6(t ) *
We use the following notations: for A =
B = (bj) and C
ABC = (dp
where dJ = E ak ab;ca. It is easy to see II ABC II 5 IIA ll IIBU II C p where 11411 = (E I ai,a 12)1/2 k,a
1,,,a
COMPARISON AND APPROXIMATION THEOREMS
510
Let t1 E [0, T]. By Theorem 111-3. 1,
I -l
E[ sup (IJI(t)II°] ostst,
r
K12E[(EtkE
(
(
n'
I ljDa lX"1
(7.80)
<
r((tntl]-I
< K l,
Y.(n
w)Ilz)°/z] n-D/z
x,
7p" n-1/2 x, w)11)
K13E[(
K,an' E
1
x, w)111211
E(IIYY(kn,x,w)Il°)
E( suP II Yn(s, x, w)il°)dt. o
J0
5ssr
As for J2(t), E[ sup 0 tst,
IIJ2(t)II °]
E suP II O
,
[ntl-1 fk/n (k+1)/n
Stst' k-0 J
[Dox(s, x, w))
- Do(Xn(n ' x, w))] dS Yn(n [nt]-I
< TD-1n2y-1E sup f 11f 0Sts[, k-0
, x, w)dkwl [D]n'
(k+l)/n
IDQ(X°(s, x, w))
k/n
DQ(X4(k , x, w))]1 ds Yn(n x, w)dkwll°] I [nt1]-I
S T D-1n2D-I E
-
E[IIJ
Do'(Xn(nk
(k+I)/n
[DQ(Xn(s, x, w))
k/n
1
, x,w))]dS
< T°-ln2D-In (D-I) rntl]7]-1 E r(ktl)/n E[II k-0 J k/n
- Do (Xn(n , x, w))}
(7.81)
Dal]-1 /`(k+l)/n
k/n
Yn(kn'x,w)dkwll°]
IDa(Xn(s, x, w))
Yn(n ' x, w)d kwll D] k
E[IIX1(s,x,w)-X4(n ,x,w)IID
x l[Yn(n , x, w)II DIIdkwIID]ds (ktI)/n
[nt1]-1
- Kisn2D
o
f
kin
s
E[IIJ k/n a(Xn(u, x, w)) dkw dull'
APPROXIMATION THEOREMS
Yn(L, x, w) II°Ildkwll°]ds
x 11
(,t1)-l (k+1)/n < K16n2D
f
3q
k ° ds E [II Y ( k
(s - -) n
r(k+i)/n
[nt17-1
= Kl6n2D
k/n
(s
J k/n
511
w) II° Ildkwll2° n , x,
_ n) k yds E[IIYn(nk , x, w)ll° E[Ildkwll2p)
[nt1J-1
<
K,7n2yn-(y+l)n y
E E k-0
IIYn(n , x, w)II°
[nt1)-1
= K17n-' E E[IlY.(n, x, w)II° k-0
S K17 f
E[OSsSt sup II YY(s, x, w)lI°)dt.
0
For J,(t) E[ sup IIJ3(t)II°I 05t5t,
r
= E I sup I I L0St ,tj
[nr7-i
f k-0
(k+1)/n k/n
Da(XX(s, x, w)) Yn(s, x, w)
- Yn(n, X, w)1dsdkw rally]
(7.82)
< TI-'n°
[n'13-I 57,
k-0
f (k+l)/n E II Da('n(s, x, w#1111 US, x, w) k/n
- Yn(n , x, w)II°Ildkwll°]ds [rat
K18 ny
If
I (`(k+l)/n J k/n
k-0
(, E US, X, W) - Y. (n - , x, w)lI°II dkwI y]l A [ll
kn <s
Da(Xn(u, x, w)) Y,,(u, x, w)dkw dull2n2
k/n
< n9lldkwli2 f'
II Da(Xn(u, x, w))ll2du
k/n
S- kIf )f
k/n
f
=
II YY(u, x, w)112du k/n
ll Yn(u, x, w)ll2du
COMPARISON AND APPROXIMATION THEOREMS
512
<
X,
)V)112
+nlldkwll2f,
k/,,
From this inequality, we obtain the following estimate:
IIUs,x,w)- Y.
(k
,x,w)I12
x, w)IIZ exp{K2onlldkw112(s
< K2olldkwll2ll
-
and consequently, Y.
(k
,x,w)II°
(7.83) < K21JI4kwl1°II Y» (n , x, w)II° exp {K2211dkw112}.
Substituting (7.83) into (7.82) yields E[ sup I1J3(t)II1] 02 tsr,
S K23n°n-'
tnri7EII
(7.84)
x,
w) I1°
x E[Ildkw112° exp {KZZI1dkw112}] W13-1
S K24 n E E II
x, w) II°
k-0
< K24f 0
0stst
w)II°]dt
since E[Ildkwll2' exp {Kzzlldkw112} ]
-fR,(2acn)-.izIxl2pexp{K22IxI2n
<
K25n-'
J'4(t) is estimated as follows: E[ sup I14(t)JID] 0stst,
Ix12}dx
for sufficiently large n.
k )j
513
APPROXIMATION THEOREMS
= E[
x, w)) Yn([ntl x, w)a(t
S K26Er sup
[nt])dc.t3wllpi
x, w)II'IldCnt7wIl']
t:5t11IY.([n]
(7.85)
-
,x,w)II'Ildkwll']
k-0 Cnt13
EE k-o
SK27n-pie
J"
< K2$(n1-p/2
0
llYn(n,x,w)II'
E[ sup II Yn(S, x, w)IIldt o5.,:!gt
+n-p/2EElIYn([n1l, x,w)II°]) In the case of J5(t), E[ sup IIJS(t)II'] 05tst1
< K29nE
sup E f[nt]/n II XX(S, x, w) - X.([nt] n
05t5t,
(7.86)
x W II °dS
x HY.([nl, x, w)II'Ildxwll] (nt1
E(IIYn()1
,x,w II'
< K30(n'-p f" E[ sup II Yn(s, x, w)II']dt 0
05.,5:
x, w)IIp,)
+ by the same estimate as for J&). As for J,(t), E[ sup IIJ6(t)II'] Ostst,
Cnt1J
(k+1)/n
f".
E[IIYn(S,x,w)-Yn(n,x,w)II'Ildkwllp ds
(nt17
< K32n' Et E[II Yn(n x, w)II' < K32(n'-p f" E[ sup1 II Ya(S, x, w)II']dt 0
05,5
+ n-PEI IIYn([n1], x, w)II'])
COMPARISON AND APPROXIMATION THEOREMS
514
by the same estimate as for J3(t). Combining (7.80), (7.81), (7.84), (7.85), (7.86) and (7.87) together, it is easy to conclude that E[stp,11YY(t, x, w)IIP]
K33(1 + J o E sup IIYY(s, x, w)II'}dt).
By a standard truncation argument, we can easily deduce from this that (7.88)
K33e'33 T.
