Stochastic Functional Differential Equations (Chapman & Hall CRC Research Notes in Mathematics Series)

Research Not s in Mathematics

S-E A Mohammed

Stochastic functional differential equations

Pitman Advanced Publishing Program BOSTON·LONDON · MELBOURNE

99

S-EA Mohammed University of Khartoum

Stochastic functional ditTerential equations

Pitman Advanced Publishing Program BOS10N ·LONDON· MBLBOURNE

PITMAN PUBLISHING LIMITED 128 Long Acre, London WC2E 9AN PITMAN PUBLISHING INC 1020 Plain Street, Marshfield, Massachusetts 02050 Associated Companies Pibnan Publishing Pty Ltd, Melbourne Pibnan Publishing New Zealand Ltd, Wellington Copp Clark Pibnan, Toronto

© S-EA Mohammed 1984 First published 1984 AMS Subject Classifications: (main) 60H05, 60Hl0, 60H99, 60J25, 60J60 (subsidiary) 35K15, 93E15, 93E20 Library of Congress Cataloging in Publication Data Mohammed, S. E. A. Stochastic functional differential equations. (Research notes in mathematics; 99) Bibliography: p. Includes index. 1. Stochastic differential equations. 2. Functional differential equations. I. Title. II. Series. QA274.23.M64 1984 519.2 83-24973 ISBN 0-273-08593-X British Library Cataloguing in Publication Data Mohammed, S. E. A. Stochastic functional differential equations.(Research notes in mathematics; 99) 1. Stochastic differential equations I. Title II. Series 519.2 QA274.23 ISBN 0-273-08593-X All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording and/or otherwise, without the prior written permission of the publishers. This book may not be lent, resold, hired out or otherwise disposed of by way of trade in any form of binding or cover other than that in which it is published, without the prior consent of the publishers. Reproduced and printed by photolithography in Great Britain by Biddies Ltd, Guildford

To Salwa and Yasar with love

Contents Preface I. PRELIMINARY BACKGROUND §1. §2. §3. §4. §5. §6. §7. §8.

Introduction Measure and Probability Vector Measures and the Dunford-Schwartz Integral Some Linear Analysis Stochastic Processes and Random Fields Martingales Markov Processes Examples: (A) Gaussian Fields (B) Brownian Motion (C) The Stochastic Integral I

II. EXISTENCE OF SOLUTIONS AND DEPENDENCE ON THE INITIAL PROCESS §1. Basic Setting and Assumptions §2. Existence and Uniqueness of Solutions §3. Dependence on the Initial Process III.MARKOV TRAJECTORIES §1. The Markov Property §2. Time-Homogeneity: Autonomous Stochastic FOE's §3. The Semigroup IV. THE INFINITESIMAL GENERATOR §1. §2. §3. §4.

Notation Continuity of the Semigroup The Weak Infinitesimal Generator Action of the Generator on ·Quasi-tame Functions

8 11 14 16 18 21 21 22 24 30 30 33 41 46 46 58 66 70 70 71 76 97

V.

REGULARITY OF THE TRAJECTORY FIELD

113

§1. §2. §3. §4.

113 114 144 149

Introduction Stochastic FDE•s with Ordinary Diffusion Coefficients Delayed Diffusion: An Example of Erratic Behaviour Regularity in Probability for Autonomous Systems

VI. EXAMPLES

165

§1. §2. §3. §4.

165 165 167 191

Introduction Stochastic ODE•s Stochastic Delay Equations Linear FDE•s FOrced by White Noise

VII. FURTHER DEVELOPMENTS, PROBLEMS AND CONJECTURES §1. §2. §3. §4. §5.

Introduction A Model for Physical Brownian Motion Stochastic FDE•s with Discontinuous lnitial Data Stochastic Integro-Differential Equations Infinite Delays

223 223 223 226 228 230

REFERENCES

2'34

I.NDEX

240

Preface

Many physical phenomena can be modelled by stochastic dynamical systems whose evolution in time i.s governed by random forces as well as intrinsic dependence of the state on a finite part of its past hi.story. Such models may be identified as stochastic (retarded) functi.onal differentia 1 equations (stochastic FOE's). Our main concern in this book is to eluci.date a general theory of stochastic FOE's on Euclidean space. I.n order to have an idea about what is actually going on in such differential systems, let us consider the simplest stochastic delay differential equati.on. For a non-negative number r > 0, this looks like dx(t)

=

x(t-r) dw(t)

(SDDE)

where w is a one-dimensional Bro~nian motion on a probability space (rl,F,P) and the solution x is a real-valued stochastic process. It is interesting to compare (SDDE) with the corresponding deterministic delay equation dy(t)

=

(ODE)

y(t-r)dt.

One can then immediately draw the followinganalogi.esbetween (SDDE) and (DOE): (a) lf both equations are to be integrated forward in time and starting from zero, then it is necessary to specify a priori an initial process {e(s): -r < s < 0} for (SDDE) and a deterministic fUcntion n~[-r,O] +R for (DOE). ln the ordinary case (r = 0), a simple application of the (lto) calculus gives the following particular solutions of (SDDE) and (ODE) i.n closed form: x(t)

=

ew(t)-it, y(t)

=

et, t € R.

For posi.tiv.e delays (r > 0), no simple closed-form solution of (SDDE) is known to me. On the other hand, (DOE) admits exponential solutions y(t) = eH, t €R, where A € l solves the characteristic equation (i )

(b) When r > 0, both (SDDE) and (ODE) can be solved uniquely through e, n, respectively just by integrating forward over steps of si.z.e r; e.g. e(O) x(t)

+

J:

= {

-r < t < 0

e(t) and n(O) y(t)

0< t < r

e(u-r)dw(u)

+

J0t n(u-r)du

0< t < r

= {

-r < t

n(t)

< 0

Similar formulae hold over the successive intervals [r,2r], [2r,3r], etc. (c) In just the same way as the continuation property fails for actual s·olutions y:[-r,m) +R of (DOE), it is clear that a Markov property cannot hold for solutions {x(t): t > -r} of (SDDE) in R when r > 0. Heuristically speaking, a positive delay upsets Markov behaviour in a stochastic delay equation. (d) To overcome the difficulty in (c), we let C: C([-r,O],R) denote the Banach space of all continuous real functions on [-r,O] given the supremum norm II • II c. For each t > 0, pick the slice of the solution paths over the interval [t-r,t] and so obtain trajectories {xt: t > 0}, {yt:t > 0} of (SDDE) and (DOE) traced out in C. It now follows from our results in Chapter Ill (Theorems (111.1.1), (111.2.1), (111.3.1)) that trajectories {xt:t > 0} describe a time-homogeneous continuous Feller process on C. (e) As functions of the initial path n € C, trajectories {nxt:t > 0}, {nyt: t > 0} of (SDDE) and (ODE) through n define a tpajectoPy field t

> 0,

and a semi-fiOIP

d. Tt.c

+

c •

n ..... nyt (ii)

t >

o

respectively. Both T~ and T~ are continuous linear, where t 2(n,C) is the complete space of all F-measurable e:n+ C such that Jnll e (w) II~ dP(w) < "" , furnished with the £ 2 semi-norm llell 2 = £

[J

Q

lle(w)ll~

dP(w)J 112 •

(Cf. Theorem (II.3.1)). However, in Chapter V §3, we show that the trajectory.field T~. t > 0, does not admit 'good' sample function behaviour. Thus, despite the fact that Borel measurable versions always exist, no such version of the trajectory field has almost all sample functions locally bounded (or even linear) on C (cf. Corollary (V.4.7.1), V §3, VI §3). It is intriguing to observe here that this type of erratic behaviour is peculiar to deZayed diffUsions (SDDE) with r > 0. Indeed for the ordinary case r = 0 it is wellknown that the trajectory field has sufficiently smooth versions with almost all sample functions diffeomorphisms of Euclidean space onto itself (Kunita [45], Ikeda and Watanabe [35], Malliavin [51], Elworthy [19], Bismut [5]). (f) At times t > r, the deterministic semi-flow T~ maps continuous paths into c1 paths, while the corresponding trajectory field T~ takes continuous paths into a-H6lder continuous ones with 0 0 x0 =

e E £ 2 (Q,C)

where the coefficient pPocess g:~ x £ 2(n,C) + £ 2(nJRn} and the initiaZ pPocess e E £ 2(n,C) are given, with z a McShane-type noise on a filtered probability space (Q,F,(Ft)t>Q,P). (Refer to Conditions (E)(i) of Chapter II) •

Chapter III essentially says that for systems of type (iii)

dx(t) = H(t,xt)dt

+

G(t,xt)dw(t), t > 0, x0 = n e:

c

e trajectory field {nxt:t > 0} describes a Feller process on the state ace C. Here the drift coefficient H:~0 x C ~Rn takes values in Rn and e diffusion coefficient G:~ x C ~ L(RmJRn) has values in the space Rm,Rn) of all linear maps ~m ~Rn. The noise w is a (standard) m-dimensional ownian motion. If the stochastic FOE is autonomous, the trajectory field a time-homogeneous diffUsion on C. In Chapter IV we look at autonomous stochastic FOE's

,d investigate the structure of the associated one-parameter semi-group 't}t>O given by the time-homogeneous diffusion on C. A novel feature of ch diffusions ~hen r > 0 is that the semi-group {Pt}t>O is never strongly •ntinuous on the Banach space Cb(C,R) =Cb of all bounded uniformly connuous real-valued functions on C endowed with the supremum norm (Theorem V. 2.2 )). Hence a weak generator A of {Pt}t>O can be defined on the .tter's domain of strong continuity C~ ; Cb and a general formula for A is ;tablished in Theorem (IV. 3.2). Due to the absence of non-trivial differttiable functions on C having bounded supports, we are only able to define ~eakly dense class of smooth functions on C which is rich enough to generate te Borel a-algebra of C. These are what we call quasi-tame functions (IV §4). 1 such functions the weak generator assumes a particularly simple and con·ete form (Theorem (IV. 4.3)). Distributional and sample regularity properties for trajectory fields of 1tonomous stochastic FOE's are explored in Chapter V. We look at two extreme tamples: the highly erratic delayed diffusions mentioned above, and the case • stochastic FOE's with ordinary diffusion coefficients viz. dx(t) = H(xt)dt

+

g(x(t))dw(t), t

>

0.

g satisfies a Frobenius condition, the trajectory field of the latter lass admits sufficiently smooth and locally compactifying versions fort> r fheorem (V. 2.1), Corollaries V (2.1.1) - V (2.1.4)). In ger.eral, the >mpactifying nature of the trajectory field for t > r is shown to persist in distributional sense for autonomous stochastic FOE's with arbitrary ip~chitz coefficients (Theorems (V. 4.6), (V. 4.7)). f

iv)

There are many examples of stochastic FOE's. In Chapters VI and VII we highlight only a few. Among these are stochastic ODE's (r = 0, VI §2), stochastic delay equations VI §3), linear FOE's forced by white noise (VI §4), a model for physical Brownian motion (VII §2), stochastic FOE's with discontinuous initial data (VII §3), stochastic integra-differential equations (VII §4), and stochastic FOE's with an infinite memory (r=~. VII §5). Chapter VII contains also some open problems and conjectures with a view to future developments. From a historical point of view, equations with zero diffusions (RFDE's) or zero retardation (stochastic ODE's) have been the scene of intensive study during the past few decades. There is indeed a vast amount of literature on RFDE's e.g. Hale [26], [27], [28], Krasovskii [43], El'sgol 'tz [18], Mishkis [56], Jones [42], Banks [3], Bellman and Cooke [4], Halanay [25], Nussbaum [62], [63], Mallet-Paret [49], [50], Oliva [64], [65], Mohammed [57], and others. On stochastic ODE's, one could point out the outstanding works of Ito [36], [37], [38], Ito and McKean [40], McKean [52], Malliavin [51], McShane [53], Gihman and Skorohod [24], Friedman [22], Stroock and Varadhan [73], Kunita [45], Ikeda and Watanabe [35], and Elworthy [19]. However, general stochastic FOE's have so far received very little attention from stochastic analysts and probabilists. In fact a surprisingly small amount of literature is available to us at present on the theory of stochastic FOE's. The first work that WP are aware of goes back to an extended article of Ito and Nisio [41] in 1964 on stationary solutions of stochastic FOE's with infinite memory (r = ~). The existence of invariant measures for non-linear FOE's with white noise and a finite memory was considered by M. Scheutzow in [69], [70]. Apart from Section VII §5 and except when otherwise stated, all the results in Chapters II-VII are new. Ce~tain parts of Chapters II, III and IV were included in preprints [58], [59], [60], by the author during the period 19781980. Section VI §4 is joint work of S.E.A. Mohammed, M. Scheutzow and H.v. Weizsacker. The author wishes to express his deep gratitude to K.D. Elworthy, K.R. Parthasarathy, P. Baxendale, R.J. Elliott, H.v. Weizsacker, M. Scheutzow and S.A. Elsanousi for many inspiring conversations and helpful suggestions. For financial support during the writing of this book I am indebted to the British Science and Engineering Research Council (SERC), the University (v)

of Khartoum and the British Council. Finally, many thanks go to Terri Moss who managed to turn often illegible scribble into beautiful typescript. Salah Mohammed Khartoum 1983.

I Preliminary background

§1.

Introduction

In this chapter we give an assortment of basic ideas and results from Probability Theory and Linear Analysi.s which make necessary prerequisites for reading the subsequent chapters. Due to limitations of space, almost all proofs have been omitted. However, we hope that the referencing is adequate. §2. Measure and Probability A measuPable space (G,F} is a pair consisting of a non-empty set Q and a a-algebra F of subsets of G. If E i.s a real Banach space, an E-valued measuPe ~on (Q,F} is a map ~:F + E such that (i} ~(fJ} = 0, (ii} ~ is a-additi~e i.e. for any disjoint count!ble famil! of sets {Ak};=t in F the series E ~(Ak} converges in E and ~( u Ak} = E ~(Ak}. When E = R, ~ is called k=1 . k=t k=t a signed measuPe and if ~(F~ it is called a positive measUPe. If sup {I~(A}I: A € F} <=,~is a finite measuPe. A positive finite measure P on (G,F} such that P(Q} = 1 is a pPobability measuPe on 0> the triple (G,F,P} is then a pPObability space. The set of all finite real-valued measures on Q is denoted by M(Q} and the subset of all probability measures by Mp(Q}. A probability space (G,F,P} is complete if every subset of a set of Pmeasure zero belongs to F i.e. whenever B € F, P(B} = 0, A~ B, then A € F. In general any probability space can be completed with respect to its underlying probability measure. Indeed let (G,F,P} be an arbitrary probability space and take Fp

= {A

u ~: A € F, ~

c

~O € F, P(~ 0 }

= 0}

to be the completion ofF undel' P. Extend P to Fp by setting P(AU~} = P(A}, ~ c ~0' P(~0 } = 0. Then (Q,FP ,P} is the smallest P-complete probability space with F c FP. Because of this property, we often find it technically simpler to assume from the outset that our underlying probability space is complete.

When n is a Hausdorff topological space and F is its Borel a-algebra, Bore1 n, generated by all open (or closed) sets, a 1neasure ll on n is :r>eguZa:r> if l.I(B) = sup {lJ(C) = inf {lJ(U)

C ~ B, C closed} B ~ U, U open}.

If n is metrizable, every l.l € M(n) is regular and hence completely determined by its values on the open (or closed) sets in n. (Parthasarathy [66], Chapter II, Theorem 1.2, p. 27). Let n be a Hausdorff space and E a real Banach space. An E-valued measure l.l on (n, Borel n) is tight if (i) sup {lu(B)I: B € Borel n} < oo, where 1·1 denotes the norm on E; and (ii) for each £ > 0, there is a compact set K£ in n such that lll(~K£)1 < £ Theorem (2.1): Let n be a PoZish space i.e. a complete separable metrizable space. Then every finite real Borel measure on n is tight. Proof: Parthasarathy ([66], Chapter II, Theorem 3.2, p. 29); Stroock and Varadhan ([73], Chapter 1, Theorem (1.1.3), pp. 9-10). c Let n be a separable metric space and F = Borel n. Denote by Cb(QJR) the Banach space of all bounded uniformly continuous functions $:n ~R given the supremum norm ll$11

c

=sup {j$(n)l: n En}.

b

The natural bilinear pairing <·. ·> ~ Cb(QJR) <$,ll> =

X

J

nEn

M(Q)

~

R

$(n) dl.l(n), $ E Cb(nJR),

lJ

E M(n),

induces an embedding M(Q) -------+ Cb(QJR) * lJ

~

<·,ll>

where Cb(nJR)* is the strong dual of Cb(QJR). sponds to the continuous linear functional

'2

Indeed each

lJ

E M(Q) corre-

because for

e~ery

' E Cb(nJR)

1J

'(n) dp(n) I < II' llcb v(lJ)(n) nEG p p where v(lJ)(n) =sup { E lll(Ak)l: Ak E F, k = 1, ••• ,p disjoint, n = u Ak' k=1 k=1 p < m} is the total variation of lJ on 0 (Dunford and Schwartz [15], Chapter III, pp. 95-155). As a subset of Cb(nJR) * give M(O) the induced weak* topology. Now this turns out to be the same as the ~eak toplogy or vague topology on measures because of the following characterizations. Theorem (2.2): Let n be a metric space and lJ, llk E M(O) fork= 1,2,3, •••• Then the following statements are all equivalent: (i)

llk

(ii)

lim J '(n)dlJk(n) = J '(n)dlJ(n), for every' E Cb(nJR); k-- nEn nEO

~

lJ as k

~min

the weak *topology of M(O);

(iii) lim sup lJk(C) < l!(C) for every closed set C in n; k

(iv)

lim inf lJk(U) > lJ(U) for every open set U in 0; k

(v)

lim lJk(B) = lJ(B) for every BE Borel n such that lJ(aB) = 0. k--

For proofs of the above theorem, see Parthasarathy ([66] Chapter II, Theorem 6.1, pp. 40-42) or Stroock and Varadhan ([73], Theorem 1.1.1, pp. 7-9). The weak topology on M(O), when n is a separable metric space, can be alternatively described in the following two equivalent ways: (a)

Define a base of open neighbourhoods of any lJ E M(O) by

k =

1,2, ••• ,p}

3

(b) Furnish M(O) with a metric p in the following manner. Compactify the separable metric space 0 to obtain "' o. Then Cb(OJR) is Banach space-isomorphic to the separable Banach space C(OJR) "' of all continuous real functions on "'0, given the supremum norm. Pick a countable dense sequence {~k};= 1 in Cb(OJR) and define the metric p on M(O) by CD p(~,v) =

1:

k=1

2kll

~kllcb

~.v

E M(O) (Stroock and Varadhan [73], Theorem 1.1.2, p. 9; Parthasarathy [66], pp. 39-52). ~ote that M(O) is complete if and only if o is so. Similarly,~(O) is compact if and only if 0 is compact. More generally compact subsets of M(O) are characterized by the well-known theorem of Prohorov given in Chapter V (Theorem '{V.4.5)). There is a theory of (Bochner) integration for maps X:O + E where E is a real Banach space and (O,F,~) a real measure space (Dunford and Schwartz [15], Chapter III §1-6). On a probability space (O,F,P) an (F, Borel E)-measurable map X:O+ E is called an E-valued random variabLe. Such a map is P-integrabLe if there is a sequence Xn:o + E, n = 1,2, ••• , of simple (F, Borel E)-measurable maps so that Xn(w) + X(w) as n +CD for a.a. wE 0 and lim J IX (w)-X (w)IEdP(w) = 0. m,n+m w€0 n m Define the expectation of an integrable random variable X:O + E by EX =

f0 ·X(w)dP(w) = lim

n+ao

j'

Xn(w)dP(w) E E.

0

This definition is independent of the choice of sequence {Xni~= 1 converging a.s. to X (Rao [67], Chapter I, §1.4; Yosida [78], Chapter V §5, pp. 132-136). For a separable Banach space E, X: 0+ E is a random variable if and only if one of the following conditions holds: (i) There is a sequence Xn:O + E, n = 1,2, ••• , of simple (F. Borel E)-measurable maps such that Xn(w) + X(w) as n + CD for a.a. w E O; (ii) X is weakLy measurabLe i.e. for each A E E*, A0 X:O +R is (F, Borel R}measurable. (Elworthy [19], Chapter I, §1(C} pp. 2-4; Rao [67], Chapter I §1.4}. Denote by £ 0 (o,E;F) the real vector space of all E-valued random variables X:O +Eon the probability space (O,F,P}. The spdce £ 0 (0,E;F} is a complete

TVS under the complete pseudo-metric

for x1• x2 € t 0 (Q,E~F). The norm in our real Banach space is always denoted by I .1 E or sometimes just I .1. A sequence {Xn};=t of random variables Xn:Q ~ E converges in probabitity to X € £ 0 (Q,E~F) if for every E > 0 lim P{w:w € Q, IXn(w)- X(w)IE > E} = 0. A random variable X:Q ~ E is

n-(Bochner) integrable if and only if the functi.on

w 1-----+

is P-integrable, in which case IJg X(w)dP(w)IE < JgiX(w)IE dP(w) i.e. lEXlE < EIXC·>IE. The space £ 1 (Q,E~F) of all integrable random variables is a complete real TVS with respect to the £1-semi-norm IIXII 1 £

=I

Q

IX(w)IE dP(w), X € t 1(Q.E;.F).

Similarly for any integer k > 1 define the complete space £k(Q,E;F) of all F-measurable maps X:Q ~ E such that JQ IX(w)lk dP(w) <~,endowed with the semi-norm

llxll

k £

=[I IX(w)l~ Q

dP(w)]1/k.

Note that the spaces £k(Q;E;F) become the real Banach spaces Lk(Q,E;F) if we identify random variables which agree on a set of full P-measure. If Xn• X € £0 (Q,E;F), n = 1,2, •••• say Xn ~X as n ~ ~ a.s. or Xn(w) ~ X(w) as n ~~for a.a. w € Q if there exist a set Q0 € F of full P-measure such that Xn(w) ~ X(w) as n ~~for all w € Q0 • The various notions of convergence for E-valued random variables are linked by the following theorem which is proved in the same way as for real random variables (cf. Dunford and Schwartz [15], Chapter III §2-4; Rao [67], Chapter I §1.4 pp. 16-29; Halmos [30]). Theorem (2.3): Let Xn• X:Q ~ E be random variables for n = 1,2, •••• and let E be a real separable Banach space. 5

(i)

If Xn +X as n + = a.s., then Xn +X as n +=in probability.

(ii) If Xn• X E tk(n,E;F), n = 1,2, ••• and Xn +X as n +=in tk (k > 0), then Xn +X as n +=in probability. (iii) If Xn +X as n + = in probability, then there is a subsequence {X }= ni i =1 of {Xn};= 1 such that xn. +X as i + = a.s. 1 (iv) Dominated Con~r:_gence: Let Xn E t 1(n,E;F), n = 1,2, ••• and XE£ 0 (n,E;F) be such that Xn +X as n +=in probability. Suppose there exists Y € t 1(nytl;F) such that, for a.a. w € n.IXn(w)IE < Y(w) for all n > 1. Then X € t 1(n,E;F) and J X(w)dP(w) = lim J X (w)dP(w). n n-- n n Chebyshev's inequality also holds for Banach-space-valued random variables X. It follows trivially by applying its classical version to the real-valued random variable IX(·)IE (Chung [7], p. 48)~ Theorem (2.4) (Chebyshev's Inequality): If E is a Banach space and X E £k (n,E;F), k > 1, then for every E > 0 P{w:w € n. IX(w)IE >

E}

<

~ f IX(w)l~ dP(w). E

n

In particular the map £k (n,E;F) X 1------;>

is continuous, for each k > 1. continuous also for k = 0.

lf E is separable, then the above map is

In a probability space (n,F,P), two events A,B € Fare independent (under P) if P(A n B) = P(A)P(B); two sub-a-algebras g1, g2 of Fare independent (under P) if P(A n B) = P(A)P(B) for all A € g1 and all B € g2; two random variables X, Y :· n + E are independent (under P) if the a-algebras a(X), a(Y) generated by X, Y respectively are independent under P. Theorem (2.5) (Borel-Cantelli Lemma) Let (n,F,P) be a probability space and nk}'!'? { •• k=1 c F (i) If P(nl<) converges, then P(lim sup nk) = 0 i.e. P(lim inf ~k) = 1; k k=1 k 0

r

00 (ii) · If W k}k=t are independent and 1: P(O k) diverges, then P(lim sup n k)=1 k=1 k i.e. P(lim inf O'Ok) = 0.

k

Proof:

Chung [7], pp. 71-74.

o

For a separable Banach space E, a random variable x~n + E on a probability space (O,F,P) and a sub~-algebra g of F, the conditional expectation of X given g is denoted by E(XIg) and is well-defined by the following theorem whose proof can be found in Elworthy ([19], Chapter I., §3, pp. 5-10). Theorem (2.6): Let E be a separable Banach space. continuous linear map

X

1

)

Then there is a unique

E(XIg)

such that

r

JA

X(w)dP(w) = J E(XIg)(w)dP(w), A

for all X E .c 1(n,E,F), A E g, with g a sub-a-algebra of F. Furthermore, IE(XIg)(w)IE < E(IX(·>klg)(w) for a.a wEn and IIE(•Ig)ll = 1. Conditional expectation given the sub-a-algebra g c F is therefore a projection (of norm one) mapping L 1(n,E;F) onto the subspace L 1(n,E;g). A detailed study of conditional expectations for E = R from this point of view is given in Rao ([67], Chapter II §2.2 pp. 44-62). Generally speaking conditional expectations have similar properties to (Bochner) integrals e.g. let E, F be separable real Banach spaces and L(E,F) be the Banach space of all continuous 1 inear maps T:E + F with the norm liT II= sup { IT(x) IF: x E E, lxiE < 1}. Then for any X E .c 1(0,E;F) and any T E L(E,F), E(ToXIg) exists and E(ToXIg) = ToE(XIg) for any sub-a-algebra g of F (Elworthy [19], §3(B), pp. 8-9). Let (O,F,P) be a probability space, g a sub-a-algebra of F and A E F. The conditional probability of A given g is denoted by P(Aig) E L1(0JR;g) and defined as

7

where XA is the characteristic function of A. Two sub-a-algebras g1 , g2 of F are conditionally independent undeP P given g if

for every A1 € g1, A2 € g2 • For real Banach spaces E, F, two random variables X:Q + E, Y:Q + F are conditionally independent undeP P given g c F if the a-algebras a{X}, a{Y} are conditionally independent given g. If the random variables X € £ 2 (Q,E;F), Y € £ 2 (Q,F;F) are conditionally independent given g and the Banach spaces E,F are separable, then we have E(X ® Ylg) = E(XIg) ® E(Yig) where ® denotes the algebraic tensor product, and E i 'IT F the compZete pPojective tensoP pPOduct (Treves [75], Chapter 43, pp. 434-445; Elworthy [19], Chapter I, §2, §3, pp. 4-13). Moreover, if X € £ 0 (Q,E;g) andY € £ 1(Q,F;F) are such that X ® Y € £ 1(Q,E in F;F), then E(X ® Ylg) = X ® E(Yig)

a.s.

If h is a sub-a-algebra of g c F, then E(E(XIg)lh) = E(XIh) for any X € £ 1(Q,E;F). §3.

If X and g are independent, then E(XIg) =EX a.s.

Vector Measures and the Dunford-Schwartz Integral

Here we review the Dunford-Schwartz (D-S) integral of a real-valued measurable function with respect to a Banach-space-valued measure. For a full exposition of the theory see Dunford and Schwartz ([15], Chapter IV, §10, pp. 318-328). Let (Q,F) be. a measurable space, E a real Banach space and ~:F + E an E-valued measure on (Q,F). Let f:Q +R be a simple jUnction viz. p

f = .~ xixA. for some x. € R, i = 1,2, ••• ,p and a partition {Ai}~= 1 c F of 1=1 1 p 1 Q. Define J f d~ = E x. ~(A. n A) for any A € F. Let llfll = max lx-1 {' 0. A

Then for any A € F

8

i =1

1

1

""

1
1

where the semi-variation llll II of lJ is defined by

a partition of A in F, p > 1} for any A € F. Though it is not a measure (it is only sub-additive!), the semi-variation is always finite. The total variation v(ll) of a vector-valued measure p is not necessarily finite. Some properties of the semi-variation are given below; (i)

lll(A) IE< llll

II (A) for all A € F;

(ii) llllii(A) < 4 sup {jlJ(B)jE:B € F, B =A}<

oo

for all A € F;

(iii)lllJII(B)< llllii(A) ifB=A, A, B€ F; (iv) llllii<.B Ai) < 1=1

.~

1=1

llll II (Ai)

if

{Ai}~= 1

c:

F;

(v) There is a positive finite measure ~ on (.G,F) such that lJ is absolutely continuous relative to ~ and ~(A) < llll II (A) for every A € F; i.e. llllii(A) = 0 if and only if ~(A) = 0. Now complete F by adjoining all subsets of sets with this completion F~ i.e.

~-measure

zero.

Call

A real-valued function f:G +R is said to be ll-measuPabte if for every B € Borel R, f- 1(8) € F~ i.e. if and only iff is ~-measurable. Say a sequence of functions fk:G +R, k = 1,2, ••• , converges lJ-atmost eve~heroe to f:G +R if fk + f ask+ oo ~-a.e. i.e. if there is a ~-null set~ € F~ such that fk(w) + f(w) ask+ oo for all w€ ~. A function f:G + R is (D-S) ll-integPabte (over G) if there is a sequence {fk}k= 1 of simple functions such that fk + f as k + oo ll-a.e. and the sequence {/A fk dlJ}k= 1 converges 9

in E for every A € F. The limit

IA

d~ = ~~ IA

f

fk

d~

€ E 00

is independent of the choice of sequence {fk}k=t and is called the (D-S) integmZ of f over A with respect to the E-vaZued measure

~.

Theorem (3.1) (FToperties of the Integm~: Let E be a real Banach space and ~an E-valued measure on a measurable space (o,F). (i) Let f,g:n +R be ~-integrable over A € F and a,a € R. Then af ~-integrable over A and

IA(

af

+

ag) d~ = a

IA

f

d~

(ii) If f:O +R is ~-measurable and grable over any A € F and

+

a

IA

g

W€A

ag is

d~.

~-essentially

1IA f d~IE < 11~ II (A) ~-essup

+

bounded, then f is

~-inte­

lf(w) I·

(iii) The map

is an E-valued measure for any

~-integrable

f:O +R.

(iv) (Dominated Convergence): Fix A € F and let fk:n +R, k = 1,2, ••• be a sequence of ~-integrable functions converging ~-a.e. to a function f:O +R. Suppose there is an integrable g:n +~such that lfk(w)l < g(w) for ~-almost all wand all k > 1. Then f is ~-integrable over A and

fA f dp = lim k-+oo

J fk d~ • A

Proof: Dunford and Schwartz [15], Theorem IV.10.8 (pp. 323-325); Theorem IV. 10.10, p. 328. c

10

§4. Some Li.nea r Ana l_Y.s i_! lf E is a real Banach space, its strong dual is E* = L(E~), the space of all continuous linear functionals A.:E -+R with II>.. II =sup ..(v)l~v E E, 1v1E =1}. The weak topology on E is obtained by taking as base of open neighbourhoods of points x E E the sets N(x;r,E) = {y:y E E, l>..(y)- >..(x)l < E, >.. E r} where r c E* is finite and E > 0. In particular a generalized sequence {x } in E a converges weakly to x E E if lim >..(x ) = >..(x) for each A. E E* (Dunford and a

a

Schwartz If E, the weak lxiE<1}

[15], §V.3). F are real Banach spaces, say a map T E L(E,F) is weakly compact if closure of the image T(B) of the closed unit ball B = {x~x E E, in E is weakly compact in F. A subset of a Banach space is weakly sequentially compact if every sequence in it has a weakly convergent subsequence. Thus T E L(E,F) is weakly compact if and only if it maps bounded sets in E into weakly sequentially compact subsets of F; e.g. a compact linear map is weakly compact. Let S be a compact Hausdorff space and C(S,R) the Banach space of all continuous functions n:S -+ R given the supremum norm lin II = sup ln(s) I· Then c sES for any real Banach space E, a weakly compact linear map T:C(S,R) -+ E corresto an E-valued measure on (S, Borel S). Indeed we have the following generalization of the Riesz Representation Theorem for continuous linear functionals due to Bartle, Dunford and Schwartz {[15], Theorems VI (7.2) (7.3), pp. 492-496). Theorem (4.1): LetS be compact Hausdorff and T:C{SJR)-+ E a weakly compact linear map into a real Banach space E. Then there is a measure ~~Borel S-+ E such that (i)

T(n) =

(ii)

IITII

=

fs

n(s) d~(s), for all n E C(S,R);

IIJ,JII(S); ~o~

(iii)

for each>.. E E*,

is a regular finite Borel measure on S;

(iv)

T*(>..) = >,.o~ for all A. E E*.

Conversely, every E-valued measure~ on (S, Borel S) which satisfies (iii) above defines a weakly compact linear map T:C{SJR) -+ E which fulfills (i), (ii) and (iv). 11

Remark The integral in (i) is the 0-S integral introduced in the previous section (§3).

A class of weakly compact 1 inear maps C(S,R)

+

E i.s provided by

Theorem (4.2): lf the Banach space E is weakly complete, then every continuous linear map C(SJR) + E is weakly compact. Proof:

Dunford and Schwartz ([15], Theorem

~l

7.6, pp. 494-496).

a

If E, F are real Banach spaces, their projecti'.Le tensor product E 9 F is 1f a real Banach space with respect to the completion of the norm

Here the infimum is taken over all possible representations p

z

=

~

i=1

x. 8 y. € E 8 F, 1

1

the algebraic tensor product of E and F. Let L(E,F,~) be the space of all continuous b·il inear maps e:E x F + R viz. all those bilinear e such that there is a constant K > 0 with IB(x,y)l < KlxiE IYIF for all x € E and all y € F. Then L(E,F~) is a Banach space when furnished with the norm liB

II =sup {IB(x,y)l : x € E, y

€

F, 1x1E < 1, IYIF < 1}.

The projective tensor product allows us to identify each continuous bilinear map Ex F +R with a continuous linear functional onE 91T F viz. Theorem (4.3):

L(E,F~)

is norm isomorphic to [E 01T F] * •

Proof: Treves [75], Proposition 43.12, pp. 443-444.

c

As an .example consider the following situation. Let S be a compact Hausdorff space and E = F = C(SJR). Then every continuous bilinear form e:C(SJR) x C(SJR) +R corresponds to a signed bimeasure on S x S viz. a set function v:Borel S x Borel S +R such that for any A,B € Borel S, v(A,·) and v(·,B) are measures on s. An integration theory for signed bimeasures has 12

been developed by Yl;nen [77]. Note that a continuous b;linear form on C{SJR) does not necessarily correspond to a measure on S x S because of the str;ct embedding C{SJR) i'lr C{SJR) ~ C(S ne

~

>

X

s. R)

{{s,s') 1--+ n<sn(s')}

(Treves [75], cf. Exerc;se 44.2, p. 450). The following theorem of Mercer w;11 prove to be useful. For a proof see R;esz and Sz-Nagy {[68], Chapter \ll, §97-98, pp. 242-246) or Courant and Hilbert {[10], pp. 122-140). Theorem (4.4) (Mercer's ~eorem): Let l cR be a closed bounded ;nterval and K:l x l +R a continuous symmetric kernel which is positive in the sense that

J1 J1 K(s,s•)n(s)n(s•)ds

ds' > 0

for all n € C{lJR). Suppose {~k};=t ;ons of the eigenvalue problem

J1 K(s,s•)~k(s•)ds•

= ~k

c

C{lJR) and {~k};= 1 c~ are solut-

~k(s), for all s

€ 1.

Then K can be expanded as a uniformly convergent series 00

on

1 x

I.

Remark

JJ

The val;d;ty of 1 1 K(s,s•)n(s)n(s•)ds ds• > 0 for all n € C{lJR) ;s equ;valent to ~k > 0 for all k > 1. Theorem (4.5) linear map T:E

For real Banach spaces E, F, every Borel-measurable F is cont;nuous.

(Douady):. +

13

Proof: §5.

Schwartz. ([11], Part U, Chapter I., §2 pp. 1.55-·1.60).

a

Stochastic Processes and Random Fi.elds

Let ( O,F,P) be a probabil i.ty space, T a complete separable metric space and E a real Banach space. An E-v.alued stochastic process x on (Q,F,P) parometl'ized by T is a map x:T x n + E such that x(t,.) e:.c~n,E;F) for all t e: T. If 0
measuroble if x is (Borel T Q F, Borel E)-measurable;

(ii) a.s. continuous (or x has continuous sample fUnctions, or continuous) if for a.a. we: n, x(.,w):T + E is continuous;. (iii) locally bounded if for a.a. we: n, x( • ,w) i.s locally bounded; (iv) square-integrable (or of class .c 2) if for each t e: T,x(t,•) e: .c 2 (~,E;F); (v) of class .Ck (k >

1)

if for each t e: T,x(t,·) e: .Ck(O,E;F).

An E-valued process x:[O,a] x n + E, parametrized by one-dimensional time T = [O,a] c~, is adapted to the filtrotion (Ft)O
C([O,a],:Rn) > x( • ,w)

i.e. as a C([O,a]JRn)-v.alued random variable (Chapter II, §2). In particular, if x:[O,a] x n +:Rn is continuous and (Ft)O
C([O,t],:Rn)

w t----> x( • ,w) 1[0, t]

(Lemma (11.2.1)). 14

I.n a filtered probability space (S'l,F,(Ft)O 1 denote k by!k the k-tuple (t 1,t 2 , ••• ,tk) E T. Then we have a measurable map x(tk,.) ~ n ~ Ek

The family of probability measures Pox(.~_k,.)- 1 E Mp(Ek) where ..tk ranges over a 11 finite subsets of T is ca 11 ed the set of finite-dimensional distributions of x. Let x1:T X n, + E, x2~T X n2 + E be stochastic processes on two probability spaces (ni,Fi,Pi), i = 1,2 with the same parameter set T and the same Banach space E as state space. Then say x1 and x2 are isonomous and write x1 ~ x2 if they have the same set of finite-~imensional distributions, i.e. if t 1, ••• ,tk E T, B1, ••• ,Bk E Borel E then

Two processes x1, x2 :T

n +Eon the same probability space (n,F,P) are versions of one another if for each t E T the random variables x1(t,.) and x2(t,•) agree on a set nt E F of full P-measure. More generally, let X:T +~ 0 (S'l,E;F) be a Borel measurable map, then a measurable version of X is a (Borel T 9 F, Borel E)-measurable process "'X:T x n + E such that "'X(t,·) = X(t) on a set nt E F of full P-measure for each t E T. The following results of Stricker, Yor, Cohn and Hoffmann-J~rgensen are found particularly useful in establishing measurable versions for the trajectory random field of a stocastic FOE. x

Theorem (5.1) (Str>icker-Yor): Let (n,F,P) be a probability space, (T,g) a measurable space and E a complete metric space. Suppose Xn:T x n + E, 15

n = 1,2, ••• is a sequence of (g 8 F, Borel E)~easurable maps such that for each t € T the sequence {Xn(t,.)};=1: con~terges in probabi.li.ty on rz. Then there is a (g 8 F, Borel E)-measurable map X:.T x. f2 -+ E such that for each t € T, X(t,·) = lim X (t,•) in probability. n--n Proof:. S.A. Elsanousi indicated to m.e that the original proof of StrickerYor ([72]) for the case E = :R does i.n fact generali.ze easily to any complete metric space E. c. Theorem (5.2):. (Cohn-Hoffman-J~rgensen) Let (fl,F,P) be a probability space, T a separable metri.c space and E a complete separable metric space. Then a Borel-measurable map X:.T -+L 0 (fl,E;F) has a (Borel T 8 F, Borel E)-measurable v.ersion T x Q-+ E if and only if the range X(T) is separable in IL. 0 (r2,E;F). ln particular, if Tis also complete, each Borel map X:.T-+ L0 (fl,E;F) admits a measurable version T f2-+ E. l.(.

Proof:. For compact E, the theorem is due to Cohn ([9]), Theorem 3). If E is a complete separable metric space, the first assertion of the theorem holds by a remark in Hoffman-J-rgensen ([34] p. 206). When T is complete separable and the map X:.T-+ l 0 (r2,E;F) is Borel, a general result of C. Moore implies that X(T) is separable ([9]) and so the second assertion of the theorem follows directly from the first. c §6.

Martingales

This is a v.ery important class of stochasti.c processes parametrized by onedimensional time. Let E be a real Banach space, a> 0 and (fl,F,(Ft)O
i.t is right-continuous i.e. has a.a. sample paths right-continuous;

(ii) for each t € [O,a], x(t,·) € £ 1 (rz,E~Ft); (iii) E(x(t2,.) 1Ft) = x(t 1 ,.) a.s. whenever 0 < t 1 < t 2
If each Ft contains all sets of P-measure zero in F, every (Ft)O
measurable (Stroock and Uaradhan [13], Lemma t.2.t~ p. 20). A real process x:[O,a] x n ~R is an (Ft)G
it is right

continuous~

(ii) x(t,•) E £ 1 (nJR~Ft) for all t E [O,a]~ (ii.i) E(x(t2 ,·)1Ft) > x(tt'•) a.s.

whene~er

1

0 < tt < t 2
Theorem (6.t) (Doob's Inequality): Suppose x~[O,a] x n ~R" is an (Ft)O 1 is an integer such that x(t,.) E £k(OJRn) for all n t E [O,a]. Let R carry the norm.

hd 1 =

n I:

i=t

lv-1 , v

=

1

(v 1, ••• ,vn) ERn.

Then the real process [O,a]xn~R

(t,w) ~ lx(t,w) I~ is an (Ft)O
P{w~w En,

sup 0
Furthermore, for every

lx(t,w)l~ >

E}

E >

0

<~ Elx(a,•) I~ E

and E{ sup lx(t,·)l~} < (k~ 1 >k Elx(a,.)l~. O
k lx(t,w)l >

1

E}

0
for every E{ ~:

E

>

Kk k < Jk Elx(a,•)l E

0, and

~

Ix(t,.) lk} < Kk2

k k (k-T)

EIx(a,·) lk •

Represent x in coordinate form 17

where each xi ~[O,a] x o + R, i = t, ••• ,n, is. a real ( Ft)O1 1 1Ft)= .E E(lxi(t 2,·>1 1Ft) t t=t t n

> i!t lxi(tp·>l = lxCt 1 ,·>1 1 a.s. Therefore lxl 1 is an (Ft)O 1, due to the convexity of the real-valued function

yo ---+yo (Stroock hold for Watanabe nom I •

t 1----+ tk and Varadhan [73], p. 21). The given inequalities are well-known to the submartingale lxl~ (Stroock and Varadhan [73] p. 21; Ikeda and [35], Theorem 6.10, pp. 33-34• Friedman [22], pp. 9-15). For any I on Rn there are constants L, M> 0 such that

Mlvl < lvl 1 < Llvl for all v ERn. .

P{w~w

Hence, for every

£ >

k lx(t,w)l >

E 0, sup

0,

£}

O
< P{w~ e 1 <~ 1

o, sup lx(t,w) I~> Mkd O
J< 1

1

Kk EIx(a,.) lk EIx(a,.) Ik1 < J<

Lk

where Kk = ,;k2 > 0. A s!milar· argument yields the final inequality of the theorem with 2 L" Kk=f.tK• D §7. Markov Processes These form another important class of Banach-space-valued stochastic processes parametrized by one-dimensional time [O,~). The following precise setting is 18

made after Dynkin ([16], Chapters 11., 1.1.1, pp. 41-1.04). A Markov process x on a real Banach space E i.s a collection {(!1,F,F~,x,Ps,n)~n € E, 0 < s < t < m}

satisfying the following

hypotheses~

For each s > 0 and n € E, (!'l,F,(F~)t>s,Ps,n) i.s a filtered probability space in whi.ch every Ps is a probabili.ty measure defined on the a-algebra s .n Fs = o{Ft~ t > s}. ( i)

m

(ii)

S

lf s' <sand t < t', then Ft

=Ft•· Sl

(iii) The process x~[O,m) x !1 + E i.s (Borel [O,m) Q F, Borel E)-measurable and x(t,.) i.s (F~, Borel E)-measurable whenever t > s. (iv.)

For fixed 0 < s < t < m , B € Borel E, the function E -->R

n t--> Ps

{w~w € Q,

.n

x(t,w)

€

B}

is Borel-measurable. (v)

(vi)

For each s > 0 and n € E, Ps,n {w~w € !1, x(s,w) = n} = t. If B € Borel E, 0 < s < t < u and n € E, then the Markov Property Ps.n(x(u,.)

€

BIF~)

=

Pt,x(t,·) (x(u,.)

€

B)

holds a.s.-Ps on !1• •n An E-v.alued Markov. process defines a family of transition probabilities {p(s,n,t •• )~n € E, 0 < s < t<m} onE gillen by p(s,n,t,B)

=

Ps

•n

(x(t,·)

€

B), B € Borel E•

These have the properties: (i)

for fixed 0 < s < t, B € Borel E, p(s,.,t,B) is

Borel-measurable~

(ii) for fixed 0 < s < t, n € E, p(s,n,t,·) is a Borel probability measure on E;. (iii) the Chapman-Kolmogorov identity

19

p(s,n,u,B) =

J

~EE

p(s,n,t,d~) p(t,~,u,B)

holds for 0 < s < t < u, neE, Be Borel E (Oynkin [t6], p. 85). A Markov process i.s time-homogeneous if its associated fami.ly of transition probabilities {p(s,n,t,·):.n e E, 0 < s < t < <»} sati.sfi.es p(s,n,t,B) = p(O,n,t-s,B) whene~er

0 < s < t, n e E, B E Borel E.

Let b(EJR) denote the Banach space of all real-valued bounded Borelmeasurable functions $on the Banach space E with the supremum norm II$ lib =sup {I$Cn>l~n E E}. Then a time-homogeneous Markov process onE with transition probabili.ties {p(O,n,t,.):. neE, t > 0} yields a contraction one-parameter semi.-group {Pt}t>O of continuous linear operators Pt:b(EJR) + b(EJR) defined by

Pt(~)(n) =

f

~EE

$(~) p(O,n,t,d~), n E E,

for each $ E b(EJR) (Dynkin [t6], Chapter 11., §t-2, pp. 47-61; Chung [8], pp. 1-12). A time-homogeneous Marko~ process on a Banach space E with semigroup Pt:.b(EJR) + b(EJR), t > 0, i.s called a FeUer process if (i) {Pt}t>O lea~es invariant the closed li.near subspace Cb(EJR) all bounded uniformly continuous functions E ·R~

c:

b(E,R) of

(ii) {Pt}t>O is weakty continuous on Cb(E,R) at t = 0 i.e. for every n E E lim Pt($)(n) = $(n) for all $ E Cb.

t.o ...

As a consequence of our results in Chapter lll (Theorem (IV. 2.2)), for infinite dimensionat E the last condition (ii) in the above definition of Feller process does not impty strong continuity at t = 0 of the semi-group {Pt}t>O on Cb(EJR). Compare this with the locally compact case in Chung ([8], pp. 48-56) where strong continuity is implied by weak continuity at t = 0. For a further study of (time-homogeneous) Marko~ processes, their transition semi-groups and infinitesimal generators, the reader may consult Dynkin ([16]).

20

§8.

Ex amp 1es

Let us wind up by looking at some useful examples of stochastic processes which are indispensible for our forthcoming discussion of stochastic FOE's. (A)

Gaussian Fields

If T is a metric space, a real stochasti.c process x~T x n + :R on a probabi 1 i ty space ( n, F, P) is a Gaussian process i.f for every t e: T the random variable x(t,.) e: i(n,R~F) and has a Gaussian distribution (Hida [31], pp. 31-43). For a Gaussian process x~T x n + R define the mean m~T + R and covariance V.~T x T +:R by setting m(t) = E x(t,.),

t

e: T

'l(s,t) = E{[x(s,.)-m(s)][x(t,.)-m(t)]}, s,t

e: T.

A Gaussian process x~T x n + R is called a Gaussian system if every finite linear combination from the family {x(t,.):~ t e: T} has a Gaussian distribution (Hida [31] §1..6). The following results concerning Gaussian systems are wellknown. Theorem (8. 1) :. (i)

Any subsystem of a Gaussian system is also a Gaussian system.

(ii) For any Gaussian system x~T x n +R the closed linear hull of {x( t, ·) :. t e: T} in .c 2 ( n,R;.F) is a Gaussian system. (iii) If x:T x n + R is a Gaussian system, then a necessary and sufficient condition for the family {x(t,.ht e: T} to be independent under Pis V(s,t) = 0 whenever s # t. (iv)

If x:z+ x n +R is a (discrete-time) Gaussian system, then the sequence {x(n,.)}~= 1 converges in probability i.f and only if i.t converges in .c 2 (n,R;F). ln this case lim x(n,.) is a Gaussian random variable.

n-~:

Hida [31], pp. 34-35.

c

lf Tis a real Banach space, a Gaussian system x:T x n +:Ron a probability space (n,F,P) will be called a Gaussian (random) field or just a Gaussian field. Gaussian fields parametrized by i.nfini.te-dimensional Hilbert or Banach spacesdo not in general admit continuous or locally bounded versions 21

which are defined on the whole of the paraQteter space (Dudley [13] and Chapter V. §3 of thi.s book). For a deeper study of sample regularity properties of Gaussian processes, the i.nterested reader could refer to the works of Dudley [13], [14], Feldman [20] and Fernique [21]. (B)

Brownian Motion

Let (n,F, (Ft)t>O,P) be a filtered probabi.li.ty spa·ce. A process w~xn +R is a one-dimensional Brownian motion (or a Wienero proocess) on (O,F,(Ft)t>O,P) if (i)

it i.s (Ft)t>O-adapted.

(ii) it i.s a Gaussian system on (n,F,P) with mean and covariance given by

E w(t,.) = 0, E w(t,.)w(s,·) =min (t,s), for all t,s > 0. A Brownian motion w can always be normalized so that w(O,.) = 0 a.s. A process w~ x n +Rm is an m-dimensional Broownian motion if it is of the form w = (w 1,w 2 , ••• ,wm) where the coordinate processes {wi}~= 1 are independent one-dimensional Brownian motions on (n,F,(Ft)t>O,P). For easy reference, we list here some of the basic properties of Browni,an motion in :Rm. These are well-understood and the reader may look at Hida [31], Chung [8], Friedman [22] and McKean [52] for proofs. Theorem (8.2)~ Let w~ x n +Rm be an m-dimensional Brownian motion on the filtered probability space (n,F,(Ft)t>O,P). Then the fo 1 lowing is true: (i)

w is an (Ft)t>O-martingale.

(ii) The sample paths of w are almost all nowhere differentiable.

