stochastic functional differential equations
3t
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© S-E A Mohammed 1984 First published 1984 AMS Subject Classifications: (main) 60H05, 60H10, 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 1. Title II. Series 519.2
QA274.23
ISBN 0-273-08593-X
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Reproduced and printed by photolithography in Great Britain by Biddles Ltd, Guildford
To Salwa and Yasar with 1 ove
Contents Preface I.
PRELIMINARY BACKGROUND §1.
Introduction
§2.
Measure and Probability
1
§3.
Vector Measures and the Dunford-Schwartz Integral
8
§4.
Some Linear Analysis
11
§5.
Stochastic Processes and Random Fields
14
§6.
Martingales
16
V. Markov Processes
18
§8.
1
Examples:
21
(A) Gaussian Fields
21
(B) Brownian Motion
22
(C) The Stochastic Integral
24
II. EXISTENCE OF SOLUTIONS AND DEPENDENCE ON THE INITIAL PROCESS
30
§1.
Basic Setting and Assumptions
30
§2.
Existence and Uniqueness of Solutions
33
§3.
Dependence on the Initial Process
41
III.MARKOV TRAJECTORIES
46
§1.
The Markov Property
46
§2.
Time-Homogeneity: Autonomous Stochastic FDE's
58
§3.
The Semigroup
66
IV. THE INFINITESIMAL GENERATOR
70
§1.
Notation
70
§2.
Continuity of the Semigroup
71
§3.
The Weak Infinitesimal Generator
76
§4.
Action of the Generator on Quasi-tame Functions
97
V.
VI.
REGULARITY OF THE TRAJECTORY FIELD
113
§1.
Introduction
113
§2.
Stochastic FDE's with Ordinary Diffusion Coefficients
114
§3.
Delayed Diffusion: An Example of Erratic Behaviour
144
§4.
Regularity in Probability for Autonomous Systems
149
EXAMPLES
165
§1.
Introduction
165
§2.
Stochastic ODE's
165
§3.
Stochastic Delay Equations
167
§4.
Linear FDE's Forced by White Noise
191
VII. FURTHER DEVELOPMENTS, PROBLEMS AND CONJECTURES
223
§1.
Introduction
223
§2.
A Model for Physical Brownian Motion
223
§3.
Stochastic FDE's with Discontinuous Initial Data
226
§4.
Stochastic Integro-Differential Equations
228
§5.
Infinite Delays
230
REFERENCES
234
INDEX
240
Preface
Many physical phenomena can be modelled by stochastic dynamical systems whose evolution in time is governed by random forces as well as intrinsic dependence of the state on a finite part of its past history.
Such models may be ident-
ified as stochastic (retarded) functional differential equations (stochastic FOE's).
Our main concern in this book is to elucidate a general theory of stochastic FDE's on Euclidean space.
In order to have an idea about what is act-
ually going on in such differential systems, let us consider the simplest stochastic delay differential equation.
For a non-negative number r > 0,
this looks like dx(t) = x(t-r) dw(t)
(SDDE)
where w is a one-dimensional Brownian motion on a probability space (R,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) = y(t-r)dt.
(DDE)
One can then immediately draw the following analogies between (SDDE) and (DDE): (a)
If both equations are to be integrated forward in time and starting from
zero, then it is necessary to specify a priori an initial process {A(s): -r < s < 0} for (SDDE) and a deterministic fucntion n:[-r,0] -* R for (DDE).
In the ordinary case (r = 0), a simple application of the (Ito) calculus gives the following particular solutions of (SDDE) and (DDE) in closed form: ew(t)-its
x(t) =
y(t) = et, t E R.
For positive delays (r > 0), no simple closed-form solution of (SDDE) is known to me.
On the other hand, (DDE) admits exponential solutions y(t) = eAt, t ER,
where A E L solves the characteristic equation
(i)
a (b)
e-ar = 0.
When r > 0, both (SDDE) and (DDE) can be solved uniquely through e, n,
respectively just by integrating forward over steps of size r; e.g. e(0) + x(t)
f t e(u-r)dw(u) JO
0 < t < r
-r
e(t) and t
n(0) + f
0 < t < r
o JO
y(t)
-r < t < 0
n(t)
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 sol-
utions y:[-r,00) -). R of (DDE), it is clear that a Markov property cannot hold
for solutions {x(t): t > -r} of (SDDE) in R when r > 0.
Heuristically speak-
ing, 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,0] given the supremum For each t > 0, pick the slice of the solution paths over the
norm 11 11 c.
interval [t-r,t] and so obtain trajectories {xt: t > 01, {yt:t > 01 of (SDDE) and (DDE) traced out in C.
It now follows from our results in Chapter III
(III.2.1), (111.3.1)) that trajectories {xt:t > 01 des-
(Theorems
cribe a time-homogeneous continuous Feller process on C. (e)
As functions of the initial path n E C, trajectories {nxt:t > 01,
{nyt: t > 01 of (SDDE) and (DDE) through n define a trajectory field
Tt:C
- t2(St,C),
t > 0,
nNnxt and a semi-flow
Tt:C -rC , n -r. nyt (ii)
t>0
respectively.
Both Tt and Td are continuous linear, where L2(c,C) is the complete space of all F-measurable 0:c2-'C such that IcIle(w)IIc dP(w) <
, furnished with
the L2 semi-norm
= [ J II 6 (w) II2 dP(w)]1/2.
11 0 11
2
(Cf. Theorem (11.3.1)).
However, in Chapter V
§3, we show that the trajec-
tory field Tt, 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 delayed diffusions (SDDE) with r > 0.
Indeed for the ordinary case r = 0 it is well-
known 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]). At times t > r, the deterministic semi-flow Tt maps continuous paths
(f)
into C1 paths, while the corresponding trajectory field T
takes continuous t
paths into a-Hdlder continuous ones with 0 < a < } (cf. Theorem (V. 4.4)). Now our discussion has so far been with reference to the rather special examples of stochastic and deterministic delay equations (SDDE) and (DDE) above.
However, this is indeed no serious restriction; it is one of our main
contentions in this-book that the observations (a) - (f) are essentially valid for a much wider class of stochastic FDE's than just (SDDE).
Thus in Chapter
II we establish existence, uniqueness and continuous dependence on the initial process for solutions to general stochastic FDE's of the form dx(t) = g(t,xt)dz(t),
x0 = 0 E
t > 0
C2(S2,C)
where the coefficient process g: R>0 x L2(S2,C) - L2(S2,Rn) and the initial process 6 E L2(Q,C) are given, with z a McShane-type noise on a filtered probability space (S2,F,(Ft)t>0,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 Here the drift coefficient H:R>0 x C - Rn takes values in Rn and
ace C.
e diffusion coefficient
G:e
x C - L(Rm,Rn) has values in the space
Rm,Rn) of all linear maps Rm + Rn. ownian motion.
The noise w is a (standard) m-dimensional
If the stochastic FDE is autonomous, the trajectory field
a time-homogeneous diffusion on C.
In Chapter IV we look at autonomous stochastic FDE's dx(t) = H(xt)dt + G(xt)d w(t)
d investigate the structure of the associated one-parameter semi-group t}t>0 given by the time-homogeneous diffusion on C.
A novel feature of
ch diffusions when r > 0 is that the semi-group {Pt}t>0 is never strongly ,ntinuous on the Banach space Cb(C,R) = Cb of all bounded uniformly con-
nuous real-valued functions on C endowed with the supremum norm (Theorem V. 2.2)).
Hence a weak generator A of {Pt}t>0 can be defined on the
.tter's domain of strong continuity Cb + Cb and a general formula for A is ;tablished in Theorem (IV. 3.2).
Due to the absence of non-trivial differ-
itiable functions on C having bounded supports, we are only able to define weakly dense class of smooth functions on C which is rich enough to generate ie Borel c-algebra of C.
These are what we call quasi-tame functions (IV §4).
i 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 itonomous stochastic FDE's are explored in Chapter V.
We look at two extreme
samples: the highly erratic delayed diffusions mentioned above, and the case stochastic FDE's with ordinary diffusion coefficients viz. dx(t) = H(xt)dt + g(x(t))dw(t), t > 0. F g satisfies a Frobenius condition, the trajectory field of the latter lass admits sufficiently smooth and locally compactifying versions for t > r Theorem (V. 2.1), Corollaries V (2.1.1) - V (2.1.4)).
In gereral, the
)mpactifying nature of the trajectory field for t > r is shown to persist in distributional sense for autonomous stochastic FDE's with arbitrary i.pschitz coefficients (Theorems (V. 4.6), (V. 4.7)). iv)
There are many examples of stochastic FDE's. highlight only a few.
In Chapters VI and VII we
Among these are stochastic ODE's (r = 0, VI §2),
stochastic delay equations VI §3), linear FDE's forced by white noise (VI §4), a model for physical Brownian motion (VII §2), stochastic FDE's with discontinuous initial data (VII §3), stochastic integro-differential equations (VII
§ 4 ) ,
and stochastic FDE'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], It6 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 FDE'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 FDE's. The first work that we are aware of goes back to an extended article of Ito and Nisio [41] in 1964 on stationary solutions of stochastic FDE's with infinite memory (r = co).
The existence of invariant measures for non-linear
FDE'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.
Certain 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. Weizsgcker, 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.
(vi)
I Preliminary background
U.
Introduction
In this chapter we give an assortment of basic ideas and results from Probability Theory and Linear Analysis which make necessary prerequisites Due to limitations of space, almost
for reading the subsequent chapters. all proofs have been omitted.
However, we hope that the referencing is
adequate.
§2.
Measure and Probability
A measurable space (S2,F) is a pair consisting of a non-empty set c and a
If E is a real Banach space, an E-valued
a-algebra F of subsets of Q.
measure u on (c,F) is a map u:F -> E such that (1) p(O) = 0, (ii) p is a-additive i.e. for any disjoint countable family of sets {Ak}k=1 in F the series M E
k=1
00
p(A ) converges in E and P( U
CO
A ) =
k=1
k
k
E p(A ). k=1 k
When E = R, u is called
a signed measure and if u(F)=J O it is called a positive measure. sup {!p(A)J: A E F} < -, p is a finite measure. P on (Q,F) such that P(Q) =
1
If
A positive finite measure
is a probability measure on 0; the triple
(S2,F,P) is then a probability space.
The set of all finite real-valued
measures on 0 is denoted by M(S2) and the subset of all probability measures
by Mp(S2). A probability space (S2,F,P) is complete if every subset of a set of P-
measure zero belongs to F i.e. whenever B E F, P(B) = 0, A c B, then A E F. In general any probability space can be completed with respect to its underlying probability measure.
Indeed let (0,F,P) be an arbitrary probability
space and take
Fp= {AUA: AEF, Ac A0EF, P(A0) =0} to be the completion of F under P. A c A0S P(A0) =0.
Extend P to Fp by setting P(A U A) = P(A),
Then (Q,Fp,P) is the smallest P-complete probability space
with F c TP
Because of this property, we often find it technically simpler to assume from the outset that our underlying probability space is complete. 1
When 0 is a Hausdorff topological space and F is its Borel c-algebra, Bore! 0, generated by all open (or closed) sets, a measure u on 12 is regular if u(B) = sup {p(C) = inf {u(U)
C c B, C closed)
:
B c U, U open).
:
If 12 is metrizable, every u E M(S2) is regular and hence completely determined
by its values on the open (or closed) sets in Q.
(Parthasarathy [66], Chap-
ter II, Theorem 1.2, p. 27).
An E-valued measure
Let 52 be a Hausdorff space and E a real Banach space.
u on (0, Borel S2) is tight if (i) sup { Iu(B) I
:
B E Borel S2} < -, where
I.
I
denotes the norm on E; and (ii) for each e > 0, there is a compact set K. in
0 such that Iu(a2,KE) I < E Theorem (2.1):
Let S2 be a Polish space i.e. a complete separable metrizable
space.
Then every finite real Borel measure on 12 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).
o
Let 0 be a separable metric space and F = Borel Q. Denote by Cb(S2,R) the Banach space of all bounded uniformly continuous functions 0:0 - R given the supremum norm
110 II Cb = sup { Iq(n) I : n E st}. The natural bilinear pairing
Cb(S2R) x M(S2) - R
(n) du(n), 0 E Cb(12,R), V E MM,
<$,p> = J TIER
induces an embedding
Cb(S2,R)
M(12)
u
>
where Cb(c,R)* is the strong dual of Cb(12,R).
sponds to the continuous linear functional
Cb(S2,R) 0
2
;-
I ----->
R
q'(n) du(n)
Indeed each u E M(12) corre-
because for every m E Cb(52,R)
if
O(n) du(n)I < II0IICb v(u)(0) nE52
p where v(u)(0) = sup { E
p lp(Ak)I: Ak E F, k = 1,...,p disjoint, 0 =
k=1
U Ak, k=1
p < -} is the total variation of u on 1 (Dunford and Schwartz [15], Chapter III, pp. 95-155). ology.
As a subset of Cb(c,R)* give M(S2) the induced weak * top-
Now this turns out to be the same as the weak toplogy or vague top-
ology on measures because of the following characterizations.
Theorem (2.2): Let S2 be a metric space and u, uk E M02) for k = 1,2,3,...
Then the following statements are all equivalent:
(i)
uk - u as k - - in the weak * topology of M(s2) ;
(ii)
lim
k-
4(n)duk() = fnER
4(n)du(n), for every $ E Cb(S2,R); JnEQ
(iii) lim sup uk(C) < u(C) for every closed set C in Sl; k
(iv)
lim inf uk(U) > U(U) for every open set U in Q; k
lim uk(B) = u(B) for every B E Borel S2 such that u(aB) = 0.
(v)
k-),w
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(c), when S2 is a separable metric space, can be
alternatively described in the following two equivalent ways: (a)
Define a base of open neighbourhoods of any u E M(2) by
U11 (019...90p's E12...sep) _ {v:v E M(2), IJok dv -J 4)k
dpi < Eks
k = 1,2,...,p}
where 41s...94p E Cb(S2sR)s EVE: 2,...,Ep > 0.
3
Furnish M(S2) with a metric p in the following manner.
(b)
Compactify the
ti
separable metric space 9 to obtain Q.
Then Cb(S2,R) is Banach space-isomorphic
to the separable Banach space C(S2,R) of all continuous real functions on S2,
given the supremum norm.
Pick a countable dense sequence {Ok}k=1 in Cb(c,R)
1
and define the metric p on M(S2) by
E
p(u,v) =
k=1 p,v
2
Ok du - J
k
11
1 I1
J
Cb C
0k dvj st
sz
E M(Q) (Stroock and Varadhan [73], Theorem 1.1.2, p. 9; Parthasarathy
[66], pp. 39-52).
$ote that M(c) is complete if and only if S2 is so.
ilarly,Mp(S2) is compact if and only if 0 is compact.
Sim-
More generally compact
subsets of M(S2) 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:S2- E where E is a real Banach space and (S2,F,u) a real measure space (Dunford and Schwartz [15], Chapter III §1-6). On a probability space (S2,F,P) an (F, Borel E)-measurable map X:f -> E is
Such a map is P-integrable if there is a
called an E-valued random variable.
sequence Xn:S2+ E, n = 1,2,..., of simple (F, Borel E)-measurable maps so that
Xn(w) -> X(w) as n -
lim J
for a.a. w E S2 and
m,n-'
IXn(w)-Xm(w)IEdP(w) = 0. wES2
Define the expectation of an integrable random variable X:52- E by EX = J S2
X(w)dP(w) = lim n
Xn(w)dP(w) E E.
J
0
This definition is independent of the choice of sequence {Xnf
.00n=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: Q- E is a random variable if and only if one of the following conditions holds: (i)
There is a sequence Xn:52 -* E, n = 1,2,..., of simple (F, Borel E)-measur-
able maps such that Xn(w) -> X(w) as n
- for a.a. w E S2;
(ii) X is weakly measurable i.e. for each A E E*, AoX:c -* R is (F, Borel R)measurable.
(Elworthy [19], Chapter I, §1(C) pp. 2-4; Rao [67], Chapter I §1.4). Denote by L°(s2,E;F) the real vector space of all E-valued random variables X:S2-> E on the probability space (0,F,P).
The space L°(S2,E;F) is a complete
TVS under the complete pseudo-metric d(X1,X2) = inf[e+P{w:w E S2,IXI(w)-X2(w)IE >E} : E > 0]
for X1, X2 E ro(S2,E;F).
The norm in our real Banach space is always denoted
by I.IE or sometimes just I.I.
A sequence {Xn}n=1 of random variables 00
Xn:S2 -* E converges in probability to X E JCO(1,E;F) if for every e > 0 lim P{w:w E S2, IXn(w) X(w)IE > E} = 0. A random variable X:1+ E is n(Bochner) integrable if and only if the function Sa
le0
W i---;. 1X(W)1E is P-integrable, in which case 1IJQ X(w)dP(w)IE <
LIX(w)IE dP(w) i.e.
E (c,E;F) of all integrable random variables is
IEXIE <
a complete real TVS with respect to the r-sem.i-norm
IIXII9
1
=
J. IX(w) IE dP(w),
X E C'(SQ,E;F).
Similarly for any integer k > 1 define the complete space rk(Q,E;F) of all F-measurable maps X:S7 -> E such that JQ
IX(w)lk
dP(w) <
, endowed with the
semi-norm
IlXllrk
=C
JQIX(w) IE dP(w)]l/k
Note that the spaces rk(S2;E;F) become the real Banach spaces ILk(S2,E;F) if we
identify random variables which agree on a set of full P-measure.
If
Xn, X E CO(Q,E;F), n = 1,2,..., say Xn -, X as n - - a.s. or Xn(w) -* X(w) as n -,.co for a.a. w E S2 if there exist a set Sto E F of full P-measure such that Xn(w) X(w) as n -> - for all w E Slo. 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:S2 + E be random variables for n = 1,2,..., and
let E be a real separable Banach space. 5
X as n 3 - in probability.
(i)
If Xn -> X as n -> - a.s., then Xn
(ii)
If Xn, X E Lk(S2,E;F), n = 1,2,... and Xn
X as n ;
in Lk (k > 0),
then Xn -r X as n -)- co in probability. (iii) If Xn -> X as n -+ oo in probability, then there is a subsequence {Xn }°° i
of {Xn}n=1 such that Xn
- X as i +
Dominated Convergence:
(iv)
Let Xn E L1(c,E;F), n = 1,2,... and X E L°(Q,E;F)
be such that Xn - X as n - - in probability.
Suppose there exists
0 ;F) such that, for a.a. w E S2,jXn(w)1E < Y(w) for all n > 1.
Y E C1(S2,J
i=1
a.s.
X E L1(S2,E;F) and f X(w)dP(w) = lim f 0
n-
Then
x (w)dP(w). S2
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
Theorem (2.4)
(Chung [7], p. 48):.
(Chebyshev's Inequality): If E is a Banach space and
X E £k(S2,E;F), k > 1, then for every e > 0
IX(w) lE
P{w:w E 0, IX(w) IE > e} < E
J
O(w).
S2
In particular the map Lk(S2,E;F) Xa
>
Mp(E)
>
P o X -1
is continuous, for each k > 1.
If E is separable, then the above map is
continuous also for k = 0. In a probability space (S2,F,P), two events A,B E F are independent (under
P) if P(A n B) = P(A)P(B); two sub-a-algebras g1, g2 of F are independent (under P) if P(A n B) = P(A)P(B) for all A E g1 and all B E g2; two random variables X, Y
:
Q - E are independent (under P) if the a-algebras O(X), a(Y)
generated by X, Y respectively are independent under P.
Theorem (2.5)
k}k
I IQ
(1)
F.
1c
If
00
z k=1
6
Let (2,F,P) be a probability space and
(Borel-Cantelli Lermna)
P(S2
k)
converges, then P(lim sup S2 k ) = 0 i.e. P(lim inf 9-S2 k ) = 1; k
k
(ii)
If
{Qk}k=t are independent and
i.e. P(lim inf a-nk) = 0.
E P(S2k) diverges, then P(lim sup 9 k)=1 00 k=1 k
k Proof:
Chung [7], pp. 71-74.
o
For a separable Banach space E, a random variable X:Q -> E on a probability
space (I,F,P) and a sub-a-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 ([191, Chapter I., §3, pp. 5-10).
Theorem (2.6):
Let E be a separable Banach space.
Then there is a unique
continuous linear map
L1(G,E.9)
X t> E(XIg) such that
I
A
X(w)dP(w) = J
E(X Ig)(w)dP(w) A
for all X E t1(S2,E,F), A E g, with g a sub-a-algebra of F.
IE(Xlg)(w) IE <
Furthermore,
E 12 and IIE(- Ig) II = 1.
Conditional expectation given the sub-a-algebra g c F is therefore a projection (of norm one) mapping L1(sl,E;F) onto the subspace LI(St,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 linear maps T:E -> F with the norm IITII = sup {IT(x) IF: X E E, IWIE < 11.
Then for any X E L1(c2,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 (l,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) EL1(St,R;g) and defined as P(AIg) = E(XAIg) 7
where XA is the characteristic function of A.
Two sub-a-algebras g1, g2 of
F are conditionally independent under P given g if P(A1 n A2I9) = P(A1I9) P(A2I9)
for every A
E g1, A2 E
92.
For real Banach spaces E, F, two random variables
1
X:c -
E, Y:Q ; F are conditionally independent under P given g c F if the
a-algebras a{X}, a{Y} are conditionally independent given g.
If the random
variables X E 92(c,E;F), Y E 92(cl,F;F) are conditionally independent given g
and the Banach spaces E,F are separable, then we have E(X ® YJg) = E(XIg) ® E(YIg)
where ® denotes the algebraic tensor product, and E ®
F the complete pro-
jective tensor product (Treves [75], Chapter 43, pp. 434-445; Elworthy [19], Chapter I, §2, §3,
pp. 4-13).
Moreover, if X E to(c,E;g) and Y E L1(c2,F;F)
are such that X 0 Y E r1(c,E ®TrF;F), then E(X ® YIg) = X 0 E(YIg)
a.s.
If h is a sub-a-algebra of g c :F, then
E(E(Xlg)lh) = E(XIh) for any X E L1(c,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 ([151, Chapter IV, §10, pp. 318-328). Let (cz,F) be a measurable space, E a real Banach space and u:F ; E an
Let f:2 - R be a simple function viz.
E-valued measure on (c,F). p
f = Q.
E i=1
xiXA
for some xi E R, i = 1,2,...,p and a partition {A1 }p-1 c F of i
Define
f du = 1 A
Then for any A E F
8
x. u(Ai n A) for any A E F. E i=1
1
Let IIflL.=
max Ix.I 0 0. 1
iEp
f d= IIII00
I
IA
t
u(An A) I < IIf 11.
Xll
1111
II (A)
where the semi-variation llull of p is defined by
IIuII(A) = sup{
I
P E i=1
aiu(Ai)IE:Iail < 1,
= 1,2,...,p,
i
{Ai}p=1
a partition of A in F, p > 1} for any A E F.
Though it is not a measure (it is only sub-additive!), the semi-variation is always finite.
The total variation v(p) of a vector-valued measure u is
not necessarily finite.
Some properties of the semi-variation are given
below:
(i) Ip(A) IE < IIu II (A) for all A E F; (ii) IIpII(A) < 4 sup {lu(B)IE:B E F, B c Al < - for all A E F;
(iii)IIpII(B) < IIuHI(A) if B c A, A, B E F;
(iv) II'II(C i=1 (v)
Ai)
P
i=1
IIu II (A1)
if {Ai}p=1
F;
There is a positive finite measure A on
such that p is absolutely
continuous relative to x and A(A) < IIu Il (A) for every A E F; i.e.
IluII(A) = 0 if and only if x(A) = 0. Now complete F by adjoining all subsets of sets with a-measure zero.
Call
this completion TX i.e.
P ={AUD:AE F,
E0
F, A(Ao) =0}
= {A U A:A E F, A c Ao E F. IIu II (Ao) = 01 A real-valued function f:SZ- R is said to be u-measurable if for every B E Borel R, f-1(B) E FA i.e. if and only if f is a-measurable.
Say a
sequence of functions fk:I -> R, k = 1,2,..., converges p-almost everywhere
to f:Q - R if fk - f as k
- a-a.e. i.e. if there is a A-null set A E FA
such that fk(w) -> f(w) as k - - for all wE a2,A. A function f:SZ - R is (D-S) p-integrable (over f2) if there is a sequence {fk}k=1 of simple functions
such that fk -* f as k +
u-a.e. and the sequence
{fA fk du}00 converges k=1 9
The limit
in E for every A E F.
f du = l im
f
fA
A
du E E 00
is independent of the choice of sequence (fk}k=1 and is called the (D-S) integral of f over A with respect to the E-valued measure U.
Theorem (3.1)
( Properties of the Integral): Let E be a real Banach space and
p an E-valued measure on a measurable space (0,F). (1)
Let
f,g:S2- R be u-integrable over A E F and a,B E R.
Then of + gg is
u-integrable over A and
(af + sg)du = JAJA g du.
JA
(ii) If f:9 -> R is u-measurable and u-essentially bounded, then f is u-inte-
grable over any A E F and
J f du1E < IIu 11 (A) u-essup If (w) I. w EA
A
(iii)
The map >
F
E
AIA f du is an E-valued measure for any u-integrable f:S2 + R. (iv)
(Dominated Convergence): Fix A E F and let fk:S'i - R, k = 1,2,... be a
sequence of u-integrable functions converging V-a.e. to a function f:c -> R. Suppose there is an integrable g:Sl -* R>' such that Ifk(w)l < g(w) for u-almost
all w and all k > 1.
f du = lim fA
Proof:
(A fk du
J
Dunford and Schwartz [15], Theorem IV.10.8 (pp. 323-325); Theorem IV.
10.10, p. 328.
10
Then f is p-integrable over A and
a
Some Linear Analysis
§4.
If E is a real Banach space, its strong dual is E* = L(E,R), the space of all
continuous linear functionals A:E -.R with 11x II = sup {1),(v) :v E E, ME =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,c) = {y:y E E, r c E* is finite and c > 0.
Jx(y) - a(x)l < c, A E r} where
In particular a generalized sequence {xa} in E
converges weakly to x E E if lim a(xa) = A(x) for each a E E* (Dunford and Schwartz [15], §V.3).
If E, F are real Banach spaces, say a map T E L(E,F) is weakly compact if the weak closure of the image T(B) of the closed unit ball B = {x:x E E, IxIE < 1} 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 IIn1IC = sup ln(s)l. Then SES for any real Banach space E, a weakly compact linear map T:C(S,R) - E corres-
to 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):
Let S be compact Hausdorff and T:C(S,R) -. E a weakly compact
linear map into a real Banach space E.
Then there is a measure u:Borel S - E
such that (i)
T(n) = JS n(s) du(s), for all n E C(S,R);
(ii)
IITh
=
Ilull(S);
(iii)
for each a E E*, )op is a regular finite Borel measure on S;
(iv)
T*(a) = aau for all a E E*.
Conversely, every E-valued measure p on (S, Borel S) which satisfies (iii) above defines a weakly compact linear map T:C(S,R) - E which fulfills (i), (ii) and (iv).
11
Remark
The integral in (i) is the D-S integral introduced in the previous section (§3).
A class of weakly compact linear maps C(SR) -* E is provided by
If the Banach space E is weakly complete, then every contin-
Theorem (4.2):
uous linear map C(S,R) -* E is weakly compact.
Proof:
Dunford and Schwartz ([15], Theorem VI 7.6, pp. 494-496).
a
If E, F are real Banach spaces, their projective tensor product E ®
F is
a real Banach space with respect to the completion of the norm p
IIzIl = inf
Ixi IE Iyi IF:
p
Z=
E
xi 0 yi}.
i=1
i=1
Here the infimum is taken over all possible representations
z=
p
E xi®yi EE®F, i=1
the algebraic tensor product of E and F. continuous bilinear maps S:E x F
Let L(E,F,;R) be the space of all
R viz. all those bilinear a such that
there is a constant K > 0 with IB(x,y)I < KIxIE IYIF for all x E E and all y E F.
Then L(E,F;R) is a Banach space when furnished with the norm
II s II
= sup {IB(x,Y)I
:
x E E, Y E F, IXIE < 1, IYIF < 1}.
The projective tensor product allows us to identify each continuous bilinear map E x F -* R with a continuous linear functional on E ® F viz.
Theorem (4.3):
Proof:
L(E,F;R) is norm isomorphic to [E 87r F]
Treves [75], Proposition 43.12, pp. 443-444.
As an.example consider the following situation. dorff space and E = F = C(S,R).
.
a
Let S be a compact Haus-
Then every continuous bilinear form
g:C(S,R) x C(S,R) -* 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 E Borel S, are measures on S. 12
and
An integration theory for signed bimeasures has
been developed by Ylinen [77].
Note that a continuous bilinear form on
C(S,R) does not necessarily correspond to a measure on S x S because of the strict embedding C(S x S, R)
C(S,R) ®,m C(S,R)
n 0 E a
>
{(s,s')
F-3'
n(s)C(s')}
(Treves [75], cf. Exercise 44.2, p. 450).
The following theorem of Mercer will prove to be useful.
For a proof see
Ri.esz and Sz-Nagy ([68], Chapter VI, §97-98, pp. 242-246) or Courant and Hilbert ([10], pp. 122-140).
Theorem (4.4) (Mercer's Theorem): Let I c R be a closed bounded interval and K:I x I - R a continuous symmetric kernel which is positive in the sense that
l I ft
K(s,s')n(s)n(s')ds ds' > 0 00k=1
for all n E C(I,R).
Suppose
c
C(I,R) and
{Ak}k=1
c R>O
are solut-
ions of the eigenvalue problem K(s,s')Ck(s')ds' = Ak Ys), for all s E I. JI
Then K can be expanded as a uniformly convergent series K(s,s') =
E k=1
AkCk(s)Ck(s'), s,s' E I,
on I x I. Remark
The validity of JI
K(s,s')n(s)n(s')ds ds' > 0 for all n E C(I,R) is JI
equivalent to Ak > 0 for all k > 1.
Theorem (4.5)
(Douady):. For real Banach spaces E, F, every Borel-measurable
linear map T:E - F is continuous.
13
Schwartz. ([71.], Part I.I, Chapter I., §2 pp. 1.55-1.60).
Proof: §5.
a
Stochastic Processes and Random Fields
Let (S2,F,P) be a probability space, T a complete separable metric space and E a real Banach space.
An E-valued stochastic process x on (0,F,P) para-
metrized by T is a map x:T x S2 - E such that x(t,.) E1t52,E;F) for all t E T. If 0 < a
-, a filtration (Ft)0
increasing family of sub-a-algebras Ft of F i.e. Ft E Ft 1
0 < t1 < t2 < a.
With this extra structure one gets a filtered probability
space (c , F, (Ft) (i)
whenever
2
O
,P) .
Say a process x .T x S2 - E i.s
measurable if x is (Borel T ® F, Borel E)-measurable;
(ii) a.s. continuous (or x has continuous sample functions, or continuous)
if for a.a. w E S2, x(.,w):T -> E is continuous; (iii)locally bounded if for a.a. w E 0,
is locally bounded;
(iv) square-integrable (or of class L2) if for each t E (v)
of class
fk
E c2(Sl,E;F);
(k > 1) if for each t E T,x(t,.) E Ek(52,E;F).
An E-valued process x:[0,a] x n
E, parametrized by one-dimensional time
T = [0,a] cR O, is adapted to the filtration (Ft)O
Quite often a continuous n-dimensional process x:[0,a] x 52 + Rn is repren)
sented as an (F, Borel C([O,a],R >
s2
)-measurable map
C([O,a],Rn)
W F-> x(-,w) i.e. as a C([0,a],Rn)-valued random variable (Chapter II, §2).
In particular,
if x:[0,a] x 52-> Rn is continuous and (Ft)0
> C([O,t],Rn)
S2
W $
>
(Lemma (11.2.1)).
14
In a filtered probability space (S2,F,(Ft)t
When the parameter set T is a separable Banach space, a stochastic process x:T x 0 + E will be called a random field. Let x:T x S2
-
E be a stochastic process and for any integer k > 1 denote
by tk the k-tuple (tl,t2,...,tk) E Tk.
Then we have a measurable map
S2 ->
Ek
w I--->
(x(t1,w),...,x(tk,w)).
The family of probability measures
E Mp(Ek) where tk ranges over
all finite subsets of T is called the set of finite-dimensional distributions of x.
Let x1:T x S21 ; E, x2:T x 22 -*'E be stochastic processes on two pro-
bability spaces (Qi,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-dimensional distributions, i.e. if t1,...,tk E T, B1,...,Bk E Borel E then
P1{w:w E 521, x1(t1,w) E B1,...,x1(tk,w) E Bk}
= P2{w:w E c2, x2(t1,w) E B1,...,x2(tk,w) E Bk}. Two processes x1, x2:T x S2 ; E on the same probability space (SZ,F,P) are
versions of one another if for each t E T the random variables agree on a set Stt E F of full P-measure.
and
More generally, let
X:T -).IL°(S2,E;F) be a Borel measurable map, then a measurable version of X is
a (Borel T ® F, Borel E)-measurable process X:T x S2 - E such that on a set Sgt E F of full P-measure for each t E T.
X(t)
The following results of
Stricker, Yor, Cohn and Hoffmann-Jorgensen are found particularly useful in establishing measurable versions for the trajectory random field of a stocastic FDE.
Theorem (5.1)
(Stricker-Yor): Let (P,F,P) be a probability space, (T,g) a
measurable space and E a complete metric space.
Suppose Xn:T x S2 - E, 15
n = 1,2,... is a sequence of (g ® F, Borel E)-measurable maps such that for converges in probability on Q.
each t E T the sequence
Then
there is a (g ® F, Borel E)-measurable map X:_T x 0 + E such that for each in probability.
lim
t E T,
nProof.
S.A. Elsanousi indicated to me that the original proof of Stricker-
Yor ([72]) for the case E = R does in fact generalize easily to any complete metric space E.
Theorem (5.2):.
a.
(Cohn-Hoffman-Jorgensen)
Let (S2,F,P) be a probability space,
T a separable metric space and E a complete separable metric space.
Then a
Borel-measurable map X.T - L°(S2,E;F) has a (Borel T ® F, Borel E)-measurable version T x S1
E if and only If the range X(T) is separable in L°(S2,E;F).
In particular, if T is also complete, each Borel map X:T - L°(S2,E;F) admits
a measurable version T x S2 - E. 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-Jorgensen ([34] p. 206).
When T is complete separable
and the map X:.T - L°(SZ,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.
§6.
o
Martingales
This is a very important class of stochastic processes parametrized by one-
Let E be a real Banach space, a > 0 and (S2,F,(Ft)k,P)
dimensional time.
a filtered probability space.
A process x:[0,a] x S2 - E is an (Ft)0
martingale if (i)
It is right-continuous i.e. has a.a. sample paths right-continuous;
(ii)
for each t E [0,a],
(iii)
)
=
E l1(SZ,E;Ft);
a.s. whenever 0 < t1 < t2 < a.
t
If each Ft contains all sets of P-measure zero in F, every (Ft)O
gale x:[O,a] x S2 - E is progressively measurable with respect to (Ft)O
16
measurable (Stroock and Varadhan [73], Lemma 1,.2.1., p. 20). A real process x:[0,a] x SZ -,- R is an
<
t)kt
-submartingale if
it is. right continuous;
(1)
E 9I(c,R;Ft) for all t E [0,a];.
(ii)
a.s. whenever 0.< t1. < t2 < a.
(ii.i)
Theorem (6.t) (Doob's Inequality): Suppose x:.[0,a] x SZ - Rn is an (Ft)ot amartingale and k > 1 is an integer such that x(t,.) E 9k(c,in) for all t E [0,a].
Let Rn carry the norm. n E
Ivl1 =
Ivil
,
v = (vt,...,an) E
n.
i-l
Then the real process
[0,a] x 12 -> R (t,W)
>
I
Ix(t,W) Ik
is an (Ft)0
sup u
P{W:W E 0,
Furthermore, for every e > 0
Ix(t,W)I1 > E} < -1
and
E{ sup
I
I
is any other norm on Rn, then there are positive constants K1, K2
independent of c, a and x such that K1
P{W:w E S2,
sup Ix(t,w)Ik 0
> E} < ek
for every e > 0, and E{
Proof:
sup O
Kk (-)k
Represent x in coordinate form
17
x = (x1,x2,...,xn) where each xi:.[0,a] x St - R, 1 = 1,...,n, is a real
t)O
Thus each 1x11 is an (Ft)0t
E
E i=1
Therefore 1xl1 is an (Ft)0
Ixl1k
for k > 1,
due to the convexity of the real-valued function
R'0
>
E--
t
R>O
tk
(Stroock and Varadhan [73], p. 21). hold for the submartingale lx1
The given inequalities are well-known to
(Stroock and Varadhan [73] p. 21; Ikeda and
Watanabe [35], Theorem 6.10, pp. 33-34; Friedman [22], pp. 9-15). norm
I-
I
For any
on Rn there are constants L, M > 0 such that
Mlvi < lvl1 < Livi for all v E Rn.
Hence, for every c > 0,
P{w:w E S1,
sup
Ix(t,w) lk > e}
0
< P{w:w E 9,
sup lx(t,w)lk 0
>
Mkc}
1
< (M * Tc k
where Kk =
-
M
> 0.
A similar argument yields the final inequality of the theorem with
Kk =
Lk
§7.
Markov Processes
2
.
o
These form another important class of Banach-space-valued stochastic processes parametrized by one-dimensional time [0,co).
18
The following precise setting is
made after Dynkin ([16], Chapters LI., I.I.L, pp. 47-1.04). A Markov process x on a real Banach space E is a collection {(Q,F,Ft,x,Ps,n):n E E, 0 < s < t
< co}
satisfying the following hypotheses:
For each s > 0 and n E E, (Sl,F,(Ft),P5,n) is a filtered probability
(i)
space in which every Ps,n is a probability measure defined on the o-algebra Fs =
a{Ft: t > s}.
(ii)
If s' < s and t < t', then Ft c Ft,.
00
(iii) The process x.[0,oo) x S2-> E is (Borel [0,ao) a F, Borel E)-measurable
is (Ft, Borel E)-measurable whenever t > s.
and (iv)
For fixed O< s < t < E
,
B E Borel E, the function
>R
n F---> Ps'n {w:w E St, x(t,w) E B} is Borel-measurable.
For each s > 0 and n E E, Ps,n {w.w E St, x(s,w) = n} = 1.
(v)
NO If B E Borel E, 0 < s < t < u and n E E, then the Markov Property E B)
E BIFt) = holds a.s.-Ps,n on S2.
An E-valued Markov process defines a family of transition probabilities E E, 0 < s < t < co} on E given by
p(s,n,t,B) =
E B), B E Borel E.
These have the properties: (i)
for fixed 0 < s < t, B E Borel E, p(s,.,t,B) is Borel-measurable;
(ii)
for fixed 0 < s < t, n E E,
is a Borel probability measure
on E; (iii) the Chapman-Kolmogorov identity
19
p(s,n,u,B) =
JSEE p(s,n,t,dE) p(t,C,u,B)
holds for 0 < s < t < u, n E E, B E Borel E (Dynkin [16], p. 85).
A Markov
process is time-homogeneous if its associated family of transition probabilE E, 0 < s < t < -1 satisfies
ities
p(s,n,t,B) = p(O,n,t-s,B)
whenever 0 < s< t, n E E, B E Borel E. Let b(E,R) denote the Banach space of all real-valued bounded Borel-
measurable functions 0 on the Banach space E with the supremum norm 11011b = sup {I4(n)I:n E E}.
Then a time-homogeneous Markov process on E
with transition probabilities
n E E, t > 01 yields a contraction
one-parameter semi-group {Pt}y of continuous linear operators defined by
Pt:b(E,R)
Pt( )(n) = JSEE 4,(F) p(O,n,t,dC), n E E,
for each 0 E b(E,R) (Dynkin [16], Chapter LI., §1-2, pp. 47-61; Chung [8], pp. 1-12).
A time-homogeneous Markov process on a Banach space E with semi-
group Pt:b(E,R) -, b(E,R), t > 0, is called a Fetter process if
{Pt}t>o leaves invariant the closed linear subspace Cb(E,R) c b(ER) of
(i)
all bounded uniformly continuous functions E -. R; (ii)
lim
{Pt}t>O is weakly continuous on Cb(E,R) at t = 0 i.e. for every n E E Pt(4)(n) = 4)(n) for all 0 E Cb*
t-+0* As a consequence of our results in Chapter IN (Theorem (IV. 2.2)), for infinite dimensional E the last condition (ii) in the above definition of Feller process does not imply strong continuity at t = 0 of the semi-group
{Pt}y on Cb(E,R).
Compare this with the locally compact case in Chung ([81,
pp. 48-56) where strong continuity is implied by weak continuity at t = 0. For a further study of (time-homogeneous) Markov processes, their transition semi-groups and infinitesimal generators, the reader may consult Dynkin ([16]).
20
Examples
§8.
Let us wind up by looking at some useful examples of stochastic processes which are indispensible for our forthcoming discussion of stochastic FDE's. Gaussian Fields
(A)
If T is a metric space, a real stochastic process x:T x S2 -).]R on a probability space (52,F,P) is a Gaussian process if for every t E T the random
variable x(t,.) E r2(2,R;F) and has a Gaussian distribution (Hida [31], pp.
31-43). For a Gaussian process x:T x S2 -. R define the mean m:T -). R and covariance V.:T x T --+ R by setting t E T
m(t) = E x(t,.), V(s,t) =
E T.
A Gaussian process x:T x Q a R is called a Gaussian system if every finite linear combination from. the family (Hida [31] §1.6).
t E T} has a Gaussian distribution
The following results concerning Gaussian systems are well-
known.
Theorem (8.1):. (i)
Any subsystem of a Gaussian system is also a Gaussian system.
(ii)
For any Gaussian system x:T x c + R the closed linear hull of t E T} in r2(S2,R;F) is a Gaussian system.
(iii) If x:.T x 0 - R is a Gaussian system, then a necessary and sufficient
condition for the family {x(t,.):t E T} to be independent under P is V(s,t) = 0 whenever s # t. (iv)
If x:Z+ x Q-). R is a (discrete-time) Gaussian system, then the sequence converges in probability i.f and only if i.t converges in C2(c2,R;F).
In this case lim x(n,.) is a Gaussian random variable. n->oo
Hida [31], pp. 34-35.
o
If T is a real Banach space, a Gaussian system x:.T x 0 + R on a probability
space (c,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 parameter space (Dudley [13] and Chapter V §3 of this book).
For a deeper study of sample regularity properties of
Gaussian processes, the interested reader could refer to the works of Dudley [13], [14], Feldman [20] and Fernique [21].
Brownian Motion
(B)
Let (c ,F, (Ft)i>0,P) be a filtered probability space. A process w:Rex S2 -R is a one-dimensional Brownian motion (or a Wiener process) on (Q,F,(Ft)i>0,P) if
it Is (Ft)t>O-adapted.
(i)
(ii) it Is a Gaussian system on (52,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(0,.) = 0 a.s. process w
A
x 0 - Rm is an m-dimensional Brownian motion if it is of the
form w = (w1,w2,...9wm) where the coordinate processes {wi}m=1 are independent one-dimensional Brownian motions on (c,F,(Ft)i
,P).
For easy reference, we
list here some of the basic properties of Brownian 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 1
Rm be an m-dimensional Brownian motion on the
x S
filtered probability space (S2,F,(Ft)t>O,P).
Then the following is true:
(i)
w is an
(ii)
The sample paths of w are almost all nowhere differentiable.
t)t>O -martingale.
ti
(iii) w has a version w such that for any a > 0 and any 0 < a < }, almost all ti
sample paths of wl[0,a] x S2 are a-Holder continuous.
(iv) For a.a. w E S2, )-i
lim sup (2t log log lim sup (2t log log
t->00 22
t
jw(t,w)j = m1/2
t)-l
jw(t,w)j = m1/2
m
where
I
denotes the Euclidean norm IvI =
I
E
v?, v = (V49 ...,v ) E R.
m
i
i=1.