E[ sup II YY(t, x, w)II'] 0StST
a such that I a I = 1.*1 Now we proceed to the case of I a I = 2. First we note that if k/n < t S (k + 1)/n, we have Thus (7.78) is proved for every
(7.89)
x, w) - X.(-L, x, w) II'] < K34n-D12
E[II XX(t,
and, also by (7.83) and (7.88), E[II Yn(t, x,
(7.90)
w) - Yn(n , x, w)II'J < K35n12,
Set
Y51y?2(t, x, w) =
a2axj2
axI
X '.(t, x, w).
Then Yi,12(t, x, w)
= k-1 E a-1 EJ
t
oa(XX(s, x, w))kYY; i''z(S, x, w)wn(s)ds
t
+ k.1- a-1 J 0 o (Xn(s, x, w))$,Yn(s, x, w)%Yn(s, x,
w)12wa(s)ds.*2
We denote the second term on the right by an(t, x, w). Then, denoting as d
r
Cnt7-1
(m+I)/n
/ x, w) = E a,(t, k.l I E [ E m/n
t
+ Cnt]/n
*11t was actually proved that sup can be strengthened to sup. : IxlaN
012
.2 aa(x) j'=
a .'(x) zL
and
dQ (x)kt =
axkax<
aQ(x)-
APPROXIMATION THEOREMS d
515
r
E (HI(t; k,1, a) + Hz(t; k,1, a)), k.1-1 a-1 HI(t, k, 1, a)
+
Cnt]-I
f M-0
Iaa\Xn(n ,
(m+l)/, m/n
X ,W))kl Yn(
n n1 ! ln(n'X°W)) [o.;(x(s,x,W))tk! -Qa V(_ ki
X Yn( 7n ,X,
n
(m+i)/n
Cntt)-1
+
m-0
f
n' X, W)h Yn( n' X, W)J2dmwa
t
n1 W)i,k Yn(-,X,W dmWan n
)J2
m
r
oa(Xn(S, x, W))k'l Yn(S, X, w) , L
m/n
k
YA(n , x, w)1 ] ds 1
I
X Y. n , x, w) dmwan J2
+
Cnt]-1
m-0
f
(m+1)/n
o (Xn(S, X, W)),"'Y,(S2 X1 WA [Y-(S' X1 W)12
m/n
- Yn(n , x, w
)
l Jds dmwan
= KI(t) + K2(t) + K3(t) + K4(t). Then, by Theorem III-3.1 and (7.88), 2l
w)I14)°/J
E[ sup KI(t) I'] < K36E[(nL (nE 1 OS T m-0
II Yn(m , x,
n
Also, by (7.89) with p replaced by 2p, E[ sup K2(t) I'] 0!5t: 5T CnT]-1
KS8n'-I
F
- Xn(n , x,
nPn-(y-1)
1
E.*
J m/n
w)I!
20]112ds
II Xn(s, x, w)
E [II n(n , x, w) 1141]"' E[I dmwa 1.1]1/4
K39 < 00 . Using (7.90), similar estimates hold for K3(t) and K4(t). Thus we obtain
E[ sup HI(t; k, 1, a) I'] < K40 < oo . 05t5T
More easily we can obtain
E[ sup H2(t; k, 1, a) I o] < K41 < oo . OScS T
COMPARISON AND APPROXIMATION THEOREMS
516
Therefore sup E[ sup IIa.(t, x, w)II'] <_ K42 < 00 . ,.x
05t5T
Using this, sup E[ sup I Y5,),(t, x, w) 1,P] < 00 n,x
05t5T
can be proved as in the case of YY(t, x, w). By continuing this process step by step we can complete the proof of Lemma 7.2. Now (X(t, x, w), Y(t, x, w), ...) is the solution of stochastic differential equation dX(t) = cr(X(t))odw(t)
dY(t) = Dai(X(t))Y(t)odw(t) X(O) = x
Y(O) = I
The coefficients of this equation are not bounded and we cannot apply Theorem 7.2 directly. But Lemma 7.2 enables us to apply a standard truncation argument as in the proof of Lemma 2.1 of Chapter V and obtain the following: for every T > 0 and N > 0, (7.91)
lim supE[sup I DaXX(t, x, w) - DaX(t, x, w) I D] = 0
n-m IxISN
O5r5T
for every p Z 2 and multi-index a. Then, as in the proof of Proposition V-2.2, we can obtain the following result. Theorem 7.3. For every T > 0 and N > 0, (7.92)
lim E[ sup sup I DaX (t, x, w) - DaX(t, x, w) I p] = 0 r»m
05t5T IxJSN
for every p > 2 and multi-index a. We have assumed that coefficients are bounded together with all of their derivatives. Now we assume only that coefficients are C°° and that the
stochastic differential equation and approximating ordinary differential
SUPPORT OF DIFFUSION PROCESSES
517
equations possess global solutions for each fixed initial value. By a standard truncation argument, we can easily deduce from (7.92) the following: Corollary. There exists a subsequence {nk} such that, with probability
one, x, w) --- DaX(t, x, w)
as k - oo compact uniformly in (t, x) for every multi-index a. *1
8. The support of diffusion processes k = 1, 2, ... , r, be bounded smooth functions on Rd with bounded derivatives*Z and consider the Let o k(x) and b`(x), i = 1, 2, ... , d,
stochastic differential equation
dX`(t) _ E ug(X(t))odBk(t) + b`(X(t))dt (8.1)
i=1,2,...,d.
X(0)=x,
Let P. be the probability law of the solution X= {X(t)}. Then as we saw in Chapter IV, the system {Px} of probabilities on Wd constitutes the unique diffusion measure generated by the operator A, where
(8.3)
all(x) = E ak(x)ok(x) k-1
and
(8.4)
P(x) = b'(x) + 2 E
k-1
(axj ak(x))ok(x)
Now the path space Wd is a Frechet space with the metric defined in Chapter IV, Section 1, or equivalently, the topology of Wd is defined by *1 The subsequence {nk] can be chosen from any given subsequence of 1, 2, n,
..
: To be precise, the following discussions will be valid if the second order derivatives of a are uniformly continuous.
,
... ,
C;(Rd), bEC;(Rd) and
COMPARISON AND APPROXIMATION THEOREMS
518
the system of seminorms {II.11 T, T > 0} where (8.5)
IlwII
T = max
I w(t) I
OSrs T
for w E Wd.
The purpose of this section is to describe the topological support S"(Px) of the measure Px, i.e., the smallest closed subset of Wd that carries probability 1. In order to do this we need to introduce the following subclasses of the space Wo = {w e W'; w(0) = 0) ;
S'C.S',C Wo where
.', = 10 E Wo ; t i- c(t) is piecewise smooth) and
So = { C Wo; t i-- ¢(t) is smooth). For 0 E.S7, and x ERd, we obtain a d-dimensional curve
=(x, ¢)
4)) by solving of the ordinary differential equation
d ,' = E a (,)Ok(t)dt +
(8
o = x,
i=1,2,...,d.
We define the subclasses Sox and So; of Wd by (8.7)
and
So; = {c(x, c);
(8.8)
e .So}.