"' such that for any a > 0 and any 0
For a.a. w € n, lim sup (2t log log{>-~ lw(t,w)l = m112 t+O.., lim sup (2t log log t)-~ lw(t,w) I = m112 t+oo

22

~.

almost all

where I • I denotes the Euclidean norm This property is called the

m 2 vi" v = (v1, ••• , vm) E Rm.

hd = .1:

\=t

of the Iterated Logarithm.

Law

(v) If t 0 > 0, then the process {w(t~t 0 ,·)- w(t 0 ,·>~ t > 0} is an m-dimensional Brownian motion on (n,F,(Ft)t>t ,P).

~ ,Rm) be the space of all conti.nuous 0 ~ .... Rm and "'F (vi) Let "' n = C(Kpaths 1\be the a-algebra generated by all cylinder sets in~ of the form {f:f E ~. (f(t 1), ••• ,f(tk)) E Bk}' t 1 , ••• ,tk E ~. Bk E Borel (Rm)k~ Then there is a unique probability measure ~w (called Wiener measure) on (rl,F) giving the finite-dimensional distributions of w viz.

I

g(t 1,t 2-t 1, ••• ,tk-tk_ 1;x 1,x 2-x 1, ••• ,xk-xk- 1)dx 1dx 2, ••• ,dxk Bk where the kernel g:(~)k x (Rm)k .... ~is given by =

g(u1, ••• ,uk;y1, ••• ,yk)

k

=

2

.rr (2w uJ.)-m/2 e-Yj/2uj

J=1

for u1, ••• ,uk E ~. y1, ••• ,yk E Rm. (vii) If 0 < s < t, then E(lw(t,·)-w(s,.)l 2k IFs)

=

(2k-2+m)(2k-4+m) ••• mlt-slk

a.s. for every integer k > 1. (viii) For 0 < t 1 < t 2 < t 3, the increments w(t 2,.)-w(t 1,.), w(t 3,.)-w(t 2,.) are conditionally independent under P given the a-algebra a{W(u,.):O < u < t 1}. (xi) w is a Feller process with stationary transition probabilities

given by

I e-1!::.\'_ I

2

p(O,x, t,B)

=

1

--m/2 (2wt)

B

--zt

dy, B E Borel Rm.

Furthermore, the associated contraction semi-group {Pt}t>O is strongly 23

continuous on b(:Rm ,R). is given by

where $~m +:R is a m 6$(x) = .~ 1=1

c2

l.ts infi.ni.tesimal generator A:.V(A) c b(:Rm ,:R) +b(:Rm ,IR)

function with compact support and 6 is the Laplacian

a2$ :-! (x). ax.1

(C) The Stochastic Integral We adopt the viewpoint of McShane ([53], Chapters ll, ll.l, 1\1) which contains adequate information for all our purposes. However, it is interesting to note that Elworthy's recent treatment ([19], Chapters 1.11, 1\1, \1) contains far-reaching generali.z.ations of McShane's stochastic integral to infinite dimensions. Other references for stochastic integrati.on include Ito [39], McKean [52], Meyer [55], Friedman [22], Gihman and Skorohod [24], Ikeda and Watanabe [35], Metivier and Pellaumail [54], and Arnold [2]. In thi.s secti.on we shall only content ourselves by giving a brief account of the stochasti.c integral including some of its most important properties. Let (n,F,(Ft)t>O,P) be a filtered probability space. For a closed interval [a,b] c:R a paPtition IT= (t 1, ••• ,tk* 1 ~T 1 , ••• ,Tk), a= t 1 < t 2 < ••• < tk = b, Ti E :R, i = 1,2, ••• ,k, is beLated if Ti < ti, i = 1, ••• ,k and is Cauchy if Ti = ti, i = 1, ••• ,k. If f:[a,b] x n + L(:Rm,IRn) and z.:[a,b] x n +:Rm are (Ft)a~ -adapted processes, form the Riemann sum S(IT) E £ 0 (n,:Rn;Fb) by setting

for all wEn.

For belated IT define

mesh IT= max {t.1 +. 1 - T.1.

~

1 < i < k}.

The stochastic integpaZ Jb f(t)dz(t) r: a

t 0 (n,:Rm~Fb)

is defined as

S(IT) J b f(t)dz(t) = lim a meshiT + 0 whenever the above limit exists in probability as meshiT + 0 24

o~er

all belated

partitions II of [a,b]. Conditions under l!lhich this li.m.i.t exists are given below, together with SO!IIe of the basic properties of the stochastic integral. For proofs see McShane [53], Elworthy [19] and Friedman [22]. Theorem (8.3):. (i) Let f;:.[a,b] X n-+ L(:R01 ,Bn), Z;,:.[a,b] X n -+R01 , i. = 1,2, be (Ft)a
j

< 2,

all exist • Then a.s.

t

{f 1(t)

t

f 1(t)d{z.1(t) "'z 2(t)} =

a

and

a

+.

f 2(t)} dz.1(t) =

t

a

t a

f 1(t)dzl(t)

f 1(t)dz.1(t) ·

t

f 2(t)dz 1(t)

t

f 1(t)dz 2(t),

-1:.

a

+.

a

Furthermore, if A € £ 0 (0JR~Fa), then the integrals

where all integrals exist.

Jb Afi(t)dzi(t), Jb fi(t)d{Azi(t)}, i = 1,2, a

a

also exist and = Jb f.(t)d{Az..(t)} =AJb f.(t)dz.(t) a.s. Jab H.(t)dz..(t) 1 1 a, 1 a, 1 fori = 1,2. (ii) Let a< c
a

+

Jb f(t)dz(t)

a.s.

c

Theorem (8.4): (Without loss of generality) suppose that each a-algebra Ft, a < t < b, contains all sets of P-measure zero in F. Let f:[a,b] x n-+ L(Rill,:R 25

be

(Ft)a~-adapted,

[a,bJ t

square-integrable and such that the map

--> icn,L(Rm ,Rn);. Fb) 1----)

f(t,·)

is continuous a.e. on [a,b] (Lebesgue measure) and bounded. Suppose z:[a,b] x n ~Rm is (Ft)a~-adapted and there is a K > 0 such that IE(z(t,.) - z(s,.)IFs)l < K(t-s) E(lz(t,•)- z(s,·>l 2 IFs) < K(t-s) t

a.s. whenever a< s < t
Elfa f(t)dz(t)l 2 < {2K(b-a) 112

=

Ja f(u)dz(u)

b

+

K112 } [fa E llf(t,·)ll 2 dt] 112

where 1 • 1 denotes Euclidean norms and II • II the standard operator norm on L(Rm ,Rn) ( §2). If, furthermore, z is a continuous (Ft)a~-martingale, then the stochastiC indefinite integral {I(t):a < t < b} has a version which is also a continuous (Ft)a 0, t

P{w:w

€

n,

sup l(w)

a~

J

a

f(u)dz(u) I

> £}

b

<

~ fa E llf(u,.) 11 2 du.

Remarks w:[a,b] x n~Rm ism-dimensional Brownian motion on the conditions of Theorem (8.4) above are automatically satisfied by z and the stochastic integral agrees with the classical Ito integral (Friedman [22], Chapter 4, pp. 55-97). (i) When z

=

(n,F,(Ft)a~·P),

(ii) McShane defines an Ito-belated integral which embraces a more general class of integrands f viz. f:[a,b] x n~L(Rm ,Rn) measurable, (Ft)a~-adapted and such that for a.a. w € n, f(.,w) is Lebesgue-square integrable over [a,b] i.e. J~ llf(t,w)ll 2 dt < oo (McShane [53], Chapter II §5, Chapter III §8). In 26

b

particular if f satisfies the stronger condi.ti.on

fa

E II f(t,· lll 2dt
Ito-belated integral fulfills the conclusions of Theorem {8.4) above. Theorem (8.5) (The Ito Integral): Let ~:.[a,b] x Q -.Rm be m-dimensional Brownian motion on the filtered probability space (Q,F,(Ft)a
Jab llf(t,w) II 2 dt < co b

for a.a. w € n.

Then the Ito integral

fa

f(t)dw(t) exists in £ 0 (SlJRn;Fb).

Furthermore if k > 1 is an integer and b 2k JaEIIf(t.·)ll dt
then the indefinite integral t {I(t) = J f(u)dw(u)~ t € [a,b]} a has a version which is a continuous (Ft)a~-martingale. E a~

t

ifa

f(u)dw(u)i 2k < Kk(m,n)(b-a)k- 1

In fact,

b

fa

E llf(u,·)lllkdu

where

Proof: In one dimension m = n = 1. see Friedman [22] (Chapter 4, Corollary 6.4 p. 87). For higher dimensions, write

where wj:[a,b] x Sl-+ R, j = 1, •••• m. are one-dimensional Brownian motions and fij:[a,b] x n -+R, i = 1, •••• n. j = 1, •••• m are the entries off with respect to canonical bases of Rm and Rn. Then

It f(u)dw(u) a

m

= (I: j=1

It f .. (u)dw.(u))._ 1 n

a lJ

J

l-

and so by H8lder's inequality 27

t n m t If f(u)dw(u) 12k < m2k--1. nk-t .1: .• 1:. If f .. (u)d~.(u) 12k a t=t J=t a lJ J

a.s. for all a< t
x

C([a,b], L(Rm ,Rn)) + L 2(n,Rn;.Fb)

by setting t

Z(t,f) = fa f(u)dw(u) for all t € [a,b] and all f € C([a,b],l(Rm,Rn)). Then Z. has a Borel-measurable version

"'Z:.[a,b]

x

C([a,b], L~,Rn))

x

n ... nn

with the following properties:. (i) the random field {Z(t,f,.):. t € [a,b], f € C([a,b], L(RmJRn))} is an n-di.mensional Gaussian system;. (ii) for each t € [a,b], the random field "'Z(t,.,.):.C([a,b], L(:RmJR)) n

x

n +R n

is linear in probability but has almost all sample functions non-linear i.e. for fixed f 1,f2 € C([a,b], L(RmJRn)) P{w:w € n, "'Z(t,6 1f 1

~

62f 2,w) = 61"'Z(t,f 1,w) + 62"'Z(t,f2,w) for all 61'62 € R} = t

while P{w:.w

€ Sl·,

"'Z(t,f1+f2,w) = "'Z(t,f,.w) ~ "'Z(t,f2,w) f 1,f2 € C([a,b], L(RmJRn))}

28

=

0.

for all

Remark Note that e~ery measurable version of ~satisfies properties (i) and (ii) of the abo~e theorem. The proof of the theorem ts included in the discussion of Sections V §3, V §4 (cf. Remarks V (3.1), Corollary V (4.7) and also Section VI §3).

29

II Existence of solutions and dependence on the initial process § t.

Basic Setti.ng and Assumptions

Let (O,F,P) be a probability space with Fa a-algebra of subsets of the measure space o and P a probability measure on o. Suppose a > 0 and let (Ft>tE[O,a] be an increasing family of sub-a-algebras ofF i.e. Ft 1 Ft 2 if

=

t 1,t2 E [O,a] and t 1 < t 2• Assume also that each Ft contains all the sets of measure zero in F. If r > 0, let J = [-r,O] and C(JJRn) the Banach space of all continuous paths y~J -+ Rn with the sup-norm IIY II C = sup IY(s) I where I· I denotes the SEJ

Euclidean norm on Rn (n > 1). As a metric space we associate with C(J,Rn) its Borel a-algebra Borel C(J,Rn) Denote by £2(o,C(J,Rn)) the space of all F-measurable stochastic processes 6:0 -+ C(J,Rn) such that the function o 3 w

,__>

J

lle(w)llcER is of class £2 i.e. 0

lle(w)l~dP(w)


Then £2(o,C(JJRn)) is complete when endowed with the semi-norm lie

II 2 l (O,C)

=

[J0 lle(w)ll~

dP(w)Jl 112 •

One may also look at the space C([O,a], ico,C(J,Rn))) oT· all i-continuous C(J,Rn)-valued processes y:[O,a] -+ £2(o,C(J,Rn)); again this is complete under the semi-norm

IIYII

C([O,a],i(o,C))

=

sup lly(t) II tE[O,a] £2(0,C)

Say a process y E C([O,a], £ 2(0,C(JJRn))) is adapted to (Ft)tE[O,a] if y(t) is Ft-measurable for each t E [O,a]. Signify by CA([O,a], £2(0,C(J,Rn))) the set of all such processes. Then CA([O,a], £ 2(o,C(J,Rn))) is closed in C([O,a], £2(o,C(J,Rn))); to see this, note that because of the continuity of the evaluation map 30

pt:.C([O,a], t. 2 (Sl,C))

~

t. 2 (G,C)

z.(t)

z.

for each t € [O,a], it is sufficient to obser~te that the space ico.C(J.Rn);.Ft) of all Ft-measurable maps ljJ:.G + C(J.Rn) is closed in s:. 2 (o,C(J.Rn)~. Similarly the space s:.:(o,C([-r,a]JRn)) of all processes belonging to t.~(Sl,C([-r,a]JRn)) and adapted to (Ft)t€[O,a] is closed in s:. 2 (o,C([-r,a]~n)). This is because the evaluation map is continuous and each Ft contains all sets of measure zero in F. When looking at stochastic (retarded) fUnationaZ differential equations, we restrict attention -for the sake of simplicity -to just one 'coefficient process' g:.[O,a] x s:. 2(n,C(JJRn)) + s:. 2(o,L(Rm.Rn)) and one 'noise process' z:.n + C([O,a]:Rm) which is m-dimensional and has continuous sample paths. For a gillen initial process (or distribution) a € ico,C(JJRn)) we seek a solution x € s:. 2(n.([-r,a]JRn)) of the stochastic FOE:.

x(w)(t)

f a(w)(O) ~ (w) Jt g(u,xu)dz.(·)(u), =l 0 S(w)(t).

t € J,

0 < t
a.a. w € o.

The stochastic integral in (I) is a McShane belated integral (§I.8(C), McShane [53]); and for each u € [O,a], xu € s:. 2(0,C(JJRn)) is defined by xu(w)(s) = x(w)(u~s) a.a. w € Sl, for all s € J. s:. 2(o,C(JJRn)) will be our basic configuration space of initial processes for the stochastic FOE (I). Obviously this choice of configuration space entails that the initial data as well as the solution process of the stochastic FOE will necessarily have almost all their sample paths continuous. The trajectory [O,a] 3 t + xt € s:. 2(o,C(JJRn)) will be sought in the space CA([O,a], s:. 2(n,C(JJRn))). (I) involves the stoahastia memory map m*:[O,a]

X s:.

(t,x)

2(o,C([-r,a]. Rn)) ~ ;-----:>

s:.

2(o,C(J, R0 ))

xt

m* is related to the detePministia memory map 31

m~[O_,a] x C{[-r_,a]_, :Rn) --..;,. C(J, Jln)

(t,a) (at(s)

=

a(t-f:s), s E J) by the equati.on

m*(t,x)

=

m(t,•)ox for all t

€

[O,a],

(1)

and all x E. £ 2(n,C{[-r,a],Jln)). Now m is continuous and m(t,.) is continuous linear for each t E [O,a] (Hale [26], Mohammed [57])~ so using this and (1), an easy argument yields that m*(t,.) is continuous linear for all t €: [O,aJ and m* is continuous. Indeed, llm*·{t,x) II 2

£ (n,c)

< llxll 2

for all x E £2 (n,C{[-r_,a],Jl n ))

£ (n,c)

and all t E [O,a]. In order to solve the stochastic FDE(l)_, we impose the following conditions of existence (cf. McShane [53], Gihman and Skorohod [24])~ Conditions (E) (of Existence) (i)

The noise process z.:.n -+C{[O_,a], Jlm) is expressible in the form z.(w)(t)

=

A(t)

-1:

z.m(w)(t) for all t E [O,a], a.a. w E n

(2)

where A~[O,a] -+ :Rm is a Lipschi.tz. function and z.m:.n-+ C{[O,a],Rm) is a (separable) martingale adapted to (Ft)tE[O,a] and is such that there is a K > 0 with IE(z.m(·)(t 2)-z.m(.)(t 1 )1Ft >I< K(t 2-t 1 ) 1

E( lzm(·)(t2)-z.m(·)(t 1 )1 2 1Ft 1 > < K(t 2-t 1)

l f

(3)

a.s. whenever t 1, t 2 E [O,a] and t 1 < t 2• (ii) The coefficient process g is continuous and is also uniformly Lipschitz in the second variable with respect to the first i.e. there exists L > 0 such that (4)

32

for all t E [O,a], and all

1j.l1,1j.l 2 €

.c 2(n,C(J,:Rn)).

(H;.) For each process y E CA([O,a], icn,C(J,:R")) the process [O,a] 3 t

r--->

g(t,y(t))

€ .c 2(n,L(Rm,

Rn))

is also adapted to (Ft)tE[O,a]• Remarks (1.1) (1) l.n condition (E)(i) the Lipschitz. function >. can be replaced by a process "'>.~n x [O,a] +Rm adapted to (Ft)tE[O,a] and whose sample paths are almost all Lipschitz. with a uniform Li.pschitz. constant i.e. there exists Jl. > 0 such that

and all t 1, t 2 E [O,a]. (2) The stochastic FDE(I.) includes both the ones with random coefficients and the (ordi.nary) stochastic differential equations (without retardation, [22], [53]);. for suppose g~[O,a] x C(J,Rn) x n + L(Rm ,Rn) is a coefficient process corresponding to a stochastic FOE with random coefficients. Assume that is .c 2 with respect to the third variable. Then we can define the coefficient process g~[O,a] x .c 2(n,C(J,:Rn)) + .c 2(n,L0Rm, Rn)) by setting

g

"'

g ( t ,1j.l }{bl ) = g ( t ,1j.l (w ) ,w)

for all t E [O,a], and all

1jJ

a •a • w

€ Q

E .c 2(n,C(J,Rn)).

§2. Existence and Uniqueness of Solutions The for

follo~ling

exi~tence

lenmas will be needed in order to establish the main theorem and uniqueness of solutions to the stochastic FDE(I).

Lemma (2.1)~ Suppose x~[-r,a] x Q +Rn is a process with almost all sample paths continuous. Assume that x![O,a] is adapted to (Ft)tE[O,a] and x(s,.) is F0-measurable for all s E J. Then the process

33

[O,a]

x

cCJ. nn>

n

(t,w) is adapted to (Ft)t€[O,a]" with almost all sample paths continuous. Proof~

Fix t € [O,a]. Since xl[-r,t] has continuous sample paths, it induces an F-measurable map xl[-r,t]:n + C([-r,t]JRn). This map is in fact Ft-measurable• to see this obser~e that Borel C([-r,t]JRn) is generated by cylinder sets of the form .H p~~(Bi) where Bi € Borel n". ti E [-r.tJ. 1=1

l

Pt. :.C([-r,t],.Bn)--> nn i.s e~aluation at ti for 1 < i < n. Thus it is suffi.clent to check Ft -measurability of xI [ -r. t] on such s~ts. With the above notation, 1

n

Cxl[-r,t])- [ n.

p~

1

(Bi)]

i

i=1

=

n

n {w:.x(ti,w) € B.}• 1

i=1

By hypotheses, if ti € J, {w:.x(ti,w) € Bi} € F0; and if ti € [O,t]. {w:x(ti,w) € Bi} € Ft.• 1

Hence {w:.x(ti,w) € Bi} €

Ft if

1

< i < n.

1

So Cxl[-r,t])

-1

n

[.n

1=1

Now xt

Ft.~

=

-1

Pt.CBi)] € Ft. 1

m(t,.)o(xl[-r,t]) and the deterministic memory

m(t,.):C([-r,t], Rn) -----> C(J, Rn) is continuous, so xt must be Ft-measurable. c Lemma (2.2)·:. Suppose that z:n + C([O,a]. Rm) is a process satisfying condition E(i). Let f:[O,a] + t 2 (n,L(Rm, Rn)) be an t 2-continuous process adapted to (Ft)t€[O,a]" Deiine the process F:[-r.a] + t 2(n,Rn) by F(t)

=

{

f f(u)dz(·)(u), t € [O,a] )i0 0

34

t € J

a.s., where the integral is McShane's belated integral off with respect to z.. Then F corresponds to a process belongi.ng to icn.c([-r,a], Rn)) and adapted to (Ft}te[O,a]" Indeed there is an M> 0 such that E(

sup te[O,a]

1ft f(u}dz.(·}(u}1 2> < M fa E( llf(u}ll 2 }du

o

o

(1}

The process [O,a] 3 t ~> Ft e t 2Cn.C(J,Rn}} is adapted to (Ft}te[O,a] with almost all sample paths continuous (i.e. it belongs to CA(O,a], t 2Cn,C(J,Rn}}}. Mis independent of f. Proof:

With the notation of Condition E(i}, F(t}(w}

=

f:

f(u}(w}d~(u} ~

(w} f: f(u)d,m(·}(u}, t E [O,a] a.a.w e n.

(2}

The first integral on the right-hand side of (2} is a Riemann-Stieltjes integral for a.a. w and is therefore continuous in t for a.a. w e Q; it thus defines an F-measurable map n + C([O.a],Rn}. As f(u} is Fu-measurable for all 0 < u < t, then J: f(u}( • }d~(u} is Ft-measurable (being a.s. a limit of Ft-measurable Riemann-Stieltjes sums}. Since zm is a martingale adapted to (Ft}te[O,a] with a.a. sample paths continuous, then so is the McShane integral on the right-hand side of (2} (§I.S(C}}. Hence by the Martingale inequality we have E(

sup te[O,a]

1J0t f(u}dzm(·}(u}l 2> < 4E(1Ja0 f(u}dzm(·}(u}l 2>

< 4C f:

E(

llf(u}(·}ll 2 }du (Theorem

(1.6.1}}

(3}

where C = 2Ka 112 +. K112 (Theorem (1.8.4}}. If 1 > 0 is the Lipschitz constant for ~. then it is easy to see that for a.a. we n lJ:

f(u}(w}d~(u}l 2 < t 2a

J: lf(u}(w}l 2du.

35

Hence E(

sup lJat f(u)(·)dA.(u)l 2) < 1 2a Jaa E(lf(u)(·)l 2)du. tE[a,a]

{4)

The inequality ( t) follows now from (3) and (4) and the fact that E(

sup lJt f(u)d,(·)(u)l 2) < 2E( sup lJt f(u)(·)dA.(u)l 2) tE[a ,a] a tE[a ,a] a +-

2E ( sup tE[a,a]

lJat f(u)d,m(·)(u)l 2>

Take M = 2(R.2a + 4C). Note that C (and hence M) is independent of f. It follows i.mmediately from ( t) that F E .c 2(n,C( [ -r ,a] ,Rn)) and from Lemma (2.1) that [a,a] 3 t~ Ft E .c 2(n,C(J,Rn)) is adapted to (Ft)tE[a,a] with the sample paths t ~ Ft(w) continuous for a.a. wEn. c Here is the main existence and uniqueness theorem for solutions of the stochastic FOE( I.)~ Theorem (2.1)~ Suppose Conditions (E) of §tare satisfied, and let e E .c 2(n,C(J,Rn)) be Fa-measurable. Then the stochastic FDE(I) has a solution x E .c 2(n,C([-r,a], Rn)) adapted to (Ft)tE[a,a] and with initial process e. Furthermore, (i) x is unique up to equi~alence (of stochastic processes) among all solutio!;'_s of 2(l) belonging ~o icn,C([-r,a],Rn)) and adapted to (Ft)tE[a,a] i.e. if x E .c (O,C([-r,a], R )) is a solution of (1) adapted to (Ft)tE[a,a] and with initial process e, then x(·)(t) = "'x(•)(t)

Ft-a.s., for all t E [a,a];

(ii) the trajectory [a,a] 3 t ~ xt E .c 2(n,C(J,Rn)) is a C(J,Rn)-valued process adapted to (Ft)tE[a,a] with a.a. sample paths continuous. (It belongs to CA([a,a], .c 2(n,C(J,Rn))). Proof:

We look for solutions of (I) by successive approximation in n ~ )). Suppose 6 E .c 2(n,C(JJRn)) is Fa-measurable. Note that this is equivalent to saying that 6(·)(s) is Fa-measurable for all s E J, because e has a.a.

-::-z::_.c (n,c ( [-r,a],

36

sample paths continuous. W.e prove by induction that there is a sequence of processes kx~[-r,a] x n +~n. k = 1,2, ••• such that each kx has the Properties P(k): (i)

kx

€

£ 2 (n,C([-r,a]~n)) and is adapted to (Ft)tE[O,a]"

For each t E [O,a], kxt E £2(n,C(JJRn)) and is Ft-measurable. k-1 (iii) llk-t.tx- kxll < (ML 2)k-t a 11 2x- 1xll , 2 £ (n,c([-r,aJ~n)) (k-1)! izn,c) (ii)

(5)

where M is the constant of Lemma (2.2). Take 1x:[-r,a] X n +Rn to be 1x(t,w)

__ { e(w)(O) 6(w)(t)

t E [O,a] t

J

€

a.s., ande(w)(O) + (w) Jt g(u, kx )dz.( • )(u) k+tx(t,w)

0

= {

6(w)(t)

t

€

u

t E

[O,a] (6)

J

a.s. Since e E £2(n,C(J,Rn)) and is F0-measurable, then 1x E £2(n,C([-r,a],Rn)) and is trivially adapted to (Ft)tE[O,a]" By Lemma (2.1), 1xtE£ 2(n,C(J,Rn)) and is Ft-measurable for all t E [O,a]. P(1)(iii) holds trivially. Now suppose P(k) is satisfied for some k > 1. Then by Condition (E)(ii), (iii) and the continuity of the stochastic memory, it follows from P(k) (ii) that the process

is continuous and adapted to ( Ft)tE[O,a]" We can therefore apply Lemma (2.2) to the right-hand side of (6) obtaining P(k+1)(i) and P(k+1)(ii). To check P(k+1)(iii), consider 37

uk•2x- k+-1xll22 s;

(n,C)

< E(

sup I<·> Jt {g(u,k•1xu)-g(u,kx )}dz(·Hul2> tE[O,a] o u

< MJa llg(u,k•tx)- g(u,kx >11 22 du,(P(k+1){ii) and LeJ:Ir.la 2.2) 0 u u £ < ML 2 Ja llk+ 1xu- kxull 22

o

s;

(n,c)

du,(Conditi.on E (ii))

( 2 k-1 JaOuk-1du <ML2.1!!:_L 112x-txll2 (k-1)! s; 2(n,c) =

2 k ak 2 (ML ) r.r. II ~:

X -

1

X

II 2

s;2(n,c)

•

Complete the proof of P(k+1)(iii) by noting that llk+2x _ k+1x 11 2 < llk+2x _ k+1x 11 2 t t s; 2 cn,c) (n,c)

i

Therefore P(k) holds for all k > 1. For each k > 1, write k- 1 i+1 i k 1 X= X+ L ( X - X). i =1 Now s;!(n,C([-r,a]JRn)) is closed in s; 2(n,C([-r,a]JRn)); so the series co

E

(i+1x - ix)

i=1 converges in s;!(n,C([-r,a], Rn)) because of (5) and the convergence of co

E

i=1

[

•

(ML2)1-1

i -1

a (i-1)!

,, /2

J

•

Hence {kx};=t converges to some x € s;!(n,C([-r,a]JRn)). Clearly xiJ = a and is F0-measurable, so we can apply Lemma (2.2) to the difference

obtaining 38

t

E(

sup t€[0,a]

1J0 g(u,

t

kxu)dz.(•)(u)- J0 g(u,xu)dz.(·)(u)! 2)

< ML 2a II k x - x 112 .c 2 (n,c) ~

0 as k

+ ....

Thus viewing the ri.ght-hand side of (6) as a process in icn,C([-r,a], Rn)) and letting k ..... , it follows from the abov.e tnat x must satisfy the stochastic FDE(l) a.s. for all t € [-r,a]. To prove uniqueness, let~ € .c~(n,([-r,a], Rn)) be also a solution of (I) with initial process e. Then it is easy to see by the Lipschitz condition (E(ii)) and Lemma (2.2) that

II xt - "'xt II 22 < ML 2 Jt II xu - "' xu II 2 2 du .c (Q,C) 0 .c (n,C)

"' = 0 for all t € [O,ala so for all t € [O,a]. Therefore we must have xt-xt "'· 2 n x = x 1n .c (n,C([-r,a], R )) a.s. The last assertion of the theorem follows imedi.ately from Lemma (2.1). a Remarks (2.1) Let 0 < t 1 < t
w(wl(O)

1 +

wlw)(t-t1)

t

(w)

J g(u,xu)dz(.)(u) t 1 < t
(II)

where the (unique) solution x € .c 2(n,C([t 1-r,t 1], Rn)). This gives a family of maps

When t 1

=

Tt

0, we define Tt' t > 0, to be =

T~:.c 2 (n,C(JJRn); F0) ~ .c 2(n,C(JJRn); Ft) 39

The following theorem on continuati.on of trajectories of a stochastic FOE is a consequence of Theorem (2.1). (2.2)~

Assume Conditions (E) of §1. If 0 < t1. < t 2 0 Tt(e)(·)(s) = { o u e(.)(t+-s) t+-S
Theorem

J

tt e(·)(O) +. (·) J g(u,Tu(e))dz(·)(u) 0 = e(·)(O) -+ (•) J:+.sg(u,Tu(e))dz(·)(u) e( • )(t+s)

+.

Jt-+s · g(u,Tu(e))dz(·)(u) t+S>t 1 t1 0 < t+s < t 1

t+s < 0

t+s = {Tt 1(e)(·)(O) + (·) J t g(u,Tu(e))dz(·)(u) 1 Tt (e)(t+s-t 1) 1

t+s > t 1 (7)

t 1-r < t+s < t 1

Putting s = 0 in (7), we can compare with (II) ness of solutions of (II) one gets a.s.

for~=

Tt (e); thus by unique1

Remark (2.2) In the case >. = 0 (when there is "no drift"), z = zm is an (Ft)D
40

§3. Dependence on the lni.tial Process use the notation of §1 and §2. Theorem (3. t) ~ Suppose the conditions of Theorem (2. t) are satisfied by g and z in stochastic FOE( l). Then each map Tt:..c 2(n,C(J,Rn);.F0 )-----+ icn,C(J,Rn);.Ft), t is Lipschi.tz.• indeed for all t 11Tt(91) - \(92>11 2

.c

E

[O,a], 91 ,9 2

(Sl,C)

E

E

[O,a],

.c 2(n,C(J,Rn);F0 ),

< 1'2" 1191 - 9211 2

.c

(Sl,C)

ML 2t e

(1)

where Mis the constant of Lemma (2.2) and L the Lipschitz constant of g.

N.B. Mdoes not depend on the coefficient process g but only on the noise z. The result follows immedi.ately from Gronwall's Lemma and the obvio•·s inequalities:

Proof~

Ig

sup 1Tt(9 1)(w)(s)- Tt(9 2)(w)(s)1 2 dP(w)

S€J

<2

In

sup l9 1(w)(s) - 92(w)(s)l 2 dP(w) sEJ

With suitable Frechet differentiability hypotheses on the coefficient process g, each Tt becomes c1 as in Theorem (3.2) below. Condition (D): The coefficient process g has continuous partial derivatives with respect to the second variable i.e. the map [O,a]

x

.c 2(n,C(J,Rn)) (t,~)

-!>

L(.c 2(n,c(J,Rn)), 11. 2(n,L(Rm, Rn))) D( 2 )g(t,~)

41

is continuous, where o( 2)g(t,ljl) is the parti.al deri.'latille of g in the second variable at (t,ljl). Lemma (3.1)~ Suppose g satisfies Condi.ti.ons (E)(iii.) and (D). processes y1, y2 E CA([O,a], t 2(o,C(J,Rn))), o( 2)g(t,y 1(t))(y2(t)) EL 2(n,L(Rm,

Then for any

Rn)~Ft), for every 0 < t
The conclusion of the lemma follows directly from the fact that, for each 0 < t
~ {g(t,y 1(t) * hy 2(t)) - g(t,y 1(t))} • h

j 0,

which belongs to l 2(o,L(Rm, R0 );.Ft) by 'li.rtue of Condition (E)(iii).

c

Theorem (3.2)~ Suppose the stochastic FDE(l) satisfies Conditions (E) and (D). Then for each t E [O,a],

Tt~ 2 (o,C(J,Rn) ;.F0) -> i(o,C(J,R 0 ) ;.Ft)

'-+-IL 2(0,C(J,Rn) ;.Ft)

. ct •

lS

Proof: Fix t E [O,a]. We first pro'le that Tt is Gateaux-differentiable on :z;::. £ (O,C(J,R n ) ,F0) 1.e. for each a,B E! 2( o,c ( J,R n) ;.F0 ) , 1 im

Tt(S+-h~)-Tt(a)

h+O

exists. Because of Conditions (D) and Theorem (2.1), the coefficient process u ~> o( 2 )g(u,Tu(a)) gives a linear stochastic FOE satisfying Conditions (E); so by Theorem (2.1), there is a unique solution a E t 2(o,C([-r,a],Rn)) of

B(·)(O) a(·)(t)

+ (·)

= {

B(·)(t) , for fixed BE t 2(0,C(J,Rn);F0 ). 42

Jt o( 2)g(u,Tu(8))(au)dz(·)(u), 0 < t
y(t) = at for all t E. [O,a]. we claim. that a Gateaux-derhati\'e

of Tt at a is given by GDTt(a)(S) = y(t) for all t E [O,a]

(2)

Using ( 1), ( 1.11) and Lemma (2.2), it i.s easy to see that if h j 0

II~

[Tt(a+ hB) - Tt(a)J- y(t>11 2 2 £

(rl,C)

t

< M J0 ll~g(u, Tu(a-~ohS))-g(u, Tu 0 is independent of ·g,a,s. Look at the integrand in (3)~ ~i,.

II~ [g(u,Tu(a-~:hS))

- g(u,Tu(a))] - D( 2 )g(u,Tu(a))(y(u))ll

< ~ llg(u, Tu(a-~:hs)) - g(u, Tu(a)) -·

o( 2)g(u, Tu(a) )(Tu(a ..hS)-Tu(a))

II

(4)

Let £ > 0 be given.

Then because of the continuity of the partial derivative

o> < o,

D( 2)g and the compactness of [O,a], one can show that there is a

0

depending on a, Band independent of u E [O,a] such that if lhl

then

<

If lhl <

o,

£

IITu(a+hB) - Tu(a)

II

for all u E [O,a].

(5)

then (3), (4) and (5) give

II~n [Tt(a-~ohS)-Tt(a)]

- y(t) 11 22

+2M Jt llo( 2)g(u,\(a)) 0

£

(Q ,C)

<

~ h~::

£2

Jt liT (a+hB)-T (a) 11 2 du 0 u u

11 2 11~[Tu(a+hB)-Tu(a)]-y(u) 11 2 2

du. (6)

£ (rl,C)

43

Now let N > 0 be such that II0( 2)g(u, Tu(6))

11 < N for all u E [O,a].

Then by Theorem (3.1),

11~ [Tt(6*he) - \(6)] - y(t) 11 22 r. (n ,C) <

£;,_2 llell 2 e2ML 2a -t 2MN 2 L~

Jt lllrT o

n~ u

(6-thB)-T (El)]-y(u) 11 22 u

r. (n,c)

du (7)

if lhl < o , t E [O,a]. Apply Gronwall's lemma to (7), obtaining 11 01

[Tt(6-thB) - Tt(6)] • y(t) II < 2Mtl/2N "''"llalleM(L2a-tN2t) . .,

for all t E [O,a] if lhl < If we let£~ 0 in (8),

(8)

o.

and so y(t) = GOTt(6)(B). By linearity of o( 2)g(u,Tu(6)) and uniqueness of solutions of (III) it follows that GOTt(e)(B) = y(t) is linear in e~ by continuous dependence (Theorem 3.1) it is also continuous in e. Thus for Tt tr be c1 it is sufficient to demonstrate the continuity of the map

So let e, ~ E r. 2(n,C(JJRn);F0). Then for any e E r. 2(n,C(JJRn);F0) the following estimate holds for all t E [O,a]~ II[GOTt(e)- GDTt(~)J(e>ll 2 < M

44

J:

II 0( 2)g ( u, Tu( 6) )(GOT u( 6)( B)) -0 ( 2) g( u, Tu(

~)) (GOTu( ~ )( B)) 11 2 du

< 2M.

-~= 2M

J: J:

II o( 2)g(u, Tu (8) )(GDTu(8 )(a))-D( 2)g(u, Tu ($) )(GDTu (8 )(a)) 11 2 du II o( 2 )g(u, Tu($)) 11 2 II GDTu(8 )(B) - GDTu($)(B) 11 2 du

( 9)

The map u ~ GDTu(8)(B) = y(u) is a solution of the stochastic FOE (III) whose Lipschit~ constant is N~ so by Theorem (3.t),

2

IIGDTu(8)(B)II

<1"2' llalleMN u for all u

€ [O,a]

(10)

Let E > 0 be given.

Then by the continuity of D( 2 )g, the compactness of [O,a] and Theorem (3.t), we can find c5 > 0 (depending on 8, E but independent

of $,u) such that ll$-811 2

t. (Q,C)

< c5 implies IID( 2)g(u,Tu($))-D( 2 )g(u,Tu(8))11

< E (11)

for all u E [O,a]. Thus if ll$-811 < c5, then (9), (tO) and (tt) ghe II[GDTt(8)- GDTt($)JCa>ll 2 < 4Me: 2a

* 4M x

II a 11 2

2 eMN a

J: {

IID( 2)g(u,Tu(8))11 2

* IID( 2)g(u,Tu(8))-o( 2)g(u,,Tu($))ll 2}

II GDTu(8)( B) - GDTu ($)(B) 11 2 du

2

2 * 4M(N 2 -~re: 2 )

< 4Me: a II a 11 2 eMN a

Jt IIGDTu(8)(B)-GDTu($)(B>II 2 du 0

for t € [O,a]. By Gronwall 1 s lemma, 2 2 2 II[GDTt(8)-GDTt($)J(B)II 2 < 4Me:2a eMN a•4M(N •e: )tlla 11 2 , t € [O,a]. Hence if

II $ -8 II

< c5, we get

2 2 2 IIGDTt(8)- GDTt($)11< 2(Ma)1/2EelMN a+2M(N +e: >t, t € [O,a]. Thus 8

~ GDTt(8) is continuous, and Tt is c1 for each t € [O,a].

a

45

III Markov trajectories

§1. The Mar·kov Property I.n this chapter we restrict ourselv.es to a class of stochastic FOE's in which the coeffi.cient process g in ( l) of Chapter ( 11) factors through a dr-ift RFDE H:.[O,a] x C(J,Rn) + :Rn and a diffusion RFDE G:.[O,a] x C(J,Rn) + L(Rm ,Rn), while the noise process takes the fonn {t +- w(t):. t E [O,a]} with wan m-dimensional Brownian motion on a filtered probability space (n,F,(Ft)O
1j!(w)(O) ..

J~

1j!(w)(t-t 1)

H(u,xu(w))du .. (w)Jt G(u,x (.))d\'l(u),t 1
Our main objecti~e here is to establish Marko~ behaviour for the trajectory {xt:.t > 0} of the stochastic FDE(l). All our results will be stated for the general stochastic FDE (1), although for the sake of simplicity all proofs are carried out for the simpler stochastic FDE ljl(w)(O) .. (w) x(w)(t) = { 1jl(w)(t-t 1)

t

]t

G(u,xu(·))dw(u), t 1 < t
without drift (H: 0). Note that in both (I.) and (II), We a1so make the fo 11 O\'li ng standing hypotheses~

1jl

(II)

E £ 2(n,C(J,Rn);Ft 1).

Hypotheses (M) (i) Let (Ft)O 0 such that 1 2 12 1 2 12 IIG(t,n )-G(t,n ) II< Llln -n lie• IH(t,n )-H(t,n ) I < L lin -n lie 46

for all t

E

[O,a] and all n 1 ,n 2

E

C(JJR 0 ).

(iii) H and G are continuous. Now the above hypotheses imply the condi.ti.ons of existence (E) of §( ll.1). To see this, observe first that the Brownian noi.se w is an (Ft)O
= 0

1

and

E( lw( • Ht 2)- w( • )(t 1) 12 1Ft ) t

=

m(t 2-t 1)

a.s. whenever 0 < t 1 < t 2
ico,c(JJRn)) ~ ico,L(Rm ,mn))

x

by

g(t,~)(w) = G(t,~(w)), t € [O,a], ~

€ s; 2(0,C(JJRn)),

w € Q,

then Condition M(ii) gives

llg(t,~ 1 )

-

g(t,~2 ) 11 2

=

JQ IIG(t,~ 1 (w))-G(t,~2 (w)) 11 2 dP(w)


(1)

for all ~ 1 .~ 2 € s; 2(0,C(JJRn)) and all t € [O,a]. Therefore Condition II E(ii) is satisfied with the same Lipschitz constant L. Thirdly, if t € [O,a] and Y € CA([O,a], s; 2(o,C(JJRn))), then g(t,y(t))(.) = G(t,y(t)(·)) is Ft-measurable. It is not hard to see from the continuity of G and the Lipschitz Condition (1) that g is continuous. Hence II E(ii) is satisfied, and the stochastic FOE (II) (or (I)) will have a unique solution. So we obtain the lipschitz maps t1 2 n 2 Tt :£ (Q,C(JJR );Ft ) ~ s; (O,C(JJRn);Ft)' t > t 1, 1

~

When t 1

'

,. xt

= 0,

47

The first main result here tells us that the trajectory of the stochast;c FOE ( 1) (or ( I.l)) can be viewed as a Marko~ process [O,a]

x

n -----+

( t,w) taking ~alues in C(J,Rn) •. The following lemmas will be needed, the f;rst of which h a well-known result for Brownian motion. Lemma (1.1):. Let gt, t € [O,a], be the a-algebra generated by {w(·)(u) -· w{•)(t)~ t < u
See Elworthy [t9],p. t9• or Friedman [22], p. 44. a

N.B. {gt}te:[O,a] is decreasing and

n

t€[0,a]

gt = {J,n}.

Con~ention

For t 1 > 0, we

ha~e

the obvious continuous linear isometry

C(J,Rn) - -1---.,> icn,C(J,Rn) ;,Ft ) 1

assoc;at;ng w;th each n e: C(J,Rn) the constant map tCn)(w) = n. for all t1 t1 . Let Tt Cn) denote Tt CtCn)) for all t > t 1• w;th the above convent;on one has

(I)

Lemma (1.2): For each n € C(J,Rn), the trajectory {T! 1(n)}t>t of (I) (or (11)) twith tp = n) ;s adapted to the filtration (Ft n gt 1)t>t 1(w;th each Ft n g 1 independent of Ft ). 1 1

Proof: For fixed t 1 > 0 def;ne the process .,t 1 : t1 (t) = w(t) - w(t 1), t > t 1•

0---+

C([t 1,a] ,Rm) by

Then wt 1 ;s adapted to (Ft n gt1)t>t • Now let n e: C(J,Rn) and nx be the solut;on of (II) w;th 48

tp =

n·

1

€

n.

t In (II) one can replace w by w 1, thus getting a unique solution through t n which is adapted to (Ft n g 1)t>t • But nx also satisfies t 1 t t > t1 n(O) + (w) Jt G(u,nxu(·))dw 1(u) 1 nx(w)(t) = { n(t-t 1) t 1-r < t < t 1, a.a. wEn.

t1 t1 Then by uniqueness of solutions {Tt (n)}t>t is adapted to (Ft n g )t>t • c 1

1

The next lemma is probably well-known for the case J = {O} (Stochastic ODE's): it essentially says that one need only solve (I) on the subspace C(JJRn) of the configuration space £ 2(n,C(JJRn)). Lemma (1.3): Suppose Hypotheses (M) are satisfied. Let~ € £ 2(0,C(JJRn);Ft) be an Ft -simple function i.e. there are nj ~ C(JJRn), j = 1,2, ••• ,k and a 1 partitio~ {O.}J~= 1 c Ft of n such that ~ = _En. Xn.• where x0 . is the J 1 J=1 J J J characteristic (indicator) function of Oj. Then t

Tt1(~) =

k (2)

E

j=1

i."e. k

1:

=

j=1

t1

Tt (nJ.)(w)x0 _(w) for a.a. wen.

J

Remark In particular, if n 0

€

Ft , then 1

t1 t1 Tt (Xg) = Tt (1)Xg 0 0

+

t1 T (O)X~ • 0

Then Xn. is Ft -measurable because n. € Ft • Solving J 1 n.J 1 the stochastic FDE(II) at nj € C(JJRn), we get a solution Jx satisfying

~:

Let 1 < j < k.

ft

n·

n· nj(O) + G(u, Jxu(·))dw(u) t > t 1 Jx(·)(t) = { tl nj(t-t 1) t 1-r < t < t 1, a.s. Since Xg is Ft -measurable, then the process u j

1

(III)'

n·

~ G(u, Jxu(·))Xg_ is J

adapted to (Ft)t>t; and so by property of the stochastic integral (Theorem 1

49

a.s.

Clearly,

for all u > t 1• Thus k n. I: Jx(·)(t)x ={ j=1 °j

Jt

k I: j=1

n.(O)x0

k . I: J=1

nJ. ( t-t 1>Xn.,

J

+

j

tl

k

I: G(u, j=1

n.

Jxu(·))dw(u), t > t 1,

J

Therefore by uniqueness of solutions to stochastic FDE(II) (Theorem (11.2.1)), we obtain k n. = I: Jx(·)(t)x0 j=1

,

t 1-r < t
=

k I: j=1

j

This implies that = Tt1 (;") ~

t

a.s. on

n.

ljlx

t

=

k nj I: xt Yj=1 -~~

Ttt1 (Tl.)Y- , J -~~

t 1 < t
a

We are now ready to prove the main theorem in this section.

50

Theorem (1.1) (The ~kov Property)~ Suppose Hypotheses (M) are satisfied by the stochastic FDE(l). Then its trajectories {nxt ~ t € [O,a], n € C(JJRn)} descri.be a t4arko'l process on C(JJRn) \'lith transition probabilities p(t 1,n,t 2,·) gi'len by p(tpn, t 2,B) =

P{w~w

t,

€ S1, Tt (n)(w) € B}

(3)

2

for 0 < t 1 < tz
1

lndeed for any

= p(tt'Tt (9)(·),t2,B) = P(Tt {9)€BITt (9)) t

2

t

(4)

holds a.s. on n. Proof~

Obser'le first that for 0 < t 1 < t 2 and e'lery n € C(JJRn), p(t 1,n,t2,·) = P o {T! 1(n) r· 1 and is therefore a probability measure on C(JJRn) because t1 2 . n Tt (n) :S1 + C(J,R) is measurable. We would like to show that, if 0
then (p{w~Tt

2

(9)(w) € BIFt })(w') = p(t 1,Tt (9)(w'),t 2,B)

(5)

t

1

for a.a. w' € n, every Borel set B in C(J,Rn) and any 9 € t 2(n,C(JJRn);F0). Now the right-hand side of (5) is equal to

JS1 x8<[T!21nt 1(e)(w'))](w)} dP(w)} for a.a.

w' €. n.

Therefore by the definition of conditional probability it is easy to see that (5) is equivalent to

JA x8(Tt 2(e)(w))dP(w)

=

J J xB{[T: 1(Tt (e)(w'))](w)}dP(w)dP(w') (6) A S1

2

1

for all A € Ft and all B € Borel C(J,Rn), where xB is the characteristic 1

function of B. To prove (6), we show first that if f:C(JJRn) +R is bounded and continuous, then 51

J f{[T: 1(Tt (e)(w'))](w)}dP(w)dP(w') JA f(Tt2(9)(w))dP(w) = JASl 2.1 for a11 A € Ft •

t

1

No~e~ since Tt (e)

By Theorem (11.2.2), Tt (e) = Tt 1(Tt (e)). 2

2

1

(7)

€

1

00

2(n,C(JJRn);ft) there is a sequence {~.}._ 1 of Ft 1-simple functions con1 2 n J Jverging to Tt 1(e) in t (Sl,C(JJR );Ft 1>; i.e.

t

~·

J

=

n. J

l: (jl .. Xg •• • Sl .. i=1 J,1 J,l J,1

€

Ft • (jl .. 1 J,l

€

n

C(J,R ).

Therefore by continuity of r! 1• we have 2

t

Tt (e)(w) = lim Tt 1 (~J.)(w) 2 j-+oo 2 nj t1 =lim .t Tt (4»J. iHw>x0 . _(w) j - 1=1 2 • J ,1 a. a. w € n. because of Lemma (1.3). t1 2 n t1 . Again, each Tt ((jlj i) €·£ (Sl,C(JJR ); Ft n g ) by Lemma (1.2); so we can write r!~(41j,i) a~ a ;imit (in t 2 or a.s.) ~f gt 1-simple functions; viz: m t1 k Tt (41 .. ) = lim t 2 J,l k- h=1

O:,h € gt 1• 1 < h < mk.

k

.•

eJk.h1 Xo* • • ·~.h

ej,i k,h

€

C(JJR">

(8)

= 1,2 •••• ; {n;,h} partition n.

Now the left hand side of (7) is equal to nj

JA j.._

lim f(

l:

i =1

nj I j-+oo A i=1

= 1im

l:

t

Tt 1(41 .. )(w)Xg 2 J •1

(w))dP(w)

j •i

t

f{Tt 1(41J,1 .. )(w)}y_ (w)dP(w) "U • • 2

nj

J ,1

mk

..

= lim lim J l: l: f(e~·~)Xo* (w)x0 (w)dP(w) j.._ k- A i=1 h=1 • k,h j,i nj mk .. = lim lim t t f(~~·~)P(Slk h n n .. n A) j-+oo k- i=1 h=1 • • l· 1 52

(9)

conversely, the right-hand side of (1) is equal to f[T: 1(1j1.(w'))(w)]dP(w)dP(w') JA Jn lim j2 J = lim

J J

j- A

n

f[T: 1(1j1.(w'))(w)]dP(w)dP(wl) 2 J

n.

t

f

=lim J J f[( Tt 1(cllj i)'Xn (w'))(w)]dP(w)dP(w') j- A n i=1 2 ' j,i =

~im J J

J- A

=lim

n

JJ

j- A

nj f[.l:

1 =1

nj l:

n i =1

t

Tt 1
'

i)(w)x0 .. <w')]dP(w)dP(w') J, 1

t

f(Tt 1(cll. i)(w))Xn 2 J'

j ,i

(w')dP(w)dP(w')

nj mk .. = 1im 1im J .t l: f( et~>xn* (w)JCn .. (w' )dP(w)dP(w') j-+cok-+co A 01=1 h=1 -l<,h J,1 nj mk . . * = lim lim J l: l: f(8~'~)P(Ok h)JCn .. <w')dP(w') , , J,1 j - k- A i=1 h=1 nj m k j i * = lim lim l: l: f(8k'h)P(Ok h)P(A n n.. ) , , J, 1 j-+cok-+co i=1 h=1

f

( 10)

Now A, nJ, ..1 € Ft 1 , soAn nJ, ..1 is independent of --k n*,h € gt 1 because of Lemma ( 1• 1); hence

* * nAn nj,i). P 1, 41a:B ·~is uniformly continuous and vanishes outside Ua, and ""l: 41a(n) = 1, for all n € B. Define the sequence {fm}== of functions 1 a=1 fm:C(JJRn) + R by setting 53

m

n EB

a~19a{n)

fm(n) = { n f. B

0

m

Clearly f (n) = 0 for all n t U Ua• so each fm is uniformly continuous and m a=1 bounded on C(JJRn). Also it is easy to see that

... l:

n € B

4> (n) = 1

= { a=1

a n

0

Xs

=

for all

t B

n E C(JJRn).