This property is called the Law of the Iterated Logarithm. (v)
t > 0} is an
If t0 > 0, then the process
m-dimensional Brownian motion on (f,F,(Ft)t>t ,P). 0
(vi)
Let
S2
=
C(R>O,Rm) be the space of all continuous paths R>O
-
Rm and f
be the a-algebra generated by all cylinder sets in SZ of the form {f:f E (f(t1),...,f(tk)) E Bk}' tl,...,tk E Rte, Bk E Borel (Rm)k
Then there is a ti
unique probability measure uw (called Wiener measure) on (S2,F) giving the
finite-dimensional distributions of w viz. uw{f:f E 2, (f(tl,...,f(tk)) E Bk}
= P {w:w E c2,(w(tl,w),...,w(tk,w)) E Bk}
=
JBk
where the kernel g:(R>O)k x (Rm)k , R>O is given by k
g(u1,...,uk;y1,...,yk) =
(2i u.)-m/2
IT
j=1
for ul,...,uk E R0, '11 ...,yk E R. (vii) If 0 < s < t, then 2k
IFs) _ (2k-2+m)(2k-4+m)...mjt-sI k
a.s. for every integer k > 1. (viii) For 0 < t1 < t2 < t3, the increments
w(t3,.)-w(t2,.)
are conditionally independent under P given the c-algebra (xi)
u < t1}.
w is a Feller process with stationary transition probabilities t > 0, x E Rm}
given by
-Ix- 12
p(0,x,t,B) =
m/2 (2,,t) m/2
e t dy, B E Borel R m. J
Furthermore, the associated contraction semi-group {Pt}t>O is strongly 23
continuous on b(Rm,R).
Its infinitesimal generator A:D(A) c b(Rm,R) -b(Rm,R)
is given by (A4)(x) = }AO(x), x E Rm,
where g:Rm
R is a C2 function with compact support and A is the Laplacian a
A0(x) =
(x).
E
i=1 8x (C)
The Stochastic Integral
We adopt the viewpoint of McShane ([53], Chapters II, III, IV) which contains adequate information for all our purposes.
However, it is interesting to
note that Elworthy's recent treatment ([19], Chapters III, IV, V) contains far-reaching generalizations of McShane's stochastic integral to infinite dimensions.
Other references for stochastic integration include Ito [39],
McKean [52], Meyer [55], Friedman [22], Gihman and Skorohod
[24], Ikeda and
Watanabe [35], Metivier and Pellaumai.l [54], and Arnold [2].
In this section we shall only content ourselves by giving a brief account of the stochastic integral including some of its most important properties. ,P) be a filtered probability space.
Let (c,F,(Ft)
For a closed inter-
val [a,b] c R a partition 11 = (tl,...,tk+1;T1,...,Tk),
a = t1 < t2 < ... < 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 52 - L(Rm,Rn) and z:[a,b] x S2 -Rm are (Ft)a
S(11)(C°) =
f(Ti,W){z(tit1,W)-z(ti,W)}
E
i=1
for all W E 0.
For belated II define
mesh II = max {ti+1
- Ti
.
1 < i < k}.
b
The stochastic integral
f(t)dz(t) r Co(SI,Rm;Fb) is defined as
J
a b
f(t)dz(t) = a
lim
S(11)
meshf - 0
whenever the above limit exists in probability as meshII - 0 over all belated
24
partitions II of [a,b].
Conditions under which this limit exists are given
below, together with some of the basic properties of the stochastic integral. For proofs see McShane [53], Elworthy [19] and Friedman [22].
Theorem (8.3):
(i) Let fi:[a,b] x S2 -> L(Rm,Rn), zi:_[a,b] x S2 - Rm, i. = 1,2, be (Ft)a
fi(t) dzi (t), 1 < i, j < 2,
J
Then a.s.
all exist .
fb
b (
a {f1(t) * f2(t)} dz.1(t) =
a
b
f1(t)dz1(t) t
Ja
f2(t)dzl(t)
and fb b
b
b
+ Ia f1(t)dz2(t),
f1(t)d{z.1(t) * z2(t)) = a
fa
Furthermore, if A E Lo(S2,R;Fa), then the integrals
where all integrals exist.
b
b Afi(t)dzi(t), Ia
fa
fi(t)d{azi(t)}, i = 1,2,
also exist and b
b a
b
Afi(t)dz.i(t) = Ia fi(t)d{Azi(t)} = A Ia fi(t)dz(t) a.s.
f
for i = 1,2. (ii)
Let a < c < b and f:[a,b] x S2 -> L(Rm,Rn) be integrable with respect to
z:[a,b] x S+ Rm; then the integrals
Ic a f(t)dz(t), f cb f(t)dz(t)
exist and c
rb
f(t)dz(t) = I J
a
Theorem (8.4):
b
f(t)dz(t) + I f(t)dz(t) c a
a.s.
(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 S+ L(RII,R 25
be (Ft)a
[a,b]
C2(fl,L(Rm,Rn), Fb)
>
t F--> is continuous a.e. on [a,b] (Lebesgue measure) and bounded.
Suppose
z:[a,b] x S2 -> Rm is (Ft)a.-adapted and there is a K > 0 such that K(t-s)
IFs) < K(t-s) t
a.s. whenever a < s < t < b.
Then the stochastic integral I(t) =
f(u)dz(u)
f
a
exists and belongs to L2(S2,Rn;Ft) for every t E [a,b]. EIJb f(t)dz(t)I2
< {2K(b-a)1/2
+ K1/2}
Indeed
E IIf(t,)IIdt]
Ja where
I.
I
denotes Euclidean norms and
II
II
the standard operator norm on
L(Rm,Rn) (§2).
If, furthermore, z is a continuous
t)a
indefinite integral {I(t):a < t < b} has a version which is also a continuous (Ft)a
P{w:w E S2,
In particular , for every E > 0,
sup I(w) a
t I
f(u)dz(u) I > E}
a
Ja E Remarks (i)
When z = w:[a,b] x 0-, Rm is m-dimensional Brownian motion on
(sl,F,(Ft)a
,P), 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). (ii) McShane defines an Ito belated integral which embraces a more general class of integrands f viz. f:[a,b] xS2-,L(Rm,Rn) measurable, (Ft)a
IIf(t,w)II2 dt < oo
Ja
26
is Lebesgue-square integrable over [a,b]
(McShane [53], Chapter II §5, Chapter III §8).
In
particular if f satisfies the stronger condi.tion
b
2dt
E II
Ja
then the,
Ito-belated integral fulfills the conclusions of Theorem.(8.4) above.
(The Ito Integral): Let w-[a,b] x St
Theorem (8.5)
-).
Rm be m-dimensional
Brownian motion on the filtered probability space (St,F,(Ft)a
Suppose
f:[a,b] x S2 , L(Rm,Rn) is measurable, (Ft)a
(b
IIf(t,w) II2 dt < oo a
J
b
for a.a. W E 2. Then the Ito integral Ja f (t)dw(t) exists in Furthermore if k > 1 is an integer and b
fa
2k dt < c
E
then the indefinite integral t
{I(t) =
f(u)dw(u): t E [a,b]}
(
a
has a version which is a continuous (Ft)a
In fact,
IIt E
sup a
f(u)dw(u)I2k
<
a
Kk(m,n)(b-a)k-1
Jb
E
du
a
where
Kk(m,n) =
nk-1(4k3m21k
`` 2k-1' Proof:
In one dimension m = n = 1, see Friedman [22] (Chapter 4, Corollary
6.4 p. 87).
For higher dimensions, write
w = (w1,w2,...,wm), f = [fij], i = 1,...,n, j = 1,...,m,
where wi:[a,b] x Q-. R, j = 1,...,m, are one-dimensional Brownian motions and fij:[a,b] x o -,. R, i
= 1,...,n, j = 1,...,m are the entries of f with respect
to canonical bases of Rm and Rn.
fij(u)dwju))i=1
f(u)dw(u)
I
a
Then
m
it
j=1
ft a
and so by Htllder's inequality 27
Ifat
n
IIt
f(u)dw(u)l2k
<
m2k l nk-t
E
i.=t j=I
a
fij(u)dwi(u)12k
Taking suprema and expectations the required estimate
a.s. for all a < t < h.
follows directly from the one-dimensional case.
a
Finally, we state a result indicating the dependence of the Ito integral on deterministic integrands viz.
Theorem (8.6):
Suppose w:[a,b] x St-, Rm is m-dimensional Brownian motion on
a complete filtered probability space (S2,F,(Ft)a<,P).
Define the map
Z:[a,b] x C([a,b], L(Rm,Rn)) - L2(c,Rn;Fb) by setting t
Z(t,f) = J
f(u)dw(u) a
for all t E [a,b] and all f E C([a,b],L(Rm,Rn)).
Then Z. has a Borel-measurable
version
Z:[a,b] x C([a,b],
L(Rm,Rn)) x S2 -, Rn
with the following properties: ti
(i)
the random field
t E [a,b], f E C([a,b], L(Rm,Rn))) is an
n-dimensional Gaussian system; (ii)
for each t E [a,b], the random field
Z(t,.,.):C([a,b], L(Rm,Rn))
x S2 -, Rn
is linear in probability but has almost all sample functions non-linear i.e. for fixed f1,f2 E C([a,b], L(Rm,Rn))
ti
ti
ti
P{w:w E 9, Z(t,61f1 + 62f2,w) = 61Z(t,f1,w) + 62Z(t,f2,w) for all 61,62 E R} = l
while
ti ti ti P{w:w E 52, Z(t,f1+f2,w) = Z(t,f1,w) + Z(t,f2,w) for all f1,f2 E C([a,b], L(Rm,Rn))} = 0.
28
Remark
Note that every measurable version of Z satisfies properties (1) and (ii) of the above theorem.
The proof of the theorem is 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 §1.
Basic Setti.ng and Assumptions
Let (0,F,P) be a probability space with F a a-algebra of subsets of the measure space S2 and P a probability measure on Q.
Suppose a > 0 and let
c Ft2 if
(Ft)tE[O,a] be an increasing family of sub-c-algebras of F i.e. Ft 1
t1,t2 E [0,a] and t1 < t2.
Assume also that each Ft contains all the sets of
measure zero in F.
If r > 0, let J = [-r,0] and C(J,Rn) the Banach space of all continuous
paths y:J - Rn with the sup-norm IIY IIC = 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 r2(S2,C(J,Rn)) the space of all F-measurable stochastic processes 6:52
-r
C(J,Rn) such that the function
S2 3 w 1-->
116()IICE R is of class L2 i.e.
r
Ile (w) IIC2 dP(w) < °° . J S2
Then C2(S2,C(J,Rn)) is complete when endowed with the semi-norm dP(w)J1/2
J" IIa(w)11c One may also look at the space C([O,a], r2(S2,C(J,Rn))) or all r2-continuous C(J,Rn)-valued processes y:[O,a] -> C2(S2,C(J,Rn)); again this is complete
under the semi-norm
Ily II
C([ O,a],r 2( 9,C))
tE[0,a]
11y(t) II
r2( 2,C)
Say a process y E C([O,a], r2(c2,C(J,Rn))) is adapted to (F
is Ft-measurable for each t E [0,a]. set of all such processes.
)
t tE[O ,a]
if y(t)
Signify by CA([O,a], C2(S2,C(J,Rn))) the
Then CA([O,a], r2(S2,C(J,Rn))) is closed in
C([O,a], r2(S2,C(J,Rn))); to see this, note that because of the continuity of the evaluation map
30
pt:.C([O,a], f2(S2,C))
t2(Q,C)
z(t)
Z.
for each t E [O,a], it is sufficient to observe that the space
L2(St,C(J,Rn);Ft) of all Ft-measurable maps W:9 - C(J,Rn) is closed in Similarly the space C22(S2,C([-r,a],Rn)) of all processes beA longing to L`(S2,C([-r,a],R")) and adapted to (Ft)tE[O,a] is closed in
£2(S2,C(J,Rn)).
L2(S2,C([-r,a]Rn)).
This is because the evaluation map is continuous and
each Ft contains all sets of measure zero in F. When looking at stochastic (retarded) functional differential equations, we restrict attention - for the sake of simplicity - to just one 'coefficient
process' g:[O,a] x rC2(St,C(J,Rn)) -> t2(9,L(Rm,Rn)) and one 'noise process' z:S2
C([O,a]:Rm) which is m-dimensional and has continuous sample paths.
For a given initial process (or distribution) e E C2(S2,C(J,Rn)) we seek a solution x E c2(S2,([-r,a],Rn)) of the stochastic FDE: t
-f e(w)(0) + (w) J 0 x(w)(t) -l
O(w)(t) ,
t E J,
0 < t < a (I)
a.a. w E Q.
The stochastic integral in (I) is a McShane belated integral (§I.8(C), McShane [53]); and for each u E [O,a], xu E c2(S2,C(J,Rn)) is defined by
xu(w)(s) = x(w)(u+s) a.a. W E 0, for all s E J. L2(S2,C(JRn)) will be our basic configuration space of initial processes for the stochastic FDE (I).
Obviously this choice of configuration space
entails that the initial data as well as the solution process of the stochastic FDE will necessarily have almost all their sample paths continuous. The trajectory [O,a] 3 t -> xt E C2(Q,C(J,Rn)) will be sought in the space CA([O,a], c2(2,C(J,Rn))).
(I) involves the stochastic memory map
m*:[O,a] x E2(S2,C([-r,a], Rn)) (t,x)
C2(S2,C(J, Rn))
xt
m* is related to the deterministic memory map
31
m:[O,a] x C([-r,a], R (t,a)
n
) . C(J, Rn) at
1
(at(s) = a(t+s), s E J) by the equation for all t E [O,a],
m*(t,x) =
(1)
and all x E.C2(S2,C([-r,a],Rn)).
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 E [O,aJ and m* is continuous.
IIm*(t,x) II
2
L
61.0
< IIxII
r
Indeed,
for all x E 92(c2,C([-r,a],Rn))
2
(52, 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:S2 ->C([O,a], Rm) is expressible in the form
z.(w)(t) = a(t) + zm(w)(t) for all t E [O,a], a.a. W
(2)
E S2
where a:[O,a] + Rm is a Lipschi.tz. function and zm:s2 + C([O,a],Rm) is a (separable) martingale adapted to (Ft)tE[O,a] and is such that there is a
K > 0 with )I < K(t2-t1)
l
f
i
(3)
) < K(t2-t1) 1
a.s. whenever tl,t2 E [0,a] and t1 < t2. (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
IIg(t,w1) - g(tov2) II 2 < L
t
32
II1p1-W211t2
(4)
(a,C)
for all t E [O,a], and all
W1,y,2 E
L2(52,C(J,Rn)).
(ii.i.) For each process y E CA([O,a], £2(S2,C(J,Rn)) the process
[O,a] 3 t F--> g(t,y(t)) E
92(S2,L(Rm, Rn))
is also adapted to (Ft)tE[O,a]'
Remarks (1.1) (1)
In condition (E)(i) the Lipschitz function A can be replaced by a proti
cess a:St x [O,a] -,.Rm adapted to (Ft)tE[O,a] and whose sample paths are almost all Lipschi.tz with a uniform Li.pschitz constant i.e. there exists
k > 0 such that Ia(w)(t2) - a(W)(ti)I < 11t t2-t11
a.a. W E 9,
and all t1, t2 E [0,a]. (2)
The stochastic FDE(I.) includes both the ones with random coefficients
and the (ordi.nary) stochastic differential equations (without retardation, ti
[22], [53]); for suppose g:[0,a] x C(J,Rn) x S2 - L(Rm,Rn) is a coefficient process corresponding to a stochastic FDE with random coefficients. Assume
that g is L2 with respect to the third variable. Then we can define the coefficient process g:[O,a] x t2(S2,C(J,Rn)) -> C2(S2,L(Rm, Rn)) by setting g(t,W)((,)) = g(t,W(w),w)
a.a. W E 0
for all t E [0,a], and all W E C2(S2,C(J,Rn)).
§2.
Existence and Uniqueness of Solutions
The following lemmas will be needed in order to establish the main theorem for existence and uniqueness of solutions to the stochastic FDE(I).
Lemma (2.1):
Suppose x:[-r,a] x S2
paths continuous.
-*
Rn is a process with almost all sample
Assume that xj[0,a] is adapted to (Ft)tE[O,a] and x(s,.)
is F0-measurable for all s E J.
Then the process
33
y C(J, Rn)
[O,a] x 2
(t,w) 0----
xt(w)
>
is adapted to (Ft)tE[O,a]' with almost all sample paths continuous.
Proof.
Since xI[-r,t] has continuous sample paths, it induces
Fix t E [O,a].
an F-measurable map xI[-.r,t]:0 - C([-r,t],Rn).
This map is in fact Ft-measur-
able; to see this observe that Borel C([-r,t],Rn) is generated by cylinder sets of the form
ptt(Bi) where Bi E Borel Rn, ti E. [-r,t],
n i.=1
pt
i
> Rn is evaluation at ti for 1 < i < n. Thus it is suff-
C([-r,t],Rn)
i.cient to check Ft-measurability of xI[-r,t] on such sets.
With the above
notation, n [ n
(xj[-r,t])-1
1
pt (Bi)] =
i=1
By hypotheses, if ti E J, {w:x(ti,w) E Bi} E Ft
is
n {w:x(ti,w) E Bi}. n i=1
{w:x(ti,w) E Bi}
E FO; and if ti E [O,t],
.
Hence {W:x(ti,w) E Bi}
E Ft c Ft if 1 < i < n. i
So n
(xl[-r,t])-1[ n i=1
Pt(Bi)] E Ft* i
and the deterministic memory
Now xt =
Rn)
> C(J, Rn)
is continuous, so xt must be Ft-measurable. Lemma (2.2): tion E(i).
Suppose that z:92
o
C([0,a], Rm) is a process satisfying condi-
Let f:[0,a] - C2(st,L(Rm, Rn)) be an t2-continuous process adapted
to (Ft)tE[O,a]' Define the process F:[-r,a] t E [0,a]
(O
0
34
,
tEJ
L2(S2,Rn) by
a.s., where the integral is McShane's belated integral of f with respect to z..
Then F corresponds to a process belonging to r2(s1,C([-r,a), Rn)) and
Indeed there is an M > 0 such that
adapted to (Ft)tE[O,a].
E(
sup tE[0,a]
IJt f(u)dz(-)(u)I2) < M Ja E( IIf(u)II2 )du 0
(1)
0
The process [O,a] 3 t t--> Ft E r2(S2,C(J,Rn)) is adapted to (Ft)tE[O,a] with almost all sample paths continuous (i.e. it belongs to CA(0,a], 92(52,C(J,Rn))). M is independent of f.
Proof:
With the notation of Condition E(i), t
rt
+ (w)
F(t)(w) = f 0 o
f(u)dzm(-)(u), t E [O,a] I
0
(2)
a.a.w E Q.
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 S2; it thus
defines an F-measurable map S2+ C([0,a],Rn).
As f(u) is Fu-measurable for
all 0 < u < t, then f(u)(-)da(u)
It
0
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) 0I.8(C)). E(
Hence by the Martingale inequality we have
sup tE[O,a]
< 4C
4E(IJa f(u)dz m
IJt f(u)dzm 0
Ja
0
)du (Theorem (1.6.1))
E(
(3)
0
where C = 2Ka1/2 + K1/2 (Theorem (1.8.4)).
If R > 0 is the Lipschitz constant for a, then it is easy to see that for
a.a. w E S2 II
f(u)(w)da(u)I2
<
X2a
If(u)(w)I2du. 10
35
Hence E(
sup tE[O,a]
IJt
k2a Ja
0
0
(4)
The inequality (1) follows now from.(3) and (4) and the fact that
E(
sup tE[O,a]
2E(
IfO t
2E (
Take M = 2(22a + 4C).
sup tE[O,a]
sup tE[O,a]
IJt 0
IJt f(u)dzm(-)(u)I2) 0
Note that C (and hence M) is independent of f.
It follows immediately from (1) that F E r2(S2,C([-r,a],Rn)) and from
Lemma (2.t) that [0,a] 3 t
Ft E C2(c,C(J,Rn)) is adapted to (Ft)tE[O,a]
with the sample paths t H Ft(w) continuous for a.a. w E Q.
o
Here is the main existence and uniqueness theorem for solutions of the
stochastic FDE(I.) : Theorem (2.1):
Suppose Conditions (E) of §1. are satisfied, and let
6 E C (S2,C(J,Rn)) be FO-measurable.
Then the stochastic FDE(I) has a solu-
tion x E C2(S2,C([-r,a], Rn) ) adapted to (Ft)tE[O,a] and with initial process Furthermore,
0.
x is unique up to equivalence (of stochastic
(i)
processes) among all solu-
tions of (I) belonging to r2(S2,C([-r,a],Rn)) and adapted to (Ft)tE[O,a] i.e. 11
if x E C2(c2,C([-r,a], Rn)) is a solution of (I) adapted to (Ft)tE[O,a] and
with initial process e, then ti
x(-)(t) = x(-)(t)
Ft-a.s., for all t E [O,a];
the trajectory [O,a] 3 ti- xt E L2(S2,C(J,Rn)) is a C(J,Rn)-valued pro-
(ii)
cess adapted to (Ft)tE[O,a] with a.a. sample paths continuous.
(It belongs
to CA(CO,a], L2(Q,C(J,Rn))).
Proof: .C
We look for solutions of (I) by successive approximation in
C(C-r,a], Rn )). Suppose 0 E L2(S2,C(J,Rn)) is F0-measurable.
Note that this is equivalent
to saying that 0(.)(s) is F0-measurable for all s E J, because a has a.a.
36
sample paths continuous.
We prove by induction that there is a sequence of processes kx:[-r,a]
x Q -).
Rn, k = 1,2,... such that each kx has the
Properties P(k): (i)
kx E r2(52,C([-r,a],Rn)) and is adapted to (Ft)tE[O,a]'
(ii)
For each t E [O,a], kxt E C2(c,C(J,Rn)) and is Ft-measurable.
IIk+1x
IIk+lxt
- kxll
ak-1
< (ML2)k-1
n
2
t (S2,C([-r,a],R ))
(k-1)! tk-1
<
2
kxt11
11 2x
(ML2)k-1
1x II
2 C
;52
C)
(5) II2x
-
1x II
(k-1)!
X (c,C)
-
2 L (92,C)
where M is the constant of Lemma (2.2).
Take 1x:[-r,a] x 0 -> Rn to be 1
9(w)(0)
t E [O,a]
O(w)(t)
t E J
x(t,w) = {
a.s., and, t
O(w)(0) + (w) J k+1x(t,w) = {
t E [O,a]
g(u, 0
(6)
O(w)(t)
t E J
a.s.
Since 0 E L2(St,C(J,Rn)) and is FO-measurable, then
and is trivially adapted to (Ft)tE[O,a]' and is Ft-measurable for all t E [O,a].
1x
By Lemma (2.1),
E 92(S2,C([-r,a],Rn)) 1xt E r2(52,C(J,Rn))
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 [O,a] 3 u
g(u, kxu) E JC2(Q,L(Rm, Rn))
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
k+1xli22
-
IIk+2x
t
< E(
a
IIg(u,k+lxu)
< M J
sup
t
tE[O,a]
(S2,C)
J0
- g(u,kxu)IIL2 du,(P(k+1)(ii) and Lerma 2.2)
o
a
<
IIk+1xu
ML2 J
- kxu1I22
t
0
du,(Condition E (ii)) (S2,C) ja
(ML2)k-1
<
112
ML2
(k-1)! k
(ML2)k kT II2
_
x - 1xII2
uk-1du 2
C (S2,C)
x-
1
0
x II22 E
(cl,C)
Complete the proof of P(k+1)(iii) by noting that IIk+2xt
k+1xt1I22
-
IIk+2x
t (SIC)
<
- k+1xlI22
r (PIC)
Therefore P(k) holds for all k > 1. For each k > 1, write kx
1x
=
+ kE1 (i+1x - ix). i=1
Now £ (S2,C([-r,a],Rn)) is closed in t2(S2,C([-r,a],Rn)), so the series Go
(i+1x
-
ix)
i=1
converges in 92(9,C([-r,a], Rn)) because of (5) and the convergence of 00
2 i-1 L(ML)
ai-1
-I1/2
r
E i=1
(i-1)! i
Hence {kx}k=1 converges to some x E LA(S2,C([-r,a],Rn)).
Clearly xIJ = 8 and is FO-measurable, so we can apply Lemma (2.2) to the difference u
obtaining
38
e
g(u, kxu) - g(u,xu)
I,0 g(u,
E(tEsupa]
kx - x1122
r
0 as k -
>
(S2,C) .
Thus viewing the right-hand side of (6) as a process in C2(St,C([-r,a], Rn)) and letting k -) co
,
it follows from the above that x must satisfy the
stochastic FDE(L) a.s. for all t E [-r,a]. ti
To prove uniqueness, let x E 9A(2,([-r,a], Rn)) be also a solution of (I) Then it is easy to see by the Lipschitz condition
with initial process 6.
(E(ii)) and Lemma (2.2) that
lixt - xtil2
(9.0
t < ML2 fo llxu
-
112
ti for all t E [O,a].
Therefore we must have xt-xt = 0 for all t E [O,a]; so
x = x in r2(s2,C([-r,a], Rn)) a.s.
The last assertion of the theorem follows immediately from Lemma (2.1). o
Remarks (2.1)
Let 0 < t1 < t < a.
process W E
Then one can solve the following stochastic FDE for any
r2(St,C(J,Rn); Ft ) at time t1:. 1
x(w)(t) = f
t
t1 < t < a
W(w)(0) + (w) Jt1
(II)
ip(w)(t-t1)
t1-r < t < t1
where the (unique) solution x E r2(I,C([t1-r,t1], Rn)).
This gives a family
of maps
W
I
. xt
When t1 = 0, we define Tt, t > 0, to be
Tt = Tt:r2(c,C(J,Rn); FO) ->
c2(Q,C(J,Rn), Ft) 39
The following theorem. on continuation of trajectories of a stochastic FDE is
a consequence of Theorem (2.1).
If 0 < t1 < t2 < a, then
Assume Conditions (E) of §1..
Theorem (2.2):
t1
-T
T
t2 =
t2
oT tt Then for S E J
Let 0 E E2(S2,C(J,Rn); F0) and t1 < t < a.
Proof:
t+s
g(u,Tu(0))dz(-)(u)
t+s > 0
f0
t+s<0 tj
t+s>t1
re(-)(0) +
t t+s
0 < t+s < t
J 0
t+s < 0
t+s t+s > t1 J =
t1
t
Tt (e)(t+s-t1)
t1-r < t+s < tt
(7)
I
Putting s = 0 in (7), we can compare with (II) for W = Tt1 (e); thus by unique-
ness of solutions of (II) one gets a.s. Tt(e) =
(0)),
Ttt l(Tt
t1
t < a.
o
1
Remark (2.2)
In the case A = 0 (when there is "no drift"), z = zm is an (Ft)0
-martin-
gale, and hence for each e E C2(S2,C(J,Rn);FO) the solution ex E L2(SZ,C([-r,a], Rn)) through 0 is also a martingale on [0,a].
the trajectory t
- ext is not in general a C(J,Rn)-valued (Ft)0<-martingale
e.g. take e HE 0 and g E 1.
40
However,
§3.
Dependence on the initial Process
Use the notation of §1 and §2.
Theorem (3.1.):
Suppose the conditions of Theorem.(2.1) are satisfied by g Then each map
and z in stochastic FDE(I.).
P
r2(52,C(JRn)>Ft), t E [O,a],
is Lipschi.tz; indeed for all t E [O,a], 61,92 E C2(S2,C(J,Rn);F0),
IITt(01) - Tt(02) II
l
2
(52,C)
< -71161 - 6211 2 (2, X
eML2t
(1)
C)
where M is the constant of Lemma (2.2) and L the Lipschitz constant of g. N.B.
Proof:
M does not depend on the coefficient process g but only on the noise z.
The result follows immediately from Gronwall`s Lemma and the obvio',s
inequalities:
sup ITt(e1)(w)(s) - Tt(e2)(w)(s)12 dP(w)
I
S2 sEJ sup le1(w)(s) - 02(w)(s)I2 dP(w)
< 2 fR
+ 2
fQ
sEJ
M
t+s
dP(w)
ssupt,O]I() fo
< 2 Ile1-021122(S2,C) +
t 2ML2
10
IITu(01)-Tu(02)1122(Q.C)du.
o
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 L2(S2,C(J,Rn) ) > L(r2(R,C(J,Rn) ), L2(S2,L(Rm, Rn)) ) (t,W)
1
D(2)9(t,W) 41
is continuous, where D(2)g(t,y,) is the partial derivative of g in the second variable at (t,W).
Suppose g satisfies Conditions (E)(iii.) and (D).
Lemma (3.1):
Then for any
processes y1, y2 E CA([O,a], L2(S2,C(J,Rn))), D(2)9(t,YI(t))(Y2(t)) E L2(S2,L(Rm, Rn);Ft), for every O< t< a.
Proof:
The conclusion of the lemma follows directly from the fact that, for
each 0 < t < a, D(2)g(t,y1(t))(y2(t)) is the L2-limit as h -> 0 of the family
I {9(t,y (t) * hy2(t)) - 9(t,yI(t))} , h 1 0, which belongs to L2(S2,L(Rm, Rn);Ft) by virtue of Condition (E)(iii).
Theorem (3.2):
o
Suppose the stochastic FDE(L) satisfies Conditions (E) and
M. Then for each t E [0,a],
Tt:t2(c2,C(J,Rn);F0)
-> L2(c,C(J,Rn);Ft) +IL2(St,C(J,Rn)1Ft)
is C1.
Proof: L
Fix t E [O,a].
We first prove that T
C(J,Rn)F0) i.e. for each 6,S
t
is Gateaux-differentiable on
E 92(S2,C(J,Rn);F0),
Tt(6+hJ)-Tt(e) lim h-0
-
exists. Because of Conditions (D) and Theorem (2.1), the coefficient process u ---> D(2)g(u,Tu(6)) gives a linear stochastic FDE satisfying Conditions (E); so by Theorem (2.1), there is a unique solution a E L2(S2,C([-r,a]Rn)) of
t
a(')(t) = {
J0
0 < t < a (III)
t E J
for fixed a E L2(0,C(J,Rn);F0). 42
Define Y E CA([O,a], L2(Q,C(J,Rn))) by
Y(t) = at for all t E.[O,a]..
We claim that a Gateaux-derivative
GDTt(6):C2(52,C(J,Rn);FO)
-+
C2(St,C(J)Rn)'Ft)
of Tt at a is given by GDTt(e)(5) = y(t) for all t E [O,a] Using (1.),
(2)
(I.I.1) and Lemma (2.2), it is easy to see that if h ' 0
III [Tt(e+ h8)
t M J
0
Tt(e)] - Y(t)II2 2 L
(Q,C)
Ilht-[g(u,Tu(e+hs))-g(u,Tu(e))]-D(2)g(uTu(e))(y(u))II2du (3)
where M > 0 is independent of-g,0,0. Look at the integrand in (3); viz.
IIF [g(u,Tu(6+h8)) - g(u,Tu(e))] - D(2)g(uTu(O))(y(u))Il
< Fi 1
IIg(u,Tu(e+hB)) -. g(u,Tu(e)) - D(2)g(u,Tu(e))(Tu(0+hB)-Tu(e))II
+ F IID(2)g(u,Tu(6))(TU(O+hB)
Let c > 0 be given.
Tu(6) - hy(u))!I
(4)
Then because of the continuity of the partial derivative
D(2)g and the compactness of [0,a], one can show that there is a 6 > 0 depending on 0, a and independent of u E [O,a] such that if IhI < 6, then IIg(u,Tu(e+h S)) - g(u,Tu(0)) - D(2)g(u,Tu(e))(Tu(6+h6) - Tu(e))II
< c IITu(e+hs) - Tu(e)II
for all u E [0,a].
(5)
If IhI < 6, then (3), (4) and (5) give
[Tt(e+h$)-Tt(e)] - Y(t) 1122
t + 2M
< (s2,C)
M E2
t JO
II2 du
IID(2)g(u,Tu(e))II2IIh1-[Tu(e+h6)-Tu(e)]-Y(u)II2L2(0,C)du. (6) 10
43
Now let N > 0 be such that
IID(2)g(u,Tu(O))II< N for all u E Ma]. Then by Theorem (3.1),
III [Tt(e+h8) - Tt(e)] - Y(t) II2 2 L
< if IhI < 6
e2ML2a
2E2 118112
,
t 2MN2
(n.C )
Jt III T (e+h8)-T (e)]-Y(u)I12 F[ u u r2
(c2,C) du
(7)
t E [0,a].
Apply Gronwall's lemma to (7), obtaining
1/2 EIIBIIeM(L2a+N2t)
II- [Tt(o+h8) - Tt(e)]
Y(t)II < 2M
2N
(8)
for all t E [0,a] if IhI < S.
If we let c -> 0 in (8), lim
IIF [Tt(e*h8) - Tt(e)] - Y(011 = 0
and so y(t) = GDTt(e)(8).
By linearity of D(2)g(uTu(e)) and uniqueness of solutions of (III) it follows that GDTt(0)(8) = y(t) is linear in 8; by continuous dependence (Theorem 3.1) it is also continuous in 8.
Thus for Tt to be C1 it is suff-
icient to demonstrate the continuity of the map
2(cI.C(J.Rn);F0) 3 0 y GDTt(8) E L(r2(SI.C(J.Rn);FO).L2(Q.C(J.Rn);Ft))
So let e, $ E r2(,1,C(J,Rn);FO).
Then for any 8 E 92(Sl,C(J,Rn);FO) the
following estimate holds for all t E [O,a]: II[GDTt(e) - GDTt(0)](8)112
< M ft IID(2)9(uTu(e))(GDTu(e)(8))-D(2)9(uTu(O))(GDTu(0)(8))II2 du 0
44
< 2M I0 IID(2)g(u,Tu(e))(GDTu(e)(0))-D(2)9(u,Tu(O))(GDTu(e) W )II2 du
- GDTu(0)(0)II2 du 2M J D IlD(2)9(u1'Tu(0))112 I1GDTu(e)(s)
(9)
The map u ;-> GDTu(e)(R) = y(u) is a solution of the stochastic FDE (III) whose Lipschitz constant is N; so by Theorem (3.1), IIa1IeMN2
IIGDTu(e)(B)II 0 be given.
U
for all u E [0,a]
(10)
Then by the continuity of D(2)g, the compactness of
[0,a] and Theorem (3.1), we can find d > 0 (depending on e, c but independent of 4,u) such that
t 2 (c,C)
< 6 implies
e (11)
for all u E [0,a].
< S, then (9), (10) and (11) give
Thus if 110-011
I1[GDTt(0) - GDTto)](a)112
2 < 4Me2a 116 112 eMN a
+ 4M Jt {IID(2)g(u,Tu(e))112 +IID(2)9(u,Tu(e))-D(2)9(u;,Tu(O))112} 0
IIGDTu(6)(0) - GDTu(4)(0)II2 du
<
4Mc2a 110112 eMN2a + 4M(N2+e2)
t IIGDTu(e)(0)-GDTu(.)(a)I12 du fo
for t E [0,a].
By Gronwall's lemma, 4Me2a
eMN2a+4M(N2+e2)t1Is112,
t E [0,a].
Hence if Iiq-eII < d, we get e}MN2a+2M(N2+e2)t,
IIGDTt(6) - GDTt(0)II< 2(MMa)1/2 a
t E [0,a].
Thus a i-> GDTt(0) is continuous, and Tt is C1 for each t E [0,a].
o 45
III Markov trajectories
§1.
Property
The
In this chapter we restrict ourselves to a class of stochastic FOE's in which the coefficient process g in (1) of Chapter (LL) factors through a drift
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 form it + w(t): t E [O,a]} with w an m-dimensional Brownian motion on a filtered probability space (St,F,(Ft)0
Thus we have the stochastic FOE 4,(W) (0) t
x()(t) =
(t H(u,x (Q) du +
1O I
(w)(t G(u,x
u
1t1u
t1-r < t < t1,
W(w)(t-t1)
a.a. W E St.
Our main objective here is to establish Markov behaviour for the trajectory All our results will be stated for the
{xt:.t > 0} of the stochastic FDE(L).
general stochastic FDE (I), although for the sake of simplicity all proofs are carried out for the simpler stochastic FDE t
t1 < t < a
yP(w)(0) * (w) J
x()(t) _
{
without drift (H
(II)
1
iy(w)(t-t1) 0).
t1-r < t < t1,
a.a. w E S2
Note that in both (L) and (II), W E C2(SZ,C(J,Rn);Ftl).
We also make the following standing hypotheses:
Hypotheses (M)
be the Brownian filtration Ft = a{w(u):O 0 such that IIG(t,n1)-G(t,n2)II< LIIn1'n2IIC, IH(t,n1)-H(t,n2)I < L IIn1'n2IIC 46
for all t E [O,a] and all r
1
,n2 E C(J,Rn).
(iii) H and G are continuous.
Now the above hypotheses imply the conditions of existence (E) of §(II.1).
To see this, observe first that the Brownian noise w is an (Ft)0
and )
= m(t2-t1)
1
a.s. whenever 0 < tI < t2 < a.
Secondly, if in (II) we define
9-[O,a] x r2(S2,C(J,Rn)) -a t2(Q.L(Rm,Rn)) by
9(t,W)(w) = G(t,W(w)), t E [O,a], W E r2(c,C(J,Rn)), w E S2, then Condition M(ii) gives JIIG(t,Ipi(w))_G(t,p2(w))II2dP(w)
119(t,W1) - 9(t,W2)112 =
<
L2 IIW1
-
(1)
W2112
for all W1,W2 E C2(S2,C(J,Rn)) and all t E [0,a].
Therefore Condition II E(ii)
is satisfied with the same Lipschitz constant L.
Thirdly, if t E [0,a] and
Y E CA([O,a], C2(S2,C(J,Rn))). then g(t,y(t))(.) = able.
is Ft-measur-
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 FDE (II) (or (I)) will have a unique solution.
So we obtain the
Lipschitz maps
Tt1:12(S2,C(J,Rn);Ft)
-
L2(2,C(J,Rn);Ft), t > t1,
1
W
t
xt
When t1 = 0,
Tt = Tt:c2(S2,C(J,Rn);FO) > l2(s,,C(J,Rn)Ft
47
The first main result here tells us that the trajectory of the stochastic
FDE (I) (or (II.)) can be viewed as a Markov process
[O,a] x 0 (t,w)
>
C(J,Rn) Tt(W)(w)
I
The following lemmas will be needed, the first of
taking values in C(J,Rn).
which is a well-known result for Brownian motion.
Lemma (1.1.):
Let gt, t E [O,a], be the a-algebra generated by t < u < a}.
Then Ft and gt are independent for all
t E [O,a].
Proof: See Elworthy [19],p. 19; or Friedman [22], p. 44. N.B.
{gt}tE[0,a] is decreasing and
n
o
gt
tE[0,a]
Convention
For t1 > 0, we have the obvious continuous linear isometry C(J,Rn)
t
> C2(Q.C(J,Rn);Ft )
associating with each n E C(J Rn) the constant map t(n)(w) = n, for all W E Il. Let Ttt 1(n)
denote Tt1(t(n)) for all t > t1.
With the above convention one has t1(n)}t>t Lemma (1.2):
For each n E C(J,Rn), the trajectory {T
of (I) (or 1
(II)) (with W = n) is adapted to the filtration (Ft n
(with each
gtl)t>t 1
Ft n gt1 independent of Ft ). 1
Proof: For fixed t1 > 0 define the process wt1
C([t1,a],Rm) by
ti
M = w(t) - w(t1), t > t1. Then w 1 is adapted to (Ft n
solution of (II) with w = n.
48
gtl)1>t1.
Now let n E C(J,Rn) and nx be the
t1,
In (II) one can replace w by w
thus getting a unique solution through
n which is adapted to (Ft n
gt1)t>t1.
But nx also satisfies
n(0) + (w) Jt1
n
t > t1
x()(t) = {
t1-r < t < t1,
l n(t-t1)
a.a. W E 11.
is adapted to (Ft n gtl)t>t
Then by uniqueness of solutions {Tt1(n)}bt 1
o 1
The next lemma is probably well-known for the case J = {0} (Stochastic ODE'S): it essentially says that one need only solve (I) on the subspace C(J,Rn) of the configuration space 92(Sq,C(J,Rn)).
Lemma (1.3):
Suppose Hypotheses (M) are satisfied.
Let W E 92(S2,C(J,Rn);Ft
be an Ft -simple function i.e. there are nj Ek C(J,Rn), j = 1,2,...,k and a
c Ft of 9 such that W = E nj X- , where Xcz is the
partition
j=1
1
characteristic (indicator) function of Saj.
Tt1(W)
Tt1(n j )Xn
E
=
j=1
j
J
Then
, t > ti
(2)
J
i.e. t Tt1(W)(M ) =
t
k
Tt1(nj)(w)Xj (w) for a.a. w E S2.
E
j=1
J
Remark In particular, if £2D E Ft , then 1
Tt1(Xn)
=
Tt1(1)X
0 Proof:
+ Tt1(0)X
.
0
0
is Ft -measurable because Sl. E F Solving n J n 1 the stochastic FDE(II) at nj E C(J,R ), we get a solution ix satisfying Let 1 < j < k.
Then X.
t1.
J
n(0) +
n
Jx(')(t)
J
t > t1 (III)'
tl
n(t-t1) Since X.
ft
t1-r < t < t1, a.s.
is Ft -measurable, then the process u
adapted to (Ft)t
-a
is J
1
> t1;
and so by property of the stochastic integral (Theorem 49
(I.8.3)(i)) we get X12
J
a.s.
Jt
Jt t1
t1
J
Using the last relation with (III)'yields
nj(0)Xn nj = {
dw(u),
+ Jt
J
1
t > t1
J
t1-r < t < t1, a.s.
nj(t-t1)X,j ,
Clearly,
k
kn
n G(u,
E G(u, Jx u j=1
for all u > ti.
E Jx u j=1
J
Thus k
k
E G(u,ix n
n.(0)X + ft t E j=1
k E
j
0
j=1
J
j
u
1
t > t , 1
j=1
k E
nj(t-t1)xa ,
j=1
t1-r < t < t1, a.s.
J
rt r
J
t1
kn G(u,
E
j=1
ix
J )dw(u),
u
t1-r < t < ti,
t > t1,
a.s.
Therefore by uniqueness of solutions to stochastic FDE(II) (Theorem (11.2.1)), we obtain k E
n.
t1-r < t < a, a.s.
j=1
J
This implies that
tt
k
Tt(W) = Wxt =
k
t
Tt1(nj)xnJ,
Jxt X0j = jEt
a.s. on Q.
n
t1 < t < a,
jE1
o
We are now ready to prove the main theorem in this section.
50
(The Mzrkov Property):.
Theorem (1.1)
by the stochastic FDE(I). {nxt
:.
Suppose Hypotheses (M) are satisfied
Then its trajectories
t E [0,a], n E C(J,Rn)}
describe a Markov process on C(J,Rn) with transition probabilities given by p(t1,n,t2,B) = P{w:w E S2, Tti(n)(w) E B}
(3)
for 0 < t1 < t2 < a, n E C(J,Rn) and B E Borel C(J,Rn).
Indeed for any
8 E L2(Q,C(J,Rn);F0) the Markov property
P(Tt2 (6) E BIFt
)
= p(t1,Tt
2,B) =
P(Tt2 A)E BITt
1
1
(0))
(4)
1
holds a.s. on 9. Proof:
Observe first that for 0 < t1 < t2 and every n E C(J,Rn),
P o
{Tt2i(n)}F1
and is therefore a probability measure on C(J,Rn) because
We would like to show that, if 0
Tt1(n) :S2 - C(J,Rn) is measurable. 2 then
(P{w:Tt2 (8)(w) E BIFt
})(w') =
p(t1,Tt1 (0)(&),t
2,B)
(5)
1
for a.a. w' E S2, every Borel set B in C(J,Rn) and any 8 E r2(St,C(JRn);F0).
Now the right-hand side of (5) is equal to Jg XB{[TtI(Tt1(0)(w'))](w)} dP(w)} for a.a. w' E. Q.