It is easy to see that the closures in Wd of .Sox and .9' coincide; i.e., Sox = .So Now we can state the main theorem due to Stroock and Varadhan [158].
Theorem 8.1. .7(Px) = Spx dt
for every x E Rd.
SUPPORT OF DIFFUSION PROCESSES
519
Proof. First we shall prove the inclusion .9'(P=) e ,7 x. For each n = 1, 2, ... and t > 0, we set t = [2°t]/2° and 1,, = ([2"t] + 1)/2". Let {B(t)} be a given r-dimensional Brownian motion with B(O) = 0 and define {B (t)} by
(t" -
t)
(tn - ta)
B(t,) +
(t -
B(t,,)
U. - tn)
E.5" . By Theorem 7.2, we can conThen B. EE S', and hence (x, clude that (x, X in Wd in probability. Hence Pz * P,, as n - 00 where Px is the probability law of (x, B.). Thus
Px(5';) > lim
=1
and consequently .So-x-= So; D V (Px). The converse inclusion .'(P.) ox will follow immediately from the following theorem. We consider the equation (8.1) on the Wiener space (Wo, PR') with respect to the canonical realization of the Wiener process; its solution is denoted by Xx=(X(t, w)).
Theorem 8.2. For every 0 e:,91, T > 0 and e > 0,
PW(IIXI-
(8.9)
(x,0)111'<EIIIw-0IIT<6) - 1
as (510.
Remark 8.1. It is well known that PR'(Ilw - oII T < 6) > 0 for every
0 ESo, T > 0 and b > 0, i.e., So(Pw) = Wo. Indeed, by the result of Chapter IV, Section 4,
Pw(IIw-oily<6)=Epw(M(w):llwlIT<6)>0 where
M(w) = exp[- E
f
o
k(s)dwk(s) - 2
Jo
I (s) 12 &J.
To prove Theorem 8.2 we first introduce several lemmas.
Lemma 8.1. There exist positive constants c, and c2 such that (8.10)
P '(IIwIIT < E)
c,
exp(- Z)
as e 10.
520
COMPARISON AND APPROXIMATION THEOREMS
Proof. Let P. be the r-dimensional Wiener measure starting at xE R'; so, in particular, Po = Ps'. Let D= {x E Rr; I x I < 1} and set o(w) = inf {t; w(t)
for w E F V.
D}
Then u(t, x)=Exf f(w(t))1,,(.)>j, x s D, t > 0, is the solution of the initial value problem in D,
at = 2 du ulaD=0
uit-o =f Consequently
u(t, x) = E e-xn' 75n(x) f 0n(Y)f(y)dy, n-1
D
where 0 < 11 < 2 < 2, < .. are eigenvalues and
{O (x)}
are cor-
responding eigenfunctions of the eigenvalue problem 2 do -{- 1o = 0
in D,
018D=0-
In particular, Pw(IIwfl, <
e) = PF'(max I w(t) I < e) = Pw( max I ew(t/e2) I < e)
PW(omrax sI
OstsT
0! tST
w(t) I < 1) = PW(a(w) > T/eZ)
e 'finT/62 95n(O) f D
3 (y)dy,
and consequently
P1(IIwIIT<e)_e12 01(0) f DOY(y)dy. Therefore (8.10) is proved with
SUPPORT OF DIFFUSION PROCESSES
C, _ 0,(0) f O,(x)dx
and
521
c2 = .1,T.
Lemma 8.2. Set (8.11)
e(t) =
2
f [w'(s)dwf(s) - w'(s)dw'(s)] o
for Q=1,2,. .
..,r.
Then we have (8.12)
lira sup P`r[ Ilillfll T > MS I IIwll T < 01= 0. Mt - o
1
Proof. Let i k j be fixed and set a(t)
4
f
[(w`)2(s) + - (wJ)2(s)]ds. o
We have remarked in Section 6 that B(t): = ij'J(a-'(t)) is a Brownian motion independent of {(w')2(t)+(wi)2(t)} (and hence, independent of the radial process { I w(t) I } and, in particular, of I I wI i T). Then P9 II11'II T > M8 I IIwI1 T < 0] = Pw[ IIB(a(t))II T > M8 I
< PR'[ max I B(s) I > M8] = = P'[ max I B(s) I > M], 05s5 T/4
0!-..j- g62 T/4
and this proves (8.12).
Corollary. Let (8.13)
'J (t) = f 'o w'(s)°dw'(s),
i, J = 1, 2, ... , r.
Then for all i, j = 1, 2, ... , r, (8.14)
lim
M6I IIwjjT<(5)=0-
Aft m o
In particular, for every e > 0, (8.15)
I IIwIIT<6)-0
as
6 1 0.
IiwIIT < 0]
COMPARISON AND APPROXIMATION THEOREMS
522
Proof. Since
Z wi(t)w'(t) -- VV) = 1i'(t), (8.14) follows at once from (8.12). The following is a key lemma for the proof of Theorem 8.2.
Lemma 8.3. Letf(x): Rd -- R be bounded and uniformly continuous. Then for all a > 0 and i, j = 1, 2, ... , r, (8.16)
PY ( 11 f' f ( X ( s, 0
w))diii(s)I I T> e I I I wI I T< fi) ---- 0
as
8 10.
Proof. First we shall assume that f e Cb(Rd). Then by Ito's formula, f ' f(X(s))ddif(s)
=.f(X(t))i'(t) - f ' fi(X(S))6ic(X(S))bi'(S)dWk(S)
-f
0
f f (X(s))QJ(X(s))w'(s)ds *1
= h(t) + 12(t) -l- 13(t) + I4(t) . Clearly it is sufficient to show that
PW(Illi(t)IIT> S I IIWIIr<6)-0 as 6 10 for every a > 0 and i = 1, 2, 3, 4. This follows from (8.15) for i = I and i = 3 and it is obvious for i = 4. So we only consider 12(t). For simplicity we set ak(x) = f (x)ak(x). Then, by Ito's formula, 12(t) = fak(X(s))Sh'(s)dwk(s) 0 zz zz ( = ak(X(t))S''(t)W"(t) - f ak.i(X(S))o (X(S))bi'(S)wk(S)dwm(S) *2 0
*1 fi = 'f
. We also adopt the usual convention for summation.
*2 ak,,(x) = axi ak(x)
SUPPORT zOF DIFFUSION PROCESSES
523
- f 0 (Aak)(X(s))S`I(s)wk(s)ds
f ak(X(s))wk(s)ddl'(s) 0
- f al(X(s))wt(s)ds - f 0
I'(s)ak.l(X(s))Um(X(S))J'-ds
S 0
f wk(s)ak,,(X(s))ai(X(s))w`(s)ds 0
= J7(t) + J2(t) + J3(t) + J4(t) + J5(t) + J6(t) + J7(t)-
Again it is sufficient to show that
PW(IIJ(t)IIT>e I IIWIIT<6)--0 as 6 l 0 for all e > 0 and i = 1, 2, ... , 7. This is no problem for i = 1, 3, 5, 6, 7. By Theorem II-7.2', there exists a one-dimensional Brownian motion B(t) such that
J2(t) = B(a(t)), where lak, l,at"](X(s))wk(s)w"(s)ds.
a(t) = <J2, J2)t = f t
Now PW( IIJZ(t)IIT> e
I
IIwIIT < a)
< PW(ll l'ilT> M6 I
IIwIIT < J)
+ PW(IIJ2(t)IIT> e, IIVIIT :! MoI IIwIIT < (5).