Therefore fm:C{JJRn) +R. m = 1.2 •••• is a sequence of -bounded and uniformly continuous functions converging pointwise to xB. We can therefore replace f by fm in (7). then pass to the limit as m + ... using Lebesgue dominated convergence theorem. We obtain (6) in this way for any open set B C(JJRn). To get (6) for any Borel set B in C(JJRn) we reason as follows. Let v:Borel C(JJRn) +R be the set function

=

v(B) =

I J

XB{[T: 1(Tt (8)(w')](w)}dP(w)dP(w'). BE Borel C(JJRn).

A Sl

2

1

(11)

where A E Ft1 is fixed. Then vis a finite measure on C(JJRn); to prove this let {Bi}7= 1 be a countable disjoint class of Borel sets. Then

I

J ml: XB {[Ttt 1(Tt (e)(w'))](w)}dP(w)dP(w') v( m U B.) = 1 i =1 A Sl i =1 i 2 1 Since

m = l: v(B.) 1 i =1

m

(12)

xB (n) +Xu B•(n) as m + ... for all n E C(JJRn). then by the domi i=t 1 inated convergence theorem we can let m + • in (12) to obtain l: i =1

~

i =1

v(B.) =lim

JJ ~

or-- A

1

J J

n ; =1

xB {[T: 1(Tt (8)(w'))](w)}dP(w)dP(w') i 2 1 t

{[Tt1(Tt (e)(w'))](w)}dP(w)dP(w') - A Sl U Bi 2 1 i=1

-

54

x...

00

= v( U Bi).

i=1

Therefore"' is a finite (Borel) measure on C(JJRn). Since C(JJRn) is metric, v is regular (See §(1.2)). The set function v*:Borel C(JJRn) +R v*(B) =

J XB(Tt (e)(w))dP(w), B € Borel C(JJRn), A

2

is also a regul.ar (finite) measure on C(JJRn);, so for any B € Borel C(JJRn), v*(B) = inf{v*(u): U ~ C(JJRn) open, B c: U} = inf {v(U): U ~ C(JJRn) open, B c: U} =

V(B).

Therefore (6) holds for all Borel sets B in C(JJRn). To see that

it is now sufficient to check that the left hand side p(t 1,Tt (9)(·),t2,B) 1 is measurable with respect to the a-algebra generated by Tt (9); this is 1 valid because the function p(t 1,·,t 2 ,B) t1 C(JJRn) 3 n J-1-~> p(t 1,n,t2 ,B) = J XB(Tt (n)(w))dP(w) n 2 is measurable: If B ~ C(JJRn) is open, let {fm}m= 1 be a sequence of uniformly bounded, uniformly continuous functions fm:C(JJRn) +R converging pointwise to XB. Therefore

. p(t 1,n,t2,B) =11m

J

fm(Ttt1 (n)(w))dP(w).

nr-t-0

If nk

2

t

+

n in C(JJRn), then Tt 1(nk) 2

fm(T! 1(n)(w)) as k +

oo

t

+

t

Tt 1(n) in £ 2(n,C); so fm(Tt1(nk)(w)) + 2

2

in probability; so by Lebesgue dominated convergence

2

theorem, for each m, 55

In f m(Ttt21(nk)(w))dP(w) +In f m(Ttt21(n)(w))dP(w) Hence n ~-+-

as k

In f (Ttt1 (n)(w))dP(w) is continuous (for each m). m

2

+

~. Therefore

p(t 1,·,t 2,B) is measurable because it is a pointwise limit of continuous functions. If B € Borel C(JJRn), then

=

p(t 1,n,t2,B) = inf {p(t 1,n,t2,U): U open in C(JJRn), B U}. But each n + p(t 1,n,t 2,u) is measurable, so p(t 1,.,t2,B) must be measurable. Therefore p(t1,Tt (9)(·),t2,B) 1

=

E[P(Tt (9) 2

BIFt )ITt (9)] 1 1

=

P(Tt (9) € BITt (9)),

€

2

1

since the o-algebra generated by Tt (9) is contained in Ft • 1

1

Finally we check the Chapman-Kolmogorov identity p(t 1,n,t2,B)

=I

~EC

p(u.~.t 2 ,B) p(t 1 ,n,u,d~)

for all B € Borel C(JJRn) 0 < t 1 < u < t 2
2

u

I xB(T!1(n)(w))dP(w) n

2

=

I

J

w'ffi

xB([T~

wEn

(T: 1(n)(w'J](w))dP(w)dP(w') 2

(6)'

Therefore,

I~EC p(u.~,t2 ,B)p(t 1 ,n,u,d~) =I~EC p(u,~,t2 ,B) d[PoT: 1 (n)J(~) =Iw'€Sl =I

t

p(u,T 1Cn)(w'),t2,B)dP(w') u

w'€Sl,w€Sl

56

t

x 8 ([T~ (Tu 1(n)(w'))J(w))dP(w)dP(w') 2

= p(t,.n,t 2,B). Transform the filtered probabili.ty space (n,F,(Ft)D
"'

X:[O,a]

x "' n+

"' = wt' "' "' C(J,Rn ), X(t,w) w € "' n,

Let Pn = P o"'(~x)-t for measure on (O,F). Also "' * wt(s) "' = Pt+s(w) "' is w 0 < t
€

0 < t < a.

each n € C(J,Rn~. Then each Pn is a probability each X(t,·) is Ft-measurable because the map "'Ft+s-measurable for each s € J. Note that for fixed C(J,Rn), B} = P{w: ll xt(w)

€

B} = p(O,n,t,B).

Hence n + P {w:X(t,w) "' "' € B} is measurable;. so i.n order to establish that X n is indeed a Markov process one needs to show that

"' = p(t 1,X(t 1),t2 ,B) a.s. (Ft "' ,Pn), Pn(X(t2) € BIFt) 1

1

forB € Borel C(J,Rn), 0 < t 1 < t 2
fA XB(X(t2))dPll = f"'A p(t 1,X(t1),t2 ,B)dPll for each "' A € "'Ft • Apply the transformation w * nx(w) and the fact that "' € Ft 1whenever "'A € "'Ft to get (ll x) -1 (A) 1

1

57

fA X8(X(t2 ,~)dPn(~)

=

J (nx)-1(A)

=

J

.

(nx)-1(A)

~

x8(nxt

(w))dP(w) 2

P(nxt E BIFt )dP 2 1

~

=

J

=

JA p(t 1 .x(t1 .~).t2 ,B)

p(t 1,nxt (w),t2,B)dP(w) (nx)-1(A) 1

~

Hence (Sl,X(t),Ft,Pn)• 0 < t
.

e(w)(t)

(IV)

t EJ

with coefficients H:C(J,Rn) +Rn, G:C(J,Rn) + L(Rm ,Rn). We adopt the following hypotheses Hypotheses (A) (i) For each t E [O,a], Ft is the a-algebra generated by {w(.)(s):O < s < t} together with all sets of P-measure zero in F = Fa. (ii) There is a constant L > 0 such that

58

IHCn1 ) - H(n 2) I < Llln1 - n2 11

c

IIGCn1 )- G(n 2>11< Llln 1 - n2 11

c

Definition: Let E be a real Banach space and T any non-empty set. (i) Two processes x1• x2:T x 0 + E are isonamous if for all

s1.s2••••• Bk e Borel

E. t 1.t2••••• tk

e T.

P(x 1(t 1.·) € B1,x 1(t2,.) € B2••••• x1(tk,•) € Bk) =

P(x 2 (t 1,.) € B1, x2(t2,.) € B2•••• ,x 2(tk,.) € Bk).

(ii) Two random variables x1• x2:o + E have the same (pPObabiZityJ dist~­ bution if for all B € Borel E

P(X 1 € B) = P(X 2

€

B).

If x1• x2 are isonomous processes. we may write x1 - x2; similarly if the random variables x1• x2 have the same distribution. write x1 - x2• Lemma (?..1): Let E,F be B-spaces (or even metric spaces). x1.x 2 :T x o + E isonomous processes. and L:T x E +Fa (Borel) measurable map. Then the processes T

X

0

(t,w)

> F

L(t,xi(t,w)). i

=

1,2.

are isonomous. ~:

Suppose B1•••• ,Bk € Borel F, t 1,t 2•••• ,tk € T. Since L(t,•) is measurable, then L(t.·)-1(B.) e Borel E for all 1 < j < k, so that by isonomy 1 2 J of x and x we have

Lemma (?..2): Suppose x1.x 2:[-r,a] x 0 +Rn have a.a. sample paths continuous. Then the following statements are equivalent: 59

( i) x1 and x2 are isonomous • (;;) The induced random 'lari.ables x1• x2 ::n distribution.

+

C([-r.a].Rn) ha'le the same

(;.; i) The processes [O.a] x (t.w)

n- -

C(J.Rn) x!(w).

i = 1.2.

are isonomous. Proof: We prove the implications (i) .. (;;) .. (iH) .. (i). ~ppose x1 - x2• Since x1• x2 have a.a. sample paths continuous. then x1• i:n+ C([-r.a].Rn) are (Borel) measurable; these wi_l_l have the same distribution if and only if the measures Po (x 1)- 1• Po (x 2)- 1 agree on cylinder sets in C([-r.a].Rn). (See Parthasarathy [66]. Theorem (2.1). p. 212). For each t € [-r.a]. let Pt:C([-r.a].Rn) "'"'""""' Rn

a

, a(t)

be evaluation at t. Let t 1.t2••••• tk Then

[-r.a]. s1.s2••••• Bk

€

k

Po (x 1)- 1[.n p~~(B.)] = P(x 1 J=1 J J

€

J

= P(x 2(.)(tj) € Bj for all 1 < j < k) 2 -1 k -1 =Po (x) [.n Pt.
Suppose now that Po (x 1)- 1 =Po (x 2)- 1• The memory x

C([-r.a].Rn) ~ C(J.Rn)

(t.a) 60

Borel Rn •

P~~(BJ.) for all 1 < j < k)

= P(x 1(·)(tj) € Bj for all 1 < j < k)

m:[O.a]

€

is continuous (and hence measurable);. so i.f B1*.•••• ,Bk* E Borel C(J,Rn ), t,. ••• ,tk E [O,a], then m(tj,.)- 1(Bj) € Borel C([-r,a],Rn) for all 1 < j < k• and so

P(x~

j

€

B~ for all 1 < j < k) = P(m(t.,•)ox 1 € B~ for all 1 < j < k) J

J

= P(x 1

€

J

* for all 1 < j < k) m(tj,•) -1 (Bj)

= P( x2 E m(t.,· )-1 (B.*> for all 1 < j < k) J

J

=P(x~ EB~forall1<j
j

Therefore (t,w) ~~ x~(w), i = 1,2, are isonomous. The implication '(iii) ~(i)' follows tri~ially from Lemma (2.1).

c

Notation E, Fare B-spaces. For any stochastic processes y 1:T x g + E, y2:T define the process y1 x y2:(T x T) x g + E x F (on T x T) by (y 1

X

x

y2 )(t1,t2,w) = (y 1 (t1,w), y2 (t2,w)) for all t1,t2 € T,

g+F

Ill

€

n.

Lemma (2.3): Let zi: 0 ~ C([O,a],R), i = 1,2, be processes satisfying condition (E)(i) of existence of McShane's integral (Theorem (1.8.4)). Suppose fi:[O,a] + t 2 (0JR~) are t 2-continuous processes adapted to (Ft)tE[O,a]" Define the processes F1 :[0,a] x n +Rn, i = 1,2, by Fi(t,w) = (w) If f 1

x

I:

fi(u)(·)dzi(·)(u),

z1 is isonomous to f 2

x

z2, then F1

t € [O,a], a.a. x

Ill

€

n.

z1 is isonomous to F2

x

z2•

Proof: We approximate each Fi by Riemann sums. For each integer k, let ~(0, ~·~ •••• ,a) be a (Cauchy) partition of [O,a] into equal subdivisions. For each t € [O,a], let m(t,k) be the greatest integer such that m(t,k)a < t < [m(t,k)+1] a

k

k

Define the processes si(nk):[O,a]

• X

g ____,. Rn by 61

for all t € [O,a], a.a. w € 0, i = 1,2. Since f 1 x z1 - f 2 x z2, then for any finite sets t 1,t2, ••• ,tk s 1,s 2, ••• ,sp € [O,a] the random variables (f 1(t 1), fi(t 2), ••• ,fi(th), zi(·)(s 1), ••• ,zi(•)(sp)),

€

[O,a],

i = 1,2,

have the same distribution, and hence so does any finite sum formed out of finite products and differences (since all these operations are continuous and hence measurable (Lemma (2.1)). Thus for any t 1,t 2, ••• ,th, cs 1(ITk)(t 1,·), ••• ,si(ITk)(th,.)), i = 1,2 have the same distribution i.e. s 1(ITk) - s 2(ITk), k = 1,2,3, •••• Now Si(If)(t.,•)-+ Fi(t.,•) as k-+oo in £. 2 for 1 < j
as k -+ oo in the weak topo 1ogy of (Bore 1) measures on Rn (h times). But we know that

x

Rn x·· .x Rn

x

Rn

k 1 k -1 2 k 2 k f1 po(S 1(rr·)(t 1,.), ••• ,S (IT )(th,.)) = Po(S (IT )(t 1,.), ••• ,S (IT )(th,.)

for all k > 1i so by uniqueness of the limit in the weak topology of measures we must have po(F 1(t 1,.), ••• ,F 1(th,.)) -1 = po(F2(t 1,.), ••• ,F 2(th,.)) -1 on Borel (Rn x ••• x Rn) (h times) {§{1.2)). Hence F1 - F2• To prove F1 x z1 -F2 xi, fix {s1' ••• ,sp) =!_and define z\~.> = {zi{s 1), ••• ,zi{sp)). Then (f 1,z 1{s)) x z1 - {fz ,z 2{s)) x z2 • .- . . . 1 2 1 2 Write {F1 ,z 1 {s)) = {F1 ,0) + (O,z 1 {s)), i = 1,2. As F - F and z - z, it follows from Lemma (2.1) that {F1 -(F2,o), {O,z\~» -{o,i{_!)) and

,o)

62

(F 1,z1(s)) = (F 1,0) + (O,z1(s)) ~ (F2,o) Thus-F 1 x z1 ,... F2 x z?. -c

+

(o,z?(s)) = (F 2,z?(s)). -

-

Lemma (2.4): Suppose G:C(J,Rn) -> L(Rm ,Rn) is Lipschitz and z 1 :n + C([O,a],Rm), i = 1,2, are processes satisfying the conditions of Existence E(i) (§11.1). For each n € C(J,Rn) let {iTt(n)}t€[O,a]' i = 1,2, be the trajectories of the stochastic FOE's: n(O) ix(w)(t)

= {

n(t)

t € J, i = 1,2,

a.a. w €

n.

If z1 and z? are isonomous, then £1Tt(n)}t€[O,a]' £2Tt(n)}t€[O,a] are also isonomous. Proof: Use the method of proof of Theorem (11.2.1). Assume inductively that there are sequences ixk:n+C([-r,a],Rn), k=1,2, ••• , i =1,2, such that (i)

· k1 · k 2 ll 1 x+ - 1 xll 2

s:. (n,c)

ix 1(w)(t)

2 k 1 ak- 1 i 2 i 1 2 <(ML)- (k- 1)illx-xll 2 ·

n(O)

t €

n(t)

t € J

s:. (n,c)

,i=1,2,

[O,a]

= {

a.a. w € n, i = 1,2. Indeed we define t €

n(t)

t € J,

[O,a]

i = 1,2, a.a. w € n.

(i) holds by the proof of Theorem (11.2.1). To prove (ii), assume it is valid for some k > 1. Therefore 1xk x z1 ,... 2xk x z 2 implies 63

tk tk ( xt (.), ••• , xt (·)) 1

x

1 2k 2k z - ( xt (.), ••• , xt (·)) 1

m

m

x

z-? (Lemma 2.2)

Hence

(G( 1 x~ (•)), ••• ,G( 1 x~ (·)}} m

1

xz.1 -

(G( 2 x~ (·)), ••• ,G( 2 x~

(·}})x z 2,

m

1

for t 1, ••• ,tm € [O,a], by Lemma (2.1)a i.e. k ) 1 G( 1X(·) X z

~

k G( 2X(·))

X

l- •

Thus 1xk+ 1 x z1 - 2xk+t x z2 because of Lemma (2.3). Obviously (;;) is valid fork = 1 because (n,z1) - (n,z 2). Therefore (i) and (ii) hold for all k. In particular (ii) implies. 1xk- 2xk for all k > 1. Thus po( 1xk)-t =Po ( 2xk)-t for all k > 1• . k . 2 Now 1 x - - > 1 x as k + oo a.s. (or in t ) from the proof of Theorem (11.2.1); soP o (ixk)- 1 ---)Po (ix)- 1, i = 1,2, ask ... oo in the weak topology of Borel measures on C([-r,a]JRn}. Hence by uniqueness of limits, we must have P o ( 1 x)- 1 = P o ( 2x)-t on Borel C([-r,a]JRn). So by Lemma (2.2), {~Tt(n)}t€[O,a] = {t ~ 1xt} is isonomous to £2Tt(n)}t€[O,a] = {t--->. xt}. o The following the9rem is our second main result; it says that the Markov process given by trajectories of the autonomous stochastic FOE (IV) is in fact time-homogeneous in the sense that each transition probability p(t 1.n,t2,.), t 1 < t 2, n € C(JJRn) depends on t 2 - t 1 (and n) only. Theorem (2.1) (~me-homogeneity): Suppose that the autonomous stochastic FOE (IV) satisfies Hypotheses (A). For 0 < t 1 < t 2 = p(O,n,t 2-t 1,·), 0 < t 1 < t 2
Let n € C(JJR n ), t 1 < t 2• Suppose t1 x:~

+

C([t 1-r,t 1JJR n ) is the

solution of n(O) ~ (·)

Jt

t

G( 1xu(·))dw(u),

tt

= {

n(t-t 1) n(O)

+. (·)

t 1-r < t < t 1

Jt-t1 G( t 1xu'+t (•))dw(u'+t 1), 0

= {

n(t-t 1) n(O)

+. (•)

1

t 1-r < t < t 1

Jt-t1 G( t 1xu'+t (.))d[w(u'+t 1)-w(t1)J, t > t 1,

={

0

n(t-t 1)

1

t 1-r < t < t 1

Then n(O) + ( )

t'

J

0

t

G( 1xu'+t (·))d[w(u'+t 1)-w(t 1)J,t'>O 1

= {

t • e: J

n( t •)

a.s. Now by property of Brownian motion, we know that the process u' ~> w(u'+t 1)-w(t 1) is isonomous to w (Elworthy [19], Corollary 3A, p. 19). So comparing the above stochastic FOE with our original one viz: n(O) + (·) nx(·)(t')

= {

n(t')

Jt'

G(n>
0 < t'
0 t' e: J

it then follows from Lemma (2.4) that the trajectories {Tt(n)}O
In particular, putting t

=

t2 - t1

~ne

gets

Remark Theorem (2.1) still holds if the Brownian noise w is repalced by an (Ft)O
martingale z satisfying Condition (I.E) and having the property that for each t 1 > 0 {t t+ z(t 1+t) - z(t 1)} - z. §3.

The Semigroup

Denote by Cb the Banach space of all bounded uniformly continuous functions :C(J,Rn) -> R furnished with the supremum norm !let> II cb = sup { l(n) I : n E C(J,Rn)}. Let 0 < t 1 < t 2 as before (§II.2), using the trajectory of the stochastic FOE (I) in §1. For each E Cb define the function P~~():C(J,Rn) ~R by setting Ptt1 ()

= E(

2

o

rtt1 (n))

=

2

Ic

(~) p(t 1 ,n,t 2 ,d~)

where p(t 1,n,t2 ,·) are the transition probabilities of (I) (Theorem (1.1)). Since is bounded, it is clear that p~1() is also bounded. Indeed 2

t

IIPt~() llcb < II llcb for all



E

cb.

t

Furthermore, each Pt!() is uniformly

continuous on C(J,Rn) by virtue of the following elementary lemma: Lemma (3.1): Let (n,F,P) be a probability space, A a metric space and E a separable Banach space. Suppose :E ~R is any bounded uniformly continuous A A 2 function on E. For each t > 0 and A E A, let ft' f E £ (Q,E;F) (§I.2). Then the following is true: (i) If the map A 3 A ~ fA E t 2(n,E;F) is uniformly continuous, then so is the map A 3

(ii) If lim e: > 0,

A I-->

In

lft(w)-fA(w)l~

A A lim P(lft-f IE> e:)

t~O+ t~+

1im

t~O+

r

Jn

(

uniformly in A E A.

66

In ct>(fA(w) )dP(w) E R. =

dP(w) = 0 uniformly in A E A, then for every 0 uniformly in A EA.

f~(w))dP(w) =

I

n

(

fA(w) )dP(w)

Moreover,

-

Proof:

( i) Let E > 0 be given. By uniform continuity of ~. there is a o• > 0 such that Whenever~ 1 .~ 2 E E and 1~ 1 -~ 2 1E < o•, then 1~<~ 1 )- ~(~ 2 ) I <E. But A~ fA is uniformly continuous, so there is a o > 0 such that if 1 A,A' E A and d(A,A 1 ) < o , then In If A(w) - f A (w) IE2 dP(w) < Eo' 2 , where d is the metric on A. Therefore, if d(A,A 1 ) < o, we get 1 P{w:w En, lfA(w)-fA 1 (w)IE > o•} < --

f

o•2 n

by the Chebyshev •s i nequa li. ty.

Jn

f A A' If -f

+

<2 II IP

II

Now if d( A,·A • < o we ha lle

l~(fA(w)- ~(fA 1 (w))ldP(w) =

where II~

lfA(w)-fA 1 (w)I 2E dP(w) < E

=

J

A

IE> o•

A A' If -f IE
l~(f (w))-~(f

sup {I~( v) I: v t-+0+

(w))ldP(w)

I
II P( IfA -fA I IE > 0 I)

(ii) Suppose lim

A1

+

EP( IfA -fA I IE < 0 I) < E( 2 II~ II

+ 1)

E E}.

Jn lftA(w)- fA(w)l~ dP(w)

= 0

uniformly in A EA.

Then the first assertion in (ii) follows immediately from the Chebyshev inequality

P(lf~-

fAIE

>E)<~ J lf~(w)- fA(w)l~ E

n

dP(w)

which holds for any E > 0, t > 0, A E A (Theorem (1. 2.4)). To prove the second assertion in (ii), use the first assertion and a similar argument to that in (i). c We now have the following theorem. Theorem (3.1): For the stochastic FOE (I) suppose Hypotheses (M) are satisfied. Then the family {p~1:o < t 1 < t 2
67

(i)

with t1 t (ii) Pt o pt2 2

=

3

t pt1' 0 < t 1 < t 2 < t 3
In particular, for the autonomous stochastic FOE (IV), the family {pt = P~:O < t
Proof:

---

2

1

+

2

t 2 E [O,a].

By Theorem (11.3.1), the map Ttt 1:C(JJRn) ~ 2

t

2(n,C(JJRn);Ft ) is 2

Lipschitz for 0 < t 1 < t 2 € cb. It is clear that 11Pt 1<,>11 cb
'E

3

{P: 1rP: 2(,)J}(n) 2

3

=

3

2

J J g

=

'{T!2rT: 1(n)(w)](w')}dP(w')dP(w) 3

g

2

t1 '[Tt (n)(w)]dP(w),

J

3

g

(Cf. proof of Theorem (1.1)). In the autonomous case, Pt

-

2

= Pt0

=

2

t1 Pt +t , t 1,t2 E [O,a], t 1 + t 2 E [O,a], 1

2

because of time-homogeneity. Thus pt opt = p~ o·p!1+t = pt +t. D 1 2 1 1 2 1 2 The semigroup {Pt}t>o of the above theorem will be studied in some detail in the next chapter. Let M(C) be the complete topological vector space of all finite Borel measures on C : C(JJRn) given the weak * (or vague) topology (§1.2, Parthasarathy [66], Schwartz [71], Stroock and Varadhan [73]). Then we have a bilinear pairing 68

<·,·>:. cb

x

M(C)---> :R, <4>,ll>

=

Ic4>(n)dJJ(n),

4> € Cb' ll € M(C). Following Dynkin [16], define the adjoint semigroup * {Pt}O
I

nEC

p(O,n,t,B)dlJ(n), B €. Borel C(J,:Rn), 0 < t
Then for any 4> € Cb' <4>,P;ll>

=

I

~€C

4>(~) d(P;ll)(~) =

f

J~EC

=

J

nEC

I

~EC

4>(~)

I

p(O,n,t,d~)dll(n)

n€C

4>(~)p(O,n,t,d~)dll(n)

Pt(cp)(n)dll(n) = .

Since {Pt}O
2

1

2

t 1,t 2 € [O,a], t 1+t 2 € [O,a]. An invariant probabiLity measure for the stochastic FOE (IV) is a probability measure llo € M(C) such that P; llo = llo for all t € [O,a]. If a = ~. a probability measure llo € M(C) is invariant if and only if lim p(O,n,t,·) = llo for some n € C(JJRn). This t-+<><>

follows easily from the fact that P; p(O,n,t',·) = p(O,n,t+t',·), t,t' > 0, n € C(JJRn). Observe also that the family of transition probabilities {p(O,n,t,.):n € C(JJRn), t > O} for (IV) is left invariant by the semigroup {p;} t>O when a = ~. It would be interesting to find generic conditions on the coefficients H, G of (IV) which guarantee the existence of a (unique) invariant probability measure. Some partial results in this connection may be found among the examples of Chapter VI (§VI.4). See also Ito and Nisio ([41]) and Scheutzow [69]).

69

IV The infinitesimal generator

§1o

Notation

For the present chapter we keep the notation, general set-up and the standing assumptions of the last chapter (§III 1,2)o In particular, we focus our attention on the autonomous stochastic FOE: n(O)

+I:

H(xu(w))du+ (w)

I:

G(xu(o))dw(o)(u)

x(w)(t) = { n(t)

t >0 (I)

t E J = [-r,O]

Very frequently, the solution x through n will be denoted by nx; and throughout the chapter we shall assume that the coefficients H:C(J,Rn)--+ Rn, G:C(J,Rn) L(Rm ,Rn) are g'LobaZZ.y bounded and Lipschitzo The driving Brownian motion w is in Rm, generating a filtration (Ft)t>O on the probability space (O,F,P)o For brevity, symbolize the above stochastic functional equation by the differential notation: dx(t) = H(xt)dt + G(xt)dw(t)

(I)

x0 = Tl € C(J,Rn) Similarly for any t 1 > 0 we represent the equation

t

n(O)+t H(xu(w))du+ (w) G(xu( o))dw( o)(u).t > t 1 x(w)(t) = { t1 t1 t 1-r < t < t 1 n( t-t 1) by the stochastic-differential notation dx(t)·= H(xt)dt + G(xt)dw(t)

t > t, > 0 }

}
(I)

1

I

xt = n € C(J,Rn) 1

Recall that at the end of the previous chapter, we constructed a contraction semigroup {pt}t>O associated with the stochastic FOE (I) and defined on the Banach space Cb of all bounded uniformly continuous functions ~:C(J,Rn) --> Ro 70

Indeed

Now, for an ordinary stochastic differential equation (Stochastic ODE, r = 0), it is well-known that the semigroup {Pt}t>O is strongly continuous on Cb with respect to the supremum norm, and its strong infinitesimal generator is a second order partial differential operator on the state space of the solution process ([16], [22]). Our first objective is to show that, when r 0, the semigroup {Pt}t>O is neve~ strongly continuous on the Banach space Cb. Furthermore, we shall derive an explicit formula for the (weak) infinitesimal generator of {Pt}t>O. §2.

Continuity of the Semigroup

For each n € C(JJRn) and t > 0, define ~:[-r,=) +Rn by "'

n(t)

=

{~ n( O)

t>O

n(t)

t € J R"

R

0

Define the shift St:Cb

~

"' st<•> = • (nt>, n

Cb, t > 0, by setting €

C(JJRn ) , •

€

cb.

The next result then gives a canonical characterization for the strong continuity of {pt}t>O in terms of the shifts {St}t>O.

71

Theorem (2.1): The shifts {St}t>O form a contraction semigroup on Cb, such that, for each n € C(J,Rn), lim St(ct>)(n) = lim Pt(ct>)(n) = cl>(n) for all t-+0+ t-+0+ 4> € Cb. Furthermore lim Pt(cl>)(n) = cl>(n) uniformly in n € C(J,Rn) if and t-+0+ only if lim St(cl>)(n) = cl>(n) uniformly in n € C(J,Rn). t-+0+ Proof:

Let t 1,t2 > 0, n € C(J,Rn),

4>

€ Cb, s € J.

Then

where

"' )(0) (nt

~

"' )t (s) (nt 1 2

=

{

"'

=

n(O)

t

(nt }(t2 + s) 1

=

+

s

< 0

"' +t )(s). (nt 1

Hence

i.e.

-r < t 2

st (St (ct>))(n) 1 2

=

2

"' +t ) ct>(nt 1 2

=

st +t (ct>)(n) 1 2

Since lim . "' nt = n , it is clear that lim St(cl>)(n) = ct>(n) for each t-+0+ t-+0+ n € C(J,Rn), 4> € Cb. Also by sample paths continuity of the trajectory {nxt:t > O} of (I) (Theorem (11.2.1)) together with the dominated convergence theorem, one obtains

72

for each ~ € Cb and n € C(JJRn). To prove the second part of the theorem, suppose K > 0 is such that IH(n) I < K and IIG(n) II< K for all n € C(JJRn). Then for each t > 0 and almost all w € Q we have t+s t+s J H(nx (w))du+(w) J G(nx (·))dw(·)(u) t+S>O 0

u

0

u

-r < t

+

= {

0

s

<

0,

s € J

Thus, using the Martingale inequality for the stochastic integral (Theorem (1.8.5)), one gets "' 2 Jt+s 2 unxt(·)- ntll 2 < 2E sup I H(nxu(·))dul £ (Q,C) S€[-t,O] 0 + 2E sup S€[-t,O]

IJ

t+s

G(nxu(·))dw(·)(u)l 2

0

< 2K 2t 2 + 2K 1 J: E IIG(nxu(·))ll 2 du < 2K2t 2 + 2K 1t, some K1

> 0.

Therefore, lim llnx - ~ II = 0 uniformly in n € C(JJRn). Using the t-+0+ t t £2(Q,C) uniform continuity of ~ € Cb it is then not hard to see that lim {E~onxt(•)-~(~t)} =0 unifom"ly inn € C(JJRn) (Cf. proof of Lemma (III. t-+0+ 3.1 )). So 1 irn {Pt(~)(n)-St(~)(n)} = 0 uniformly in n € C(JJRn). Finally, \'lriting t-+0+

the second assertion of the theorem is now obvious. c Let C~

Cb be the set of all ~ € Cb such that lim Pt(~) = ~(= lim St(~)) t-+0+ t-+0+ in Cb. Then C~ is a closed linear subalgebra of Cb which is invariant under the semigroups {pt}t>O, {St}t>O. Both {pt}t>O and {St}t>O restrict to strongly continuous semigroups on C~ (Dynkin [16], pp. 22-26). c

Theorem (2.2): The semigroup {pt}t>O is not strongly continuous on Cb with respect to the supremum norm. ~:

It is sufficient to find~

€

Cb(C(JJRn)JR) such that St(~) ~ ~as 73

t +0+ i.n Cb;. (but St(lll)(n) + lll(n) as t + 0+ for each n €. C(J,Rn).) Let B c: C(J,Rn) be the closed unit ball. Fi.x any -r < s0 < 0 and define lii:C(J,Rn) ~ R by n(s 0) lll(n) = {

lin

II

< t

1 lin II >

n<s 0) lin II

t

Clearly 111 is continuous; indeed 111 is globally Lipschitz (and hence unifonmly continuous) on C(J,Rn). To prove this let n. n• E C(J,Rn) and consider the following cases (i)

n,n• e B.

( ii )

n, n•

t.

Then llll(n) - lll(n•)

i nt B• i.e.

II n II >

t,

= ln<s 0) - n•(s 0) I
II n• II >

1•

Write

I

n(s 0) n'(s 0) J llll(n) - lll(n•) I = - - • - lin II lln'll n(s 0) n(s 0) J n(s 0)