Therefore by the definition of conditional probability it is easy to see that (5) is equivalent to
f A x (Tt 2
for all A E Ft
(e)(w))dP(w) =
XB{[Ttl(Tt (8)(w'))](w)}dP(w)dP(w') (6)
j
fA
S2
2
1
and all B E Borel C(J,Rn), where XB is the characteristic
function of B.
To prove (6), we show first that if f:C(J,Rn) - R is bounded and continuous, then
51
f(Tt (e)(w))dP(w) = J 2
JA
A
for all A E F
t
j
f{[T tt(Tt (e)(w'))J(w)}dP(w)dP(w') 2.
S2
ti
t1
By Theorem (11.2.2), T t2 (e) = T t2 (Tt1 (e)).
(e) E
Now since T
t1
j=1 of Ft1-simple functions con-
there is a sequence
92(S2,C(J,Rn);Ft
(7)
1
verging to Tt1(e) in r2(S2,C(J,Rn);Ft1); i.e. n.
J,i E Ftt, iE1 'j.i Xs2j.i'
Wj
Therefore by continuity of
T
nj E
= lim
, we have
Ttt2
t2 (e)(w) = lim j-° Tt2
j-' ° i=1
E C(J,Rn).
ill
J t
Tt 2 ( )()X J'
J,i
1
(w)
a. a. W E S2, because of Lemma (1.3). t1)
Again, each
by Lemma (1.2); so we can
E rC2(S2,C(J,Rn); Ft2 n g
write T4 (4j j) as a limit (in L2 or a.s.) of g 1-simple functions; viz: 2
M
S2k,h E
ek'h , *
E
lim
2
ek,h E C(J,Rn)
k,h
h=1
(8)
9t I, 1 < h < mk, k = 1,2,...; {S2k,h} partition 12.
Now the left hand side of (7) is equal to n Ttt
j
E
lim f(
A j.
i=1
J,i.(w))dP(w)
2
n
= lim J
j-'°°
E
A i-1
J,i.(w)dP(w)
2
nj
Mk
E f(e')* (w)X()O(w)
E
= lim lim
j-,°° k-i=1 h=1 JA
= lim lim j-
52
nj
Mk
E
E h=1
k- i=1
k,h
j,i
f(ek:h)P(nk,h n ni'i n A)
(9)
Conversely, the right-hand side of (7) is equal to
J
A
J
lim f[Tt1(Wj(w'))(w)]dP(w)dP(w') 2
S2 j-
f[Tt1(W (w'))(w)]dP(w)dP(w`)
= lim j-wO fA
= 1
t2 J
J S2
f[( E
mm
j-HO JA
i=1
fn
= Jim
j
f[ E fA
J S2
A
fil
= lim
j
Tt1(0j,i)XQ 2
2
f(Tt1(4j,i)(w))Xn(w')dP(w)dP(w')
E
i=1
j,i
2 nj
Mk
E
E h=1
r
r
= lim lim 1 J
A
j-4 k4
j(w'))(w)]dP(w)dP(w')
(w')]dP(w)dP(w')
1
i=1
,i
S2 i=1
f(Ak'h)X * (w)X 'k,h
(w')dP(w)dP(w')
j,i
Mk
n. r
= lim lim J j-+o
A nj
= lim lim
E
E
i=1
h=1
Mk
E
k- i=1
j ,i (w')dP(w')
E
'
h=1
n 52.
.)
(10) t1
Now A, j,i E Ft1, so A n Q
j,i
is independent of S2k*'h E g
because of Lemma
(1.1); hence
P9t*h)P(A n sJJ
=
P(Stk,h n A n sl 'j).
Thus comparing (9) and (10), it follows that (7) must hold. The next step is to establish (6) for all open sets B in C(J,Rn); so let B c C(J,Rn) be open.
Since C(J,Rn) is separable and admits uniformly by con-
tinuous partitions of unity, we can choose a countable uniformly by continuous partition of unity {(U(x,Oa)}a=1 of B where U. c B are open for all a > 1,
a:B - R' is uniformly continuous and vanishes outside Ua, and 00
E a(n) = 1, for all n E B. a=1
fm:C(J,Rn)
Define the sequence
100
of functions
R by setting 53
m
nEB fm(n)
n¢.B
0
Clearly fm(n) = 0 for all n ¢
Ua, so each fm is uniformly continuous and
VU
a=
Also it is easy to see that
bounded on C(J,Rn).
0
E 4(n)=1 a
lim
TI EB
a=1
fm(n) _
M+W
n 4 B
0
XB(n) for all n E C(JRn).
=
is a sequence of bounded and uniformly
Therefore fm:C(J,Rn) -r R, m = 1,2,...
continuous functions converging pointwise to XB.
We can therefore replace f
by fm in (7), then pass to the limit as m - co using Lebesgue dominated con-
We obtain (6) in this way for any open set B c C(J,Rn).
vergence theorem.
To get (6) for any Borel set B in C(J,Rn) we reason as follows.
Let
v:Borel C(J,Rn) - R be the set function
v(B) = IA IS, XB{[Tt2(Tt1(e)(w')](w)}dP(w)dP(w'), B E Borel C(J,Rn),
(11) where A E Ft1 is fixed.
Then v is a finite measure on C(J,Rn); to prove this
let {Bi=1 be a countable disjoint class of Borel sets.
Then
m
v( U
Bi) = I
i=1
E
A I
XB {[Tt1(Tt (6)(w'))1(w)}dP(w)dP(w')
i=1
i
2
1
m =
(12)
E v(Bi) i=1
m Since
E XB (n)-* X U B,(n) as m i=1 i i=1 1
for all n E C(J,Rn). then by the dom-
inated convergence theorem we can let m -
in (12) to obtain
00
E
i=1
v(Bi) = lim I
mA
I
E
n i=1
XB {[Tt1(Tt (e)(w'))](w))dP(w)dP(w')
i
2
1
{[Tt1(T
X
fA
tt
U
i=1
54
Bi
t2
(e)(w'))](w)}dP(w)dP(w')
t1
= V( U
Bi).
i=1
Therefore v is a finite (Borel) measure on C(J,Rn).
Since C(J,Rn) is metric,
v is regular (See §(I.2)).
The set function v*:Borel C(J,Rn) - R
v*(B) = JA XB(Tt2(e)(w))dP(w), B E Borel C(J,Rn),
is also a regular (finite) measure on C(J,Rn); so for any B E Borel C(J,Rn), * v (B) = inf{v*(U): U c C(J,Rn) open, B c U}
= inf {v(U): U c C(J,Rn) open, B c U}
= v(B). Therefore (6) holds for all Borel sets B in C(J,Rn). To see that P(t1,Tt1(e)(.),t2,B) = P(Tt2(e) E BIT
ti
(0)) a.s.
it is now sufficient to check that the left hand side p(tl,T
ti
(e)(-),t
29
B)
is measurable with respect to the a-algebra generated by Tt (e); this is valid because the function
1
C(J,Rn) 3 n f------ p(t1,n,t2,B) = JQ x (Tti(n)(w))dP(w) is measurable: If B c C(J,Rn) is open, let {fm}m=1 be a sequence of uniformly bounded, uniformly continuous functions fm:C(J,Rn) -r R converging pointwise to XB.
Therefore t
f m (Tt1(n)(w))dP(w).
P(t1,n,t2,B) = lim J m
2
t
If nk + n in C(J,Rn), then Tt1(nk) -T t1(n) in Z2 (0,C); so f (Tt1(nk)()) m t2
2
2
t
fm(Tt1(n)(w)) as k - - in probability; so by Lebesgue dominated convergence 2
theorem, for each m,
55
f (Ttl(n)(w))dP(w) as k
f (Ttl(nk)(w))dP(w) Sm t2
J
m
J
t2
Hence n - fJ fm(Tt2(n)(w))dP(w) is continuous (for each m).
Therefore
is measurable because it is a pointwise limit of continuous functions.
If B E Borel C(J,Rn), then
p(t1,n,t2,B) = inf {p(t1,n,t2,U): U open in C(J,Rn), B c U}.
But
must be measurable.
each n - p(ti,n,t2,U) is measurable, so Therefore P(t1,Ttl
E[P(Tt2 (0) E BIFt1 )ITti (B)]
= P(Tt2 (0) E BITti (0)),
since the o-algebra generated by T
t1
(0) is contained in F
t1
Finally we check the Chapman-Kolmogorov identity P(t1,n,t2,B) = JCEC P(u,C,t2,B)
for all B E Borel C(J,Rn) 0 < t1 < u < t2 < a, and then construct a canonical
Markov process on C(J,Rn) with the same distributions as the trajectory {nxt:t E [0,a], n E C(J,Rn)}.
Note that if 0 < t1 < u < t2 < a, then
Tt1(n) = Tt (Tul(n)) a.s. and as in (6) we get 2
2
t
J 52
XB(Ttl(n)(w))O(w) = J t2
,
w ES2
XB([Tt (T u1(n)(w'))](w))dP(w)dP(w') 2 LE92 (6)'
Therefore, t
CEC
SEC
56
P(u,c,t2,B)
J
J
p(u,Tu1()('),t2,B)dP(w')
=
L.
-
Jw' Ec,wES2
X B([Tut2 (Tt1(n)(w'))](w))dP(w)dP(w') u
XB(Ttt2(n)(w))dP(w)
= P(t1,n,t2,B).
Transform the filtered probability space (S2,F,(Ft)
,P) using the family
of solutions nx:.S2+ C([-r,a],R") to the stochastic FDE(LI.) at n E C(J,Rn), n,
=
viz.. let
C([-r,a],R"), F = Borel Q. pu:C([-r,a],R") - R", U E [-r,a], be
evaluations at u and Ft = a{pu:-r < u
and Ft c Ft
Then Fa = F
when 0 < t1 < t2 < a. If A ={w :w(u.) E A., j = 1,...,k} for J 2 u. E [-r,t], j = 1,...,k, is a cylinder set in Ft. then (nx)(A) _
tiJ
1
{w:nx(w)(u
E A.., j = 1,...,k) E Ft because uj < t, j = 1,...,k.
nx:(S2,Ftti (Si,t
Hence
is measurable for each 0 < t < a. Define the process
{(St,X(t),Ft,Pn):0 < t < a, n E C(J,R")} on C(J,R") through the memory n,
n,
n
n,
n,
n,
X:[0,a] x S2 + C(J,R ), X(t,w) = wt, w E St, 0 < t < a. C(J,R").
Let Pn = P ate("x)-1 for each n E measure on (92,F). n,
t
n,
n,
Then each Pn is a probability
Also each X(t,.) is F -measurable because the map
w .} wt(s) = Pt+s(w) is Ft+s-measurable for each s E J.
Note that for fixed
0< t< a and B E Borel C(JR"), Pn {w:X(t,ai) E B} = P{w: nxt(w) E B} = p(O,n,t,B). n,
Hence n - Pn {w:X(t,w) E B} is measurable; so in order to establish that X is indeed a Markov process one needs to show that Pn(X(t2) E BIFt ) = p(t1,X(t1),t2,B)
a.s. (Ft1,Pn),
for B E Borel C(J,R"), 0 < t1 < t2 < a. For this we need only prove that JA XB(X(t2))dPn = f
,t,
P(t1,X(t1),t2,B)dPn
A
for each A E Ft
n (x)
-1 n,
(A) E F
.
1
t1
Apply the transformation w y nx(w) and the fact that n,
n,
whenever A E F
t1
to get
57
XB(X(t2,w))dPn(w) = IA
XB(nxt2 (w))&(w) Jflx)_1CA)
E BIF
P(nx
i A
t2
)dP ti
1(x) p(ti, xt(),t2,B)dP(w) I(flx)_1(A)
= IA p(t1,X(t1,w),t2,B) ti
ti
0 < t < a, n E C(J,Rn), is a Markov process in the
Hence
sense of Chapter I §7 (cf. Dynkin [16], pp. 47-104). proof of the theorem.
§2.
This completes the
o
Time-Homogeneity: Autonomous Stochastic FDE's
Here we look at the autonomous stochastic FDE
t
t
e(w)(0) + Jo H(xu(w))du+ (w) JG(xu())dw(.)(u), 0
x(w)(t) =
(IV)
tEJ
19M(t)
with coefficients H:C(J,Rn) + Rn, G:C(J,Rn) + L(Rm,Rn).
We adopt the follow-
ing hypotheses Hypotheses (A) (i)
For each t E [0,a], Ft is the a-algebra generated by {w(.)(s):0 < s < t}
together with all sets of P-measure zero in F = Fa.
(ii) There is a constant L > 0 such that IH(ni) - H(n2) I < LIIni IIG(ni) - G(n2) II < for all ni, n2 E C(J,R').
58
Llln1
n211 C
- n2II
c
Definition: (i)
Let E be a real Banach space and T any non-empty set.
Two processes x1, x2:T
x52- E are isonomous if for all
B1,B2,...,Bk E Borel E, tilt 29...,tk E T,
E Bk).
E
E B1t
=
(ii)
E Bk)
E
E
Two random variables X1, X2:St- E have the same (probability) distri-
bution if for all B E Borel E P(X1 E B) = P(X2 E 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
Lemma (2.1):
-
X2.
Let E,F be B-spaces (or even metric spaces), x1,x2:T x 0 - E
isonomous processes, and L:T x E - F a (Borel) measurable map.
Then the pro-
cesses
T x S2
F
(t,w)
L(t,xi(t,w)),
>
i
= 1,2,
are isonomous.
Proof:
Suppose B1,...,Bk E Borel F, tl,t2,...,tk E T.
measurable, then
Since L(t,.) is
E Borel E for all 1 < j < k, so that by isonomy
of x1 and x2 we have E =
E Bk)
E
E
E
E
E Bk).
E
Lemma (2.2):
Suppose x1,x2:[-r,a] x S2
0
Rn have a.a. sample paths continuous.
Then the following statements are equivalent: 59
(i)
and x2 are isonomous .
x1
(ii)
have the same
The induced random variables x , x2 :S2 -+
distribution.
(iii) The processes
[O,a] X 9 -- C(J,R") (t,w) I, xt(),
i
= 1,2,
are isonomous.
We prove the implications (i) -*
Proof:
(ii) ->
(iii) * (i).
Since x1, x2 have a.a. sample paths continuous, then
Suppose x1 - x2.
- C([-r,a],Rn) are (Borel) measurable; these will have the same
x1, x2:n
distribution if and only if the measures P o (x1)-1, P o (x2)-1 agree on cylinder sets in C([-r,a],R"). 212).
(See Parthasarathy [66], Theorem (2.1), p.
For each t E [-r,a],let Pt:C([-r,a],Rn)
-,
R"
Let t1,t2,...,tk E [-r,a], B1,B2,...,Bk E Borel R".
be evaluation at t. Then k
P
(x 1)-1[ n
j=1
pt1(Bj)] = P(x1 E Pt1(B.) for all 1 < j < k) J
j
=
E Bj for all 1 < j < k)
=
E Bj for all 1 < j < k) k
= P o (x2)-1
[ n Pt1(B)M .
j=1
Therefore P
(x1)-1
j
= P o (x2)-1.
Suppose now that P o (x1)-1 = P o (x2)-1.
m:[O,a] x C([-r,a],R') -> C(J,Rn) at 60
The memory
is continuous (and hence measurable)} so i.f B1.9 ...,Bk E Borel C(J,Rn),
tl,...,tk E [O,a], then
E Borel C([-r,a],Rn) for all
1<j
P(x1
E B. for all 1 < j < k)
J
= P(x1 E
for all
1(B
= P(x2 E
1 < j < k)
for all 1 < j < k)
= P(x2
B
for all
1 < j < k).
J
Therefore (t,w)
t-> xt(w), i = 1,2, are isonomous.
The implication '(iii) = (i)' follows trivially from Lemma (2.1).
a
Notation
E, F are B-spaces.
For any stochastic processes y1:T x 2 -*- E, y2:T x 0 -, F
define the process y1 x y2:(T x T) x S2+ E x F (on T x T) by (y1
x y2)(t1,t2,w) = (y1(t1,w), y2(t2,w)) for all t1,t2 E T,
Lemma (2.3):
W E 0.
C([O,a],R), i = 1,2, be processes satisfying
Let z1: S2
condition (E)(i) of existence of McShane's integral (Theorem (1.8.4)). Suppose f':[O,a]
C2(c,Rn) are t2-continuous processes adapted to (Ft)tE[O,a]'
Define the processes F1:[O,a] x S2 - Rn, i = 1,2, by
t F'(t,w) = (w) J
t E [O,a],
a.a. W E s2.
0
If f1 X z1 is isonomous to f2 x Z2, then F1 X z.1 is isonomous to F2 x
Proof: n
We approximate each FI by Riemann sums. a
z2.
For each integer k, let
2a
- (0, k, -k-,...,a) be a (Cauchy) partition of [0,a] into equal subdivisions.
For each t E [O,a], let m(t,k) be the greatest integer such that
m(t,k)a < t <
[m(tk
a .
Define the processes S'(nk):[O,a] x SZ --> Rn by 61
m(t,k)-1 S'(IIk)(t,w
=
E
f i(ja )(w)[z
i
(j+1)a
i
(w)(
jar)]
j=0
+
f'(mT a)(w)[zi(m )(t)
for all t E [O,a], a.a. W E 9, i = 1,2. Since f1 x z1 - f2 x z2, then for any finite sets tilt 2,...,tk E [0,a], sits 2,...,sp E [0,a] the random variables i = 1,2,
(ft(t1), f'(t2)....,f'(th),
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 tilt 21...Ith,
i
= 1,2
have the same distribution i.e. S1(IIk) ^- S2(IIk), k = 1,2,3,... Now J
k
F'(t.,-) as k+°° in l2 for 1 < j < h. J
.
Thus
ask-- in
f2
This implies that
(i = 1,2).
PO(S'(IIk)(ti
as k +
in the weak topology of (Borel) measures on Rn X Rn
(h times).
k
x Rn
But we know that
> 1; so by uniqueness of the limit in the weak topology of measures
we must have
on Borel F1
x z1
(Rn x
... X Rn) (h times) (§(I.2)).
Hence F1 - F2.
To prove
F2 x z2, fix (s ,...,sp) = s and define z'(s) = (z'(s),...,z'(sp)). 1
Then (f1,z1(s)) x z1 - (f',z2(s)) x z2. Write (F',z'(s)) = (F',0) + (O,z'(s)), i = 1,2.
As F1
F2 and z1
z2,
it follows from Lemma (2.1) that (F1,0) '- (F2,0), (O,z1(s)) - (O,z2(s)) and
62
(F1,z1(s)) = (F1,O) + (O,zt(s)) ti (F2,O) + (O,z2(s)) = (F2,z2(s)). F2 x z2. G Thus F1 x zI
-
Suppose G:C(J,Rn) -> L(Rm,Rn) is Lipschitz and
Lemma (2.4):
z1:St- C([O,a],Rm), i = 1,2, are processes satisfying the conditions of For each n c C(J,Rn) let {1Tt(n)}tE[O,a]' i = 1,2,
Existence E(i)
be the trajectories of the stochastic FDE's: rt
n(O) + (w) J
t E J, i= 1,2,
n(t) If z1 and z2
0 < t < a
(u)
0
r 'x(w)(t) = j
a.a. W E Q.
are isonomous, then {1Tt(n)}tE[0,a]' {2Tt(n)}tE[0,a] are also
isonomous.
Proof:
Use the method of proof of Theorem (11.2.1).
there are sequences ixk:11
(i)
i k+1
x
-
i k
2
x
< (ML
2
ak-1
2 k-1
)
!
i
x
2_i
lxk x zl .., 2xk
x
2 x II2 1
z2,
i = 1,2, 1
L 01,C)
t (11,C)
(ii)
Assume inductively that
C([-r,a],Rn), k = 1,2,..., i = 1,2, such that
where
n(0)
t E [O,a]
n(t)
t E J
ix1(w)(t) = {
a.a. w E 0, i = 1,2. Indeed we define t
n(0) + (w) j
G(
i
tE
i
E S2.
(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
z2 implies 63
x
(1xt
z1
.
x z2
(2xk
m
1
(Lemma 2.2)
m
1
Hence (G(1xt
x
1-1
(G(2xt
)x z1 ti G(2xk,))
,,, 2xk+1 x Thus 1 xk+1 x z1
(.)))x z2, m
1
for t1,...,tm E [O,a], by Lemma (2.1); G(Ixk
-
m
1
i.e.
z2.
z2 because of Lemma (2.3).
Obviously (ii) is valid for k = 1 because (n,z1) - (n,z2).
Therefore
(i) and (ii) hold for all k. In particular (ii) implies 1xk _ 2xk for all = P o (2xk)-1 k > 1. Thus for all k > 1. Po(Ixk)-1
Now ixk
>
ix as k -)...o a.s. (or in L2) from the proof of Theorem (lxk)-1
(11.2.1); so P o
> P o (1x)-1, i = 1,2, as k - w in the weak
topology of Borel measures on C([-r,a],Rn). x)-1
we must have P o
(1
(2.2), {1Tt(n)}tE[O,a]
{t -> 2xt}.
_
Hence by uniqueness of limits,
= P o (2x)-1 on Borel C([-r,a],Rn).
{t E-->
So by Lemma
1xt} is isonomous to {2Tt(n)}tE[O,a]
_
o
The following theorem is our second main result; it says that the Markov process given by trajectories of the autonomous stochastic FDE (IV) is in fact time-homogeneous in the sense that each transition probability p(t1,n,t2,.), t1 < t2, n E C(J,Rn) depends on t2 - t1 (and n) only.
Theorem (2.1)
(Time-homogeneity):
FDE (IV) satisfies Hypotheses (A).
Suppose that the autonomous stochastic For 0 < t1 < t2 < a, n E C(J,Rn) let
p(t1,n,t2,.) be the associated transition probabilities of trajectories of (IV) (as given by Theorem (1.1)).
Then the trajectory Markov process {nxt:0 < t < a, n E C(J,Rn)} is timehomogeneous, i.e. 0 < t1 < t2 < a, n E C(J,Rn). t1x:;2
Proof: 64
Let n E C(J,Rn), t1 < t2.
Suppose
->
C([t1-r,t1],Rn) is the
solution of t
n(0) +
G(t
t > t1
j
t1
t1
X%.) _ { t1-r < t < t1
n(t-t1) t-t1
t > t1
G(
n(O) + j
ii
t-r
{n(t-t) 1
t-t1
t
n(O) +
G(
Jo 1
1
1
Then to
n(0) + (
)
t
to E J
n(t') a.s.
Now by property of Brownian motion, we know that the process
u'1-> w(u'+t1)-w(t1) is isonomous to w (Elworthy [19], Corollary 3A, p. 19). So comparing the above stochastic FDE with our original one viz: Jt,
n(0) +
X(-)(t')
0 < to < a
= {
0
to E J
n(t')
it then follows from Lemma (2.4) that the trajectories {Tt(n)}0
1
In particular, putting t = t2 - t1 one
are isonomous.
{tlxt+t }0
gets P o
{Tt1(n)}-1
P 2
1
=
o
2
Remark Theorem (2.1) still holds if the Brownian noise w is repalced by an (Ft)0
65
martingale z satisfying Condition (I.E) and having the property that for
each t1 > 0 {t s-' z(t1+t) - z(t1)} - Z. U.
The Semigroup
Denote by Cb the Banach space of all bounded uniformly continuous functions
$:C(J,Rn) -> R furnished with the supremum norm II4 II Cb = sup {I4(n)I
Let 0 < t < t2 < a, n E C(J,Rn) and define
n E C(J,Rn)}.
Tt2(n) E £2(c,C(J,Rn);Ft2) as before (§II.2), using the trajectory of the
stochastic FDE (I) in §1.
For each 0 E Cb define the function
Pt2(O):C(JRn) - R by setting t = E(4) c Tt1(n)) = J 0(&) P(t1,n,t2,dC) 2 C
Pt1(4)) 2
are the transition probabilities of (I) (Theorem (1.1)).
where
Since 0 is bounded, it is clear that IIPt'(0)IICb < II0IICb for all 0 E Cb.
Ptt1(4))
t2
is also bounded.
Indeed
Furthermore, each
is uniformly
continuous on C(J,Rn) by virtue of the following elementary lemma:
Let (S2,F,P) be a probability space, A a metric space and E a
Lemma (3.1):
separable Banach space.
Suppose 4':E - R is any bounded uniformly continuous
For each t > 0 and A E A, let ft, fA E t2(c,E;F) (§I.2).
function on E.
Then the following is true: (i)
If the map A 3 A - fA E C2(2,E;F) is uniformly continuous, then so
is the map
A 3 a F-> (ii) If lim
4)(fX(w))dP(w) E R.
Ift(w)-fA(w)IE dP(w) = 0 uniformly in a E A, then for every
J
t-'0+ C > 0,
J
A
lim
0 uniformly in A E A.
t-'0+ lim
t-'0+
I
uniformly in A E A.
66
t
4)(fX(w))dP(w) = J
JQ
4)(fA(w))dP(w)
n
Moreover,
Proof : (i)
By uniform continuity of 0, there is a 6' > 0 such
Let c > 0 be given.
that whenever E1,E2 E E and A2-
'
1-E2IE < d', then I4(
I
1)
- 0(&2)I < c
But
fx is uniformly continuous, so there is a d > 0 such that if
X,A' E A and d(A,A') < d , then t- IfX(w) - f
(w)12 dP(w) < c6'2, where d
Therefore, if d(A,A') < d, we get
is the metric on A.
If'(w)-fX1(w)IE
> d'} < _6L-2 J
P{w:w En,
If'(w)-f' (w)IE dP(w) < E S2
by the Chebyshev's inequality.
f9
Now i.f d.(A,A' < d we have
I0(f'`(w) - 4(fX1(w)) (w))IdP(w)
j1f_fA
=
IdP() CIE>d1
I
IdP(w)
1
E
EP(IfX- fAlIE
<2110 11
where 11011
< d') < E(2IIfII + 1)
= sup {I4,(v)I: V E E}.
(ii) Suppose lim
A(w)12
JQ Ift(W) -
dP(w) = 0 uniformly in A E A.
Then the first assertion in (ii) follows immediately from the Chebyshev inequality P(Ift -
fAIE > E) <
J" Ift(w) - fA(w)12 dP(w) 1
which holds for any e > 0, t > 0, A E A
(Theorem (I. 2.4)).
To prove the second assertion in (ii), use the first assertion and a similar argument to that in (i).
o
We now have the following theorem.
Theorem (3.1): isfied.
For the stochastic FDE (I) suppose Hypotheses (M) are sat-
Then the family {Pt2:0 < t1 < t2 < a} is a contraction semigroup on
Cb i.e.
67
Each Pt1:Cb
(i)
t2*
- , Cb is a continuous linear operator
I -4 Pt1(0) t2 IIPt2II < 1 for all 0 < t1 < t2 < a;
with
ti
(ii)
Pt2 = Pt1, 0 < t1 < t2 < t3 < a. t t
o
t2
3
3
In particular, for the autonomous stochastic FDE (IV), the family {Pt = Pt :0 < t < a} is a one-parameter contraction semigroup on Cb i.e. Pt
o
Pt2
=
Pt1
if t1,t2 E [0,a], t1 + t2 E [0,a].
+t 2
1
By Theorem (11.3.1), the map Tt1:C(J,Rn)
Proof:
t2
>
t2(c2,C(J,Rn);F
)
t2
is
But 0 is uniformly continuous, so from the
Lipschitz for 0 < ti < t2 < a.
above lemma it follows that Pt2(4)) is uniformly continuous on C(J,Rn).
Thus
Pt (0) E Cb. It is clear that IIPt (O) II Cb < II 4II Cb for all 0 E Cb i.e. 2
2
each Pt1:Cb - Cb is continuous linear with IIPt1II < 1. 2
2
For 0 < ti < t2 < t3 < a, the semigroup property
Pt1 2
from the relation Tt1 = Tt2 t3
{Pti[Pt2(m)]}(n) =
3
3
t2
I
t2 t3
Pt2 = Pt1 follows
Tt1 (cf. Theorem (II. 2.2)) and
o
t3
o
4){Tt2[Tti(n)(w)](W')}dP(w')dP(w)
I
SZ
t3 t2
Sl
t =
J
4)[Tt3(n)(W)]dP(w),
(Cf. proof of Theorem (1.1)).
t
Pt+t
Pt
In the autonomous case, Pt 2
ti,t2 E [0,a], t1 + t2 E [0,a],
=
'
2
1
2
because of time-homogeneity.
Thus Pt
o Pt
0
o
= Pt
1
1
2
The semigroup {Pt}i>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(J,Rn) given the weak * (or vague) topology (§I.2, Partha-
sarathy [66], Schwartz [71], Stroock and Varadhan [73]). bilinear pairing 68
Then we have a
Cb X M(C) - > R, <$,u> = jC 0(n)du(n), E Cb, P E M(C).
Following Dynki.n [16], define the adjoint semigroup
Pt*1(kt
the measure (Ptu)(B) =
p(O,n,t,B)du(n),
J
B E.Borel C(J,Rn), 0 < t < a.
nEC
Then for any 0 E Cb,
d(Ptu)() = f&EC
«,Ptu> = f CEC EEC
JW
nEC
p(O,n,t,d&)du(n)
fnEC
0(C)p(O,n,t,dE)du(n)
'nEC Pt(4)(n)du(n) = .
Since {Pt}0
o 1
= Pt
Pt 2
+t 2
1
An invariant probability measure for the
t1,t2 E [O,a], t1+t2 E [O,a].
stochastic FDE (IV) is a probability measure u0 E M(C) such that Pt u0 = u0 for all t E [O,a].
invariant if and only if lim
If a
a probability measure u0 E M(C) is
p(O,n,t,.) = u0 for some n E C(J,Rn).
This
t-, follows easily from the fact that Pt n E C(J,Rn).
t,t' > 0,
Observe also that the family of transition probabilities
E C(J,Rn), t > 0} for (IV) is left invariant by the semigroup
{Pt}t>0
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
§1.
Notation
For the present chapter we keep the notation, general set-up and the standIn particular, we focus our
ing assumptions of the last chapter (§III 1,2). attention on the autonomous stochastic FDE: r
f
n(0) + 10 H(xu(w))du + (w) x(w)(t)
t > 0
JO
(I)
1
t E J = [-r,0]
n(t)
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) G:C(J,Rn)
-
L(Rm,Rn) are globally bounded and Lipschitz.
>Rn,
The driving
Brownian motion w is in Rm, generating a filtration (F t)1>0 on the probability space (SZ,F,P).
For brevity, symbolize the above stochastic functional equa-
tion by the differential notation. dx(t) = H(xt)dt + G(xt)dw(t)
t > 0 1 (I)
x0 = n E C(J,Rn)
J
Similarly for any t1 > 0 we represent the equation n(0)+Jt
H(xu(w))du + (w)
XMM n(t-t1)
t1
Jt
t1-r < t < t1
by the stochastic-differential notation dx(t) = H(xt)dt + G(xt)dw(t) xt
t > t1 > 0
= n E C(J,Rn)
Recall that at the end of the previous chapter, we constructed a contraction semigroup {PtIt>O associated with the stochastic FDE (I) and defined on the Banach space Cb of all bounded uniformly continuous functions c:C(J,Rn) --.> R. 70
Indeed
Pt(m)(n) = E[ a
n E C(J,Rn),
t > 0.
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>0 is never strongly continuous on the Banach space Cb. Furthermore, we shall derive an explicit formula for the (weak) infinitesimal
generator of {Pt}y.
§2.
Continuity of the Semigroup
For each n E C(J,Rn) and t > 0, define n:[-r,co) ; Rn by
ti
n(0)
t > 0
n(t)
t E J
n(t) = t
1
-r
t-r
0
t
Define the shift St:Cb s Cb, t > 0, by setting StM (N ) _ ,(nt), n E C(J,Rn), 0 E Cb.
The next result then gives a canonical characterization for the strong continuity of {Pt}
t>O
in terms of the shifts {St}t>0.
71
The shifts {St}t>0 form a contraction semigroup on Cb, such
Theorem (2.1):
t-,o+
t-'0+
t
P (4))(n) = $(n) for all
Pt(0)(n) = $(n) uniformly in Ti E C(J,Rn) if and
Furthermore lim
0 E Cb.
t
S MN N ) = lim
that, for each n E C(J,Rn), lim
St(4))(n) = ON uniformly in n E C(J,Rn).
only if lim
t-0+ Let t1,t2 > 0, n E C(J,Rn), 4) E Cb, S E J.
Proof:
Then
ti (St2 (4)))(n)
St
$C(nt1 )t2 ]
=
1
where ti
t2 + s > 0
n(0)
,..
(nt )t (s)
ti
2
1
-r< t2 + S < 0
(nt1)(t2 + s)
t2+s>0
In(0)
ti =
t1 + t2 + s > 0
nt1 (0) = n(0)
ti nt (t2+s) = n(t1+t2+s)
-r < t1 + t2 + s < 0
1
_ (nt1 +t )(s). 2
Hence I,
St1 (Si. (4)))(n) = (nt +t ) = St1 +t 2
1
2
MN
i.e.
St2 =
St1
St1+tz.
ti
n = n , it is clear that lim S ($)(n) = (n) for each t-$.0+ t-*o+ n E C(J,Rn), 0 E Cb. Also by sample paths continuity of the trajectory Since lira
t
t
{nxt:t > 0} of (I) (Theorem (11.2.1)) together with the dominated convergence theorem, one obtains
lim YON t ) = lim t-,0+ 72
J (nx S2
t
(w))dP(w) _ 4)(n)
for each 4) E Cb and n E C(J,Rn).
To prove the second part of the theorem, suppose K > 0 is such that Then for each t > 0 and
IH(n)I < K and IIG(n)II< K for all n E C(J,Rn).
almost all w E 0 we have Jt+sH(nx
r
u
0
(w))du+(w)
Jt+sG(nx
0
t+sa0
u
nxt(w)(s)-nt(s) = J
l
-r
0
Thus, using the Martingale inequality for the stochastic integral (Theorem (I.8.5)), one gets II
Ilnxt(-)
-
H(nxu(.))dul2
< 2E sup
ntII22 L
sE[-t,0]JJ0
Irt+s G(nxu(-))dw(-)(u)I2 + 2E sup SEE-t,01
0
Jt
<
2K2t2 + 2K1
du < 2K2t2 + 2K1t, some K1 > 0.
E 0
ti Therefore, lim
t-0+
Ilnxt - ntll
n
= 0 uniformly in Ti E C(J
2
).
Using the
t 01,C)
uniform continuity of 4) E Cb it is then not hard to see that
=0 uniformly in n E C(J,Rn) (Cf. proof of Lemma (III.
lim
t+0+ 3.1)). So lim
{Pt(4))(n)-St(0)(n)} = 0 uniformly in n E C(J,Rn). Finally, writing
Pt(O N - ON = [Pt($)(n) - St(O)(n)] + [St(4')(n) - O(n)], the second assertion of the theorem is now obvious.
o
Let C°° c C b be the set of all 0 E Cb such that lim Pt (0) = 4)(= lim St(0)) b t-+0+ t-,0+ in Cb. Then Cb is a closed linear subalgebra of Cb which is invariant under
the semigroups {Pt}
{St}1>0.
Both {Pt}1>0 and {St}1>0 restrict to strongly continuous semigroups on Cb (Dynkin [16], pp. 22-26). ,
Theorem (2.2): The semigroup {Pt}
t>O
is not strongly continuous on Cb with
respect to the supremum norm.
Proof:
It is sufficient to find W E Cb(C(J,Rn),R) such that St(W) -f-> P as 73
t +0+ in Cb; (but St(W)(n) - W(n) as t - 0+ for each n E. C(J,Rn).) Let B c C(J,Rn) be the closed unit ball.
Fix any -r < s0 < 0 and define
W:C(J,Rn) _ R by
IIn II < t
n(s0) W(n) _ { n(s0)
IInII
>t
IInII Clearly W is continuous; indeed W is globally Lipschitz (and hence uniformly To prove this let n, n' E C(J,Rn) and consider the
continuous) on C(J,Rn). following cases (1)
n,n' E B.
= ln(s0)
Then 19)(n) - TV(n')
n'(s0)I < IIn - n'll
(ii) no' !E int B, i.e. IInII > 1, IIn' II > 1. 1
- W( n ')I = I
4,01 )
n(s 0)
n(s0) I
n(s0)
In(s0)1 IIn II IIn' 1
IIn' II
Illn'11
-
IInII I
+
III
On' - nil
Iln' II (iii) n E int B, n' ¢ B; i.e.
74
I
n(s0)
+
11
II n'Il I
11n II
-L n'( s )
-
lln
<
Write
+
n'(s0)
Iln'II In(s0) - n'(s0) I IIn' II
IIn - n'II < 2Iln-n'II
1
Iln' II linIi
< 1,
lln'11 > 1.
Find n" E aB
(where 8B is the boundary of B) such that n" lies on the line segment joining n and n' i.e. find A0 E [0,1].such that n" = (1-A0)n + A0n' and IIn"II
= 1.
Define the function f:[0,1] --> R by f(A) = 11(1-A)n + an'Ii-1, A E [0,1].
Then f is clearly continuous. Also f(0) = IIn II - t < 0 and f(1) = III II -1 >0. Hence by the Intermediate-Value Theorem there exists a0 E (0,1) such that f(A0) = 0 i.e. take n" = (1-A0)n + a0n' and IIn"II
W(n) - W(n')l
= 1.
Hence
IW(n) - W(n")I t IW(n") - W(n')II
IIn_n"11 +211n"-n'II =[IIn -n"11 +
IIn"- n' II
]+ IIn"-n' II
= IIn -n'II + IIn" -n'II < 211n -n'II Therefore IW(n) - W(n')I < 211n - n'11 for all n,n' E C(JRn). Let {nn}rt=1 be a sequence in C(J,Rn) looking like this:
)
-r
b
s
s,+(1/n)
0
i.e.
s
0
nn(s) ={
+1<s<0 n
[s0+n-s]n 1
Note that IIn"II
0
-r<s<s0
< 0; i.e. nn E 3B for all n.
= 1 for all n, such that s0 +
Consider for each fixed n
s0<s<s0+1
n
75
0 < t < - s0.
St (W)(nn) = P(nt9 = nn(t + s0),
Therefore lim
t-'0+
t
nn(t + s)
S (W)(nn) = lim
0
t-0+
=
nn(s) = W(nn) 0
(*)
Now (*) is not uniform in n; for if (*) were uniform in n, given 0 < e < 1, there is a 0 < 6 < - s0 (6 independent of n) such for all 0 < t < 6 and for all n > 1.
that Inn(t+s0)-nn(s0)I < e
Fix 0 < t0 < 6 and choose n0 such that
n0 > t i.e. t0 + s0 > s0 +-. Thus no
0
nn0(t0 + s0) - nn0(s0) = 10 - 11 = I.
But, by choice of 6, Inn0(t0 + s0) - nn0(s0)l < e . tradicts the choice of e.
Hence 1 < c, which con-
o
Therefore W f Cg.
Remarks (i)
One could have chosen W* E Cb in the above proof such that 11n 11
1
11n II < 1 11n 11
It can be shown that W* f Cb.
>1
However, note that, while W* is only contin-
uous, W is smooth in any small neighbourhood of 0. (ii) Later on, in §4, we construct a concrete class of smooth functions on
C(J,Rn) lying entirely in C. tions.
These functions are called quasi-tame func-
The class of quasi-tame functions is sufficiently rich to generate
Borel C(J,Rn), in exactly the same fashion as the tame functions do. (iii) Note that the domain of strong continuity Cb of {Pt}t>0 is independent of the coefficients H, G and the Brownian motion w.
As yet, a complete
characterization of Cb 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>0 (Dynkin [16] Vol.
1
pp. 36-43).
Define a weak topology on Cb as follows: 76
Let M(C(J,Rn)) be the Banach
space of all finite regular measures on Borel C(J,Rn) given the total variaThen there is a continuous bilinear pairing
tion norm.
defined by
«,u> =
0 E Cb, u E M(C(J,Rn))
4)(n) du(n), J C(JRn) ,
Say a family {4)t:t > 0} in Cb converges weakly to 4) E Cb as t -
lim «t,u> = 0,0 for all u E M(C(J,Rn)). t-'O+ (Dynkin [16], p. 50):
Proposition (3.1)
0 = w-lim
0+ if
Write this as 4) = w-lim
4t.
t-'O+
For each t > 0 let 4)t, 0 E Cb.
Then
0t if and only if {110j: t > 0} is bounded and 4t(n) --- ON
t-,0+
as t - 0+ for each n E C(JRn). Proof:
For each n E C(J,Rn) let do be the Dirac measure concentrated at n,
defined by 1
nEB
,
dn(B) = {
0, n0B for all B E Borel C(J,Rn).
u E M(C(J,Rn))
ot(u) =
If 0 = w-lim
Define t E [M(C(J,Rn))]* by
t
4), then 4)(n) = <4),6 > = lim
t
<0 ,d > = lim
4 (n) for all
n E C(J,Rn), and the set {0t(u):t > 0} is bounded for each u E M(C(J,Rn)).
By the uniform boundedness principle {110j: t > 0} is bounded.
lI4t II =
""Ot 11
for each t > 0, so { 11
'1
:
But
t > 0} is bounded.
Conversely suppose {110j: t > 01 is bounded and st(n) > 4)(n) as t -> 0+, for all n E C(J,Rn).
«t,u> =
By the dominated convergence theorem
Ot(n)du(n) -J
C(J,Rn)
as t - 0+, for all u E M(C(J,Rn)).
I
C(J Xn) O(n)du(n) = «,u>
Thus 0= w-lim
0t.
o
t+0+ The weak infinitesimal generator A:V(A) c Cb - Cb of {Pt}t>0 is defined by Pt(4T) - 4)
A(4)) = w-lim
77
where D(A) is the set of all 4) 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($) _ 4) for every 0 E Cb.
following properties can be found in Dynkin ([16] Vol.
1
Moreover the Chapter I §6, pp.
36-43).
Theorem (3.1):
V(A) c C.
(i)
D(A) is weakly dense in Cb, Pt(D(A)) c D(A) for all t > 0.
(ii) If 4) E D(A), then the weak derivative
d Pt(0) = w-lim
1F [Pt+h(4)) - Pt(0)]
t > 0
h-0
d Pt(y) = A(Pt($)) = Pt(A(4))), r0 t
Pt(y) -
4) = 1
Pu(A (4)))du for all t > 0.
(iii) A is weakly closed i.e. if {Ok}k=1 00
c
D(A) is weakly convergent and
is also weakly convergent, then w-lim 0k E D(A) and {A($k)}k=1 k-w-lim A(4k) = A(w-lim 0k). (iv)
For each A > 0, aid - A is a bijection of V(A) onto Cb.
(Aid - A)-1(W) =
e -AtPt(W)dt
Indeed
for all W E Cb.
TO
The resolvent RA _ (aid - A)-1 is bounded linear and (v)
for all A > 0.
w-lim ARA(W) = W for every W E Cb. In order to derive a general formula for the weak infinitesimal generator
A, one needs to augment the state space C(J,Rn) 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{0}where v E Rn and X{0}:J 78
->
R is the characteristic function of {0}.
Clearly
C(J,Rn) n Fn = M. Form the direct sum C(J,Rn) ® Fn and give it the complete norm
p In(s)I + IvI , n E C(J,Rn), V E Rn
un + vX{0}Ij
sup Indeed C(J,Rn) ® Fn is the space of all functions E:J - Rn which are contin-
uous on [-r,0) 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.
Let a E C(J,Rn)
Lemma (3.1):
.
Then a has a unique (continuous) linear
extension a:C(J,Rn) ® Fn > R satisfying the weak continuity property if {&k}k=1 is a bounded sequence in C(J,Rn) such that Ck(s) _
(w
for all s E J for some & E C(J,Rn) is Fn, then a(l;k) -.
as k
Vs) a(i;) as
k i 00
The extension map e
:
> [C(J,Rn) ® Fn]*
C(J,Rn)*
a I- a
is a linear isometry into.
We prove the lemma first for n = 1.
Proof:
Suppose a E C(J,R)*.