By (8.14), we can choose M > 0 for any given r/ > 0 such that the first term on the right is less than 17 for all 1 Z S > 0. Clearly I T < M6 and Ilwll T < 6 imply that a(t) < a(T) < c304,* and hence PW(I1J2(0IIT> e, T MS, IIwIIT < 6) < P''( max I B(t) I > s) = PW( max I B(t) I > e/ 0st c,64
ogtsJ
* in the following, c3, c4, ... are positive constants independent of S.
b2)
COMPARISON AND APPROXIMATION THEOREMS
524
=2
f31ne a-2
z
z
exp [-
I
] dx < ca exp [- c,,
2
as
.
By Lemma 8.1,
P(IIWIIT<6)>_ c6exp[-c,a21.
Hence Y
P '(IIJ2(t)II TT > s, (ISf'II T :! Moo I IIwII T < 6) c10321
< c8 exp I - c9c2
--- 0
as
as
610.
Consequently,
1iiPR'(IIJ2(t)IIT>E1 IIwtIr<0) and since ?I is arbitrary, we obtain the desired conclusion. Next we consider J4(t).
J4(t) = -
f' ak(X(s))wk(s)w`(s)dwJ(s) 0
iJ f a,, (X(s)) Wk (s) dw o
= K1(t) + K2(t). Then it is obvious that for any a > 0
PW(IIK2(t)IIT>e I IIwIIT<6)-.0
as 6 10.
K1(t) is a martingale with
= f 0
[ak(X(S))Wk(S)W1(s)]2dS.
Therefore, if IIwIIT < 0 then T < c1184. By repeating the same argument as above we can conclude that Pw (IIKI(t)II T > e I
IIWIIT <
(5)
<- c12 exp L- C1382
8a
C1452]J
--0
0. Thus (8.16) is proved for f E Cb(Rd). Now suppose that f is f uniuniformly continuous and choose f E=-CZ(Rd) such that f as (5
525
SUPPORT OF DIFFUSION PROCESSES
formly. Set
Y.(t) = f 'f(X(S))dc'J(S) - J 0fn(X(s)) d'J(s) 0 I('
J O (({
-fn)(X(S))w'($)dwf(s) + 1 1 (f -fn)(X(s)) O
.
Then, for any given e > 0, we have by the same argument as above that PW(IIYyIIT> e I IIwIIT < S)
< PR'(II f o (f -fn)(X(S))w'(S)dwf(S)II T > 2 I
< C15 eX p _
C16E2
lI
IIf
< Cls exp
CI6 E
11W117. < a) z
I-T,
-ffIIZa2J
and 0 < 6 < 1. Hence
for all large n
P"(IffXSd(S)IIT> s I
IIwIIT < b)
< P"'( II f ofn(X(S))d J(S)lI T >
+ PW(IIYnIIT>
2
I
2
1
IIIIT < a)
IIwIIT < a)
For given ij > 0, we can choose n such that the second term is less than
rl for all 0 < 6 < 1. Then letting 8 j 0 the first term tends to 0. Consequently ME 610
P'(II f
' 0
T>eI
IIwIIT <
_ 11, &)<
and since ?7 is arbitrary, the proof of (8.16) is now complete.
Proof of Theorem 8.2. First we shall prove (8.9) when 0(t) - 0. In
this case, , =
d, = X.
Now
0)) is the solution of
526
COMPARISON AND APPROXIMATION THEOREMS
xv) - t(t) = E f k-1
t 0
ak(X(s))°dwk(s) +
f' [bt(X(s)) - b`( t)lds. 0
We shall prove that t
(8.17)
PW( II f ak(X(s))°dWk(s)II T > E I II WII T < a) ~ 0 0
as 8 j 0 for every s > 0. By Ito's formula f' ae(X(s))°dwk(s) 0
= ak(X(t))Wk(t) -
= aX(t))wk(t) -
ft 0
Wc(s)°d(ak(X(S)))
t
f ak.,(X(s))o,,(X(s))wk(s)°dwm(s) f ak,(X(s))b`(X(s))wk(s)ds t
= 11(t) + 12(t) + I3(t)-
It is sufficient to show that for every e > 0 and i = 1, 2, 3,
P'(III1(t)II >E I I1wIIr
as 6 j0.
Only the case of I = 2 needs to be examined. Then I2(t)
_ - f a k.t(X(s))am(X(s))-d km(s) - f' ak.1(X(s))am(X(s))dSkm(S) 0
zz
0
2
0 ax°
[ak.la»,l (X(s))aa(X(s))wk(s)S*mds
J1(t) + J2(t),
and we can conclude that for every r > 0 and i = 1, 2
P`'(IIJJ(t)Ilr>a I IIWIIT
as 61 0 .
SUPPORT OF DIFFUSION PROCESSES
527
Indeed, it is obvious for i=2 and the case i=1 follows from Lemma 8.3. On the set {Wi 1If ' 6k(X(s))odWk(S)IIT < E},
we have
I IIWIIT<6)--0 as
S 10 for every e > 0. Now we consider the case of general 0 E=-,91. Set
M(w) = exp { k f ok(s)dwk(s) - 2 f
I
I (s)zds}
o
and define the measure p on W,' (T): = the restriction of Won the interval [0, T) by
dPW
M(w).
Then by Theorem IV-4.1, w(t) = w(t)-O(t) is an r-dimensional Brownian motion for P, and X(t) satisfies
X(t) = x + f
a(X(s))odw(s) + f o b(s, X(s))ds o
where b(s, x) = b(x)+o (x)¢(s). * Since c _
fl
0)) satisfies
dot = b(t, ,)dt 1o=x,
we can conclude from the above that for every e > 0, * Thus we need to consider the case that the drift b dependent on t and b (-= C'([0, oo) x Rd). All the above results remain valid in such a case with an obvious modification of the proof.
528
COMPARISON AND APPROXIMATION THEOREMS
(8.18)
P(IIX(t)-Srllr>EIIIwIIT<(5)---0
as
6 10.
Noting that
M(w) = exp {kE k(i )wk(T) - k -1 J 0
wk(S)Ok(S)ds
f0 2
I
I (s)Zds}
0, then M(w)
is continuous in w (i.e., if IIw - wll (8.18) implies that IIIw-6IIT<
610
5)
PW(IIX(t)-S,IIT <s,
lim 65 X
11w-oiIT
Ew(M:IIw-oilT<6) PW(llw-0IIT<(5)
---1.
Example 8.1. Let Lo, L1, ... , L, be vector fields on Rd whose coefficients (in the Euclidean coordinates) are bounded and smooth with bounded derivatives. Let Xx = (X(t, w)) be the solution of L1(X(t))odw'(t) + Lo(X(t))dt
dX(t) X(0) = X.