< =

<

Iw-~~~ I In<s >I 0 llln• llnlllln•ll

-L

lin•- nil

lln•ll (iii) n e int B, n•

74

+

II -

n'(s 0)

~~~-M~-

lin II I

+

I

- 1-ln<s 0) - n•(s 0) I

lln'll

+

-L

lin- n•ll < 2lln- n•ll

lln'll

t.

B;. i.e. lin II < 1,

lin•

II>

1.

Find n" e

as

II

(where 38 is the boundary of B) such that n" lies on the line segment joining nand n• i.e. find >.0 E [0,1].such that n" = (t->.0 )n.., >.0n• and lln"ll = 1. Define the function f:[0,1]---> R by f(>.) = 11(1->.)n • >.n'll-1, >. E [0,1]. Then f h clearly continuous. Also f(D) = lin II- t < 0 and f(1) = llrlll-1 >0. Hence by the Intermediate-Value Theorem there exists >. 0 E (0,1) such that f(A 0 ) = o i.e. taken"= (1-><0)n ... >. 0n• and lln''ll = t. Hence lljl(n) - ljl(n') l < lljl(n) - ljl(n")

I ...

lljl(n") - ljl(n')

II


R"

-r

i.e.

0 nn(s) = { [s 0 • 1

*-

s]n

so

1 +-

<s< 0

n

s0 < s < s0 +

-r < s < s 0

Note that llnnll = t for all n. such that s 0 • Consider for each fixed n

*

*

< 0; i.e. nnE 38 for all n.

75

St(~)(nn)

= ~<~{' = nn(t + s 0>. o < t

s 0•

< -

Therefore lim St(~)(nn) t-+o+

= lim

nn(t + s 0)

t-+o+

= nn(s 0) = ~(nn)

(*)

Now(*) is not unifo~ inn; for if (•) were uniform inn. given 0 < E < 1, there is a 0 < 6 <- s0 (6 independent of n) such that lnn(t+s 0)-nn(s 0) I <E for all 0 < t < 6 and for all n > 1. rtx 0 < t 0 < 6 and choose n0 such that 1 1.e. 1 t 0 + s 0 > s 0 + n-· Th us · n0 > t: 0

0

no

no

n (t 0 + s0) - n

(s 0) = lo- 1 I = 1.

But, by choi.ce of 6, lnn°(t0 + s 0) - nno(s 0) I < tradicts the choice of E. Therefore ~ t cg.

E •

Hence 1 <

E,

which con-

a

Remarks (i)

One could have chosen

~· €

Cb in the above proof such that

II n II <

1

~*(n) =

II n II

> 1

It can be shown that ~· t c~. However, note that, while ~· is only continuous. ~ is smooth in any small neighbourhood of 0. (ii) Later on, in §4, we construct a concrete class of smooth functions on C(JJRn) lying entirely in C~. These functions are called quasi-tame fUnctions. The class of quasi-tame functions is sufficiently rich to generate Borel C(JJRn). in exactly the same fashion as the tame functions do. (iii) Note that the domain of strong continuity C~ of {Pt}t>O is independent of the coefficients H. G and the Brownian motion w. As yet, a complete characterization of C~ is unknown to me. §3. The Weak Infinitesimal Generator In this section we obtain a general formula for the.weak infinitesimal generator of {Pt}t>O (Dynkin [16] Vol. 1 pp. 36-43). Define a weak topology on Cb as follows: Let M(C(JJRn)) be the Banach 76

space of all finite regular measures on Borel C(JJRn) given the total variation norm. Then there is a continuous bilinear pairing <·,·>:CbxM(C(JJRn))~ defined by

<~.~>

=

fC(JJRn) $(n) d~(n),

$ € Cb'

~

€ M(C(JJRn))

Say a family {$t:t > O} in Cb conve~es ~eakLy to $ € Cb as t ~ 0+ if lim <~t.~> = <~.~> for all ~ € M(C(JJRn)). Write this as $ = w-lim ~t• t~+

t~+

Proposition (3.1) (Dynkin [16], p. 50): For each t > 0 let ~t' ~ € Cb. Then $ = w-lim ~t if and only if £11~tll: t > 0} is bounded and $t(n) ~ ~(n) t~+

as t ~ 0+ for each n € C(JJRn). Proof: For each n € C(JJRn) let on be the Dirac measure concentrated at n, defined by 1 ,

n € B

o, nt B Define ~t € [M(C(JJRn))]* by

for all B € Borel C(JJRn).

$t(~) = <~t'~>. ~ € M(C(JJRn)).

If$= w-lim

t~+

~t'

then

~(n) =<~,on>

"'

= lim

t~+

<~t.on>

=lim

t~+

~t(n)

for all

n € C(JJRn), and the set {$t(~):t > O} i! bounded for each~ € M(C(JJRn)). By the uniform boundedness principle {II $til : t > O} is bounded. But ll~tll = ll~tll for each t > 0, so £11~tll : t > O} is bounded. Conversely suppose {II $til : t > O} is bounded and ~t(n) -;:. $(n) as t ~ 0+, for all n € C(JJRn). By the dominated convergence theorem

J

n C(JJR )

as t ~ 0+, for all ~ € M(C(JJRn)). The

~ak infinitesimaL generatoP

A($) = w-lim t~+

~(n)d~(n)

Thus ~ = w-lim $t.

= <~.~>

D

t~+

A:V(A)

c

Cb

~

Cb of {pt}t>O is defined by

Pt($) - $ t 77

where O(A) is the set of all ' for which the above weak limit exists. By continuity of the sample paths and the dominated convergence theorem it is easy to see that w-lim Pt(') = 'for ev.ery ' € Cb. Moreover the t~+

following properties can be found in Dynkin ([16] Vol. 1 Chapter I §6, pp. 36-43). Theorem

(3.1)~

(i) O(A)

=C~.

O(A) is weakly dense in Cb, Pt(O(A)) ~ O(A) for all t >

o.

(ii) If ' € O(A), then the weak derivative

exists and

~ Pt(') = A(Pt(,)) = Pt(A(,)), Pt(') - ' =

J:

Pu(A ('))du for all t > 0.

(iii) A is weakly closed i.e. if {'k};= 1 c O(A) is weakly convergent and {A(,k)};= 1 is also weakly convergent, then w-lim 'k € O(A) and w-lim A(,k) = A(w-lim 'k). k~ k~

(iv)

k~

For

each~>

(~id

-

0,

A)- 1 (~)

~id-A

=

is a bijection of O(A) onto Cb.

J: e-~tpt(~)dt

for all

~

Indeed

E cb.

The resolvent R~ = (~id- A)-1 is bounded linear and IIR~II< {for all ~ > 0. (v) w-lim

~~

~R~(~)

= ~ for every

~

E Cb.

In order to deriv.e a general formula for the weak infinitesimal generator A, one needs to augment the state space C(JJRn) by adjoining a canonical n-dimensional direction. The generator A will then be equal to the weak infinitesimal generator S of the shift semigroup {St}t>O of §2 plus a second order partial differnetial operator along this new direction. The construction is as follows. Let Fn be the vector space of all simple functions of the form vx{O}where vERn and x{O}:J ~ R is the characteristic function of {0}. Clearly 78

c(JJRn) n Fn = {0}. plete norm lin

+

Form the direct sum C(JJRn) • Fn and give it the com-

vx{o}ll = sup ln(s) I s€J

+

lv I

• n e: C(JJRn). v € Rn

Indeed C(J,Rn) • Fn is the space of all functions ~:J +Rn which are continuous on [-r,O) and possibly with a finite jump discontinuity at 0. The following lemma shows that the above construction admits weakly continuous extensions of linear and bilinear forms on C(J,Rn) to C(J,Rn) • Fn. Lemma (3.1): let a € C(J,Rn)*. Then a has a unique (continuous) linear extension a:C(J,Rn) • Fn ---~ R satisfying the weak continuity property (w 1) if {~k};= 1 is a bounded sequence in C(J,Rn) such that ~k(s) --">· ~(s) as k +=for all s € J for some ~ € C(J,Rn ) • Fn• then a(~ k) ---. a(~) as The extension map e : C(J,Rn) *---> [C(J,Rn) • FnJ*

is a linear isometry into. Proof: We prove the lemma first for n = 1. Suppose a € C(J,R)*. By the Riesz representation theorem (Dunford and Schwartz [15], §IV.6.3 p. 265) there is a (unique) regular finite measure ~= Borel J +R such that a(n) = Jt -r

n(s)d~(s)

for all n € C(J,R).

Define a:C(J,R) e F1 ~ R by a(n

+

v1x{O}) = a(n)

+ v 1 ~{0},

n € C(J,R), v1 € R.

a is

clearly a continuous linear extension of a. k let {~k};= 1 be a bounded sequence in C(J,R) such that ~ (s) + n(s) + v 1 ~ 0 js) as k +=for all s € J where n € C(J,R), v1 € R. By the dominated convergence theorem,

By

79

The map e:ceJ,R) * -~ rceJ,R) e F1J* is clearly linear. a

•->

a

Higher dimensions n > 1 may be reduced to the 1-dimensional situation as follows: Write a E ceJ,Rn)* in the form n . aen) = E a 1 en.) 1 i=1 where n = en,_ ••• ,nn) E ceJ,Rn), ni E ceJ,R), 1 < i < n, aien*) = aeo,o, ••• ,O,n*,O, ••• ,O) with n* E ceJ,R) occupying the i-th place. Hence ai E ceJ,R)* for 1 < i < n. Write Fn=F~, where F1 = {v*x{O}:v* ER} by taking vX{O} = ev 1x{O}'"" ., vnx{O}) for each v = ev 1, ••• ,~n) ERn, vi E R 1 < i < n. Let al E [CeJ,R) e F1J* be the extension of a 1 described before and satisfying ew 1). I_t is easy to see that ceJ,Rn) e Fn = rceJ,R) e F1J

x ••• x

rceJ,R) e F1J en copies)

i.e. n + VX{O} = en 1 + v1x{O}'"""'nn + VnX{O}) Define a E [CeJ,Rn) e FnJ* by n

•

aen+ VX{O}) = i~ 1 a 1 eni + ViX{O}) when n = en 1 ,.~··nn), v = ev 1, ••• ,vn). Since ai is a continuous linear~ extension of a 1 , then a is a continuous linear extension of a. Let {~k}k= 1 be bounded in ceJ,Rn) such that ~kes) ~ ~es) as k + ~ , for all s E J, n k k k k where~ E ceJ,R) e Fn. Let~ = e~ 1 ••••• ~n), ~ = e~ 1 ••••• ~n), ~i E ceJ,R), ~i E ceJ,R) e F1, 1 < i < n. Hence {~~};= 1 is bounded in C(J,R) and 80

t~(s) ~ l;;i(s) as k k

-+

Q), s

€

n

i

J, 1 < i < n. Therefore k

n

-i

lim a(l;; ) = lim E a (l;;i) = E a (l;;i k.._, k.._, i=1 i=1 ·so

)

-

= a(l;;).

a satisfies

(w 1). To prove uniqueness let "' a € [C(JJRn) e Fn] * be any continuous linear extension of a satisfying (w 1). For any vx{O} € Fn choose a bounded sequence {t~};= 1 in C(JJR) such that l;;~(s) ---) YX{O}(s) as k -+ Q), for all s € Ji e.g. take. (ks + 1)v, -

r1 < s < 0

= {

-r < s < -

0

r1

-r

Note that lll::~ll = lvl for all k > 1. Also by (w 1) one has

-

for all n € C(JJRn). Thus "' a = a. Since the extension map e is linear in the one-dimensional case, it follows from the representation of a in terms of the a; that e:C(JJRn) * --~ a

;,

a 81

is also linear. But a is an extension of a, so llall > llall. Conversely, let ~ = n +- vx{O} € C(J,Rn) e Fn and construct {~~};= 1 in C(J,Rn) as above. Then

a(~) = lim·a(n k-+OO

+

~~).

But Ia( n

+

~~ > I < II a II II n .., ~~ II < II a II [ II n II = II a II

+

II ~ ~ II J

[ II n II .., IvI J = II a II II ~II for a11 k > 1•

Hence Ia(~) I= lim la(n +-~~)I< llall II~ k-+OO

II for every~

€

C(J,Rn) e Fn.

Thus llall < llall. So lliill = llall and e is an isometry into.

o

Lemma (3.2) :. Let B:C(J,Rn) x C(J,Rn) ~ R be a continuous bilinear map. Then B has a unique (continuous) bilinear extension S:[C(J,P.n) eFn] x [C(J,Rn) E&Fn] +F. satisfying the weak continuity property: (w2) if {~k};= 1 , {nk};= 1 are bounded sequences in C(J,Rn) such that ~k(s)--!> ~*(s), nk(s)--!> n*(s) as k + oo, for all s € J, for some ~*.n* € ·c(J,Rn) e Fn, then B(~k.nk)--+ B(~*.n*) ask+ oo. Proof: Here we also deal first with the 1-dimensional case: Write the continuous bilinear map B:C(J,R) x C(J,R) --+ R as a continuous linear map B:C(J,R) --!> C(J,R)*. Since C(J,R) *is weakly complete (Dunford and Schwartz [15], §IV. 13.22, p. 341), B is weakly compact as a continuous linear map . C(J,R) ~ C(J,R) * (Theorem (1.4.2); Dunford and Schwartz [15], §VI. 7.6, p. 494). Hence there is a unqiue measure A:Borel J ---~ C(J,R)* (of finite semi-variation IIA II (J) < oo ) such that for all ~ € C(J,R)

B(~) = ~~r ~(s)

dA(s)

(Theorem (I.4.1)). Using a similar argument to that used in the proof of Lemma (3.1), the above integral representation of B implies the existence of a unique continuous 82

nnear extension a:C(J,R) C9 F1 ~ C(J,R)* satisfying (w1). To prove this, one needs the dominated convergence theorem for vector-valued measures (Dunford and Schwartz [15] §IV.10.10, p. 328; Cf. Theorem (1.3.1)(iv)). Define a:C(J,R) e F1 -> [C(J,R) e F1J* by B = e o 13 where e is the extension isometry of Lemma (3.1); i.e.

Clearly a gives a continuous bilinear extension of 13 to [C(J·,R) • F1] X [C(JJR) • F1]. To prove that a satisfies (w2) let kco kco k {~ }k= 1' {n }k= 1 be bounded sequences in C(J,R) such that~ (s) + ~*(s), nk(s) + n*(s) as k +co for all s € J, for some~·. n* € C(J,R) e F1• By ,.. ,.. * k (w 1) for 13 we get 13(~ ) = lim 13(~ ). Now for any k, k--

,..

But by (w 1) for

13(~*)

we have

lim la(~*)(nk)- a(t*)(n*)l = 0 k-Since {llnkll };= 1 is bounded, it follows from the last inequality that lim 13(~k)(nk) = ~(~*)(n*). k-When n > 1, we use coordinates as in Lemma (3.1) to reduce to the 1-dimensional case. Indeed write any continuous bilinear map 13:C(J,R)n x. C(J,Rn) ~ R as the sum of continuous bilinear maps C(J,R) x. C(J,R) + R in the following way 13((~1•···•~n),(n1, ••• ,nn>> =

n

..

E 13,Jc~ .• n.) i,j=1 1 J

~~re (~ 1 ••••• ~n>• (n 1, ••• ,nn) € C(J,Rn), ti,ni € C(J,R), 1 < i < n,

13 1 J:C(J,R)

x

C(J,R) +R is the continuous bilinear map 83

aij(~'.n'> = a((o,o, ••• ,o.~·.o, ••• ,o), (o,o, ••• ,o.n·.o •••• ,o)), for ~·, n' € C(J,R) occupying the i-th and j-th places respectively, 1
where~·= (~1·····~~), n* = (n; ••••• n~) € C(J,Rn) • Fn, ~i· nj € C(J,R) • F1, 1 < i, j < n. It is then easy to see that is a continuous bilinear extension of a satisfying (w2). Finally we prove uniqueness. Let e:C(J,Rn) e Fn ~ [C(J,Rn) e Fn]* be any continuous bilinear extension of a satisfying (w2). Take~· = ~ + v1x{O}' n* = n + v2XV} € C(J,Rn) e Fn' where~. n € C(J,Rn), v1, v2 € Rn. Choose k k n) such th:t ~ k( s ) --+ v x{O} ( s ) , b~unded sequences {~O}k=t' {n 0}k=t 1n CkJ,R 1 0 n0(s) - > v2x{O}(s) as k~oo, s € J; ll~oll = 1v 1 1. llnoll = 1v 2 1 for all k > 1. koo koo . k k k k Let~ = ~ + ~ 0 • n = n + n0• Then{~ }k= 1' {n }k= 1 are bounded sequences 1n C(J,Rn) such that ~k(s) ~ ~*(s), nk(s) ~ n*(s) as k ~ oo, for all s € J. So by (w2) for "'a and -a one gets

a

00

00

•

(

B(~*)(n*) =lim a(~k)(nk) = B(~*)(n*) k~

Thus ~ =

s.

c

For each n € C(J,R0 ) let nx € t 2(n,C([-r,a],Rn)) be the solution of (I) through n. and ~t € C(J,Rn) be defined as in §2 for each t € [O,a]. Lemma (3.3):

There is a K > 0 (independent of t,n) such that

II{E(nxt - ~t) II C < K for all t Also if a

€

lim

t~+

Proof: 84

>

0 and n

€

C(J,Rn).

C(J,Rn)*, then 1 nxt- "' tEa( nt) = a(H(n)x{O}) for each n

€

C(J,Rn).

Denote by E the expectation for Rn or C(J,Rn)-valued random variables.

Let K > 0 be such that IH (n)l < K for all n € C(J,Rn). Now t+s Jt+s n ~ { J H(nxu(w))du + (w) G(nxu(·))dw(·)(u), t+S>O xt(w)(s)-nt(s) = 0 0 0 t + s < 0, for each t E [O,a], a.a.w € o, s € J. Since evaluation at each s € J is a continuous linear map C(J,Rn) then it commutes with the expectation i.e.

+

Rn,

1n ~ 1n ~ [Eft< xt- nt)}](s) =Eft£ xt(•)(s) - nt(s)]}, s € J n 1 Jt+s = { t 0 E(H( xu))du, 0

t +s>0

t + s <

,

o.

t > 0

using the Martingale property of the Ito integral. Thus lim t J: E(H(nxu))du lim [E{t(nxt- ~t)}](s) = { t-+0+ 0 -r < s < 0 t-+0+ =

s =0

H(n)x{O}(s), for all s € J.

We prove next that lit E(nxt - ~t) lie is bounded in t

>

0 and n. Clearly

Jt

~ 1 Jt+s IE(H(nxu))ldul
< K for all t

>

Therefore lit E(nxt - ~t) lie < K for all t If a

0 and n

>

C(J,Rn).

€

o and n

€

C(J,Rn).

C(J,Rn) *, then

€

~ ) 1 (nxt - nt tEa = t1 a ( E(nxt - ~nt)) = a[t1 E( nxt - ~nt)].

By the weak continuity property (w1), one gets .

1

n

~

-

11m tEa( xt - nt) = a(H(n)x{o}>· t-+0+

a

85

Lemma (3.4): For each t

w~(w)(s)

>

0 and a.a. w €

n,

define wt(w)

1 [w(w)(t+s) - w(w)(O)] -= {

€

C(JJRn) by

-t < s < 0

If -r < s < -t

0

Let a be a continuous bilinear form on C(JJRn). Then 1 nxt - "' lim [tEa( nt, nxt - "' nt) - Ea(G(n) t-+0+

wt, G(n)

o

o

wi)l =

o.

Proof: We prove first that (1)

Observe that 1 Jt+s H(nx )du + -1 Jt+s G(nx )dw(u) -If 0 u If 0 u 1 Jt+s dw(u)], -t - G(n)[ -If 0

<

s<0

-r < s < -t

0

almost surely.

for t > 0, and some K1 > 0. Since Hand G are Lipschitz with Lipschitz constant L, then almost"surely IH(nxu) 12 < 2IH(n) 12 + 2L 2 11 nxu -nil~, u and

86

IIG(nxu) - G(n) 11 2 < L2 llnxu -nil~, u

>

0.

>

0;

Hence (3)

and (4)

Now by the main existence theorem (Theorem (11.2.1)), the map [O,a] ?u~o-+ xuEi(n,C(J,.r.n)) is conttnuous;. so Hm Ellnxu-nii 2=0. Therefore the u...O-~: last two inequalities (3) and (4) imply that {EIH(nxu>l 2: u E [O,aJ} is bounded and Hm Ell G(nxu) - G(n) 11 2 = 0. u-+0+ Letting t-+ 0+ in (2) yields (1). Since a is bilinear,

' - a(G(n) o wi• G(n) o wi> tt a( nxt - " nt•' n xt " - nt) a( __ 1 (nxt - ~t) - G(n) o wi• __ 1 (nxt - ~t) - G(n) o wi> If If 1 (nxt-~t)-G(n)owi•G(n)owi) + a(G(n)owi• -l{nxt-~t)-G(n)owt). + a(-If If

=

Thus, by continuity of a and H6lder•s inequality, one gets It Ea(nxt- ~t• nxt- ~t)- Ea(G(n) o wt• G(n) o wi>l < llall

Ell~ (nxt- ~t) - G(n) o wi11 2

1 + 211a II [EII-If for all t > 0.

(nxt-~t) - G(n)owi11 2J 112 [E IIG(n)owi11 2J 112 •

(5)

But E IIG(n) o wi11 2 < E sup I w(t+s~w(O)I 2 IIG(n) 11 2 sE[-t,O] = =

t E SlAP lw(t+s)-w(O) 12 IIG(n) 11 2< t t IIG(n) 11 2 sE[-t,OJ IIG(n) 11 2 • for all t > 0. (6) 87

Comb;n;ng (6) and (5) and lett;ng t + 0+ g;ves the requ;red result.

a

Lemma (3.5): Let ;n~n + Fn be the ;somorph;sm ;n(v) = vx{O}' v € Rn, and G(n) x G(n) denote the l;near map

(v,.v 2)

~

(G(n)(v 1), G(n)(v 2)).

Then for any cont;nuous b;l;near form Bon C(J,Rn) Hm t+O+

t EB(nxt-~t•nxt-~t) = trace [B

o

(;n x in)

o

(G(n)

x

G(n))]

for each n € C(J,Rn), where B ;s the cont;nuous b;l;near extens;on of B to C(J,Rn) • Fn (Lemma (3.2)). Indeed ;f {ej}J= 1 ;s any bas;s for Rm, then . "' nxt-nt) "' = .Em B(G(n)(ej)X{o}•G(n)(ej)X{o}>· 11m t1 EB( nxt-nt• t+O+ J=1

Proof:

In v;ew of Lemma (3.4) ;t ;s suff;c;ent to prove that m

1;m EB(A t+O+

o

wt• A o wt) = _E B(A(eJ.)X{O}' A(eJ.)X{O}) J=1

(7)

for any A € L(Rm ,Rn). We deal f;rst w;th the case m = n = 1, v;z. we show that 1;m EB(wi• wi> = B(x{o}•X{o}> t+O+

(7)'

for one-dimensional Brown;an mot;on w. If t, n € C = C(J,R), lett 8 n stand for the funct;on J x J +R def;ned by (t 8 n)(s,s') = t(s)n(s')for all s, s' € J. The projective tensor product C 8~ C is the vector space of all functions of the form E~= 1 ti 8 ni where ti,ni € C, i = 1,2, ••• ,N. It carries· the norm N N llhll 8 = inf { E llt-11 lin; II :h = E ti 8 ni,ti,ni €C, i=1,2, ••• ,Nl. i =1 ~ i =1 1 The infimum is taken over all possible finite representations of h € C 8~ C. Denote by C i~ C the completion of C 8~ C under the above norm. It is well known that C ~~ C is continuously and densely embedded in C(J x J,R), the Banach space of all continuous functions J x J +R under the supremum norm 88

(Treves [75], pp. 403-410; §1.4). Since C is a separable Banach space, so is C in C. . countable dense subset of C. Then the countable set N

Y8 Y = { t

~-

i=1

1

For let Y c C be a

8 n. : ~ 1.,n 1• E Y, i = 1, ••• ,N, N = 1,2, ••• } 1

is dense in C en C and hence in C in C. The continuous bilinear form Bon C corresponds to a continuous ZineaP ,.. C] * (Treves [75] pp. 434-445; Cf. Theorem 1.43). functional "'BE [C ew Now let ~,.~2 e t2(g,c). The map

cec---+ cenc (~.n)

~~en

is clearly continuous bilinear. Thus ~, ( •)

e ~2 (.)

:

n - > c in c w1

is Borel measurable.

for almost all wE

>

~ 1 (w) 8 ~ 2 (w)

But

n;

hence by HOlder's inequality the integral

fn 11~ 1 (w) 8 ~2 (w) 11 8

dP(w) exists and

n

From the separability of C •n C the Bochner integral (§1.2) E

~ 1 (.) 8 ~2 (·)

=

J0 ~ 1 (w) 8 ~2 (w)dP(w)

exists in C•n C. Furthermore, it commutes with the continuous 1inear functional "'B; viz.

89

Fix 0 < t < r and cons;der

t E[w(·)(t+s) - w(•)(O)][w(·)(t+s') - w(•)(O)],

s,s• € [-t,O]

= {

s

0 1 +

[-r,-t) or s•

€

t m;n (s,s•)

s,s•

€

€

[-r,-t).

[-t,O]

= {

s

0

[-r,-t) or s•

€

€

[-r,-t) (9)

Def;ne Kt:J x J +R by lett;ng Kt(s,s•) = [1 +

t m;n (s,s')]X[-t,O](s)x[-t,O](s•)

s,s• € J

(10)

;.e. Kt

=

E[wt(•)

€ £ 2(n,C),

S;nce wt

8

wt(•)].

;t ;s clear from (8) that Kt

ES(wt(•), wt(·))

=

"'B(Kt)

€

C e~ C and

•

( 11)

In order to calculate 1;m ~(Kt)• we shall obta;n a ser;es expans;on of Kt. t+O+ We appeal to the follow;ng class;cal techn;que. Note that Kt ;s cont;nuous; so we can cons;der the e;genvalue problem

t

-r

Kt(s,s•)~(s')ds'

=

'-~(s)

s

€

J

(12)

s;nce the kernel Kt ;s symmetr;c, all e;genvalues '-of (12) are real. (10) rewr;te (12) ;n the form t

J0-t ~(s')ds' + J-t0 m;n(s,s•)~(s')ds' '-~(s) =

Therefore 90

0

= '-t~(s)

s € [-t,O] (;)

s € [-r,-t)

(;;)

} (13)

t J0 E;(s')ds'.., Js s'E;(s')ds' -t -t

+-

s J0 E;(s')ds' = UE;(s) s

sE[-t,O] ( 14)

Differentiate ( 14) with respect to s, keepi.ng t fixed, to obtain

t

( 15)

E;(s')ds' = U E;'(s), s E (-t,O]

s

Differentiating once more, -E;(s) = ).t E;"(s) ,

s

E

(-t,O]

(16)

When).= 0, choose E;~~J +R to be any continuous function such that ~~(s) = 0 for all s E [-t,O] and normalized by

f-r-t t;0(s) 2 ds = t. t

Suppose)."/ 0. Then (13)(ii) implies that E;(s) = 0 for all s E [-r,-t)

( 17)

In (14) put s = -t to get ~(-t)

(18)

= 0·

In (15) put s = 0 to obtain ~·co> =

o

( 19)

Hence for). I 0, (12) is equivalent to the differential equation (16) coupled with the conditions (17), (18) and (19). Now solutions of this are given by

A,et-h--~is

+

A2e-t-h-~is • s E [-t,O]

E;(s) = {

(20) s E [-r,-t), i = r-T

0

Condition (19) implies immediately that A1 = A2 = 1, say. e

-t-~A-~it

+

e

t-~A-~it

From (18) one gets

= 0.

Since the real exponential function has no zeros, it follows that ).~ cannot 91

he

imaginary i.e. A > 0.

Being a covariance function, each kernel Kt is non-

negative definite in the sense that

tt

Kt(s,s')t(sH;(s')ds ds' > 0 for all t

€

c.

Using (18), we get the eigenvalues of (12) as solutions of the equation t(-t) = 2 cos [ - 1 (-t)] = 0 •

liT

Therefore the eigenvalues of (12) are given by k = 0,1,2,3, •••

(21)

and the corresponding eigenfunctions by t (2)i (2k+1)~s tk(s)= t x[-t,OJ(s) cos [ 2 t J,

s € J, k = 0,1,2, •••

(22)

after being normalized through the condition t 2ds = 1, k = 0,1,2, ••• JJ tk(s)

Now, by Mercer's theorem (Courant and Hilbert [10] p. 138, Riesz and Sz-Nagy [68] p. 245), the continuous non-negative definite kernel Kt can be expanded as a uniformly and absolutely convergent series 00

Kt(s,s') = E Ak t~(s) t~(s'), k=O 00

E

k=O = {

0

8

~2(2k+1)2

(23)

s,s' € J

cos [(2k+1)~s] cos [(2k+1)~s·J 2t 2t s,s' € [-t,O]

(24)

s € [-r,-t) or s' € [-r,-t)

But from the definition of Kt' one has Kt(O,O) = 1 for every t putting s = s' = 0 in (24) we obtain

>

0. Thus (25)

From the absolute and uniform convergence of (24), it is easy to see that Kt can be expressed in the form 92

(26) where ~

~k(s) =cos [

(2k+1)ns zt Jx[-t,O](s),

s € J.

Note that the series (26) converges (absolutely) in the projective tensor ,.. product norm on C en C. Hence we can apply "'B to (26) getting from (11) the equality EB(wi(·),wt(.)) =

=

"" 8 "' '!1: E 2 2 B(~k k=O n (2k+1)

'!1:

8 ~k)

""E

8 B(~ ~t) k' k k=O n2(2k+1)2

(27)

But II~ II< 1 for all k > 0 and all 0 < t < r; so the series..,(27) is uniformly convergent in t, when compared with the convergent series E 2 8 2• ~ k=O n (2k+1) Moreover, for each s € J, ~k(s) ~ x{O}(s) as t + 0+, k = 0,1,2, •••• Thus if we lett+ 0+ in (27), we obtain

""

= E 2 8 2 S(x{O}' X{O}) k=O n (2k+1)

using (25) and Lemma 2. This proves (7)'. For dimensions n > 1, write e:C(JJRn) X C(JJRn) +R in the form n

..

.

.

e<~1.~2> = . ~ e,J<~~.~~> 1 ,J=1 - ( 1 n) lJ. ( ) ( ) where ~ 1 -- ( ~ 11 ••••• ~ n) 1 , ~ 2 - ~ 2 •••• ,~; 2 and each B .C JJR x C JJR ~ R is continuous bilinear. Let A € L(Rm, Rn) and {ek}~= 1 ' {ei}~= 1 be the canonical bases for Rm and Rn, respectively; i.e.

93

Write m-dimensi.onal Brownian motion w in the form w = (w1,w2,. ••• ~) where wk (t) = ~(t),ek>' k = 1, ••• ,m, are independent one-dimensional Brownian motions. Then

Letting t

=

n 1: i ,j=1

=

n 1: i ,j=1

+

0+ and using (7)' gives

lim Ea(Aowt,Aowt) t+O+

=

m 1: k=1

To obtain the final statement of the lemma, take A = G(n) and note that the last trace term is independent of the choice of basis in Rm. c Let V(S) c C~ be the domain of the weak generator S for the shift semigroup {St}t>O of §2. We can now state our main theorem which basically says that if' € V(S) is sufficiently smooth, then it is automatically in V(A). Furthermore, A is equal to S plus a second order partia~ differential operator on C(J,Rn) taken along the canonical direction Fn. The following conditions on a function ':C(J,Rn) + R are needed. Conditions (OA): (i )

' € 1)( s)t

( i i) ' is c2; (iii) 0,, o2' are globally bounded; (iv) o2, is globally Lipschitz on C(J,Rn). Theorem (3.2): Suppose ':C(J,Rn) ~ R satisfies Conditions (OA). Then 94

• €

O(A) and for each n

C(JJR")

€

A( 4> )(n) = S( l!>)(n)+( Dl!>(n )o in)(H(n))+Hrace[D24>(n) o( in Xin) o(G(n) XG(n))]

where~. D24>(n) denote the canonical weakly continuous extensions of o•(n) and D24>(n) to C(JJR"> e Fn' and in~" ~ Fn is the natural identification v t--> vx{O} • Indeed if {ej}J= 1 is any basis for R", then n :2:::'

-

A(l!>)(n) = S(l!>)(n)+DII>(n)(H(n)x{O})+-i j: 1o l!>(n)(G(n)(ej)X{o}•G(n)(ej)X{o}>· ~: Fix n € C(JJR") and let nx be the solution of the SRFDE (I) through n. Suppose 4> satisfies (DA). Since 4> is c2, then by Taylor•s Theorem (Lang [47]) we have

where (28)

Taking expectations, we obtain

Since 4>

€

O(S), then (30)

In order to calculate lim following two limits

t~+

t E[ll>(nxt) - l!>(n)], one needs to work out the

"' n ~+ t1 E04>(nt)( xt - "' nt)

lim

t~O+

t ER2(t)

We start by considering (31). that

(31) (32)

From Lemma (3.3), there exists a K > 0 such

95

Hence

Let t

+

0+ and use the cont;nu;ty of 0' at n to obta;n

"-n ' = 1;m t1 EO,(n)( n " nt) ' 1;m t1 EO,(nt)( xt " - nt) xt-

t~+

t~+

(33) by Lenvna (3.3). Secondly, we look at the l;m;t (32). Observe that ;f K ;s a bound for H and G on C(JJRn) and 0 < t < r, then

< 8K4t 4 + 8K2t

J:

Ell G(nxu) 11 4 du

< "'K(t4 + t 2 ), some K2, "'K > 0,

(34)

where we have used Theorem (1.8.5) for the Ito ;ntegral. Furthermore, ;f u € [0,1] and 0 < t < r, then t 2'(nt+u( "' nxt-nt))( "' nxt-nt, "' nxt-nt) "' Il£0

<

t 2'(n)( nxt-nt, "' nxt-nt) "' 1 - l£0

t E llo2'(~t + u(nxt-~t)) - o2,(n) II

II nxt-~tll2

< [E llo2,(?) + u(nx -~ ))-o2'(n) 112J 1/2 [ 1 Ell nx -~ 114J 1/2 tt t2" tt t < ~1/2(t2+1)1/2[E llo2'<~t+u(nxt-~t))- o2'(n)112J1/2. 96

(35)

But o2$ is globally Lipschitz., ~i.th Li.pschi.tz..constant L say;. so E IID 2 $(~t""u(nxC~t)) - D2$(n) 11 2 < L2E( II ~t-nll

+

II nxt-~tll >2

< 2L 2 11~t - n11 2 ... 2L 2 [~1nx t - ~t 11 4 J 112 < 2L 2 ll~t - n II 2 ... 2L 2~ 1 1 2 t(t 2 because of the inequality (34). . tED 1 2$(nt "' 11m

+

t+O+

+ 1) 112

Letting t

+

(36)

0+ in (35) and (36), we obtain

u( nxt- "' nt))( nxt- "' nt• nxt- "' nt) (37)

uniformty in u E [0,1].

1 =~

n

:r.;:-

.r o $(n)(G(n)(eJ.)X{o}•G(n)(eJ.)X{o}>·

J=1

(38)

Since $ E V(S) and has its first and second derivatives globally bounded on C(JJRn). it is easy to see that all three terms on the right hand side of (29) are bounded in t and n. The statement of the theorem now follows by letting t + o... in (29) and putting together the results of (30), (33) and (38).

c

It will become evident in the sequel that the set of all functions satisfying Condition (DA) is weakly dense in Cb. Indeed within the next section we exhibit a concrete weakly dense class of functions in Cb satisfying (DA) and upon which the generator A assumes a definite form. §4. Action Of the Generator on Quasi-tame Functions The reader may recall that in the previous section, we gave the algebra Cb of all bounded uniformly continuous functions on C(JJRn) the weak topology induced by the bilinear pairing ($,lJ). ~ J n $(n)dlJ(n), where C(JJR ) ~ € Cb and lJ runs through all finite regular Borel measures on C(JJRn). Moreover, the domain of strong continuity C~ of {pt}t>O is a weakly dense proper 97

subalgebra of Cb. Our aim here is to construct a concrete class Tq of smooth functions on C(JJRn), viz. the quasi-tame fUnctions, with the following properties: (i)

Tq is a subalgebra of C~ which is weakly dense in Cb;

(ii)

Tq generates Borel C(JJRn);

(iii) Tq c V(A), the domain of the weak generator A of {pt}t>O; (iv)

for each ~ € Tq and n € C(JJRn), A(~)(n) is a second-order partial differential expression with coefficients depending on n.

Before doing so, let us first formulate what we mean by a tame function. A mapping between two Banach spaces is said to be cP-bounded (1 < p < m) if it is bounded, cP and all its derivatives up to order p are globally bounded; e.g. Condition (DA) implies c2-boundedness; and c3-boundedness implies (DA)(ii), (iii), (iv). Definition (4.1)

(Tame FUnction):

A function ~:C(JJRn) +R is said to be tame if there is a finite set {s 1,s 2, ••• ,sk} c Janda em-bounded function f:(Rn)k +~such that ~(n) = f(n(s 1), •••• n(sk)) for all n € C(JJRn).

(*)

The above representation of ~ is called minimal if for any projection p:(Rn)k + (Rn)k- 1 there is no function g:(~n)k- 1 +R with f =gop; in other words, no partial derivative Djf:(Rn)k + L(RnJR), j = 1, ••• ,k, off vanishes identically. Note that each tame function admits a unique minimal representation. Although the set T of all tame functions on C(JJRn) is weakly dense in Cb and generates Borel C(JJRn), it is still not good enough for our purposes. due to the fact that •most• tame functions tend to lie outside c~ (and hence are automatically not in V(A)). In fact we have Theorem (4.1): (i) The set T of all tame functions on C(J,Rn) is a weakly dense subalgebra of Cb' invariant under the shift semigroup {St}t>O and generating Botel C(JJRn). 98

(ii)

Let

~

E T have a minimal representation

~(n) = f(n(s 1), ••• ,n(sk))

where k > 2.

-

Proof:

Then

~ ~

n E C(JJRn)

0

Cb.

For simplicity we deal with the case n = 1 throughout.

lt is easy to see that T is closed under linear operations. We prove the closure ofT under multiplication. Let ~,.~ 2 E T be represented by $1(n) = f1(n(s 1), ••• ,n(sk)), $2(n) = f 2 (n(s1)~ ••• ,n(s~ )), for all n E C(JJR), where f 1 ~k -+R, f 2 :Rm -+Rare C -bounded functions and s 1, ••• ,sk' s,, ••• ,s~ E J. Define f 12 :Rk+m -+R by (i)

f 12 cx 1, ••• ,xk, x,, ••• ,x~) = f 1(x 1, ••• ,xk)f 2 (x1•····x~) for all x1, •.• ,xk, x,, ••• ,x~ E R.

00

Clearly f 12 is C -bounded and

Thus ~,~~ E T, and T is a subalgebra of Cb. It is immediately obvious from the definition of St that if ~ E T factors through evaluations at s 1, ••• ,sk E J, then St($) will factor through evaluations at t + sj < 0. So T is invariant under St for each t > 0. Next we prove the weak density of T in Cb. Let T0 be the subalgebra of Cb consisting of all functions $:C(JJR) -+R of the form ~(n) =

f(n(s 1), ••• ,n(sk)), n E C{JJR)

( 1)

when f:Rk -+R is bounded and uniformly continuous, s 1, ••• ,sk E J. Observe first that T is (strongly) dense in T0 with respect to the supremum norm on Cb. To see this, it is sufficient to prove that if c > 0 is given and f:Rk -+R is any bounded uniformly continuous function on Rk, then there is a k C -bounded function g:R k -+R such that lf(x) - g(x)l < c for all x E R. We Prov.e this using a standard smoothing argument via convolution with a C bump function (Hirsch [32], pp. 45-47). By uniform continuity of f there is a 0 > 0 such that lf(x 1)- f(x 2 )1
00

R

99

Define g~k +R by

the integral being a Lebesgue one on Rk. g(x) = J, k h(y)f(x-y)dy R

=I

8(0,15)

h(y)f(x-y)dy, x ERk.

(2)

By choice of 6 and property of h, it follows that lf(x) - g(x) I

= I JRk

h(y)f(x)dy - IRk h(y)f(x-y)dyl


E

J, k h(y)dy R

= E

for all x € Rk. To prove the smoothness of g, use a change of variable y• = x-y in order to rewrite (2) in the form g(x)

=-

J

B(x,IS)

h(x-y')f(y')dy'

Now fix x0 € Rk and note that B(x,IS) Therefore g(x)

= -

I

B(x 0 ,2&)

c

for all x

€

Rk.

(3)

B(x 0,2&) whenever x € B(x 0,&).

h(x-y')f(y')dy'

for all x € B(x 0,&).

(4)

Since f is continuous and the map x ~ h(x-y•) is smooth, it follows from (4) that g is em on B(x 0 ,&) and hence on the whoLe of Rk, because x0 was chosen arbitrarily. Indeed g and all its derivatives are globally bounded on Rk, for II oPg(x) II <

fB(x ,2&) II oPh(x-y• >II

lf(y• >ldy'

0

< V.

supk IIDph(z) 11. supk lf(z) I

z~

z~

=

N, say,

for all x € Rk, where N is independent of x0 and V = J dy' is the 8(0,26) . volume of a ball of radius 215 in Rk. Secondly we note thatTOis weakly dense in Cb. Let ITk:-r = s 1 < s 2 < s 3 < ••• < sk = 0, k = 1,2, ••• , be a sequence of partitions of J such that mesh nk + 0 as k + m. Define the continuous linear embedding Ik:Rk + C(JJR) by letting lk(v 1,v 2, ••• ,vk) be the piece-wise 1inear path = (s-sj_ 1) v. + (sj-s) lk(v 1, ••• ,vk)(s) vJ._ 1 , s € [sJ._ 1,sJ.] sj-sj_ 1 J sj- 5 j-t 100

joining the points

\L,. ... ,'lk E R,

j

= 2~ ••• ,k.

R

-r-s,

Denote by !k the k-tuple (st•···•sk) E Jk, and by.p!k the map e(J,R) ~ Rk n

~

(n(st), ••• ,n(sk)).

Employtng the unifo.rm. continuity of .each n E. e(J,R) on the compact interval J, the reader may easily check that lim (Ik o.p k)(n)

!

k.....

=n

in e(JJR).

(5)

Now if' E eb' define 'k~e(J,R) +R by 'k = 'olk 0 Psk• k = 1,2, •••• Since' is bounded and uniformly continuous, so is ' o lk~ Rk + R. Thus each 'k E T0 and lim 'k(n) = '(n) for all n E e(J,R), because of (5). Finally k.....

note that II 'k II e < b

II' lie for a11 k > 1. Therefore ' = w-Hm 'k and b

k-

T0

is

weakly dense in eb. From the weak density of T in T0 and of T0 in eb' one concludes that T is weakly dense in eb. Borel e(J,R) is generated by the class {,- 1 (u)~U c:R open, 'E T}. For any finite collection !k = (s 1, ••• ,sk) E Jk let Psk ~ e(JJR) +Rk be as befor Write each' E Tin the form'= f o P5 k for some-em-bounded f~k +R.

It is 101

a: 8

,}~--~~r·----,~~----------------------,0 I I I

. r---------------

1

I I

.a

:.

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

.e....

+~ ~

.a + i

• .6

-------------------

~

.aI

~

102

well-known that Borel C{JJR) is generated by the cylinder sets {psk -1 (U 1 x ••• xUk):Ui =:R open, i = 1, ... ,k,!k=(s 1, ... ,sk) e:J k, k=1,2, ... } (Parthasarathy [66] pp. 212-213). Moreover, it is quite easy to see that Borel Rk is generated by the class {f- 1(U) : U=:R open, f:Rk ..... R C00-bounded} (e.g. from the existence of smooth bump functions on Rk). But for each open set U =R, $- 1(U) = P;~ [f- 1{U)]. Therefore it follows directly from the above that Borel C(JjR) is generated by'· (ii) Let $ e: < have a minimal representation $= f o Psk where !k = (s 1, ••• ,sk) e: Jk is such that -r < s 1 < s 2 < ••• < sk <-0, f:Rk ..... R is Ceobounded and k > 2. Take 1 < jo < k so that -r < Sj 0 < 0. Since the representation of $is minimal, there is a k-tuple (x 1, ••• ,xk) e: Rk and a neighbourhood [xj 0-E0 ,xj 0 + E0 J of xj 0 in R such that oj 0f(x 1, ••• ,xj 0_1,x,xj 0+1 , ••• ,xk) '0 for all x e: [x.

- E0 , x. + E0], with EO> 0. Jo Jo + E0 ] ..... R by

Define the function

g:[x. -E 0 , x. Jo Jo g(x) = f(x 1 , ••• ,x. _1,x,x. 1 , ••• ,xk) for all x e: [x. -E 0 ,x. +E 0]. Jo Jo+ Jo Jo co

Hence Dg(x) ' 0 for all x e: [x. -E 0 , x. + E0] and g is a C diffeomorphism Jo Jo onto its range. Therefore, there is a A > 0 such that lx' - x"l

< Alg(x')- g(x")l for all x', x"e:[x. -E 0 ,x. +E 0] Jo Jo

Pick o0 > 0 so that o0 <Eo and the intervals are mutually disjoint. In the remaining part with no loss of generality, that all integers struct a sequence {nn} in C(JJR) looking like

(6)

(sj-2o 0 , sj+2o 0 ), j =1,2, ••• ,k, of the argument we may assume, n are such that~< o0 • Conthe picture opposite

103

viz.

xj

sj-oO < s

- -0) 010 x.(s-s.-o J J

xJ.

sj+o 0 < s < sj+2o 0• j

~

~ x.(s-s.+o0) + xJ.

sj-2o 0 < s < sj-o 0, j

- -01o x.Jo (s-s.Jo -o 0)+x.Jo

s. +o0 < s < s. + 2o0 Jo Jo

0

nn(s) =

+-

J

J

j0 ~

jo

(7)

0

Suppose, if possible, that ' € Cb. Then lim St(,)(nn) = lim f(nn(t+s 1), ••• ,nn(t+sk))=f(nn(s 1h···•nn(sk)) t-+0+ t~+ unifo~ty in n.

But, for j ~ jo and 0 < t < o0, nn(t+sj) = xj fo~ att n.

Therefore

unifo~ty

inn; i.e. for any E > 0, there is a 0 < o < o0 such that

lg(nn(t+sJ. )) - g(nn(sJ. ))I < E for all n, all 0 < t < o. . 0 0

(8)

Now suppose 0 < t < o0• If*< t < o0, nn(t+sj 0) - xj 0 = o. If o < t < *· then lnn(.t+s.)- x. I= l-nE 0t + x. + .E0 - x. I = E. 0 (1-nt) <Eo· Hence Jo Jo Jo Jo nn(t+s. ) € [x. -E0,x. +Eo] for all 0 < t < o0• Applying (6) and (8), we Jo Jo Jo get for 0 < t < o , 104

Note that 6 is independent of n. ln the above inequality, fix t < 6 and choose n0 large enough such that~< t. Then s. * < t + s. < s. + 60 , n

and n °(t

+-

s. )

Jo

lnn°(t

+-

=

x. ~so

i-0

Jo

0

Jo

Jo

Jo

s. ) - nn°(s. ) I

Jo

Jo

= lx.

Jo

-(x.

Jo

+- £ 0)

which clearly contradicts the arbitrary choice of

I = £0 < l£

£.

Therefore

<1>

t

C~.

a

Definition (4.2) (Quasi-tame FUnctions)~ A function ~C(JJRn) +R is quasi-tame if there is an integer k > 0, C~-bounded maps fj~n +Rn, 1 < j < k-1, h:(Rn)k +Rand piecewise c1 functions gj:J +R, 1 < j < k-1, such that (n)

=

h( J:r f 1(n(s))g 1(s)ds, ••• ,

J~r

fk_ 1(n(s))gk_ 1(s)ds,n(O))

for all n € C(J,Rn). Each derivative gj is assumed to be absolutely integrable over J. Denote by Tq the set of all quasi-tame functions on C(J,Rn). Theorem (i)

(4.2)~

Tq

c

V(s)

m(n)

=

c~.

c


-r

Indeed if

=

hom € Tq where

f 1 (n(s))g 1(s)ds, ••• ,

t

-r

fk_ 1(n(s))gk_ 1(s)ds,n(O))

1, ••• ,k-1, are as in Definition (4.2), then k-1 S()(n) = j:1 Djh(m(n)){fj(n(O))gj(O)-fj(n(-r))gj(-r)

and h.,fj ,gj, j

=

- J0 fj(n(s))gj(s)ds} -r

for all n € C(J,Rn). The function h:(Rn)k +R is considered as one of k n~dimensional variables and Djh are the corresponding partial derivatives (J = 1, ••• ,k).

(ii) Tq is invariant under the shift semigroup {Sf t>O" 105

(iii)

Tq is a weakly dense subalgebra of Cb generating Borel C(JJRn).

Proof: In proving statements (i) and (ii) of the theorem, we shall assume for simplicity that $ = hom where h:(Rn) 2 +R is C~-bounded and m(n) = cf0 f(n(s))g(s)ds, n(O»for some C~-bounded map f~n +Rn and a -r

(piecewise) c1 function g:J +R. Let 0 < t < r and consider the expression { [$(~t) - $(n)J

={ =

[h(m(~t)) - h(m(n))]

J: Dh(z~){¥m

using the Mean-Value Theo~em for h, with z~ But {rm
- m(n)J}du =

(10)

(1-u)m(n) + um(nt), 0 < u < 1.

f-r-t f(n(t+s))g(s)ds + i Jo-t f(n(O))g(s)ds 0

-{ J

f(n(s))g(s)ds,O)

-r

=

cf 0

t-r

f(n(s)) t{g(s-t)-g(s)}ds + f(n(O)) t

J-t0 g(s)ds

t-r - { J-r f(n(s))g(s)ds,O).

Now f is bounded and g is (piecewise) c1, so all three terms in the above expression are bounded in t and n. Moreover letting t + 0+ we obtain lim t+O+

¥m<~t)-m(n)J

=

<-J 0 f(n(s))g'(s)ds+f(n(O))g(O)-f(n(-r))g(-r),O) -r ( 11)

As Z~ is bounded inn and continuous in (t,u), it follows from (10), (11) and the dominated convergence theorem that $ € V(S) and S($)(n)

106

=

Dh(m(n)){lim ¥m<~t) - m(n)J} t+O+

=

D1h(m(n)){f(n(O))g(O)-f(n(-r))g(-r) -

J~rf(n(s))g'(s)ds}.

For •q to be invari.ant under {St}t>O it i.s suffi.cient to prove that it is invariant under St• 0 < t < r, due to the semigroup property. So let $ be as above and t E [O,r]. Then St($)(n) = h(J0 f(n(s))g(s-t)ds t-r

+-

f(n(O)) J0 g(s)ds,n(O)) -t

for all n E C(J,R'n). Define "' g:.J -+:R by "' { g(s-t) t-r < s < 0 g(s) = 0 -r < s < t-r Then clearly ~is piecewise c1 with g' absolutely integrable over J. also ~:C(J,Rn) -+ (:Rn) 2• F:(:Rn) 2 -+ (:Rn) 2 and ~:.(:Rn) 2 -+ :R by

~(n) = <J0 f(n(s))~(s)ds,n(O)) = <J0 -r

F(x,y) = f(y)

t-r

Define

f(n(s))g(s-t)ds,n(O)),

J-t0 g(s)ds, h(x,y) = h[F{x,y)

+

(x,y)],

for all x,y ERn, n E C(J,Rn). Since f and hare c=-bounded, so is F and hence "' h. Therefore St($) = "'"' hom E •q· We show that •q is a weakly dense subalgebra of Cb. It is an easy matter checking that •q is closed under addition and multiplication of functions in Cb. To prove weak density of •q in Cb" it is sufficient to show that •q is weakly dense in • u •q• because the set of tame functions • is already weakly dense in Cb (Theorem (4.1)(i)). Let$ E • have the representation $(n) = f(n(s 1) ••••• n(sk))

n E C(J,P.n)

where f:(:Rn)k -+:R is c=-bounded and s 1•••• ,sk E J. For each 1 < j < k, construct a sequence {gj}== 1 of piecewise linear functions on J with the property that gj(s)-+ X{s.}(s) as m-+ =and lgj(s)l < 1 for all s E J, for all m> 1; e.g.

J

107

-----------

1

0

-r

Also choose a sequence {6m}== 1 of C~ bump funct;ons on Rn such that lxl < m lxl >m+1

0

and l6m(x)l < 1 for all x € Rn. Def;ne the sequence 1 and n E C(JJRn), ltm(n(s)) gj(s)l < ln(s)l for all s E J. So by the dom;nated convergence theorem n(sj) = 1;m J 0 tm(n(s))gj(s)ds, j = 1, ••• ,k nr-- -r

(12)

Therefore '(n) = 1;m f(J 0 JII'+OO

-

r

tm(n(s))g~(s)ds, ••••

J0 -r

tm(n(s))g~(s)ds)

(13)

Denot;ng the expression under the l;m;t;ng operat;on ;n (13) by 'm(n), ;t ;s clear that·'m € Tq for each m and l'm(n)l <sup
p!k:C ( JJRn)

+ ( Rn)k ,

! k = (s 1, ••• ,sk ) , sj E J, J. = 1, ••• ,k, k=1,2, •••

=

. aut this is agai.n generated by sets of the form {n:.n(sj) E C} where C Rn is closed. Now each closed set in Rn is a countable intersection of complements of open balls, so we need only prove that if BeRn is any open ball and B' is its complement, then {n:.n(sj) E B'} E a(Tq) for any sj E J. Suppose B has radius b > 1 and for each integer p > 0 let Up be the complement of the concentric closed ball of radius b - ~- Let the sequences {~}m= 1 and {gJ}m= 1 be as before, so that (12) is satisfied. Let ~mE Tq be given by

~m(n)

J-r ~(n(s))gj(s)ds,

= 0

We contend that

m> 1, n E C(JJRn).

... n

p=t

lim inf

m

{n:~m(n)

E UP}

(14)

To see this, let n(sj) E B'. Then n(sj) E UP for all p > 1. Since n(sj) = lim ~m(n), then for each p there is an m0 > 0 such that ~m(n) E UP for all

m--

m> m0• Hence, for each p > 1, belongs to the set on the right lim inf {n:~m(n) E UP} for each m m0 > 1 such that ~m(n) E Up for

n belongs to lim inf {n:~m(n) E UP} i.e. n m hand side of (14). Conversely, let n be in p > 1. Then for every p > 1, there is an

all m > m0• Taking m + '"' gives n(sj) E Up for all p > 1 i.e. n(sj) E Up. But B' = Up, so n(sj) E B'. Thus p=1 p=1 our contention is proved. As the sets {n:~m(n) E UP} are clearly in a(Tq), it follows directly from (14) that {n:n(sj) E B'} E a(Tq). Therefore a(Tq) =Borel C(J,Rn). a

n

n

The final result in this chapter asserts that every quasi-tame function is in the domain O(A) of the weak generator A of {Pt}t>O. Theorem (4.3): Every quasi-tame function on C(J,Rn) is in the domain of the weak generator A of {pt}t>O. Indeed if ' E Tq is of the form '(n) = h(m(n)), n E C(J,Rn), where

109

then A{~){n)

k-1 = j;1 Djh{m{n)){fj{n{O))gj{O)-fj{n{-r))gj{-r)

- J0

-r

+

Dkh{m{n)){H{n))

+

fj{n{s))gj{s)ds}

t trace [D~h{m{n))o(G{n)

x

G{n))]

for all n E C{JJRn). Here again D.h denote the partial derivatives of h:{Rn)k +R considered as a functi~n of k n-dimensional variables. Proof: To prove that Tq c V{A), we shall show that each ~=homE Tq satisfies®Conditions {DA) of §3. First, it is not hard to see that each ~ E Tq is C • Also by applying the Chain Rule and differentiating under the integral sign one gets

--

D~{n){~)

+

= Dh{m{n)){J 0 -r

o

Dh{m{n)){ J

-r

Df 1 {n{s)){~{s))g 1 {s)ds, ••• ,

2

D f 1 {n{s)){~ 1 {s).~ 2 {s))g 1 {s)ds, .•• ,

for all n, ~. ~,.. ~ 2 E C{JJRn). Since all derivatives of h, f., 1 < j < k-1, are bounded, it is easy to see from the above formulae thatJD~ and o 2 ~ are bounded on C{JJRn). By induction it follows that~ is C -bounded. Hence Codnitions {DA){ii), {iii), {iv) are automatically satisfied. Condition {DA){i) is fulfilled by virtue of Theorem {4.2){i). From the above two formulae we see easily that the unique weakly continuous extensions D~(n), o 2 ~{n) of D~{n) and o 2 ~{n) to ®

110

c(J,Rn) a Fn are given by D$(n)(vx{O}) = Dh(m(n))(O, ••• O,v) = Dkh(m(n))(v), ~

D $(n)(v 1x{o}•v 2x{o}> = o2h(m(n))((O, ••• ,o,v 1), (O, •••• o,v 2)) 2 = Dkh(m(n))(v 1,v 2),

for all v, v1,v 2 ERn. The given formula for A($)(n) now follows directly from Theor~~ (3.2) and Theorem (4.2). a Definition (4.3) (Dynkin [16]): Say n° E C(J,Rn) is an absoPbing state for the trajectory fietd {nxt:t > 0, n € C(J,Rn)} of the stochastic FDE(I) if p{w:w En, noxt(w) = n°} = 1 for all t > 0, where we take a =~here, i.e. p(O,n°,t, {n°}) = 1 for all t > 0. The following corollary of Theorem (4.3) gives a necessary condition for n° E C(J,Rn) to be an absorbing state for the trajectory field of the stochastic FDE (I). Corollary: Let n° E C(J,Rn) be an absorbing state for the trajectory field of the stochastic FOE (1). Then (i) n°(s) = n°(0) for all s E J i.e. n° is constant; (ii) H(n°) = 0 and G(n°) = 0. ~:

Let n° E C(J,Rn) be an absorbing state for {nxt:t > 0, n E C(J,Rn)}. For each t > 0 and s E J, define the Ft-measurable sets 0

0

nt = {w:w E n, n xt(w) = n } 0

Qt(s) = {w:w En, n xt(w)(s) = n°(s)}. Then nt c Qt(s) for all t > 0, s E J and since P(nt) = 1, it follows that P[Qt(s)] = 1 for t > 0, s € J. Suppose if possible that there exist s 1,s 2 E J such that n°(s 1) ~ n°(s 2). 0 Without loss of generality take -r < s 1 < s 2 < o. For each wEn, n x5 _ 5 (w)(s 1) = n°(s 2) ~ n°(s 1) and so ns _5 (s 1) = 2

1

~.

2

1

Hence P[Q5 -s (s 1)J = 0, which contradicts P[Qt(s)] = 1, 2

1

111

t > 0, s € J. So n° must be a constant path. Th;s proves (;). To prove that n° sat;sf;es (;;), note that the absorb;ng state n° must sat;sfy A(~)(n°) = 0 for all ~ € V(A) (Dynk;n [16], Lemma (5.3), p. 137). In part;cular A(~)(n°) = 0 for every quas;-tame ~:C(J,Rn) +R. Note f;rst that s;nce n° ;s constant then so ;s the map t + ~ and so S(~)(n°) = 0 for every~ € V(S). Take any ~:Rn +R C~-bounded and def;ne the (quas;)-tame funct;on ~:C(J,Rn) +R by ~(n) = ~(n(O)) for all n € C(JJRn). Then by Theorem (4.3), ~ € V(A) and

A(~)(n°) = D~(n°(0) )(H(n°))

+

i trace [D ~Cn°(0) ) (G(n°) 2

0

x

G(n°) )] = 0

The last ;dent;ty holds for every C~-bounded ~:Rn +R. Choose such a~ w;th the property D~(n°(0)) = 0 and o 2 ~(n°(0)) = <·,·>, the Eucl;dean ;nner product on Rn e.g. take "'~ to be of the form "'~(v) = lv - no(0) I2 ;n some ne;ghbourhood of n°(0) ;n Rn. Then

for any basis {eJ. }mJ·= 1 of Rm. Therefore G(n°) = o. Thus D~(n°(0)) (H(n°)) = 0 ~ n ~ 1\1 ~~~~ for every C -bounded ~:R +R. Now p;ck any C -bounded A such that ~(v) = for all v ;n some ne;ghbourhood of n°(0) ;n Rn. Then DA(n°(0))(H(n°)) = IH(n°)1 2 = 0; so H(n°) = 0 and(;;) ;s proved. c Remarks (;) We conjecture that cond;t;ons (;) and (;;) of the corollary are also suff;c;ent for n° to be an absorb;ng state for the trajectory f;eld {nxt:t > 0, n € C(J,Rn)}. It ;s perhaps enough to check Lemma (5.3) of Dynk;n ([16], p. 137) on the weakly dense set of quas;-tame funct;ons ;n V(A). (;;) An absorb;ng state n° corresponds to the o;rac measure onO be;ng ;nvar;ant under the adjo;nt sem;group {p~}t>O assoc;ated w;th the stochast;c FOE (I). Thus a necessary cond;t;on for the existence of ;nvar;ant o;·rac measures ;s that the coeff;c;ents H, G should have a common zero (cf. §III.3).

112

V Regularity of the trajectory field

§1.

Introduction

Given a filtered probability space (n,F,(Ft)G
=

H(xt)dt + G(xt)dw(t)

x0

=

n Ec

= C(JJRn)

0 < t < a

}

(I)

with coefficients H:C ~Rn, G:C ~ L(RmJRn) and m-dimensional Brownian motion w. A version of the trajectory field of (I) is a measurable process x:n x [O,a] x C ~ C such that X(·,t,n) = nxt a.s. for all t € [O,a] and n € c. In order to deal with the question of the existence of versions of the trajectory field with reasonable sample function behaviour one needs to isolate two qualitatively different examples of stochastic FOE's. The first of these is when the diffusion coefficient G is independent of the past history, in which case we show that the trajectory field has a version whose sample functions are almost all compactifying (See Theorem (2.1)). In the second example, where G incorporates explicit dependence on the past (e.g. as in stochastic delay-equations), we shall see that all versions of the trajectory field are almost surely highly irregular (See §3). Such erratic behaviour is essentially due to the Gaussian-type nature of delayed diffusions coupled with infinite-dimensionality of the state space C. Under these considerations we are lead naturally to investigate regularity in probability of the trajectory field for the general system (1). This is done in §4 where the compactifying nature of the trajectory field is still shown to persist, though in a distributional sense, as in Theorem (4.6). It is perhaps worth noting here that for ordinary stochastic DE's (r = 0) satisfactory results on the sample function behaviour of the trajectory field are known. See Remark (i) of §3.

113

§2. Stochastic FOE's with Ordinary Diffusion Coefficients Consider the stochastic FOE dx(t) = H(xt)dt

+

g(x(t))dw(t)

x0 = n € c = C(J,Rn),

0 < t
(II)

J = [-r,O].

The coefficients H:C +~n. g:~n + L(Rm,Rn) are Lipschitz maps and w is m-dimensional Brownian motion adapted to the filtered probability space (Q,F,(Ft)O
m

€ R •

For any 0 1 lin II a= sup { a : t 1,t2 € I, t 1 f. t 2} + llnllc· c lt 1 - t 2 1 Ca(I,Rm) and Ca(I,Rm xRn) are similarly defined. The following lemmas will be needed in the sequel. If h € Ca([O,a],R), then the map [r,a] 3 t ~ ht € Ca(J,R) is

Lemma (?..1): continuous.

Proof: Suppose first that the lemma holds for any c1 map h € c1([0,a],R). Let£> 0 and h € Ca([O,a],R). Since c1([0,a],R) iS dense in Ca([O,a],R), there is an h0 € c1([0,a],R) such that llh-h 0 llc < £/4. By uniform continuity of the map [r,a] 3 t ~h~ € Ca(J,R), there exists 6 > 0 such that if t 1,t 2 >rand lt 1-t 2 1 < 6, then llh~ - h~ II a<E/2. Now if s 1,s 2 € J, t>r, .

114

1

2

c

and "' h € ca([O,a]JR). we have

; .e.

for all t > r. If t 1.t2 >rand lt 1-t 2 1 < 6. then one has llht -ht II < llht -h~ II + llh~ -h~ II . + llh~ -ht II 1 2 ca 1 1 ca 1 2 ca 2 2 ca

< 2 llh-h 0 II

ca

+ llh~ -h~ II < £. 1 2

Thus we need only prove the lemma for h € c1([0,a]JR). If so, let r < t 1 < t 2 < a, -r < s 1 < s 2 < 0 and consider s2 (ht -ht )(s 2)-(ht -ht )(s 1) = J (ht -ht ) 1 (u)du 1 2 1 2 s1 1 2 s2 ~ J [h 1 (t 1+u)-h•(t 2+u)]du. s1 By the uniform continuity of hand h 1 , if£> 0 is given there exists 60 > 0 such that lh(u)- h(v)l < £/ 1+r1-a and lh•(u)- h•(v)l< £/ 1+r1-a whenever u,v € [O,a], 1u-v1 < 6 0• Hence if lt 1-t 2 1 < 60• then

So llht -ht II a= sup j(ht -ht )(s 2)-(ht -ht )(s 1>1·1<s 2-s 1 1-QI 1 2 C s 1,s 2EJ 1 2 1 2 s1 '# s2 + llht -ht II 1

2

c

But llht -ht 11 = sup lh(t 1+s) - h(t 2+s) 1 < £ 1 2 C sEJ 1+rl-a 115

Therefore e:rt-a II ht - ht II < 1 2 1+r 1-a

+

e: - e: t+rl-a -

This completes the proof of the lemma. Lemma (?..2): Suppose F:RP Lipschit~. Then the map

-+

a

Rn is a C1 map with DF:RP

-+

L(RP ,Rn) locally

is Lipschit~ on every bounded set in Ca(J,RP). Proof~ Let s 1,s 2 € J, ~ 1 .~ 2 € Ca(J,RP). For each u € [O,t] define y(u), z(u) € iP by y(u) = u~ 1 (s 2 ) + (t-u)~ 2 <s 2 ), z(u) = u~ 1 (s 1 )+(1-u)~ 2 (s 1 ). By the Mean-Value Theorem for F, the expression

= F(~ 1 (s 1 )) -

F(~ 2 (s 1 ))

-

[F(~ 1 <s 2 ))

-

F(~ 2 <s 2 ))]

=I: DF(z(u))(~ 1 (s 1 )-~ 2 (s 1 ))du- I: DF(y(u))(~ 1 (s 2 )-~ 2 (s 2 ))du =It 0

DF(z(u))[(~ 1 -~ 2 )(s 1 )-(~ 1 -~ 2 ><s 2 )Jdu

+

1

+

J0 [DF(~(u))-DF(y(u))] (~ 1 (s 2 )-~2 (s 2 ))du.

and lz(u)I 0 such that Now for each u

€

[0,1], ly(u)l < 11~ 1 11c

+

11~ 2 11c

IIDF(z(u))- DF(y(u))ll< K1 1z(u)-y(u)l for all Letting K2 = sup { IIDF(v) II : v 116

€

RP, lv I < 2}, we get

u€

[0,1].

11~1- ~211 a -11~1- ~211cJ ls1-s21a

IE(s1,s2)1 < K2[

+ K1

c

11~ 1 - ~ 2 11 c

<

K2[11~1 -~211ca- ~~~1

+

K1

<

K2[H~1 -~211

11~ 1 - ~ 2 11 C

+

< K3

~~~1

-

~211

~211

lz(u)-y(u) ldu

C] ls1-s21a

J: {ui~ 1 Cs 1 >-~ 1 Cs 2 )1

a -

c

-

J:

11~1-~211

+

(1-u)I~ 2 Cs 1 )-~ 2 <s 2 )1}du

+

11~ 2 11 ca> ls 1-s 2 1a

cJ Is1-s21a

K111~1-~211c
cals1-s21a

where

The last estimate clearly implies that there is a K4 > 0 such that

"' "'F(~2>11ca
F(Q,x) = x for all ! E ~m, x E ~n. Furthermore {F(,!, ·) =! E ~m} is a group of c 2 diffeomorphisms of ~n i.e. F(,!1 •• ) oF(~,.) = F(!1 +·~··) F(,!.·)- 1 = F(-,!,•)

for all ,!1 • ~ E ~m for all t E ~m. 117

Proof~ We follow Dieudonne {[1.1], Chapter X, pp. 303-307). Consider the family of Lipschitz. c2 vector fields {g(·)(_!):! e: nm} on n". Using the theory of existence, uniqueness and dependence of solutions of ODE's, on initial conditions and parameters, we get a unique c2 map 4>:R x Rn x Rm -+ Rn such that

D14>(s,x,t)(t)

cj>(O~x,!)