By the Riesz representation theorem (Dunford and
Schwartz [15], §IV.6.3 p. 265) there is a (unique) regular finite measure
R such that
u: Borel J
t
a(n) = (-r n(s)du(s)
for all n E C(J,R).
1
Define a:C(J,R) ® F1 > R by a(n + vix{0}) = a(n) + v1u{0},
n E C(J ,R), V1 E R.
a is clearly a continuous linear extension of a. {Ek}k00
Let k(s)
-1 -
be a bounded sequence in C(J,R) such that
n(s) + v1X{0}(s) as k
for all s E J where n E C(J,R), v1 E R.
By
BY the dominated convergence theorem,
79
1im a(Ck) = lim JO k-
-r
Ck(s)du(s) =
O n(s)du(s) f-r
* JO
r
v1X{0}(s)du(s)
= a(n) + v1u{0} = a(n + v1X{0})
The map e:C(J,R)* -> [C(J,R) ® F1]* is clearly linear.
#> a
a
Higher dimensions n > 1 may be reduced to the 1-dimensional situation as follows:
Write a E C(J,Rn)* in the form n
a(n) =
E i=1
where n = (ni,...,nn) E C(J,Rn), ni E C(J,R), 1 < i < n,
al(n*) = a(O,O,...,O,n*,O,...,O) with n* E C(J,R) occupying the
i-th place.
Hence ai E C(J,R)* for 1 < i < n. Write Fn=FI, where F1 ={v*X{0}:v* E RI by taking vX{0}
for each v = (vi,...,vn) E Rn, vi E R 1 < i < n.
_ (v1X{0},...,vnX{0})
Let aI E [C(J,R) ® F1]*
be the extension of a' described before and satisfying (w1).
It is easy to
see that
C(J,R') ® Fn = [C(J,R) ® F1] x ... x [C(J,R) ® F1]
(n copies)
i.e.
n + vX{0} _ (Ti + V1X{0},...,nn + VnX{0}) Define & E [C(J,Rn) ® Fn]* by n
E
a(n+ vX{0}) =
a'(ni + viX{0})
i=1
when n = (ni,...,nn), v = (vi,...,vn).
Since V is a continuous linear
extension of a1, then a is a continuous linear extension of a.
be bounded in C(J,Rn) such that Ck(s) --> C(s) as k where C E C(J,Rn) ® Fn.
Let Ck =
Ci E C(J,R) ® F1, 1 < i < n.
80
Hence
{Ck}k=1
, for all s E J,
(C19...9Cn {i}k=1
Let
C E C(J,R),
is bounded in C(J,R) and
k C(s) ---s &i (s) as k
-, s E J, 1 < i < n.
lim a(Ck) = Jim E i=1 k-ow k-
Therefore
a'(Ci) = aW.
E
i=1
So & satisfies (w1). ti
To prove uniqueness let a E [C(J,R extension of a satisfying (w1 O}k=1
k.
® Fn]* be any continuous linear
For any VX{O} E Fn choose a bounded sequence
in C(J,R) such that CO(s) -> vX{0}(s) as k
for all s E J; e.g.
take.
(ks + 1)v, -k<s<0 0k
(s)
-1/k
Note that
II&kII
= IvI for all k > 1.
A4
Also by (w1) one has
1%j
1%,
k
a(n + vX{01) = a(n) + a(vX{0}) = a(n) + kim a(C0)
-O
= a(n) + a(vX{0}) = a(n + vX{0})9 ti
for all n E C(J,Rn) .
Thus a = a.
Since the extension map e is linear in the one-dimensional case, it follows from the representation of & in terms of the a' that 4,
e:C(J,Rn)*
[C(J,Rn) ® Fn]*
81
is also linear. let
Conversely,
But & is an extension of a, so II&ll > Ilall.
= n + VX{0} E C(J,Rn) ® Fn and construct
{C0 }k=1
in C(J,Rn) as above.
Then
k-'
a
But
Ia(n + 40k)I < IIaII Iln + doll <
= Ilall [Ilnll Hence 1&(E) l
+
,all
[Ilnll + Il4kll] for all k > 1.
Iv I] = Ilail
= lim la(n + 4o) I < IIall
II& II
for every E E C(J,Rn) ® Fn.
kThus
Hall < IIall. So IIaII
=
Ila ll and e is an isometry into.
Lemma (3.2):.
o
X C(J,Rn) > R be a continuous bilinear map.
Let
Then a has a unique (continuous) bilinear extension
S:[C(Jjpn) $ Fn] X [C(J,Rn) @F n] +F. satisfying the weak continuity property:
(w2) if {&k}k=1' {nk}k=1 are bounded sequences in C(J,Rn) such that k(s)
---->
*(s), nk(s) --> n*(s) as k
C*,n* E'C(J,Rn) ® Fn, then B(&k,nk)
Proof:
,
for all s E J, for some as k + - .
Here we also deal first with the 1-dimensional case:
Write the con-
tinuous bilinear map B:C(J,R) X C(J,R) -3 R as a continuous linear map
B:C(J,R) -9 C(J,R)*. C(J,R) -p. 494).
Since C(J,R) * is weakly complete (Dunford and Schwartz
p. 341), a is weakly compact as a continuous linear map
[15], §IV. 13.22,
C(J,R)* (Theorem (1.4.2); Dunford and Schwartz [15], §VI. 7.6,
Hence there is a unqiue measure A:Borel J -> C(J,R)* (of finite
semi-variation IIXII(J) <
such that for all C E C(J,R)
0
B() = J-
E(s) dA(s) r
(Theorem (1.4.1)).
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
linear extension 8:C(J,R) 0 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 (I.3.1)(iv)).
Define S:C(J,R) ® F1 > [C(J,R) ® F1]* by B = e o 8 where e is the extension isometry of Lemma (3.1); i.e. 8(n + v1X{0}) = 8(n + v1X{0}), n E C(J,R), V1 E R. Clearly B gives a continuous bilinear extension of 8 to [C(J,R) ® F1] x [C(J,R) ® F1]. To prove that B satisfies (w2) let k}k-1, {nk)k-1 be bounded sequences in C(J,R) such that ck(s) - *(s), nk(s) -' n*(s) as k -'
for all s E J, for some &*, n* E C(J,R) is F1.
(w1) for 8 we get 8(§*) = lim 8Rk).
By
Now for any k,
I8(Ck)(nk) - s(&*)(n*)I < IS(Ck)(nk) -
Is(n*)(n k) - a( *)(n*)
.<
II6(Ek)
- s(E*)II
II71 k1I
+
IW*)(nk) - 8(*)(n*)I.
But by (w1) for 8(F*) we have
lim IS(*)(nk) k-'°°
(C*)(n*)I = 0
Since {IInkII}k=1 is bounded, it follows from the last inequality that lim k-'°°
8(Ck)(nk) = R(*)(n*).
When n > 1, we use coordinates as in Lemma (3.1) to reduce to the 1-dimensional case.
Indeed write any continuous bilinear map 8: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 n
8((19 ...,Y,019 ...9nn)) =
E
8iJ(&i,nj)
i,j=1
where (E1,...' n), 012 ...971 n) E C(J,Rn), i,ni E C(J,R), 1 < i < n, Bij:C(J,R) X C(J,R) - R is the continuous bilinear map
83
0ij W .T11) = S((O,O,...,O,C',0,...,0), (O,O,...,O,n',0,...,0)),
for C', n' E C(J,R) occupying the i-th and j-th places respectively, 1 < i,j < n.
Now extend each 8
continuously to a bilinear map Sij: Then define
[C(J,R) ® F1] x [C(J,R) ® F1] + R satisfying (w2). S:[C(J,Rn) ® Fn] x [C(J,Rn) ® Fn] - R by n
;Q
-ij
a W" TI j
T
I
i,j=1
where
E C(J,Rh) ® Fn, Et, n E C(J,R) ® F11
n* =
1 < i, j < n.
It is then easy to see that S is a continuous bilinear
extension of a satisfying (w2). Finally we prove uniqueness.
Let S:C(J,Rn) ® Fn
continuous bilinear extension of g satisfying (w2).
[C(J,Rn) ® Fn]* be any
Take * = C + v1X{0}'
n* = n + V2XtO} E C(J,Rn) ® Fn, where C, n E C(J,Rn), v 1 , v2 E Rn.
Choose
bounded sequences {Co}k=1 , {nO}k=1 in C(J,Rn) such that C0 (s) --p v1X{0}(s)°
no (s) -> v2X{0}(s) as
s E J; II CO II = Iv1 I, IInOII
k'
k.
nkk= n +
Let Ek = C +
= 1v21 for all k > 1.
{nk}k00
Then {Ck}k=
=1 are bounded sequences in
C*(s), n (s) - n*(s) as k - , for all s E J.
C(J,R ) such that C (s) -
So
ti
by (w2) for S and
one gets
S(C*)(n*) = lim
s(Ek)(nk) = S(C*)(n*)
k-
Thus W = S.
a
For each n E C(J,Rn) let nx E 92(9,C([-r,a],Rn)) be the solution of (I) ti
through n, and nt E C(J,Rn) be defined as in §2 for each t E [O,a].
Lemma (3.3):
There is a K > 0 (independent of t,n) such that
IIt E(nxt - nt)IIC < K for all t > 0 and n E C(J,R").
Also if a E C(J,Rn)*, then ti
lim
t Ea(nxt - nt) = a(H(n)X{0}) for each n E C(J,Rn).
t-O+ Proof:
84
Denote by E the expectation for Rn or C(J,Rn)-valued random variables.
Let K > 0 be such that IH (n)I < K for all n E C(J,R').
t+s
t+s H(x (w))du + (w)
J
nxt(w)(s)-nt(s) ={
Now
u
0
G(nx
J0
t+s>O u
t+s<0,
0
for each t E [0,a], a.a.w E 2, s E J.
Since evaluation at each s E J is a continuous linear map C(J,Rn) +
Rn,
then it commutes with the expectation i.e. It,
CE{t nxt - nt))](s) =
E{t
nt(s)]), s E J
t+s f.1
E(H(nxu))du,
t + s > 0
0
0
t + s <0, t>0
,
using the Martingale property of the Ito integral.
Thus
t
lim
lim
ntt-O+
[E{t(nxt
0
t fo
E(H(nxu))du
s = 0
-r < s < 0
H(n)X{0}(s), for all s c J. ti
We prove next that IIt E(nxt - nt)IIc is bounded in t > 0 and Ti.
Clearly
t+s
I[t E(nxt-nt)](s)I < tl f
IE(H(nxu))Idul < t JIE(H(nxu))Idu O
< K for all t > 0 and n E C(J,Rn). ti
Therefore IIt E(nxt - nt)IIC < K for all t > 0 and n E C(JRn)
If a E C(J,Rn) *, then 1
A, I Ea(nxt - nt) = t a(E(nxt - nt)) = aCt E(nxt - nt)]
By the weak continuity property (w1), one gets
1im t Ea(nxt - nt) = a(H(n)X{0}).
o
85
For each t > 0 and a.a. W E 11, define wt(w) E C(J,Rn) by
Lemma (3.4):
[w(w)(t+s) - w(w)(0)]
-t < s < 0
wt() (s)
-r<s< -t
0
Then
Let B be a continuous bilinear form on C(J,Rn).
[t ER(nxt - nt, nxt - nt) - ER(G(n) o wt, G(n) o wt)] = 0.
lim
Proof:
We prove first that
lim
E II
(nxt - nt) - G(n) o wt II 2 = 0
(1)
Observe that
1
t+s
(n x H(x )du + -
r0
u
I0
n
t+s
G( x )dw(u)
I
u
t+s
dw(u)],
G(n)[
-t < s < 0
t to
-r<s< -t almost surely.
Therefore
I tnxt - nt)(s) - G(n) o wt(s)I2 E sup t+s
1
< 2
t
E
sup SEE-t,03
I
fo
n
2
t+s 1
+2 tE
sup sE[-t,0]
n
[G(xu)-G(n)]dw(u)I
II
0
+2 K1 It EIIG(nxu)
<
Jt EIH(nxu)I2du
- G(n)II2 du
(2)
0
for t > 0, and some K1 > 0. Since H and G are Lipschitz with Lipschitz constant L, then almost surely 2IH(n)12 + 2L211 nxu - n,12,
u > 0;
and
IIG(r'xu) - G(n) 112 < L2 IInxu -nII
86
, u > 0.
2
Hence
EIH(nxu)I2 < 2IH(n)12
2L2E Ilnxu -n112
+
(3)
and
2 11C E IIG(nxu) - G(n)112 < L E I I nxu _n
(4)
Now by the main existence theorem (Theorem (11.2.1)), the map
[0,a] 3u- xuEr2(St,C(J,Fn)) is continuous; so uin
Ellnxu-nfl2=0. Therefore the
last two inequalities (3) and (4) imply that {EIH(nxu)12: U E [0,a]} is bounded and lim EII G(nxu) - G(n)II2 = 0. u-O+ Letting t - 0+ in (2) yields (1). Since B is bilinear,
8(71 xt
- nt, nxt - nt) - B(G(n) ° wt, G(n) ° wt) 11,
= B( (nxt - nt) - G(n) ° wt,
+ B(
(nxt
1(nxt-nt)-G(n)°wt,G(n)°wt)
-
nt) - G(n) ° wt)
+ B(G(n)°wt, *(nxt-nt)-G(n)°wt). rt
/t-
Thus, by continuity of B and Holder's inequality, one gets It E8(nxt - nt, nxt - nt) - EB(G(n) ° wt, G(n) ° wt)I
< 11811
Ell -L (nxt - nt) - G(n) ° w*ll2
+ 2118 11 [Ell t (nxt-nt) - G(n)°wtII2] 1/2 [E IIG(n)°wtII2] 1/2,
(5)
for all t > 0. But
E IIG(n) ° wtll2 < E
sup
I
SE[-t,0]
lw(t±s)-w(0)12 IIG(n)112<
t EsE[yp = IIG(n)
112
r
w(t+s)-w(0)12 IIG(n)112
,
for all t > 0.
t
t IIG(n)112
(6)
87
Combining (6) and (5) and letting t - 0+ gives the required result.
Lemma (3.5): Let in:Rn
o
Fn be the isomorphism in(v) = vX{0}, v E Rn, and
G(n) X G(n) denote the linear map
Rn
x
Rn
(v1,v2)
Rn X J n
--> (G(n)(v1), G(n)(v2)).
Then for any continuous bilinear form a on C(J,Rn) 1im
t
ES(nxt-ntnxt-nt) = trace
(in x in) ° (G(n) X G(n))]
for each n E C(J,Rn), where ; is the continuous bilinear extension of 0 to C(J,Rn) ® Fn (Lemma (3.2)).
Proof:
m E
ES(nxt-nt,nxt-nt) =
lim
t-0+
Indeed if
{e}M=1
is any basis for Rm, then
8(G(n)(ej)X{o},G(n)(ej)X{0}).
j=1
t
In view of Lemma (3.4) it is sufficient to prove that m lim
t-o+
ES(A ° wt, A ° wt) =
Z
R(A(ej)X{0}, A(ej)X{0})
(7)
j=1
for any A E L(Rm,Rn).
We deal first with the case m = n = 1, viz. we show that lim
(7)'
ES(wt, wt) = B(X{0},X{0})
t- O+ for one-dimensional Brownian motion w.
If C, n E C = C(J,R), let C 0 n stand
for the function J X J -; R defined by (C ® n)(s,s') = C(s)n(s')for all s, s' E J.
The projective tensor product C ®a C is the vector space of all
functions of the form
EN=1
Ci ® ni where Ci,ni E C,
i
= 1,2,...,N.
It carries the norm N
N
IIhII®r = inf {1 E1 IIEiII Ilni II :h = E1 Ci ® ni,Ci,ni EC, The infimum is taken over all possible finite representations of h E C 0 Denote by C
C the completion of C aff C under the above norm.
C.
It is well
known that C e,,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 an C. countable dense subset of C.
For let Y c C be a
Then the countable set
N
0 ni : Ci,ni E Y,
Y 0 Y = { E
i
= 1,...,N, N = 1,2,...}
i
i=1
is dense in C ®n C and hence in C Off C.
The continuous bilinear form B on C corresponds to a continuous linear functional a E [C &r C]* (Treves [75] pp. 434-445; Cf. Theorem 1.43).
The map
Now let W1,Vr2 E C2(S2,C).
C®C (E,n)
CO,R C
I -->
®n
is clearly continuous bilinear.
Thus
> C ®n C
st
W I--> W1(w) 0 q)2 (W) is Borel measurable.
But
II'P1(w) a w2(w) II On
< II'P1(w) II
IIW2(w) II
for almost all w E St; hence by Holder's inequality the integral
f
IIW1 (w) ®'P2(w) II
®7r
dP(w) exists and
SZ
JSI
11V1(w) ® V2 (W) II
ar dP(w) < [ Jii1 (w) 112 dP(w)]l[ JnIIV2 (w)II2dP(w)]
From the separability of C of C the Bochner integral (§I.2) W1(w) ® W2(w)dP(w)
E
JSZ
exists in Cti®,R C.
Furthermore, it commutes with the continuous linear
functional S; viz.
0
(8)
89
,
Fix 0 < t < r and consider
E [-t,0]
t s E [-r,-t) or s' E [-r,-t).
0
1
+
min (s,s')
s,s' E [-t,0]
t
s E [-r,-t) or s' E [-r,-t)
0
_ [1 + t min (s,s')]X[_t,0](s)
X[_t,0](s'),
(9)
s,s' E J
Define Kt:J x J - R by letting Kt(s,s') = [1 +
min (s,s')]X[_t,01(s)X[_t,01(s')
s,s' E J
(10)
t i.e.
Kt = E C2(c,C), it is clear from (8) that Kt E C ®7r C and I'll
S(Kt)
wt
(11)
.
ti
In order to calculate lim
8(Kt), we shall obtain a series expansion of Kt.
t-'0+ We appeal to the following classical technique.
Note that Kt is continuous; so we can consider the eigenvalue problem 0
s E J
-r Kt(s,s')E(s')ds' = Xe(s)
(12)
Since the kernel Kt is symmetric, all eigenvalues A of (12) are real.
Using
(10) rewrite (12) in the form
t fO
-t
+ fO
-t
(s')ds' = Ate(s)
XE(s) = 0 Therefore
90
s E [-t,0] (i)
s E [-r,-t)
(ii)1
(13)
S
t JO
t
F(s')ds' + f
-t
s'E(s')ds' + s
E(s')ds' = Ate(s)
D
fs
S E [-t,0]
(14)
Differentiate (14) with respect to s, keeping t fixed, to obtain D
(= fs')ds' at &'(s),
s E (-t,0]
(15)
s E (-t,0]
(16)
S
Differentiating once more,
-&(s) = At
"(s)
,
When A = 0, choose ED:J - R to be any continuous function such that E0(s) = 0 for all s E [-t,0] and normalized by
-t
J_r0(s)2ds=1.
Suppose A # 0.
Then (13)(ii) implies that
c(s) = 0 for all s E [-r,-t)
(17)
In (14) put s = -t to get &(-t) = 0
(18)
In (15) put s = 0 to obtain '(0) = 0
(19)
Hence for A # 0, (12) is equivalent to the differential equation (16) coupled with the conditions (17), (18) and (19). Aet-a'iis
+
Now solutions of this are given by
A2e-t'ia-his,
s E [-t,0]
c(s) = {
(20)
s E [-r,-t),
0
i = -T
= A2 = 1, say.
Condition (19) implies immediately that A
From (18) one gets
1
e-t-x-iit
+
etx-iit
= 0.
Since the real exponential function has no zeros, it follows that a' cannot be 91
imaginary i.e. A > 0.
Being a covariance function, each kernel Kt is non-
negative definite in the sense that
JJ JJ
Kt(s,s')E(s)C(s')ds ds' > 0 for all E E C.
Using (18), we get the eigenvalues of (12) as solutions of the equation
2 cos [ 1 (-t)] = 0 VTF Therefore the eigenvalues of (12) are given by 4t
k = 0,1,2,3,...
(21)
and the corresponding eigenfunctions by
k(s)=(t)
X[-t,03(s) cos
(2k+l
2T -
s],
s E J, k = 0,1,2,...
(22)
after being normalized through the condition Ck(s)2ds = 1, k = 0,1,2,...
J J
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') =
Xk &t (s) Ck(s'),
E
s,s' E J
(23)
k=0 E
8
k=0
Tr (2k+I)Z
cos
[ (2k+1)uts ]
cos
[ (2k+1)"SI
2t
]
s,s' E [-t,0]
{
(24)
s E [-r,-t) or s' E [-r,-t)
0
But from the definition of Kt, one has Kt(0,0) = 1 for every t > 0.
Thus
putting s = s' = 0 in (24) we obtain E
k=0
8
2-
1
(25)
it (2k+1) 2k+1)
From the absolute and uniform convergence of (24), it is easy to see that Kt can be expressed in the form 92
Kt = k=E 0
8 11-t n2(2---Z Ek ® Ck
(26)
where
k(s) = cos
)ns]X[-t,0](s).
C( 2k
S E J.
Note that the series (26) converges (absolutely) in the projective tensor product norm on C ®n C.
Hence we can apply S to (26) getting from (11) the
equality
8 E
z
,
k=0
7r
t Wk a Y ''
(2k+1) ti
=
E
k=0 But (I kJI<
( k'
w (2k+1
(27)
k
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
8
k=0 (2" E
00
Moreover, for each s E J, k(s) -- X{0}(s) as t - 0+, k = 0,1,2,...
.
Thus
if we let t- 0+ in (27), we obtain
lim k=0
(2
tlim k+1)2 -'0+ S( k.
k)
00
_
8
E
S(X{0}
k=0
(2
k+1)2
X{0})
= a(X{0}.X{0})
using (25) and Lemma 2.
This proves (7)'.
For dimensions n > 1, write S:C(J,Rn) x C(J,Rn) - R in the form S(C1.C2) =
E
01I(C14)
i,j=1
where C1
C2 =
=
is continuous bilinear.
(C12
and each 51
C(J,R) x C(J,R) --> R
Let A E L(Rm, Rn) and {ek}k=1' {eiI%
be the
canonical bases for Rm and Rn, respectively; i.e. ek =
ik)m=1. ei = (6ji)J=1. k = 1,...,m, i = 1,...,n.
93
Write m-dimensional Brownian motion w in the form w = (w1,w2,...,wm) where wk(t) = <w(t),ek>, k = 1,...,m, are independent one-dimensional Brownian
Then
motions.
n
EB(A ° wt, A ° wt) = E
S'J((A ° wt)', (A ° wt)J)
Z
t,j=1 m E = E ''j-1
m
k*
w
t
, k
i
E w h=1
h* t
h
>)
m
n E i
E (k=1
Sij
E
,j =1
n
m
E
E k=1
i,j=1
ES'3(wt*, wt*)
i>
k , h=1
k*
k*
Letting t - 0+ and using (7)' gives
m lim
ES(A°wt,A°wt) =
t-,0+
n
E
E
k=1
i,j=1
B1-J(X{0},X{0})
m
E k=1
S(A(ek)X{0}' A(ek)X{0}) = trace
(in x in) o (A x A)].
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.
o
Let V(S) c Cb 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 0 E 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 (DA): (i)
0 E V(S);
(ii)
0 is C2;
(iii) DO, D24 are globally bounded; (iv)
D20 is globally Lipschitz on C(J,Rn).
Theorem (3.2): Suppose q:C(J,Rn) --? R satisfies Conditions (DA).
94
Then
D(A) and for each n E C(J,Rn)
A(4)(n) = S( )(n)+(DO(n)oin)(H(n))+}trace[D2 (n)o(inxin)o(G(n) xG(n))] 2
where Di(n), D fi(n) denote the canonical weakly continuous extensions of DO(n) and D20(n) to C(J,Rn) ® Fn and in:Rn - Fn is the natural identification
v i.--> vX{0} .
is any basis for Rn, then
Indeed if
n
AM(n) = S($)(n)+DO(n)(H(n)X{0})+7j
E C(J,Rn) and let nx be the solution of the SRFDE (I) through
Suppose 0 satisfies (DA).
n.
Since 0 is C2, then by Taylor's Theorem (Lang [47]) we have 116
(nxt) - ON = (nt) -O(n)+ D$(nt)(xt n
- nt) + R2(t)
a.s.,
t > 0,
where
R2(t) = j1 (1-u)D24[nt + u(nxt - nt)](nxt-nt,nxt-nt)du
a.s.
(28)
Taking expectations, we obtain
t E[O(nxt)-O(n)] = t St(4)(n)-4(n)] + t DO(nt)(nxt-nt) + t R2(t) (29) Since 0 E D(S), then
lim !-[S M(n) - 4(n)] = SM(n)
t-,0+t
t
In order to calculate lim
t-'0+
t
(30)
t
1 E[O(nx) - (n)], one needs to work out the
following two limits
lim u0+ t ED4(nt)(xt lim
1
t-,0+t
- nt)
ER (t) 2
(31)
(32)
We start by considering (31).
From Lemma (3.3), there exists a K > 0 such
that
Il t E(nxt - nt) II
.
K
for all t > 0. 95
Hence It EDO(n t)(nxt - nt) - t ED4(n)(nx t -
ID$(nt)[tE(xt - nt)]
-
nt)I
DO(n)[t E(nxt - nt)]I
for all t > 0.
K IID$(nt) - DO(n)II
Let t - 0+ and use the continuity of DO at n to obtain
lim t
EDO(nt)(nxt - nt) _ 1imm+ t ED$()(nxt - nt
= D$(n)H(n)Xf0))
(33)
by Lemma (3.3).
Secondly, we look at the limit (32).
Observe that if K is a bound for H
and G on C(J,Rn) and 0 < t < r, then ti
4
EIIIt - ntll
< 8E
IJt+s
= E
sup SE[-t,0]
n
n 4 H(xu)du + jt+s G(xu)dw(u)I
0
t+s t+s n 4 n 4 + 8E sup H( xu)duI G( x )dw(u)I sup If Ij sE[-t,O] 0 0 u SEC-t,0]
< 8K4t4 + 8K2t
ft
Ell G(nxu)
114du
0
some K2, K > 0,
< K(t4 + t2),
(34)
Furthermore, if
where we have used Theorem (1.8.5) for the Ito integral. u E [0,1] and 0 < t < r, then 1ED20(n
t
t+u(nx-n t t))(nxt -n
< t E IID2$(nt < [E
IID241 (nt +
+
t t
nx -n )
t,
u(nxt-nt)) -
-
D2O(w ) II
t
t
jED2O(n)(nx -n
t,
II nxt-ntII2
u(nxt-nt))-D2m(n)II2]1/2[_! Eli nxt ntlI4]1/2
t
< K1/2(t2+1)1/2[E IID24(nt+u(nxt-nt)) - D20 (n)II271/2. 96
t t
nx -n )I
(35)
But D2$ is globally Lipschitz, with Lipschi.tz constant L say; so
-
E II D2$(nt+u(nxt-nt)
D2$(n) II 2 < L2E( II nt-n II +
II nxt-ntll) 2
< 2L2 Ilnt - nll2 + 2L2[Ellnxt - ntll4]1/2 < 2L2 lint - n112 + 2L2k1/2t(t2 + 1)1/2 because of the inequality (34).
lim t
Letting t - 0+ in (35) and (36), we obtain
ED2$(nt + u(nxt - nt))(nxt -
= lim ED2$(n)(nxt t-)-0+
uniformly in u E [0,1].
(36)
-
nt'
nxt
-
nt
nt, nxt - nt)
(37)
From this and Lemma (3.5), one gets 1
lim
t_0+
t 1
ER (t) = 2
1 ED2$(n)(nx -n ,nx -n )du
0
_ 7 E
D
t t t t
t
(1-u) lim
J
n)(G(n)(e])X{0},G(n)(e])X{0}).
(38)
j=1
Since $ E U(S) and has its first and second derivatives globally bounded on C(J,Rn), 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 -.,0+ in (29) and putting together the results of (30), (33) and (38).
a
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.
M.
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(J,Rn) the weak topology induced by the bilinear pairing ($,u).
i
'
JC(J,Rn)
$(n)du(n), where
$ E Cb and u runs through all finite regular Borel measures on C(J,Rn).
More-
over, the domain of strong continuity Cb of {Pt}t>0 is a weakly dense proper 97
subalgebra of Cb.
Our aim here is to construct a concrete class Tq of smooth functions on C(J,Rn), viz. the quasi-tame functions, with the following properties: (i)
Tq is a subalgebra of Cb which is weakly dense in Cb;
(ii)
Tq generates Borel C(J,Rn);
(iii) Tq c 12(A), the domain of the weak generator A of {Pt}t>O; (iv)
E Tq and n E C(J,Rn), A(4)(n) is a second-order partial
for each
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 < co) 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 0:C(J,Rn) - R is said to be tame if there is a finite set {s1,s2,...,sk} c J and a C°'-bounded function f:(Rn)k -* R such that
O(n) = f(n(s1),...,n(sk)) for all n E C(J,Rn).
(*)
The above representation of 4 is called minimal if for any projection p:(Rn)k i (Rn)k-1 there is no function - R with f = gop; in other g:(Rn)k-1
words, no partial derivative Djf:(Rn)k -- L(Rn,R), j = 1,...,k, of f vanishes identically.
Note that each tame function admits a unique minimal represen-
tation.
Although the set T of all tame functions on C(J,Rn) is weakly dense in Cb and generates Borel C(J,Rn), it is still not good enough for our purposes, due to the fact that 'most' tame functions tend to lie outside Cb (and hence are automatically not in D(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
Borel C(J,Rn). 98
Let 4) E T have a minimal representation
(ii)
ON = f(n(s1),...,n(sk)) where k > 2.
Proof:
n E C(J,Rn)
Then 4 ¢ Cb.
For simplicity we deal with the case n =
1
throughout.
it is easy to see that T is closed under linear operations.
(r
the closure of T under multiplication.
We prove
Let 01,02 E T be represented by
01(n) = fl(n(sl),...,n(sk)), 02(n) = f2(n(sj'),...,n(sm )), for all
n E C(J,R) , where fl :Rk -' R, f2:Rm -' R are C°°-bounded functions and
s1,...,sk, sj,...,sm E J.
Define
f12:Rk+m
i R by
f12(xl,...,xk, x,..... xm) = fl(xl,...,xk)f2(xj,...,xm) 00
for all xl,.,.,xk, x,...,xr'1 E R.
Clearly f12 is C -bounded and for all n E C(J,R).
Thus
E T, and T is a subalgebra of Cb.
It is immediately obvious from the definition of St that if 0 E T factors through evaluations at s1,...Isk E J, then St(0) 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 T° be the subalgebra of
Cb consisting of all functions 4:C(J,R) - R of the form
ON = f(n(s1),...,n(sk)), n E C(J,R)
(1)
when f:Rk -' R is bounded and uniformly continuous, s1,...Isk E J.
Observe
first that T is (strongly) dense in To 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
Cam-bounded function g:Rk -' R such that If(x) - g(x)l < e for all x E Rk.
We
co
prove this using a standard smoothing argument via convolution with a C function (Hirsch [32], pp. 45-47).
d > 0 such that If(x1) - f(x2)I
Let B(x,6) be the open ball centre x E Rk and radius 6. function h:R k
bump
By uniform continuity of f there is a .
Choose a Coo bump
+ R>O with the properties that supp h c B(0,6) and JR kh(y)dy = 1, 99
the integral being a Lebesgue one on Rk.
Define g:Rk - R by
g(x) = JRk h(y)f(x-y)dy = J B(0,6) h(y)f(x-y)dy,
x E Rk.
(2)
By choice of 6 and property of h, it follows that If(x) - g(x)1 =
< for all x E Rk.
IJRk h(y)f(x)dy
-
Jk h(y)f(x-y)dyl
JB(O,d) h(y)If(x) - f(x-y)Idy < e I Rk h(y)dy = e
To prove the smoothness of g, use a change of variable
y' = x-y in order to rewrite (2) in the form g(x)
for all x E Rk.
h(x-y')f(y')dy'
J
(3)
B(x,d)
Now fix x0 E Rk and note that B(x,6) c B(x0,2d) whenever x E B(x0,d). Therefore for all x E B(x0,d).
h(x-y')f(y')dy'
g(x) = - J
(4)
B(x0,2d)
Since f is continuous and the map x - h(x-y') is smooth, it follows from (4) that g is C** on B(x0,6) and hence on the whole of Rk, because x0 was chosen arbitrarily.
Indeed g and all its derivatives are globally bounded on Rk
for IIDpg(x)II < J
If(y')Idy'
IIDph(x-y')II
B(x0,2a)
< V.
sup k IIDph(z) II. sup
k
If(z) I = N, say,
volume of a ball of radius 26 in Rk. in Cb.
< ... < S
Let 11 k:-r = s1 < s2 < S 3
V = J
dy' is the B(0,26) Secondly we note that Tois weakly dense
for all X E Rk, where N is independent of x0 and
k
= 0, k = 1,2,..., be a sequence
of partitions of J such that mesh lIk -". 0 as k -
Define the continuous
linear embedding Ik:Rk - C(J,R) by letting Ik(v1,v2,...,vk) be the piece-wise linear path (s-s 1) (s -s) vi_1, vi + Ik(vl,...,vk)(s) = s-sj-1 si_sj-1 100
S E CSJ-1,sj]
joining the points V1,...,vk E R, j = 2,...,k.
R N,
--8t
st
s,-0
s,_,
sa
Denote by sk the k-tuple (st,...,sk) E Jk, and by. Psk the map
C(J,R)
Rk
n F---> (n(s1),...,n(sk)). Employing the uniform. continuity of each n E C(J,R) on the compact interval
J, the reader may easily check that
lim (Ik °Psk)(n) = n
in C(J,R).
(5)
k-*w
Now if 4 E Cb, define k:C(JJO - R by
k =
is bounded and uniformly continuous, so is
410 1 k°psk, k = 1,2,...
o Lk: Rk
R.
.
Since
Thus each
Finally T° and lim 4k(n) = (n) for all n E C(J,R), because of (5). k-' note that II0II IICb for all k > 1. Therefore 0 = w-lim 0k and To is k Cb < 110
k E
--
weakly dense in Cb.
From the weak density of T in T° and of T° in Cb, one
concludes that T is weakly dense in Cb. Borel C(J,R) is generated by the class {$ 1(U):U c R open, 0 E T}. any finite collection sk = (s1,...,sk) E Jk let P
sk
For
C(J,R) + Rk be as befor
CCO
Write each 0 E T in the form 0 = f ° Psk for some
-bounded f:Rk - R.
It is 101
N
aO
well-known that Borel C(J,R) is generated by the cylinder sets
{psk-1(U1X... XU ):U k
E R open, i = 1,...,k, sk=(s1,...,sk) E Jk i
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 c R open, f:Rk -' R COO-bounded}
:
(e.g. from the existence of smooth bump functions on Rk). set U c R,
(U) = psk [f1(U)].
But for each open
Therefore it follows directly from the
above that Borel C(J,R) is generated by T. (ii)
Let 4) E T have a minimal representation
(s1,...Isk) E Jk is such that -r < s1 < S
2
t= f - Psk where sk =
< ... < S
k
< O, f:Rk
R is Coo-
Take 1 < jO < k so that -r < sj0 < 0. Since the represen-
bounded and k > 2.
tation of 0 is minimal, there is a k-tuple (xl,...,xk) E Rk and a neighbourhood [xj0-E0,xj0 + EO] of xj0 in R such that Djof(xl,...,xj0_1,x,xj0+1,...,xk) # 0 for all x E [xj
-
E0, xj0 + EO], with EO > 0.
Define the function
0
g:[xj0-E0, xj0 + E0]
R by
g(x) = f(xl,...,xj0_1x,xjo+l,...,xk) for all x E [x j0-EO,xjo+EO + EO] and g is a C00 diffeomorphism
Hence Dg(x) # 0 for all x E [xj0-E0, xj 0
onto its range. Ix' - x"I
Therefore, there is a A > 0 such that < AIg(x') - g(x")I for all x', x" E [x.0-EO,x.0 +EO]
Pick d0 > 0 so that d0 < are mutually disjoint.
E0
(6)
and the intervals (sj-260, sj+260), j =1,2,...,k,
In the remaining part of the argument we may assume,
with no loss of generality, that all integers n are such that
< d0.
Con-
struct a sequence {nn} in C(J,R) looking like the picture opposite
103
Viz.
r
sj-60 < s < sj+60.
x
J 0 JO
i
- d xj(s-sj-60) + xj
sj+60 < s < sj+260, j 0 jo
0
xj(s-sj+60) + xj
X11
sj-260 < s < sj-60.
j 0 j0
0
x
1
60
sj0 +60<s<s.0 +260
-6 )+x
(s-s
j0
j0
j0
0
(7)
n n(s) =
d xj (s-sj +60)+xj 0
0
0
0
0
0
sj-60<s<sj-I,sj+n<s<sj
xj
0
0
0
+c
ne0(s-s. ) + xj 0
0
0
0
0
0
0
+
< s < sj
S.
+e0
+60
sj - n < s < sj
0
-ne0(s-sj )+xj
0
0
n
otherwise
0
Suppose, if possible, that lim
sj -260 < s < sj -60
E Cb.
St($)(nn) = lim
Then
f(nn(t+sl),...,nn(t+sk))=f(nn(sl),....nn(sk))
t-.0+
uniformly in n.
But, for j 0 j0 and 0 < t < 60, nn(t+sj) = xj for all n.
Therefore f(x1,x2,...,xj0-l,nn(t+sj0).xj0+l.... .xk
lim
= f(xl,...,xj0-i.
+ e0.
xj0+l"...xk)
xj0
uniformly in n; i.e. for any c > 0, there is a 0 < 6 < 60 such that Ig(nn(t+sj0))
-
Now suppose 0 < t < 60. then
Inn(t+sj0
)
-
g(nn(sj0M < e for all n, all 0 < t < 6.
If
n<
t < 60, nn(t+s
xj0 I =-ne0t + xj0
0
+ e0 - xj0l
nn(t+sj0) E [xj0-e0,xj0+e0] for all 0 < t < 60.
get for0
)
(8)
- x.0= 0. If 0 < t < = e.0(1-nt) < co.
n,
Hence
Applying (6) and (8), we
Inn(t+sjo)-.nn(sjo)I < XIg(nn(t+s j0 ))-g(nn(sj0))I < aE for all n.
Note that 6 is independent of n.
choose n0 large enough such that no
andn(t+s J.o )
In the above inequality, fix t < 6 and
n0
< t.
Then s
xJ.O;so
n
In 0(t +
sj0 )
(9)
+
j0
L < t n0
+ s
j0 <
sj
+
60
o
n - n 0(s.o)I = Ix. -(x .o + E0)I = ED < AC o
which clearly contradicts the arbitrary choice of c.
(Quasi-tame Functions):
Definition (4.2)
Therefore
£ C.
o
A function :C(J,Rn) _R is
quasi-tame if there is an integer k > 0, Co-bounded maps fj:Rn + Rn, 1 < j < k-1, h:(Rn)k - R and piecewise C1 functions gj:J -i R, 1 < j < k-1, such that for
f0
f1(n(s))g1(s)ds,...,
fi(n) = h(
fk-1((s))gk-1(s)ds,n(0))
Each derivative g' is assumed to be absolutely inte-
for all n E C(J,Rn). grable over J.
Denote by Tq the set of all quasi-tame functions on C(J,Rn).
Theorem (4.2): (i)
Tq c D(s) c Cb.
Indeed if $ = hem E Tq where 1-
m( n) _
Lr
(n(s))gi(s)ds,...,
r
1
and h,,fj,gj, j = 1,...,k-1, are as in Definition (4.2), then S($)(n) =
k-1 E j=1
D.h(m(n)){f.(n(0))gj(0)-fj(n(-r))gj(-r) 0
--r for all n E 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 {Si
t>o.
105
,
(iii)
Proof:
Tq is a weakly dense subalgebra of Cb generating Borel C(J,Rn).
In proving statements (i) and (ii) of the theorem, we shall assume
for simplicity that 4 = hom where h:(Rn)2 -' R is C°°-bounded and 0
m(n) _ (J
f(n(s))g(s)ds, n(0))for some Cam-bounded map f:Rn -' Rn and a -r
(piecewise) C
function g:J - R.
Let 0 < t < r and consider the expression
[(I) - 4(n)] = t Ch(m(nt)) - h(m(n))]
= J' Dh(zut){t m(nt)
(10)
- m(n)]}du
0
using the Mean-Value Theorem for h, with zu = (1-u)m(n) + um(nt), 0 < u
1.
But J_t
iCm(nt)-m(n)] = (t
f(n(0))g(s)ds
f(n(t+s))g(s)ds + t JD -r
t
- t 1_r f(n(s))g(s)ds,0) (JO
f(n(s)) t{g(s-t)-g(s)}ds + f(n(O))
_
t-r
t-r -
t
D
g(s)ds
t f-t
sds,0
J_r
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
0
lim [m(n)-m()] _ t t+0+
t
(-J-
r
f(n(s))g'(s)ds+f(n(O))g(O)-f(n(-r))g(-r),O) (11)
As Zu is bounded in n and continuous in (t,u), it follows from (10), (11) and the dominated convergence theorem that 0 E U(S) and
Dh(m(n)){lim f[m(n ) - m(n)11 t-O+
t
0
= D1h(m(n)){f(n(0))g(0)-f(n(-r))g(-r) - Jf(ri(s))g'(s)ds}. -
106
For Tq to be invariant under {St}
it is sufficient to prove that it is
t>O
invariant under St, 0 < t < r, due to the semigroup property.
So let 0 be
Then
as above and t E [O,r].
0
r0t-r
f(n(s))g(s-t)ds + f(n(0)) Et g(s)ds,n(O))
St(4)(n) = h(1
ti
for all n E C(J,R'n). Define g:J +R by
t-r<s<0
- rg(s-t)
1%,
g(s)
-r < s < t-r
0
ti Then clearly g is piecewise C1 with g absolutely integrable over J. (Rn)2, F:(Rn)2 . (Rn)2 also m:C(J,Rn) and h:(Rn)2 + R by
m(n) _ (JO r
Define
f(n(s))g(s-t)ds,n(0)), f(n(s))g(s)ds,n(0)) _ (JD t-r 0
ti
F(x,y) = f(y) J -t g(s)ds, h(x,y) = h[F(x,y) + (x,y)],
for all x,y E Rn, n E C(J,Rn). hence h.
Since f and h are C°°-bounded, so is F and
Therefore St(d) = hom E Tq.
We show that Tq is a weakly dense subalgebra of Cb. It is an easy matter checking that Tq is closed under addition and multiplication of functions in Cb.
To prove weak density of Tq in Cb, it is sufficient to show that Tq is
weakly dense in T U Tq, because the set of tame functions T is already weakly dense in Cb (Theorem (4.1)(i)).
ON = f(n(s1),...,n(sk))
Ti E
Let
E T have the representation
C(J,.n)
where f:(Rn)k i R is C°°-bounded and s1,...,sk E J.
construct a sequence
For each 1 < j < k,
of piecewise linear functions on J with the
property that gm(s) -* X{5J}(s) as m
and Igm(s)I < 1 for all s E J, for
all m > 1; e.g.
107
Also choose a sequence {dm}m=1 of Coo bump functions on Rn such that
-1
1xI<m
6m(x)
lxi >m+1
0
and 1dm(x)I < 1 for all x E Rn.
Define the sequence {fm}m=1 of C°°-bounded
functions on Rn by fm(x) = 6m(x)x for all x E R. n E C(J,Rn), Ifm(n(s)) gT(s)I < In(s)l
Then for each m > 1 and
for all s E J.
So by the dominated
convergence theorem 0
n(s.) = lim J
J
nt-- -r
fm(n(s))gm(s)ds,
j = 1,...,k
(12)
J
Therefore fi(n) = lliimm
f(J0
f (n(s))gm(s)ds,..., J0
fm(n(s))gk(s)ds)
(13)
Denoting the expression under the limiting operation in (13) by %(n), it is
clear that% E Tq for each m and 1%(n)1.4 sup {If(z)j: z E (Rm)k} for all n E C(J,Rn).
So Tq is weakly dense in T U Tq, and hence in Cb.