Let Px be the probability law of Xx. Theorem 8.1 implies that
s°(P.)_ {cx,0);0ESo . (x, 0) is the solution of the dynamical system L0(,).
dtt = ,E 0 = X.
It is known (cf. [95]) that
that Z(?71)
J
dt ?7o = x,
+ Lo(t7r)
c E So contains all curves 1, such
SUPPORT OF DIFFUSION PROCESSES
529
where Z is an element of the Lie algebra 2(L,, L2, ... , L,) generated by
L), L2, ... , L,. In particular, if £(L,, L2, ... , L,) has rank d at every 0 E So = WX: = {w E Wd; w(0) = x} and consepoint, quently .So (Px) = Wx
Example 8.2. (maximum principle). Let A be the differential operator defined by (8.2). A function defined in a domain D e Rd is said to be Asubharmonic in D if it is upper semicontinuous and Au > 0 (in a certain weak sense to be specified later). A classical maximum principle for the Laplacian asserts that any subharmonic function in D which attains its maximum in D must be a constant. If the operator A is degenerate, however, such a maximum principle no longer holds in general. For example,
ifd=2,
2
A = x; + axe ,
x = (xi, x2)
and D is any domain intersecting the x,-axis, the function u(x) defined by
if if
x2 > 0 x2 < 0
is a nonconstant A-subharmonic function which attains its maximum in D.
We are interested in the following problem. For a given domain D and a point x E D, we want to determine a (relatively) closed subset D(x) of D having the following properties: (8.19) for any A-subharmonic function u(y) in D such that u(x) = max yED(x)
u(y), it holds that u(y) = u(x) for every y e D(x), (8.20) D(x) is maximal with respect to the property (8.19): i.e., if z E D\D(x), then there exists an A-subharmonic function u(y) in D such that u(x) = max u(y) and u(z) < u(x). ye D(x)
It is clear that D(x) is uniquely determined if it exists. The support theorem 8.1 enables us to describe the set D(x). Before proceeding, however, we shall first make precise the notion of A-subharmonicity. For simplicity we shall restrict ourselves to locally bounded functions. Let P. be the probability law of the solution of (8.1). Let D be a domain in Rd and be a compact exhaustion of D: D. are bounded subdomains of
COMPARISON AND APPROXIMATION THEOREMS
530
D such that D. c
and U D. =D. Let o =
inf {t; w(t)
,wEWd. Definition 8.1. A function u(x) defined in D is said to be A-subharmonic in D if (i) it is locally bounded and upper semicontinuous and (ii) for each n = 1, 2, ... and x E D, t u(x) is a u(w(t A Pz submartingale. If u is in C2(D), then u(w(t A a ))-u(x) = a martingale + ^ (Au)(w(s))ds. From this it is easy to see that u is A-subharmonic if
f0
and only if Au > 0 in D. Now we shall describe the set D(x). For simplicity, we shall assume that D has the property that t sup Ey[ f I,r(w(s))ds] < 00 yeD 0
for every compact set K C D, where z = -r(w) = inf {t; w(t)D}. Choose a compact exhaustion of D. Then the above assumption clearly implies that sup o for every n. Set Ye D
(8.21)
D(x) = {y; 3p1 E .7, 3t0 > 0 such that y = t(x, ¢)(t0)
and (x, O)(t); t E [0, to]) c D fl D. Theorem 8.3.* D(x) possesses the properties (8.19) and (8.20). Proof. First we shall prove that the set D(x) defined by (8.21) possesses
the property (8.19). Let y E D(x). It follows from Theorem 8.1 that for every neighbourhood U of y there exists an n such that Px(cr < cr) > 0,
where o = o (w) = inf {t; w(t) E U1. Let u(x) be an A-subharmonic function in D such that u(x) = max u(z). Then zCD(x)
u(x) < EE(u(w(o- A Q.)));
moreover, Theorem 8.1 implies that Px(w(a , A a.) E D(x)) = 1. Conu(x)) = 1. In particular, u(w(6 )) = u(x) on sequently, P(u(w(cr the set {au < c,,}. From this it is easy to find a sequence z ED such
that z - y and
u(x). Then
*_This was first proven by Stroock-Varadhan [158], but they only discussed the case of operators of parabolic type.
SUPPORT OF DIFFUSION PROCESSES
U(X) > u(y)
531
hm u(Z,) = u(x),
that is, u(y) = u(x). Next we shall prove that D(x) defined by (8.21) possesses the property (8.20). For this we need the following lemma.
Lemma 8.4. If f is a nonpositive continuous function in D with compact support, then the function u defined by u(y) = E,, [ f f(w(s))ds]
is a bounded A-subharmonic function in D. Assuming this lemma for a moment, we shall now complete the proof of Theorem 8.3. Let z E D\D(x). Then by Theorem 8.1 we can easily find a bounded neighbourhood U of z such that P.,(a,, <,r) = 0. We choose a
nonpositive continuous function fin D such that,f(z) = -1 and f(y) = 0 if y
U. Set
u(y) = EY[ f rf(w(s))ds]
Then u is a bounded nonpositive A-subharmonic function in D such that u(x) =0 and u(z) < 0. This proves that D(x) possesses the property (8.20). Proof of Lemma 8.4. u is bounded because of the above assumption on D. First we shall prove that it is upper semicontinuous. We saw in the proof of Proposition V-2.1 that if X(t, x) is the solution of (8.1), then for every t > 0, Rd D X , X(t, X) E Wd is continuous a.s. Then we can easily see that
is lower semicontinuous a.s. and hence x'-' I+r
is upper semicontinuous a.s. The upper semicontinuity of
x
u()c) = Ex( f o.f'(w(s))ds] = f E[f(X(t, o
follows at once from Fatou's lemma.
dt
COMPARISON AND APPROXIMATION THEOREMS
532
Next we shall prove that t -- u(w(t A v )) is a Pz submartingale for u(x) for every every n. It suffices to prove that EE[u(w(c A
stopping time v. But this is a consequence of the following formula (Dynkin's formula) which is immediately obtained from the strong Markov property :
Ex[ulw(cAo ))] - u(x) _ -Ex[
I0
Jlw(s))ds1
9. Asymptotic evaluation of the diffusion measure for tubes around a smooth curve
Consider a non-singular diffusion process X on a manifold M. We are sometimes interested in the following questions: given two smooth curves starting at the same point, which one is more probable for the diffusion process, or, among all possible smooth curves connecting two given points, which one is most probable for the diffusion process? One way to answer these questions is to evaluate the measure of tubes around a smooth curve ([154]). As we know from Chapter V, Section 4, we may assume that M is a Riemannian manifold and the diffusion is generated by the operator A where
A= 2 d -{- b.
Here d is the Laplace-Beltrami operator and b is a vector field. The Riemannian distance p(x, y) is defined (if x and y is sufficiently near) as the minimum length of a geodesic curve from x toy. For a given smooth curve
0: [0, T] -- M such that 0(0) = x, we want to evaluate p (0) = Px {w; p(X,(w), 0(t)) < a for all
t
[0, T]} .