=

g{cj>{s,x,t))(t)

=

X

-

-

(1)

}

for all s e: R, x e: Rn, t e: Rm. Note that for each t e: Rm, cjl{•,•,t):RxRn-+Rn is a-one-parameter c2 flo\'1 on Rn~ Define F:RmxRn-+Rn by F(!,x)

=

4>( 1,X,!), !

€

:Rm, X € Rn.

Clearly F is c2• As g(.)(O) = 0, then cj>{s,x,O) = x for all s e: R. Thus F(O,x) = 4>(1,x,0) = x for ;11 X € Rn. -Since g is c1, we can differentiate both sides of (1) with respect to! getting D 1 [D 3 c~>(s,x,!)(v)]{1) =

{Dg{cjl{s,x,!))[D34>(s,x,!)(v)]}{!)+g(cl>(s,x,!))(v) (2)

D34>(0,x,!)(v) = 0 for all s € R, x € Rn, !•V e: Rm• Fix v,! e: Rm and x e: Rn; define u:R -+ Rn by u(s)

=

D3 4>(s,x,!)(v) - sg(cl>(s,x,!))(v)

for all s e: R.

Differentiating with respect to s gives

- s{Dg{cjl{s,x,!))[D 14>(s,x,!)(1)]}{v) = {Dg(cl>(s,x,!))[D 3 4>(s,x,!)(v)]}{!)-s{Dg(cl>(s,x,!))[D 1 4>(s,x,!)(~)}{v) =

{Dg(cl>(s,x,!) )(u(s) )}(!) + sDg(cl>(s,x,!) )[g(·:P(s,x,!) )(v)] - s{Dg(cl>(s,x,!))[g(cl>(s,x,!))(!)J}(v)

=

118

{Dg(cl>(s,x,!))(u(s))}(!)

(3)

for all s E R, because of (1), (2) and (Fr). Also by definition of u, u(O) = D3cll(o,x,_!)(v) = o. Since g is c1, the map s 1+ Dg(cll(s,x,_!) )( • )(_!) is continuous and therefore the linear differential equation y 1 (s)

=

Dg(cll(s,x,_!))(y(s))(!), s E R

y(O)

=

0

}

has y = 0 as its unique solution. Since u satisfi.es (4), then u(s) all s E R. ln particular, putting s = 1 one gets

(4) = 0

for

D1F(_!,x)(v) - g(F(_!,x))(v) = D3cll(t,x,_!)(v) - g(cJl(t,x,_!))(v)

= u(1) = 0 for all v E Rm, i.e. D1F(_!,x)

=

g(F(_!,x))

for a11 ! E Rm, x E Rn. Note that for each tERm, F(t,•) = cll(1,·,t) is a c2 diffeomorphism of Rn. To prove the grou; property~ let !t, ~ ~ Rm and define z,'i:Rm + Rn by z(_!)

=

F(_!,F(!1,x)), "'z(_!)

where x ERn is fixed.

=

F(!

+

,!1,x) for a11 ! E Rm,

Then

Dz(_!)

= g[F(_!,F(_!1,x))] = g(z(_!)),

Dz(_!) "'

= D1F(!

+

_!1,x)

= g[F(!

+

_!1,x)]

= g(z(_!)) "'

for all! E Rm, and z(~) = ~(Q,F(_!1 ,x)) = F(_!1,x) = "'z(~). But, for each fixed ! E Rm, the maps s 1-+' z(s_!), s >+ z(s_!) are clearly both solutions of h 1 (s) = g(h(s))(_!), s E R, h(O) = F(_!1 ,x). So by uniqueness of solutions, z(st) = "'z(st) for all s E R. In particular - + _! ,x)- for all! E Rm, all x ERn. c Z(!) = "'z(_!) i.e. F(_!,F(_!1,x)) = F(t 1 Remark ( 2 .1) If the c2 condition on g is weakened to g being c1 but with Dg locally 119

Lipschitz, a unique c1 flow F~m x Rn +Rn for g still exists. Furthermore DF is locally Lipschitz due to the following argument. Use the notation in the proof of the lemma. Let B1 eRn, B2 cRm be bounded sets. By continuity of'· the setS= {'(s,x,!) : s € [0,1], x € B1, ! € B2} is relatively compact in Rn. Since Scan be covered by a finite number of arbitrarily small balls and Dg is locally Lipschitz, it follows that Dg is Lipschitz on S with Lipschitz constant K1 , say. Let x,x• € B1 , !• !' € B2• Then from differentiating (1) with respect to x we obtain ID 2 '(~.x.!)(y)- o 2 '(~.x',!')(y)l

= 1J:{(Dg(,(s,x,!))[D2,(s,x,!)(y)])(!) - (Dg(,(s,x',!'))[D 2,(s,x',!')(y)])(!)}dsl < K2 +

J:

IIDg(,(s,x,!)) II ID 2,(s,x,!HY) - o2,(s,x • •!' )(y) Ids

K2 J: IIDg(,(s,x,!))- Dg(,(s,x',!'»lllo2,(s,x•,!•)(y)lds

< K2 K3 J: ID2,(s,x,!)(y)- o2,(s,x',!')(y)lds +

IYI K4 K1 J: l'(s,x,!)- '(s,x',!')lds,

~

€ [0,1],

where y € Rn, K2 =max [sup lxl. sup I!IJ. x€B 1 !EB2 K3

=sup {IIDg(,(s,x,!»ll: s € [0,1], x € B1, ! € B2},

and K4 .=sup

{IID 2,(s,x,!)ll: s € [0,1], x € B1 , ! € B2}.

But ' is c1 and therefore Lipschitz on the compact set [0,1] there is a K5 > 0 such that l'(s,x,!) - '(s,x',!') I< K5[1x-x'l Therefore 120

+

x

s1

x

B2 i.e.

1!- !'IJ for all s € [0,1].

ID 2 ~(~,x.!)(y)- o 2 ~(~,x',!')(y)

< K2K3 +

I

J: ID2~(s,x,!HY)- o2~(s,x',!')(y)lds K1K4K5 IYI Clx-x' I

+

I!-!' IJ for all

~

E [0,1].

By Gronwall's lemma,

for all

~

E [0,1].

In particular,

IID2F(!,x)- o2F(!',x'>ll = IID2 ~(1,x,!)- o 2 ~(1,x',!'>ll KK < K1K4K5 e 2 3 clx-x'l + I!- !'IJ for all x, x' E B1, !•!' E B2• Hence D2F is Lipschitz on bounded sets in Rm x Rn. Also o1F = goF is Lipschitz on bounded sets in Rm x Rn. Therefore so is OF. Theorem (2.1): In the stochastic FOE (II), suppose His Lipschitz and g is a c2 map satisfying the Frobenius Condition. Then the trajectory field {nxt:t E [O,a], n E C(JJRn)} has a version x:n x [O,a] x C(JJRn) + C(JJRn) having the following properties. For any 0 < a < i, there is a set na c n of full P-measure such that, for every wE na, (i)

the map X(w,.,.):[O,a]

X

C(JJRn)

+

C(JJRn) is continuous•

(ii) for every t E [r,a] and n E C(JJRn), X(w,t,n) E Ca(JJRn)i (iii) X(w,.,.):[r,a]

x

C(JJRn)

+

Ca(JJRn) is continuous•

(iv) for each t E (r,a], X(w,t,.):C(JJRn) + Ca(JJRn) is Lipschitz on every bounded set in C(JJRn), with a Lipschitz constant independent oft E [r,a]. ln particular each X(w,t,•) : C(JJRn) + C(JJRn) is a compact map.

~: Suppose H is Lipschitz and g is a c2 map satisfying the Frobenius Condition. Employing a technique of Sussman ([74]) and Doss ([12]), we show that a 121 .

version X:fl x [O,a] x C(J~n) + C(J,Rn) of the trajectory {nxt:t € [O,a], n € C(J,Rn)} of (II) can be constructed, sample-function-wise, by first solving a suitably defined random family "' H(.,.,w):[O,a] x C(J,Rn) +Rn of retarded FOE's, and then deforming their semi-flows into X using the flow F of g parameterized by Brownian time w(t). The random family "' H and the flow F are sufficiently regular to guarantee the required sample function properties of X. Indeed, by Lemma (2.3), we let F:Rm x Rn +Rn be the c2 flow of g, viz: D1F(!_,x)

=

g(F(!_,x)) ( 5)

F(Q_,x)

= X

for all t € Rm, x € Rn. For a;y ~ € C(J,Rm) and n € C(J,Rn), define the map Fo(~,n):J +Rn by [Fo(~.n)](s) = F(~(s),n(s))

for all s € J. For each! € Rm, F(!,·> is a diffeomorphism of Rn; so for any x € Rn the linear map o2F(!,x):Rn +Rn is invertible. Define the 'Brownian motion' w0 :fl + C([-r,a]~m) by w(w)(t) - w(w)(O) w0 (w)(t) = {

0

t € [O,a] t € J

(6)

for a.a. w € fl. Recall that w~:fl + C(J,Rm) is the slice of w0 at t € [O,a], i.e. w~(w) € C(J,Rm) for a.a. w € fl. We now define a random family of retarded FOE's on Rn by letting H:[O,a] x C(J,Rn) x fl +Rn be the map

H( t,_n,w)

=

[D2F(w0 (w)( t) ,n( 0)) ]- 1{H[Fo (w~(w) ,n)]

-!trace (Dg[F(w0 (w)(t),n(O))]og[F(w0 (w)(t),n(O))])} for a.a. w € fl, t € [O,a], n € C(J~n). We shall show that, for a.a. w € fl, there is a map n~(w):[-r,a] +Rn continuous on J and c1 on [O,a] which is a solution of the RFDE: 122

(7)

n~(w)•(t) = H(t,n~t(w),w)

0 < t
}

(8)

n~o(w) = n

For this purpose, we contend that for a.a. w € n the RFDE (t,n) ~ H(t,n,w) is (jointly) continuous and satisfies a Li.pschi.tz. condi.tion in n over bounded sets of C(JJRn), uniformly with respect tot € [O,a]. We also contend that the above map satisfies a linear growth condition of the form I"'H(t,n ,w) I < "'K(w)( 1

lin

+

II c>

for all t € [O,a], n € C(JJRn), where "'K(w) > 0 i.s some constant. First of all, note that for a.a. w € n every term appearing on the right-hand side of (7) is Lipschitz inn over bounded subsets of C(JJRn) uniformly with respect to t € [O,a]. Referring to the proof of Lemma (2.3), let

o , x ,.!) = x,

s

€

R, x

€

Rn, t

€

Rm

}

Suppose g:Rn-+ L(RmJRn) has the same global Li.pschitz constant L1 of H. Then integrating (9) with respect to s gives I
< lx-x•l

+

L11!1

+

(9) >

0 as that

IJ: {g(
J:lcp(s,x,!}-cp(s,x•,!)lds

for all A € [0,1].

Using Gronwall •s lemma and putting A= 1, we get L1 1tl IF(!,x)- F(!,x•)l I
( 10)

for all! € Rm, x,x• € Rn. Now, by the remark following the proof of Lemma (?..3), the map (!,x) ~ D2F(!,x) is Lipschitz on bounded sets in Rm x Rn. But the group property of Lemma (2.3) gives [D 2F(!,x)] -1 = o2F(-!,F(!,x)) for all ! € Rm, x € Rn. As (t,x) ~ F(t,x) is Lipsxhitz in x uniformly with respect tot in -1 a bounded subset of Rm, it follows by composition that the map x ~ [D 2F(t,x)] 123

;s Lipschitz on each bounded subset of :Rn. uniform.ly wi.th respect to ! in a bounded subset of Rm. Moreover, if IIDg II = sup { IIDg(x) II !X ERn}, then from the equation

o2 '(~.x.!)(y)

=y

~

J:

(Dg(,(s.x,!))[D2,(s,x,!)(y)])(!)ds,

x,y E Rn, ! E Rm, we obtain

ID 2'<~.x,!HY>I

< IYI

Apply Gronwall's lemma and

~ IIDglll!l

put~=

J:

ID 2,<s.x,!HY) ds.

~

~.s >

E R,

0.

1 to get for a11 x E :Rn, ! E :Rm

( 11)

Therefore

For the sake of brevity. and unless otherwise indicated, all subsequent 'unqualified' statements involving w may be taken to hold for att wE n0, the set of all those wE n for which w(w) is continuous on [O,a]. Lett E [O,a] and n1,n 2 E C(J,Rn). Then by (tO) I[Fo(w~(w),n 1 )J(s)- [Fo(w~(w),n 2 )J(s)l

=IF(w0 (w)(t+s),n 1(s)) - F(w0 (w)(t+s),n 2(s)) <e

L1 lw0 (w)(t+s) I

ln 1(s)- n2(s)l

for all s E J.

Letting llw0 (w>ll = sup { lw0 (w)(t) I: t E [-r,a]}, one gets L1llw0 (w) II 0 IIFo.(w~(w),n 1 )- Fo(wt(w),n 2>11c < e lln 1-n 2 11 C

(12)

In particular.

By the global Lipschitz properties of H and g, it follows immediately from the last two inequalities that the maps (t,n) ~ H[Fo(w~(w),n)] and 124

(t,n) ~ g[F(w0 (w)(t),n(O))] are globally Lipschitz in n uniformly with respect to t € [O,a]. Furthermore, o2F and Dg are Lipschitz on bounded sets; therefore the maps (t,n) ~ [D2F(w0 (w)(t),n(O))J- 1 , (t,n) ~ Dg[F(w0 (w)(t),n(O))J are continuous and Lipschitz in n over bounded subsets of e(JJRn) uniformly with respect to t € [O,a]. An elementary argument, exploiting the continuity of the maps (t,n) ~ (w~(w),n), F and the trace, shows that (t,n) ~ H(t,n,w) is continuous and Lipschitz in n over bounded subsets of e(JJRn) uniformly with respect to t € [O,a]. To establish the linear growth property of "'H(.,.,w), note that IH(n) I < IH(O) I + L1 lin

II e for all n

€

e(JJRn).

(14)

Also if s € J, n € e(JJRn) and t € [O,a], then L llw0 (w) II I[Fo(w~(w),n)J(s)l < IF(w0 (w)(t+s),O)I + e 1 II nile i.e. L llw0 (w) II IIFo(w~(w),n>ll< IIF(w0 (w)(•),O)II + e 1 II n lie where IIF(w0 (w)(·),O>II =sup {IF(w0 (w)(t),O)I: t

€

( 15)

[-r,a]}.

Using (14) and (15) gives L llw0 (w) II IH[Fo(w~(w),n)JI < IH(O)I + L1 IIF(w0 (w)(·),O)II + L1e 1 llnlle (16) for all t € [O,a], n € e(JJRn). Similarly, by the inequality ( 17)

we get L llw0 (w)lj lg[F{w0 (w)(t) ,n(O) )] I < llg(O) II + L1 IIF(w0 (w)( ·) ,O)II+L 1e 1 In lie ( 18) Finally, use (7), (11)', (16) and (18) to deduce that

125

"' IH(t,n,w) I < "'K(w)(t "" lin

II ), n €. C(JJl">, t E [O,a]

eta)•

where "' K(w)

II 1111 o II Lllwo(w)ll e Dg w (w) max{L 1e 1 , IH(O) I "" L1 IIF(w0 (w)( ·) ,0) II ,

=

L llw0 (w) II ~ llogll ( llg(O) II+ L1 IIF(w0 (w)(.),O) II>.~ llogll L1e 1 }. Thus for a.a. wE g and every n E C(JJRn), the RFDE (8) has a un;que solut;on n~(w) def;ned on [-r,a] and start;ng at n.

Def;ne the random f;eld ~:0

~(w,t,n)

x

[-r,a]

x

C(JJRn)

+

Rn by

F(w0 (w)(t), n~(w)(t))

=

(19)

for a.a. wE o, all t E [-r,a], and all n E C(J,Rn). Note that each x(•,t,n) ;s def;ned on a set of full P-measure ;n g wh;ch ;s independent of "' t and n e.g. 00 • Also s;nce g ;s c2, then so ;s F because of Lemma (2.3). Therefore we can apply Ito•s formula (Elworthy [19], p. 77) to (19) gett;ng, for each f;xed n E C(J,Rn), d~(·,t,n)

=

D 1 F(w0 (·)(t),n~(·)(t))dw0 (.)(t)

+ D 2 F(w0 (·)(t),n~(·)(t))dn~(·)(t) + ~ D~F(w0 (·)(t),n~(.)(t))(dw0 (.)(t), dw0 (·)(t)), t

>

0 (20)

But for a

and 126

>

t

>

0 we have

dw0 (.)(t)

=

dw(·)(t)

dn~(·)(t)

=

"' H(t, n~t(•),·)dt

D~F(w0 (·)(t),n~(·)(t))(dw0 (·)(t),dw0 (·)(t)) =trace D~F(w0 (·)(t),n~(·)(t))dt. so (20) yields

"' dx(.,t,n) = D1F(wo(.)(t), n~(·)(t))dw(.)(t)

"'

~ o2 F(w0 (.)(t),n~(·)(t)){H(t,n~t(•),·)dt +

i trace D~F(w (•)(t),n~(·)(t))dt, 0

a> t > 0.

(21)

Now in (5), keep x fixed and differentiate with respect to! € Rm; then

D~F(!,x) = Dg(F(!,x)) o D1F(!,x) = Og(F(!,x))og(F(!,x)) for all!

€

Rm, x

€

Rn.

In particular,

trace D~F(w0 (w)(t),n~(w)(t)) = Dg[F(w0 (w)(t), n~(w)(t))] o g[F(w0 (w)(t),n~(w)(t))] =

Dg[x(w,t,n)] o g[x(w,t,n)] "' "'

for t € [O,a], and a.a. w €

(22)

n.

From> ( 7) we see that o 2 F(w0 (w)(t),n~(w)(t)){H(t,n~t(w),w)} = H[F o (w~(w),n~t(w))]

-i trace (Dg[F(w0 (w)(t),n~(w)(t))] o g[F(w0 (w)(t),n~(w)(t))]) =

"' H[xt(w,•,n)]

1 -~trace

"' "' Dg[x(w,t,n)] o g[x(w,t,n)], 0 < t
using the fact that (24)

a.s., for all t € [O,a], n € C(JJRn). From (21), (22) and (23), it follows that, for each n € C(JJRn), ~(.,•,n) 127

is a solution version of the stochastic FOE ~

~

dx(•,t,n) = H[xt(•,•,n)]dt

~

+

g[x(•,t,n)]dw(.)(t), 0 < t
(25)

Moreover,

~(w,s,n) = F(!,nt(w)(s)) = F(!,n(s)) = n(s) for all s € J, a.a. w € o. Therefore ~(.,•,n) is the unique solution of the stochastic FOE (II), and the random field X:O x [O,a] x C(JJR"> + C(JJRn) X(w,t,n)

=

.

~

n

xt(w,·,n), a.a. w € O, t € [O,a], n € C(JJR )

is a version of the trajectory {nxt:t € [O,a], n € C(JJRn)} of (II), viz. x(.,t,n) = nxt a.s., for all t € [O,a], n € C(JJRn). We now verify that X has the stated regularity properties. is continuous on [-r,a]; hence the map

For a.a. w €

w0 (w)

t

w~(w)

>

is continuous for a.a. w € o. Also by continuous dependence of solutions to RFOE's on the initial condition, it is clear that the semi-flow [O,a]

x

C(JJRn) --> C(JJRn)

(t,n) is continuous for a.a. w € 0 (Hale [26]). Therefore, composing with the continuous f)ow F, it is easy to see that, for a.a. w € 0, X(w,·,·): [O,a]

x

C(JJRn) ~ C(JJRn)

(t,n) is also continuous. Let Oa = {w:w € 0, w(w) € cfl([O,a]JRm)}. Then P(Oa) = 1 since 0 < a < l. For· the rest of this proof, fix w € Oa. Let r < t < a and n € C(JJRn). 128

o,

since w0 (w) € Ca([-r,a],Rm), then w~(w) € Ca(J,Rm). But F:Rm x R" ~Rn is c1, and since n~t(w) is c1 on J, it follows that the composite map X{w,t,n) , fo(W~{w), n~t{w)) must be in Ca{J,Rn). This proves {ii) of the theorem. we now go on to prove assertion {iii) of the theorem. the continuity of the map

=

First we establish

[r,a] x C(J,Rn) {26)

(t,n)

~.-~

Fix anyn€ C{J,Rn) and let r < t 1 < t 2
Then

[n~t {w) - n~t {w)]{s2) - [n~t {w) - n~t {w)](s1) 2

1

2

= l{n~{w){t 2 +s 2 )

1

- n~{w){t 2 +s 1 )}-{n~{w){t 1 +s 2 )~n~{w)(t 1 +s 1 )}1

"' IH{s+t 2, n~s+t {w),w)- "'H{s+t 1 , n~s+t {w),w)lds 2

2

s

+

Js12"'IH{s+t 1, n~s+t2{w),w) - "'H{s+t 1, n~s+t 1{w),w)lds

{27)

Now by continuity of the slicing"'map the set {n~t{w):t € [O,aJ} is compact in C{J,Rn). Furthermore, each map H{t,•,w) is Lipschitz in each bounded set of C{J,Rn) with a Lipschitz constant independent of t € [O,a]; in particular the map C{[O,a],Rn)

"'H( •• ~,w) is continuous.

"' n~t{w),w);t € [O,a]} is compact in Hence the set {H{·,

129

C([O,a]JRn), and therefore equicontinuous due to Ascoli's theorem i.e. given € > 0, there is a o > 0, depending only on n and w, such that, if lt 1-t 2 1 < o , then "' IH(s+t 2, n~s+t (w),w) - "'H(s+t 1, n~s+t (w),w) I 2

for all s

J.

€

2

<

€

Indeed because of the uniform continuity of

[O,a]

C(JJRn)

t

n~t(w)

(28)

o is so chosen that II n~s+t 1 < w) _n~s+t (w)ll C < € all sEJ, whenever lt 1-t 2 1 0 the Lipschitz2 constant of

we may suppose that the above ~or

H(t,•,w) on the compact set {n~t(w):t € [O,a]}. lt1-t21 < 0 ,

Then (27) gives, for

l[n~t (w) - n~t (w)](s2) - [n~t (w) - n~t(w)](s1) I 2 <

1

€1s1-s21 + ~1(w)

2

1

52

J un~s+t 51

(w)2

n~s+t

(w)llcds 1

Therefore

<

€ + €(1 + "'L1(w))r 1-a.

whenever t 1,t 2 € [r,a] and 1t 1-t 2 1 <

o.

Thus for each n € C(JJRn) the map

[r,aJ _ ___...,. (29) t

1----~

is continuous. Let n1,n 2 € C(JJRn), r < t 1 < t 2
and the continu;ty of the map (29) with n = n2, i.t is easy to see that the maP (26) will be (jointly) continuous provided we pro'le that for each fixed t € [r,a] the map C(J,Rn) --~ (30)

n

is continuous. ln fact we shall prove next that the above map is Lipschitz on bounded sets of C(J,Rn) (with Lipschitz. constants independent of t € [r,a]). Let n € C(J,Rn). Then lnt(w)(t)l
I

"' II C + t0 IH(u,ntu(w),w)ldu, t

< lin II C + "'K(w)t

+

€ [O,a]

"'K(w) It0

due to the linear growth property (18)'. Therefore

II ntt(w) lie.< lin II c

+

K(w)t

+

K(w)

J:

llntu(w) II cdu, t E [O,a]

Applying Gronwall's inequality, we obtain

"' II ntt(w) II C < [lin II C + K(w)t]eK(w)t, t

€

[O,a], n

€

C(J,Rn)

(31)

Let B c C(J,Rn) be any bounded set. By (31) the set ~(w) ={ntt(w):t € [O,a], n € C(J,R")} is also bounded in C(J,Rn). Let L2(w) > 0 be the Lipschitz constant of H(t,·,w) on B(w). Take any n1.n 2 € B, t € [r,a] and -r < s 1 < s2 < 0. Then l[n 1tt(w)- n2tt(w)](s 2)- [n 1tt(w)- n2tt(w)](s 1>1

(32)

131

for all t' € [O,a]. Therefore (32) g;ves

l[n1~t(w)-n2~t(w)](s2)-[n1~t(w)-n2~t(w)](s1) I

"'L 2 (w)~

< lln 1-n2 11c e

L2(wl-ls 1-s 2 1

(34)

Th;s last ;nequal;ty clearly shows that the map (30) ;s L;psch;tz on B un;formly w;th respect to t € [r,a]. Now w € na, so the map [O,a]

-~

--+:. w~(w)

t

1-1

;s cont;nuous (Lemma (2.1)). [r,a]

By cont;nu;ty of (26), the map

x C(J,R") ~ Ca(J,Rm) x Ca(J,R") ;; Ca(J,Rm+n)

(t,n)

~

(w~(w), n~t(w))

;s also cont;nuous. Thus, compos;ng w;th the cont;nuous map

(~,n)

132

Fo(~,n)

(36)

__,

(Lemma (3.2)), one gets the continuity of the sample function X(w, ••• )~[r,a] x C(JJRn)

ca(J.Rn>" Fo(w~(w), ntt(w)).

(t,nJ

To prove the final assertion (iv) of the theorem, fix t E [r,a] and compose the isometric embedding --:>

Ca(J,.Rm)

x

Ca(J,.Rn)

(w~(w),n)

n

with the map (30), deducing that the map C(JJRn) --~ Ca( J ,.Rm)

x

Ca( J ,.Rn)

n a-----> (w~(w), ntt(w))

is Lipschitz on each bounded set in C(J,.Rn) with Lipschitz constant independent of t E [r,a]. Applying Lenma (3.2) once more shows that X(w,t,• ):.C(J.Rn) -+ ca(J,.Rn) is Li.pschitz. on every bounded set in C(J.Rn), with Lipschitz constant independent of t E [r,a]. The compactness of X(w,t,·):C(J.Rn)-+ C(J,.Rn) is then a consequence of the last statement together with the compactness of the embedding ca(JJRn) ~> C(JJRn) (viz. Asco 1i •s Theorem) • a. The reader may easily check that the construction in the above proof still works when the drift coefficient is time-dependent i.e. for the stochastic FOE dx(t)

H1(t,xt)dt + g(x(t))dw(t) 0 < t
f_orollary (2.1.1): Suppose g is a c2 Lipschitz map satisfying (Fr), and the map (t,n) f-> H1(t,n) in the stochastic FOE (Ill) is continuous, Lipschitz in n over bounded subsets of C(JJRn) uniformly with respect to t E [O,a]. Assume also that H1 satisfies a linear growth condition: 133

•

for all t• E [O,a]. Therefore (32) gives l[n 1tt(w)-n 2tt(w)](s 2 )-[n 1tt(w)-n2tt(w)](s 1)1

"'L 2 (w)~

< II n 1-n2 11 c e

l 2 (wJ-Is 1-s 2 1

(34)

This last inequality clearly shows that the map (30) is Lipschitz on B uniformly with respect to t E [r,a]. Now w E ga' so the map

--+,

t

t-1

w~(w)

is continuous (Lemma (2.1)). [r,a]

By continuity of (26), the map

x C(J,Rn) ~ Ca(J,Rm) x Ca(J,Rn) ;; Ca(J,Rm+n)

(t,n)

,

(w~(w), ntt(w))

is also continuous. Thus, composing with the continuous map

(t,n)

132

Fo(t,n)

(36)

(Lemma (3.2)), one gets the continuity of the sample function X(w, ••• )~[r,a] x C{JJRn)

__,

Ca{J,Rn). Fo(w~{w), n~t{w)).

{t,n)

To prove the final assertion {iv) of the theorem, fix t the isometric embedding

n

1-----i»

€

[r,a] and compose

{w~{w) ,n)

with the map {30), deducing that the map C{JJRn) --~ Ca{J,Rm)

x

Ca{J,Rn)

n ~----i> {w~{w), n~t{w))

is Lipschitz on each bounded set in C{J,Rn) with Lipschitz constant independent of t € [r,a]. Applying Lemma {3.2) once more shows that X{w,t,.):.C{J;Rn) + ca{J,Rn) is Li.pschit:z. on every bounded set in C{J;Rn), with Lipschitz constant independent of t € [r,a]. The compactness of X(w,t,·):C{J;Rn) + C{J,Rn) is then a consequence of the last statement together with the compactness of the embedding Ca{JJRn) ~> C{JJRn) {viz. Ascoli's Theorem). o The reader may easily check that the construction in the above proof still works when the drift coefficient is time-dependent i.e. for the stochastic FOE dx{t) x0 with H1:[O,a]

= H1{t,xt)dt +

=

n E C{JJRn)

x

C{JJRn)

+

g{x{t))dw{t)

Rn, g:Rn

0

< t
+

L{Rm JRn) given maps.

(III) Indeed one has

Corollary {2.1.1): Suppose g is a c2 Lipschitz map satisfying {Fr), and the map {t,n) ~> H1{t,n) in the stochastic FOE (III) is continuous, Lipschitz in n over bounded subsets of C(JJRn) uniformly with respect to t € [O,a]. Assume also that H1 satisfies a linear growth condition: 133

IH 1(t,n)l < K(1.., lin II ), n e: C(J,Rn), t e: [O,a],

.

(37)

c

for some K > 0. Then all the conclusions of Theorem (2.1) hold for (III). Unfortunately, ;t ;s not poss;ble to deal w;th t;me-dependent d;ffus;ons as above, except perhaps ;n the followtng rather spedal case: Corollary (2.1.2):

In the stochast;c FOE

dx(t) = H1(t,xt)dt .., g1(t,x(t))dw(t),

0 < t
x0 = n e: C(J,Rn) let H1:[0,a] x C(J,Rn) +Rn sat;sfy the cond;t;ons of Corollary (2.1.1) and 91 :[O,a] X Rn + L(Rm ,Rn) have a c2 extens;on 91 :R X Rn + L(Rm ,Rn) w;th Dg1 bounded and fulf;11;ng the Froben;us cond;t;on

for all t e: R, x e: Rn, v1,v 2 e: Rm. Then the trajectory f;eld {nxt:..t e: [O,a], n e: C(J,Rn)} of (IV) has a v.erston X:..nx[O,a] x C(J,Rn) +C(J,Rn) wtth all the propert;es stated ;n Theorem (2. t). Proof: To el;m;nate expl;c;t t;me-dependence ;n the d;ffus;on coeff;c;ent of (IV) we borrow the follow;ng techn;que from determ;n;st;c OoE•s. Embed (IV) ;n a stochast;c FOE of type (Ill) but def;ned on R x Rn, v;z. dy(t) = "'H(t,yt)dt

Yo

+

"'g(y(t))dw(t)

(O,n) e: C(JJR

=

Indeed let p2:C(J,R ject;on

x

x

(I II)

Rn)

R0 ) + C(J,Rn) be the natural cont;nuous l;near pro-

.

n

p2(y,n) =n, y e: C(J,R), n e: C(J,R ). Def;ne H:[O,a] ~

H(t,~)

x

C(J,R

=

(1,H 1 (t,p 2 (~))),

x

Rn) + R

"' g(t,x) = (O,g- 1(t,x)),

134

x

Rn, g:R

x

.

t e: [O,a],

Rn + L(Rm ,R ~

t e: R, x e: Rn•

e: C(J,R

x

x

R0 ) by n

R );

By virtue of (38), it is easily seen that g and~ satisfy all the conditions of Corollary (2.1.1). Hence there is a continuous version v:n X [O,a] X C(J,R X Rn) ~ C(JJR X Rn) of the trajectory field {E:yt:t € [O,a], E: € C(J,R X Rn)} for cdi.) which satisfies Theorem (2.1). Defi.ne x:.n X [O,a] X C(J,Rn) ~ C(J,Rn) by X(w,t,n) = p2[Y(w,t,(O,n))], w € Oa (0
n €

where nx is the unique solution of (IV). Then, for t € (O,a), dy(t)

=

(dt, dnx(t))

=

(dt, H1(t,nxt)dt

+

g1(t,nx(t))dw(t))

= (1,H 1(t,p 2(yt)))dt + (O,g 1(y(t)))dw(t) =

"'H(t,yt)dt ~

+ "'~ g(y(t))dw(t)

i.e. y is a solution of (III) and by uniqueness one gets a.s. y = (O,n)y, the "' at (O,n) € C(J,R x Rn ). Therefore Y(·,t,(O,n)) = Yt ~ a.s. solution of (III) and so X(•,t,n) = p2(yt) = nxt a.s. fort € [O,a], n € C(J,Rn). This proves our claim, and it is then easily checked that X is the required version satisfying Theorem (2.1). o Remark Condition (38) is apparently rather strong; it is satisfied by c2-limits of mappings [O,a] x Rn ~ L(Rm ,Rn) of the form (t,x) ~---+ A(t)g(x) where A:[O,a] ~R is such that A(O) ~ 0 and g:Rn ~ L(Rm,Rn) is of Frobenius type. It is not known to me whether this condition, or the original Frobenius condition for that matter, can be relaxed with Theorem (2.1) still true. However, see the next section on delayed diffusions (§3), and also Corollaries (2.1.3) and (2.1.4) below. 135

Corollary (2.1.3h ln the stochastic FOE (U.l) suppose H1 and g satisfy all the conditions of Corollary (2.1.1), with x~n x [O,a] x C(JJRn) + C(J,Rn) the version of the trajectory field of (Ill) obtained therein. Assume also that H1 has a partial derivative o2H1(t,n) i.n the second variable such that (t,n) ---> o2H1(t,n) is continuous. Then for all w € na (0
n

X(w, t,n)

is c1• Let its derivative be o3x(w,t,n), and for any~ the process a:n x [-r,a] +Rn by o 3 x(w,t,n)(~)(O)

€

C(J,Rn) defi.ne

0< t < a

a(w,t) = { ~(t)

t

€

J,

w €.

na

Then a € £0 (n,C([-r,a],Rn)) and is the unique solution of the linear stochastic FOE da(t)

=

o2H1(t,X(.,t,n))(at)dt

~

Og(X(.,t,n)(O))(a(t))dw(t) 0 < t
for any n, ~ € C(J,Rn).

In particular, for a fixed n € C(J,Rn), the map is a continuous version of the trajectory field of (0 III)i and for each w € na, t € [r,a], the map C(J,Rn) 3 ~ ~ o 3 x(w,t,n)(~) € C(JJR") is compact linear. (w,t,~)

~

o 3 x(w,t,n)(~)

Proof: For any n € C(J,Rn) and w € Qa(O
=

H(t, n~t(w),w)

n~o(w)

=

n

where H:[O,a]

136

x

C(J,Rn)

x

0 < t


n +Rn is defined by

}

(39)

H(t,n,w)

=

[D 2 F(w0 (w)(t),n(O))J- 1 {H 1 [t,Fo(w~(w),n)J

-i trace (Dg[F(w (w)(O) ,n(O) )]og[F(w (w)(t) ,n(O) )]) 0

0

}(40)

and F, w0 are as in the proof of Theorem (2.1) (cf. (5),(6) and (7)), t E [O,a], n E C(JJRn), wE Qa· Since F is c1 , it is easy to see that the map

n

1------'l>

Fo(w~(w) ,n)

is also c1, for each fixed t E [O,a]. Now g is c2 and H1 has a continuous partial derivative with respect to the second variable, so it follows from (40) that for fixed w the map (t,n) ~ H(t,n,w) has a continuous partial derivative with respect to n € C(JJRn). We may then apply Theorem (11.3.2) with Q = {w}, z(t) = t (or Hale [26], p. 46), to deduce that the map C(JJRn) 3 n ~ n~t(w) E C(JJRn) is c1 for each t,w. Noting that X(w,t,n) = Fo(w~(w), n~t(w)) it is clear by composition that each map X(w,t,.) is c1 on C(JJRn). Fix n, ~ E C(JJRn) and let 0 ~ h E R. Because X is a version of the trajectory field for (III), one has d[X(.,t,n+h~)(O)J = H 1 (t,x(.,t,n+h~))dt

d[X(•,t,n)(O)] for 0 < t < a.

=

+

g(X(·,t,n+h~)(O))dw(t),

H1(t,X(•,t,n))dt + g(X(•,t,n)(O))dw(t),

Subtracting and dividing by h, we get

d[k{X(·,t,n+h~)(O) - x(.,t,n)(O)}J 1

= ~H 1 (t,x(.,t,n+h~))-H 1 (t,X(·,t,n))Jdt

+ ~[g(X(•,t,n+h~)(O))-g(X(•,t,n)(O))]dw(t),O < t
(41) h~

a

137

lim ~H 1 (t,X(w,t,n+h~)) - H1(t,X(w,t,n))J

h~

exists for all wE na. Furthermore, H1(t,·) and X(w,t,•) are Lipschitz on bounded sets of C(JJRn) uniformly with respect to t € [O,a]; thus for each wE na the set {~H 1 (t,X(w,t,n+h~)) - H1[t,X(w,t,n))]:t E [O,a], h ; O} is bounded. Therefore by the dominated convergence theorem, we have

lim~ [H 1 (t,X(w,t,n+h~)) - H1(t,X(w,t,n))]dt

h~

In order to deal with the second term on the right hand side of (41) ash+ 0, it is enough to show that lim Jt Eh(u)dw(u) = 0

h~

(43)

0

in probability (uniformly fortE [O,a]), where Eh(u) = ~ [g(X(.,u,n+h~)(O)) - g(X(·,u,n)(O))-Dg(X(.,u,n)(O))(a(u))l (44)

for all u E [O,a]. To prove (43) write u E

[O,a], h ; 0

with Ah(u)

=

h1

[g(X(·,u,n+h~)(O))

-

- g(X(·,u,n)(O))

Dg(X(•,u,n)(O))(X(•,u,n+h~)(O)

- X(·,u,n)(O)}]

and Bh(u) = Dg(X(·,u,n)(O)){~ [X(.,u,n+h~)(O) - X(•,u,n)(O}] - a(u)}. Fix wE na (0
is compact in Rn. 138

{X(w,u,n)(O) +

AX(w,u,n+h~)(O):

Since g is c2, let

A,h E [0,1], u E [O,aJ}

(45)

Also by Taylor's formula for g, one gets sup IID 2 g(~) II IX(w,u,n-t.ht)(O)-X(w,u,n)(O) 12 \l€S(w,n,F;) L1(w)IX(w,u,n-t.ht)(O)- X(w,u,n)(O)I 2 •

IAh(u)(w)l< ~-~ =

t

Suppose L2(w) > 0 is the uniforml lipschit~ constant of X(w,u,•) over any bounded set in C(JJRn) containing the segment {n-t.ht! h € [0,1]}. Then IAh(u)(w)l < L1(w)L 2(w)hlltll 2 ,

h € (O,t], w € fla,u€[0,a] (46)

and one has

~~

J:

l~(u)(w)l 2 du

(47)

= 0 for all w € na.

Furthermore, for each w € na, la(u)(w)l =lim I~ [X(w,u,n+ht)(O)- X(w,u,n)(O)JI h-+0

< lim h-+0

t

llhtll =

L2 (w)

L2 (w)

lit II ,

u € [O,a],

i.e.

la(u)(w)l < L2Cw>lltll

for all u € [O,a].

Let

L3(w) = sup
and we deduce also that lim Ja 1Bh(u)(w)l 2du = 0 h-+0

w € Qa

(48)

0

Clearly (47) and (48) yield lim Ja 1Eh(u)(w)l 2du = 0 for all w € Oa,so that h-- 0

limP( sup 1Jt Eh(u)dw(u)l t€[0,a] 0

> €)

= 0.

h~

139

(Theorem (I.8.5); or Friedman [22], Vol. I. p. 70).

In other words,

lim~ [g(X(•,t,n+h~)(O)) - g(X(·,t,n)(O))]dw(t)

h-+0

=

Dg(X(•,t,n)(O))(a(t))dw(t)

almost surely, and a is indeed a solution of the linearized system (D III). To see that a is the only solution of (D III) in .c 0 (n,C([-r,a]JRn)) we proceed as below. Suppose for the time being that D2H1 is bounded as a map [O,a] x C(JJRn) 3 (t,n)-+ o2H1(t,n) e: L(C(JJRnlJR") by a number IID2H1 11 > 0. Represent the coefficients of (D III) as maps H:n x [O,a] x C(JJRn) -+Rn, g:n x [O,a] x C(JJRn)-+ L(RmJRn) defined by ~

~

H(w,t,~) = D 2 H 1 (t,X(w,t,n))(~). g(w,t,~) = Dg(X(w,t,n)(O))(~(O))

for we: na, t e: [O,a], ~ e: C(JJRn), with n e: C(JJRn) fixed. a,.a2 e: .c 2(n,c(JJRn)), then ~ J)H(w,t,B 1(w))-

••

If

2 2 2 H(w,t,B2(w))l dP(w) < IID 2H1 11 IIB 1-B2 11 2

~

.c cn,c)

2

n

~

for all t e: [O,a]. Also for any Be: .c (n,c(JJR );Ft) the map w•-+ H(w,t,B(w)) is Ft-measurable because it is almost surely a limit as h-+ 0 of the Ft~asurable maps w ~ ~ [H(X(w,t,n) + hB(w)) - H(X(w,t,n))]. The map g satisfies similar conditions. Hence Condition (E) of Chapter II is satisfied and by Theorem (II.2.1), equation (D III) has a un;que solution in .c 2(n,C([-r,a]JRn)). Assume now that o2H1 is continuous but not necessarily bounded on [O,a] x C(JJRn). Take any two (Ft)O
~

sup llog(X(w,u,n)(O)) II< k, ~

sup 11a(w)(u)l
12a(w)(u)l

< k}.

O
Since o2H1 is continuous and X(.,u,n), 1a(.)(u), 2a(.)(u) are all Ft-measurable 140

for 0 < u < t, then g~ € Ft for each k > 0 and t E [O,a]. Moreover, for a.a. w each of the expressions sup IID2H1(u,X(w,u,n)) 11.

O
sup IIDg(X(w,u,n)(O)) IL

O
is finite; and so the Ft-measurable set k

00

gt = u ~ k=1 has full P-measure for each t Then for a.a. w € ga we have 1a(w)(t) - 2a(w)(t) = + (w)

J:

€

[O,a].

Fix any k

>

0 and let t

€

[O,r].

J: o

2H1(u,X(w,u,n))[ 1au(w) - 2au(w)]du

Dg(X(•,u,n)(0))( 1a(•)(u) - 2a(·)(u))dw(u)

and so xg.k<w>

11 1at<w>- 2at<w>

II~

t

where xgk is the characteristic function of g~·

If 0 < t• < t, g~

=~· and

t

X k is Ft.-measurable. In particular, by Theorem (I.8.3)(i) (or Friedman gt [22], Lemma (2.9), p. 65), one has t+s X k(·) Dg(X(.,u,n)(0))( 1a(.)(u) - 2a(·)(u))dw(u) gt 0 t+s 1 2 (50) = x k(.)Dg(X(.,u,n)(O))( a(.)(u) - a(·)(u))dw(u) 0 gt

J

J

a.s. on g, for all s € [-t,O]. Thus if we take expectations in (49) and 141

apply the t1arti.ngale inequality (Theorem (1..8.5)) we get K1

Ink

11 1at(w)

- 2at(w) 11 2 dP(w) <2k 2

t

+ 2K 1

r 0

< 2k 2

I: {I sf

I:

>

0 such that

{Ink 11 1au(w)- 2au{w) 11 2 dP(w) }du t

IQI< IIDg(X(w,u,n)(O)) 11 2 11a(w)(u)- 2a(w)(u) 12dP{w)du t

11 1au{w)- 2au(w) 11 2 dP(w)

}du

u

+ 2K1k2J:Id<

11a(w)(u)-

2a(w)(u)J 2dP(w)du

(51)

u

because ntk c gk for all 0 < u < t. - u setting f(t) =

J k 11 1at(w)

Now define the function f:[O,r] +~by

- 2at(w) 11 2 dP(w), t

[O,r],

€

Qt

and deduce from (51) that f(t) < 2k 2(1+K 1) It f(u)du

for all t

€

[O,r].

0

This clearly implies that

Jk

11 1at(w)

- 2at(w) 11 2 dP(w) = f(t) = 0 for all

Qt t € [O,r]. Hence 1at(w) = 2at{w) for a.a. w € n~. ~As this holds for every k > 0, we obtain 1at(w) = 2at(w) for a.a. w € Qt = u Q~ and P(Qt) = 1. In k=1 1 2 particular, take t = r to obtain a set n* c Qr such that a(w)(t) = a(w)(t) for all w € n*, 0 < t < r, with P(Q*) = 1. On the interval [r,2r] a similar argument yields 1a(w)(t) = 2a(w)(t) for a.a. wand all r < t < 2r. Therefore 1a(w) = 2a(w) for a.a. w € Qa• To see the compactness of the map ~ ~ o 3 x(w,t,n)(~) for fixed w € Qa' t € [r,a], n € C{JJRn), note the compactness of the c1 map X(w,t,•) and the fact that the convergence

D3X(w,t,n)(~) = lim~ [X{w,t,n+h~) - X(w,t,n)] h+O

is uniform for~ in bounded subsets of C{JJRn).

142

c

Remarks (2.2) (i) lf, in addition to the hypotheses of Corollary (2.1.l, we assume that o2H1 is bounded, then the solution a of (0 1.1.1.) belongs to icn,C([ -r,a] ,Rn)). (ii) The localization technique used above to establish uniqueness of solutions to (D Ul) in .c 0 (n,C([-r,a]~n)) is adapted from McShane ([53], pp. 154160). The reader may check that it can further be used to prove existence and uniqueness in .c 0 (n,C([ -r,a] ,Rn)) of soluti.ons to the stochastic FOE :t:

dx{w)(t) = H(w,t,xt(w))dt x0

=

~

%

G(w,t,xt(w))dw(t) (V)

e E .c 0 (n,c(J,Rn)) :t:

%

with random coefficients H~n x [O,a] x C(J,Rn) -+Rn ,Ga"~><[O,a]xC(J,Rn}-+L(Rm ,Rn) satisfying the conditions (a) (b)

:t:

%

For a.a. wEn, H(w,.lt.) and G(w,.,.) are conti.nuous;. IIf(., t,n 1 ) - ~( ., t,n 2 > I P ( sup sup < oo) = 1, tE[O,a] n1n2EC lln 1 - n2 11 n1 # n2

sup l~(.,t,O) I < oo) = 1, and a similar condition for~tE[O,a] % % (c) For each t E [O,a], H(·,t,.) and G(·,t,.) are Ft 0 Borel C(J,Rn)-measurable. Indeed, the reader may note that in (V) one can replace w by a semimartingale z:n-+ C([O,a],Rm) fulfilling the Conditions (E) of Chapter II; P (

(iii) lt follows from (ii) above that, under the conditions of Theorem (2.1), X(.,t,n) E .c 2(n,C(JJRn)) fortE [O,a], n E C(JJRn). Alternatively this fact may be proved more directly by looking at the definition

of the version X using the notation of the proof of Theorem (2.1). Indeed the process {el1(wo(.)(t) l:t E [-r,a]} is a submartingale because the function A. ~ e 1>.. I is convex; so by Ooob' s inequality L 1 Iw0 ( • )( t ) I L 1 Iw0 ( • )( a ) I L llw0 ( ·) II e <Ee
143

L 1llw0 (w)

II

the random variable w ~ e belongs to .cP(n,R) for all p > 1. Similarly by the growth properties of F, all the maps below w ,...____> F

o

(w~(w) ,n)

w ~ "' H(t,n,w)

are of class .cP for any p > 1, for fixed t € [O,a], n € C(J,Rn). Therefore X(•,t,n) € .cP(n,C(J,Rn)) for all p > 1. Just as for stochastic ODE•s (Sussman [74], Doss [12]) the Frobenius Condition is not required when the noise w is one-dimensional. Corollary (2.1.4): If w is one-dimensional Brownian motion (m = 1), the Frobenius Conditions (Fr) and (38) may be dropped in Theorem (2.1) and its corollaries. Proof:

Note that every c1 map g~n ~ L(R,Rn) satisfies the identity

§3. Delayed Diffusion: An Example of Erratic

Behaviou~

In the previous section (§2) it was established, in some detail, that for o~inary diffUsion coefficients a stochastic FOE may admit a nice version of its trajectory field. It is our intention in the present section to show that such continuous versions of the trajectory random field neve~ exist if the diffusion coefficient depends on the past. To see this let us consider the one-dimensional linea~ stochastic delay diffe~ential equation (SDDE): dx(t) = x(t-1)dw(t)

0
(VI)

x0 = n E C([-1,0l.R) Denote by C the Banach space C([-1,0],R), given the usual supremum norm, and by {nxt:t € [0,1], n € C} the trajectory field of (VI) generated by the unique 144

solution nx € £ 2(n,C([-1,1JJR)) of (VI) through n € C. Both nx and the onedimensional Brownian motion w in (VI) are carried by a complete filtered probability space (n,F,(Ft)O
"x(t)

= {

n(O)

+

J:

n(u-1)dw(u)

n(t) a.s. on

0< t < 1

t

€

[-1,0)

n.

Suppose, if possible, that the trajectory field {nxt:t € [0,1], n € C} admits a version x:n x [0,1] x C ~ C with X(w,t 0 ,.) continuous (or even locally bounded) for a.a. w € n and a fixed t 0 € (0,1]. Without loss of generality, we may take t 0 = 1 and suppose that X(w,1,·) is continuous for a.a. w € n. Define the process v:n X c ~R by Y(w,n) = X(w,1,n)(O) - n(O)

a.a. w €

n, n € c.

(2)

Clearly, Y(·,n) = nx(1) - n(O) =

J:

n(u-1)dw(u)

a.s.

Since the Brownian motion {w(u):u € [0,1]} is a Gaussian system (§I.8(A), §I.8(B); or Hida [31], Definition (1.6), p. 31), then so are the partial sums defining the stochastic integral J~ n(u-1)dw(u). But these converge in £ 2 sense to the Gaussian r.v. Y(.,n) for each n € C (Theorem (I.8.1)(ii), Hida [31], Proposition 1.11, p. 34). Therefore {Y(·,n): n € C} is a Gaussian system (cf. Dudley [13]). It is well-known that Gaussian processes parametriized by function spaces never allow continuous (or even locally bounded) sample functions on the whole parameter space; (see Dudley [13]). To see this in a rather more elementary fashion we proceed as follows. Note first that if n € c1([-1,0]JR), then we can integrate by parts to obtain Y(w,n) = n(O)w(w)( 1) - n( -1)w{w)(O) -

J:

w(w)(u)n' (u-1)du

(3)

for all w € n0 , where n0 c n is a set of full P-measure independent of nand such that w(w) is continuous for all w € n0 • Therefore for each w € n0 , Y(w,.): c1([-1,0]JR) ~JR is a linear map. Now these maps are, for a.a. w€fl 0 , discontinuous everywhere in the supremum norm topology of C([-1,0],R). 145

Indeed we contend that for any £ P{w:w

E

>

0,

Q0 , sup [IY(w,n)l: n E C1([-1,0],R), lin IIC < £] < oo} = 0 (4)

So define the sequence {nk};= 1 in C1([-1,0],R) by nk(s) =£sin 2nks for all s E [-1,0], k > 1. Clearly llnkllc =£for all k > 1. Note also that each Gaussian random variable v(.,nk) has mean zero and variance

a~=

EIY(.,nk) 12 = EIJ' 1 nk(u-1)dw(u) 12 = J0 0

Jo

= £2

lnk(s)l 2ds

-1

. 2nks] 2ds = ~1 £2 , k > 1. [s1n

-1

In fact for any integers k,h

>

0 one has

E v(.,nk) Y(•,nh) = E[J: nk(u-1)dw(u) J: nh(u-1)dw(u)] = J: nh(u-1)nk(u-1)du =

Jo

-1

h

k

1

2

n (s)n (s)ds = ~ £ okh

where okh is the Kronecker Delta (Friedman [22], Theorem (4.2.5), p. 63). This implies that the Gaussian random variables {y(.,nk)};= 1 are mutually independent (Theorem (I.8.1); Hida [31]). We then con~ider the set 00

00

{w:w E f20 , sup IY(w,nk)l 1 N=1 k=1 obtaining

°

P{w:w E f20 , sup IY(w,n k>1 I< N}).(S) k>1 N-+00 k=1 00

°

But by independence P(

~ {w:wEf20 ,jY(w,nk)l
for each integer N > 0. 146

Now

(6)

P{w:w € n0 ,1 Y{w,nk) I < N} -- I21T t t • ok

J

N -N

=-

/if

dx

JN/E

-N/E

=ln.

e

-x2

dx for all N > 0.