Finally we prove that Tq generates Borel C(J,Rn).
For this purpose it
is enough to show that the a-algebra a(Tq) generated by Tq coincides with that generated by the family of evaluations 108
psk:.C(J,Rn) + (Rn)k, sk = (s1,...,sk), sJ E J, j = 1,...,k, k=1,2,...
But this is again generated by sets of the form {n:n(s is closed.
E C} where C c Rn
Now each closed set in Rn is a countable intersection of comple-
ments of open balls, so we need only prove that if B c Rn 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 - 1 .
Let the sequences {f }m=1 and
{gy}m=1 be as before, so that (12) ispsatisfied.
Let Wm E Tq be given
by 0
IPM(n) = j_r f (n(s))(s)ds,
m > 1, n E C(J,Rn).
We contend that n p=1
{n:n(s.) E B'} =
To see this, let n(s lim
E B'.
Iim inf {n:W (n) E U } m p m
(14)
Then n(sj) E Up for all p > 1.
Since n(sj) _
Wm(n), then for each p there is an m0 > 0 such that Wp(n) E Up for all
m-
Hence, for each p > 1, n belongs to lim inf {n:Wm(n) E Up} i.e. n m belongs to the set on the right hand side of (14). Conversely, let n be in
m > m0.
lim inf {n:Wp(n) E Up} for each p > 1. Then for every p > 1, there is an m m0 > 1 such that Wm(n) E Up for all m > m0. Taking m + - gives n(sj) E Up CO
But B' =
for all p > 1 i.e. n(s.) E n p=1
Thus
As the sets {n:Wm(n) E Up} are clearly in a(Tq),
our contention is proved.
it follows directly from (14) that {n:n(s a(Tq) = Borel C(J1Rn).
Up, so n(s.) E B'. 1
E B'} E a(Tq).
Therefore
o
The final result in this chapter asserts that every quasi-tame function is in the domain V(A) of the weak generator A of {Pt}t>0.
Theorem (4.3):
Every quasi-tame function on C(J,Rn) is in the domain of the
weak generator A of {Pt}t>0.
Indeed if $ E Tq is of the form
4(n) = h(m(n)), n E C(J,Rn), where
109
0
m(n)
I0
f-r f1(n(s))g1(s)ds.... ...,
fk-1(n(s))gk-1(s)ds,n(0)),
then k-1 E D.h(m(n)){f.(n(0))g.(O)-fj(n(-r))g.(-r) J j=1 0 -
J
-r
Dkh(m(n))(H(n)) + 1 trace [D2h(m(n))°(G(n) x G(n))] for all n E C(J,Rn).
Here again Djh denote the partial derivatives of
h:(Rn) k -' R considered as a function of k n-dimensional variables.
Proof:
To prove that Tq c D(A), we shall show that each 0 = hom E Tq satis-
fies Conditions (DA) of §3. is Cam.
First, it is not hard to see that each 0 E Tq
Also by applying the Chain Rule and differentiating under the
integral sign one gets f0
Df1(n(s))(E(s))g1(s)ds,...,
D4(n)(E) = Dh(m(n))(1_ r
I°- r
Dfk-l(n(s))(E(s))gk-1(s)ds, E(0)),
D24(n)(E1,E2) = D2h(m(n))(Dm(n)(E1), Dm(n)(E2))
+ Dh(m(n))(JO
D2f1(n(s))(E1(s),E2(s))g1(s)ds,..., r 0 D2fk-1(n(s))(E1(s),E2(s))gk-1(s)ds,0).
J_r for all n,
E1., E2 E C(J,R').
Since all derivatives of h, f.,
1
j < k-1, are bounded, it is easy to
see from the above formulae that DO and D20 are bounded on C(J,Rn). induction it follows that 0 is C°°-bounded.
(iv) are automatically satisfied. of Theorem (4.2)(i).
Hence Codnitions (DA)(ii), (iii),
Condition (DA)(i) is fulfilled by virtue
From the above two formulae we see easily that the
unique weakly continuous extensions D4'(n), D m
110
By
)
of D4(n) and D24(n) to
C(J,Rn) ® Fn are given by
Dh(m(n))(O,...O,v) = Dkh(m(n))(v),
D2h(m(n))((O,...,O,v1), (0,...,0,v2))
D2h(m(n))(v1,v2),
=
for all v, v1,v2 E Rn.
The given formula for A(4)(n) now follows directly
from Theorem (3.2) and Theorem (4.2).
Definition (4.3)
o
Say n° E C(J,Rn) is an absorbing state for
(Ihynkin [16]):
the trajectory field {nxt:t > 0, n E C(J,Rn)} of the stochastic FDE(I) if P{w:w E 0, n0xt(w) = n°}
1 for all t > 0, where we take a =
=
here, i.e.
p(0,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 FDE (I). (i)
n°(s)
=
Then
n°(0) for all s E J i.e. no is constant;
(ii) H(no) = 0 and G(n°) = 0.
Proof:
Let no 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
Qt = {w:w E 9, n xt(w) = no} 0t(s) = {w:w E 52,
n
0
xt(w)(s) = n°(s)}.
Then Sgt c Stt(s) for all t > 0, s E J and since P(SZt) = 1, it follows that
Suppose if possible that there exist
P[Qt(s)] = 1 for t > 0, s E J.
s1,s2 E J such that n°(s1) # n°(s2).
-r< s1 < s2 < 0.
For each w E S2,
Without loss of generality take o xs2-s ()(sl) = n°(s2) # n°(s1) and so 1
(s1) =
Ds 2-s 1
0.
Hence P[S2s2-s (si)] = 0, which contradicts P[Qt(s)] = 1, 1
111
So n° must be a constant path.
t > 0, s E J.
This proves (i).
To prove that n° satisfies (ii), note that the absorbing state no must
satisfy A(O(n°) = 0 for all 0 E D(A) (Dynkin [16], Lemma (5.3), p. 137). In particular A($)(n°) = 0 for every quasi-tame 4':C(J,Rn) -IR. that since
n
is constant then so is the map t
every 0 E D(S).
Take any A:Rn
Note first
- nt and so S(q)(n°) = 0 for
R Coo-bounded and define the (quasi)-tame
function P:C(J,Rn) - R by W(n) _ A(n(0)) for all n E C(J,Rn).
Then by
Theorem (4.3), W E D(A) and
A(W)(n) = DA(n(0))(H(n°)) +
trace [D2A(n(0))0(G(n°) xG(n°))] =0
2 The last identity holds for every Cam-bounded A:Rn - R.
Choose such a A with
the property DA(n°(0)) = 0 and D2a(n°(0)) = <.,.>, the Euclidean inner product on Rn e.g. take A to be of the form M(v) = Iv - n°(0)12 in some neigh-
bourhood of n(0) in Rn.
Then
Y trace [D2a(n°(0)) ° (G(n°) x G(n°)] =
for any basis
of Rm
for every Cam-bounded A:Rn ; R.
E j=1
Therefore G(n°) 0.
IG(n°)(ej)12 = 0
=0
Thus
Now pick any C -bounded A such that A(v) _
for all v in some neighbourhood of n°(0) in Rn.
Then
Da(n°(0))(H(n°)) = IH(n°)12 = 0; so H(n°) = 0 and (ii) is proved.
o
Remarks (i)
We conjecture that conditions (i) and (ii) of the corollary are also
sufficient for no to be an absorbing state for the trajectory field {nxt:t > 0, n E C(J,Rn)}.
It is perhaps enough to check Lemma (5.3) of
Dynkin ([16], p. 137) on the weakly dense set of quasi-tame functions in D(A).
(ii) An absorbing state no corresponds to the Dirac measure dno being invariant under the adjoint semigroup {Pt} associated with the stochastic t>O FDE M. Thus a necessary condition for the existence of invariant Di-rac measures is that the coefficients H, G should have a common zero (cf. §111.3).
112
V Regularity of the trajectory field
§1.
Introduction
Given a filtered probability space (Q,F,(Ft)0.< ,P) our main concern in this chapter is to look at various regularity properties of the trajectory field {nxt:t E [0,a], n E C(J,Rn)} generated by the autonomous stochastic FDE: dx(t) = H(xt)dt + G(xt)dw(t)
0 < t < a
l
(I)
J
x0=nEC=C(J,Rn) with coefficients H:C w.
Rn, G:C
L(Rm,Rn) and m-dimensional Brownian motion
A version of the trajectory field of (I) is a measurable process
X:S2 x [0,a] x C - C such that n E C.
nxt a.s. for all t E [0,a] and
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 FDE'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 con-
siderations we are lead naturally to investigate regularity in probability of the trajectory field for the general system (I).
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
Stochastic FDE's with Ordinary Diffusion Coeffici.ents
§2.
Consider the stochastic FDE 0 < t < a
dx(t) = H(xt)dt + g(x(t))dw(t)
1
(II)
J=[-r,0].
x0 = n E C= C(J,Rn),
J
The coefficients H:C - Rn, g:Rn -' L(Rm,Rn) are Lipschitz maps and w is
m-dimensional Brownian motion adapted to the filtered probability space
(c,F,(Ft)O<,,P).
Suppose 0 < r < a in all that follows.
The following condition on the diffusion coefficient g will be required in the proof of the main theorem in this section.
The Frobenius Condition (Fr): A map
g:Rn -' L(Rm,Rn) is said to satisfy the Frobenius Condition if it is
L(Rm,Rn)) globally bounded, locally Lipschitz and such that
C1 with Dg:Rn
{Dg(x)[g(x)(v1)]}(v2) _ {Dg(x)[g(x)(v2)]}(v1) for all x E Rn v1,v2 E Rm.
For any 0 < a < 1 and a closed interval I a R, form the Banach space Ca(I,Rn) of all a-Htllder continuous paths n:I - Rn furnished with the a-H8lder norm In(t1) - n(t2)1 11n II
= sup { Ca
It
1
t2
:
t1,t2 E I, t1 # t2} + IIn 11C .
'CL
Ca(I,Rm) and Ca(I,Rm x Rn) are similarly defined. The following lemmas will be needed in the sequel.
If h E Ca([O,a],R), then the map [r,a] 3 t & ht E C°1'(J,R) is
Lemma (2.1): continuous.
Proof:
Suppose first that the lemma holds for any C1 map h E C1([O,a],R).
Let e > 0 and h E Ca([O,a],R). there is an h
0
Since C1([O,a],R) is dense in Ca([O,a],R),
E C1([O,a],R) such that
IIh-h°IIC < e/4.
By uniform continuity
of the map [r,a] 3 t)-'ht E Ca(J,R), there exists 6 > 0 such that if t1,t2 > r and
It1-t21
< 6, then
- ht 11 Ca <E/2.
Ilht 1
114
2
Now if s1,s2 E J, t >r,
ti
and h E Ca([0,a],R), we have
i.e.
I I ht I I
Ca
< II h I I
for all t > r. Ca
If t1,t2 > r and Iti-t2I < 6, then one has
Ilht -ht 1
< Ilht -ht
II
2
CO'
1
II
i
C
< 2 IIh-h°II
a
Ca
+ Ilht -ht 1
II
2
C
a+
Ilht -ht 2
II
2
Ca
+ IIht1-ht2II < e.
Thus we need only prove the lemma for h E C1([O,a],R).
If so, let
r < t1 < t2 < a, -r < s1 < s2 < 0 and consider s2
(ht -ht )(u)du
(ht -ht )(s2)-(ht -ht )(s1) = 2
1
1
2
/
s1
1
2
fs2 [h'(t1+u)-h'(t2+u)]du. si
By the uniform continuity of h and h', if c > 0 is given there exists 60 > 0
such that Ih(u) - h(v)I< a/i+r1-a and Ih'(u) - h'(v)I< C/i+r1-a whenever Hence if
u,v E [0,a], Iu-vl < d0.
It1-t21
< d0, then
I(ht1-ht2)(s2) - (ht1-ht2)(si)I < Ei_a Is2-s1l So I- a Ilh tl
-ht
=
II
2
Ca
)(s )I.I(s -s sup I(h h h )(s )-(h t1 t2 2 1 2 t1 t2 1 s1,S2EJ
si <
1
# s2
erl-a ia
+
IIht1-ht211 C
+ IIht1-ht2II C.
But C
lh(t1+s) - h(t2+s)I <
IIht1-ht211C
sup
1+r
-a
115
Therefore
1-a
IIht1 - ht2II < 1+r1
+ i+r7
This completes the proof of the lemma.
Lemma (2.2):
if Itt-t2I < S0.
=e o
Suppose F:R' - Rn is a C1 map with DF:P
- L(Rp,Rn) locally
Then the map
Lipschitz.
-- Ca(J,n)
Ca(J,RP)
F :
is Lipschitz on every bounded set in Ca(J,RP).
Proof:
Let s1,s2 E J, 19C2 E Ca(J,P ).
For each u E [0,1] define
Y(u), z(u) E 1Rp by Y(u) = u&1(s2) + (1-u)C2(s2), z(u) = uC1(s1)+(1-u)C2(s1).
By the Mean-Value Theorem for F, the expression ti
ti
ti
ti
- F(E2)](s1) - [F(C1) - F(C2)1(s2
E(s1,s2) = CF(E
= F(C1(s1)) - F(C2(s1)) - [F(E1(s2)) 1
fl0
DF(z(u))(C1(s1)-C2(s1))du - IO J,
DF(z(u))[(&1 Y(s1) (C1 E2)(s2)]du + 0
I1 [DF(z(u))-DF(Y(u))] (C1(s2)-Ys2))du. 0
Now for each u E [0,1], Iy(u) I < II& II C + 1
IIC21I C and Iz(u) I
Since DF is locally Lipschitz, it is Lipschitz on every bounded set in RP; in particular, if I I &111 < 1 and 2 1 1 Ca < 1, then there is a K1 > 0 such Co,,
that
11&
IIDF(z(u)) - DF(y(u))II< K1Iz(u)-y(u)I for all u.E [0,1].
Letting K2 = sup{IIDF(v)II: v E RP, IvI < 2}, we get
116
IE(s1,s2)I < K2[ IK1 - Y1Ca - II&1 - &211C] Is1-s21a + K1 IIE1 - E211 C J 1 Iz(u)-y(u) Idu 0
< K2[II&1 -E21ICa- IIE1 - E211 C] Is1-s21a
+ K1IIE1 - C211C ID
< K2[I1
1
(1-u)1&2 (s1)-&2(s2)I}du
-&211 Ca - 11&1-&211
C]Isi-s21
+ Ki
Isi-s2Ia
11&211 Ca)
1s1-s21at
< K3 II&1 - C211 Ca
where
K3 = max (K2, 2K1) . The last estimate clearly implies that there is a K4 > 0 such that
IIF(E1) - F(C2) II Ca < K4 II&i - &211 Ca for all C1,E2 E Ca(J,Rp) with
a <
II&1II
n,
C
1,
a < 1.
IIE21I
By linearity of
C
F a similar argument works for every bounded set in Ca(J,RP).
Lemma (2.3) (Frobenius):
o
Suppose g:Rn - L(Rm,Rn) is a C2 Lipschitz map
satisfying the Frobenius condition.
Then there exists a unique C2 map
F:Rm x Rn - Rn such that D1F(t,x) = g(F(t,x)) F(O,x) = x for all t E Rm, x E Rn.
diffeomorphisms of Rn F(t1,.) o
Furthermore {F(t,.):t E Rm} is a group of C2 i.e. F(t1
for all t1, t2 E Rm for all t E Rm. 117
Proof:
We follow Dieudonne ([1.1], Chapter X, pp. 303-307).
family of Li.pschitz. C2 vector fields
Consider the
t E Rm) on Rn.
Using the
theory of existence, uniqueness and dependence of solutions of ODE's, on Rm -*Rn initial conditions and parameters, we get a unique C2 map O:R x Rn x such that D1q(s,x,t)(1) = g(4(s,x,t))(t) 0(0,x,t) = x Note that for each t E Rm,
for all s E R, x E Rn, t E Rm.
:R xRn _Rn is a one-parameter C2 flow on Rn.
Define F:Rm xRn
by
F(t,x) _ O(1,x,t), t E Rm, X E Rn. Clearly F is C2.
As
0, then O(s,x,O) = x for all s E R.
Thus
F(O,x) = 4(1,x,0) = x for all x E Rn. Since g is C1, we can differentiate both sides of (1) with respect to t getting
D1[D3c(s,x,t)(v)](1) = {Dg(O(s,x,t))[D30(s,x,t)(v)]}(t)+g(O(s,x,t))(v)
D30(0,x,t)(v) = 0 for all s E R, X E Rn, t,v E Rm. Fix v,t E Rm and x E Rn; define u:R -. Rn by
u(s) = D3O(s,x,t)(v) - sg(4(s,x,t))(v)
for all s E R.
Differentiating with respect to s gives u'(s) = D1[D30(s,x,t)(v)](1) - g(O(s,x,t))(v)
- s{Dg(4(s,x,t))[D1O(s,x,t)(1)]}(v) = {Dg(O(s,x,t))[D30(s,x,t)(v)]}(t)-s{Dg(O(s,x,t))[D14(s,x,t)(1)}(v) = {Dg(O(s,x,t))(u(s))}(t) + sDg((P(s,x,t))[g(q(s,x,t))(v)]
- s{Dg(4(s,x,t))[g(O(s,x,t))(t)]}(v) _ {Dg(O(s,x,t))(u(s))}(t) 118
(3)
for all s E R, because of (1), (2) and (Fr). u(O) =
0.
Also by definition of u,
Since g is C1, the map s l-*
is
continuous and therefore the linear differential equation y'(s) = Dg(4(s,x,t))(y(s))(t),
S E R
I has y = 0 as its unique solution. all s E R.
(4)
J
Y(O) = 0
Since u satisfies (4), then u(s) = 0 for
In particular, putting s = 1 one gets
D1F(t,x)(u) - g(F(t,x))(v) = D3W,x,t)(v) - g(4(1,x,t))(v)
= u(1) = 0 for all k E Rm, i..e.
D1F(t,x) = g(F(t,x)) for all t E Rm, x E Rn.
Note that for each t E R, F(t,.) = Rn.
is a C2 diffeomorphism of
To prove the group property, let t1, t2 E Rm and define z,i:Rm - Rn by z(t) = F(t,F(t1,x)), z(t) = F(t + t1,x) for all t E Rm,
where x E Rn is fixed.
Then
Dz(t) = g[F(t,F(t1,x))] = g(z(t)), Dz(t) = D1F(t + t1,x) = g[F(t + t1,x)] = g(i(t)) ti
for all t E Rm, and Z(O) = F(O;F(t1,x)) = F(t1,x) = z(0).
But, for each
fixed t E Rm, the maps s..z(st), s H z(st) are clearly both solutions of h'(s) = g(h(s))(t), s E R,
h(0) = F(t1,x).
So by uniqueness of solutions, z(st) = i(st) for all s E R.
In particular
z(t) = i(t) i.e. F(t,F(t1,x)) = F(t + t1,x) for all t E Rm, all x E Rn.
a
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:Rm x Rn - Rn for g still exists. DF is locally Lipschitz due to the following argument. the proof of the lemma.
pact in Rn.
Use the notation in
Let B1 c Rn, B2 c Rm be bounded sets.
of 0, the set S =_ {o(s,x,t)
:
Furthermore
By continuity
s E [0,1], x E B1, t E B2) is relatively com-
Since S can 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' E B1, t, t' E B2.
Then from differ-
entiating (1) with respect to x we obtain ID24(A,x,t)(y) - D20(A,x',t')(y)I
X IJ0{(Dg(O(s,x,t))[D20(s,x,t)(y)])(t)
- (Dg($(s,x',t'))[D20(s,x',t')(y)])(t))dsI
< K2 J X IIDg(O(s,x,t))II ID20(s,x,t)(y) - D24(s,x',t')(y)Ids 0
+ K2 JDX IIDg(m(s,x,t)) - Dg(O(s,x',t'))IIID20(s,x',t')(y)Ids
< K2 K3 J
D20(s,x',t')(y)Ids 0 a
+ IyI K4 K1 JO Im(s,x,t) - (s,x',t')Ids,
where y E Rn, K2 = max [ sup xEB1
K3 = sup
IxI,
A E [0,1],
sup It I],
tEB2
S E [0,1], x E B1, t E B2),
and
K4 = sup {IID2O(s,x,t)II: S E [0,1], x E B1, t E B2). But 0 is C1 and therefore Lipschitz on the compact set [0,1] x B1 x B2 i.e. there is a K5 > 0 such that
I4(s,x,t) - (s,x',t')I < K5[Ix-x'I + It - t'I] for all s E [0,1]. Therefore
120
ID2X(X,x,t)(y) - D20(A,x',t')(y)I
o ID20(s,x,t)(y) - D20(s,x',t')(y)Ids K2K3 J0
+ K1K4K5 Iyl [Ix-x'I + It - t'I]
for all A E [0,1].
By Gronwall's lemma,
I[D2W,x,t) - D24(A,x',t')](y)I < K1K4K5IYI[Ix-x'I+It-t'I]e for all A E [0,1].
KKA 2 3
In particular,
IID2F(t,x) - D2F(t',x')II = IID2m(1,x,t) - D24(1,x',t')II
< K1K4K5 e
2 3
for all x, x' E B1, t,t' E B2. Rm x Rn.
[Ix-x'I + It - t'I]
Hence D2F is Lipschitz on bounded sets in
Also D1F = goF is Lipschitz on bounded sets in Rm x Rn.
Therefore
so is DF. Theorem (2.1): a C2
In the stochastic FDE (II), suppose H is Lipschitz and g is
map satisfying the Frobenius Condition.
Then the trajectory field
{nxt:t E [0,a], n E C(J,Rn)} has a version X:St x [O,a] x C(J,Rn) _ C(J,Rn)
having the following properties.
For any 0 < a < }, there is a set SZa c St
of full P-measure such that, for every w E Sla, (1)
the map
(ii)
for every t E [r,a] and n E C(J,Rn), X(w,t,n) E Ca(JRn);
(iii) (iv)
x C(J,Rn) _ C(J,Rn) is continuous;
x C(J,Rn) - Ca(J,Rn) is continuous; for each t E [r,a], X(w,t..):C(J,Rn) _ Ca(J,Rn)
is Lipschitz on every
bounded set in C(J,Rn), with a Lipschitz constant independent of t E [r,a]. In particular each X(w,t,.) : C(J,Rn) - C(J,Rn) is a compact map.
Proof:
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:2 x [O,a] x C(J,Rn) - C(J,Rn) of the trajectory Oxt:t E [O,a],
n E C(J,Rn)} of (II) can be constructed, sample-function-wise, by first solving a suitably defined random family
x C(J,Rn)
Rn of
retarded FDE's, and then deforming their semi-flows into X using the flow F ti
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(t,x) = g(F(t,x)) (5)
F(O,x) = x
for all t E Rm, x E Rn.
For any C E C(J,Rm) and n E C(J,Rn), define the map Fo(C,n):J + Rn by [F-(C,n)](s) = F(E(s),n(s)) for all s E J.
For each t E Rm, F(t,.) is a diffeomorphism of Rn; so for any x E Rn the linear map D2F(t,x):Rn - Rn is invertible. Define the 'Brownian motion' w°:0 - C([-r,a],Rm) by w(w)(t) - w(w)(0) w°(w)(t)
for a.a. w E 0.
t E J
0
Recall that wt:c -
t E [O,a] (6)
C(J,Rm) is the slice of w° at t E [O,a],
i.e. wt(w) E C(J,Rm) for a.a. W E 92.
We now define a random family of retarded FDE's on Rn by letting ti
H:[O,a] x C(J,Rn) x Q ; Rn be the map 116
H(t,n,w) = [D2F(w°(w)(t),n(O))]-{H[F-(wt(w),n)] 1
-
trace (Dg[F(w°(w)(t),n(O))]og[F(w°(w)(t),n(O))])}
for a.a. W E 2, t E [O,a], n E C(J,Rn). We shall show that, for a.a. W E Q, 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)
0
Em-(t) = H(t,nEt(w),w)
1
(8)
nE0(w)
=n
For this purpose, we contend that for a.a. W E 0 the RFDE (t,n) N H(t,n,w) is (jointly) continuous and satisfies a Lipschitz condition in n over bounded sets of C(J,Rn), uniformly with respect to t E [O,a].
We also contend that
the above map satisfies a linear growth condition of the form
IH(t,n,w)I < K(w) (1 + IIn II c) for all t E [0,a], n E C(J,Rn), where
'K(w)
> 0 i.s some constant.
First of
all, note that for a.a. w E 0 every term appearing on the right-hand side of (7) is Lipschitz in n over bounded subsets of C(J,Rn) uniformly with respect to t E [0,a].
Referring to the proof of Lemma (2.3), let 4:R x Rn x Rm
Rn
be the C2 map solving the equation
D14,(s,x,t)(1) = g(4(s,x,t))(t) 1
O(o,x,t) = x,
s E R, X E Rn, t E Rm
(9)
I
Suppose g:Rn 3 L(Rm,Rn) has the same global Lipschitz constant L1 > 0 as that of H.
Then integrating (9) with respect to s gives I0(A,x,t) - 4(A,x',t)I < Ix-x'I + IJ0 {9(4(s,x,t))(t)-g(c(s,x',t))}ds
< Ix-x'I + L1It1
for all A E [0,1].
x j0 I0(s,x,t)-q(s,x',t)Ids
Using Gronwall's lemma and putting A = 1, we get
It1
L
IF(t,x) - F(t,x')l = I4(1,x,t) - 4(1,x',t)I < e
1
Ix-x'I
(10)
for all t E Rm, x,x' E Rn.
Now, by the remark following the proof of Lemma (2.3), the map (t,x) H D2F(t,x) is Lipschitz on bounded sets in Rm x Rn.
But the group
property of Lemma (2.3) gives [D2F(t,x)]-1 = D2F(-t,F(t,x)) for all t E Rm X E Rn.
As (t,x) H F(t,x) is Lipsxhitz in x uniformly with respect to t in
a bounded subset of Rm, it follows by composition that the map x r* [D2F(t,x)] 123
1
is Lipschitz on each bounded subset of Rn, uniformly wi.th respect to t in a bounded subset of Rm.
Moreover, if IIDgII
= sup { IIDg(x)II.x E Rn}, then
from the equation D2$(X,x,t)(y) = y +
(Dg(4(s,x,t))[D24(s,x,t)(y)])(t)ds, A,s E R, 0
ID2$(X,x,t)(y)I < Iyl + IIDgII ItI J A ID2$(s,x,t)(y) ds, A > 0. 0
Apply Gronwall's lemma and put A = 1 to get
IID2F(t,x) II < e IIDgII III
for all x E Rn, t E R"`
Therefore
II[D2F(t,x)]-1 II = IID2F(-t,F(t,x)) II < e IIDgII III, t E0, X E Rn (11)' For the sake of brevity, and unless otherwise indicated, all subsequent 'unqualified' statements involving w may be taken to hold for all w E SaO, the set of all those w E S2 for which w(w) is continuous on [0,a].
Let t E [0,a] and ni,n2 E C(J,Rn).
Then by (10)
I[F°(wt(w),n1)](s) - [F°(wt(w),n2)](s)I
=IF(w°(w)(t+s),n1(s)) - F(w°(w)(t+s),n2(s)) L Iw°(w)(t+s)I
< e
1
In1(s) - n2(s)I
for all s E J.
Letting IIw°(w)II = sup {Iw°(w)(t)I: t E [-r,a]}, one gets
L1IIw° () IIF°(wt(w),n1) - F°(wt(w),n2)IIc < e
II
(12)
11nt-n211
In particular, 0 (w)II
IF(w°(w)(t),n1(0)) - F(w°(w)(t),n2(0))I < eL1IIw
IIn1-Ti 21Ic(13)
By the global Lipschitz properties of H and g, it follows immediately from the last two inequalities that the maps (t,n)'-> H[F°(wt(w),n)] and
124
(t,n) - g[F(w°(w)(t),n(O))] are globally Lipschitz in n uniformly with respect to t E [O,a].
Furthermore, D2F and Dg are Lipschitz on bounded sets; there-
fore the maps (t,n) N [D2F(w°(w)(t),n(O))]-1, (t,n) N Dg[F(w°(w)(t),n(O))] are continuous and Lipschitz in n over bounded subsets of C(J,Rn) uniformly with respect to t E [O,a].
An elementary argument, exploiting the continuity ti
of the maps (t,n) 1+ (wt(w),n), F and the trace, shows that (t,n) w H(t,n,w)
is continuous and Lipschitz in n over bounded subsets of C(J,Rn) uniformly with respect to t E [O,a]. ti
To establish the linear growth property of H(.,.,w), note that IH(n)I < IH(n)I + L1 IInIIC for all n E C(J,Rn).
(14)
Also if s E J, n E C(J,Rn) and t E [O,a], then 0
I[F-(wt(w),n)](s) I < IF(w°(w)(t+s),O) I
eL1 IIw (w) I I
+
II nllc
i.e.
0
IIF.(wt(w),n) II < IIF(w°(w)( ),0) II + where
eL1 IIw (w) II
(15)
II n IIC
sup {IF(w°(w)(t),O)I: t E [-r,a]}.
Using (14) and (15) gives 0 II
IH[F°(wt(w),n)]I < IH(O) I + L1
II
IIIIC (16)
for all t E [O,a], n E C(J,Rn). Similarly, by the inequality
II9(x)II
(17)
we get
I9[F(w°(w)(t),n(O))]I < 119(0)11+ L1
Ilw °
(w)IIIniIc (18)
Finally, use (7), (11)', (16) and (18) to deduce that
125
IH(t,n,w)I <eIIDgII IIw°(w)II[IH(O) +
II+L1eLtllw°(InIIC]
+L1.
eIIDgII IIw°(w)II[m IlDgll (IIg(O) II + L1
This last inequality implies that
(18)'
IH(t,n,w)I < K(w)(1 + IIn II ),n E. C(J,Rn), t E [O,a] where
K(w)
=
eIIDgII IIw°(w) II
max{L1eLtllw°(w) II
, IH(O) I + L1
II
,
0
IIw (w)11
eIIDgII ( IIg(0)II + L1 IIF(w
L1eL1
Thus for a.a. W E SZ and every n E C(J,Rn), the RFDE (8) has a unique solution
ne(w) defined on [-r,a] and starting at n.
Define the random field x:S2 x [-r,a] x C(J,R") - Rn by x(w,t,n) = F(w°(w)(t), nv w)(t)) for a.a. w E S2, all t E [-r,a], and all n E C(J,Rn).
(19)
Note that each
ti
is defined on a set of full P-measure in S2 which is independent of t and n e.g. 92 o.
Also since g is C2, then so is F because of Lemma (2.3).
Therefore we can apply Ito's formula (Elworthy [19], p. 77) to (19) getting, for each fixed n E C(J,Rn),
(')(t) +
I
t > 0 (20)
But for a > t > 0 we have
dnV -)(t) = and
126
trace
So (20) yields
a > t > 0.
I trace
(21)
Now in (5), keep x fixed and differentiate with respect to t E Rm; then D2F(t,x) = Dg(F(t,x)) o D1F(t,x) = Dg(F(t,x))og(F(t,x))
for all t E Rm, X E Rn.
In particular,
trace D2F(wo(w)(t),nE(w)(t))
=
Dg[F(w (w)(t), TE(w)(t))] o g[F(wo(w)(t),'E(w)(t))]
ti
ti
= Dg[x(w,t,n)] o g[x(w,t,n)]
(22)
for t E [0,a], and a.a. W E 0.
From (7) we see that
D2F(wo(w)(t),%(w)(t)){H(t,%t(w),w)} = H[F o (wo(w),ntt(w))]
-
trace (Dg[F(wo(w)(t),%(w)(t))] o g[F(wo(w)(t),n&(w)(t))])
=
-7 trace Dg[x(w,t,n)] o g[x(w,t,n)], 0 < t < a
(23)
using the fact that 1XI
T
(
) )
F o (wo(w),
(24)
a.s., for all t E [0,a], n E C(J,Rn). ti
From (21), (22) and (23), it follows that, for each n E C(J,Rn),
127
is a solution version of the stochastic FOE 0 < t < a
(25)
Moreover, x(w,s,n) = F(O,nE(w)(s)) = F(0,n(s)) = n(s) for all s E J, a. a. W E S. ti
is the unique solution of the stochastic FDE (II),
Therefore
and the random field X:S2 x [0,a] x C(J,Rn) - C(J,Rn) a.a. W E SZ, t E [0,a], n E C(J,Rn)
X(w,t,n) =
is a version of the trajectory {nxt:t E [O,a], n E C(J,Rn)} of (II), viz. nxt a.s., for all t E [0,a], n E C(J,Rn). We now verify that X has the stated regularity properties.
For a.a. w E S2,
w°(w) is continuous on [-r,a]; hence the map > C(J,Rm)
[O,a] t
wt (w)
i
is continuous for a.a. w E 0.
Also by continuous dependence of solutions
to RFDE's on the initial condition, it is clear that the semi-flow
[O,a] x C(J,Rn)
C(J,Rn)
(t,n) i- > n t(w) is continuous for a.a. W E 0 (Hale [26]).
Therefore, composing with the
continuous flow F, it is easy to see that, for a.a. W E 9, [0,a] x C(J,Rn)
(t,n)
.
C(J,Rn)
i----> X(w,t,n) = Fo(wt(),nct())
is also continuous.
since 0 < a < For the rest of this proof, fix w E Sta. Let r < t< a and n E C(J,Rn). Let 0a = {w:w E S2, w(w) E Ca([O,a],Rm)}. Then P(fa) =
128
1
Since w°(w) E Ca([-r,a],Rm), then wt (w) E Ca(J,Rm). C1
,
and since nEt
But F:Rm x Rn - Rn is
is C1 on J, it follows that the composite map X(w,t,n) _
(M )
Fo(wt(w), r'C (w)) must be in Ca(J,Rn).
This proves (ii) of the theorem.
We now go on to prove assertion (iii) of the theorem.
First we establish
the continuity of the map Ca(J,Rn)
[r,a] x C(J,Rn)
(26)
%t(w)
(to) 1--
Fix any n E C(J,Rn) and let r < t1 < t2 < a, -r < s1 < s2 < 0.
net
()](s2) - [n&t2 (w) - nEt (w](sl)
[nEt2 (w) -
1
1
= I{n,(w)(t2+s2) - n (w)(t2+s1)}-{TIE(w)(tl+s
tl+s2
t2+s2 ti = If 't2+s1
t1+s1
nEs+t
(w),w)dul
n
H(s+t1' &s+t (w),w)dsl
(w),w)ds 2
s1
u
fS2
H(s+t2'
n&(w)(tl+sl)}1
,,
H(u, n
H(u,n&u (w).w)du -
s2 ,
= JJ
Then
sl
1
s2 ti
n
ti
< Js
n
s+t2(M).w) - H(s+tl. Cs+t2(w).w)Ids
IH(s+t2, 1
+ J sl
TI IH(s+tl,n& "' s+t ()'w) - H(s+tl, &s+t (w),w)Ids
2
(27)
1
Now by continuity of the slicing map the set {n&t(w):t E [0,a]} is compact in C(J,R").
Furthermore, each map
is Lipschitz in each bounded set of
C(J,Rn) with a Lipschitz constant independent of t E [0,a]; in particular
the map C(J,Rn)
> C([O,a].Rn) ti
ti
is continuous.
Hence the set
E [0,a]} is compact in
129
C([O,a],Rn), and therefore equicontinuous due to Ascoli's theorem i.e. given e > 0, there is a 6 > 0, depending only on n and w, such that, if It1-t21 < 6 , then IH(s+t2,
for all s E J.
1Es+t2
(w),w) - H(s+t1, nEs+t (w),w)I < e 2
Indeed because of the uniform continuity of
[0,a]
C(J,Rn)
t I--- nEt(w)
(28)
we may suppose that the above 6 is so chosen that II11Es+t1(w) nEs+t2 w)IIC < e for all s E J, whenever
It1-t2I < 6.
Denote by Li(w) >0 the Lipschitz constant of
on the compact set {nEt(W):t E [0,a]}. It1-t21 < 6
Then (27) gives, for
,
t2(w) - 'Et1(w)](s2) - [nEt2(w) - nEt (w)](s1)I
I['
1
S
< cls1-s21 + L1(w)
fS1 IInEs+t
(w) - nEs+t1 (w)IICds 2
ti
< elsl-s21 + ELi(w)Is1-s21 Therefore
I
I "Et2(w) - nEti (w) II ca
CO +
whenever ti,t2 E [r,a] and It1-t21 < d. [r,a]
t
L1(w))r1-a
Thus for each n E C(J,Rn) the map
C (J,Rn) (29) i
y
nEt(w)
is continuous.
Let ni,n2 E C(J,Rn), r < ti < t2 < a. IITI
130
Noting the inequality
1Eti(w) -712Et2(w)II0
and the continuity of the map (29) with n = n2, i.t is easy to see that the
map (26) will be (jointly) continuous provided we prove that for each fixed t E [r,a] the map
C(J,Rn)
>
Ca(J,IItn)
n i> N(w) is continuous.
(30)
In fact we shall prove next that the above map is Lipschitz
on bounded sets of C(J,Rn) (with Lipschitz constants independent of t E[r,a]). Then
Let n E C(J,Rn).
In,(w)(t) I < IIn II C + J0 IH(u,n&u(),w) Idu, t E [0,a] 11
+ K(w)t + K(w) Jt
< IIn II
II Cdu,
t E [0,a],
0
due to the linear growth property (18)'. II
nyw) III < IIn II C +
Therefore
K(w)t + K(w) J0t
II Cdu,
t E [0,a]
Applying Gronwall's inequality, we obtain ti II
nCt(w)
II
C < [IIn 11 C +
K(w)t]eK(w)t,
Let B c C(J,Rn) be any bounded set.
t E [0,a], n E C(JRn)
(31)
By (31) the set
B(w) _ {nCt(w):t E [0,a], n E C(J,Rn)} is also bounded in C(J,Rn).
Let
ti
L2(w) > 0 be the Lipschitz constant of t E [r,a] and -r < s1 < s2 < 0.
I[n1Ct(w) -
[n1Ct(w)
-
(w),w)du
t+s1 s2
u
[ti
n1
H(t+s,
IJ
-
n2Ct(w)](s1)1
- rt+s2 ,
Ti 1
H(u,
=
Take any n1,n2 E B,
Then
n2Yw)](s2)
rt+s2 ti
on B(w).
H(u,
J
t+s1 ti
Ct+s(w),w) - H(t+s,
s1
n2
C (w),w)dul u
n2
Et+s(w),w)]dsl
fS2
< L2(w)
11n1Ct+s(w)
n2Ct+s(w)IIC ds
(32)
s1
131
But again, II
n1Et.(w) -
n2CtI() IIC < IIn 1-n2IIc+L2(w) J0IIfuu(w)2u(w)IIcdu
for all t' E [O,a]; so by Gronwall's lemma, one gets
ti
ti
n1
II
EtI() -
for all t' E [O,a].
L2(w)t'
n2
Et'(w)IIC < IIn1-n2IlCe
L2(w)a
< IIn1-n2IICe
(33)
Therefore (32) gives
[n1Et(w)-n2Et(w)](s2)-[n1&t(w)-n2Et(w)](s1)I
L2(w)8,
< IIn1-n2IIC e
L2(w)-Is1-s21
From (33) and (34), we deduce that II n1&t(w) - n2Et(w) II
Ca
(34)
ti
< II
n1Ct(w)-n2Et(w)
ti L (w)a II c+ rt-a L2(w)e 2 IInj n2IIC
ti e
'L 2M] IIn1-n2IIC
(35)
This last inequality clearly shows that the map (30) is Lipschitz on B uniformly with respect to t E [r,a].
Now W E %, so the map
[O,a]
> Ca(J)Rm)
t I---) wt(w) is continuous (Lemma (2.1)).
By continuity of (26), the map Ca(J,Rm+n)
[r,a] x C(J,Rn)
Ca(J,Rm) x Ca(J,Rn) =
(tin) I-- (wt(w), is also continuous.
t())
Thus, composing with the continuous map
F':Ca(J,Rm) x Ca(J,Rn)
Ca(J,Rn)
(,Ti) CI> FoR,n)
132
(36)
(Lemma (3.2)), one gets the continuity of the sample function X(w,.,.):.[r,a] x C(J,Rn)
>
Ca(J,Rn)
F°(wt(). %t(w)).
(tin)
To prove the final assertion (iv) of the theorem, fix t E [r,a] and compose the isometric embedding
> Ca(J,m) x Co`(J,n)
Ca(J,Rn) T1
> (wt(),n)
i
with the map (30), deducing that the map
C(J,Rn)
Ca(Jm) x Co`(JRn)
n r---> (wt(). TIEt()) is Lipschitz on each bounded set in C(J,Rn) with Lipschitz constant independent of t E [r,a].
Applying Lemma (3.2) once more shows that
X(w,t,.):.C(J,Rn) - Ca(J,Rn) is Lipschitz on every bounded set in C(J,Rn),
with Lipschitz constant independent of t E [r,a].
The compactness of
C(J,Rn) is then a consequence of the last statement together with the compactness of the embedding Ca(J,Rn) Ascoli's Theorem).
---> C(J,Rn) (viz.
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 FDE
dx(t) = H1(t,xt)dt + g(x(t))dw(t)
0 < t < a (III)
x0
= n E C(J,Rn)
With H1:[0,a] x C(J,Rn) _Rn, g:Rn -> L(Rm,Rn) given maps.
Corollary (2.1.1): map (t,n)
Indeed one has
Suppose g is a C2 Lipschitz map satisfying (Fr), and the
F.-> H1(t,n) in the stochastic FDE (III) is continuous, Lipschitz
in n over bounded subsets of C(J,Rn) uniformly with respect to t E [0,a]. Assume also that H1 satisfies a linear growth condition:
133
IH1.(t,n)I < K(1 + 11nIIC), n E C(J,Rn), t E [O,a], for some K > 0.
(37)
Then all the conclusions of Theorem (2.1) hold for (III).
Unfortunately, it is not possible to deal with time-dependent diffusions as above, except perhaps in the following rather special case:
Corollary (2.1.2):
In the stochastic FDE
dx(t) = HI(t,xt)dt + g1(t,x(t))dw(t),
0 < t < a (IV)
x0
=
}
n E C(J,Rn)
let H1:[O,a] x C(J,Rn)
Rn satisfy the conditions of Corollary (2.1.1) and
g1:[O,a] x Rn + L(Rm,Rn) have a C2 extension 51:R x Rn -> L(Rm,Rn) with D51
bounded and fulfilling the Frobenius condition {D 291(t,x)[91(t,x)(v1)}(v2) = {D 2g1(t,x)[91(t,x)(v2)]}(v1)
(38)
Then the trajectory field
for all t E R, X E Rn, v1,v2 E Rm.
{nxt*t E [O,a], n E C(J,Rn)} of (LV) has a version X:c2x[O,a] x C(J,Rn) +C(J,Rn)
with all the properties stated in Theorem (2.1). Proof:
To eliminate explicit time-dependence in the diffusion coefficient
of (IV) we borrow the following technique from deterministic ODE's.
Embed
(IV) in a stochastic FDE of type (III) but defined on R x Rn, viz. dy(t) = H(t,yt)dt + g(y(t))dw(t) y0
= (0,n) E C(J,R x R
Indeed let p2:C(J,R x Rn)
a > t > 0 (III)
n )
J
C(J,Rn) be the natural continuous linear pro-
jection p2(Y,n) = n,
Y E
C(J,R), T1 E C(J,Rn).
Define H:[O,a] x C(J,R x Rn) - R x Rn, g:R x Rn ; L(Rm,R x Rn) by H(t,E) = (1,H1(t,P2(C))), t E [O,a], C E C(J,R x Rn g(t,x) = (O,g1(t,x)),
134
t E R, X E R n
By virtue of (38), it is easily seen that g and HA' satisfy all the conditions of Corollary (2.1.1).
Hence there is a continuous version
Y:S2 x [O,a] x C(J,R x Rn) -' C(J,R x Rn) of the trajectory field
ti { yt:t E [O,a], C E C(J,R x Rn)} for (I.I.I.) which satisfies Theorem (2.1).