If for any two such curves ¢ and yr, llm elo
exists and can be expressed as exp [ T L(O(s), 0(s))ds - f L(yr(s), i(s))ds] J
0
o
by some function L(z, x) on the tangent bundle TM, the above questions
ASYMPTOTIC EVALUATION OF DIFFUSION MEASURE
533
can be answered in terms of the function L Such a function is called the Onsager-Machlup function by physicists (cf. [20], [41], [56] and [134]. For simplicity, we discuss here the case of M = Rd and obtain the function L: we refer the reader to (199] and [227] for the results in general manifolds. So let M = Rd and let {Px} be the diffusion measure (on Wd) generated
by the operator
We shall assume that bi(x) E Cb(Rd), i=1,2,... , d. Let Pw be the d-dimensional Wiener measure (on Wd) starting at 0. By Lemma 8.1, we know that ,FT]
(9.1)
where A, is the first eigenvalue of the eigenvalue problem in D= {x E Rd; I x I < 11 given by
1 d0+20=0 (9.2)
OIaD=0, and c = 0,(0) f, q3,(x)dx, O,(x) being the normalized eigenfunction for A,. In the following, T > 0 and x E Rd are arbitrary but fixed.
Theorem 9.1. Let 0: [0, T] -- Rd be a smooth curve*2 such that 0(0) = x. Then we have (writing b(x) = (b'(x), bz(x), ... , bd(x)) Px(w IIw - 0IIT < a) (9.3)
exp
[- 2 f
I b(a(s)) - c(s) I gds o
-2f
(div b)(O(s))ds] P'(II wll r < a)
T
c exp
(- 2 f o I b(O(s)) - (s) I z ds o
T - 2 f (div b)(O(s))ds) exp [- E1 T o
" 11 wll7 = max I w(t) 1, w e W. Osi T
It is sufficient to assume that 0 e C2([O, Tl - Rd).
as
a 10.
COMPARISON AND APPROXIMATION THEOREMS
534
(div b) (x) = E
Here
ax`
b'(x).
Corollary. The Onsager-Machlup function L(x, x) is given, up to an additive constant, by
L(z, x)
(9.4)
2 12 - b(x)
Z I
-
(div b)(x).
Proof. By the transformation of drift discussed in Chapter IV, Section 4-1, we have T
PP(B) = Ew(exp [f0 b(x + w(s))dw(s)
(9.5)
2
f
o I
b(x + w(s)) I Zds]; B)
for BER(Wd). Similarly, PW(B)
O(s)
Ew[exp [-f o (s)dw(s)
2
(9.6)
f
I
I
Zds]
o
:(w;[w+¢-x]E B)].*i By combining (9.5) and (9.6), we have
P.(IIw-01IT<E)
= Ea'[exp [- J 0 (s)dw(s)
(9.7)
+f
o
2
f
o
I (s) I Zds
b(w(s) + O(s))d[w(s) + c(s)] - i f o I b(w(s) + O(s)) I Zds]
IIwIIT<E].
If w
satisfies IIwIIT < e, then
I f O(s) dw(s) I = I O(T) w(T) o
-f
w(s) (s)ds
A,c, *2
o
` For w e Wd and x e R , w + x E Wd is defined by (w + x](t) = w(t) + x. i2 All A21 ... and K Ks, ... are positive constants independent of e.
ASYMPTOTIC EVALUATION OF DIFFUSION MEASURE
535
b(o(s))c(s)ds I< AZE I f T b(w(s) + O(s))q$(s)ds - f T 0 0
and
I f I b(w(s) + ¢(s)) I zds - f I b(o(s)) I zds I< A,E. 0
0
We immediately have from this that
exp [- 2 f0 I c(s) - b(a(s)) I zds - (A, +A2+A3) < PP(II w -1II'T < s) {E1'(exp[ f o b(w(s) + 0(s))dw(s)]: IIII T < E]} -1
(9.8)
< exp [- 2
Jo
It(s) - b(o(s)) 12ds + (Al+Az+A3)E] .
Consequently it is sufficient to show that
Ew[exp[f o b(w(s) + ¢(s))dw(s) + 2
(9.9)
f
(div b) (O(s))ds] 111w11 r < E] o
as Ej0. Now
f
b(w(s) + O(s))dw(s) + 2 0
=f
o
+
f
(div b)(O(s))ds 0
b(¢(s))dw(s) +f b`J(¢(s))w1(s)dw'(s) .
f
o
(div b)(i(s))ds + Y; f
0t(s,
w)dw°(s)
o
where
0,(s,w) = b`(w(s)+O(s)) - b`(g(s)) - Z
(9.10)
Since
* b'j=a b' ax
COMPARISON AND APPROXIMATION THEOREMS
536
f
if
b(¢(s))dw(s) I < Ac
IIWIIT < e,
o
(9.9) is equivalent to d
E w exp {
1 i f ro b`,(¢(s))w'(s)dw`(s) p`(s, w)dw`(s)} I IIwIIr
(9.11)
+ 2 f o (div b)(O(s))ds + 571 fo
< EJ
as e10. Note that
t, J-1
f
T
b1,(c(s))w'(s)dw'(s) + 2 fo (div b)(O(s))ds 0
4J-1
f
0
b`j(O(s))
[wJ(s)dwi(s)
+ 2 a,, ds]
r
_ r, -1 f 0 b`,(O(s))dc "(s) where
V" (t) = fa w'(s)edw`(s) Generally, if random variables Y1, Y2 satisfy lim Ew[ e°'"r
I
11wIIr < e] < 1,
810
i= 1,2
for every real constant c, then
limE"[exp[Y1+ YZ] I IIWIIr<E] 810
< lim 810
{Ew( exp[2Y1] I !IwIIr
< c) Ew( exp[2Y2] I IIwIIT < e)} lie
< 1, and similary
lim E'(exp(-Y,-Y2) I IIwIIr < E) < 1. 810
ASYMPTOTIC EVALUATION OF DIFFUSION MEASURE
537
But
E`"(exp[Y, + Yz] I Ilwllr < E)ER'(exp[-Y,-Yz] I Ilwllr < e)
JEW(exp[Y 2 Yz]eXP[-Y2 Yz] Illwllr<s)}2=1, and hence lim EW(exp[Y, + Yz] I Ilwll r < e)
-ia
{alienE}(exp(-Y, - Y2) IIIwIIr<e)}-'> 1. In conclusion, we have lira EW( exp[Y, + Yz] I Ilwllr < e) = 1. alo
This result can be extended easily to an arbitrary number of random variables: if Y1, Yz,
... , Y.
satisfy
lien EW(exp[cY1] I Ilwllr < e) < 1
for every real constant c and i = 1, 2, ... , n, then lien EW( exp[Y,+Yz+
I
alo
Ilwllr < e) = 1.