JN

2

2

e-y /2ok dy

-N

e

2 2 -y /E

1 JN/E dy =e lif -N/E

Then PN,E <

-i dx.

because

Using {5) and {6) we deduce that

P{w:w € f2 0 , sup IY{w,nk)l <~}=lim limP: E = 0. 1<>1

N-+00 M-+00

•

In fact P{w:w € n, sup IY{w,nk)l < N} = 0 for each N > 0. 0

1<>1

These clearly

imply that {4) holds for any E > 0, and so the trajectory field {nxt:t € [0,1], n € C} of {VI) cannot admit a locally bounded version. Remarks {3.1) {i) The above observations constitute a significant departure from what is known to hold for stochastic ODE's {r = 0) on Euclidean space or finitedimensional manifolds where sufficiently smooth versions of the trajectory field always exist via Kolmogorov-Totoki theorem. Indeed, for such systems, the sample functions are almost all diffeomorphisms of the underlying state space onto itself {Kunita [45], [46], Elworthy [19], Malliavin [51]}. It is perhaps worth noting here that the Kolmogorov-Totoki theorem {Elworthy [19], p. 17) does not hold for the field {Y{•,n): n € C} of {VI) which is parametrized by the infinite-dimensional space C even though it satisfies

for all n1,n 2 € c. {ii) One can always~enla~ge the filtered probability space {fl,F,{Ft)O
the evaluations P( t , n ) :.

"' S)

____..

________,

c

The transition probabtlities {p(O,n,t,•): t E [0,1], n E C} of the stochastic DOE (\ll) give the product probability measure "'P = II p(O,n,t,·) on the t€(0, tJ

"' "' ne:C measurable space (S'l,F). As we ha~e already seen in Chapter lV, the map n 1-> p(O,n,t,.) is (uniformly) continuous i.nto the narrow (weak *) topology ·of probabili.ty measures on C. So i.f we set "'X(w,t,n) = w(t,n), "' "' "'wE "'n, "' -t t E [0,1], n E C,th~ because of the fact that Po p(t,n) = p(O,n,t,.) one can easily see that X is isonomous to the trajectory field {nxt:t €"'[0,1], n E C} and is continuous in probability on [0,1] x C. Notice that X(.,t,n) is measurable with respect to the a-algebra "'Ft ~ o{~(u,n):u E [O,t], n E C} for each t E [0,1], n E c.· Of course for a.a. wE S'l and all t E (0,1] the map X(~,t,.) is discontinuous everywhere on C. (iii) A further discussion of stochastic DOE's will course of our next chapter on examples of stochastic see that a measu~bZe version X:.S'l x [0,1] x C ~ C of {nxt:t E [0,1], n E C} always exists. lt is howev.er i.e. for each t E [0,1] P{w:.w En,

be carried out in the FDEas. There we shall the trajectory field highly non-linear on C,

X(w,t,An 1 +~n 2 ) = AX(w,t,n 1 )~~X(w,t,n 2 ),

for all

n1 .n 2 E C, >•• ~ e: R} = 0

despite the linearity of the equation (\ll) and the fact that P{w:.w E S'l, X(w,t,An 1

*

~n 2 )

=>< X(w,t,n 1) +-

~X(w,t,n 2 )} =

E R, t E [0,1], n1, n2 E C. See §Vl.3. (iv) In the next section we shall show that on the interval [1,2] the map n ~ p(O,n,t,•) is locally compact into the weak* topology of all regular Borel probability measures on C. So by a theorem of Prohorov (Schwartz [71], Theorem 21, p. 74• Rao [67], p. 125) we can construct a proces~ isonomous to {nxt:t E [1,2], n e: C} on the probability space (C[ 1•2] x C.~. ~) with for all

148

><.~

~

= a{p(t,n) :.t E [1,2], n E C} and ~a r>adon pr>obability measur>e on c[ 1•2Jxc. It is not clear, however, if the trajec~ory field {nxt~t E [1,2], n e: C} is then isonomous to a continuous process F x [1,2] x C +C. §4.

Regularity in Probability for Autonomous Systems

So far we have been looking for the trajectory (random) logical example of §3 kills x:n x [O,a] x C + C for the general stochastic FOE

x0 = n e: c

at the question of establishing suitable versions field of a stochastic FOE. The simple but pathoall hope in obtaining a sample-continuous version trajectory field {nxt~t E [O,a], n E C} of our

= C(JJRn),

J = [-r,O].

}

(I)

On the other hand, Theorem (2.1) of §2 indicates that the trajectory field admits good compactifying versions in the case of ordinary diffusion coefficients. It therefore seems necessary in dealing with regularity of (I) to shift our viewpoint towards distributional properties of the trajectory field rather than sample function behaviour. Indeed we motivate ourselves by the conclusions of Theorem (2.1) to show that their parallels in pr>obabiZity hold good under fairly general assumptions on the coefficients of the stochastic FOE (I) above. In particular these conclusions are valid for the delayed diffusion example (VI) in §3 as well as the ordinary 'Frobenius type' one (II) of §2. To fix ideas, recall that (I) is defined on the filtered probability space (n,F,(Ft)G1 llnll a= llnllc +sup { a: s 1, s 2 E J, s 1 # s 2}. C I s 1 - s2 1 According to the notation of §(III.1}, let {p{O,n,t,.):t € [O,a], n € C} stand for the transition probabilities of the Markov trajectory field of (I) i.e. 149

p(O,n,t,B) = P(nxt E B), B E Borel C, n E C, t E [O,a]. For the rest of this section we always take a > r. Our study of the regularity properties of these transition probabilities will turn on the following result which is taken from. Stroock and Varadhan [73] (Garsia, Rodemick and Rumsey [23])~ Theorem (4.1): Let E be a real Banach space and y~Q x [O,a] + E an (F e Borel [O,a], Borel E) measurable process with almost all sample paths continuous. Suppose that for each t E [O,a], y(.,t) E £ 2k(Q,E;F) in the Bochner sense and there is a number C = C(a,k) > 0 such that Ely(·,t 1)- y(.,t 2) l~k < Clt 1-t 2 1k for all t 1,t 2 E [O,a]. Then for every 0
t>

and any real N > 0,

Proof: The result follows directly from Corollary 2. 1..4 and Exercise 2.4.1 of Stroock and Varadhan {[73], pp. 47-61), noting that J

a Ja

( ) It -t Ik 1- 2a - 2 dt dt 1 2 0 0 1 2

=

2a k(1-2a) , 0 < a < i. k(1-2a)[k(1-2a)-1]

c

The first main result tells us that the sample paths of the solution nx of (I) are almost all a-H6lder continuous on [O,a] for any 0 < a < i. Thus the trajectory field segment {nxt:t E [r,a], n E C} on [r,a] will always lie in Ca{JJRn) for 0 < a < i. Before proving this we need estimates on higher order moments of the trajectory field. Such estimates are obtained from the following two theorems which are well-known for stochastic ODE's (r = 0) (Gihman and Skorohod [24], pp. 44-50; Friedman [22], pp. 102-107). The reader is invited to make the obvious modifications in the classical proofs to obtain Theorem (4.2) (LocaZ Uniqueness) Suppose Hi:Q x C +Rn, Gi:Q x C + L(Rm,Rn), i = 1,2, are such that H (·,n), Gi(.,n) are F0 -measurable for each n E C and 150

there is a constant L

>

0 so that

1Hi(w,n 1)- Hi(w,n 2>1 < Llln 1 -n 2 11, 11Gi(w,n1)- Gi(w,n2)11
=

Hi(.,x~)dt

x0i = n E c,

+

Gi(.,x~)dw(t)

0 < t
= 1,2,

i

with random coefficients. Suppose th~t nTr :Q ~ [O,a], i = 1,2, are the first times that the trajectories {nx~:t E [O,a]} hi.t C....U viz: inf {t:t > 0, nTr(w)

n . x~(w)

t U} if infimum E [O,a]

= {

a

otherwise

Then for any n E U one has

and P{w:w E Q,

sup O
llnx~(w) - nx~(w) II

=

O}

=

1.

Remark ( 4• 1 ) Note that nT~ is a stopping time for the process {nx~:t E [O,a)}, (Dynkin [16], p. 106, Stroock and Varadhan [73]). Theorem (4.3): tions:

Suppose H:Q

x

C ~Rn, G:Q

x

C ~ L(Rm,Rn) satisfy the condi-

(i)

For each n E C, H(·,n) and G(·,n) are F0 -measurable;

(ii)

There is a constant K > 0 such that IH(w,n) I <

K(1 +

lin

1,2;

II c>· IIG(w,n)ll <

K(1 +

lin

II c>

for a.a. wE Q, all n E C; 151

(iii)

For each N > 0, there exists LN > 0 such that

for a.a. wE o, all n1,n 2 E C with lln 1 11 c < N, lln 2 11C < N. Then for each n E C the stochastic FOE dx(t)

=

H(•,xt)dt

x0

=

n E

+

G(.,xt)dw(t)

0 < t
c

has a (pathwise) unique solution nx in the sense of Theorem (11.2.1). Furthermore, nx E £ 2k(Sl,C([-r,a]JRn)), nxt E £ 2k(Sl,C) for every integer k > 0, and each t E [O,a]. Indeed there is a constant C~ > 0 depending only on K, k, a (independent of n E C) such that Ell nxt 11 2k < C~(1

c

+

lin II ~k) for all n E C, t E [O,a], k

=

1,2,3, ••••

We now go on to prove our first main result in this section Theorem (4.4)~ Let n E C and 0
Then the solution nx of (I) satis-

and P{w~w E Sl, nxt(w} E Ca for all r < t 1 2k where k is any positive integer and nx E £ 2k(Sl,C([-r,a]JRn}} is the unique soluti.on of {I} at n. Note that H, G satisfy the conditions of Theorem (4.3}. Indeed we have t2 t2 Elnx(t1}-nx{t2}12k < 22k-1EIJ H(nxu}dul2k+22k-1E1J G(nxu}dw(u} 12k t1 t1 t2 t2 < 22k-t It-t 12k-1 J EIH{nx ) 12kdu + 22k-1K It-t lk-1J EIG(nx >12~ 1 2 t u k,1 1 2 t u

1

152

1

by H6lder's inequality and Theorem (1.8.5), where Kk,t > 0 is a constant depending only on k, m and n (independent of n, H, G). Using the linear growth condition on H, G together with Theorem (4.3) it follows that there are posi.tive constants Kk, 2' Kk, 3' Kk, 4 ' Kk,S depending on K, k, a, m, n and independent of n, t so that t2 Elnx(t 1)- nx(t 2>1 2k < Kk, 2 1tct2 12k- 1 Jt (1 ""E llxull 2k)du t

"" Kk '31 t 1-t2 Ik-1

t2 Jt {1 "" E II xu 112k ) du 1

< Kk, 4(t k-t.1} lt 1-t2 1k < Kk,S(tk-t.1}(1

+

1 * Kk, 4(t k-t.t} Itct2I k-1 .Ck(h llnll 2k ) lt 1-t 2 1

lln11 2k) lt 1-t2 1k

(1)

Now apply Theorem (4.1) to the continuous process nxi[O,t], choosing k > 1 sufficiently large such that 0 ~ thus P{w:w

€

Inx(w)( t1 )-nx(w)( t2) I

n,

lt1 - t21a

k 2k < Kk,S(t +1)(1 + lin II ) =

Kk, 6 (tk+1)(1 + lln11 2k>

>

8(ka+1) a

4

ik

N}

2ak( 1- 2a)

1

k{1-2a)[k(1-2a)-1]

N2K

-~----

:k N

(2)

2K ak( 1-2a) k,S , independent oft € [O,a], for every N > 0, where Kk 6 = ' k(1-2a)[k(1-2a)-1] n E C. Define the sequence {~};=t of subsets ~ of n by ~ = {w:w €

Then ~~ € F, ~+t

c

n,

sup O
lnx(w)(t 1)-nx{w)(t 2>1

> M}

lt 1 - t 2 1

~ for every integer M > 0 and by (2)

K

P(~) < ~, M

M> 1

(3)

153

2k co where K = s2k (ka+ 1) (ak+1). As the series E k,Z a2k M=1 by the Borel Cantelli lemma (Theorem (I.2.5)(i))

K k, 7 converges, then

M2!<

P(lim inf n~) = P( ~ n~) = 1. M M=1 But lnx(w)(t 1)-nx(w)(t 2>1 u n~ = {w:w e: n, sup _ _ _....:_ _ _ _=-- < M} M=1 M=1 O
u

t,~t2

= {w:w e:

n,

sup O
= {w:w e: n, nx(w) I[O,a] e: cho,a] ,Rn)l Therefore the first assertion of the theorem holds. Finally, the second assertion of the theorem is implied by the first because lxt(w)(s 1)-xt(w)(s 2)1 lx(w)(t 1)-x(w)(t2)1 _;;..__...;....___;;---:--;;;...__ < sup a ls 1 - s 2 1a t 1,t 2e:[O,a] lt 1 - t 2 1 t, ~ t2 for all t e: [r,a].

a

Corollary (4.4.1): If n e: C, then for any 0
lnx(w)(s)-nx(w)(s)l t1 t2

---'---~;;.......-

It, - t21a

=

n, then

sup sup se:J t 1t 2e:[r,a] t ~ t 1 r

2

lnx(w)(t 1)-nx(w)(t 2)1 lt 1-t 2 1a 154

•

D

Recall that M(C) is the complete separable topological 'lector space of all finite regular Borel measures on C given the vague (or weak *) topology (§1.2~ Parthasarathy [66]; Stroock and Varadhan [73]). Since C is complete separable, the probability measures MP(C) form a closed convex subset of M(C). Compact subsets of M(C) are characterized by the classical theorem of Prohorov: (~ohorov)~

Theorem (4.5)

(i) A subset r c: M(C) is relatively weakly * compact if and only if it satisfies the following two conditions (a)

sup

{~(C)~~

€ r} <

~;

(b)

for any E > 0, there is a compact set KE c: C (depending only on E and r) such that sup {~(C~E)~ ~ € r} < £. When this condition is fulfilled we say r is uniforml.y tight. (ii) A subset r uniformly tight.

MP(C) is relatively weakly * compact if and only if it is

See Parthasarathy [66], pp.

Proof~

10.

c:

47-49~

Stroock and Varadhan [73], pp. 9-

c

Our second main result below uses Prohoro'l's theorem to assert the compactness in probability of the trajectory field {nxt~n € C, t € [r,a]} on [r,a]. Theorem (4.6): Suppose H, G are Lipschitz on bounded subsets of C and have linear growth. Let n € C and 0 1} 2a there is a constant C~ > 0 depending only on k, H, G, m, n and independent of n € C, t € [r,a] such that (4) for all real N > 0, t [r,a]

x

(t,n)

€

[r,a].

C

In particular, the map

Mp(C) 1,--~

p(O,n,t,·) 155

takes bounded sets into relatively compact sets. Proof: Fix n E C, 0
<

1 and t E [r,a].

Since nxt(w) E Ca(J,Rn) for lnx(w)(t+s 1)-nx(w)(t+s 2>1

1st - s21a for a.a. w E n, and therefore for any N > 0 we

ha~e

P{w:.w E o, unxt(w) II ·> N} < P{w:.w E o, unxt(w) II c > N/2} ca + P{w:w E O,

lnx(w)(t 1)-nx(w)(t2)1 N sup _ _ _...:.,.._ _ _ _=-->~} O
because of Chebyshev's inequality (Theorem (1.2.4)) and (2) of the proof of Theorem (4.4). Therefore, using Theorem (4.3), we get 2k P{w:w En, unxt(w) I~>N} < ~~(t+llnl~k) +Kk,7(tk+t)(t+llnll~k)·~ 2k 2k 4(ka+t) 2k . . where Kk 7 = Kk 6 8 .2 • 1s 1ndependent of t,n. Choosing 2k • • a C~ = Kk, 7 + 2 2 kc~ we immediately obtain assertion (4) of the theorem. We next prove that for any bounded set B c C, the set of probability measures {p(O,n,t,.): nEB, t E [r,a]} is uniformly tight and hence relatively weakly * compact in Mp(C) by Prohorov•s Theorem. Let £ > 0 and choose M, N£ > 0 sufficiently large so that B c {n:n E C, II n lie<' M} and C~(ak+t)(t+M 2 k) ~ < i£· Define the set K£ c C by K£ ={~:~ E Ca(JJRn), N£ 11~11 Ca < 2N£}. By Ascoli's theorem, K£ is clearly compact in c. Moreover, for any n E B and t E [r,a], we have

156

= P{w:w

€

Q, llnxt(w) II ca

< P{w:w

€

Q, llnxt(w) II a> NE}

2NE}

c

< c2(ak+1)(1+M2k>· -!.. < IE. k N'K E

Therefore sup {p(O,n,t,C~E):n € B, t € [r,a]} p(O,n,t,•) takes bounded sets into relatively weakly * compact sets.. c Denote by Mp(Ca) the space of all Borel probability measures on Ca(JJRn) (0 < a < I) given the weak * topology. We shall show below that for t > r the transition probabilities p(O,n,t,.) € Mp(Ca) for 0 r. By separability, Borel Ca is generated by a countable collection of closed ballsi so for Borel Ca to be contained in Borel C we need only check that every closed ball {n:n € Ca, lln-n0 11 < 6} inCa belongs to Borel C. For simplicity, ca and without loss of generality, it is sufficient to show that the closed unit ball Ba = {n:n € Ca, II nil a< 1} inCa is a Borel set in c. Let {si}7= 1 be an enumeration of all thecrationals in J = [-r,O]. Then ln(s.)-n(s >I Ba = {n:n € C, sup ln(si)l + sup J k < 1} i>1 j ,k>t IsJ.-sk I a j~k

= {n:n €

c,

sup [ ln(si) i ,j ,1<>1

ln(sj)-n(sk) I

I + _.....,.__.....;..;;..._.] IsJ.-sk Ia

j~k

< 1}

For each i,j,k > 1, j ~ k, define the function $ijk:C +~by ln(sj)-n(sk)l for all n € c. lJ , Isj-sk 1a Then each $. "k is continuous and hence (Borel C, Borel R)-measurable. $ .. k(n) = ln(s.) I + lJ

fore B = n {n:n E c, $1.J.k(n) < 1} a l,J. . . 1<>1

€

There-

Borel C.

j~k

157

Recall that the spaces t 0 (0,C) and t(O,Ca) carry the pseudo-metrics d , d o a defined by d ce,,e2)

=

0

inf [E E>O

+

P{w:w € o, lle,(w)-e2(w)llc >

+

P{w:w € 0, lle,(w)-62(w) II > ca

E}]

for all e1,e2 € t 0 (0,C) and d ca,,e2) a

=

inf [E E>O

E}]

for all e1,e2 € t 0 (0,Ca). Un~er these pseudo-metrics and a global Lipschitz condition on the coefficients H, G of the stochastic FOE (I), we get our last main result concerning the regularity of the map n ~> nxt into t 0 (0,C) or t 0 (0,Ca): Theorem (4.7): Suppose H:C +Rn and G:C + L(Rm,Rn) in (I) are globally Lipschitz with a Common Lipschitz constant L > 0. Let 0 N} t 1 ,t2€[0,a] It, - t21a t, ~ t2 <

N~3k lln 1 -n 2 11~k

, N>0

(5)

n n c 2k P{w:w € 0, II 1xt(w)- 2xt(w) II a>N} < :21(11n 1-n 2 11 , N> 0, t € [r,a], 3

C

N

(6)

n1 n2 2k/2k+.1 4 d0 ( xt' xt)
(7)

n2 n2 2k/Zk+.1 5 da( xt' xt) < c lln 1-n 2 11c .t € [r,a].

(8)

and In particular the maps C ~ t 0 (0,C),

t € [O,a], C ~ t 0 (0,Ca), t € [r,a] n

158

are uniformly HlHder continuous with H61der exponent the maps [O,a]

x

C + Mp(C), [r,a]

(t,n)

~>

p(O,n,t,·)

x

(t,n)


-a)-t.

Furthermore

C + Mp(Ca) ~>

p(O,n,t,·)

are continuous.

~: Let n1, n2 E C and 0
+

1(1 -

t>·

Define the measurable process

C([-r,a]JRn) by y(w)(t)

=

n1

x(w)(t) -

n2

x(w)(t),

t

€

[-r,a], a.a. w € n.

t

2 E IIYull ~k du t1 where K1 = K1(k,m,n,a), K2 = K2(k,L,m,n,a) are positive constants independent of n1,n 2,t 1,t 2• Similarly, there are positive constants Ki = Ki(k,L,m,n,a), i = 3,4,5,6,7 independent of n1,n 2,t 1,t 2 such that < K2 1t 1-t 2 1k-t

( )

< 22k-1 lin-n 112k+ 22 2k-1 E sup 1

2

C

sEJ

Jt+s [H( n1x )-H( n2x )]dul2k

I

0

u

u

t+S>O 159

<22k-111n-n 112k ..,22(2k-1)t2k-1L2kJtEIIn1x- n2x 112kdu 12C O u uC + 22(2k-1 )K L2ktk-1 Jt Ell n,x - n2x II 2k du 3 0 u u c < 22k- 1

lln 1 -n 2 11~k

+ K4 J: E

IIYull~kdu,

for all t € [O,a];

and hence, by Gronwall's Lemma, one gets K t E IIY II 2k < 22k-1 lin -n II 2k e 4 < K lin -n 112k tC 12C 5 12C

for all t € [O,a].

(9)

Therefore

Ely(·Ht 1)-y(·Ht2 >12k < K2K5 1t1-t2 1k

lln 1 -n 2 11~k= K6"1t 1 -t 2 1klln 1 -n 2 1~ck (10)

for 0 < t 1 < t 2 < a, where K6 = K2K5 • Now apply Theorem 1 to get a constant c3 = c 3(k,a,L,m,n,a) independent of n 1,n 2 , N > o such that P{w:w €

n,

ly(·Ht 1)-y(·Ht2>1 ----=--1-=a____.;~ lt 1-t 2 ,

sup

t 1,t2€[0,a]

>

N}

<

c3 ::21(

N

t, '1: t2 for all real N > 0.

Hence (5) is satisfied.

To prove the inequality (6), let r < t 0 and consider p{w:w €

n n n, II 1xt(w) - 2xt(w) II a> N}

c

n

< P{w:w €

n

n, II 1xt(w) - 2xt(w) II c > ;}

n1

+ P{w:w €

n,

sup

s 1 ,s 2€J

s,

160

'1: 52

I

x(w)(t+s 1)-

n2

x(w)(t+s 2) a

ls 1 - s2 1

I

N

> ~}

2k

lln 1-n 2 11 C

<

22k

2

2k

2 3

2k < 2 Ks·

1 2k 2 "c tfK lln,-n211c .. """N'2K

2k lln,-n211c

2k

K7

=

22kc3

tfK E IIYt II ck .. ""'tfK II n,-n2 II c

tfK lin,- n2ll c

Where we have used Chebyshev's inequality (Theorem (1.2.4)) and (9)• K7 = K7(k,a,L,m,n,a) = 22kK5 + 22kc3 is independent of.n 1,n2, Nand t € (r,a]. This proves (6) with c3 replaced by K7 • For the inequality (7), note that, if t € (O,a], then d0 (

n1 n2 xt' xt)

< inf (E E>O

+

=

inf (E £>0

+

P{w:w

€ 0,

n1

n2

II xt(w)- xt(w) lie > E}]

K -:kE IIYtll 2kJ < inf (E +-=* lln 1-n2 11 2kJ. E £>0 E c

Define the Cm function f~>O +R by f(E)

=

Ks 2k £ + -:2k lln 1-n211c for all E > o. E

Then f attains its absolute minimum value at EO > 0 where f'(E 0) -

i.e.

d0 (

o, f"(E 0) > o i.e.

2k Ks Eo

Eo 2k+t

Therefore

=

n1

2k+l =

2k

lln 1-n 2 11 c

=

o

2k K5 lln 1-n 2 11c2k.

n2

xt' xt) < inf f(E) E>O

= f(E 0) = E0(t + ~)

1 =

1

c iK>

K5"2K+f (2k)"2K+f 1 +

-n

lln 1 2 11

2k

F

1

Hence (7) holds with c4

=

(2K5k)"2K+T(1 + ~). 161

1 Choose and f;x an ;nteger k0 = k0(a) such that k0 > T=2a • Then by a s;m;lar argument to the above one, we can prove that

1 5

3

'2I<()+T(2k )'2'k"(i+T(1

where C = (C ) 2k s;nce - -0- > 2k 0+1

0


lin-T' II_. .

5

C (k 0(a),a,L,m,n,a)

=

5

C (a,L,m,n,a).

0 1,

then { lln 1-n 2 11

2k0

"2k(i+··

1

+ ~) =

<

c 1

lln 1 -n 2 11~a

;f

lln 1-n 2 11c <

1

and hence the f;rst of the last two assert;ons of the theorem holds. The f;nal statement of the theorem follows from the un;form (L;psch;tz) cont;nu;ty ;n the second var;able of the maps

(t,n)

(t,n)

~----+

and the cont;nu;ty of the maps _____,._ Mp (C) • ..."0 (",Ca) - -_, M. p (Ca) ..." 0 (" •• • C) ---,.. ..

(See Elworthy [19], Append;x C, p. 300). Th;s completes the proof of the theorem.

c

Remark (4.2) Let P~:M(C) + M(C), 0 < t r. As a consequence of the above theorem, one can always select measurable 162

versions n X [O,a] X c -+ C, n X [r,a] X c -+ ca of the trajectory field {nxt:t E [O,a], n E C}, {nxt:t E [r,a], n E C} for 0 < a < i. The proof of this result was shown to me by K.D. Elworthy; I am very grateful to him. Corollary (4.7): Suppose in the Stochastic FOE (I) the coefficients H, G are globally Lipschitz on C. Then the trajectory field {nxt:t E [O,a],n E C} has a measurable version n x [O,a] x C -+ C, and for any 0 < a < i the segment {nxt:t E [r,a], n E C} admits a measurable version n x [r "~] x C -+ca. Proof: Since the constant c4 in (7) is independent of t E [O,a], it follows, from the continuity of the map

for each n E C, that the map

(t,n)

)

is jointly continuous (in the metric d0 ) and hence Borel-measurable. But [O,a] x C is complete separable metric in the product metric; so by a result of C. Moore (Cohn [9]), the map Y has separable range in £ 0 (n,c). Applying a general theorem of Cohn we immediately get an (F ®Borel [O,a] ®Borel C, Borel C)-measurable version X:n x [O,a] x C-+ C for Y. (Theorem (I.5.2)). A similar argument works for the continuous map

( t, n)

noting that Cohn's theorem above does hold for processes with values in the complete separable Banach space Ca • c Remark ( 4. 3) (i) I do not know if the trajectory field {nxt:t E [O,a], n E C} has a cValued separable measurable version. If one exists, then by the Borel-Cantelli 163

Lemma it is not hard to see that this separable measurable version "' X will have the property that there is a set ga € F of full P-measure, ~hosen independentLy of n € C, so that, for all w € g and any n € C, X(w,t,n) € C, "' a X(w,t,n) € Ca if t € [O,a] and t E [r,a] respectively. (ii) Both Theorem (4.7) and its corollary hold for random coefficients H~g x C +Rn, G~g x C + L(Rm,Rn) satisfying the conditions: (a) For each n € C, H(·,n) and G(·,n) are F0-measurable; (b) There exists l > 0 such that

for a.a. w €. g, all n1,n 2

164

€

C.

VI Examples

§1.

Introduction

In this chapter we illustrate our general results of Chapters II,III,IV and V on various examples of stochastic FOE's. These examples include stochastic delay equations (§3) and linear FOE's which are forced by white noise (§4). The latter class of stochastic FOE's corresponds to equations of the form dx(t)

=

H(xt)dt

+

g(t)dw(t), t > 0

x0 = n E C : C(J,Rn) where H:C +Rn is a continuous linear drift and g:R + L(Rm,Rn) is locally integrable. Section 4 is joint work of the author with Henrich Weizs!cker and Michel Scheutzow : the asymptotic behaviour of trajectories to the above equation is analysed along the stable and unstable subspaces in C of the deterministic linear system dx(t)

=

H(xt)dt, t > 0.

Applying well-known results of J. Hale ([26] pp. 165-190) we find that, if g is constant, the forced system is globally asymptotically stochastically stable (§3) whenever the unforced linear system is globally asymptotically stable. The stochastically stable case corresponds to the existence of a limiting invariant Gaussian measure under the stochastic flow. On the other hand, when g is periodic and the unforced system is asymptotically stable, the transition probabilities converge to a periodic family of Gaussian measures on c. §2. Stochastic ODE's These correspond to the case when our system has no memory i.e. r = 0, thus J = {O} and C(J,Rn) is just the Euclidean n-dimensional space Rn. Our basic stochastic FOE (I) of Chapter II then takes the simple differential form 165

dx(t) = g(t,x(t))dz(t),

t>O (I)

x(O) = v € t 2(nJRn)

Such stochastic ODE's were first studied by Ito ([36]) in 1951 and has since been the subject of intensive research. Indeed there are now excellent texts on the subject such as the works of Gihman-Skorohod [24], Friedman [22], Stroock-Varadhan [73] on Euclidean space,and Ikeda and Watanabe [35] and Elworthy [19] on differentiable manifolds. We therefore make no attempt to give any account of the behaviour of trajectories to (1), but content ourselves by noting that Theorems (11.2.1), (11.2.2), (11.3.1), (111.1.1) (111.2.1), (111.3.1) are all well-known to hold for stochastic ODE's of type (I) or its autonomous versions dx(t) = g(x(t))dz(t) ,

t

> 0

(II)

x(O) = v dx(t) = h(x(t))dt

+

g(x(t))dw(t), t > 0

(III)

x(O) = v

for z a continuous semi-martingale, w m-dimensional Brownian motion and coefficients h:Rn ~Rn, g:Rn ~ L(RmJRn). The associated semigroup {pt}t>O on Cb(RnJR) for (III) is, however, always strongly continuous in the supremum norm of Cb(RnJR). A strong infinitesimal generator A:V(A) c Cb ~ Cb can therefore be computed to get A($)(v) = D$(v)(h(v))

+

~ trace D2$(v)

o

(g(v)

x

g(v)),

v € Rn

(*)

for$ c2-bounded on Rn. The reader may check that this agrees formally with the conclusion of Theorem (IV.3.2) (or that of Theorem (IV.4.3)). See [22] and [24] for a classical derivation of formula (•). The trajectory field {vx(t):t ·> 0, v € Rn} of (III) has a measurable version consisting a.s. of diffeomorphisms in case h and g are sufficiently smooth (Elworthy [19], Kunita [46], Malliavin [51]). Furthermore, if h and g are linear maps, then selecting a separable measurable version x:n x ~ x Rn ~Rn of the trajectory field implies that for a.a. w € n the map X(w,t,.):Rn ~Rn is a linear bijection for all t > 0. This sharply contrasts the corresponding behaviour for linear delayed diffusions of §(IV.3). 166

§3.

Stochas~1c

Delay Equations

Let (n,F,(Ft)t>O,P) be a filtered probabili.ty space with z~n -+ C(~ ,Rm) a continuous Bm-~alued martingale adapted to (Ft)t>O and satisfying E(z(.)(t 2) - z(.)(t 1) 1Ft ) < K(t 2-t 1) t

E(lz(.)(t 2)-z(.)(t 1)1 21Ft) < K(t 2-t 1)

}

(E)(i)

t

a.s. whenever 0 < t 1 < t 2, for some K > 0. Suppose hj:Bn -+Bn, j = 1, ••• ,p, g1 :Rn-+ L(Bm,Rn), i = 1, ••• ,q are Lipschitz maps. Let there be given p ~ q delays as random variables r.~n-+ ~. j = 1, ••• ,p, di~n ..... ~. i = 1, ••• ,q, which are P-essentiall~ bounded and F0-measurable. Define r =

max (essup rj(w), essup di(w)) 1<j

Given a E t 2(n,C(J,Rn); F0), J = [-r,OJ, consider the stochastic delay equation (with finite random delays):

dx(t) = x0 =

p I:

h.(x(t-r.))dt

j=1 J

J

9 +

I:

i=1

gi(x(t-di))dz(t),

t > 0 (IV)

a

The random delays rj, di, j = 1, ••• ,p, i = 1, ••• ,q, in (IV) may not all be essentially bounded away from zero; and so a step-by-step direct integration of (IV) is not in general possible. Nevertheless we can still apply our basic Existence Theorem (11.2.1). First we need to check that (IV) does indeed satisfy Conditions (E) of Chapter II. Define h:£ 2(n,C(J,Rn)) --> .c 2(n,Rn), ~:£ 2 (n,c(J,Rn)) -~> icn,L(Bm,Rn)) by "' h(~)(w)

= I:p h.[~(w)(-r.(w))J, "'g(~)(w) = 9I: j=1 J J . i=1

gi[~(w)(-di(w))J

E £ 2 (!l,C(J,R n)), a.a. wEn. To see that "'h, "'g are globally Lipschitz, let L > 0 be a common Lipschitz constant for all the hj's and gi's. Then, if ~,.~ 2 E t 2(n,C(J,Rn)), we have

for~

167

llh(1j1t)-h(1j12)1122 n t. (O,.R )

.~ J lh .[1j1 1 (w)(-rJ.~))-hJ.[1j12 (w)(-rJ.(w))] 12 dP(w) J=1 n J < p L2 ~ J 11j11(w)(-r.(w))- 1j12(w)(-r.(w)l 2 dP(w) j=t n J J < p

< p2 L2 111jl1 - 1jl2112 2 t. (O,C)

g

Similarly is Lipschitz with Lipschitz constant qL. It remains now to verify the adaptability condition E(iii) of §(li.1). To do this it is sufficient to show that for each 1jl E t. 2(O,C;Ft), "' h(ljl) and "'g(ljl) are Ft2 measurable, fort> 0. Let 1jl E t. (0,C) be Ft-measurable, and P:J x C ~Rn the evaluation map (s,n) ~ n(s), s E J,n E C. Then P is continuous, and since each rj is F0-measurable, it follows that

W~> ~(ljl)(w) = ~ h.[P(-r.(w),ljl(w))] j=1 J

J

~s Ft-measurable. Therefore h(ljl)

E t. 2(n,.Rn;Ft); similarly 2 g(ljl) E r. (n,L(Rm,Rn);Ft). Hence all the conditions of Theorem (II.2.1) are satisfied and so a unique strong solution ex E t. 2(n,C([-r,a],Rn)) of the stochastic ODE (IV) exists with initial process e. The trajectory {ext:a > t > 0} is defined for every a > 0, is (Ft)-adapted and has continuous sample paths. Moreover each map

icr~.C;Ft), t > e

e

o.

xt

is globally Lipschitz, by Theorem (II.3.1). Now su.ppose in (IV) that z is m-dimensional Brownian motion w adapted to (Ft)O
dx(t) = E h.(x(t-r.))dt j=1 J J x0 = n E C(J,Rn) 168

q +

E g.(x(t-d.))dw(t),

i=1

1

1

t

>

0 (V)

with deterministic initial cond;tion n. Note that the coefficients in (V) ~an also be viewed as F0 ®Borel C measurable maps H~n x C(JJR"> ~ Rn, G:n x C(J,lRn) ~ L(Rm,Rn) given by "'

=

H(w,n)

p

E h.(n(-r.(w))) , "' G(w,n)

j=1

J

J

=

q

E g.(n(-d.(w))) 1

i=t

1

a.a. w € n, all n € C(J,lRn). So (V) becomes the stochastic RFDE with random coefficients: dx(t)

=

"'H( • ,xt)dt + "'G( • ,xt)dw(t),

x0

=

n

0
I

Because of the randomness in the coefficients of (V)' we cannot apply theresults of Chapter III to obtain a Markov property for the trajectory field of (V). However, if the delays rj, di are all independent of the Brownian a-algebra F: = o{w(·)(t): 0 < t
dx(t) = h(x(t-r 0))dt x0

=

n

€

+

g(x(t-d 0))dw(t),

o<

t
C(J,Rn)

Where h:Rn ~ Rn, g:Rn ~ L(Rm ,Rn) are Lipschitz maps. !!mma (3.1):

Suppose the delays r 0, d0 € £~(nJR>OiF0 ) are simple functions 169

with respect to the deLay a-aL,gebra F0 = a{r0,d 0} c F0 i.e. there are k 0 0 < r m' , dm' < r, m' = 1, ••• ,k and {0m' }m'= 1 c F such that r0 =

k I:

m1=1

m' r X...m l

•

~r

do =

m1=1

For each m' = 1, ••• ,k, let "xm 1:n 1

I

I

m' d X...Ill I •

k I:

dxm (t) = h(xm (t~rm ))dt

+

+

~r

k

u

m1=1

n

ml

= n.

C([-r,a]JRn) be the unique solution of ml m1 g(x (t-d ))dw(t). 0 < t
xm~ = n k

Then x(t) _

(VI(m1)) I

"xm (t)x...m 1, -r < t < a, gives the unique solution of (VI)

I:

m1 =1 ~r in t 2(n,C([-r,a]JRn)) starting at n € C(JJRn) and adapted to (Ft)O
Proof: By the definition of x, k ml xt(·) = I: "xt <·>x...~~~~<·> for all t > 0. m'=1 ~r Then

x0 =

k I: nx...~~~.<·>

m1 =1

~r

= n.

Also since r 0 , d0 are simple functions, h(x(t-r0 )) =

k I:

I

m1=1

I

h(nxm (t-rm »x...~~~ .. g(x(t-d 0 ))= ~,-

k I:

m'=1

I

I

g(nxm (t-dm ))x...JD~; ~'

and so by Theorem (I.8.3)(i) for the stochastic integral one has Lh( 1xm 1(u-rm'»x 1du + ~ x ~{J\cnxm 1 (u-dm 1 ))~")} x(t) = { m'=1 0 rP' m1=1 rP' 0 0< t < a n(t) t EJ n(O)

·={n(O)

+

+

~

t J h(x(u-r0 ))du

+

0

n(t) n(O)

={

n(t)

170

Jt

k n I I x...~~~~g( xm (u-dm ))dw(u),O
t EJ +

J: h(x(u-r0))du t EJ

+

J: g(x(u-d 0))dw(u), 0 < t
1 w As each nxm is (Ft)O
Hence x is the unique solu-

For a.a. W1 E n, let nxw 1 E .c 2(n,c([-r,a]JRn}) be the unique solution adapted to (F~)O
+

g(x(t-d 0(w 1 )))dw(t)

0 < t
=n

with fixed delays r 0(w 1 ) , d0(w 1 ) € [O,r]. The trajectory fields I wl {nxt:t E [O,a], n E C, u.f En therefore correspond to a random family {X :w 1 En} of continuous time-homogeneous Feller processes on C(J,Rn) adapted to (F~)O
p(w 1 ,0,n,t,B) = and

E pm 1 (0,n,t,B) x 1 (W 1 } m1 =1 nm

a.a.

W1

En

Jn p(w 1 ,0,n,t,B)dP(w 1 ) = p(O,n,t,B) - p{w:w En, nxt(w) E B}. ~

Note that for each BE Borel C, the map (w 1 ,n,t) F0 ® Borel C & Borel [O,a]-measurable. ~oof:

~>

p(w 1 ,0,n,t,B) is

By Lemma (3.1), the trajectory field of (VI) is given by nxt(w) =

k

1

E nxmt (w)x

m1 =1

1

rfl

(w)

a.a. wEn, t E [O,a],

n E C. The rfl form a partition of n and are all independent of nx~ 1 • Let 1

171

w• € Sl. Then there is an m• such that w' €. dm•. Hence nxw' = nxm• and

an•

and so r(w') = t'm' • d(w•)

=

p(w•,o,n,t,B) -= P{w:nxt w• (w) €. B} = P{w:w € Sl, nxtm• (w) € B} m• = p (O,n,t,B)

for every B € Borel C. Thus p(w',O,n,t,B) =

k

m•

E p (O,n,t,B)x-m.(w•) m•=t ~r

for a11 w • € Sl. Since each pm'(o,.,.,B) is Borel C 8 Borel [O,a]-measurable (Theorem (111.1.1)), it is clear that p(.,o,.,.,B) is F0 8 Borel C 8 Borel [O,a] measurable. Also, by independence of xd"' and nxtm• for t € [O,a], we get

fSl

p(w•,o,n,t,B)dP(w•) =

=

k

E

m•=t

m• ...m• p (O,n,t,B)P(~,- )

Jg xB(nxt(w))dP(w)

= P{w:w € Sl, nxt(w) € B} for every B € Borel C.

c

The following is a general approximation lemma for stochastic DDE•s: Lemma (3.3): Suppose the coefficients h, gin (VI) are globally bounded and "" ' {dk}k=t "" ->0-;FD) • Lipschitz. Let {rk}k= be increasing sequences in£""(Sl~ 1 0 such that rk ~ r 0 and dk ~ d0 as k ~""in £ 2 (Sl~;F~). Let nxk and nxk,w denote solutions of the stochastic DDE 1 s: dxk(tk) = h(x(t-rk))dt

x0 172

=

n

+

g(x(t-dk))dw(t),

0 < t
(VIk)

and dxk,w 1 (t) k w•

x0 •

=

h(xk,w 1 (t-rk(w•)))dt

=

n

+

g(xk,w•(t-dk(w•)))dw(t) O
respectively, fork = 1,2,3, ••• , a.a. w• € n. Then the map n 3 w• ~ 11xw• € l 2(n,C([-r,a]JRn)) is in lm(n,l 2(n,C([-r,a]JRn));F0),

and

f

1 im k--

W 1 €0

{fW€0

11 11 xk,w• (w) - nxw• (w) II~ dP(w) }dP(w•)

= 0

Remark For each n € C(JJRn), the random family {11 xw• :w• € 0} has a unique version nx ~ l 2(n x n, C([-r,a]JRn); ~ e Fa) such that for a.a. w• € n, 11x(w•,.) = nxw (·) a.s. Proof: Suppose h and g a·re bounded by a conmon bound K > 0 and have a conmon Lipschitz constant L > 0. Let rk ~ r 0, dk ~ d0 as k ~ m in l 2(nJR>0;F0). Fix n € C(JJRn) and take any integer k > o. Consider J: {h( 11 xk(u-rk)) - h( 11x(u-r0))}du t

+

J 0 {g( 11xk(u-dk))-g( 11x(u-d 0))}dw(u), O
t € J

By Doob•s inequality, we have, for any t E

sup

D
0

4E sup 1Jv {h( 11 x(u-rk))-h( 11 x(u-r 0))}dul 2 D
t

+

[O,a],

111xk(v)-11x(v)l 2 < 4E sup 1Jv{h( 11 xk(u-rk))-h( 11x(u-rk))}dul 2

-I"
€

4K 1 J 0

0

Ell g( 11xk(u-dk))-g( 11 x(u-dk))ll 2du 173

t

+

4K 1 J0 Ell g(nx(u-dk}}- g(nx(u-d0}}11 2 du, some K1

>

o,

where II All denotes the operator norm sup {lA( e) I:e € Rm, leI < A E L(RmJRn). Therefore, by the Lipschitz condition, one gets E

sup

lnxk(v)-nx(v)l 2 < 4aL 2

-~

+

+

1}

for each

Jt Elnxk(u-rk)-nx(u-rk)l 2du 0

Jt

4aL 2 0 Elnx(u-rk)- nx(u-r0>1 2du

4K 1L2 J:Einxk(u-dk)-nx(u-dk)l 2du

+

4K 1L2 J:Einx(u-dk)-nx(u-d 0)1 2du (1)

for t € [O,a]. Clearly, Elnxk(u-rk)-nx(u-rk>l 2 < E sup

lnxk(v)- nx(v) 12

(2)

lnxk(v)- nx(v)l 2

( 3)

-r
and Elnxk(u-dk)-nx(u-dk)l 2 < E sup -l"
for all u € [O,a]. We must estimate the second integrand Elnx(u-rk) - nx(u-r0) 12 in (1) for sufficiently large k, unifo~ly in u € [O,a]. To do this, let E > 0 and note that by uniform continuity of non J there is a 0 < o < 1, o = o(E,n) such that for s 1 ,s 2 € J, ls 1-s 2 1 < o , we have ln(s 1)-n(s 2>1 2 <E. Let E' = o2E, so by convergence of {rk};= 1 in £ 2(nJR>0;F0) there exists k0 = k0(E,o) such that Elrk-r0 12 < E' = o2E for all k > k0• Suppose k > k0 and for any u E [O,a] denote by

the characteristic functions of the sets {w:w € G, r 0(w) < u}, {w:w € G, r 0(w) > u.l, {w:w € G, rk(w) < u}, {w:w € G, rk(w) > u} in F0 and A= {(w,v):w € G, v € [O,a], v + r 0(w) > u}, Bk = {(w,v):w € G, v € [O,a], v + rk(w) < u} in F0 Q Borel [O,a], respectively. Using Theorem (1.8.3) for the stochastic integral, we may write 174

+

=

J:

n{O)-n{u-r0), ;f rk < u and r 0 > u.

X(ro
+

J: X
,a

+

J0 +

+

X( rk tJ) \k ( ·, v)h(nx( v-r0 ) )dv Ja x

x X (•,v)g (nx(v-d 0 ))dw(v) (rk < u) (r >u) Bk 0 0

[n{O) - n(u-r0 )Jx(rktJ)

[n{u-rk)-n(u-r 0 )J x(r~)

+

Therefore,

+

+

Ja

n

6Ein(O)-n(u-r 0)1

2

2

X(rkO)

< 12K aEx(ro
2

6K 1 0 EX(rktJ)XBk ( • ,v) llg( x(v-d 0)) 11 dv

2

+

+

2

6Ein(u-rk)-n(u-r 0) I X(r~)

2

12K 1K Ex(rkO)(u-rk)

6Ein(O)-n(u-r 0>1 X(rkY)

+

2

6Ein(u-rk)-n(u-r 0)1 X(r~)

(4)

175

Now

2

Eln(O) - n(u-r0 >1 x(rk
J

+

J

2

ln(O)-n(u-r 0 ) I x(r < u)X(r >u)dP lu-r 0 1<6 k 0

lu-r0 1>6

ln(O)-n(u-r 0 ) 12x(r < u)x(r >u)dP k o

< E + 4lln 11 2 P{w:WE Sl, (r0 (w) - rk(w))

c

< E

+~ 6

because k > k0 •

llnl1 2 Elr0-rkl 2 < E + 4lln11 2 E,

c

Similarly, for k > k 0 , we get

+

Combining

J

Ir.0-rk 1>6

+~lin 1~2

(4),

(5)

c

Eln(u-rk)-n(u-r0 ) 12x(r >u) = k

< E

> 6}

J

lr 0-rkl<6

ln(u-rk)-n(u-r 0 ) 12x(r >u)dP k

ln(u-rk)-n(u-r0 ) 12x(r >u)dP k

Elr0-rkl 2 < (1 + 4llnllc2 )E

(6)

(5) and (6), one gets

Elnx(u-rk)-nx(u-r0 ) 12 <.12K2(a+K 1)Eir0-rk 12 + 12(1 + 4lln < 12K2(a+K 1)6 2E + 12(1 +

1~2 )E

4lln1~2 )E

< 12[K2(a+K 1) + 1 + 4lln11lJE

(7)

for all u € [O,a], k > k0 • Note that k 0 is independent of u € [O,a]. A similar argument applied to the last integrand in (1) yields for every E > 0, a k 0 > 0 such that (8)

for all u € [O,a], k > k0• (7) and (8) to obtain 176

Now put together the inequalities (1), (2),

(3),

E sup -I"
lnxk(v)-nx(v)l 2 < (4aL 2+4K 1L2) JtE sup lnxk(v)-nx(v)l 2 du 0 - r
for k > k0 and for all t E [O,a]. if k > k 0 , then E sup -I"
2 lnxk(v)-nx(v)l 2 < 48aL 2(a+K 1)[K2(a+K 1)+1+4IIn II~JEe4 L (a+K1)t

for all t E [O,a].

In

By Gronwall's lemma, it follows that,

In particular, take t = a to get 2

llnxk(w)-nx(w)ll~dP(w)<48aL 2 (a+K 1 )[K 2 (a+K 1 )+1+4IIn II~JEe 4 L (a+K1)a

for all k > k0 ; i.e. {nxk};= 1 converges to nx in

£ 2(n,C([-r,a]JRn)).