Define X:S2 x [O,a] x C(J,R")
C(J,Rn) by
X(w,t,n) = p2[Y(w,t,(O,n))], w E Pa (0 < a < n E C(J,Rn).
), t E [O,a],
We claim that X is a version for the trajectory field
{nxt:t E [O,a], n E C(J,Rn)} of (IV).
To see this define the process y on
[-r,a] by (t,nx(t))
t E [0,a]
(0,n(t))
t E J
y(t) _ {
where nx is the unique solution of (IV). dy(t)
Then, for t E (0,a),
= (dt, dnx(t))
= (dt, H1(t,nxt)dt + g1(t,nx(t))dw(t))
= (1,H1(t,P2(yt)))dt + (0,g1(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 solution of (III) at (0,n) E C(J,R x Rn). and so
Therefore
P2(yt) = nxt a.s. for t E [0,a], n E C(J,Rn).
yt a.s.
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 X:[O,a] - R is such that A(0) # 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 (2.1.3):
In the stochastic FOE (LI.L) suppose H1 and g satisfy all
the conditions of Corollary (2.1.1), with X:S2 x [0,a] x C(J,Rn) _ C(J,Rn) the version of the trajectory field of (LLI.) obtained therein.
Assume also that
H1 has a partial derivative D2H1(t,n) In the second variable such that (t,n)
--> D2H1(t,n) is continuous.
Then for all w E QM (0 < a < }) and
t E [0,a], the map X(w,t,.): C(J,Rn)
> C(J,Rn) X(w,t,n)
n
is C1.
Let its derivative be D3X(w,t,n), and for any C E C(J,Rn) define
the process a:S2 x [-r,a] -> Rn by rD3X(w,t,n)(C)(0)
0 < t < a
a(w,t) = j
l
t E J,
C(t)
W E. Sta
Then a E t°(S2,C([-r,a],Rn)) and is the unique solution of the linear stochastic FDE da(t) =
0
In particular, for a fixed n E C(J,Rn), the map
(w,t,C) s--; D3X(w,t,n)(C) is a continuous version of the trajectory field of (D III); and for each w E S1, t E [r,a], the map C(J,Rn) 3 C i--> D3X(w,t,n)(C) E C(J,Rn) is compact linear.
Proof:
For any n E C(J,Rn) and w E Sa(0 < a < }), let nC(w) E C([-r,a],Rn)
be the unique solution of the retarded FDE nC(w)'(t) = H(t, nCt(w),w) nC0(w)
0 < t < a
(39) = n
where H:[0,a] x C(J,Rn) x SZ- Rn is defined by
136
s
H(t,n,w) = [02F(wo(w)(t),n(0))]-1{H1[t,Fo(wt(),)]
trace (Dg[F(wo(w)(0),n(0))]og[F(wo(w)(t),n(0))])}(40)
-
and F. w
0
are as in the proof of Theorem (2.t)(cf. (5),(6) and (7)),
t E [0,a], n E C(J,Rn),
to E Sta.
Since F is C1, it is easy to see that the
map
> C(J,Rn)
C(J,Rn) n
> F-(wt(w),n)
is also C1, for each fixed t E [0,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 to the map (t,n) ma H(t,n,w) has a continuous partial derivative with respect to n E C(J,Rn).
We may then apply Theorem (11.3.2)
with St = {w}, z(t) = t (or Hale [26], p. 46), to deduce that the map C(J,Rn) 3 n
I
;.
Et(w) E C(J,Rn) is C1 for each t,w.
Noting that
X(w,t,n) = Fo(wt(w, nCt()) it is clear by composition that each map X(w,t,.) is C1 on C(J,Rn). Fix n, E E C(J,Rn) and let 0 0 h E R.
Because X is a version of the
trajectory field for (III), one has g(X(-,t,n+hC)(0))dw(t),
for 0 < t < a.
Subtracting and dividing by h, we get
dCh{X(-,t,n+h&)(0) -
W t < a.
We shall take h to 0 in (41) by arguing as follows.
(41)
Note that H1 has a
continuous partial derivative in the second variable, so that
137
H1(t,X(w,t,n+hE)) - H1(t,X(w,t,n))]
h_imm
= D2H1(t,X(w,t,n))(D3X(w,t,n)(&)) exists for all w E S1a.
Furthermore,
and
are Lipschitz on
bounded sets of C(J,Rn) uniformly with respect to t E [O,a]; thus for each
w E 0a
the set
is bounded.
{h H1(t,X(w,t,n+h&))
- H1[t,X(w,t,n))]:t E [O,a], h # 0}
Therefore by the dominated convergence theorem, we have
h_imm F [H1(t,X(w,t,n+h&)) - H1(t,X(w,t,n))]dt
= D2H1(t,X(w,t,n))(D3X(w,t,n)(E))dt, 0 < t < a.
(42)
In order to deal with the second term on the right hand side of (41) as h - 0, it is enough to show that t
lim h-O
Eh(u)dw(u) = 0
j
(43)
0
in probability (uniformly for t E [0,a]), where Eh(u) = 1Fi
(44)
for all u E [O,a].
To prove (43) write
Eh(u) = Ah(u) + Bh(u),
u E [0,a], h # 0
with
Ah(u) = h [g(X(,u,n+h&)(0)) - g(X(,u,n)(O))
- Dg(X(,u,n)(0))(X(,u,n+hE)(0) - X(,u,n)(0))] and
Bh(u) = Dg(X(,u,n)(0)){- [X(,u,n+hE)(0) - X(,u,n)(0)] - a(u)}. Fix W E na (0 < a < 3), then by continuity of X(w,,) the set S(w,n,&) = {X(w,u,n)(0) + AX(w,u,n+hE)(0): A,h E [0,1], u E [0,a]} is compact in Rn.
138
Since g is C2, let
L1(w) =
sup { IID2g(v) II : v E S(w,n,C)},
(45)
w E 52a .
Also by Taylor's formula for g, one gets IAh(u)(w)I<
F
L1(w)IX(w,u,n+h&)(0) - X(w,u,n)(0)I2
Suppose L2(w) > 0 is the uniforml Lipschitz constant of
over any
bounded set in C(J,Rn) containing the segment {n+h&: h E [0,1]}. IAh(u)(w)I < L1(w)L2(w)hII&II2,
h E (0,1],
Then
w E Qa,u E[0,a]
(46)
and one has
ra lim h-O
IAh(u)(M )I2du = 0 for all w E 11a 1
(47)
0
Furthermore, for each w E 0a (a(u)(w)I = lim IF [X(w,u,n+hF)(0) - X(w,u,n)(0)]I
h_imm F L2(w) i.e.
Ia(u)(w)I < L2(w)IICII
Let
L3(w) = sup
11K II
= L2(w) II & I
I
for all u E [0,a].
{IIDg(X(w,u,n)(0)II : u E [0,a]},
IBh(u)(w)I <
<
, u E [0,a],
so
IX(w,u,n+h)(0)-X(w,u,n)(0)I + L3(w)Ia(u)(w)I
2L3(w)2I I C I I
and we deduce also that a
lim
w E S2a
IBh(u)(w)I2du = 0
(48)
JO a
Clearly (47) and (48) yield lim J IEh(u)(w)I2du = 0 for all w E sta,so that h-"O 0 IJt
lim P(
h-0
Eh(u)dw(u)I > e) = 0.
sup
tE[0,a]
0
139
(Theorem (1.8.5); or Friedman [22], Vol. I. p. 70).
In other words,
lim 1 h-'O =
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 t°(S2,C([-r,a],Rn)) we proceed as below.
is bounded as a map
Suppose for the time being that D2 H 1
[0,a] x C(J,Rn) 3 (t,n) - D2H1(t,n) E L(C(J,Rn),Rn) by a number IID2Hill > 0.
Represent the coefficients of (D III) as maps In x [0,a] x C(J,Rn) -' Rn,
g:SZ x [0,a] x C(J,Rn) - L(Rm,Rn) defined by H(w,t,F) = D2H1(t,X(w,t,n))M, g(w,t,C) = for w E na, t E [0,a], E E C(J,Rn), with n E C(J,Rn) fixed.
If
S19S2 E l2(D,C(J,Rn)), then
JQIH(w,t,s1(w))
-
H(w,t,R2(w))I2dP(w)
<
IID2H1II2 1181-a2IIL2(RIC) ti
for all t E [0,a].
Also for any S E t2(S1,C(J,Rn);Ft) the map wig H(w,t,O(w))
is Ft-measurable because it is almost surely a limit as h - 0 of the Ft-
measurable maps w - F [H(X(w,t,n) + hB(w)) - H(X(w,t,n))].
The map
ti
g satisfies similar conditions.
Hence Condition (E) of Chapter II is satis-
fied and by Theorem (11.2.1), equation (D III) has a unique solution in r2(Q,C([-r,a],Rn)).
Assume now that D2H1 is continuous but not necessarily
bounded on [0,a] x C(J,Rn).
Take any two (Ft)O<-adapted solutions
1a, 2a E t°(SI,C([-r,a],Rn)) of (D III) at C E C(J,Rn).
For each positive
integer k and t E [0,a], define Sgt
= {w:w E 2a,
sup
11D2H1(u,X(w,u,n))II < k,
0
sup IIDg(X(w,u,n)(0))II< k, 0
Since D2H1 is continuous and X(.,u,n), 1a(.)(u),
140
I2a(w)(u)I < k}.
are all Ft-measurable
Moreover, for
for 0 < u < t, then SZt E Ft for each k > 0 and t E [O,a].
a.a. w each of the expressions
IIDg(X(w,u,n)(0)) II,
sosuup`t IID2H1(u,X(w,u,n)) II1
I1a(w)(u)I,
sup
o
I2a(w)(u)I
sup Oau
is finite; and so the Ft measurable set
k
S2
t
=
u Pk k=1
t
has full P-measure for each t E [0,a].
Fix any k > 0 and let t E [O,r].
Then for a.a. W E Sta we have 1a(w)(t) - 2a(w)(t) = fo D2H1(u,X(w,u,n))[1 u(w)
-
2au(w)]du
t
+ (w) J
0
and so
Xw) II1at(w)_2at(w) IIc nk(
t
< 2 JD + 2
where
x k
XPk
(w) II D2H1(u,X(w,u,n)) II 2 II1 au(w) - 2a u w) II 2du
sup SEE-t,0]
IXS2 k(w) t
(49)
JO
is the characteristic function of Sgt.
If 0 < t' < t, Sgt EE Sgt, and
t
XOk
In particular, by Theorem (I.8.3)(i) (or Friedman
is Ft,-measurable.
t
[22], Lemma (2.9), p. 65), one has t+s !0 t _
t+s JO x
(50) t
a.s. on 9, for all s E [-t,0].
Thus if we take expectations in (49) and 141
apply the Clarti.ngale inequality (Theorem (I..8.5)) we get K1 > 0 such that
Jslk II1(yw) - 2at(w) II2 dP(w) <2k2 Io {
Jck 111a (w)_2a (w) II2 dP(w) }du
t
+ 2K1 Jt
JSjk IIDg(X(w,u,n)(0))II2 I1a(w)(u)-2a(w)(u)I2dP(w)du
a
< 2k2 J{J
t
t
II1au(w)-2au(w)112 dP(w)}du u
2K1k2JtOJSZk
+
(51)
Ila(w)(u) - 2a(w)(u)I2dP(w)du u
because
SZt
c
52 for all 0 < u < t.
Now define the function f:[0,r]
by
setting f(t) = Jok II1at(w) - 2at(w)II2 dP(w), t E [O,r], t
and deduce from (51) that t
f(t) < 2k2(1+K1)
f(u)du
J
for all t E [O,r].
0
This clearly implies that I
k
II1at(w) - 2at(w)II2 dP(w)
= f(t) = 0 for all
t
t E [O,r].
Hence 1at(w) = 2at(w)
As this holds for every
for a.a. w E fit
In k > 0, we obtain lat() = 2at(w) for a.a. w E Sgt = U Qt and P(Slt) = 1. * k=1 2 1 particular, take t = r to obtain a set Q c S2r such that a(w)(t) = a(w)(t)
for all w E 2*, 0 < t < r,
with P(S*) = 1.
On the interval [r,2r] a similar
argument yields 1a(w)(t) = 2a(w)(t) for a.a. w and all r< t < 2r.
Therefore
la(w) = 2a(w) for a.a. w E 0a.
To see the compactness of the map E e-a D3X(w,t,n)M for fixed w E Pa, t E [r,a], n E C(J,R ), note the compactness of the C fact that the convergence D3X(w,t,n)(
)
= lim F [X(w,t,n+hE) - X(w,t,n)] h-O
is uniform for & in bounded subsets of C(J,Rn).
142
o
map
and the
Remarks (2.2) (i)
If, in addition to the hypotheses of Corollary (2.1.3, we assume that
D2H1 is bounded, then the solution a of (D I.I.I.) belongs to L2(SZ,C([-r,a],Rn)).
(ii) The localization technique used above to establish uniqueness of solutions to (D I.II.) in L°(St,C([-r,a],Rn)) is adapted from McShane ([53], pp. 154-
The reader may check that it can further be used to prove existence
160).
and uniqueness in L°(2,C([-r,a],Rn)) of solutions to the stochastic FDE R;
1%
dx(w)(t) = H(w,t,xt(w))dt + G(w,t,xt(w))dw(t)
(V)
x0 = 0 E l°(SZ,C(J,Rn)) ti
with random coefficients
H:SZ
x [0,a] x C(J,Rn)
+Rn,G:Sax[O,a]xC(J1Rn)->L(Rm,J
)
satisfying the conditions Pd
(a)
For a.a. W E Q, H(w,.
(b)
P
*and G(w,.,.) are continuous; I
(
sup
sup
< CO)
tE[O,a] n1n2EC
= 1,
IIn1 - n211
n1 #n2 P (
sup tE[0,a]
IH(.,t,0)I < a) = 1, and a similar condition for G. Pd
(c)
able.
For each t E [0,a],
H%
(.,t,.) and G(.,t,.) are Ft ® Borel C(J,Rn)-measur-
Indeed, the reader may note that in (1l) one can replace w by a semi-
martingale z:Sl-> C([0,a],Rm) fulfilling the Conditions (E) of Chapter II;
(iii) It follows from (ii) above that, under the conditions of Theorem (2.1), X(.,t,n) E r2(2,C(J,Rn)) for t E [0,a], n E C(J,Rn).
Alternatively this fact
may be proved more directly by looking at the definition X(.,t,n) =
of the version X using the notation of the proof of Theorem (2.1). Indeed the process {eL1(w0(.)(t)I:t E [-r,a]} is a submartingale because the function
A ---, elal is convex; so by Doob's inequality
Ee
L1
<E sup
e
tEe
<
tE[-r,a]
143
()II L111w°belongs the random variable w e
;.
e
to r p(S2,R) for all p > 1.
Similarly by the growth properties of F, all the maps below
w F---->
F ° (wo(w) n) D2F(w°(w)(t),n(O))-1
w P--->
ti
w'---0 H(t,n,w) W I-# nCt(w) are of class rp for any p > 1, for fixed t E [O,a], n E C(J,Rn).
Therefore
X(.,t,n) E rp(c,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:Rn - L(R,Rn) satisfies the identity o
{Dg(x)[g(x)(v1)]}(v2) _ {Dg(x)[g(x)(v2)]}(v1), X E Rn, v1,v2 E R.
U.
Delayed Diffusion: An Example of Erratic Behaviour.
In the previous section (§2) it was established, in some detail, that for
ordinary diffusion coefficients a stochastic FDE 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 never exist if the diffusion coefficient depends on the past.
To see this let us consider
the one-dimensional linear stochastic delay differential equation (SDDE): dx(t) = x(t-1)dw(t)
0 < t.< 1
(VI)
x0 = n E C([-1,0],R)
Denote by C the Banach space C([-1,O],R), given the usual supremum norm, and by {nxt:t E [0,1], n E C} the trajectory field of (VI) generated by the unique 144
solution nx E c2(c2,C([-1,1],R)) of (VI) through n E C.
Both nx and the one-
dimensional Brownian motion w in (VI) are carried by a complete filtered probability space (c2,F,(Ft)0
The SDDE (VI) is directly solved by
'integrating' over steps of lengths 1 viz: t
0 < t < 1
n(0) + J0 n(u-1)dw(u) 1x(t)
t E [-1,0)
n(t)
a.s. on 0.
Suppose, if possible, that the trajectory field {nxt:t E [0,1], n E C} admits a version X:2 x [0,1] x C - C with
locally bounded) for a.a. w E Q
continuous (or even
and a fixed to E (0,1].
generality, we may take to = 1 and suppose that
Without loss of is continuous for
a.a. w E 2 . Define the process Y:SZ x C ; R by Y(w,n) = X(w,1,n)(0) - n(0)
a.a. W E
2,
n E C.
(2)
Clearly, 1
1x(1) - n(0) = J
a.s.
n(u-1)dw(u)
0
Since the Brownian motion {w(u):u E [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 Jo n(u-1)dw(u). But these converge in L2 sense to the Gaussian r.v. Y(.,n) for each n E C (Theorem (I.8.1)(ii),
Hida [31], Proposition 1.11, p. 34). system (cf. Dudley [13]).
Therefore {Y(.,n): n E C} is a Gaussian
It is well-known that Gaussian processes parametri-
ized by function spaces never allow continuous (or even locally bounded) sample functions on the whole parameter space; (see Dudley [13]). this in a rather more elementary fashion we proceed as follows.
To see
Note first
that if n E C1([-1,0],R), then we can integrate by parts to obtain J1
Y(w,n) = n(0)w(w)(1) - n(-1)w(w)(0) -
w(w)(u)n'(u-1)du
(3)
0
for all w E Q0, where Q0 c :Q is a set of full P-measure independent of n and
such that w(w) is continuous for all w E 00. Y(w,.): C1([-1,0],R) - R is a linear map.
Therefore for each w E 00,
Now these maps are, for a.a. W E s
discontinuous everywhere in the supremum norm topology of C([-1,0],R). 145
o'
Indeed we contend that for any c > 0,
P{w:w E st°, sup [ IY(w,n) I: n E C1([-1,0],R), IIn IIC < c] < -} = 0 (4) So define the sequence In
in C 1([-1,0],R) by
k }00k=1
for all s E [-1,0], k > 1.
nk(s) = c sin 2rrks
Clearly IInkIIC = e for all k > 1.
Note also that each Gaussian random
has mean zero and variance
variable
EIJ1 nk(u-1)dw(u)I2 =
°k =
0
f-1
0
Ink(s)I2ds
0 = e2 J-1 [sin 2Trks]2ds
= Y e2, k > 1.
In fact for any integers k,h > 0 one has 1
J1
n (u-1)dw(u)]
nk(u-1)dw(u)
E[
E
JO
0
J1
nh(u-1)nk(u-l)du
= 0
fo1 =
nh(s)nk(s)ds =
1 e26kh
where 6kh is the Kronecker Delta (Friedman [22], Theorem (4.2.5), p. 63). k Go are mutually This implies that the Gaussian random variables {Y(.,n )} k=1
independent (Theorem (1.8.1); Hida [31]). {w:w E % , sup IY(w,nk)I< oo} =
tl
n 00
{w:w ESt ,IY(w,nk)I < N},
N=1 k=1
k>1
°
We then consider the set
°
obtaining P{w:w E Sd , sup IY(w,nk)I < -} = lim P( n00 {w:w E Q ,IY(w,nk)I
°
k>1
N->°°
< N}).(5)
°
k=1
But by independence 00
P( n {w:w ESl k=1
°
for each integer N > 0.
146
m P{w:w ESt , IY(w,nk) I < N} Y(w,nk) < N}) = l im II
in o k=1
Now
°
(6)
2
N
P{w:w E 0o
I Y(w.rlk) I< N} = t t I v ak -N
f WE Let PN,E -
1
e
'-N/c
-x2
-y2/E2
IN
1
dy =
e
F,/2
2
e-y 12ak dy
-N
dx for all N > 0.
Then
<
N/c
I -N/E
e
-x2
dx.
1 because
2
J.
e-x
dx = Tr.
P{w:w E S2
o
Using (5) and (6) we deduce that
sup IY(w,nk)I < 00} = lim lim Pm = 0. N,c k>1
N -m-
In fact P{w:w E S2o, sup IY(w,nk)I
N} = 0 for each N > 0.
These clearly
k>1
imply that (4) holds for any c > 0, and so the trajectory field {nxt:t E [0,1], n E 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 E C} of (VI) which is parametrized by the infinite-dimensional space C even though it satisfies 11n1-n2112
for all n1,n2 E C. (ii)
One can always enlarge the filtered probability space (I2,F,(Ft)O
ti ti ti
to get a new space (c2,F,(F)
< <1,P) on which a measurable process
X:S2x[0,1] x C -C isonomous to the field {nxt:tE[0,1],nEC} can be constructed.
This is very easily done by applying the well-known Kolmogorov-Bochner Theorem (Rao [67], pp. 8-16):
Let 0 = C[0'1] x C, the space of all mappings
[0,1] x C+ C endowed with the a-algebra F = o{p(t, ):t E [0,1],n E C} generated by 147
the evaluations ti
:
p(t,n)
--
C
t E [0,1.], n E C} of the stochastic
The transition probabilities
Pti
DDE (VI.) give the product probability measure
measurable space (0,F).
II on the tE[0,1] nEC As we have already seen in Chapter IV, the map
=
is (uniformly) continuous into the narrow (weak *) topology
n _-->
titi
of probability measures on C.
ti
ti
ti
So if we set X(w,t,n) = w(t,n), w E SZ,
t E [0,1], n E C,then because of the fact that P
c
1,
p-1
(t,n)
one
=
can easily see that X is isonomous to the trajectory field {nxt:t E [0,1], ti
n E C} and is continuous in probability on [0,1] x C.
Notice that X(.,t,n)
ti
is measurable with respect to the a-algebra Ft for each t E [0,1], n E C.
E [0,t], n E C} = a{p(u,n):u Of course for a.a. w E St and all t E (0,1] the
ti ti
map X(w,t,.) is discontinuous everywhere on C. (iii)
A further discussion of stochastic DDE's will be carried out in the
course of our next chapter on examples of stochastic FDE's.
There we shall
see that a measurable version X:O x [0,1] x C + C of the trajectory field It is however highly non-linear on C,
{nxt:t E [0,1], n E C} always exists.
i.e. for each t E [0,1]
P{w:w E 0, X(w,t,Xnl+un2) = XX(w,t,n1)+pX(w,t,n2), for all n1,n2 E C, ',u E R} = 0
despite the linearity of the equation (VI) and the fact that P{w:.w E S2, X(w,t,Xn1 + un2) =A X(w,t,n1) + pX(w,t,n2)} =
for all X,u E R, t E [0,1], Till n2 E C. (iv) n f--4
1
See §VI.3.
In the next section we shall show that on the interval [1,2] the map 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 process isonomous to {nxt:t E [1,2], n E C) on the probability space (C[1,21 x C,W,
148
) with
= a{p(tin):t E [1,2], n E C} and P a radon probability measure on
C[1'2]XC
It is not clear, however, if the trajectory field {nxt:t E [1,2], n E C} is then isonomous to a continuous process F x [1,2] x C -r C.
M.
Regularity in Probability for Autonomous Systems
So far we have been looking at the question of establishing suitable versions for the trajectory (random) field of a stochastic FDE.
The simple but patho-
logical example of §3 kills all hope in obtaining a sample-continuous version X:52 x [O,a] x C - C for the trajectory field {nxt:t E [O,a], n E C} of our
general stochastic FDE dx(t) = H(xt)dt + G(xt)dw(t)
0 < t < a 1 t
x0 = n E C = C(J,Rn), J = [-r,0].
(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 probability hold good under fairly general assumptions on the coefficients of the stochastic FDE (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
(S2,F,(Ft)Oa,P).
For simplicity of exposition assume that the coefficients
H:C -> R n, G:C - L(Rm,Rn) are Lipschitz on the Banach space C = C(J'Rn) given
the supremum norm
II
Let 0 < a <
IIC.
and Ca = C(,(J,Rn), the Banach space
of all a-Holder continuous paths n:J - Rn with the a-Holder norm
In(s1)-n(s2)I IIn II Ca = 11n II
C
+
sup {
Is 1
- s 2 Ia
According to the notation of §(III.1), let
' s1'
s2 E J, s1 # s2}. E [O,a], n E 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:c2 x [0,a] 3 E an
(F ® Borel [O,a], Borel E) measurable process with almost all sample paths C2k(c,E;F) in the E continuous. Suppose that for each t E [O,a], Bochner sense and there is a number C = C(a,k) > 0 such that CIt1`t2lk
for all t1,t2 E [O,a].
Then for every 0 < a < }(1 - k) and any real N > 0,
one has
P(
sup t1,t2E[O,a]
k a
It1 - t2I
2Cak(1-2a)
1
k(1-2a)[k(1-2a)-1] N
ti # t2 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 (a
0
fa It1-t2Ik(1-2a)-2
dtldt2 =
2a
k(1-2a)
0 < a <
.
o
k(1-2a)[k(1-2a)-1]
0
The first main result tells us that the sample paths of the solution nx of (I) are almost all a-Hdlder continuous on [0,a] for any 0 < a < J.
Thus the
trajectory field segment {nxt:t E [r,a], n E C} on [r,a] will always lie in Ca(J,Rn) for 0 < a < J.
Before proving this we need estimates on higher order
moments of the trajectory field.
Such estimates are obtained from the follow-
ing 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) i
(Local Uniqueness)
= 1,2, are such that H
150
Suppose H1:0 x C }
Rn,
G1:c2 x C i
n),
L(Rm JR
are Fo-measurable for each n E C and
there is a constant L > 0 so that
IH'(w,n1) - H'(w,n2) I < L11nl-n211 , IIG'-(w,nt) - G'(w,n2)II
Let U
C be open such that H1(w,n) = H2(w,n),
G1(w,n) = G2(w,n) for a.a. W E 0, all n E U.
Denote by
nx' E 92(St,C([-r,a], Rn)) the unique solutions of the stochastic FDE's
0 < t < a
dx'(t) =
x0
= n E C, i = 1,2, Suppose that nT P :9 - [0,a], i = 1,2, are the
with random coefficients.
first times that the trajectories {nxi:t E [0,a]} hi.t CJU viz:
nTU(w)=f
inf {t:t > 0,
nxi(w) a U} if infimum E [0,a]
otherwise
l a
Then for any n E U one has P{w:w E 0,nTU(w) = 'IT U(W)
and sup
P{w:w E 0,
0 A
IInx1(w) u
-
nx2(w)I1= 0} = 1. u
TU(w)
Remark (4.1)
Note that nTV
is a stopping time for the process {nxt:t E [0,a)}, i = 1,2;
(Dynkin [16], p. 106, Stroock and Varadhan [73]).
Theorem (4.3):
Suppose H:St x C -Rn, G:S2 x C -} L(Rm,Rn) satisfy the condi-
tions: and G(.,n) are F0-measurable;
(i)
For each n E C,
(ii)
There is a constant K > 0 such that
IH(w,n) I < K(1 + IIn 11 C),
IIG(w,n)II < K(1 + IIn 11 C)
for a.a. W E 0, all n E C;
15t
(iii)
For each N > 0, there exists LN > 0 such that
IH(w,n1)-H(w,n2) I < LNIIni n2II , IIG(w,n1)-G(w,n2) II < LN IIn1-n211
for a.a. WE S2, all n1,n2 E C with IIn111 C <
IIn2IIC < N.
N,
Then for each n E C the stochastic FDE 0 < t < a
dx(t) =
xO=nEC has a (pathwise) unique solution nx in the sense of Theorem (11.2.1). C2k(S2,C([-r,a],Rn)),
more, nx E
nxt E C2k(St,C)
Further-
for every integer k > 0, and
Indeed there is a constant Ck > 0 depending only on K, k, a
each t E [0,a].
(independent of n E C) such that
EII
nxt
2k 11
<
Ck(1 + Mn II
C
2k) for all n E C, t E [0,a], k = 1,2,3,... . C
We now go on to prove our first main result in this section
Theorem (4.4):
Let n E C and 0 < a < J.
Then the solution nx of (I) satis-
fies P{w:w E S2, nx(w)I[O,a] E Ca([O,a],Rn)} =
1
and P{w:w E S2, nxt(w) E Ca for all r < t < a} = 1.
Proof:
Let n E C, 0 < a < }, t E [0,a], 0 < t1 < t2 < t.
We shall first
obtain an estimate for EInx(t1) - nx(t2)12k where k is any positive integer nx Note that and E r2k(S2,C([-r,a],Rn)) is the unique solution of (I) at n. H, G satisfy the conditions of Theorem (4.3). Elnx(ti)
nx(t2)I2k
<
22k-1 Elft2 t1
<
22k-1Itl-t212k-1
Indeed we have H(nxu)du12k+22k-1EIJt2G(nxu)dw(u)12k
ti
(t2EIH(nxu)12kdu + 22k-1Kk,lIt 1-t21k-1(t2EIG(nxu)I2j t1
t1
152
by H8lder's inequality and Theorem (L.8.5), where Kk,1 > 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 positive constants Kk,2' Kk,3' Kk,4' Kk,5 depending on K, k, a, m, n and independent of n, t so that Elnx(t1) - nx(t2)I2k
<
Kk,2It1-t2I2k-1
E IIxII2k)du
':: (1
Kk,3lt1-t2lk-1
+
t2 ft Cl + E IIxuII2k)du t
< Kk,4(tk+1)Itl-t2Ik
<
Kk,4(tk+1It1-.t2Ik-1.Ck(1+
+
IInII2k)
It1-t2I
(1)
Kk,5(tk+1)(1 + IInhI2k)It1-t2lk
Now apply Theorem (4.1) to the continuous process nxI[O,t], choosing k > 1 sufficiently large such that 0 < a < }
k); thus
(1
Inx(w)(t1)-nx(w)(t2)I
P{w:w E St,
sup
t2la
0
It1
>
8(ka+1) 4 ikN}
a
-
t1#t2
W
K k,S(tk+1)(1 + IInII2k)
k(1-2a)[k(1-2a)-1] =
Kk,6(tk+1)(1 + IInII2k) -1
NN
(2)
N 2Kkak(1-2a)
,5
for every N > 0, where Kk,6 =
independent of t E [O,a],
k(1-2a)[k(1-2a)-1]
n E C.
Define the sequence {cZn}M=1 of subsets QM of S2 by 1M S1M
= {w:w E St,
sup
O
Inx(w)(t )-nx()(t )I 2 It1 - t2l 1
. M}
t1#t2
Then 9t
E F,
1M+1
c S2M for every integer M > 0 and by (2)
K
P(S2M) < M
R-
M>1
(3)
153
)2k
(ak+1).
(ka+21 k
where Kk,z = 8
As the series
00
K
E
-
a 2k M=1 by the Borel Cantelli lemma (Theorem (I.2.5)(i))
converges, then
M
00
P(l im i of S2't2M) = P( U M
= 1.
S2_2M)
M=1
But n
00
U
S2-S2
U
M
M=1
Inx(w)(tl) nx(w)(t2)I
00
=
{w:w E I2,
sup
< M}
Oct1,t2
M=1
It1 - t2 a
t1#t2 Inx(w)(t1) nx(w)(t2)I
= {w:w E
Sup
2,
O
< CO}
It1 - t2la
t1#t2
= {w:w E 0, nx(w)I[O,a] E a [O,a],Rn)}
Therefore the first assertion of the theorem holds.
Finally, the second
assertion of the theorem is implied by the first because Ixt(w)(si)-xt(w)(s2)I
sup s1,s2EJ s l
Is1 -
Ix(w)(t1)-x(w)(t2)I
<
s21a
t1,t2E[O,a]
# s2
for all t E [r,a].
Corollary (4.4.1):
sup It1
- t2l a
t1 # t2 a
If n E C, then for any 0 < a < } tha trajectory
{nxt:t E [r,a]} is a process S2 X [r,a] -> C with almost all sample paths a-Holder continuous.
Proof:
Note that if nx(w)I[O,a] is a-Holder continuous for any w E 0, then
so is the path [r,a] 3 ti--> nxt(w) E C because sup sup t1,t2E[r,a] sEJ t1 # t2
Inxtw)(S) nxtw)(S)I 1
It1 - t2101
Inx(w)(t1+S)-nx(w)(t2+5)I I(t1+s)_(t2+s)Ia
2
sup sEJ tIt2E[r,a] t1 # t2
sup
t1,t2E[O,a] t1#t2
154
sup
Inx(w)(ti)-nx(w)(t2)I t1-t2 a
o
Recall that M(C) is the complete separable topological vector space of all finite regular Borel measures on C given the vague (or weak *) topology (§I.2; Parthasarathy [66]; Stroock and Varadhan [73]).
Since C is complete
separable, the probability measures Mp(C) form a closed convex subset of Compact subsets of M(C) are characterized by the classical theorem of
M(C).
Prohorov:
Theorem (4.5)
(Prohorov):
A subset r c M(C) is relatively weakly * compact if and only if it
(i)
satisfies the following two conditions (a)
sup {p(C):. u E r} < -;
(b)
for any e > 0, there is a compact set KE c C (depending only on c and
r) such that sup {N(C-.KE): P E r} < c.
When this condition is fulfilled we
say r is uniformly tight. (ii)
A subset r c Mp(C) is relatively weakly * compact if and only if it is
uniformly tight.
See Parthasarathy [66], pp. 47-49; Stroock and Varadhan [73], pp. 9-
Proof: D
10.
Our second main result below uses Prohorov's theorem to assert the compactness in probability of the trajectory field {nxt:n E C, t E [r,a]} on [r,a].
Theorem (4.6):
Suppose H, G are Lipschitz.on bounded subsets of C and have
linear growth.
Let n E C and 0 < a < J.
Then for every integer k >
1
1-2a there is a constant Ck > 0 depending only on k, H, G, m, n and independent
of n E C, t E [r,a] such that
P{w:w E 0, Ilnxt(w)II
C> N) < C
(tk+1)(1+IIn 112k)
(4) N1
for all real N > 0, t E [r,a]. In particular, the map [r,a] x C (t,n)
Mp(C) i
>
155
takes bounded sets into relatively compact sets.
Proof:
Since nxt(w) E Ca(J,Rn) for
Fix n E C, 0 < a < } and t E [r,a].
a.a. w E 2, then 11 r'xt(w) I I a= II"xt(w) II C+
sup
Inx(w)(t+st ) rlx(w)(t+s2 )I a
s1,s2EJ
C
Is1 - s2I
Si 0 S2
for a.a. w E S2, and therefore for any N > 0 we have
> N) < P{w.w E Q, llnxt(w) II C > N/2}
P{w:w E S2, Ilnxt(w) II Ca
+ P{w:w E 12.
Ir'x(w)(t1) nx(w)(t2) I
sup
N
>7} C.
O
It1 - t21
tt # t2 2k
2k
Ell nxtll ck +
Kk,6(tk+1)(1+llnhl2k).82k
C
2
2k
NN
because of Chebyshev's inequality (Theorem (1.2.4)) and (2) of the proof of Therefore, using Theorem (4.3), we get
Theorem (4.4).
P{w:w E Q, Ilnxt(w) IICa>N} <W 2k
2k
4(ka+1)2k
F
is independent of t,n. Choosing a Ck = Kk,7 + 22kC1 we immediately obtain assertion (4) of the theorem.
where Kk,7 = Kk,6 8
.2
We next prove that for any bounded set B c C, the set of probability n E B, t E [r,a]} is uniformly tight and hence relatively
measures
weakly * compact in Mp(C) by Prohorov's Theorem.
Let c > 0 and choose
M, NE > 0 sufficiently large so that B c {n:n E C, II nllC <'M} 1
)N--
IIEIICa < 2NE}.
< }E.
By Ascoli's theorem, KE is clearly compact in C.
for any n E B and t E [r,a], we have p(O,n,t,C-.KE) = P{w:w E S2, nxt(w) 0 KE}
156
and
Define the set KE c C by KE ={E:& E Ca(J,Rn), Moreover,
= P{w:w E St, Ilnxt(w) II Ca
2NE}
NE}
< P{w:w E 0, Ilnxt(w) II Ca
< Ck(ak+1)(1+M2k)
-
>
< }E.
NE
, and the map
Therefore sup {p(O,n,t,C-,IZE):n E B, t E [r,a]} < is < c (t,n)
i ->
sets.
o
takes bounded sets into relatively weakly * compact
Denote by Mp(Ca) the space of all Borel probability measures on Ca(J,Rn) We shall show below that for t > r
(0 < a < }) given the weak * topology.
the transition probabilities p(O,n,t,.) E Mp(Ca) for 0 < a < i i.e. they are all supported on Ca rather than C.
This will follow easily if we prove that
Borel Ca is a sub-Q-algebra of Borel C.
Indeed if Borel Ca a Borel C, then
by Theorem (4.4) there is a set 52a E Ft of full P-measure such that nxt(w) E Ca
for all w E a and so
n
x
t
By separability,
E ro(S2,Ca;Ft) for all t > r.
Borel Ca is generated by a countable collection of closed balls; so for Borel Ca to be contained in Borel C we need only check that every closed ball {n:n E Ca,
Iln-nollCa < d} in Ca belongs to Borel C.
For simplicity,
and without loss of generality, it is sufficient to show that the closed unit ball Ba = {n:n E Ca, IIn1ICa < 1} in Ca is a Borel set in C.
Let {si
be
Then
an enumeration of all the rationals in J = [-r,0].
I(s .)-n(sa)I
Ba = {n:n E C, sup In(si)I + i>1
< 1}
sup j,k>i j#k
Is
k
In(s
= {n:n E C,
sup [In(si)I + i,j,k>l j#k
s
-n( s
k
)I
] < 1}
Is.-ska
For each i,j,k > 1, j # k, define the function
ijk:C - R>O by
HSs j )-n(a )I for all Ti E C.
H= I(si)I +
Isj-Ski Then each ijk is continuous and hence (Borel C, Borel R)-measurable. fore B. =
There-
{n:n E C, $ijk(n) < 1} E Borel C. n i,j,k>1 j#k 157
Recall that the spaces L°(c,C) and c(S2,Ca) carry the pseudo-metrics do, da
defined by d0(01,02) = inf [e + P{w:w E 2, I1e1(w)-02(w)IIC > e}] e>O
for all 01,02 E c°(52,C) and da(61,02) = inf [e + P{w:w E S2, 1161(w)-02(w)II e>O Ca
> e}]
Under these pseudo-metrics and a global Lipschitz
for all 01,02 E CO(S2,Ca).
condition on the coefficients H, G of the stochastic FDE (I), we get our last main result concerning the regularity of the map n ---> nxt into t°(c,C) or
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. integer greater than
Let 0 < a < J and k be any
Then there are positive constants
.
C4 = C4(L,m,n,a), C5 = C5(a,L,m,n,a) such that for any
C3 = C3(k,a,L,m,n,a),
ni,n2 E C we have
P{w:w E 0,
I(nlx(w)- 2x(w))(t 1 )-(n1x()-n2x(w))(t 2 )I
sup t1,t2E[0,a]
It1 -
t2Ia
> N}
t1 0 t2 IIn l -n 2 II Ck
P{w:w E
2, Il n
1xt()- n 2xt(w) II a Al
(5)
N>0
, 3
-IIn1-n2II 2k ,
<
C
N
> 0, t E [r,a],
N
(6)
n do(
1
(7)
2k/2k+1,
xt, 2xt)
t E [O,a],
C4 IIn 1-n211 C
and
da( 2xt,
2
2k/2k+1,t E [r,a].
x ) < C5 I1 ni-n2IIC t
In particular the maps C
T1
158
r°(S2,C),
nxt
t E [0,a], C --> £C°(52,Ca), t E [r,a] n
1
;.
nxt
(8)
are uniformly Holder continuous with Holder exponent (7
-a)-1
Furthermore
the maps
[O,a] x C - Mp(C), [r,a] x C - MP(Ca) (t,n) -->
(to) F--> are continuous.
Proof:
Define the measurable process
Let n1, n2 E C and 0 < a < }(t - k
y:S2 - C([-r,a],Rn) by Y(w)(t) = n1x(W)(t) -
t E [-r,a], a.a. W E St.
2x(W)(t),
Then, for 0 < t1 < t2 < a, we have
22k-lEIt2
[H(n1x)-H(n2x)]dul2k u u
Jt
2
1
1
t2 + 22k-1Elf
[G(n 1xu)-G(n
2xu)]dw(u)l2k
t1
t
<
22k-1L2kIt1-t2I2k-1
2
ftl
E llI xu-n2xull2k du
JtZ + 22k-1K1It1-t2Ik-1L2k
Ellnlxu n2xullCkdu
t t
<
22k-1L2k(It1-t2lk
K1)It1-t2lk-1
+
2
E
du
ftl
t2 < K2It1-t2Ik-1 fti
E
2k du
where K1 = K1(k,m,n,a), K2 = K2(k,L,m,n,a) are positive constants independent of n1,n2,t1,t2. i
Similarly, there are positive constants Ki = Ki(k,L,m,n,a),
= 3,4,5,6,7 independent of n1,n2,t1,t2
such that
E IlytIl2Ck = E sup sEJ
<
22(2k-1)E
22k-1
IIn1-n2II2k +
sup sEJ
Ifot+s
[H(n1xu)-H(n2xu)]dul2k
t+s. 0
159
Jt+s 22(2k-1)
+
<
22k-1
E
sup sEJ t+s>0
a nd
t
2k
E Ilnlxu - n2x u
fo
22(2k-1)K3L2ktk-1
22k-1 llnl2II 2k
xu )]dw(u)I
0
t
J0
<
G(112
) -
22(2k-1)t2k-.1L2k
1111-12 112k +
+
[G(111xu
I
+ K4 Jt E Ilyu II
n2xullck du
-
EIIn1xu
2k
112kdu
du, for all t E [0,a];
hence, by Gronwall's Lemma, one gets
E Ilytll
Ck
< 221-1
for all t E [0,a].
11111-11211 c
eK4t < K IIn1-n21121
(9)
Therefore K2K51t1-t2lk IIn1-n2I12k= K6lt1-t2 IkIIn1-n2llrk (10) Now apply Theorem 1 to get a constant
for 0 < t1 < t2 < a, where K6 = K2K5.
C3 = C3(k,a,L,m,n,a) independent of n1,n2, N > 0 such that
Iy(')(t
P{w:w E 0,
sup t1,t2E[0,a]
2 1 i
It1-t 2,
s
3
)l
> N} <
Iln1-n2ll2k
N
ti # t2
for all real N > 0.
Hence (5) is satisfied.
To prove the inequality (6), let r < t < a, N > 0 and consider P{w:w E ,, Ilnlxt(w)
-
a > N}
n2xt(w)II C
< P{w:w E 0, Ilnlxt(w) - n2xt(w)IIC
>
2}
In1x(w)(t+s )-n2x (w)(t+s2)l
+ P{w:w E S,
sup s1,s2EJ si
160
i S2
Is1 -
s2a
>
N}
-2k 3
2k
< NW
E IlytllC
+ 2 Zk
m2UCk
II n1
2k 3
2k+
22kK5. MW
2
2k
-
IIn1-n2IIck
_
where we have used Chebyshev's inequality (Theorem (L.2.4)) and
(9);
K7 = K7(k,a,L,m,n,a) = 22kK5 + 22kC3 is independent of n1,n2, N and t E [r,a]. This proves (6) with C3 replaced by K7.
For the inequality (7), note that, if t E [O,a], then
do(nlxt,n2xt) = inf [c + P{w:w E St, IInlxt(w) n2xt(w) IIC e>O
E
IIytII2k] < inf [C
c
c>O
IIn1-n2II2k]
+
c>O
Define the Coo function f:R>0 - R by K
f(c) = c + - 5 IIn1-n2IIck for all e > 0. Then f attains its absolute minimum value at CO > 0 where f'(c0) = 0, f"(e0) > 0
i.e.
2k V 1
i.e.
-
Zk+1
IIn1-n2IIC k = 0
c02k+1 = 2k K5 IIn1-n2IIc k
Therefore
do(n1xt,n2xt) < inf f(c) = f(c0) = c0(1 + c>0 =
2k
1
1
K52T (2k)
(1 +
IIn1-n2"Ck+'f
)
1
Hence (7) holds with C4 =
,a).