Noting this fact, (9.11) will follow if we can show that for every real
constant c and i, j = 1, 2, ... , d, (9.12)
lien E W(exp [c f o b;,(O(s))dcfl(s)]
I
I I wl l r< e) S 1
and
(9.13)
11lm E"'[exp[c I o cP (s, w)dw'(s)] I Ilwllr < e] < 1.
First we shall prove (9.13). Since I co,(s, w) I < Asez
on the set
{w; Ilwllr < e} ,
COMPARISON AND APPROXIMATION THEOREMS
538
we have
PIVI
ft 0
cO,(s, w)dw'(s)lIr > 6 111w;!r < E)
< A6exp [ L
-
A162
4 A8EZ1
< Klexp
J
E
rE 1-K2
Sa]
if 0 < E < K38
by a standard estimate as in the proof of Theorem 8.2. From this it is easy to conclude that lim E w( exp [ I c f 0#1 w)dw`(s) I]
(9.14)
810
I
I I W1 I T< e) = 1.
o
Indeed, for each 80 > 0 and 0 < E < K3S0,
Ew(exp[Ic f o 0t(s, w)dw'(s)I] I
< eao + Kl f ee exp [- K4 2] dd + Kiebo exp [- K;ao] ao
(9.14) follows by first letting a } 0 and then letting So 10. Finally we shall prove (9.12). By writing cb'f = c(x) it is sufficient to show that limo E'(exp[ f c(O(s))ddf`(s)] o 81
(9.15)
I
lIwlIr < E) < 1
for every c(x) E Cb(Rd). Now
f
r
r 0
c(O(s)) ddf'(s) = c(O(T ))cf r(T) -
f0 f'(s)d [c(qS(s))]
Also cf'(t) = 2 w'(t)w'(t) - raf(t) where i7"(t) is defined by (8.11). As we saw in the proof of Lemma 8.2, i7'1(t) = B(
f
[(w`)2(s) + (w')2(s)]ds) o
where B is a Brownian motion independent of the radial process (and hence independent of IIwjjr). Then
w(t) I }
ASYMPTOTIC EVALUATION OF DIFFUSION MEASURE
E"'(exp[
<
Jo
eKgea
c(c(S))d `'(S)]
I
E '(exp [K5 max
539
IIWIIT < e) I B(t) I ])
0
=
eKgez
JoV
as e j 0.
exp
This completes the proof. In Chapter V, Section 4, we saw that the diffusion {P,} is symmetrizad
ble if and only if the differential 1-form co defined by w= E bt(x)dxt is
given as co = dF for some F e C°°(Rd --- R), or equivalently the line integral f r o) vanishes along every closed smooth curve. Using Theorem 9.1, this condition can be restated as follows. Theorem 9.2. The diffusion {Px} is symmetrizable if and only if (9.16)
lim PX(I I W -
011T < E) = 1 eloPx(IIW-0-IIT<E)-
for every x and every smooth curve 0: [0, T] -. Rd such that 0(0) _ 4(T) = x. Here the curve 0_ is defined by (9.17)
¢_(t)=O(T-t),
0
Proof. By Theorem 9.1, the limit in (9.16) is equal to exp(2
J0
exp(2
J
o(0).
Thus (9.16) holds if and only if f d co = 0. A similar probabilistic characterization of the symmetry was obtained by Kolmogorov [87] in the case of a Markov chain.
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Index
absolute boundary condition, 310 absorbing barrier Brownian
horizontal-, 298, 312 Lobachevskii-, 347 occupation time of-, 146 pinned-, 243 reflecting (barrier)-, 119, 209, 313
motion, 211 adapted, 21
(_97)--,21 admissible control, 442 admissible system, 442 affine connection, 278 approximation of a Wiener process,
-with drift, 208 -with random absorbtion, 208 bundle of linear frames, 267
-of orthonormal frames, 282 -of unitary frames, 345
480
approximation theorem, 480 arcsine law, 134 Ascoli-Arzala theorem, 17 asymptotic expansion of Wiener functional, 384
Cameron-Martin subspace, 351 canonical isomorphism, 423 canonical decomposition, 98 characteristic measure, 43 Chern polynomial, 432 Christoffel symbol, 282 comparison theorem, 437
Berezin formula, 434 Bessel diffusion, 237, 347 Borel
-cylinder set, 16
-for one-dimensional projection
-field, I
of diffusion processes, 452 compensated sum, 63 compensator, 60 complete, I complete probability space, I complex Brownian motion, 155 complex manifold, 341 component of connection, 278 conditional expectation, 12 conformal mortingale, 155 connection
-isomorphic, 13 -measurable, I
-set, I
boundary condition, 217 bounded variation part, 98
boundary operator of the Wentzell type, 217 Brownian bridge, 243 Brownian excursion, 240 Brownian functional, 349 Brownian motion, 40
affine-, 278
absorbing barrier-, 211 canonical realization of-, 40 complex-, 155 d-dimensional-, 40, 207 elastic barrier-, 211
-compatible with the Riemannian metric, 281
component of-, 278 -of Levi-Civita, 282 Riemannian-, 282 symmetric (torsion free)-, 281
(3')--, 41
551
INDEX
552
conservative, 203 convergence
-almost everywhere (almost surely), 8
-in law, 8
exponential quasimartingale, 149 extension, 89
standard-, 89 exterior derivative, 277 exterior product, 277
-in probability, 8 coordinate neighborhood, 247 holomorphic-, 3 41 countably determined, 14 counting measure, 43 covariant derivative, 279 covariant differentiation, 278 curvature
Gauss total-, 432 scalar-, 423 space of constant-, 462 tensor, 297
-transform, 461 differential one (p) form, 277 diffusion, 204 A --, 205, 213
(A, L)--, 218 Bessel-, 237 Gaussian-, 232
Fernigue's theorem, 402
Feynman-Kac -formula, 274, 383 -type weigh, 306 finite dimensional distribution, 17 first hitting time, 23, 316 Fisk integral, 97, 101 flow of diffeomorphisms, 254 frame, 267
unitary -, 345 Fubini-type theorem (for stochastic integrals), 116 fundamental solution, 420, 435 fundamental tensor field, 251
Gaussian diffusion, 232 Gauss-Bonnet-Chern theorem, 432 generalized expectation, 365 generalized Wiener functional, 364
Kahler-, 344 linear-, 232
geodesic, 462
-measure, 204 -measure generated by A, 205, 213 -measure generated by (A, L), 218 -process, 205
Girsanov transfomation, 192
boundary distinguished. 349 Doob-Kolmogorov's inequality, 28 Doob-Meyer's decomposition theorem, 34 drift
-term, 160 transformation of-, 192, 225 -vector field, 284 d -system, 22
elastic barrier Brownian motion, 211 equi-integrable, 30 equivariant, 280 Euler characteristic, 425 excursion, 123, 325
-formula, 328 -interval, 123, 325 -of Brownian motion, 123 existence theorem, 167 expectation, 11
conditional-, 12 generalized, 356 explosion time, 172, 173, 248 exponential mapping, 462 exponential martingale, 149
-polar coordinate, 463 harmonic form, 292 heat equation, 269
-for tensor field, 297 heat kernel, 391 Hilbert-Schmidt norm, 360 holomorphic coordinate neighborhood, 347 Hermite polynomial, 150, 354 Hermitian metric, 343 horizontal, 279