To demonstrate the F0-measurability of the map n 3 w' + nxw' E £ 2 (n,C([-r,a]JRn)) assume for the moment that the delays rk' dk are simple ones of the form nk ".. r m' ,k Xnm ' , k , rk m'=1 for some nk > 0, 0 < rm',k, dm',k < r, nm',k E F0 , 1 < m' < nk, k = 1,2,3, ••• Denote by nxm',k E £ 2(n,C([-r,a]JRn)) the unique solution of the stochastic DOE

k xm' 0 ' =n

with fixed (deterministic) delays r m' ' k , dm' ' k , for each m' = 1, ••• ,nk, k = 1,2,3, •••• Clearly the approximations {nxk,w' :w' E S'2} are then given by PO)

for a.a. w, w' En, k = 1,2,3, •••• Since the simple delays can always be chosen so that rk < rk+ 1 , dk < dk+ 1 for all k > 1 and rk(w') + r 0{w'), dk(w') + d0(w') as k +~for a.a. w' En, it follows from what has already 177.

been proved that n>f' = Hm nxk,w' in ico,C([-r,a],Rn)) for a.a. w•

€

o.

k-+co

But (10) implies that the map o 3 w• + nxk,w' € ico,C([-r,a],Rn)) is F0measurable and 0 x 0 3 (w',w) + nxk,w'(w) € C([-r,a],Rn) is F0 ~Fa-measurable for each k > t. By the Stricker-Yor Lemma (Theorem (1.5.1)) the random fami.ly {n>f' :w' € O} admits an F0 8 Fa-measur~ble version nx:oxo+C([-r,a],Jfl) such that for a.a. w• € 0, nx(w', ·) = nxw (·) a.s. Moreover the map o 3 w' + nxw' €. ico,C([-r,a],Rn)) is F0-measurable and is in fact essentially bounded because, for a.a. W1 € o, E sup lnxw'(t)1 2 < 3llnlll

+

3 E sup 1Jt h(nxw'(u-r(w')))dul 2 O
-f'
0

+ 3E sup I Jt g(nxw'(u-d(w')))dw(u)l 2 O
0

< 3llnll~ + 3iK2 + 3K1aK 2 = 3CIInlb2 + iK2 + K1aK2). Therefore by Theorem 17 (p. 198) of (Dunford and Schwartz [15]) the measurable version nx of {nxw' :w• € O} is uniqueZy determined up to a set of P x Pmeasure zero in 0 x o. Now assume once more that {rk}~= 1 , {dk}~= 1 are our original increasing sequences in £=(oJR>0;F0)·converging to r 0, d0 in £2• The reader may use a very similar argument to the one used in deriving (9) to prove that for every E > 0, there is a ko = k0(E,n) > 0 such that sup fw'€0 E-r
lnxk,w'(v)- nxw'l 2dP(w')

< (4aL 2 + 4K 1L2) Jt {J 0

w'€0

E sup lnxk,w'(v)-nxw'(v)l 2dP(w')}du -r
+ 12a(4aL 2 + 4K 1L2)[K2(a + K1) + 1 + 4lln II~JE for all k > k0• So, by Gronwall's lemma, sup fw'€0 E-r
lnxk,w'(v)-nxw'l 2dP(w'} 2

< 48aL 2(a+K 1)[K2(a+K 1) + 1 + 4lln II~]Ee 4 L (a+K1)t

178

for all t € [O,a].

In particular, the final assertion of the lemma holds. c

We can now prove the following theorem which says that the distribution of the trajectory field {nxt: t € [O,a], n € C} to the stochastic DOE (VI) is the expectation of the associated family of transition probabilities {p(w•,o,n,t,.): w• En, n E c, t € [O,a]} of (VI(w•)). Theorem (3.1): In (VI) suppose the coefficients h, g are globally Lipschitz on Rn. Let the delays r 0 , d0 E £=(nJR>O) be independent ofF: =o{(w(·)(t): 0 < t < a}. Then P{w:w € n, nxt(w) € B} = Jn p(w•,o,n,t,B)dP(w•) for every n € C(J,Rn), t € [O,a] and B € Borel C. Furthermore the trajectory field is a time-homogeneous continuous process on C(J,Rn) viz. PoTt -t (n)-1 t1 -1 n t1 2 1 if 0 < t 1 < t 2
t 1 < t
Yt = n • 1

Proof: We first prove the theorem under the additional assumption that the coefficients h, g are gLobaLLy bounded and Lipschitz. Choose increasing sequences of simple delays {rk};= 1' {dk};= 1 independent of F: and coverging almost surely to r 0 , d0 respectively. Use the notation of Lemma (3.3) to denote ~Y nx the version in £2(n x n,C([-r,a],Rn);F0 Q Fa) of the random family {nxw ;w• € Q} of solutions to VI(w•), w• € n. Let {nxk};= 1' {;:
x

n,C([-r,a],Rn); F0 ~F), nxk(w•,w) = nxk,w•(w), a

for a.a. w,w• € n, and each k = 1,2,3, •••• Because of Lemma (3.3), one 1 = 2 n k = can select a ~ubsequence {n1 Xk 1 }k•= 1 of the£ -convergent sequence {X }k= 1 such that nxk (w•,w) = nxk ,w (w) + nx(w•,w) as k1 + = for a.a. w,w• € n. N = 1 of {n~ k }k= = 1 converges 1n . £2 to nx ote also that the subsequence{ nxk• }k•= as k 1 + =. In view of Lemma (3.1), write nx~ , nx~•,w• in the form 179

nkl

nxtk1(w) = nx

kl

nxtm1' k 1(w) X ml ,.k ~w),

l:

m1 =1

Sl ·

( 11)

nkl

I

,w (w) =

E

m1 =1

t

nkl I k1 D for a.a. w,w 1 € 0, t € [O,a], where Om ' € F , rk1 = l: m1 =1 nkl m1 k dk1 = E d ' X_m 1 kl and m1 =1 ~& ' I kl I kl I kl I kl I kl dxm ' (t) = h(xm ' (t-rm ' ))dt + g(xm ' (t-dm ' ))dw(t) 0 < t < a

(12)

k xm = n 0 ' 1

1

for 1 < m1 < nk1' k1 = 1,2, •••• If' € Cb is any bounded (uniformly) continuous real-valued function on C(JJRn), consider '( nxtm ' k (w) )x m1 k 1(w)dP(w) 1

I

w€0

=

I I w1 ffi

0

'

,(nxk 1,w 1(w) )dP(w)dP(w 1)

(13)

t

w€0

n m1

1

k1

and X ml,kl for each k1 > 1, 1<m 1
due to the independence of'" xt'

= 1im k 1 .....,

180

I I w 1 ffi

w€0

,(nx~ 1,w 1(w) )dP(w)dP(w 1)

=

{J

r

Jw•Eo

for every $

€

Cb, n

€

w€0

C(JJRn), t

$C1 x~· (w) )dP(w) }dP(w•) €

( 14) I

[O,a].

Now let B c C(JJRn) be open. Use a un;formly cont;nuous part;t;on of un;ty ;n C(JJRn) to construct a sequence {$ 1 }7.. 1 c Cb such that $1 (n) + xB(n) as 1 + ... for each n € C(JJRn) and l$ 1 (n) I < 1 for all n € C(JJRn), 1 > 1 (Cf. Proof of Theorem (111.1.1)). Put$= $1 ;n (14) and pass to the l;m;t as 1 + ... v;a the dominated convergence theorem, thus obta;n;ng p{w:w

€

0, nxt(w)

€

B} =

J

P{w:w

J

P{w:w € O, nxt001 (w) € B}dP(w•)

J

p(w•,o,n,t,B)dP(w•).

€

0, nXt(w•,w)

€

B}dP(w•)

w•€g

=

w•€g

=

( 15)

w•Eo

s;nce th;s holds for every open set B ;n C(JJRn) and Borel C ;s generated by all open sets, ;t follows by un;queness of measure-theoret;c extens;ons that (15) must be true for all B € Borel C. Hence the f;rst assert;on of the theorem. Suppose now that the coeff;c;ents h, g are L;psch;tz but not necessarily globally bounded on Rn. We ma;nta;n that (14) 1 still holds ;n th;s case. The ;dea of the proof ;s to approx;mate the stochast;c DDE•s (VI), (VI)(w•) by sequences of stochast;c DDE 1 s whose coeff;c;ents are globally bounded and L;psch;tz. For each ;nteger N > 0, and v € Rn, set h(v) hN(v) = { h(v)(2 - lijl> 0

g(v) gN(v) = { g(v)(2 0

Jwl>

lvl < N N < Ivi;< 2N lvl > 2N !vi< N N < lvl < 2N lvl > 2N.

Then clearly hN, gN are globally bounded on Rn by

sup lh(v)l, sup ig(v)l Iv I< 2N Iv I<2N and are also Lipsch;tz. with the same Upschi.tz constant L as that of h,g. Fer each N > 0, let nxN € £ 2(0,C([-r,a]JRn)), nxN,w• , a.a. w• € 0, be the un;que 181

solutions of the stochastic ODE's dxN(t)

=

hN(xN(t-r0))dt + gN(xN(t-d 0))dw(t),

o

xN 0 = nEc dxN,w'(t)

=

x0N•w'

hN(xN,w'(t-r0{w')))dt+gN(xN,w'(t-d0 {w' )))dw(t), O
=

n

View (VI)N and (VI)N(w') as stochastic FOE's with random coefficients HN:Q x C + Rn, GN:Q x C + L(Rm ,Rn) viz. dx N(t)

=

N + GN(·,xt)dw{t), N HN(·,xt)dt U < t
x0N = n dx N•w' (t)

=

N w' )dt + GN(w' ,xt' N w' )dw(t), 0 < t < a HN(w' ,xt'

x0N•w'

=

n

where HN(w,n) = hN(n(-r0 (w))), GN(w,n) all n E c. It is easy to see that

for a.a. wE such that

fl,

=

gN(n{-d 0{w))), for a.a. wE f2 and

all n1,n 2 E C; and there is a K > 0 (independent of w,n,tl)

for a.a. wEn, all n E C. So HN, GN satisfy all the conditions of Theorem (V.4.2). Indeed if BN = {n: n E C, lin II < N} for each N > 0, then HN(w,·)IBN = HN,(w,.)IBN, GN(w,·)IBN = GN,(w,·)IBN if N' > N, a.a. wE fl; so the trajectories {nx~}t>O, {nx~'}t>Oagree up to the time they leave the ball BN. Using the uniform estimates E llnx~llc2 <

C~(1 + lin llc2>

E llnx~·w'llc2 < C~(1 + lin llc2> 182

for all N > 0, all t E [O,a], and a.a. w E n, it ;s not hard to see that nxN + nx as N + m a.s. in £ 2(n,C([-r,a]JRn)). s;m;larly, for a.a. w' En, nXN' w' + nXw' as N + m a.s. ;n £ 2(n,C([-r,a]JRn)). Therefore, ;f ~ E Cb, we have by the dom;nated convergence theorem

fwEn $(nxt(w))dP(w)

=

J

$(nx~(w))dP(w)

f

$(nx~•w'(w))dP(w)

1;m

wEn

N--

and

fwEn $(nX~ 1 (w))dP(w)

1;m

=

N-- wEn

a.a. w' En.

By the remark follow;ng Lemma (3.3), the random fam;ly {nxN,w' :w' E n} has a vers;on ;n £ 2(n x n, C([-r,a]JRn); F0 e Fa) for each N > 0; so each map w'

~ J

wEn

$(nxN,w' (w) )dP(w) t

;s F0-measurable, by Tonell;'s Theorem (Dunford and Schwartz [15], p. 194).

~(nx~'(w))dP(w) ;s also F0-measurable. Therefore, WEn we can once more apply the dom;nated convergence theorem to get Hence the map w' r-->

J

fw'En fWEn~(nx~' (w))dP(w)dP(w')

=

Hm

J J

~(nx~•w' (w) )dP(w)dP(w')

N-- w'En wEn

Now, by the f;rst part of the proof,

JwEn ~(nx~(w))dP(w)

=

J

w'En

J

WEn

~(nx~•w'(w))dP(w)dP(w')

for each N > 0; so (14)' holds by lett;ng N + m ;n the last equal;ty. The relat;on (15) then follows as before. To prove t;me-homogene;ty note that by the above result and Theorem (III. 2.1) we have (PoT! 1(n)- 1)(B) 2

=

J

p(w',;,n.t 2 ,B)dP(w')

=

J

p(w',O,n,t2-t 1,B)dP(w')

=

(PoTt -t (n)

W'EQ W'EQ

2

1

-1

)(B)

for every B E Borel C, 0 < t 1 < t 2 < a, n E C. 183

This completes the proof of the theorem.

c

As a consequence of the above theorem one can show that the trajectory field of the stochastic DOE (V) (or VI) possesses all the distributional regularity properties of §(V.4). In particular, we get Corollary (3.1.1): If the coefficients hj' gi, j = 1, ••• ,p, i = 1, ••• ,q in (V) are Lipschitz, then the trajectory field of (V) satisfies all the conclusions of Theorems(VI.3.1), (V.4.4), (V.4.6), (V.4.7). Proof: These results can in fact be proved using the methods of Chapter V. However, they follow more or less directly from the identity (15) of the theorem. c Definition (3.1): The trajectory field {nxt:t > 0, n € C} of a stochastic FOE is asymptotically stochastically stable if lim po(nxt)-1 exists in the space MP(C) of all probability measures on c. t~ The next corollary says that in order for the stochastic ODE (V) to be asymptotically stochastically stable it is sufficient to verify this property for every stochastic DOE with fixed delays of the type p

0

dx(t) = .E hJ.(x(t-rJ.))dt J=1

x0

=

+

q

0

.E gi(x(t-di))dw(t), 1=1

t >0 (VII)

nEc

and aPbitrary {rj}j= 1• {d~}~= 1 c [O,r]. A similar result for the case gi = 0, i = 1, ••• ,q, hj linear, j = 1, ••• ,p has been established by Lidskii ([48], 1965). Note however that Lidskii allows the delays rj to be timedependent, Markovian in general. Corollary (3.1.2): Suppose that for every choice of fixed deterministic delays rj, d~, j = 1, ••• ,p, i = 1, ••• ,q the system of stochastic ODE's (VII) is asymptotically stochastically stable. Then so is the stochastic DOE (V) for all random delays rj, di € £~(Q,[O,r];F 0 ), j = 1, ••• ,p, i~1 •••• ,q which are independent of the Brownian a-algebra Fw = u Fwt• ~

t>O

Proof: Take rj = rj(w'), d~ = di(w') for a.a. w' € Q in (VII). 184

Then by

hypothesis, lim p(w 1 ,0,n,t,•) = ~(w 1 ), a.a. w1 € n. where {~(w 1 ):w 1 € n} is t-a random family of probability measures on C. If {nxt:t > 0, n € C} is the trajectory field of (V), then the identity {Po(nxt)- 1}(B) =

J

w1 ffi

p(w 1 ,0,n,t,B)dP(w 1 ) , B € Borel C,

the dominated convergence theorem, and Theorem (I.2.2)(v) imply that lim Po(nxt)- 1 = ~ t-+co

~(w 1 )dP(w 1 ) in M (C).

=J

w1 €Q

c

P

Remarks (i) In.our next section we shall give some sufficient conditions in the linear case under which the system with fixed delays (VII) becomes asymptotically stochastically stable. (ii) We conjecture that the first assertion of Theorem (3.1) also holds if in (V) we allow the delays rj' di, j = 1, ••• ,p, i =1, ••• ,q to be continuous stochastic processes, each independent of w. (iii) Although the trajectory field {nxt:t > 0, n € C} of (V) may not in general be isonomous to a Markov process on C, we can still apply the results of Chapter IV to obtain the random family of weak infinitesimal generators Aw :V(Aw ) c Cb + Cb associated with the family of semigroups {~ }t>O defined by the systems I

I

I

p

q

dx(t) = E h.(x(t-r.(w 1 ))dt j=1 J J

x0

=

n

E

+

E gi(x(t-di(w 1 ))dw(t)

i=1

t > 0 (V(w 1 ) )

c

for a.a. w1 € n. Using the notation of Chapter IV and the results of Theorems (IV.3.2), (IV.4.2), we observe that when hj' gi are globally bounded the weakly dense subalgebra Tq of all quasi-tame functions in Cb is determined independently of the choice of the random parameter w1 in V(w 1 ) . In parti1 cular, Tq c V(Aw ) for a.a: w1 € n. ~o suppose ~ € Tq and using functional calculus write formally~ (~) = etAW (~) a.a. w1 € n , t > 0. Then by (14) 1 of the proof of Theorem (3.1) we get

fw€Q ~(nxt(w))dP(w) = Jw €Q ~~(~)(n)dP(w 1 ) = Jw €QetAw 1 (~Hn)dP(w 1 ) 1

1

185

where

Recall that {St}t>O and nxt(w) € 8} function x8

S is the weak infinitesimal generator of the shift semigroup {ek}~= 1 is any basis for ~m. To get the distribution P{w:w € n, of the trajectory field to (V), approximate each characteristic weakly by a directed family {$ 1 } in Tqi thus

P{w:w €

n,

nxt(w) € 8} =lim 1

J

w'€n

w'

etA ($ )(n)dP(w'). 1

As our final topic in this section, we now turn to the question of selecting a measurable version of the trajectory field to the stochastic DDE(V). If the coefficients hj, gi are all Lipschitz, it is easy to check that Remark (V.4.3)(ii) is fulfilled and so the trajectory fields {nxt:t € [O,a],n € C}, {nxt:t € [r,a], n € C} admit measurable versions n x [O,a] x C + C, n X [r ,a] X c + c'X, for any 0 < a < l. Note hePe that we do not pequi'I'e the delay

a-a~ebra F0 to be independent of F:.

Alternatively, and in relation to our Remark (V.3.1)(iii) on delayed diffusions, it is perhaps instructive to indicate here how a measurable version may be obtained for the trajectory field of the one-dimensional quadratic delay equation dx(t) = [x(t-1)] 2dw(t)

0 < t < 1

(VI II) x0 = n € c = C([-1,0JJR) Note that in this case r = 1, m = n = 1 and the diffusion coefficient g:R +R, g(v) = v2, v € R, is clearly not globally Lipschitz on~. In fact g is clearly locally Lipschitz but does not satisfy a linear growth condition and so none of the regularity theorems in §(V.4) can be directly applied to the stochastic DOE (VIII). Nevertheless, we proceed to find a measurable version nx [0,1] x C + C of the trajectory field {nxt:t € [0,1], n € C} as follows. It is sufficient to find a measurable version for the field {nxt- ~t: t € [0,1], n € C} where 186

~t(s)

n(O)

s e: (-t,O]

n(t~s)

s e: (-1,-t), t

= { €

(0,1]

as in §(1V.2). For each n e: C define the sequence {nk}~= 1 of piecewise linear approximations nk e: C([-t,O]JR) by nk(s) = [k(s~1)-j]n(-t + jkt) ~ [j + 1 - k(s~1)Jn(-1 +f) 1"f

se: [ .. 1

j 1 *-y-:sJ= j+ 1J . +1(·-

0,t,2, ••• ,k, k = t ,2,3, •••

R

-2/lc

-1/k

0

Clearly nk + n as k + m in C, and each nk is piecewise c1 and hence of bounded variation on [-1,0]. Define the processes Yk:g x [0,1] x C + C, k = 1,2, ••• , by setti.ng t+s - 2 0 w(w)(u)nk(u-1)(nk)'(u-1)du, s e: [-t,O] s e: [-1,-t)

J

0

for all t e: [0,1], n e: C and all we: g0, where go is a set of full P-measure such that w(w) e: C([ 0,1]JR) for each we: go· Therefore for all we: go• t e: [0,1], n e: Cit is clear that Yk(w,t,n) e: C for every k > 1. Moreover each process Yk is (F 8 Borel [0,1] 8 Borel C, Borel C)-measurable. To see 187

this, note that, since Borel C is generated by evaluations at s E [-1,0], it is sufficient to check that for any fixed s E [-1,0] the map (w,t,n) ~ Yk(w,t,n~(s) is meas~ra~le. Now for each k, (nk)'(u-1) = kn(-1 + 4f!> kn(-1 + t> for u E . j = 0,1,2, •••• k. Fix any t 0 E [0,1] and let o < jo < k be such that t 0 E [j 0/k,(j 0+1)/k]. Then

J0tow(w)(u)nk(u-1)(nk)'(u-1)du j 0-1 E

j=O

. k[n(-1 + 1fl>

+ k[n(-1 +

=

.

- n(-1 + t>J

j +1 4-> - n(-1

fj~t)/k

j/k

j Jto + k0)] .

w(w)(u)nk(u-1)du

w(w)(u)nk(u-1)du.

Jolk

But the maps n ~ n(-1 + j/k), (u,n) ~ nk(u-1) are continuous and (w,u) ~ w(w)(u) is measurable, so the integrals (w,n) r->

(j+1)/k k w(w)(u)n (u-1) du, (w,n) j/k

J

depend measurably on the pair (w,n). that the map (w,n) r->

Jto >+

j 0/k

w(w)(u)nk(u-1)du

From the preceding equality, it follows

J: w(w)(u)nk(u-1)(nk)'(u-1)du

is (F & Borel C}measurable for every t E [0,1]. Since this indefinite integral is continuous in t for each w E n0• n E C, it is easy to see that the map

is (F & Borel [0,1] & Borel C)-measurable. From the definition of Yk, it follows that (w,t,n) ~ Yk(w,t,n)(s) is measurable for each s E [-1,0]. Thus Yk is (F@ Borel [0,1]@ Borel C, Borel C)-measurable. Moreover, using integration by parts (Elworthy [19] p. 79), one has a.s. Jt+s [nk(u-1)] 2dw(u) Yk(·.t.n)(s) = { 188

s E [-t,O]

0

0

s E [-1,:t)

i.e. E [0,1], n E C, a.s.

t

Since nk(s)-+ n{s) as k-+ oo uniformly in s E [-1,0], it follo\1S easily from ooob's inequality for the stochastic integral (Theorem (1.8.5)) that Yk(.,t,n) -+ nxt- ~t as k-+ oo in £ 2 (n,c). Hence by the Stricker-Yor lemma for C-valued mappings we get a measurable version Y: n ~ [0,1] x c-. c for . "' } the f1eld {nxt-nt:t E [0.1], n E C. A very similar argument to the above gives a measurable version for the trajectory field of the one-dimensional polynomial delay equation dx(t)

=

[x{t-1}] 1 dw(t).

0

<

t

<

1 (IX)

x0

=

n E C = C{[-1,0]JR}

where 1 E z>O is a positive integer. Furthermore. if 1 is odd, then by the method of §(V.3) it is not hard to see that the trajectory field does not admit a locally bounded (or continuous) version n x [0,1] x C-+ C. In particular, when 1 = 1, every measurable version x:n x [0,1] x C -+ C for the trajectory field of dx(t) = x(t-1 )dw(t) ,

0 < t < 1

is a.s. non-linear on C, i.e. for every t E [0,1] the set

n~ = {w: wEn, X(w,t,•):C-+ C is linear} is contained in a set of P-measure zero in F. For simplicity, take t = 1 and assume that the probability space (n,F,P) is complete; so it is sufficient to prove that P(n~) = 0. To do this, let w0 E n~. From the measurability of X, X(w0,1,·)(0):C -+R is measurable linear and by a theorem of Douady (Theorem (1.4.5)) it must be continuous. So w0 must belong to the set {w:w En, X(w,1,•)(0):C -+R is continuous} which is of P-measure zero by §(V.3).

By completeness of P, n~ E F and 189

P(~) = 0. Note, however, that linearity in probability (or in distribution) always holds in this case i.e. for any n1,n2 E C, u,v E R, t E [0,1], we have

Indeed this last result is vali.d for li.near stochastic FOE's t > 0

x0

=

n

with coeffici.ents H E L(C,Rn), G E L(C,L(Rm ,Rn)) due to (pathwise) uniqueness of solutions (Theorem (ll.2.1)). lt is interesti.ng to observe here that for stochasti.c 1inear ODE's on Euclidean space Rn dx(t) = h(x(t))dt

~

g(x(t))dw(t),

t > 0

(X)

x(O) =vERn, hE L(Rn), g E L(Rn,L(Rm,Rn)) the trajectory field {vx(t);t > 0, vERn} possesses a measurable version X;Q x ~ x Rn ~Rn which is a.s. linear on Rn i.e. for a.a. wEn, all t > 0, X(w,t,·) E L(Rn). This follows from the easily-verifiable fact that for a measurable field n x Rn ~Rn linearity in probability is equivalent to almost sure linearity. For the simple one-dimensional linear stochastic ODE dx(t) = x(t)dt

~

cx(t)dw(t),

t

>

0

x(O) = v E R. 2

Ito's formula shows that the process X(w,t,v) = ve< 1-ic )t~c[w(w)(t)-w(w)(O)] a.a. w E n, t > 0, v E R, gives a measurable version of the trajectory field which is a.s. linear in the third variable v E R. More generally a measurable version for the trajectory field of the linear system (X) can be constructed by solving the associated fundamental matrix equation e.g. as in Arnold ([2], pp. 141 -144) •

190

§4. Linear FOE's Forced by. White Noise As before, we take (n,F,(Ft)tSR'P) to be a filtered probability space satisfying the usual conditions. Note that here we require the filtration (Ft)tSR to be parametrized by all time, with an m-dimensional standard Brownian motion w:n + C(RJRm) adapted to it. Let H:C = C(JJRn) +Rn be a continuous 1inear map and g:R + L(Rm JRn) be measurable such that llg( ·)II is locally integrable over R, where II · II is the operator norm on L(Rm JRn). Consider the fo~ced tinea~ system dx(t) = H(xt)dt

x0

=

+

g(t)dw(t),

t > 0 (XI)

nEc

as opposed to the unforced deterministic linear RFDE: dy(t)

=

H(yt)dt

Yo

=

n E c.

(XII)

The dynamics of (XII) is well-understood via the fundamental work of J. Hale ([26], Chapter 7). In particular, the state space C splits in the form ~

c = u • s.

( 1)

The subspace u is finite-dimensional, sis closed and the splitting is invariant under the semigroup Tt:C + C, t > 0, Tt(n) = Yt for all n € C, t > 0. (Hale [26] pp. 168-173, c.f. Mohammed [57] pp. 94-104). According to Hale [26] the subspace U is constructed by using the generalized eigenspaces corresponding to eigenvalues·with non-negative real parts of the infinitesimal generator AH to {Tt}t>O viz.

s € [-r,O) s =

o.

For convenience, identify the spaces L(RmJRn), L(Rn) with the corresponding 191

spaces of n x m and. n x n rea 1 matri.ces. From the Ri.esz. representation theorem, there is a (unique) L(mn)-v.alued measure lJ on J such that

H(~)

= J:r

~(s)dl..l(S)

for all

~

E C(JJRn).

It is therefore possible to extend H to the Banach space "'C = "'C(JJRn) of all bounded measurable maps J +Rn, given the supremum norm. We denote this extension also by H. Solving the li.near FOE (XU.) for initial data in "'C, we can extend the semigroup {Tt}t>O to one on "'C denoted also by Tt:C"' + "'C, t > 0. The splitting (1) of Cis topological, so the projections nU:c + U, rrs:C + S are continuous linear maps. Since dim u < oo, rr 0 has a representation by an L(Rn,U)-valued measure p on J v.iz.. rr 0 (~) = ! 0 r ~(s)dp(s). This formula gives a natural extension to a continuous linear map rr 0 :c"' + u. Defining "'S = {~~~ E "'C, rr u(~) = 0}, we see that "'C has a topological splitting

-

"'c = u • "'s.

(2)

The projection rr~"' ~C + "' s is conti.nuous 1inear, being given by

"' = ~ - rru(~) for all ~ E "'C. rrS(~)

"' by ~U and ~s"' respectively. The following When ~ E "'C, denote rr 0 (~) and ITS(~) lemma shows that the spli.tti.ng (2) is invariant under the semigroup {Tt}t>O. Lemma (4.1):

For each

~

E "'C, and t > 0,

\'le

have

"' [Tt(~)Ju = Tt(~u), [Tt(~)Js"' = Tt(~s). Proof: For E C, the result follows directly from the well-known invariance ---of the splitting ( 1) under {Tt}t>O. To prove it for E "'C, the ~

~

cons~der

following definition of weak continuity for linear opet·ators on C. A linear operator B~C + is '!..eakly continuous if whenever {~k}k=t is a uniformly bounded sequence in C with ~k(s) + 0 as k + oo for each s E J, then B(~k)(s) + ( ask+ oo for each s E J (cf. the 'weak continuity property (w 1)• of Lemma

C

(IV.3.1)).

The Riesz representation theorem implies that every continuous linear map C + U has a unique weakly continuous extension "'C + u. Hence for the first assertion of the lellllla to hold, it is enough to show that rr 0oTt and Ttorr0 are 192

both weakly continuous for all t > 0. lt is clear from the definition of nU:c"' -+ u that it is weakly conti.nuous. As the composition of weakly continuous linear operators on "'C is also weakly conti.nuous, it remains to show that each Tt:C"' -+ "'C is weakly continuous. This is so by the following lemma. The second assertion of the lemma follows from the first because

[Tt(~)]S"'

=

Tt(~) - [Tt(~)]U

=

Tt(~) - Tt(~U) t

> 0,

"'c.

~ €

D

Lemma (4.2): For each t > 0, Tt:.C"' -+ "'C i.s weakly continuous. Proof: Let v(~) be the total variation measure of the L(Rn)-valued measure ~on J representing H. Fix~ E and define ~f:R>O -+R by

C

~f(t) = J~r ITt(~)(s)ldv(~)(s) Now

Tt(~)(s) =

r

(O) + J

t+S

+

ITt(~)(O)I,

H(Tu(~))du

t > o.

t+s > 0

0

~(t+S)

-r < t+S

<

0

Thus

~f(t)

=J

+

max(-r,-t) -r l~(t+s)ldv(~)(s)

1~(0)

0 Jt+S +J 1~(0) + H(Tu(~))duldv(~)(s) max(-r,-t) 0 + J:

H(Tu(~))dul

max(-r,-t) < J l~(t+s)ldv(~)(s) +

[v(~)(J)

-r

max(-r,-t)

< J -r

+

1](1~(0)1+

Jt

IH(Tu(~))ldu)

0 l~(t+s)ldv(~)(s)

· + [v(~)(J)

+1JI~(O)I

193

+-

[\l(J.~}(J)

< F;h(t)

+.

t](Jt ITu(F;)(O)Idu + Jt J0 ITu(F;)(s) ldv(J.~}(s)du)

+-

0

-r

0

C J: f;f(u)du

where f; h(t) = Jmax(-r,-t) IF;(t+-S) ldv(J.~}(s)

+.

[\l(lJ}(J) + 1JIF;(O) I, t

> 0,

-r

and

C = v(J.~)(J)

+

1.

By Gronwall's lemma, we obtain f;f(t) < f;h(t) +- C J: f;h(u)eC(t-u)du, for all t > 0.

C

Now let {f;k};=t be a uniformly bounded sequence in converging pointwise to 0; then the sequence {~kh}~=l is uniformly bounded on [O,t] and f;kh(t) ... 0 ask ... ~ for each t > 0, by the dominated convergence theorem. The last estimate then implies again by dominated convergence that f;kf(t) ... 0 as k ... for each t > 0. ln particular, Tt(f;k)(O) ~ 0 as k ~~for every t > 0. But for each s. € J, F,;k(t+s) -r < t+-s < 0 Tt(F;k)(s) = { k t+-s > 0, k > 1, Tt+s (F; )( 0)

=

so Tt(F;k)(s) ~ 0 as k ~ ~ for each s € J, t >

o.

c

By analogy with deterministic forced li.near FOE's, our first objective is to deriv.e a stochastic variation of parameters formul-a for the forced linear system (XI). The main idea is to look for a stochastic interpretation of the determintstic variation of parameters formula corresponding to non-homogeneous linear systems dy(t) = H(yt)dt + g(t)dt,

Yo

= n

Ec

t

>

0 (XIII)

(cf. Hale [26] pp. 143-147• Hale and Meyer [29]). To start with, we require some notati.on. Denote by t:.:J ~ L(~n) the map 194

= X{O}I• wh~re l E L(~n) is the i.dentity ~ x n matrix. Also, for any linear map B:C{JJRn) + C{JJRn) and any A E C(J,LORmJRn)),

ll

let where and s If

A(s) = (a 1(s), a2(s), ••• ,am(s)), s EJ ~ m n BA = (B(a 1),B(a 2), ••• ,B(am)) E C(J,LOR JR )) a.(s) h the j-th column of then x m matrix A(s) for each 1 < j < m J ~ n ~ m n E J. Thus~each aj E C(JJR ) and BA E C(J,L(~ JR )). F:[a.b] + C(J,L(RmJRn)) is a map, define the stochastic integral

1: F(t)dw(t)

by

[Jb F(t)dw(t)](s) = Jb F(t)(s)dw(t), s E J, a

a

whenever the Ito integral /: F(t)(s)dw(t) ERn exists for every s E J. This will exist for example if F is measurable and IIF(t) II~ dt < oo. In case

1:

1: F(t)dw(t)

c

E C{JJRn) a.s., its transform under a continuous linear map C{JJR"> +Rn is described by ~~

Lemma (4.3): Let L:C(JJRn) +Rn be continuous linear and suppose L:C(JJRn) + Rn is its canonical continuous linear extension using the Riez representation theorem. Assume that F:[a,b] + C(J,L(RmJRn)) is such that /b F(t)dw(t) E n b ~ a C(JJR ) a.s. Then /a LF(t)dw(t) exists and L(Jb F(t)dw(t)) = Jb LF(t)dw(t) = ~ Jb L(f.(t))dw.(t) a a j=1 a J J a.s., where fj(t) is the j-th column of F(t) and wj(t) is the j-th coordinate of w(t), j = 1, ••• ,m. ~:

Represent L by an L(RmJRn)-valued measure on J via the Riesz representation theorem; then use coordinates to reduce to the one-dimensional case m = n = 1. Namely, it is sufficient to prove that if~ is any finite positive measure on J and f E t 2([a,b] x JJR;dt 0 d~). then Jb J 0 a

-r

f(t,s)d~(s)dw(t)

= J 0 Jb

-r a

f(t,s)dw(t)d~(s)

a.s.

(3)

195

Suppose first that f

=

X[a,a]x[y, 6] , the characteristic function of the

=

rectangle [a,a] x [y,6] [a,b] x J. Then (3) holds trivially. Also, by linearity of the integrals in f, (3) is true for all simple functions on [a,b] x J with rectangular steps. Since these are dense in iua,b] x J, dt 8 d\1), we need only check that each side of (3) is continuous in f € £ 2([a,b] x JJR,dt 8 d~). But this is implied by the easy inequalities: EIJb J 0 a -r <

f(t,s)d~(s)dw(t)l 2

~(J)

=

Jb [J0 a -r

Jb J 0 lf(t,s) 1 2 d~(s)dt = a -r

f(t,s)d~(s)J 2dt

~(J)

llfll 22 £

and

c

Remark Since f is £ 2 , there is a version of the process s ~ J~ f(t,s)dw(t) with almost all sample paths in £ 2 (JJR;d~). Next we shall need the following result in 'differentiating• a stochastic integral with respect to a parameter. Lemma (4.4): Assume that f:[O,a] x [O,a] + LORmJRn) is continuous on {(t,u):O < u < t
=

Jt f(t,u)dw(u), 0

t € [O,a].

Then dz(t) viz.

196

=

f(t,t)dw(t)

+ {

J0t ata f(t,u)dw(u)}dt

It f(t,u)dw(u) = It f(u,u)dw(u) * It 0

Jv.

a 0 { 0 ~ f(v,u)dw(u)}dv

0

(4)

for all t e [O,a], a.s. Proof~

Suppose first that "' w is a process on [O,a] with almost all sample

---paths piecewise C1• Define "'z by

"'z(t) = It f(t,u)dw(u) "' = It f(t,u)w"'I (u)du, 0

0

t e [O,a],

a.s. Then almost all sample paths of "'z are differentiable and a.s.

"' (t) = f(t,t)w"' (t) * It a f(t,u)w"' (u)du Z1

0~

1

I

for all t e (O,a). Thus dz(t) "' = "'Z 1 (t)dt = f(t,t)w" ' I (t)dt

"'

= f(t,t)dw(t)

It a f(t,u)w"' •(u)du]dt It a f(t,u)dw(u)]dt "'

+ [

+ [

0

0

~

~lC

i.e. (4) holds if w is replaced by the piecewise c1 process ~If w is Brownian motion on Rm, define piecewise linear approximations of was follows. Let IT~O = t1 < t 2 < t 3 < ••• < tk =a be a partition of [O,a]. Define the process wiT in Rm by IT ~.w w (t) = w(t.) + ~ (t-t.), tJ. < t < tJ.+ 1, j

J

J

where

Suppose G: ox [O,a] -+ L(Rm ,Rn) is an (Ft)O
!

)'

t

2(0,L(Rm ,Rn))

G(t): G(•,t)

is (uniformly) continuous. Let

£

>

0.

Then there is a 61

>

0 such that 197

1 < -x-r. uj'"

-~ .ll.t = i £ (£/.a 1

J

ra

since lt-tjl < ltj+ 1-tjl <mesh rr < o1 for all t E [tj, tj+ 1]. Applying Lemma 2 of Elworthy ([19], Chapter III, §2, pp. 25-28), we obtain k-1 Ja El .E G(tJ.)llJ.wG(t)dwrr(t)l 2 J=1 0 k-1 1 Jtj+1 2 < .E lljt IIG - n G(t)dtll 2 J=1. J t.J s:. <

1 £2 k-1 1 2 E ll.t = 4 £. 4a j=1 J

But, by property of the stochastic integral, there is a o2 k-1 2 1 2 a E I J 0 G(t)dw(t) - E G(t.)fl.wl < 4 £ j=1 J J 198

>

0 such that

if mesh IT< o2 • So if mesh IT< Eli: G(t)dw(t)

o =min (o 1,o 2),

-I: G(t)d~(t)l 2

a k-1 < 2El I G(t)dw(t)- .~ G(tJ.)~J.wl 2 0 J=1 <

1 2 2.4e:

+

one gets

+

k-1 Ia 2El .~ G(tJ.)~J.w- G(t)d~(t)l 2 J=1 0

1 2 2 2.4e: = e:.

Thus we have proved that lim Elia G(t)dw(t)- Ia meshiT +0 0 0

G(t)d~(t)l 2

= 0.

(5)

In particular, since the functions u ~ f(t,u), u ~ f(u,u), u ~ :v f(v,u) are continuous for fixed t and v, we get

I: f(t,u)d~(u) ~ I: t

Io f(u,u)dwil(u)

t

~ J 0 f(u,u)dw(u)

J: ;v f(v,u)dwn(u) in £ 2 as mesh IT

I:

+

f(t,u)dw(u)

~

I: a~

f(v,u)dw(u)

0, for each t € [O,a], v € [O,t]. Therefore the equality

f(t,u)dwil(u) = J:

f(u,u)d~(u)

+

J: {J: :v

f(v,u)d~(u)}dv

(6)

will yield the required assertion (4) of the lemma, if we let mesh IT + 0 and show that the second integral on the right-hand side of (6) tends to J~ {!~ :v f(v,u)dw(u)}dv in £ 2 as mesh IT + 0. To see this consider ElJ:

{I:

:v

f(v,u)d~(u)}dv-


f(v,u)d~(u)

J:

-I:

{J:

:v f(v,u)dw(u)}dvl 2

:v f(v,u)dw(u)l 2dv

for each t € [O,a], by HOlder's inequality and Fubini's Theorem. From the dominated convergence theorem, the right-hand side will tend to zero as mesh IT + 0 if we can prove that the expression 199

v v E1J 0 :v f(v.,u)dwil(u) - 0 :v f(v,u)dw(u)l 2

J

is bounded in v and rr. To see this write

v v El 0 a~ f(v.,u)dwil(u) - J 0 :v f(v,u)dw(u)l 2

J

< 2EI J:

aav f(v,u)dwil(u) 12 + 2K 1 J:

11

a~

f(v,u) 11 2du

for all v E [O,a] and some K1 > 0 (depending on m,n,a). Let M=sup £11 aav f(v,u)ll: 0 < u < v
<

k, then

E I J: :v f(v,u)dWU(u)l 2 k'-1 1 ( Jtj+1 a ) 1 (Jv a \ 2 I: W av f(v,u)du t:.jw + ~t av f(v,u)du}t:.k,wl j:1 J tj k tk 1 k' 2 2 2 < M I: t:.J.t = Mtk'+ 1 < Ma j=1 = El

for every partition rr of [O,a]. Here we have again used Lemma 2 of Elworthy ([19], pp. 25-28). Thus

for all v E [O,a] and all partitions rr of [O,a]. This completes the proof of the lemma. c We are now in a position to state and prove a stochastic variation of parameters formula for the trajectory of the forced system (XI). By virtue of the splitting (2) of "'C, the formula gives a convenient representation for the projections of the flow onto the subspaces u and S of c. Theorem (4.1): In the stochastic FOE (XI) suppose H:C -~>-Rn is continuous linear and g~ + LORmJRn) is locally integrable. Then the trajectory {xt:t E [O,a]} of (XI) through n E C satisfies 200

~ J: Tt-u ~g(u)dw(u),

(7)

xt

=

Tt(n)

x~

=

Tt(nu) + J: Tt-u

~u g(u)dw(u),

(8)

xs

=

Tt(ns)

~

~s g(u)dw(u),

(9)

Jt Tt u 0

-

for all t € [O,a], where~ matrix.

=

x{O} I and I € L(Rn) is the identity n

x

n

Remark (4.1) It is evident from the following proof that the stochastic variation of parameters formula (7) (together with (8) and (9)) still holds if n is replaced by any a € £ 2 (n,C(JJRn);F0 ). Proof: We prove the formula first for g~ y:n x [-r,a] + Rn by Tt(n)(O) + J: [Tt-u y(t)

+

LORmJRn} c1 • Define the process

~g(u)](O)dw(u),

0

< t
= {

n(t)

t

€

J

Since gi[O,a] is c 1-bounded, a straightforward integration by parts implies that the process t

~

y(t) - Tt(n)(O)

=

J: [Tt-u

~g(u)](O)dw(u)

has continuous sample paths and belongs to £ 2 (n,c([O,a]JRn}). Thus y has a.a. sample paths continuous and belongs to t 2(n,c([-r,a]JRn}). Fix t € [O,a] and s € J. Then [Tt-u ~g(u)](s)

_f [T t-u+s -l 0

~g ( u)] ( 0) , -r

<

t-u+s >

t-U+S <

0

0.

Hence

201

t+s Tt+s(n)(O) + J 0 [Tt+s-u~g(u)](O)dw(u), t+s yt(s)

=

y(t+s)

= {

n(t+s),

f

=

> 0

Tt(n)(s) +

· n ( t+s) ,

-r < t +s < 0

[Tt u ~g(u)](s)dw(u) 0 -r < t+s < 0

Jt

t+s > 0

i.e. Yt

=

Tt(n) +

Jt0 Tt- u ~g(u)dw(u),

t > 0.

a.s.,

We prove next that y solves the stochastic FOE (XI). f~[O,a] ~ [O,a] ~ L(RmJRn) by [Tt-u f(t,u)

~g(u)](O)

Defining

0< u< t < a

= {

u > t

0

we see that~ f(t,u) = H(Tt-u ~g(u)) for 0 < u < t
=

{Jt Tt(n)(O)}dt t

+ {J0 =

a

]t

+ (Tt-t~g(t))(O)dw(t)

[Tt-u ~g(u)](O)dw(u)Jdt

H(Tt(n))dt + g(t)dw(t) + {J: H(Tt-u

~g(u))dw(u)}dt.

Now

Jt Tt u ~g(u)dw(u) 0

-

=

n

Yt - Tt(n) E C(JJR ) a.s.,

so by Lemma (4.3), dy(t)

=

H(Tt(n))dt + g(t)dw(t) + H[J: Tt-u

=

H[Tt(n) + Jt Tt-u 0

=

202

~g(u)dw(u)]dt

H(yt)dt + g(t)dw(t)

0 < t
~g(u)dw(u)]dt

+ g(t)dw(t)

;.e. y is a solution of (Xl) in £ 2(g,C([-r,a]JRn)). By the uniqueness theorem (Theorem (11..2.1)), it follcms that for a.a. w € Q, y(w) = x(w) in C([-r,a]JRn). Hence xt

=

Yt

=

Tt(n) *

J:

Tt-u 6g(u)dw(u), 0 < t
a.s. l.f g is just locally integrable, approximate gi[O,a] by a sequence {gk};= 1 of c1 maps gk~[O,a] + L(RmJRn) such that

J: Let xk FOE

€

llg(t) - gk(t) 11 2 dt

0 as k

+

+

oo

•

iCQ,C([-r,a]JRn)) denote the uni.que solution of the stochastic dxk(t) = H(x~)dt + gk(t)dw(t), 0

xk0

=

<

t

<

a

n

for each k > 1. Suppose£> 0. Then there is a k0 > 0 such that ~ llg(t) - gk(t) 11 2 dt < £ for all k > k0 • For k > k0 , consider Ell x~- xtll ~ < E sup

lxk(t+s) - x(t+s)l 2

S€J

t+-s>O < 2E sup S€J

J

t+S

I0

k

{H(xu) - H(xu)}dul

2

t+s>O

J

t+S

+ 2E sup sEJ t+s>O

I 0 [gk(u) - g(u)]dw(u)l 2

< 2a IIHII 2 Jt E llx~- xull 2 du + K1 Jt llgk(u) - g(u)ll 2 du

c

0

<

2a II HII 2 Jt E II xk - x 0

u

u

0

II 2 du + K1£ c

Where K1 > 0 is some positive constant (Theorem (1.8.5)). lemma, we obtain

By Gronwall's

203

E llxk - x 112 < Ke: e2a IIHII2t t t c for all t E [O,a] and all k > k0• This implies that x~ ~ xt as k + t 2 (~.C(JJRn)), uniformly for t E [O,a]. But xtk

=

Tt(n)

We let k +

m

to get

xt = Tt(n)

+

+

ft0 Tt- u hgk(u)dw(u),

0 < t
0

where the limit exists in t 2 (~.C(JJRn)).

1,2, •••

0 < t < a,

lim Jt Tt-u hgk(u)dw(u),

k......

=

m

Indeed

EIJt(Tt u hgk(u))(s)dw(u)- Jt (Tt u hg(u))(s)dw(u)l 2 0

-

0

-

< K1 Jt IITt u[hgk(u) - hg(u)](s) 11 2 du 0

c

-

< K1 Jt IITt u h(gk(u) - g(u)) 11 2 du, 0 -

s

E

J.

Since {Tt}t>O is a strongly continuous semigroup on "'C, there and M> 0 such that

for 0 < u < t, k

=

exist.~>

1,2, ••• (Hale [26] p. 180). Hence

EIJ:(Tt-u hgk(u))(s)dw(u) - J: (Tt-u hg(u))(s)dw(u)l 2

.

K 1 M2e2 ~a

t

<

K 1 M2 e 2 ~a

J: llgk(u) - g(u) 11 2 du, k

<

ll_gk(u) - g(u) 11 2du

0

=

1,2, ••• ,

and each s E J. But the last expression tends to zero as k + lim Jt (Tt-u

k......

0

in t 2 for each s E J. 204

~gk(u))(s)dw(u)

=

Jt

0

(Tt-~hg(u))(s)dw(u)

m ,

so

0,

in

Therefore xt

=

Tt(n)

+-

J:

Tt-u 6g(u)dw(u), 0 < t
a.s., which pro~es the stochastic ~ariation of parameters formula (7) for locally integrable g. I.t remains to prove fonJI.Ilae (S)~and (9). This is indeed quite simple. Just apply the projections nu and~ to both sides of (7). Indeed

x~

= llu(xt) = nuTt(n) + nu = Tt(nu) +-

Jt

= Tt(n u) +-

Jt

= Tt(nu) +-

Jt

J:

Tt-u 6g(u)dw(u)

u 0 n Tt-u 6g(u)dw(u)

u 0 Tt-u n 6g(u)dw(u)

.

0

Tt-u 6ug(u)dl!l(u), 0 < t
because of LemmBS(4.1) and (4.3). Similarly for (9). a Remark (4.2) By a slight modification of the above argument, it follows that for any t 0 € R the unique solution of the stochastic FOE dx(t)

=

H(xt)dt

+

g(t)dw(t) , t > t 0

x0 = n E c satisfies xt

=

+I: s It to It o

Tt-t 0(n)

Tt-u 6g(u)dw(u)

0

sg(u)dw(u)

xts

=

Tt-t0(n )

+

Tt-u

xtu

=

Tt-t (nu )

+

Tt-u 6u g(u)dw(u)

t0

~

a.s. for all t > t 0• The next step in our analysis is to use the representations (8) and (9) of Theorem (4.1) in order to study the asymptotic behaviour as t +=of the 205

u s projections {xt}t>O' {xt}t>O to the flow onto the subspaces u and

s.

To begin with, recall that u has finite dimension d. Therefore it is possible to thi.nk of {x~}t>O as the solution of an unstable stochastic ODE (without delay) on :Rd. We make this more preci.se by appealing to the following considerations which are taken from Hale ([26], pp. 173-190). Define * where lRn * is the Eucli.dean space of all n-dimensional row c* = C([O,r],Rn) vectors. The continuous linear map H:.C -+:R0 defines a continuous bilinear pairing c* X c -+ R:. (a,~)

= a(O)~(O)

+

fo-r Jos'

a(s-s')d~(s')~(s)ds

(10)

where ~ is the L(Rn)-valued measure on J representing H, a E C* and ~ E C. With reference to this bilinear pairing, the generator AH of {\}t>O possesses a (formal) adjoint A*11 :.V(A*H) c: C*-+ c* defined by the relations

-a' ( t),

(A*Ha)( t) = {

r -r

V(A*H) =

{a~a

0 < t < r

a( -s)d~(s), t = 0;.

f

E C*, a is c 1, a'(O) = 0 -r

a(-s)d~(s)}.

Furthermore, o(AH) = o(A*H) and the spectra are discrete consisting only of eigenvalues with finite multipliciti.es. Both o(AH) and o(A*H) are invariant under complex conjugation~ and the multiplicities of the eigenvalues coincide. Construct u* c: c* using the generalized eigenspaces of A*H which correspond to eigenvalues with non-negative real parts. Then dim u* = dim U =d. Take_a basis~= (~ 1 ••••• ~d) for u and a basis - ( ljl.1 \

1¥-\:) ljld

for u* such that (lJij'~i) = oji' i,j = 1,2, ••• ,d. The basis~ of U defines a unique matrix representation BE L(lRd) of AH IU i.e. AH~ =~B. A*HI¥ = Bl¥, where AH~. ~B. A*HI¥, Bl¥ are all formally defined like matrix multiplication •. 206

Note that the eigen~alues of B are preci.s.ely thos.e A e: o(AH) with ReA > o. The reader s.hauld. also obs.er~e here that the s.pli.tting (t) of c is realized bY the bi.li.near pairi.ng (tO) through the fortll.lla ~U

=

~(~o~)

for all ~

€

C.

( t 1)

The results quoted in this paragraph are all well-known for linear FOE's and proofs may be found in Hale [26]. We would like to extend formula (11) so as to co~er all ~ e: "'C. First note·that the bilinear pairing (tO) extends to a continuous bilinear map c* x "'C + R defi.ned by the same formula. So the right hand side of ( 11) makes sense for all ~ e: "'C. But both sides of ( 1t) are continuous with respect to pointwise convergence of uni.farmly bounded sequences in C, because of the dominated con~ergence theorem and the weak continuity of rrU~~ + u. As "'C is closed under pointwise limits of uniformly bounded sequences, (tt) holds for all ~ e: "'C. ln view of the abo~e considerations we may now state the following corollary of Theorem (4.1). Corollary (4. 1.1) :. Define ~ c: u, ~ c: u* and B e: L(Rd) as above. Let {xt~t e: [O,a]} be the trajectory of (Xl) through n e: C. Define the process z~n X [O,a] +Rd on Rd by z(t) = c~.xt)• 0 < t
=

z(O)

Bz(t)dt

+ ~(O)g(t)dw(t), 0

< t
= (~.n)

}

(XIV)

~:.