161
Choose and fix an integer k0 = k0(a) such that k0 >
a .
Then by a similar
argument to the above one, we can prove that da(nixt.r)'xt)
inf [e + e>0
1
where C5 = (C3)
0
e
2k0 2k +1
II n1-n2II 2k0] = C51In1-n2IIC
0
C
1
(2k0)
0
(1 +
0
C5(k0(a),a,L,m,n,a) = C5((x,L,m,n,a).
= 0
Since
2k0
> (4 - a)-1, then
2k0+1 2k
I I n-r
IIn1-Ti 2II
0
II ^21
if I1n1-n211C > 1
C
1
<
1
a if IIn1-n211C< 1
IIn1-n21I
and hence the first of the last two assertions of the theorem holds. The final statement of the theorem follows from the uniform (Lipschitz) continuity in the second variable of the maps
[0,a] x C --> L°(2,C), (t,n)
>
E°mCa)
[r,a] X C (t,n)
nxt
nxt
and the continuity of the maps
t°(S2,C)
-p Mp(C),
0 i->
t°(c2,Ca)
Poe-1
-->
0
Mp(Ca)
po0-1
(See Elworthy [19], Appendix C, p. 300). This completes the proof of the theorem.
o
Remark (4.2)
Let Pt:M(C) - M(C), 0 < t < a, be the adjoint semigroup associated with the stochastic FDE (I) ((§III.3)). P E M(C).
If r < t < a, then PtpE M(Ca) for all
In fact each map PP:M(C) - M(Ca), r < t < a, is continuous linear
and the space of probability measures Mp(Ca) on Ca is invariant under the adjoint semigroup Pt for t > r.
As a consequence of the above theorem, one can always select measurable
162
versions 2 x [O,a] x C -* C, Q 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 < }.
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 FDE (I) the coefficients H, G
are globally Lipschitz on C.
Then the trajectory field {nxt:t E [0,a],n E C}
has a measurable version Q x [O,a] x C + C, and for any 0 < a < } the segment
{nxt:t E [r,a], n E C} admits a measurable version 0 x[r a] 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
[0,a] -> c2(c,C)
t )-;$
cow,C)
nxt
for each n E C, that the map Y:[0,a] x C
(to)
; t°o,C)
t-- nxt
is jointly continuous (in the metric d0) and hence Borel-measurable.
But
[0,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 t°(Q,C).
Applying
a general theorem of Cohn we immediately get an (F ® Borel [0,a] ® Borel C, Borel C)-measurable version X:c2 x [0,a] x C + C for Y.
(Theorem (1.5.2)).
A similar argument works for the continuous map
[r,a] x C -> L°(Q,Ca)
(t,n) r- >
nxt
noting that Cohn's theorem above does hold for processes with values in the
complete separable Banach space C.
o
Remark (4.3) (i)
I do not know if the trajectory field {nxt:t E [0,a], n E C} has a C-
valued separable measurable version.
If one exists, then by the Borel-Cantelli 163
1
Lemma it is not hard to see that this separable measurable version X will have the property that there is a set Sa E F of full P-measure, chosen
independently of n E C, so that, for all w E "a and any n E C, X(w,t,n) E C, X(w,t,n) E Ca if t E [O,a] and t E [r,a] respectively. (ii)
Both Theorem (4.7) and its corollary hold for random coefficients
H:Sl x C -, Rn, G:SE x C - L(Rm,Rn) satisfying the conditions: and G(.,n) are F0-measurable;
(a)
For each n E C,
(b)
There exists L > 0 such that
IH(w,n1)-H(w,n2) I < L IIn1-n2IIC,IIG(w,n1)-G(w,n2) II< L 11n1-n2IIC
for a.a. w E.
164
Q,
all n,02 E 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 FDE's.
These examples include stochastic
delay equations (93) and linear FDE's which are forced by white noise (§4). The latter class of stochastic FDE's corresponds to equations of the form
dx(t) = H(xt)dt + g(t)dw(t),
t > 0
x0=nEC EC(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 Weizskcker
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 = {0} and C(J,Rn) is just the Euclidean n-dimensional space
n.
Our basic
stochastic FDE (I) of Chapter II then takes the simple differential form
165
t > 0
dx(t) = g(t,x(t))dz(t),
(I)
x(0) = v E C2(St,Rn) 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 (I), but content ourselves by noting that Theorems (11.2.1), (11.2.2), (II.3.1), (III.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
x(0) = v
dx(t) = h(x(t))dt + g(x(t))dw(t),
t > 0
x(0) = v
for z a continuous semi-martingale, w m-dimensional Brownian motion and coefficients h:Rn - Rn, g:Rn
L(Rm,Rn).
The associated semigroup {Pt}t>O
on Cb(Rn,R) for (III) is, however, always strongly continuous in the supremum norm of Cb(Rn,R).
A strong infinitesimal generator A:D(A) c Cb -). Cb can
therefore be computed to get A(4,)(v) = D4(v)(h(v)) + } trace D2O(v) o (g(v) x g(v)), for c C2-bounded on Rn.
v E Rn
(*)
The reader may check that this agrees formally with
the conclusion of Theorem (IV.3.2) (or that of Theorem (IV.4.3)). and [24] for a classical derivation of formula (*).
See [22]
The trajectory field
{vx(t):t > 0, v E 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:Q x R>0 x Rn _ Rn of the trajectory field implies that for a.a. W E S2 the map X(w,t,.):R
jection for all t > 0.
Rn is a linear bi-
This sharply contrasts the corresponding behaviour
for linear delayed diffusions of §(IV.3).
166
n
§3.
Stochastic Delay Equations
) be a filtered probability space with z:c
Let (S2, F, (Fd t>0 9
->
C(R>,Rm) a
continuous Rm-valued martingale adapted to (Ft)t>0 and satisfying ) < K(t2-t1)
E(z(.)(t2) -
1
1
I
) < K(t2-t1)
t 1
a.s. whenever 0 < t1 < t2, for some K > 0. Suppose hj:Rn + Rn, i = 1,...,p, gi:Rn - L(Rm,Rn), i = 1,...,q are Lipschitz maps.
Let there be given p + q delays as random variables
rj:S2-
R>0
j = 1,...,p, di.Q - R', i = 1,...,q, which are P-essentially bounded and Define
F0-measurable. r =
(essup r.(w),
max
W
7
J
essup d.(w))
W
1
Given 0 E r2(SZ,C(J,Rn); F0), J = [-r,0], consider the stochastic delay
equation (with finite random delays): p
dx(t) =
q
h.(x(t-r.))dt +
E
j=1
E i=1
g (x(t-d ))dz(t), i
t> 0
i
x0=0 The random delays r3, 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. basic Existence Theorem (11.2.1).
Nevertheless we can still apply our First we need to check that (IV) does ti
indeed satisfy Conditions (E) of Chapter II. 92(Q,Rn), 9:c2(Sj,C(J,Rn)) ti
h(W)(w) =
p E
j=1
>
Define h:L2(S2,C(J,Rn)) ---
C2(c,L(Rm,Rn)) by ti
h.[w(w)(-r.(w))], 9(W)(w) = J
3
for W E L2(S2,C(J,Rn)), a.a. w E 0.
q E
9i[w(w)(-di(w))]
i=1
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 W11W2 E C2(S2,C(J,Rn)), we have
167
Ilh(W1)-h(W2) 1Ir2(p.Rn)
< P
E
Ihj[w1(w)(-rj(w))-h.[P2(W)(-r.(W))]I2 dP(W)
J
j=1
Q
p E
p L2
J
j=1
< p2 L2
0
1k)1
lip (w)(-r.()) - W (w)(-r.(w)12 dP(W)
-
2
J
1
3
w2112 2
r (,C)
Similarly g is Lipschitz with Lipschitz constant qL.
It remains now to
verify the adaptability condition E(iii) of §(II.1).
To do this it is
11
sufficient to show that for each measurable, for t > 0.
It,
E L(Q,C;Ft), 2 h('P) and g(W) are Ft-
2 n Let W E L (2,C) be Ft-measurable, and P:J X C -> R
the evaluation map (s,rl) Ea n(s), s E J,n E C.
Then p is continuous, and
since each rj is F0-measurable, it follows that
W
>
h('P)(w) _
[P(-r .(W),w(W))]
h
j=1
j
J
ti
is F -measurable. ti
Therefore h(W) E r2(D,Rn;F ); similarly
t
g(W) E 92(SI,L(Rm,Rn);Ft).
Hence all the conditions of Theorem (11.2.1) are
satisfied and so a unique strong solution
ex
E C (c,C([-r,a],Rn)) of the
stochastic DDE (IV) exists with initial process e.
The trajectory
{ext:a > t > 01 is defined for every a > 0, is (Ft)-adapted and has con-
Moreover each map
tinuous sample paths. Tt:c2(c,C;F0)
`
-'
e
c2(c,C;Ft),
t > 0,
ext
is globally Lipschitz, by Theorem (11.3.1). Now suppose in (IV) that z is m-dimensional Brownian motion w adapted to
(F t)0., where each Ft is suitably enlarged by adding all sets of measure zero, i.e. we get the stochastic DDE of It; type q
p
dx(t) =
Z
j=1
h.(x(t-r .))dt + J
x0=nEC(J,Rn) 168
J
E
i=1
g (x(t-d.))dw(t),
t > 0
(V)
with deterministic initial condition n.
Note that the coefficients in (V)
ti
can also be viewed as F0 ® Borel C measurable maps H:S2 x
G:Q x C(J,Rn) ---> H(w,n) =
n
L(Rm,Rn) given by p E
ti
n
C(J,R ) ---> R
j=1
q
ti
h(n(-r.(w)))
,
G(w,n) =
i
E i=1
1
1
So (V) becomes the stochastic RFDE with random
a.a. w E sl, all n E C(J,Rn).
coefficients:
ti
ti
0 < t< a
dx(t) =
x0=n Because of the randomness in the coefficients of (V)' we cannot apply the results 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 0 < t < a}, we shall show that the trajectory field
a-algebra Fa E { n xt:0 < t < a,
n E C} of (V) is generated in a natural way by a random
family of continuous time-homogeneous Feller processes on C, each of which adapted to the Brownian filtration
u < t})0<. First,
(Ft
note that when the delays are all deterministic the coefficients in (V)' become non-random and Theorems (111.1.1), (111.2.1) imply that the trajectory field gives a continuous time-homogeneous Feller process on C(J,Rn) adapted to (Ftw)O
Secondly, when all the delays rj, di are simple functions, an
associated random family of continuous time-homogeneous Feller processes on C can easily be constructed (Lemma (3.2)) and one can then approximate the general equation (V) by a sequence of stochastic simple-delay equations (Lemma (3.3)).
For simplicity of exposition, and without loss of generality, we deal with the case of only two F0-measurable delays
r0,d0:52- R which are independent
of Fa, viz: dx(t) = h(x(t-r0))dt + g(x(t-d0))dw(t),
0 < t < a
x0 = n E C(J,Rn)
where h:Rn -' Rn, g:Rn Lemma (3.1):
L(Rm,Rn) are Lipschitz maps.
Suppose the delays r0, d0 E LOO(Q J o ;FD) are simple functions 169
with respect to the delay a-algebra F0 = a{r0,d0} c F0 i.e. there are dm,
c FD such that
< r, m' = 1,...,k and {c
0 < rm',
k
E rmX
r0 =
m'=1
e
m =1
St
k
E dm X is
d0 =
For each m' = 1,...,k, let nxmo:0
,
U
= S2.
Stm
m.=1
s C([-r,a],Rn) be the unique solution of 0 < t < a
dxm'(t) = h(xm'(t-rm'))dt + g(xm'(t-dm'))dw(t),
(VI(m'))
xm' = n Then x(t)
nxm'(t)X
E
m' =1
2
-r < t < a, gives the unique solution of (VI)
SP
n
n
in t (S2,C([-r,a],R )) starting at n E C(J,R ) and adapted to (Ft)0
Proof:
By the definition of x, x
E
for all t > 0.
in
m' =1
Stm
k
Then
x
E nx
=
0
m'=1
n.
f
Also since r0, d0 are simple functions,
0 h(x(t-r)) =
E
h(nxm'(t-rm'))X
,, g(x(t-d0))=m Eg(nxm'(t-dm'))X 1
,;
m' =I
and so by Theorem (I.8.3)(i) for the stochastic integral one has n(0) +
x(t) = {
E fo h(1xm,(u-rm,))x d m'=1
h(x(u-r 0))du +
n(0) +
ln(t)
170
J
k E
0 m'=1
0
m
g(nxm'(u-dm'))dw(u)} 0
Jot
0 < t < a
,
X
S'
g(nxm (u-dm ))dw(u),0
t E J
n(t)
-J
t
rt J
E x m'=1
t E J
n(t)
-rn(o) + -{
du +
h(x(u-r0))du +
t E J
J0t
g(x(u-d0))dw(u),
0 < t < a
nxm'
is (Ft)a-adapted, then so is x.
As each
tion of (VI) in £2(Sj,C([-r,a],Rn)).
For a.a. W' E S, let
nxw'
Hence x is the unique solu-
o
E c2(Q,C([-r,a],Rn)) be the unique solution
adapted to (Ft)o
dx(t) = h(x(t-r0(w'))dt + g(x(t-d0(w')))dw(t)
0 < t < a (VI(w'))
x0
= n
with fixed delays r0(w'), d0(w') E [0,r].
The trajectory fields
{nxt:t E [0,a], n E C, of E 0 therefore correspond to a random family
{Xw
:w' E S2}
of continuous time-homogeneous Feller processes on C(J,Rn) adapted to (Ft)O
w' E sl,n E C, t E [0,a]}
Let
denote the associated random family of transition probabilities of the Xw''s. Po(nxt)-1 for the distribution
Note that we retain the notation
of the trajectory field {nxt: t E [0,a], n E C} of (VI).
In (VI) suppose the delays r0, d0 are simple, independent of
Lemma (3.2):
Fa and satisfy the notation of Lemma (3.1).
t E [0,a],
Let
n E C, m = 1,...,k} be the family of transition probabilities associated with the trajectory fields {nxt': t E [0,a], n E C, m' = 1,...,k} for (VI(m')), m' = 1,...,k.
Then for any B E Borel C, n E C, t E [0,a], we have k E
P(w',O,n,t,B) =
,
pm (O,n,t,B) x
,(w')
a.a. W' E S2
m'=1 and
JQ p(w',O,n,t,B)dP(w') = P(0,n,t,B) = P{w:w E Sl, nxt(w) E B}.
Remark Note that for each B E Borel C, the map (w',n,t) +--> p(w',O,n,t,B) is FD
0 Borel C 0 Borel [0,a]-measurable.
!_roof:
By Lemma (3.1), the trajectory field of (VI) is given by k
nx (w) =
t
n E C.
The
E
m'=1 SZm'
nxm
t
,
(w)x
,(w)
a.a. W E sl, t E [O,a],
Stn
form a partition of S2 and are all independent of nxt'.
Let 171
w' E Q. dm
.
Then there is an m' such that w' E. 52 and so r(w')
Hence
nxw
= nxm
d(w') _
and
P(w',O,n,t,B) = P{w:nxtw"(w) E B} = P{w:w E R,nxtm'(w) E B)
=
for every B E Borel C.
pm (O,n,t,B) k E
Thus p(w',O,n,t,B) =
,
pm (O,n,t,B)X
m' =1
for all w' E 12.
,(w')
S2m
is Borel C ® Borel [0,a]-measurable (Theorem
Since each
it is clear that p(.,O,.,.,B) is FD ® Borel C 0 Borel [O,a] measurable.
fQ
Also, by independence of xand nxtm' for t E [O,a], we get k E
p(w',O,n,t,B)dP(w') =
,
pm (O,n,t,B)P(cim )
m' =1
k
E
m '=I fQ k =
1
j
XB(nxtm'(w))dP(w)
`
X
Qd
(w)O(w)
XB(nxtm,(w))XOM,(w)dP(w) nxtm,
=
JXB( m'=l
(w)X
.(w))O(w)
XB(nxt(w))dP(w) = P{w:w E S2, nxt(w) E B} for every B E Borel C.
o
The following is a general approximation lemma for stochastic DDE's:
Suppose the coefficients h, g in (VI) are globally bounded and
Lemma (3.3):
Go
Lipschitz.
00
C'(SIjbo-u;
Let {rk}k=1, {dk}k=1 be increasing sequences in
such that rk -' r0 and dk
d0 as k -'
in E2(SIJ ° ;FD).
FD
Let nxk and nxk'w
denote solutions of the stochastic DDE's: dxk(t) = h(x(t-rk))dt + g(x(t-dk))dw(t),
0 < t < a (VIk)
xk = n 0
172
and
dxk,w'(t)
=
h(xk,w'(t-rk(w')))dt + g(xk'w8(t-dk(w')))dw(t)
xk,w' = n
0
} (VIk(w' )
0
respectively, for k = 1,2,3,..., a.a. W E Q.
Then the map
0 3 w' -> nxw E C2(a.C([-r,a],Rn)) is in rOO(S2,r (S2.C([-r.a1 .Rn));FD
lim J
Ilnxk(w) - nx(w)II2 dP(w)
k-
= 0
and
lim
Ilnxk'w'(w) - Ixw'(w)II2 dP(w)}dP(w') = 0
{J
J
k-c
w' ES2
wES2
Remark For each n E C(J,Rn), the random family {nxw':w' E Sl} has a unique version nX E 9 2(S2 x St, C([-r,a],Rn); FD 0 Fa) such that for a.a. w' E S2,
nxw
a.s.
Suppose h and g are bounded by a common bound K > 0 and have a common
Proof:
Lipschitz constant L > 0.
Let rk -' r0, dk -' d0 as k
Consider
Fix n E C(J,Rn) and take any integer k > 0.
{h(nxk(u-rk)) - h(nx(u-r0))}du
Jt0 nxk(t)
{g(nxk(u-dk))-g(nx(u-d0))}dw(u), 0
0
nx(t) =
in r2(SI;FD).
ft
t E J
0
By Doob's inequality, we have, for any t E [O,a], IJv{h(nxk(u-rk))-h(nx(u-rk))}du E
sup Inxk(v)nx(v)l2 < 4E sup 0
l2
0
v
sup
+ 4E O
I
{h(nx(u-rk))-h(nx(u-r0))}dul2
J 0
+ 4K1 Jt
Ell g(nxk(u-dk))-g(nx(u-dk))II2 du
0
173
0t
+ 4K1
Ell g(nx(u-dk)) - g(9x(u-d0))II2 du, some K1 > 0,
I
denotes the operator norm sup {IA(e)l:e E Rm, lel < 1} for each
where IIAII
A E L(Rm,Rn).
Therefore, by the Lipschitz condition., one gets t
E
sup -r
Inxk(v) nx(v)l2
< 4aL2 J
Elnxk(u-rk)-nx(u-rk)I2du 0
Jt
+ 4aL2
Elnx(u-rk) - 9x(u-r0)12du 0
t + 4K1L2
J
OElnxk(u-dk)-nx(u-dk)I2du + 4K1L2 J OElnx(u-dk)-nx(u-d0)I2du (1)
for t E [O,a]. Clearly,
Elnxk(u-rk)-nx(u-rk)I2 < E
sup -r
Inxk(v) - nx(v)I2
(2)
Elnxk(u-dk)-nx(u-dk)I2 < E
sup -r
Inxk(v) - nx(v)I2
(3)
and
for all u E [O,aJ.
We must estimate the second integrand Elnx(u-rk) - nx(u-r0)l2 in (1) for sufficiently large k, uniformly in u E [0,a].
To do this, let E > 0 and note
that by uniform continuity of n on J there is a 0 < 6 < 1, 6 = 6(E,n) such that for s1,s2 E J, Is1-s2I < 6 , we have In(si)-n(s2)I2 < E. so by convergence of {rk}k=1 in that Elrk-r0 I2 < c' = 62E for all
Let E' = 62E,
there exists k0 = k0(c,6) such k
> k0.
Suppose k > k0 and for any
u E [0,a] denote by
X(r0u), X(rku), XA, XBk the characteristic functions of the sets {w:w E St, r0(w) < u}, {w:w E Q, r0(w) > u.}, {w:w E Q, rk(w) < u}, {w:w E Q, rk(w) > u} in F
D
and
A = {(w,v):w E SI, V E [O,a], v + r0(w) > u}, Bk = {(w,v):w E 0, v E [0,a],
v + rk(w) < u} in FO 0 Borel [O,a], respectively. the stochastic integral, we may write
174
Using Theorem (1.8.3) for
r
rk
ulu--r r0g(nx(v-d0))dw(v)
fu--r0
h(nx(v-r0))dv + if r0 < u
-rkh(nx(v-r0))dv nx(u-rk)-nx(u-r0)
_
u
u0-rk
+
+ n(0)-n(u-r0), if rk < u and r0 > U.
n(u-rk) - n(u-r0) ,
if rk > u
a
J X(r0
+
(nx(v-d0))dw(v)
10 X(r0
a
+ Jo X
(-'v)9 (nx(v-d0))dw(v)
X
(rk u) Bk
/0
[n(u-rk)-n(u-r0)]
+ En(O) - n(u-r0)]X(rk
X(ro-u)
Therefore, a
Elnx(u-rk) - nx(u-r0)I2 < 6a
( EX(r 0
+ 6K1 Ja EX
(r
0
0
k
II9(nx(v-d0 ))II2 dv k
+ 6a
a
II9(nx(v-d0))II2dv
J
+ 6EIn(0)-n(u-r0)I2 X(rku) +
<
6EIn(u-rk)-n(u-r0)I2X(rk>u)
12K2aEX(r0u)(u-rk)
+ 6EIn(0)-n(u-r 0)I2X(rku) + 6EIn(u-rk)-n(u-r0)I2X(rk>u)
(4)
175
Now
Eln(O) - n(u-r0)12X(rku)
1iu-r0i<6 In(O)-n(u-r
0)12X(rk < u)X(r0 >u) dP
fIu-roI>6 In(O)-n(u-r 0)I2X(rk < u)X(r0 +
< c + 411n II 2 P{w:wE S2 ,
>u)
dP
(r0(w) - rk(w)) > 6)
C
< E + d IIn IIC EIrO-rkl2 < E + 4IIn II2 E, because k > k0.
(5)
Similarly, for k > k0, we get
EIn(u-rk)-n(u-r0)I2X(rk
> u)
I<6In(u-rk)-n(u-r0)I2X(rk
I Ir -r 0
+
0
>U)
dP
k k1>6
< E + s IIn III E IrO-rkl2 < (1 + 4IIn I102)E Combining
dP
k1<6
In(u-rk)-n(u-r0)I2X(rk
f1r-r
>u)
(6)
(4), (5) and (6), one gets <.12K2(a+K1)EIrO-rk12
Elnx(u-rk)-nx(u-r0)I2
+ 12(1 + 4IInII2 )E
< 12K2(a+K1)62E + 12(1 + 4IInhIc )E
< 12[K2(a+K1) + 1 + 4IIn IIc ]E for all u E [O,a], k > k0.
(7)
Note that k0 is independent of u E [O,a].
A similar argument applied to the last integrand in (1) yields for every
E>0, a k 0 > 0 such that Elnx(u-dk) nx(u-d0)I2 for all u E [O,a], k > k0. (7) and (8) to obtain
176
<
12[K2(a+K1) + 1 + 4IInhlc ]E
(8)
Now put together the inequalities (1), (2), (3),
t E
sup Inxk(v) nx(v)I2 -r
<
(4aL2+4K1L2)
sup IN k(v)-nx(v)I2 du
E
f
0 -rtv
+ 12a(4aL2 + 4K1L2)[K2(a+K1)
for k > k0 and for all t E [O,a].
+ 1
+ 411n II2]c
(9)
By Gronwall's lemma, it follows that,
if k > k0, then E
Inxk(v) nx(v)I2 sup -r
for all t E [0,a].
<
]ee4L2(a+K1)t
48aL2(a+K1)[K2(a+K1)+1+4IInIIC2
In particular, take t = a to get
xk(w) nx(w)II2dP()< 48aL2(a+K1)[K2(a+K1)+1+4IIn1I2]ce4L2(a+K1)a
J II
for all k > k0; i.e. {nxk}k-1 converges to nx in 92(S2,C([-r,a],Rn)).
To demonstrate the F
D_measurability
of the map SZ 3 w' - nxwe E
2(52,C([-r,a],Rn)) assume for the moment that the delays rk, dk are simple
ones of the form rk =
rm1'k Xm',k,
E
m=l
dk =
Ek
dm',k
m=1 rm',k, dm' k
for some nk > 0, 0 <
< r,
52m',k E FD,
1 < m' < nk, k = 1,2,3,...
Denote by nxm',k E l2(S2,C([-r,a],Rn)) the unique solution of the stochastic DDE
dxm" k(t) = h(xm'k(t-rm',k))dt + g(xm''k(t-dm'k))dw(t), xm',k 0
= n
with fixed (deterministic) delays rm k = 1,2,3,...
.
0
,k, dm
k,
for each m' = 1,...,nk,
Clearly the approximations Ox 'w :w' E
2}
are then given
by nxk'wo()
E
=
1
nxm''k(w)Xek(w)
(10)
m for a.a. w, w' E 52, 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') - r0(w'),
dk(w') - d0(w') as k -> - for a.a. w' E 52, it follows from what has already
177
been proved that
= lim nxk,w
nxw
in t (S2,C([-r,a],Rn)) for a.a. w' E 0.
k,
But (10) implies that the map S2 3 w' , ; nx w E L 2 (c,C([-r,a],Rn)) is FDmeasurable and 0 x S2 3 (w',w) -' 'Ix kw (w) E C([-r,a],Rn) is FD ® Fa-measurable for each k > 1. By the Stricker-Yor Lemma (Theorem (1.5.1)) the random family {nxw' :w' E Q j admits an FD 0 Fa -measurable version nX:S2 x S2 - C([-r,a],Dr) ,
such that for a.a.
E S2,
w'
nxw
Moreover the map
a.s.
S2 3 w' + nxw' E L2(SZ,C([-r,a],Rn)) is F D_measurable and is in fact essent-
ially bounded because, for a.a. w' E S2, t E
lnxw (t)I2 < 3IInII2 + 3 E
sup
-r
< 311n 11C
sup
h(nxw,(u-r(w')))dul2
I
0
sup I1t g(nxw'(u-d(w')))dw(u)I2 0[t
a2K2 + K1aK2).
+ 3a2K2 + 3K1aK2 =
Therefore by Theorem 17 (p. 198) of (Dunford and Schwartz [15]) the measurable {nxw,:w' nX version of E c2} is uniquely determined up to a set of P x P-
measure zero in S2 x Q. CO
00
Now assume once more that {rk}k_1 D)
sequences in £O(S2,R ";F
,
{dk}k=1
are our original increasing
-converging to r0, d0 in L
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 k0 = k0(e,n) > 0 such that
J w'ES2
E
sup Inxk,w,(v) -r
<
(4aL2 + 4K1L2) J
-
r
{1
0
nxw'(v)I2dP(w')
E
sup Inxk9 wl(v)-nxw'(v)I2dP(w')}du
W'ES2 -rN
+ 12a(4aL2 + 4K1L2)[K2(a + K1) + 1 + 411njI2]e C
for all k > k0. E
So, by Gronwall's lemma, sup
fw'En -r'
< 178
Inxk'w'(v)-nxw'(v)I2dP(w')
48aL2(a+K1)[K2(a+K1) + 1 + 411njI2lee4L2(a+K1)t
for all t E [O,a].
In particular, the final assertion of the lemma holds. o
We can now prove the following theorem which says that the distribution of the trajectory field {nxt: t E [O,a], n E C} to the stochastic DDE (VI) is the expectation of the associated family of transition probabilities w' E S2, n E C, t E [O,a]} of (VI(w')).
Theorem (3.1):
In (VI) suppose the coefficients h, g are globally Lipschitz
Let the delays r0, d0 E LOO(c,R
on Rn.
0 < t < a}.
) be independent of Fa
Then
p(w',O,n,t,B)dP(w')
P{w:w E 52, nxt(w) E B} = J S2
for every n E C(J,Rn), t E [O,a] and B E Borel C.
Furthermore the trajectory
field is a time-homogeneous continuous process on C(J,Rn) viz. PoTt _t (n)-1 2
1
PoTt1(n)-1
if 0 < t1 < t2 < a, n E C(J,Rn), where Tt1(n) = nyt
=
2
and
ny is
2
2
the unique solution of dy(t) = h(y(t-r0))dt + g(y(t-d0))dw(t),
t1 < t < a,
y t1 = n We first prove the theorem under the additional assumption that the
Proof:
coefficients h, g are globally bounded and Lipschitz. 00
00
Choose increasing sequences of simple delays {rk}k-1, {dk}k=l independent of Fw and coverging almost surely to r0, d0 respectively.
Use the notation
of Lemma (3.3) to denote by nX the version in L2 (S2 x S2,C([-r,a],Rn);FD ® Fa)
of the random family {nxw ;w' E c2} of solutions to VI(w'), w' E St. {nxk}°°_
,
{n;(k,w'}°°_
Let
, w' E c, stand for the corresponding approximations for
and1nxwo.
nx
Each family {nxk'w':w' E c} corresponds to a map nXk E C2(S2 x Q,C([-r.a].Rn);
FO 0 Fa), nXk(w',w) = nxk.w'(w).
for a.a. w,w' E 52, and each k = 1,2,3,...
.
Because of Lemma (3.3), one
can select a subsequence {nXk'}k'=1 of the L2-convergent sequence {nXk}k-1
such that nXk (w',w) = nxk ,w (w) -> 1X(w',w) as k'-- - for a.a. w,w' E S2. Note also that the subsequence{nxk }k'=1 of {nxk}k=1 converges in L2 to nx as In view of Lemma (3.1), write nxt" nxt,,w, in the form 179
n
nxk(M ) t
=
k
,
nxt ,k (w) X m,.k+(w),
E
m,=1
ni nxkw1(w) t
nxml'k'()X
E
=
m ,k ,(w')
t
m'=1
nk,
k
for a.a. w,w' E 11, t E [0,a], where 0m
E
FD,
rk, =
m'=1
nk,
dk, _
E m=1
rm
E
k
,
X Stm
, k"
k' and
dm''k x f?'
dxml'ke(t) = h(xm" V(t-rm',k'))dt + g(xm.k'(t-dm" ki))dw(t)
(12)
0
k'
'
'
for 1 < m' < nk
= n
k' = 1,2,...
If 0 E Cb is any bounded (uniformly) con-
.
tinuous real-valued function on C(J)Rn), consider n r
(n x
k'
(w))dP(w) = q(nx kI(w))X Jwec2
m =1
J
m,,X,(w)dP(w)
S2
nk E'
m'=1
o(n
J J
w' ES2
wEc
t
(w))X m,,k,(w')dP()dP(w')
J'(nxk''w'(w))dP(w)dP(w')
-
J w' ES2
t
w2
(13)
n
due to the independence of 4° xt '
and X m,'k' for each k' > 1, 1.<m'
nxt, nXk'
Furthermore, since nxt,
. nXt as k' -
k"
in t2 and 4) is
bounded uniformly continuous, we can therefore take k' to - in (13) so as to obtain
IwEa $(nxt(w))dP(w) =
urn
f w E S2 4(nxt' (w))dP(w) 0(nxk''w'(w))dP(w)dP(w')
= lim k'_11
=
180
fw'C2
LEO LO
1
wES2
(TXt(w',w))dP(w)dP(W')
(14)
=
O(nxt'(w))dP(w)}dP(w')
{J
I
J w' ES7
(14)'
wES2
for every 0 E Cb, n E C(J,Rn), t E [O,a]. Now let B c C(J,Rn) be open.
Use a uniformly continuous partition of
unity in C(J,Rn) to construct a sequence {4l}i=1 c Cb such that 4)(n) -r XB(n)
as i -
for each n E C(J,Rn) and I01 (H)I < 1 for all n E C(J,Rn), i > 1
Put $ = 0t in (14) and pass to the limit
(Cf. Proof of Theorem (III.1.1)). as i
-'
via the dominated convergence theorem, thus obtaining P{w:w E S2, nXt(w',w) E B}dP(w')
P{w:w E sl, nxt(w) E B} = J
w'EQ nxtw
(w) E B}dP(w')
P{w:w E 2,
fQ W'
p(w',O,n,t,B)dP(w').
(15)
E
Since this holds for every open set B in C(J,Rn) and Borel C is generated by all open sets, it follows by uniqueness of measure-theoretic extensions that (15) must be true for all B E Borel C.
Hence the first assertion of
the theorem.
Suppose now that the coefficients h, g are Lipschitz but not necessarily globally bounded on Rn.
We maintain that (14)' still holds in this case.
The idea of the proof is to approximate the stochastic DDE's (VI), (VI)(w') by sequences of stochastic DDE's whose coefficients are globally bounded and Lipschitz.
For each integer N > 0, and v E Rn, set h(v)
hN(v) = { h(v)(2
Ivy < N
--i)
N < IvI'< 2N
0
IvI > 2N
g(v)
!vl < N
gN (v) = { g ( v )(2 - -
I
0
)
N <
I
v
I
<
2N
IvI > 2N.
Then clearly hN, gN are globally bounded on Rn by
sup
Ivk2N
ih(v)i,
sup
ig(v)i
Ivk<2N
and are also Lipschitz.with the same Lipschi.tz constant L as that of h,g. Fcr 2 nxN,w each N > 0, let nxN E L , a.a. W' E SI, be the unique (c2,C([-r,a],Rn)),
181
solutions of the stochastic DDE's dxN(t) = hN(xN(t-r0))dt + gN(xN(t-d0))dw(t), 0 < t < a
x
=nEC
0
dxN,w'(t) = hN(xN'w'(t-r0(w')))dt+gN(xN')S(t-d0(w')))dw(t),
xN'w' - n
0 < t < a NO NW)
U
View (VI)N and (VI)N(w') as stochastic FDE's with random coefficients
HN:S2 X C - Rn, GN:S2 x C - L(Rm,Rn) viz. U < t < a
dxN(t) =
x0 = Ti dxN'w1(t) = HN(w',xt'w1 )dt + GN(',xt'we )dw(t), 0 < t < a w'
x0'
= n
where HN(w,n) = hN(n(-r0(w))), GN(w,n) = 9N(n(-d0(w))), for a.a. w E St and
It is easy to see that
all n E C.
IHN(w,nl)-HN(w,n2)I < LIIn1-n211C' IGN(w,nl)-GN(w,n2)I <
LIIn1-n2II
for a.a. W E 0, all n1,n2 E C; and there is a K > 0 (independent of w,n,Pi)
such that
IHN(w,n)I < K(1 + IInII C), IGN(w,n)I < K(1 + Mn IIC) for a.a. W E Q, all n E C. (V.4.2).
So HN9 GN satisfy all the conditions of Theorem
Indeed if BN = {n: n E C,
IInII
< N} for each N > 0, then if N' > N, a.a. w E St;
up to the time they leave the
so the trajectories {nxt}1,0, ball BN.
Using the uniform estimates 112
nxt'w'
I
EIl
182
I IC2 < C2 (1 +
I n 11C
)
for all N > 0, all t E [0,a], and a.a. w E 0, it is not hard to see that nxN -, nx as N - - a.s. in L2(s2,C([-r,a],Rn)). Similarly, for a.a. w' E 2, nxN,w + nxw as N - - a.s. in L (c,C([-r,a],Rn)). Therefore, if 0 E Cb, we have by the dominated convergence theorem 4(nxt(w))dP(w) = lim JwES2
c(nxt(w))dP(w) J
wESt
and
w xw,(w))dP(w) = lim J
t
J WE52
N-"°
t
4(nxN'w'(w))dP(w) LEO
a.a. W' E S2.
By the remark following Lemma (3.3), the random family {nxN'w :w' E S2} has a version in L2(52 x 12, C([-r,a],Rn); FD ® Fa) for each N > 0; so each map o(nx'wI())dP(w)
w' 4-> LEO is F
D_measurable,
t
by Tonelli's Theorem (Dunford and Schwartz [15], p. 194).
Hence the map w' t-->
O(nxt (w))dP(w) is also F D_measurable. fWEQ we can once more apply the dominated convergence theorem to get c(nxw'(w))dP(w)dP(w')
J
w' ES2
fwEQ
t
= lim
J
I
Therefore,
(nxN'wI())dP(ww')
N-'°° J w' ES2 WEST
t
Now, by the first part of the proof,
WxN(w))dP(w) JWES7
t
= J
J
W' 2
t
4(nxN,
LEO
for each N > 0; so (14)' holds by letting N
(w))dP(w)dP(w')
in the last equality.
The
relation (15) then follows as before.
To prove time-homogeneity note that by the above result and Theorem (III. 2.1) we have (P°Tt12 N()-1)(B)
P(w'.t ,n.t2,B)dP(w')
=
p(w',O,n,t2-t1,B)O(w')
= J1 Q
= (P°Tt2-t1 (n)-1)(B)
for every B E Borel C, 0 < t1 < t2 < a, n E C. 183
This completes the proof of the theorem.
o
As a consequence of the above theorem one can show that the trajectory field of the stochastic DDE (V) (or VI) possesses all the distributional regularity properties of §(V.4).
Corollary (3.1.1):
In particular, we get
If the coefficients h3, 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 a
theorem.
The trajectory field {nxt:t > 0, n E C} of a stochastic
Definition (3.1):
FDE is asymptotically stochastically stable if lim Po(nxt)-1 exists in the space Mp(C) of all probability measures on C. The next corollary says that in order for the stochastic DDE (V) to be asymptotically stochastically stable it is sufficient to verify this property for every stochastic DDE with fixed delays of the type q
p
dx(t) =
E
h(x(t-r°))dt +
j=1
i
i
g (x(t-do))dw(t), i=1
i
t > 0
i
(VII)
xO=Ti EC and arbitrary gi = 0,
i
{do}Q-1 c [0,r].
A similar result for the case
= 1,...,q, hj linear, j = 1,...,p has been established by Lidskii
([48], 1965).
Note however that Lidskii allows the delays rj to be time-
dependent, Markovian in general.
Corollary (3.1.2): delays
Suppose that for every choice of fixed deterministic
do, j = 1,...,p, i = 1,...,q the system of stochastic DDE's
(VII) is asymptotically stochastically stable.
Then so is the stochastic
DDE (V) for all random delays rj, di E cO`(c,[O,r];FO), j = 1,...,p, i=1,...,q
which are independent of the Brownian a-algebra Fw
U
Ft.
t>O Proof:
184
Take r = rj(w'), d0 = di(W') for a.a. w' E Q in (VII).
Then by
hypothesis, lim
u(w'), a.a. w' E S2, where {u(w'):w' E S2} is
ta random family of probability measures on C.
If {nxt:t > 0, n E C} is the
trajectory field of (V), then the identity {Po(nxt)-1}(B) =
P(w',O,n,t,B)dP(w'), B E Borel C, J
wES2
the dominated convergence theorem, and Theorem (I.2.2)(v) imply that lim t+00
Po(nx
)-1
t
= u - J
p(w')dP(w') in M (C).
a
P
w' ES2
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, dill j = 1,...,p, i =1,...,q to be continuous
stochastic processes, each independent of w. (iii) Although the trajectory field {nxt:t > 0, n E 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(e) c Cb
Cb associated with the family of semigroups {Pt }b0 defined
by the systems p
dx(t) =
q
h.(x(t-r.(w'))dt +
E
j=1
E g (x(t-d1(w'))dw(t) i=1
x0=nEC for a.a. w' E S2.
t > 0 (V(w'))
Using the notation of Chapter IV and the results of Theo-
rems (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 w' in V(w'). cular, Tq c V(Aw
In parti-
for a.a. w' E S2. w3o suppose 0 E Tq and using functional
calculus write formally Pw- (4) = etA
a.a. w' E 0 , t > 0.
Then by (14)'
of the proof of Theorem (3.1) we get O(nxt(w))dP(w) = J
J wES2
etAw,(0)(n)dP(w')
PW'($)(n)dP(w') w ES2
Jw ES2
185
where
p S($)(n) +
+ 3
E
E
k=1
i,i =l
E j=1
DO(n)(h(n(-r(w')))X(0})
D O(n)(gi(q(-di(w')))(e
Recall that S is the weak infinitesimal generator of the shift semigroup {St}t>O and {ek}k=1 is any basis forIRm.
To get the distribution P{w:w E Q,
nxt(w) E B} of the trajectory field to (V), approximate each characteristic function XB weakly by a directed family {fit} in Tq; thus W'
t
P{w:w E S2, nx (w) E B} = lim
t
e J W'ES2
to ( )(n)dP(w'). t
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 E [O,a],n E C}, {nxt:t E [r,a], n E C} admit measurable versions 2 x [O,a] x C i C, 2 x[r,a] x C ; Ca, for any 0 < a < J.
Note here that we do not require the
delay a-algebra FD 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
x0 = n E C = C([-1,0],R)
Note that in this case r = 1, m = n = 1 and the diffusion coefficient 2
g:R -)- R, g(v) = v , v E IR, is clearly not globally Lipschitz on R.
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 DDE (VIII). version St x[0,1] x C follows.
Nevertheless, we proceed to find a measurable
C of the trajectory field {nxt:t E [0,1], n E C} as
It is sufficient to find a measurable version for the field
{nxt - nt: t E [0,1], n E C} where
186
as in §(IV.2).
TI(O)
S E [-t,0]
n(t+s)
s E [-1,-t), t E [0,1]
For each n E C define the sequence {n
k }k00
=1
of piecewise
linear approximations nk E C([-1,0],R) by n k(s) _ [k(s+1)-j]n(-1 +
if s E [--i +
) + [j + 1 - k(s+1)]n(-1 + k)
i
-1 + 3--], j = 0,1,2,...,k, k = 1,2,3,...
Clearly nk i n as k - - in C, and each nk is piecewise C1 and hence of bounded variation on [-1,0].
Define the processes Yk: S,
X [0,1] x C - C,
k = 1,2,..., by setting w(w)(t+s)nk(t+s-1)2 - w(w)(0)nk(-1)2 t+s
Y (W,t,n)(s) =
- 2 J
k
w()(u)nk (u-1)(n k )'(u-1)du, s E[-t,0]
0 0
s E [-1,-t)
for all t E [0,1], n E C and all w E SI0, where 2
0
such that w(w) E C([ 0,1],R) for each w E 20.
is a set of full P-measure
Therefore for all w E SI0,
t E [0,1], n E C it is clear that Yk(w,t,n) E C for every k > 1. Moreover each process Yk is (F a Borel [0,1] 0 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) '--a Yk(w,t,n)(s) is measurable.
Now for each k, (nk)'(u-1) = kn(-1 + = t) -
kn(-1 + k) for u E (k,
j = 0,1,2,..., k.
Fix any t0 E [0,1] and let
0 < j0 < k be such that t0 E [j0/k,(j0+1)/k]. Then t
oW(w)(u)nk(u-1)(nk)'(u-1)du = f0
+1)/k
j0-1
k[n(-1 +) - n(-1 + k)]
E
w(w)(u)nk(u-1)du J j/k
j=0 t0
j +1
j
+ k[n(-1 +) - n(-1 + k )] Jj0/k w(w)(u)nk(u-1)du. But the maps n E--> n(-1 + j/k), (u,n) h-->
(w,u) --a
nk(u-1) are continuous and
w(w)(u) is measurable, so the integrals k t0 k w(w)(u)n (u-1)du, (w,n) - Jj0/k w(w)(u)n (u-1)du
(j+1)/k (w,n) F--> J
J/k
depend measurably on the pair (w,n).
From the preceding equality, it follows
that the map t
(w,n) -a
w(w)(u)nk(u-1)(nk)'(u-l)du
J 0
is(F ® Borel C}measurable for every t E [0,1].
Since this indefinite inte-
gral is continuous in t for each w E P09 n E C, it is easy to see that the map t (w,t,n) T-->
w(w)(u)nk(u-1)(nk)'(u-1)du
J 0
is (F ® Borel [0,1] ® Borel C)-measurable.
follows that (w,t,n),->
From the definition of Yk, it
Yk(w,t,n)(s) is measurable for each s E [-1,0].