-Brownian motion, 298 canonical-vector fields, 280 -Laplacian, 298 -lift, 279, 280 -subspace, 279 hypercontractivity (of OrnsteinUhlenbeck semigroup), 367 hypoellipticity problem, 416 image measure, 2, independent
mutually-, l l
-of a a-ifield, II induced measure, 2 the regularity of, 375 inequality Doob-Kolmogorov's-, 28
INDEX
Jensen's-, 12 moment-for martingales, 110 initial distribution, 40, 205 inner regular, 3 integrable ( p-th, square), I1 integrable increasing process, 35
natural-, 35 invariant measure, 290 lt$'s formula, 66
-in the complex form, 157 Ito process, 64 Ito's stochastic integral, 45
Jensen's inequai lity, 12
553
exponential-, 149
local-, 52 localconformal-, 155 locally square-integrable-, 52
-part, 98 -problem, 74 square-integrable-, 47 -term, 160
-transform, 25 -with reversed time, 31 maximum principle, 529 measurable
-mapping, I -process, 21
K5hler -diffusion, 341, 344 -manifold, 344 -metric, 344 Knight's theorem, 86,92, 156 Langevin's equation, 233 Laplace-Beltrami operator, 285 Laplacian
de Rham-Kodaira's-, 292 horizontal-, 298 last exit formula, 328 last exit time, 316 Levy
-Khinchin formula, 65 -measure, 65 -process, 65 life time, 205 linear diffusion, 232 Lobachevskii Brownian motion, 347 local conformal martingal, 155 local coordinate, 247
-conplex-, 342 local martingale, 52 local time, 113, 220
-of Brownian motion, 113
-space, I universally-, 1 Meyer's theorem (on the equivalence of Soboldv norms), 365 minimal - A-diffusion, 290 -Brownian motion, 211 modification, 266
right continuous-, 33 mollifier, 485
moment inqualities for martingales,
110
motion with random acceleration, 202
multiplicative operator functional (MOF), 318, 321 natural, 35 non-degenerate, 288 non-sticky, 221 normal component (of d -form), 309
normal coordinate mapping, 464 normal derivative, 309 normally reflecting diffusion process, 313, 325, 341
number operator, 356
Lp-multipliplier theorem, 369 Malliavin convariance, 375
Malliavin's stochastic calculus of variation, 349 manifold, 247
complex-, 341
Kahler-, 344
Riemannian-, 281
-with boundary, 308 Markovian system, 204
one-dimensional diffusion process, 446 Onsager-Machlup function, 533 optimal control problem, 441 optional sampling theorem, 26, 34 optional stopping, 25 Ornstein-Uhlenbeck
-'s Brownian motion, 233 -operator, 356 -process, 351 -semigroup, 356
strongly-, 204 Markov time, 22 martingale, 25
conformal-, 155
'tr-system, 22
parallel displacement, 279
stochastic-, 297
554
INDEX
pathwise uniqueness, 162 piecewise linear appoximation, 484 Pitman's theorm, 140 plane section, 461 polynominal functional, 351 point function, 43 point process, 43
characteristic measur of-, 43
(3 )-adapted-, 59
-of Brownian excursions, 125 -of the class (QL), 59 Poisson-, 43, 140
a-finite-, 59 stationary-, 43 Poisson point process, 43
Ricci curvature, 461 Ricci tensor, 302 Riemannian
-connection, 282 -manifold, 281 -metric, 281 scalarization, 280 sectional curvature, 461 semi-martingale, 64
continuous martingale part of-, 64 representaion theorem of-, 84 a-field, I right-continuous-, 20
topological-, I
(,9;)-, 60
Poisson random measure, 42 intensity mesure of-, 43
mean measure of-, 43 predictable, 21,25 probability, I
-space, I -distribution, 2 -law, 2 process
adapted-, 21 bounded-, 21 continuous-, 16 measurable-, 21 -of time change, 102, 197 predictable-, 21 right-continuous-, 21 well-measurable-, 21 proper reference family, 27
pull-back of Schwartz distributions (under a Wiener mapping); 379
skew product, 472 Skorohod equation, 121 Sobolov norm of Wiener functionals, 362
Sobolov space of Wiener functionals, 364,365 smooth functional, 352 sojourn time density, 113 solution, 161
-admitting of explosions, 172 strong-, 163 uniqueness of-, 162, 221 standard measurable space, 13 state space, 203 sticky, 221
non--, 221 stochastic area, 470 stochastic differential, 97, 98 stochastic differential equation, 159 -, Markovian type, 161
-of the jump type, 244 quadratic variational process, 53 quasimartingale, 98 canonical decomposition of-, 98
bounded variation part of-, 98 exponential-, 149 martingale part of-, 98
time homogeneous-, 161 -, time independent Markovian type, 161
-with respect to Poisson point process, 244
-with respect to quasimartingale, 103
random variable, 2
complex-, 2 d-dimensional-, 2
real-, 2
reference family, 20
proper-, 27 reflecting (barrier), Brownian motion, 119, 209, 314
regular
-condiuitnal probability, 13, 15 -submartingale, 38 relatively compact, 7
stochastic integral
ItO's-, 45 -with respect to the Brownian motion, 49
-with respect to a martingale, 55, 57
-with respect to point processes, 59
stochastic line integral, 468 stochastic moving frame, 298, 424 stochastic parallel displacement, 297 stopping time, 22, 204
INDEX Stratonovich integral, 97, 101
strong Markov property -of Brownian motion, 78 -of Poisson point process, 78 strongly Markovian system, 204 strong renewal property, 78 subharmonic, 529, 530 submartingale, 25
regular-, 38 -of class (DL), 35
555
transformation of drift, 192, 225 transition probability, 204 transition semigroup, 290 uniqueness
-in the sense of probability law, 162
-of the regular conditional probability, 13
-of the solutions, 162, 221
successive approximation, 180
universally measurable, I
sum, 77 supermartingale, 25 supertrace, 425 suppprt, 517
variance, I I vector field, 248
symmetric multiplication, 97, 100 symmetrizable, 290
locally-, 291 tangent component (of p-form), 309 tangent space, 247 tangent vector, 247 tensor, 276. 297 tensor field, 277
fundamental-, 281 scalarization of-, 280 terminal point, 203 tight, 7 time change, 102, 197, 225
process of-, 102, 197 torsion free, 281 torsion tensor, 281 total, 206
transformation of Brownian motion, 224
basic-, 280 C° --, 248 system of canonical horizontal-, 280
weakly convergent, 4 WeitzenbSck's formula, 301 well-measurable, 21 Whiteney's imbedding theorem, 253 Wiener
-functional, 349 genesalized-functional, 350, 364
-'s homogeneous chaos 354 mapping, 375
-martingale, 100 -measure, 40 -process, 40 Williams's decomposition theorem, 135, 146
Williams's description of Brownian excursion law, 144