Use the definition of z, the stochastic variation of parameters formula and Lemma (4.3) to obtain z.(t)

=

c~.Tt(n))

+I: c~.Tt-ullg(u))dw(u)

a.s.

for all t e: [O,a]. Take differentials and use properties of the bilinear Pairing and the generator AH of {Tt}G
207

d~(t)

=

* =

(A

*H

+-

(~.~g(t))dw(t)

I: ~ (~,Tt-u~g(u))dw(u)dt

~.Tt(n))dt

+-

=

(~,AH(Tt(n)))dt

{It

0

(~.

(A*H~,Tt(n))dt + {

I0 (A*H t

*

(~.~)g(t)dw(t)

;t Tt +-

(Lernna (4.4))

-

u~g(u))dw(u)}dt

~(O)g(t)dw(t)

~.Tt-u~g(u))dw(u)}dt

= (B~,Tt(n))dt + ~(O)g(t)dw(t)

-~: (B~. =

(B~,Tt(n)

I: ~g(u)dw(u))dt I: Tt-u~g(u)dw(u))dt Tt-u

+-

(Lellllla (4.3)) +

~(O)g(t)dw(t)

= (B~,xt)dt + ~(O)g(t)dw(t) = B(~,xt)dt + ~(O)g(t)dw(t) =

Bz(t)dt

+ ~(O)g(t)dw(t),

0 < t
Thus z is the unique solution of the stochastic ODE (XIV) through (~.n)ERd. The relation x~ = ~~(t), t € [O,a], follows directly from (11). This completes the proof of the corollary. o The next theorem specifies the asymptotic beha~iour as t ~ ~ of the stable projection {x~}t>O of our stochastic FOE (XI). ln particular the trajectory {x~}t>O is asymptotically close to a continuous Gaussian process. Theorem (4.2): Let H:C ~Rn be continuous linear, g~ ~ L(Rm,Rn) c1-bounded and {xt:t > 0} the trajectory of the stochastic FOE (XI) through n € C. Then there is a sample continuous Gaussian process x:n x [-r,~) ~Rn satisfying:

208

(;)

~xt = J~ Tt-u 6~ g(u)dw(u)

a.s. for all t > 0&

(U)

There are constants K > 0, a.

< 0

(;H)

~x

such that

is the unique solution of the stochastic FOE

~x(t) = H(~xt)dt

~x 0

0

"'

J_~ T_u aSg(u)d\'l{u)

€

t

>

0

.c 2(rl,Ch

{X\l)

is constant, then ~x is stationary•

(;v)

lf

g

(v)

lf

g ;s

~th

=

"' * ~(O)g(t)dw(t)

period;c with per;od k

> 0, ~x

is peroiodic in distroibution

;.e.

pePiod k

Suppose g and ;ts der;vative g• are both globally bounded on R. We def;ne the process ~x:Sl x [-r,~) +Rn by Proof~

~x(t)

Jt = {

J~

"'

{Tt-u 6s g{u)){O)dw{u),

"'

t

> 0

{12)

-~ {T_u 6s g{u)){t)dw{u).

t

€ J

To see that ~x ;s well-def;ned, we note the existence of the l;m;t t

t

"'

"'

J (Tt-u 6s g(u))(O)dw(u) J {Tt-u 6s g{u)){O)dw(u) = 1;m v- v ~

a.s. lndeed the map u r-> (Tt-u 6s"' g(u))(O) ;s c1 and so ;ntegrat;ng by Parts g;ves the class;cal pathw;se ;ntegral

J:

(Tt-u

6~ g(u))(O)dw(u)

=

6l{O)g(t)w(t) - (Tt-v

-J: ~

6~ g(v))(O)w(v)

{(Tt-u 6l g(u))(O)}w(u)du 209·

a.s. for all ~ < t. Now by Hale ([26], p. 187) there are constants M> 0, B < 0 such that

"'

IITt lls g(u) II< MeBt llg(u) II

(13)

for all t > 0, u € R. But the law of the interated logarithm for Brownian motion (Theorem (l.8.2)(iv)) i.mpHes that there is a constant L > 0 such that for a.a. w € Sl there exists T(w) < 0 with the property IW(w)(t)l for all t

<

T(w).

<

L(ltl log log ltl)i ln particular, we

(14)

ha~e

a.s.

"' )(O)w(v:) I < M~ im eB(t-~) llg(~) lllw(v) I Hml (Tt-~llsg(~) vv-oo <MLIIgllc Hm eB(t-~) 1 v-oo =

ML llgllc eBt Hm e-B~( 1~1 log log 1~1> 1 v-

< ML llgllc eBt lim lvl e-B'l vwhere

llgllc

=

sup { llg(t) II:. t

€

=

0

R}.

Next we consider the existence of the a.s. limit lim 'l-

t

"'

v

.

J ~ {(Tt-u lls g(u))(O)}w(u)du.

To prove its existence, i.t i.s sufficient to demonstrate that t

"'

J_ool.~ {(Tt-u llsg(u))(O)}w(u) ldu

<

oo

a.s. Note first that

"'

~ {(Tt-u lls g(u))(O)} for u < t;. so 210

=

"'

"'

-H(Tt-u llsg(u)) ~ (Tt-u lls g•(u))(O)

Ia~ {(Tt-u 6s"' g(u))(O)}w(w)(u)l

"'

< IIHII IITt-u 6s g(u) II lw(w)(u) I < IIHII

M

llgllc eB(t-u)lw(w)(u)l

IIHIIIIgllc

= M(

+

< LM( IIHIIIIgllc

t

J-m

+

Mllg'llceB(t-u)lw(w)(u)l

llg'llc>e 6t lui e-Bu

o. Since B < 0 the integral

< m and hence so is "'

I~ {(Tt-u 6s g(u))(O)}w(u)l du

min (T(w),-t)

=

+.

llg'llc>e 6t e-Bulw(w)(u)l

for all u <min (T(w),-t) and a.a. wE

J~lule-Budu

"'

lltt-u 6s g'(u) II lw(w)(u) I

+

J-m

"'

I~{(Tt-u6sg(u))(O)}w(u)ldu

t

9

.

J.

mtn(T(w),-t)

"'

I~{(Tt-u6s g(u))(O)}w(u)ldu

for t > 0. From the definition of mx the reader may easily verify that the trajectory {mxt}t>O satisfies assertion (i) of the theorem. Thus we can look at the difference

(x~"' - mxt)(s)

=

"'

Tt(nS)(s) *

0 f_m

"'

{Tt-U*S 6S g(u)}(O)dw(u)

for all s E Jandt> r. Let At:n x J +Rn be the process At(s)

=

0

"'

J_m
for each fixed t > r. Then integration by parts gives

211

~

At(s)

=

~

{Tt+s ~s g(O)}(O)w(O) • 1~ {Tt-u~s~s g(u)}(O)w(u) u-

0

-lUn 'l-

J

~

[·H(Tt-u+s

~

~s g(u))+ {Tt-u~s ~s g•(u)}(O)]w(u)du

'l

a.s. for t > r, s € J. From the subsequent discussion we shall see that At is (F0 8 Borel J, Borel mn}-measurable and has a.a. sample paths continuous on J. Futhermore, one has the following esti.mates satisfied a.s. lA {s) I < MeB{t~s) llg{O) II lw{O) I + M1im eB{t-u+s) llg{u) II lw(u) I t u-m

~

f4

J0 eB(t-u-~:s)

IIH II lim.

.., M 1im 'l-m

llg{u) II lw(u) ldu

V

'l-m

J0 eB(t-u+s)

llg• (u) II lw{u) ldu

V

< MeB{t-r) llg(O}IIIw(O)I ~ MeB{t-r) llgllclim e-Bulw(u)l u-

for t > r, s € J. In the above inequalities, all limits exist a.s. because for a.a. w € n,

for all u

<

lim

min{T{w),-1).

u-m

e-Bu lw(u)l = 0

and so if we let

r v

for a.a. w E n, we get 212

In particular, a.s.

e-Bu lw{w)(u) ldu

sup IAt(s>l < MeB(t-r) llg(O)II lw(O)I

se:J

+ M(

+ 119'

lie >~ 1

eB(t-r)

> r.

a.s. for all t Thus a.s.

llx~

IIHII ll9llc

- coxtll c <Meat

II nsllc

... sup IAt(s) I

s.e:J

< Meatllnsllc + MeB(t-r) llg(O) II lw(O) I .., M(

IIHII II gil c.., llg' llc>K1 eB(t-r)

for t > r. I.n particular,

To establish assertion (i.i.) of the theorem, note that by cominated convergence and Fubini•s theorem one gets

E~~ = E(J: e-sulw(u)~) 2 < =m

J: e-BuEiw(u)l 2du

J~co e-Budu

J~co (-u)e-Sudu J~co e-audu

m = - :J .

s

Thus E llx~ for t > r.

-coxtl~

<

3.1 e26t

2 llnsll ~ - 3 M6

f<

IIHII llgllc + llg'llc> 2 e2S(t-r)

Now take

and a = 2S < 0.

Then

213

We next

s~ow

that mx is the solution of the 0

~tochastic

FOE (XV) through

"'

e = J~ T_u As g(u)dw(u). By our main existence theoren (Theorem (11.2.1)), the stochastic FDE(XV) will have a unique solution in .c 2(o,C([-r,a],Rn)) for any a > 0 if e e:.c 2 (o,C). ln fact, we can write mx in the form. Tt(e)(O) *

t

"'

J [Tt-u As g(u}](O)dw(u),

t>O

0

mx(t) = {

(15)

t e: J

e(t)

provided we can show that e e: .c 2(o,C). Now the stochastic integral depends measurably on parameters, so the process e(s)'

=

0

"'

J~ [T-u As g(u}](s)dw(u),

s e: J,

has a measurable version e~o x J +Rn according to a result of Stricker and Yor ([72], Theoreme t, §5, p. 119). Indeed e i.s sample continuous. To see this, use integration by parts to write e(s) = =

Js

-m

"' [T_u+s As g(u}](O)dw(u)

"' s S' "' AS(O)g(s)w(s)-lim J [-H(T_U+SA g(u))+-{T_U+SASg'(u)}(O}]w(u)du (16) y+-m

V

a.s. for all s e: J. Since g, g• and ware continuous, thP. processes s

1---+

AS'(O)g(s)w(s) s

"'

"'

s ~ J [-H(T_U+S AS g(u)) + {T_U+S AS g'(u)}(O)]w(u)du v are sample continuous on J. Thus for e to have continuous sample paths one needs to check that the a.s. limit on the right-hand side of (16) is uniform for s e: J. Now, for v < s < 0,

!fvs [-H(T_U+S AS"' g(u)) + {T_U+S AS"' g'(u)}(O)]w(u)du

214

- J_~s [-H(T_u+-s 6s"' g(u) + T-u+s 6s"' g'(u)]w(u)dul <

(}M IIHIIIIgllc

ea(-u+s) +-

Mllg'llc

ea(-u+s)}lw(u)!du

'l

< M( IIHIIIIgllc +-llg"llc> e-ar J~ e-aulw(u)ldu

(17)

\l

s;nce

Hm v...-~

J

e-aulw(u)!du = 0, it follows illlltediately from (17) that

-~

the Hm;t ;n th·e right-hand side of ( 16) ;s uniform ;n s € J and so a(w) € C for a.a. w E n. As the map J 3 s --~ a(w)(s) € mn is Borel-measurable for a.a. wEn, it follows that a E t 0 (n,C). Indeed, starting from (16), ;t ;s easily seen that

a.s. for all s

€

J. Taking suprema and expectations g;ves

Hence a € t 2(n,C). In fact, by a s;milar argument, one can show that ~xt € t 2(n,C) for all t > 0. Note also that th;s ;s a consequence of assert;on (ii) of the theorem fort> r. Since {w(t)~t ER} is a Gaussian system (§§I.B(A),(B)), the right-hand side of (16) ;s also a Gaussian system. Thus a:n x J +Rn is a Gaussian process (Theorem (1.8.1)(;i)). In a similar fash;on ~x:n x [-r,~) +Rn, ~xt:Q x J +Rn are Gaussian for each f;xed t > 0. The stochastic FOE (XV) for ~x may be der;ved d;rectly from (15) by tak;ng stochastic differentials as in the proof of the stochastic variation of parameters formula (Theorem (4.1)). Thus d~x(t)

=

{J.t Tt(a)(O)}dt

+

(6s"' g(t))(O)dw(t)

215

t "' ~ {J0 ~ [Tt-u~s g(u)]{O)dw(u)}dt

={H(Tt(e)) = H( 00Xt)dt

+

J0t

"'

H(Tt-u ~ s g(u))dw(u)}dt ~

~ ~~(O)g(t)dw(t) •

t

~

ll

(O)g(t)dw(t)

> 0.

Let us now check assertion (v) of the theorem. Suppose g is periodic on R with period k > 0. Consider 00 X(t~k) for t > -r. Then a.s. 00

x(t~k) =

=

Jt+k ~ -oo Tt+.k-u ~ g(u)dw(u)

ft Tt-u' aS"' g(u'+.k)dw{u'~k)

-

(u'

=

u-k)

But g(u'+k) = g(u') and {w(u'):u' E R}• {w(u'+k)-w(k):u' E R} are isonomous, so by isonomy properties of the stochastic i.ntegral (Lemma (111.2.3)) we get X(t+k)

00

= J:oo Tt-u'

~~g(u')dw(u'+.k)- J:00Tt-u'A~g(u')dw(u')

= X(t) 00

for every t > -r. Thus Xj[O,oo) is periodic in distribution with period k i.e. Po x(t)- 1 = Po X(t+k)- 1 for all t > -r. In particular if g is constant, one can take the 'period' k to be arbitrary and so Po x(t)- 1 is independent of t > -r. Hence X is stationary. This completes the proof of the theorem. c 00

00

00

00

Corollary (4.2.1): Let H:C + Rn be continuous linear and g:R c1-bounded. Suppose that the deterministic RFDE dy(t)

=

H(yt)dt,

+

L(RmJRn) be

(XII)

t > 0

is globally asymptotically stable i.e. Re ~ > 0 for every ~ E a (AH). where AH is the infinitesimal generator of {Tt}t>O. Then the process x: g x [-r,oo) + Rn given by 00

00

xt

=

Jt-oo Tt-u ~g(u)dw(u),

t>O

is a sample continuous Gaussian solution of the stochastic FOE 216

dx(t)

=

H(xt)dt

+

g(t)dw(t), t > 0

such that for any trajectory {xt:t > 0} K > 0, a < 0 with CD 2 at E II xt - xt II C < Ke

c t

(XI)

2(g,C) of (XI) there are constants

for a11 t > r.

If g is constant, CDx is stationary; if g is periodic with period k, CDx is periodic in distribution with period k. Proof: If a(AH) lies to the left of the imaginary axis, the splittings (1) and (2) reduce to the trivial case C=

S ,

U =

"' "'

{0}, C = S , ITS

=

ide,

"'

ITS =

ide.

Hence x~ = xt for all t > 0; so all conclusions of the corollary follow immediately from the theorem. a Corollary (4.2.2): Suppose all the conditions of Corollary (4.2.1) hold and let g be constant i.e. g(t) = G € L(RmJRn) for all t € R. Then the stochastic FOE dx(t)

=

H(xt)dt

+

Gdw(t)

t

>

0

(XVI)

is (globally) asymptotically stochastically stable i.e. for every n € C lim Po(nxt)- 1 exists in Mp(C) and is an invariant Gaussian measure for the t-+oo

stochastic FOE (XVI), (§(III.3)).

~: As CDx is stationary, let Po(CDxt)- 1 = ~O = Poe- 1 for all t > 0. We show that the transition probabilities {p(O,n,t,.) = Po(nxt)- 1:t > 0, n € C} of (XVI) converge to ~O in Mp(C) as t +CD for every n € C. Thus it is sufficient to prove that lim J

t-+oo

~EC

$(~)p(O,n,t,d~)

=

J

~€C

$(~)d~ 0 (~)

for every bounded uniformly continuous function $ on C. But by Corollary (4.2.1) above, limE unxt- CDxt11 2 = 0 for every n € C; so it follows from t-+oo

c

217

the proof of Le11111a ( 111.3. 1)(H) that lim t-i.e. lim t--

[J [J

,(nxt(w))dP(w) g

J ,(~xt(w))dP(w)]

= 0, ' € Cb.

g

~EC

'(~)p(O,n,t,d~)

-

J

~E.C

'(~)d~(~)J .

= 0, 'E cb.

Thus lim p(O,n,t,•) = ~ 0 , a Gaussian measure on C. This implies that t-P~(~o> = ~ 0 for all t > 0, where {P~}t>O is the adjoint semigroup on M(C) associated with the stochastic FOE (XVL) (§(111.3)). a Remark (4.3) Note that the invariant measure ~O of Corollary (4.2.2) i.s uniquely determined independently of all i.nitial conditions n € C (or in .c 2(n,C)). 1n contrast with the asymptotically situation when a(AH) has some elements case it turns out that the variance of nx~ in U di\lerges to -~: ~ exponentially have

stable case, we finally look at the with positive real parts. In this every one-dimensional projection of as t + ~ for each n € C. Indeed we

Theorem (4.3)~ Define~ c u*, B € LORd), ~~g x ~ +Rd as in Corollary (4.1.1). For the stochastic FOE (XVI) (constant G), let C = ~(O)GELORm,Rd). Suppose {Aj = aj -~: ibj~1 < j < p} is the set of all eigenvalues of B where aj > 0 and bj € R, j = 1,2, ••• ,p. Denote by<·,·> the (Euclidean) inne~ product on Rd i.e. for u = (u 1, ••• ,ud), v = (v 1, ••• ,vd) € Rd, = _E uivi. 1=1

(i) Assume that the pair (B,C) is contzoottabte viz. rank (C, BC, B2C, ••• ,Bd-1 C) =d.

(18)

Then for every v € Rd, there exists 1 < j < p and t 0 > 0 such that given£ >0 we can find o1, o2, o3 > 0 with the property

o1e

2a.t J

for all t > t 0• 218

-

o2 <

Ell 2 < o3e

(2a.+£)t J

(19)

(ii) lf the rank condition (18) is not satisfied, then either the inequalities (19) hold or else Ell 2 = 0 for all t > 0 and any v E Rd. The following argument is taken directly from (Mohanuned, Scheutzow, Weizs!cker [61]). Fix t > 0. Since z. satisfies the stochastic ODE Proof~

dz{t) = Bz{t)dt

+

it follows that z.(t) = et 8z.(O)

+

Cdw(t)

J:

e
and Ez{t) = e tB z(O) = etB (',n), t > 0. Let v = (v 1,v 2, ••• ,vd) E Rd , w = (w 1,w2, ••• ,wm>· Then Ell 2 = Ell 2 = E[

~

i=1

v.(Jt e
Jt (e (t-u)B C) .. dw.(u)] 2 = E[ dE v. m E 1 1J J i=1 j=1 0

(20)

Now the entry (e(t-u)Bc) .. of the matrix e
where C = (Cij), i = 1, ••• ,d, j = 1, ••• ,m; therefore using (Hirsch and Smale [33], §6.5, Theorem 1) we can write P mk ak(t-u) ( t )B (e -u C) .. = E (t-u) e [aiJ"ksin bk{t-u) + SiJ"kcos bk(t-u)] 1J k=1 for some real constants aijk' Sijk and integers mk > 0, i = 1,2, ••• ,d, j = 1,2, ••• ,m, k = 1,2, ••• ,p. If we insert the above expression into (20) 219

e get Ell 2 d m p Jt mk ak(t-u) (t-u) e [viaiJ'ksin bk(t-u) = E[ t t t i=t j=t k=t 0 +

= E{

m Jt p

t (t-u)

t

j=t 0 k=t

p t

k=1

(t-u)

2

mk ak(t-u) d e [ t v.a .. k sin bk(t-u) i=t 1 1J d t

v.e .. k cos bk(t-u)]dw.(u)} i =1 1 1J J

+

= m t Jt j=1 0

vi Bijk cos bk(t-u)]dwj(u)]

2

mk ak(t-u) d e [ t v.a. 'k sin bk(t-u) i=1 1 1J +

d

because of the independence of wj• j = 1.2 ••••• m. gjk~ ~R be defined by d

2

v.B .. k cos bk(t-u)J} du i=t 1 1J t

Now let the functions

d

g.k(u) = t v.a .. k sin bku J i=1 1 1J

+

t

i=1

v.B. 'k cos bku. 1 1J

u> 0

Then Ell 2 = d

m (21)

t

j=1 d

If all .t viBiJ'k = 0 and if for all k with bk ~ 0 one has .t viaiJ'k = o. 1=1 1=1 2 then El 0. According to (Zabczyk [79]) this cannot happen if (B.C) is controllable. We therefore assume that m 2 m 2 .t lgjk(u)l ~ 0 for some k. Now among all k with .t lgjk(u)l ~ 0 pick J=1 J=1 . one for which ak is largest. and if this is not unique take one with the highest exponent ~· Call this index k0 • Let jo be any index with gj k (u) ~ 0. Then from (21) 00 Jt p m au 2 El 0 { t u k e k g. (u)} du. (22) k=t Jok 220

Suppose y ;sa pos;t;ve constant such that I= {u:lg. k (u)l > y} 1 ~and Jo o let Mk, k = 1,2, ••• ,p be constants so that 19Jok(u)l < Mk for all u > 0, all k. Note that th;s ;s poss;ble because eac.h gj k ;s globally bounded. 0 Then there ex;sts t 0 > 0 such that mko akou rnk aku u e Y > 2E lu e Mkl for all u > t 0 , where the sum ;s taken over all k 1 k0 for wh;ch lgj 0k(u)l 2 1 0. Thus J

p mk aku J mko ak u mk aku 2 2 u e gj k(u)} du > {u e 0 gj k (u)-Eiu e Mkl} du o k=1 o [t 0 ,tJni oo t

{ E

>! J lu~oeakou Yl2du > D e2akot- D 4 [to,tJni 1 2 for some constants o1,o2 > 0, .s;nce gj k ;s per;od;c. Hence by (22) we have 0 0 2 2akot Ell > D1e - D2 for all t > t 0 • On the other hand Ell

2

m Jt p p mk+m1 (ak+a 1)u = E E E u e gjk(u)gj 1(u)du j=1 0 k=1 1=1

p p Jt ~+m 1 (ak+a 1)u m < E E u e E lgjk(u) I lgj 1(u) ldu, t > 0. k=1 1=1 0 j=1 Def;n;ng k0 , jo as before and lett;ng m

Mki. =sup { E lg.k(u)l lgJ.1 (u)l : u > O}, j=1 J we can f;nd u1 > 0 such that 2~ 0

u

2ak0u _ ~+m 1 (ak +a1 )u e y > 2 E u e Mkt (k,t)

for all u > u1, where summat;on ;s taken over all k,t 1 k0 w;th Mkt 1 0, and y = max {Mk 1:1 < k,t < p}. Therefore 221

2 3 Jt 2mko 2akou El<'l,z.(t)-Ez.(t)>l < D+ '2' u e y du u1

for some D> 0 and all t > u1• But for any such that

£ >

0, there exists a

D> 0

= (2ak 0+£)t 3 Jt 2111ko 2ak 0u _ 3 Jt 211\ 2ak 0u _ '2' u Oe ydu< 2 u e ydu
u,

0

So (2ak ~£)t 2 0 Ell < o3 e for all t > 0 and some constant 03 theorem. a

>

0. This completes the-proof of the

Remark (4.4) ln the hypePbotic case when no element of o(AH) lies on the imaginary axis, it is easy to see that there are constants M, A > 0 such that I Ez(t) 1 >MeAt for all t > 0 (Hirsch and Smale [33], Chapter 7 §1, Theorem 2, pp. 144-150).

222

VII Further developments, problems and conjectures §1.

Introduction

The greater part of the discussion in this chapter will be sketchy and often speculative in nature. Our purpose is to introduce the reader to some further examples and open problems in stochastic functional differential equations. The examples include a generaLised Ornstein-UhZenbeck pPocess (§2), stochastic FDE's with discontinuous initiaZ data (§3), stochastic integPo-diffePentiaZ equations (§4) and stochastic DE's with infinite deZays

(§5). The latter class of equations was first systematically studied by K. Ito and M. Nisio [41] in 1964. §2. A Model for Physical Brownian Motion The classical problem of describing the random motion of a molecule in a gas 'heat bath' goes back to Einstein [17], Ornstein and Uhlenbeck [76]. In 1966 R. Kubo [44] proposed the follO\'Iing modification of the Ornstein-Uhlenbeck process d~(t) =

v(t)dt } t (I) mdv(t) = -m[J e(t-t')v(t')dt']dt * y(~(t),v(t))dw(t), t > t 0•

to

The molecule is of mass m and moves under no external forces; B represents a frictional (viscosity) coefficient function having compact support, and y~3 x ~ 3 ~~is a function giving the random gas forces on the molecule. The position and velocity of the molecule at time t > t 0 are denoted by ~(t), v(t) E R3 respectively. Without loss of generality we may assume that t 0 = 0, supp Bc [O,r] for some r > 0 and m ~ 1. It is clear that one needs to specify viJ, J = [-r,O], for equations (I) to make sense. As usual w stands for 3-dimensional Brownian motion. Now (I) is a stochastic FOE. To see this note that 223

I0t S(t-t')v(t')dt' = Io-t S(-s)vt(s)ds, and define the mappings H:Jt'l 3 3 p 0 :C(J,R ) + R by ,

"'

H=

for

11

=

( o 0

€ C(JJR3 ).

C(J,R3 )

+

R3 ,

> 0,

H':JtO

x

C(J,R6 ) ~

R6 ,

0

-I

-t

H(t,n)

x

t

S(-s)n(s)ds

0

f_r

S(-s)n(s)ds

< t

0

< r

t>r

Po \

H )

0 Setting x(t) = (f;(t)) v(t) € R6 , t > -r, "'w(t) = ( w(t) ) € R6 ,

it is easily seen that equation (I) is equivalent to the stochastic FOE dx(t)

=

"'H(t,xt)dt

+

"' y(x(t))dw(t),

t

> 0

(II)

in R6 • Note that this stochastic FOE is time-dependent for 0 < t < r but becomes autonomous for all t > r. If a € t 1([0,r],R), it follows that Hand "'H are continuous and Lipschitz in the second variable uniformly with respect tot € ~. In fact H(t,•), H(t,•) are continuous linear maps with norms IIH(t,•)ll, IIH(t,•)ll uniformly bounded in t € ~. Now xiJ is specified by v.IJ and so from Theorem (11.2.1), (III.1.1) the stochastic FOE (II) has a unique Markov trajectory {xt}t>O in C(JJR6 ) with given viJ. The Markov process {xt}t>O is time-homogeneous for t > r. In contrast to the classical Ornstein-Uh.lenbeck process, observe here that the pair {(F;(t), v(t)):t > -r} does not correspond to a Markov process on R6 , yet the trajectory {(f;t,vt): t > O} in C(J,R6 ) does have the Markov property. We would like to consider the velocity process {v(t):t > -r} in the simple case when the noise coefficient Y is identically constant i.e. let y(x,y) = mr0 € R for all x,y € R3 • Then v satisfies the autonomous stochastic FOE (III)

224

fort> r, where H0 :C(JJR3 ) +R3 is given by H0(n)

= -

Jo-r S(-s)n(s)ds,

3

n E C(JJR ).

By Theorem (\ll.4.1), write vt

=

Tt-r<~r) ~

Jt Tt-u 6yo dw(u), r

t > 2r

a.s., where {Tt}t>O is the semigroup of the deterministic linear drift FOE

> r.

t

Now suppose J 0 S(-s)ds < TI/2r.

We show that if A E o(AH 0 ), the spectrum H of the generator of {Tt}, then Re A < 0. Write A = A1 + u 2 € o(A 0), for some A1,A 2 E R. Suppose if possible that A1 > 0. Using Hale [26] (LeDIIla (2.1), p. 168), A satisfies the characteristic equation o S(-s)eAsds = 0 ( 1) A+ -r

J-r

Hence. A1

Jo-r S(-s)e 1s cos A2s ds A

+

and

= 0

(2)

(3)

But from (3), IA 2sl < IA 2 1r < r

Jo

-r

As S(-s)e 1 lsin A1slds

J-r

< r 0 S(-s)ds < TI/2

for all s € J. Therefore cos A2s > 0 for all s € J and is positive on some open subinterval of J. So from (2), A1 =- f~r S(-s)eA1S cos A2s ds < 0 where S is assumed positive on a set of positive Lebesgue measure in J. This is a contradiction and ReA must be less than zero for all A E o(AH 0 ). Therefore, according to Corollary (VI.4.2.1), we obtain 225

Theorem (2.1): In the system (I) assume that y is constant (:my 0) ,a has compact support in [O,r], a E .c 1([o·,r]_rQ) and 0 r} of (I) and positive real numbers K,a such that m~(t) = m~(r)

(i)

+

Jt mv(u)du r

mvt

J~ Tt-u

=

t > r,

AyO dw(u)

a.s., (ii) for every solution

(~,v)

of (1),

'E ~~~t - m~t~~2 < Ke-at E llvt - mvtlt2 < Ke-at for all t > 2r, (iii) my is stationary and m~ has a.a. sample paths c1• Remark Physically speaking, the above theorem implies that the 'heat bath' will always eventually stabilize itself into a stationary Gaussian distribution for the velocity of the molecule. §3.

Stochastic FOE's with Discontinuous Initial Data

This is a class of stochastic FOE's with initial process having a.a. sample paths of type .c 2 allowing for a possible finite jump discontinuity at 0. These equations were studied by T.A. Ahmed, S. Elsanousi and S.E.A. Mohammed and can be formulated thus: dx(t)

=

x(O)

=

H(t,x(t),xt)dt v E .c 2(n,Rn)

x(s) = e(s)

+

G(t,x(t),xt)dz(t), t

> 0

}

for all s E [-r,O).

In (IV) the initial condition is a paiP (v,e) where v E .c 2(n,Rn) and 226

(IV)

a € t 2(g,£ 2(JJR")). Note that here we confuse £2 with L2, the Hilbert space of all equivalence classes of (Lebesgue)-square integrable maps J +R". The trajectory of (IV) is then defined as pairs {(x(t),xt):t > O} in R"xt 2(JJR"). We assume that the coefficients

are measurable with the maps H(t,.,.), G(t,.,.) globally Lipschitz on R" x t 2(JJR") having their Lipschitz constants independent oft € ~. The noise process z·:R x g +Rm is a sample continuous martingale on the filtered probability space (g,F,(Ft)t>O,P) with z(t,•) € t 2(g,Rm;Ft) for all t € ~ and satisfying McShane's Condition II(E)(i). Using the method of successive approximations (cf. Theorem (11.2.1)), it can be shown that there is a unique measurable solution x:[-r,m) x g +Rn through (v,e) € i(g,JR";F0 ) x t 2(g,i(J.R");F0 ) with a continuous trajectory {(x(t),xt):t>O} adapted to (Ft)t>O (Ahmed [1]). From the point of view of approximation theory, a Cauchy-Maruyama scheme can be constructed for the stochastic FOE (IV) in the spirit of McShane ([53], Chapter V, §§3,4, pp. 165-179). For more details on this matter see [1]. In addition we would like to suggest the following conjectures: Conjectures (i) In the stochastic FOE (IV), suppose the coefficients H,G satisfy the conditions of existence mentioned above. Let z = w, m-dirnensional Brownian motion adapted to (Ft)t>O. Then the trajectory {(x(t),xt):t > 0} corresponds to a Feller process on R" x t 2(JJR"). If H, G are autonomous viz dx(t) = H(x(t),xt)dt

+

G(x(t),xt)dw(t), t

>

0,

(V)

then the above process is time-homogeneous. The transition probabilities {p(t 1,(v,n),t 2,.):0 < t 1 < t 2, v € R", n € t 2(JJR")} are given by p(t 1,(v,n),t2,B)

=

P{w :w € g, ((v,n)x(w)(t 2 ), (v,n)xt (w)) € B} 2

where B € Borel OR" x t 2(JJR")} and (v,n)x is the unique solution of (IV) through (v,n) € R" x t 2(JJR") at t = t 1• 227

(ii) Let Cb = Cb(Rn x £ 2(JJRn)JR) be the Banach space of all uniformly continuous and bounded real functions on Rn x £ 2(JJRn). Define the semigroup {Pt}t>O for the stochastic FOE (V) by t > 0, ~

€ Cb.

Define the shift semigroup St:Cb St(~)(v,n)

=

"'

~(v,nt),

Cb, t > 0, by setting

+

t > 0,

for each ~ € Cb. The semigroups {pt}t>O and {St}t>Owill have the same domain of strong continuity C~ ~ Cb (cf. Theorem (IV.2.1)), but it is not cleaP if C~ ~ Cb in this case (cf. Theorem (IV.2.2)). However it is easily shown that both semigroups are weakly continuous. Let A,S be their respective weak infinitesimal generators (cf. IV §3); then we conjecture the following analogue of Theorem (IV.3.2): Suppose~ € V(S).~ c2, D~ globally bounded and o 2 ~ globally bounded and globally Lipschitz. Then ~ € V(A) and A(~)(v,n)

=

S(~)(v,n) + o 1 ~(v,n)(H(v,n))

+

2

1 n L

j=1

2

D 1 ~(v,n)(G(v,n)(e.),

J

G(v,n)(e.)) J

where o 1 ~. D~~ denote the partial derivatives of ~ with respect to the first variable and {ejlj= 1 is any basis for Rn. Remark In contrast with the non-Hilbertable Banach space C(JJRn), the state space Rn x £ 2(JJRn) carries a natural real Hilbert space structure and so Cb(Rn x £ 2(JJRn)JR) contains a large class of smooth (non-zero) functions with bounded supports. By a result of E. Nelson (Bonic and Frampton [6]), a differentiable function on C(JJRn) with bounded support must be identically zero. §4. Stochastic Integra-Differential Equations In the stochastic integra-differential equation (SIDE)

228

dx(t)

,o

{J_r

=

h(s,x(t~r(s)))ds}dt ~

Jo

{ -r

g(s,x(t~d(s)))ds}dz(t),t >

0

(VI) x(t)

= e(.)(t),

= [-r,O]

t € J

z:n + c(ltD ,.Rm) is a continuous Rm-valued marti.ngale on a filtered probability space (n,F,(Ft)t>O,P), satisfying the usual conditions of McShane (Conditions E(i) of Chapter Il). The coefficients h:J x Rn +Rn, g:J x Rn + L(RmJRn) are continuous maps which are globally Lipschitz in the second variable uniformly with respect to the first. Denote their common Lipschitz constant by L 0. The delay processes r,d:J x n + J are assumed to be (Borel J 8 F0 • Borel J)measurable and the initial condition e € £ 2(n,C(JJRn)•F0 ). To establish the existence of a unique solution we shall first cast the stochastic IDE (VI) i.nto the general format of Chapter ll §1. lndeed, let us define the maps ~:£ 2 (n,C) + £ 2 (n,.Rn), g:£ 2 (n,C) + £ 2(n,L(Rm JRn)) as follows:

"'h(~)(w) = Jo-r h(s,~(w)(r(s,w)))ds "'g(~)(w) = Jo-r g(s,~(w)(d(s,w)))ds for all FOE

~

€ £ 2 (n,C), a.a. w € n.

dx(t)

=

Observe now that (VI) becomes the stochastic

"'h(xt)dt + "'g(xt)dz(t),

t >0

Note also that the coefficients "'h, "'g are globally Lipschitz because if 2 ~,.~ 2 € £ (Sl,C), then "' "' llh<~, )-h(~2> II 2 2 £

=In

n

(SlJR )

0

1J_r {h(s,~ 1 (w)(r(s,w)))- h(s,~ 2 (w)(r(s,w)))}dsl 2 dP

< rL 2 Jn

1~ 1 (w)(r(s,w))- ~ 2 (w)(r(s,w))l 2 dP

< rL 2 11~1

- ~2112 2 £

(Sl,C)

•

229

A similar inequality holds for "' g. To check that "'h, "' g sati.sfy the adaptability condition E(iii) of Chapter l i §1, notice that the processes (s,w) 1------7> h(s,liJ(w)(r(s,w)), (s,w) ~ g(s,ljl(w)(d(s,w))) are (Borel J 8 F.J-measurable whenever liJ € £ 2(n,C;Ft), fort> 0. Thus by Theorem (11.2.1) the stochastic IDE (Vl) has a unique sample continuous trajectory {xt~t > 0} in C(JJRn) through e. The trajectory field of (VI.) describes a time-homogeneous Feller process on C if z = w, m-dimensional Browni.an motion and the delay processes r, d are just (detenninisti.c) continuous functions r,d~J .... J. According to Theorem (11l.3.2), the weak generator A of the assaci.ated semigroup {Pt}1>0 is given by the formula A(~)(n) = S(~)(n)

+

•

-

D~(n1

( Jo-r h(s,n(r(s)))ds X{O})

i J=1.~ o2~

where ~ € Cb = Cb(CJR) satisfies the conditions of Theorem (IV.3.2), and the notation is the sarne as that of I.V §3. If the delay processes r, d:J x Q .... J are assumed to be independent of the Brownian motion w, we believe that the trajectory field of the stochastic IDE (V1) corresponds to a random family of ~~rkov processes on C(JJRn) in very much the same spirit as that of Chapter Ill §3 (Theorem (VI.3.1) for stochastic DOE's). For deterministic delays r, d:J -+- J and Brownian noise z = w the coefficients H:C -+- Rn, G:C -+- L(Rm JRn), H(n)

= J0

-r

h(s,n(r(s)))ds, G(n)

= J0

-r

g(s,n(d(s)))ds, n E C,

are clearly globally Lipschitz and so all the regularity properties of Section ll §4 hold in this case viz. Theorems (V.4.2), (V.4.3), (V.4.4), (V.4.6) and Corollaries (V.4.4.1), (V.4.7.1). It is not clear, however, if the trajectory field admits versions with continuous or locally bounded sample functions. §S.

Infinite Delays

The problem of determining sufficient conditions for the existence (and 230

uniqueness) of stationary solutions to stochastic FOE's was first considered by K. Ito and M. Nisio ([41], 1964) in the context of an infinite retardation time (r =co). We quote here solll! of their results without proofs. For further details the reader should consult [41]. For simplicity, we only consider the one-dimensional case. The state space is the Polish space C = C((.co,O]JR} of all real-~alued continuous functions (-co,O] .... R furnished with the lll!tri.c co lln-t;lln p(n,E;) = ~ 2-n n=1 1 + lin -t;ll n where lln-t;lln = FOE

ln(s) - E;(s) I, n, E;

sup

-n<S
dx(t) = H(xt)dt

+

G(xt)dw(t),

t

>

E

C. Consider the stochastic

0.

(VII)

for a one-dimensional Brownian motion won a filtered probability space (n,F,(Ft)t>O,P) and continuous coeffici.ents H,G:C .... :R. Assume the following~

Theorem (5.1) (Ito-Nisio):

(i) Equation (VLl) has a solution °x:R x n +:R such that 0 x(s) = 0 a.s. for all s < 0 and there is a constant a> 0 with El0 x(t) 14 < a for all t > 0;. (ii) There is a number that

f~

> 0 and a finite positive measure

lJ

on (-co,O] so

0

IH(n)l

+

IG(n)l < M+ J_coln(s)ldll(S)

for all n E c. Then the stochastic FOE (VII) has a stationary solution. Ln [41] se~eral conditions on the coefficients H, G are given in order to guarantee the existence of a stationary solution e.g. Theorem (5.2) (Ito-Nisio): {i)

Under the assumptions

H is of the form H(n) = -H0{n)n(O)

+

H1(n), n

E

c, 231

for some continuous H0 ,H 1 :c

+~;

(ii) G is continuous; (iii) There are constants m, M, M1, M2 > 0 and finite positive measures on (-m,O] such that for all n € C

~1 •

~2

m < H0 (n)

<

M

IH 1Cn>1 4 < M1 +

J~m

IG(n)1 4 < M2 +

J~m ln(s)l 4 d~ 2 (s)

ln(s)l 4

d~ 1 (s)

where

the stochastic FOE (VII) has a stationary solution. Theorem (5.3) (It~-Nisio): H(n)

=

Suppose H, G can be written in the form

-H 0 (n)n(O) + H1(n), G(n)

=

-G 0 (n)n(O) + G1(n), n € c,

with H0 , H1 ,G 0 ,G 1 :c +~all continuous and bounded on C. positive constant m (> 0) such that

If there is a

2H 0 Cn) + IG0 Cn>l 2 < -m for all n € C, then the stochastic FOE (Vll) has a stationary solution. Now in (VII) assume that the coefficients H, G are linear or rather affine of the form. H(n)

= M1 +

G(n)

=

M2 +

J~m n(s)d~ 1 (s) J~m n(s)d~2 (s),

n

€

C,

for M1, M2 € ~.and ~,.~ 2 finite signed measures on (-m,O]. Let 1~ 1 1, denote the total variation measures of ~ 1 • ~ 2 • Define the constants c 232

1~ 2 1 € ~.

: 1 ,c 2 > 0 by

c = - jump of c1 = c2

~,

at 0,

1~ 1 1(-co,O),

= 1~ 2 1c-~,o].

Then the stochast;c FDE (VII) will follow;ng cases: (;) c1

+

lc~

<

ha~e

a stationary solution ;n each of the

c, provided ~,, ~ 2 ha~e compact supports&

}cc:


(;;) c1 + {c~ + + ac 1 c~) 1 1 2 In th;s case the stat;onary solut;on ;s un;que among those w;th sup Elx(t)l 2 < ~. ([41] §11 pp. 47-56).

tER

Tak;ng the further spec;al case G(n) = 1 for all n € C ;.e. M2 = 1 and ~2 = 0, ItO and N;s;o ([41], §12, pp. 51-56) also proved the ex;stence of a un;que stat;onary solution to dx(t) = H(xt)dt under the cond;tions

+

dw(t),

~,c-~,OJ

<

t >

0 and

o,

J~~lsl dl~ 1 1(s)

<

1. Note here that

;n the case when~, has compact support (;.e. a f;n;te retardat;on t;me) the last two cond;t;ons ;mply that the character;stic equat;on

~(A)

=A - J0 eAs -~

d~ 1 (s) = o

of H has all ;ts roots to the left of the ;mag;nary ax;s (Lemma (12.1), p. 54, ;n [41]). Therefore th;s last result of [41] ;s ;ndeed a spec;al case of our results ;n VI §4 v;z. Corollar;es (VI.4.2.1), (VI.4.2.2). F;nally, ;n v;ew of our analys;s ;n Chapter Ill and the fact that C ;s a Pol;sh space, it is tempt;ng to believe that for the general non-linear stochastic FDE (VII) the trajectory field {nxt:t > 0, n € C} also describes a time-homogeneous Markov process on C.

233

References

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239

Index

Absorbing state 111 Adapted 14, 30 Adjoint semi-group 69 a-H6lder continuous (iii) norm 114 Algebraic tensor product 8 Almost sure convergence 5 linearity 190 Approximation of stochastic DOE's 172 Asymptoti.c behaviour 165 fixed point theorems 236 stochastic stability 184

Cauchy-Maruyama scheme 227 cP-bounded function 98 Characteristic equation (i) function 49 Chapman-Kolmogorov identity 19 Chebyshev's inequality 6 Coefficient process (iii), 31 Compact Hausdorff space 1 map 121 Compactifying version 113 Compactness in probability 155 Complete filtered probability space 28 Completion of a probability space 1 Condition (D) 41 (DA) 94 Backward stochastic DE's 237 Banach space (ii) Condition (E) 32 Banach-space valued random Conditional expectation 7 independence 7 variable 4 Bilinear pairing 2 probability 7 Configuration space 31 Bochner Integration 4 Continuation property (ii) integrable random variable 4 of trajectories 40 Borel-Cantelli lemna 6 Borel measurable version (iii) Continuity of the semi-group 71 a-algebra (iv), 2 in probability 148 Continuous bilinear form 12 Browni.an filtration 46 motion, one-dimensional (i), 22 bilinear maps 12 dependence (iii) m-dimensional 22 Feller process {ii) Brownian a-algebra 169 function (ii) time 122 linear functional 2 Bump function 99 240

Continuous path (iii) Contraction semi-group 20, 67 Controllable 218 Convergence in £k 6 in probability 5 Convolution 99 Covariance 21 function 92 Cylinder sets 23 Decomposition of solutions 236 Delay (i) processes 229 a-algebra 170 Delayed diffusions (iii) Deterministic delay equation (i) memory map 31 Deviating argument 235 Diffeomorphism (iii), 117 Differential system (i) difference equations 234 notation 70 Differentiation of stochastic integral with respect to parameters 196 Diffusion coefficient (iv) RFDE 46 Dirac measure 77 Direct sum 79 Discontinuous initial data (v), 226 Discrete-time Gaussian system 21 Distribution 59 Distributional regularity (iv) Domain of strong continuity (iv), 76 Dominat~d convergence 6 Doob's inequality 17

Drift coefficient (iv) RFDE 46 Dunford-Schwartz (D-S) integral 8, 10 (D-S)-~-integrable 9 Eigenfunctions 92 Eigenvalues 92 Eigenvalue problem 13 Erratic behaviour 144 Estimates on higher order moments 150 Euclidean inner product 218 norm 23 space (i) Evaluation map 30 Existence (iii), 33 Expectation 4 Feller process (ii), 20 Filtration 14 Filtered probability space (iii), 14 Finite-dimensional distributions 15 jump discontinuity 79 memory (v) First hitting time 151 Fluctuation-dissipation 236 Forced linear system 191 Formal adjoint 206 Frdchet differentiability 41 Frictional coefficient function 223 Frobenius condition (Fr) (iv), 114 Functional calculus 185 Gateaux differentiable 42 Gaussian distribution 21 241

Gaussian process 21 random fi.eld 21 random variable 21 system. 21 Generalized eigenspace 191 Ornstein-Uhlenbeck process 223 Generic condi.tions 69 Globally asymptotically stable 216 stochastically stable 165 Globally Lipschitz. 46 Group of diffeomorphisms 117 property 119 Growth of solutions 238 Hausdorff topological space 2 Heat bath 223 Hyperbolic case 222 Hypotheses (A) 58 (M) 46 Increments of Brownian motion 23 Independence of events 6 random variables 6 sub-a-algebras 6 indicator function 49 Infinite memory (or delay) (v), 239 lnfi.nitesimal generator, strong 24 weak 71 lnitial data 31 lnitial path (H) process (i) lnv.ariant Di.rac measures 112 Gaussian measure 165 lnv.ariant probability measure 69 Isonomous 15 lto· belated integral 26 242

,.. Ito calculus (i) Ito's formula 126 Ito integral 27

Kolmogorov-Bochner theorem 147 Kolmogorov-Totoki theorem 147 Kronecker delta 146 Laplacian 24 Law of the iterated logarithm 23 t 2-continuous process 34 Linear drift 165 FOE's with white noise 191 growth condition 123 maps (iv) stochastic DOE 144 Linearity in probability 28, 190 Lipschitz coefficients (iv) Local uniqueness 150 Localization technique 143 Locally bounded (iii) compactifying version (iv) 1 . 5 t -sem1-norm t 2-semi-norm (iii) ~-almost everywhere convergence 9 Markov behaviour (iii) process 19 property (ii), 19, 51 trajectories 46 Martingale 16 inequality 35 McShane belated integral 31 type noise (iii) Mean 21 Measure 1

Measure, v.ector-valued 1 finite 1 positive 1 signed 1 Measurable space Measurable versi.on 15, 148 Mercer's theorem 13 Mesh 24 ~-essentially bounded 10 Minimal representation 98 1,1-measurable 9 FOE's 235 Noise (iii) coefficient 224 Noise process 31 Non-existence of continuous versions 144 locally bounded versions 145 measurable linear versions 148 Non-negative definite kernel 92

Neut~al

One-dimensional polynomial delay equation 189 quadratic delay equation 186 One-parameter semi-group (iv) Ordinary diffusion coefficient (iv) Oscillations 235 Parameter space 22 Partition 24 belated 24 Cauchy 24 of unity 53 Periodic family of Gaussian measures 165

Periodicity in distribution 209 Physical Brownian motion (v), 223 Piece-wise linear approximations 187 path 100 Polish space 2 Position 223 Probability measure Probability space (i), Product probability measure 148 Progressively measurable 16 Prohorov's theorem 155 Projective tensor product 8 Pseudo-metric 5 Quasi-tame function

(iv), 105

Random coefficients 33 delays 167 family of Feller processes 169, 171 field 15 gas forces 223 infinitesimal generators 185 RFDE 122 retardations 237 semi-groups 185 transition probabilities 171 Regular measure 2 Regularity properties 113 in probability 113, 149 Resolvent 78 Retardation (v) Retarded FOE's (v) Riemann-Stieltjes integral 35 sums 35 Riemann sum 24 243

representation theorem. tt Right continuous 16 Ries~

a-addithe 1 a-algebra 1 Sample function (iii) regularity ( iv) Semi -group property 68 Semi-martingale 143 Semi-flow ( U) Semi-nriaUon 9 Separable martingale 32 measurable version 163 Shift 71 semi-group 78 Signed bimeasure 12 Simple delay 169 Simple function 8 Slice (ii) Solution process 31, Splitting of C 191 Stable projection 208 subspace 165 State space 15 Stationary transition probabilities 23 Stationary solutions (v), 231 Stochastic DOE (i), 167 dynamical system (i) memory map 3.1 ODE's (v), 49, 165 Stochastic process (i) a.s. continuous 14 continuous 14 locally bounded 14 of class .ck 14. 244

square-integrable 14 vector-valued 14 with continuous sample functions. 14 Stochastic indefinite integral 26 integral (McShane) 24 integra-differential equations (v), 228 RFDE (i). 31 variation of parameters 194 Stopping time 151 Stricker-Yor lemma 15 Strong continuity (iv), 20 dual 2 infinitesimal generator 166 Submartingale 17 Successive approximation 36 Supremum norm ( i i) Symmetric kernel 13 continuous 13 positive 13 Tame function 98 Taylor's theorem 95 Tight measure 2 Time-homogeneous (ii), 20, 64 -lags 235 Tonelli's theorem 183 Total variation measure 3 norm 77 Trajectories (ii) Trajectory field (ii), 111 Transition probabilities 19 Uniform Lipschitz constant 33 Uniformly continuous (iv) Lipschitz 32

Uniform.ly tight 155 Uniqueness (11i). 33 Unstable stochastic ODE 206 subspace 165

Wiener process 22 measure 23

topology 3 Vector fie 1ds 118 Velocity process 224 Version (iii) Borel measurable 15 discontinuous 145 ~ague

smooth

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non-linear measurable 148 Weak continuity of linear operators 192 property (w1) 79 property (~) 82 Weak convergence in Banach space 11 in cb 77 Weak Derivative 78 generator (tv) sequential compactness 11 Weak * topo1ogy 3 Weak (or narrow) topology 3 Weak topology on Banach space 11 on cb 77 Weakly closed 78 Weakly compact map 11 complete 12 continuous extensions 79 continuous semi-group 20 dense (iv.) measurable 4 White noise (v) 245