Thus Yk is (F 0 Borel [0,1] 0 Borel C, Borel C)-measurable.
Moreover, using
integration by parts (Elworthy [19] p. 79), one has a.s. t+s
[nk(u-1)]2dw(u)
{ 188
s E [-t,0]
f0
=
0
s E [-1,-t)
i.e.
ti Yk(-,t,n) = nxt
k
- nt
t E [0,1], n E C, a.s.
,
Since nk(s) -> n(s) as k - - uniformly in s E [-1,0], it follows easily from
Doob's inequality for the stochastic integral (Theorem (1.8.5)) that -ti
nxt
2
nt as k - - in r (c,C).
Hence by the Stricker-Yor lemma
for C-valued mappings we get a measurable version Y:S2 x- [0,1] X C -
C for
the field {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
x0 = n E C = C([-1,0],R)
where k E 1"0 is a positive integer.
Furthermore, if k 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 St x [0,1] x C - C.
In
particular, when I = 1, every measurable version X:S2 x
C for
[0,1] x C -
the trajectory field of dx(t) = x(t-1)dw(t) ,
0 < t <
1
x0=nEC is a.s. non-linear on C, i.e. for every t E [0,1] the set Sgt
= {w: w E S2, 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 (Q,F,P) is complete; so it is sufficient to prove that ability of X,
0.
To do this, let w0 E Sly.
From the measur-
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 E 0,
R is continuous}
which is of P-measure zero by §(V.3).
By completeness of P, SI E F and
189
P(q)
= 0.
Note, however, that linearity in probability (or in distribution)
always holds in this case i..e. for any nt,n2 E C, u,v E R, t E [0,1], we have P{w:.w E 0, X(w,t,un1+vn2) = uX(w,t,n1) + vX(w,t,n2)} = 1.
Indeed this last result is valid for linear stochastic FDE's dx(t) = H(xt)dt + G(xt)dw(t),
t > 0
x0 = n with coefficients H E L(C,Rn), G E L(C,L(Rm,Rn)) due to (pathwise) uniqueness of solutions (Theorem (11.2.1)). It is interesting to observe here that for stochastic linear ODE's on Euclidean space Rn
dx(t) = h(x(t))dt + g(x(t))dw(t),
t > 0 (X)
x(0) = v E Rn, h E L(Rn), g E L(Rn,L(Rm,n)) the trajectory field {vx(t):t > 0, v E Rn} possesses a measurable version
X:SI X R' x Rn i Rn which is a.s. linear on Rn i.e. for a.a. w E S2, all t > 0,
E L(Rn).
This follows from the easily-verifiable fact that
for a measurable field S2 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.
Ito's formula shows that the process X(w,t,v) =
ve(1-}c2)t+c1w(w)(t)-w(w)(0)1
a.a. w E St, 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 FDE's Forced by. White Noise
As before, we take (S2,F,(Ft)tm,P) to be a filtered probability space satisfying the usual conditions. to be parametrized
Note that here we require the filtration (Ft)tER
by all time, with an m-dimensional standard Brownian
w:S2- C(R,Rm) adapted to it.
motion
Let H:C = C(J,Rn) - Rn be a continuous
linear map and g:R -+ L(Rm,Rn) be measurable such that
integrable over R, where
is locally
is the operator norm on L(Rm,Rn).
Consider
the forced linear system dx(t) = H(xt)dt + g(t)dw(t),
t > 0
x0=nEC as opposed to the unforced deterministic linear RFDE: dy(t) = H(yt)dt
y0
(XII)
=nEC.
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
It
C=U$S.
(1)
The subspace U is finite-dimensional, S is closed and the splitting is invariant under the semigroup Tt:C - C, t > 0, Tt(n) = yt for all n E 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
AH(D)
tlt>O viz.
= lim t
E D(AH
t-+0+ V(AH) = {&
:
E E C1([-r,O],Rn),
V(S)
S E [-r,0)
H(E)
s = 0.
&,(0)
= H(E)}.
AHW(s) For convenience, identify the spaces L(Rm,Rn), L(Rn) with the corresponding 191
spaces of n x m and. n x n real matrices.
From the Ri.esz representation
theorem, there is. a (unique) L(1Rn) valued measure u on J such that r0
H(E) = 1-
(s)du(s) for all E E C(J,1Rn). r ti
.
It is therefore possible to extend H to the Banach space C = C(J,R") of all We denote this
bounded measurable maps J ->1Rn, given the supremum norm.
ti
extension also by H.
Solving the linear FOE (XI.I.) for initial data in C, we ti
can extend the semi.group {Tt}
t>O
ti
The splitting (1) of C is topological, so the projections IIS:C
Al
to one on C denoted also by Tt:C -> C, t > 0.
+ S are continuous linear maps.
Since dim U <
IIU:C - U,
IIU has a representa-
E(s)dp(s).
tion by an L(1Rn,U)-valued measure p on J viz. 11U(E) _ fo
This
formula gives a natural extension to a continuous linear map IIU :C - U. 1, IL R, U Defining S = {&: E C, II M = 0}, we see that C has a topological splitting ti
ti
C=U®S.
(2) ,
The projection II :C
IIS(C)
S is continuous linear, being given by
IIU() for all C E C.
ti
ti
ti
When C E C, denote IIU (C) and R (E) by
and CS respectively.
The following
lemma shows that the splitting (2) is invariant under the semigroup
t>O*
ti
Lemma (4.1):
For each C E C, and t > 0, we have
CTt(E)]u = Tt(CU), [Tt(C)]S = Tt(CS
Proof:
For C E C, the result follows directly from the well-known invariance ti
of the splitting (1) under {Tt}tom.
To prove it for C E C, consider the
following definition of weak continuity for linear operators on C. operator B:C - C is weakly continuous if whenever bounded sequence in C with Ck(s) -> 0 as k ; as k -
A linear
k}k=1 is a uniformly
for each s E J, then B(Ck)(s)
- for each s E J (cf. the 'weak continuity property (w1)' of Lemma
(IV.3.1)).
The Riesz representation theorem implies that every continuous linear map C - U has a unique weakly continuous extension C -i U.
Hence for the first
assertion of the lemma to hold, it is enough to show that IIU°Tt and Tt°lIU are 192
Lt is clear from the definition of
both weakly continuous for all t > 0.
As the composition of weakly con-
IIU:C - U that it is weakly continuous. ti
tinuous linear operators on C is also weakly continuous, 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 ti
CTt(C)]S = Tt(E) - ETt(C)]U = Tt(E) - Tt(&U)
= Tt(& -
U)
=
Tt(ES),
t > 0,
C E C.
o
Lemma (4.2): For each t > 0, TtC - C is weakly continuous. Let v(u) be the total variation measure of the L(Rn) -valued measure Ef:R'0 R by y on J representing H. Fix E E C and define Proof:.
Cf(t) = j_r ITt()(s)Idv()(s) + ITt()(0)I, O Now
t > 0.
rt+s
&(0) +
H(Tu(g))du J
Tt(9)(s) =
Vt+s)
t+s > 0
0
-r < t+s < 0
Thus max(-r,-t) (t+s)Idv(u)(s)
Cf(t) = 1_r
(t+s H(Tu(E))duldv(U)(s) + J0 max(-r,-t)
R(0) + 0
t
+ IC(0) + JO H(Tu(C))dul
t max(-r,-t) IC(t+s) Idv(u)(s) + Cv(u)(J) + 1](IE(0) I+ J < J -r 0
Idu)
max(-r,-t)
J -r
IC(t+s) Idv(u)(s) + WOW +1] W0) I 193
ft
f0
+ [v(u)(J) + t](J O ITu(C)(O)Idu +
ITu(&)(s)Idv(u)(s)du)
0
0
-r
t
<
Ch(t) + C
Cf(u)du fo
where
h(t) =
max(-r,-t)
IC(t+s)Idv(u)(s) + [v(u)(J) +1]IC(0)I, t > 0,
J_r
and
C = v(u)(J) + 1.
By Gronwall's lemma, we obtain t
Ef(t)
<
Eh(u)eC(t-u)du,
Ch(t) + C J 0
for all t > 0. Now let
be a uniformly bounded. sequence in C converging pointwise
to 0; then the sequence
is uniformly bounded on [O,t] and kh(t) -, 0
as k -. co for each t > 0, by the dominated convergence theorem.
The last
estimate then implies again by dominated convergence that Ckf(t) -1.0 as k - for each t > 0.
In particular, Tt(Ek)(0) - 0 as k -
for every t > 0.
But
for each s E J, k(t+s)
Tt+s(Ek)(0)
-r < t+s < 0 t+s > 0, k > 1,
so Tt(Ek)(s) -> 0 as k - - for each s E J, t > 0.
o
By analogy with deterministic forced linear FDE's, our first objective is to derive a stochastic variation of parameters formula for the forced linear system (XI).
The main idea is to look for a stochastic interpretation of the
deterministic variation of parameters formula corresponding to non-homogeneous linear systems
dy(t) = H(yt)dt + g(t)dt,
t > 0 (XIII)
YO =n EC (cf. Hale [26] pp. 143-147; Hale and Meyer [29]). To start with, we require some notation. 194
Denote by t:J - L(Rn) the map
A = X{U}I, where I. E L(IR") is the identity n x n matrix. "U
^U
Also, for any
nu
linear map B:C(J,Rn) + C(J,Rn) and any A E
C(J,L(Rm,Rn)),
A(s) = (a1(s), a 2(s),...,am(s)),
s E J
BA = (B(aI),B(a2),...,B(am)) E C(J,L(Rm,Rn))
let
where a.(s) is the j-th column of the n x m matrix A(s) for each 1 < j < m Thus each aj E C(J,Rn) and BA E C(J,L(IR R )).
and s EJJ.
,
If F:[a,b] - C(J,L(Rm,Rn)) is a map, define the stochastic integral
fa F(t)dw(t) by b
[Jb
F(t)dw(t)](s) = J a
F(t)(s)dw(t),
S E J,
a
whenever the Ito integral tb F(t)(s)dw(t) E Rn exists for every s E J. This will exist for example if F is measurable and fa IIF(t)Ik. dt < -.
In case
fa F(t)dw(t) E C(J,Rn) a.s., its transform under a continuous linear
map C(J,Rn) + Rn is described by
Lemma (4.3):
"'ti
Rn be continuous linear and suppose L:C(J,Rn) -'
Let L:C(J,Rn)
Rn is its canonical continuous linear extension using the Riez representation ti
theorem.
Assume that F:[a,b] -> C(J,L(Rm,Rn)) is such that to F(t)dw(t) E b
Then to LF(t)dw(t.) exists and
C(J,R ) a.s. b
m
b ti
F(t)dw(t)) = J
L(J a
LF(t)dw(t) =
E
j=1
a
b ti
L(f(t))dw.(t)
J a
i
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.
Proof:
Represent L by an L(Rm,Rn)-valued measure on J via the Riesz repre-
sentation theorem; then use coordinates to reduce to the one-dimensional case m = n = 1.
Namely, it is sufficient to prove that if u is any finite positive
measure on J and f E C2([a,b] x J,R;dt 0 du), then fl
Jb
a
-r
0
f(t,s)du(s)dw(t) = f
r
Jb f(t,s)dw(t)du(s)
a.s.
(3)
a
195
Suppose first that f = X[a,R]M[y,6]
,
rectangle [a,B] x [y,5] c [a,b] x J.
the characteristic function of the 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
r2([a,b] x J. dt ® du), we need only check that each side of (3) is continuous in f E C2([a,b] x J,R,dt ® du). Elrb a
But this is implied by the easy inequalities: fb [f0
0
f(t,s)du(s)dw(t)I2 =
f-r
a
u(J)
fb fl _
a
r
-r
f(t,s)du(s)]2dt
lf(t,s) I2du(s)dt = u(J)
111,112
t2
and ElJ0
b f(t,s)dw(t)du(s)I2 < u(J)EJO lJbf(ts)dw(t)I2du(s) -r a
r fa
JO
Jb lf(t,s)I2dt du(s) = u(J) IIfII22.
= u(J) r
r
a
°
Remark
Since f is r2, there is a version of the process s N Jb f(t,s)dw(t) with almost all sample paths in r2(J,R;du). 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 [0,a] - L(Rm,Rn) is continuous on
{(t,u):O < u < t < a} with partial derivative (t,u) - at f(t,u) continuous on {(t,u):O < u < t < a}.
Let z be the process
t
f(t,u)dw(u),
z(t) = J
t E [O,a].
0
Then
dz(t) = f(t,t)dw(t) + {JO at f(t,u)dw(u)}dt
viz.
196
v
It f(t,u)dw(u) = It f(u,u)dw(u) + It 0
0
0
f(v,u)dw(u)}dv
(4)
{fo
f or all t E [0,a], a.s. ti
Proof:
Suppose first that w is a process on [0,a] with almost all sample ti
paths piecewise C1. Z(t) =
Define z by
f(t,u)dw(u) =
f(t,u)w'(u)du,
I0t
t I0
t E [0,a],
ti a.s.
Then almost all sample paths of z are differentiable and a.s.
t
IV
z'(t) = f(t,t)w'(t) + IO
for all t E (0,a).
a Tt-
f(t,u)w'(u)du '\j
Thus
dz(t) = z'(t)dt [It
f(t,t)w1(t)dt
f(t,u)w'(u)du]dt
+
=
0
t =
f(t,t)dw(t) + [ IO t f(t,u)dw(u)]dt ti
i.e. (4) holds if w is replaced by the piecewise C1 process w. If w is Brownian motion on Rm, define piecewise linear approximations of Let 11:0 = t1 < t2 < t3 < ... < tk = a be a partition of [0,a].
w as follows.
Define the process w11 in Rm by .
(t-tj), ti < t < tj+1
= w(ti) +
w11 (t)
i
where A t i
= tj+1-tj,
A w = w(tj+1)-w(tj), j = 1,2,...,k. i
Suppose G: $Z x [0,a] -> L(Rm,Rn) is an (Ft)0
each t E [0,a],
[0,a] ---a t
.
E C2(g,L(Rm,Rn)) and the map t2(SI,L(Rm,Rn)) G(t) =
is (uniformly) continuous.
Let e > 0.
Then there is a 51 > 0 such that 197
IIG(u1) - G(u2)IIr 2 < jE/,ia whenever u1, u2 E [O,aj,
mesh
lu1-u2I
< S.
Suppose
II < d1. Then k-1 E
a
G(t)W(t)
G(t.)A.w -
j=1
J
J
I
0
k -1
j=1
t
{G(t.) -
= kE1
r J+1 1
j=1
AJw G(t)dt
tj
j=1
J
J
Jtj1
k-1 E
G(t )A w
E
a
J
f
G(t)dt}Aw.
tj
J
Now E(AjwlFt.) = 0, E(IAjwI2IFt ) = Ajt a.s. for j = 1,2,...,k-1 and
J
J
tj+l
J+1
IIG(tj) -
J
2=
G(t)dtII
tj
4
II J
J
tj
{G(tj)-G(t)}dtll
2
jtJ+1
<
IIG(tj) - G(t) II
tj
1
J
2
dt
t=je
<
J
since
l
J
Va-
It-tjI < Itj+1-tjI < mesh n < d1 for all t E [tj, tj+1].
Applying
Lemma 2 of Elworthy ([19], Chapter III, §2, pp. 25-28), we obtain
Elk E1
G(t.)AJ w - Ja G(t)dw11(t)
j=1
<
0
J
A.t IIG(t.) _
j=1 j it < a Ek-1 Ajt=E kE1
1
12
E2
1
J+G(t)dtll .
22
t
2
j=1
But, by property of the stochastic integral, there is a 6 kE1
G(tj)Ajwl2 < 1 E2
I Ja G(t)dw(t) 0
198
j=1
2
> 0 such that
if mesh II < 62.
So if mesh II < 6 = min (61,62), one gets a
Ja G(t)dw(t) - f G(t)dw"(t)l 2 0 0
El
k -1
a < 2EJ fo G(t)dw(t) -
2.4 e2 + 2.4
<
£2
=
k -1
2
=1
2
Iira
+ 2E1 E G(t )D W_
E G(t .)A wl j=1 i J
.
.
J
J
G(t)dw1(t) 1 0
e2
Thus we have proved that Ja
lim
ElJa G(t)dw(t) -
meshII -0
G(t)dwII(t)l 2 = 0.
0
(5)
0
In particular, since the functions us-* f(t,u), u H f(u,u), ur av f(v,u) are continuous for fixed t and v, we get t
f(t,u)dw11(u)
-#
f0
rt
f(t,u)dw(u) J
0
t
ft f(u,u)dw1(u) > J
(
f(u,u)dw(u) 0
0
fvo
v f(v,u)dw(u)
(0 av f(v,u)dO(u)
in t2 as mesh n - 0,
for each t E [O,a], V E [O,t]. t
t f0
v
f(u,u)dw1(u) + ft {J
f(t,u)dw1(u) = J
Therefore the equality
0
0
0
a f(v,u)dw1(u)}dv av
(6)
will yield the required assertion (4) of the lemma, if we let mesh II + 0
and show that the second integral on the right-hand side of (6) tends to f0 {f0 av
f(v,u)dw(u)}dv in £2 as mesh II
o
To see this consider
0.
a f(v,u)dw(u)}dv12
t {Jv
El fo
-
_§ _V
f(v,u)dwII(u)}dv - ft {Jv 0 0 av
a f0 t Ejjv av f(v,u)dw1(u) - JO
0
WV
f(v,u)dw(u)l 2dv
for each t E [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 n -* 0 if we can prove that the expression 199
a f(v,u)dw1(u)
EIJD
is bounded in v and R.
.
ElJD
J0 a
-
f(v,u)dw(u)l 2
To see this write
f(v,u)dwII(u)
- JO av
< 2EJf0 a f(v,u)dww(u)1 2
f(v,u)dw(u)1 2
+ 2K1
f0 11a f(v,u)l,2du
for all v E [0,a] and some K1 > 0 (depending on m,n,a).
M = sup {11- f(v,u)ll: 0 < u < v < a).
E
I JO
a
k' < k, then
f(v,u)dw1(u) 12
k'-1
tj+1
1
= ElJE1
Let
If tk, < v < tk,+1
A t
ft
v
a
a f(v,u)duN A w + A i
i
k'
a
av
t`Jt
f(v,u)duIAk,wl
k'
k'
<
M2 JE1 A t = M2tk,+1
<
for every partition II of [O,a].
Elworthy ([19], pp. 25-28).
M2a
Here we have again used Lemma 2 of
Thus
fVO
EIJO av f(v,u)dw1(u) -
av f(v,u)dw(u)l 2
for all v E [0,a] and all partitions II of [0,a].
of the lemma.
<
M2a + 2K1M2a
This completes the proof
o
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
ti
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 FDE (XI) suppose H:C -> Rn is continuous
linear and g:R - L(Rm,Rn) is locally integrable. {xt:t E [0,a]) of (XI) through n E C satisfies 200
Then the trajectory
2
t
xt = Tt(n) +
(7)
Tt-u Ag(u)dw(u). fo
t x
= Tt (nu) + fo
t
Tt-u Au g(u)dw(u).
t
x S = Tt(ns) + J 0
(8)
AS Tt-u
(9)
g(u)dw(u).
for all t E [0,a], where A= X{0} I and I E L(Rn) is the identity n x n matrix.
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 9 E 92(52,C(J,Rn);F0).
Proof:
Define the process
We prove the formula first for g:R i L(Rm,Rn) C1.
y:52 x [-r,a] - Rn by Tt(n)(0) + Jt [Tt-u Ag(u)](0)dw(u),
0 < t < a
0
t E J
n(t)
Since gl[0,a] is C1-bounded, a straightforward integration by parts implies that the process t
t -> y(t) - Tt(n)(0) = J0 [Tt-u Ag(u)](0)dw(u) has continuous sample paths and belongs to 92(Q,C([O,a],Rn)).
Thus y has
a.a. sample paths continuous and belongs to 92(S2,C([-r,a],Rn)). Fix t E [O,a] and S E J.
Then
[Tt-u+s
Og(u)](0),
t-u+s > 0
[Tt-u og(u)](s) _ {
0
.
-r < t-u+s < 0.
Hence
201
t+s
[Tt+s-u Ag(u)](O)dw(u),
Tt+s(n)(0) + J0
t+s > 0
yt(s) = y(t+s) = {
n(t+s),
-r < t +s < 0
t
Tt(n)(s) + J0 [Tt_u Ag(u)](s)dw(u)
t+s > 0
-r < t+s < 0
n(t+s), i.e. t
t > 0.
yt = Tt(n) + JO Tt-u Ag(u)dw(u), a.s.,
We prove next that y solves the stochastic FDE (XI).
Defining
f:[O,a] x [0,a] - L(Rm,Rn) by [Tt_u Ag(u)](0)
0 < u < t< a
r
f(t,u)
l
u >t
0
we see that t f(t,u) = H(Tt_u Og(u)) for 0 < u < t< a, and so f satisfies the conditions of Lemma (4.4).
Hence
dy(t) = { d Tt(n)(0)}dt + (Tt-tAg(t))(O)dw(t)
{JO +
71-
[Tt-u Ag(u)](0)dw(u)Jdt t
= H(Tt(n))dt + g(t)dw(t) + {J0 H(Tt_u Ag(u))dw(u)}dt. Now ft 0
Tt-u
Ag(u)dw(u) = yt - Tt(n) E C(J,Rn)
a.s.,
so by Lemma *(4.3),
t dy(t) = H(Tt(n))dt + g(t)dw(t) + H[J
Tt_u Ag(u)dw(u)]dt 0
t
= H[Tt(n) + J
Tt-u Ag(u)dw(u)]dt + g(t)dw(t) 0
= H(yt)dt + g(t)dw(t)
202
0 < t < a
i.e. y is a solution of (XI) in C2(St,C([-r,a],Rn)).
By the uniqueness
theorem (Theorem (11.2.1)), it follows that for a.a. W E 9, y(w) = x(w) in Hence
C([-r,a],Rn).
t
xt = Yt = Tt(n) + JO Tt-u og(u)dw(u),
0.4 t < a,
a.s.
If g is just locally integrable, approximate gI[0,a] by a sequence {gk}0=1 of C1 maps gk_[O,a] - L(Rm,Rn) such that
JO II9(t) - 9k(t) II2 dt - 0 as k -
.
Let xk E C2(St,C([-r,a],Rn)) denote the uni.que solution of the stochastic FDE
dxk(t) = H(xt)dt + gk(t)dw(t), 0 < t < a k
x0 = n for each k > 1.
Then there is a k0 > 0 such that
Suppose c > 0.
II9(t) - gk(t)II2 dt < e
For k > k0, consider
for all k > k0.
Ixk(t+s) - x(t+s)12
Ell xt - xtII 2 < E sup
sEJ t+s>0
t +s
< 2E
sup sEJ t+s>O
I
{H(x k ) - H(x u ))dul
J
2
0
IJt+s[9k(u)
+ 2E
sup sEJ t+s>O
-
9(u)]dw(u)12
0
< 2a IIHII2 JO E IIxu -
<2aIIHII2
112
du + K1 JO II9k(u) - 9(u)II2du
JtEIIxk-x1112 du+K1e 0
where K1 > 0 is some positive constant (Theorem (1.8.5)).
By Gronwall's
lemma, we obtain 203
E Ilxt - xtll2 < Ke
e2a IIHII2 t This implies that xt _->
for all t E [0,a] and all k > k0.
92(SZ,C(J,Rn)), uniformly for t E [0,a].
xt as k
But
0 < t < a, k = 1,2,...
xt = Tt(n) + fo Tt-u Agk(u)dw(u),
We let k -> - to get
t xt = Tt(n) + lim k-'°°
f0
0 < t < a,
Tt-u tgk(u)dw(u),
where the limit exists in E2(S2,C(J Rn)).
EIJO(Tt-u Agk(u))(s)dw(u)
-
Indeed
Jt (Tt-u 09(u))(s)dw(u)I2
< K1 Jt IITt-u[Agk(u) - A9(u)J(s)I12 du C 0 s E J.
< K1 Jt IITt-u A(9k(u) - 9(u))II2 du, 0
ti
Since ITt}ta0 is a strongly continuous semigroup on C, there exist A > 0, and M > 0 such that IITt-ut(9k(u) - 9(u))IIC <
I[A(9k(u) - 9(u))11 C
for 0 < u < t, k = 1,2,... (Hale [26] p. 180).
Hence
EIJO(Tt-u Agk(u) )(s)dw(u) - Jt (Tt-u A9(u))(s)dw(u)I2 t
< KM e2 a J I19k(u) -
9(u)112
du
0
a
<
KIM2e2aa J
119k(u) - 9(u)II2du,
k = 1,2,...,
0
and each s E J.
But the last expression tends to zero as k
t l
in
204
c2
mm
t
f0 (Tt-u Agk(u))(s)dw(u) = JO (Tt-uAg(u))(s)dw(u)
for each s E J.
so
in
Therefore t
xt = Tt(n) + J0 Tt-u og(u)dw(u),
0 < t < a,
a.s., which proves the stochastic variation of parameters formula (7) for locally integrable g. It remains to prove formulae (8) and (9). This is indeed quite simple. U Just apply the projections 11 and to both sides of (7). Indeed
n
t
xt =
lU(xt )
nUTt(n) + ll
=
J0 Tt-u Ag(u)dw(u)
ft
RUTt-u Ag(u)dw(u)
= Tt(nU) + 0
t =
Tt(nU) + J
Tt-u
HU Ag(u)dw(u)
0 f
= Tt(nU) +
J
0
Tt-u
AU g(u)dw(u),
because of Lemmas(4.1) and (4.3).
0 < t < a,
Similarly for (9).
a
Remark (4.2)
By a slight modification of the above argument, it follows that for any t0 E R the unique solution of the stochastic FDE dx(t) = H(xt)dt + g(t)dw(t) ,
t > t0
xo=nEC satisfies
Ag(u)dw(u) xt = Tt-t0 (n) + it t 0Tt-u xt = Tt-t (ns) + Tt-u Jt 0 t0
AS g(u)dw(u)
t
xt = Tt-t
(nu) 0
+ fto Tt -u
Au
g(u)dw(u)
a.s. for all t > to.
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
projections {xt}c, {xt}
to the flow onto the subspaces U and S.
To begin with, recall that U has finite dimension d. possible to think of {xt} (without delay) on Rd.
Therefore it is
as the solution of an unstable stochastic ODE
We make this more precise by appealing to the follow-
ing considerations which are taken from Hale ([26], pp. 173-190).
Define
C* = C([O,r],R"*) where Rn* is the Euclidean space of all n-dimensional row
vectors.
The continuous linear map H:C - Rn defines a continuous bilinear
pairing C* x C - R: a(s-s')du(s')&(s)ds
(a,&) = a(0)&(0) +
(10)
where u is the L(Rn)-valued measure on J representing H, a E C* and C E C. With reference to this bilinear pairing, the generator AH of {Tt} a (formal) adjoint A*E{:D(A*H) c C*
t>O
possesses
C* defined by the relations
(a,AHC) = (A*Ha,E), C E D(AH), a E D(A*H)T
-al(t),
(A*Ha)(t)
0 < t < r
r0
-
a(-s)du(s), t = 0 ;
J r to
D(A*H) = {a:a E C*, a is C1, a'(0) = J -
a(-s)du(s)}. r
Furthermore, a(AH) = a(A*H) and the spectra are discrete consisting only of eigenvalues with finite multiplicities.
Both a(AH) and a(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. = d.
Take a basis 0 =
Then dim U* = dim U
for U and a basis
W1
'Pd
for U* such that (Wj.0i) = 6ji, i,j = 1,2,...,d.
The basis (D of U defines a
unique matrix representation B E L(Rd) of AHIU i.e.
AH.D
= OB, A*H T = BT,
where AHD, OB, A*H'V, B'' are all formally defined like matrix multiplication.. 206
Note that the ei.genvalues of B are precisely those A E a(AH) with ReX > 0. The reader should. also observe here that the splitting (1) of C is realized
by the bilinear pairing (10) through the formula &U
for all E E C.
= IP(Y'oC)
The results quoted in this paragraph are all well-known for linear FDE's and proofs may be found in Hale [26]. ti
We would like to extend formula (1i) so as to cover all E E C.
First
note-that the bilinear pairing (10) extends to a continuous bilinear map C* x
C + R defined by the same formula.
So the right hand side of (11)
ti
makes sense for all E E C.
But both sides of (11) are continuous with
respect to pointwise convergence of uniformly bounded sequences in C, because ti
of the dominated convergence theorem. and the weak continuity of 11 :.C - U. ti
As C is closed under pointwise limits of uniformly bounded sequences, (11) ti
holds for all & E C.
In view of the above considerations we may now state the following corollary of Theorem (4.1).
Corollary (4.1.1):.
Define 4) c U, 'Y c U* and B E L(Rd) as above.
{xt:t E [0,a]} be the trajectory of (XL) through n E C.
Let
Define the process
z:St x [0,a] - Rd on Rd by z(t) = ('Y,xt), 0 < t < a. Then xt = Oz(t) for all t E [O,a] and z is the unique solution of the stochastic ODE: dz(t) = Bz(t)dt + 'Y(0)g(t)dw(t), 0 < t < a 1
I
z(0) = ('',n) Proof:.
Use the definition of z., the stochastic variation of parameters
formula and Lemma (4.3) to obtain t
z(t) = (`Y,Tt(n)) +
(`Y,Tt-uAg(u))dw(u)
a.s.
0
for all t E [O,a].
Take differentials and use properties of the bilinear
Pairing and the generator AH of {Tt}O
ation:
207
dz(t) _ MAH(Tt(n)))dt + (T,Ag(t))dw(t) Jt
(T,Tt_uAg(u))dw(u)dt
+
(Lemma (4.4))
0
= (A*H*v,Tt(n))dt + (W,A)g(t)dw(t)
+ ( t OF.
t-fu WE TAg(u))dw(u))dt
0
= (A*HT,Tt(n))dt + T(O)g(t)dw(t)
+ {f 0
(A*HT'Tt-uAg(u))dw(u)}dt
= (BT,Tt(n))dt + T(O)g(t)dw(t) t
+ (B', to Tt-u Ag(u)dw(u))dt
(Lemma (4.3))
t
Tt-uAg(u)dw(u))dt + T(O)g(t)dw(t)
= (BT,Tt(n) + JO
= (BT,xt)dt + T(O)g(t)dw(t)
= B(',xt)dt + T(O)g(t)dw(t)
= Bz(t)dt + '(0)g(t)dw(t),
0 < t < a.
Thus z is the unique solution of the stochastic ODE (XIV) through (If,n) E Rd.
The relation xt = 'z(t), t E [O,a], follows directly from (11). completes the proof of the corollary.
This
a
The next theorem specifies the asymptotic behaviour as t i projection {xt}t>O of our stochastic FDE (XI).
of the stable
In particular the trajectory
{xt}t>O is asymptotically close to a continuous Gaussian process.
Theorem (4.2):
Let H:C - Rn be continuous linear, g:R -
L(Rm,Rn) C1-bounded
and {xt:t > 0} the trajectory of the stochastic FDE (XI) through n E C.
Then
there is a sample continuous Gaussian process x:12 x [-r,00) - Rn satisfying:
208
t (i)
as. for all t > 0;
Tt_u AS g(u)dw(u)
xt = J
(ii)
There are constants K > 0, a, < 0 such that
E jjxt -
CoxtIIC 2
eat for all t > r;
(iii) °°x is the unique solution of the stochastic FDE
ti d°°x(t) = H(°°xt)dt + &'(0)g(t)dw(t)
t>0
ti
°°x0
=
T_u 'Sg(u)dw(u) E C2(St,C),
O
(X1()
J (iv) (v)
If g is constant, then °°x is stationary; If g is periodic with period k > 0, °°x
is periodic in distribution
with period k i.e.
Po0°x(t)-1 Proof:
Po°°x(t+k)-1 for all t > 0.
=
Suppose g and its derivative g' are both globally bounded on R.
We define the process °°x:0 x [-r,oo) - Rn by ti
t J
AS g(u))(0)dw(u), (Tt-u
t > 0
(12)
x(t) ti
f_00 (T_u AS g(u))(t)dw(u).
t E J
To see that °°x is well-defined, we note the existence of the limit
1t
(Tt-u
ti
a.s.
vv
AS g(u))(0)dw(u) = limn ft (Tt-u AS g(u))(O)dw(u)
Indeed the map u F-->
(Tt_u
AS
g(u))(0) is C1 and so integrating by
parts gives the classical pathwise integral t lv
(Tt-u
AS g(u))(0)dw(u) =
-
A (0)g(t)w(t) - (Tt-v
Jv
{(Tt-u
9(v))(0)w(v)
A g(u))(0)}w(u)du 209
a.s. for all v < t.
Now by Hale ([26], p. 187) there are constants M > 0,
a < 0 such that ti
IITt AS g(u) II < Mest 11g(u) II
(13)
But the law of the interated logarithm for Brownian
for all t a 0, u E R.
motion (Theorem (I.8.2)(iv)) implies that there is a constant L > 0 such that for a.a. W E sl there exists T(w) < 0 with the property
IW(w)(t) I
< L(ItI log log Itl)'
for all t < T(w).
(14)
In particular, we have a.s. ti
liml(Tt-vASg(v))(0)w(v)I
ML IIglIC lim es(t-v)(Ivi
M lim
es(t-v) II9(v)IIIw(v)I
log log Ivl)'
V-),-CO
= ML II911C
eat lim
e-av(Ivl log log IvI)i
v+-oo
where
<MLIIgIIC eat lim
Ivi a-sv=0
I1911C = sup { 119(t) II
. t E R}.
Next we consider the existence of the a.s. limit ft
lira
v-1-
V
a
{(T t_o AS g(u))(0)}w(u)du.
To prove its existence, it Is sufficient to demonstrate that ti
t
f -00 a.s.
{(Tt-u OS g(u))(0))w(u)Idu < co
Note first that
ti u {(Tt-u AS g(u))(0)} = -H(Tt-u ASg(u)) + (Tt-u AS g'(u))(0)
for u < t; so
210
ti
I2u {(Tt-u AS 9(u))(0))w(w)(u)I
IIHII
IITt-u AS g(u) II
M
IIHII C
Iw(w)(u) I ` IITt-u AS g,(U)Illw(w)(u)l
e6(t-u) Iw(w)(u) I + M II9' IIC e6(t-") Iw(w)(u) I
= M( IIHIIIlgI1C + II9` IIC)eat e-sulw(w)(u)l < LM(
IIHII II9II
c
+II9' IIC)
eat
Iul
e-Bu
for all u < min (T(w),-1) and a.a. w E Q.
Since a < 0 the integral
Jiule -Bu du < - and hence so is ft
ti
-,
I'u {(Tt-u AS g(u))(0)}w(u)I du It,
min (T(w),-1)
9(u))(0)}w(u)Idu
I_co
ti
t +
g(u))(0)}w(u)ldu fmin(T(W),-I)
for t > 0.
From the definition of x the reader may easily verify that the trajectory {00
xt}t>0 satisfies assertion (i) of the theorem.
Thus we can look at the
difference 0
(xt - xt)(s) = Tt(ns)(s) +
for all s E J and t > r.
J_
ti
{Tt-u+s AS g(u)}(0)dw(u)
Let At:a X J - Rn be the process
At(s) = JO {Tt-u+s AS g(u)}(0)dw(u), for each fixed t > r.
s E J,
Then integration by parts gives
211
ti
ti AS
g(0)}(0)w(0) - li.m.{Tt-u+/AS g(u)}(o)w(u)
At(s) = {Tt+s
u
ti
0
-lim
v-Ta
ti
C-H(Tt-u+s AS g(u)) + {Tt-u+s AS g`(u)}(0)]w(u)du Jv
From the subsequent discuss ion we shall see that
.s. for t > r, s E J.
At is (F0 0 Borel J, Borel ]Rn)-measurable and has a.a. sample paths continuous on J.
Futhermore, one has the following estimates satisfied a.s.
IA(s) t I < Mes(t+s) IIg(°) II Iw(0) I + M 1 im + M IIHII
li.m.
+ M lim
°
J°
es(t-u+s)
v es(t-u+s)
fv
es(t-r)
+ M IIHII IIg II C
+ 14 IIg' II C
v-*-OD
I
(u)Idu
Mes(t-r) IIg II C 1 im e-Ou Iw(u) I 0
lim
e-Ou IW(u) Idu
J
V-0-00
e0(t-r) lim
for t > r, s E J.
+
v
0
e-su lw(u) Idu
J v
In the above inequalities, all limits exist a.s. because
for a.a. wESt, e-sulw(w)(u)I
Lue-13u
<
for all u < min(T(w),-1). lim
In particular,
e-Ou Iw(u)I = 0
a.s.
u-'-°° and so if we let K (w) = lim 1
V-).-GO
for a.a. W E 9, we get
212
IIg(u) II Iw(u) I
Ilg(u) II Iw(u) Idu
Iig'(u)II
< Mes(t-r) IIg(°) II Iw(0) I
es(t-u+s)
° e-SuIw(w)(u)Idu v
Mes(t-r)
SUP JA t (s)I < sEJ
IIg(O)II Iw(0)I es(t-r)
+ M( IIHII II9IIC + II9' IIC )K1 a.s. for all t > r. Thus a.s. S IIxt - xtII C < Meat II nSllC + sup IAt(s) I CO
SEJ
< MeatllnSllC+ Mes(t-r) IIg(0)II Iw(O)I + M( IIHII II9IIC+ IIg'IlC)K1 eO(t-r)
for t > r. In particular, "xtIIC
lim Ilxt -
a.s.
= 0
tTo establish assertion (ii) of the theorem, note that by cominated convergence and Fubini`'s theorem one gets
EK1= E(J0
e-6uIw(u)kiu)2 <
-00
- 00
_IW
0
(-u)e-Bu du f
= in
e-auEIw(u)I2du Je_81du
J0
JO00
a-Sudu
00
m
e20t E Ilxt xtl2 < 3M2 for t > r.
M2m
ii S112C
-3
(
-By
IIHII II9IIC+ IIg'IIC)2
e20(t-r)
Now take
K = 3M2 IInSII2 - 3m
(IIHII IIgIIC + II9'IIC)2
a-2gr
>0
3
and a = 28 < 0.
Then
E Ilxt - xtii
< Keat for all t
> r. 213
We next show that °°x is the solution of the stochastic FDE (XV) through ti
0
T_u AS g(u)dw(u).
e = J 111-°,
By our main existence theorem.(Theorem (IL.2.1)), the stochastic FDE(XV) will have a unique solution in r2(52,C([-r,a],,n)) for any a > 0 if e E t2(S2,C).
In fact, we can write x in the form ti
AS
Tt(e)(0) + J 0t
I.
g(u)](0)dw(u).
ETt-u
t > 0
°°x(t) =
(15) t E J
e(t)
provided we can show that 0 E 92(S2,C).
Now the stochastic integral depends
measurably on parameters, so the process ti
0
[T
e(s)' = J
_u
s E J.
AS g(u)](s)dw(u),
has a measurable version 0:12 x J - Rn according to a result of Stricker and
Yor ([72], Theorgme 1, §5, p. 119).
Indeed a i.s sample continuous.
To see
this, use integration by parts to write ti
s
9(s)
AS g(u)](0)dw(u)
= J -00 [T_u+s
ti
s
= AS(0)g(s)w(s)-lim
v-p-
A (0)g(s)w(s)
ti
s J
s
ti
Since g, g' and w are continuous, the processes
a.s. for all s E J.
s I---
ti
J [-H(T_u+sOSg(u))+{T-u+sASg'(u)}(0)]w(u)du (16) v
v
ti
AS
[-H(T-u+s
are sample continuous on J.
AS
g(u)) + {T_u+s
g'(u)}(0)]w(u)du
Thus for a to have continuous sample paths one
heeds to check that the a.s. limit on the right-hand side of (16) is uniform Now, for v < s < 0,
for s E J. s
f v [-H(T-u+s
214
ti
ti
AS
AS
g(u)) + {T-u+s
g'(u)}(0)]w(u)du
JS
-
-00
^L
1\,
[-H(T-u+S AS g(u) + T_u+S A 9'(u)]w(u)dul
eB(-'+s) + M Ilg'II, es(-u+s)}Iw(u)Idu
11H
v
< M( IIHII II911C + IIg IIC ) e Sr J
e-au i
w(u) du
(17)
V
Since
lim f e-Sulw(u)Idu = 0, it follows immediately from (17) that 00
the limit in 00 the right-hand side of (16) is uniform in s E J and so 6(w) E C for a.a. w E Q.
As the map J 3 s -> 6(w)(s) E Rn is Borel-measurable for
a.a. W E S2, it follows that 0 E to(c,C).
Indeed, starting from (16), it is
easily seen that
I0(s)2 < 2 II9II 2 Iw(S) I2 a.s. for all s E J.
+ 2142(11.H 111b
II C + II9' II C )2
a-2Br K1
Taking suprema and expectations gives
E sup I0(s)I2 < 2 II9II2CE sup 1w(s)12+ 2M2(IIHII sEJ
SEJ < 2 11911
E sup Iw(s) I2 - 2m M2( IIHII I19I1C + 119' IIc)2 a-20r sEJ s
<00 Hence 6 E c2(52,C).
In fact, by a similar argument, one can show that
00
xt E c2(c,C) for all t > 0.
Note also that this is a consequence of assert-
ion (ii) of the theorem for t > r.
Since {w(t):t E R} is a Gaussian system (§§I.8(A),(B)), the right-hand side of (16) is also a Gaussian system. process (Theorem (L.8.1)(ii)).
Thus 6:52 x J - Rn is a Gaussian
In a similar fashion °°x:S2 x [-r,-) - Rn,
xt:52 x J -Rn are Gaussian for each fixed t > 0. The stochastic FDE (XV) for x may be derived directly from (15) by taking stochastic differentials as in the proof of the stochastic variation of parameters formula (Theorem (4.1)).
Thus ti
d x(t) = {Uf Tt(0)(0))dt + (AS 9(t))(0)dw(t)
215
+
{[Tt-JAS JO
= {H(Tt(e)) +
g(u)](0)dw(u)}dt
J0t H(Tt-u A
g(u))dw(u)}dt + A (O)g(t)dw(t)
t > 0.
= H(°°xt)dt + o (0)g(t)dw(t),
Let us now check assertion (v) of the theorem. with period k > 0.
Consider°°x(t+k) for t > -r.
Suppose g is periodic on R Then a.s.
t+k
"x(t+k)
=
J_00 Tt+ku
_
Jt
g(u)dw(u)
ti Tt_u' OS 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 integral (Lemma (111.2.3)) we get
x(t+k) = J t'O Tt-u' A g(u')dw(u'+k) - t. flt_u.A 9(u')dw(u') = x(t) for every t > -r.
Thus xj[O,oo) is periodic in distribution with period k
x(t)-1
i.e. Po
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 Pa x(t)-1 is independent of t > -r. This completes the proof of the theorem.
Hence °°x is stationary.
o
Corollary (4.2.1): Let H:C - Rn be continuous linear and g:R -). L(Rm,Rn) be C -bounded.
Suppose that the deterministic RFDE
dy(t) = H(yt)dt,
t > 0
(XII)
is globally asymptotically stable i.e. Re x > 0 for every X E a(AH), where
AH is the infinitesimal generator of {Tt}. Then the process
z :St x [-r,00)+ Rn
given by
Jt
CO
xt =
00
Tt-u og(u)dw(u),
t>0
is a sample continuous Gaussian solution of the stochastic FDE 216
dx(t) = H(xt)dt + g(t)dw(t),
t > 0
such that for any trajectory {xt:t > 01 c r2(sl,C) of (XI) there are constants K > 0,
a < 0 with
E JJxt - xt J J 2 < Ke°''t
for al l t > r.
If g is constant, "x is stationary; if g is periodic with period k, x is periodic in distribution with period k.
Proof:
If o(AH) lies to the left of the imaginary axis, the splittings (1)
and (2) reduce to the trivial case
ti
C=S, U={0}, C=S, IIS=idC,IIS=idC. Hence xt = xt for all t > 0; so all conclusions of the corollary follow immediately from the theorem.
Corollary (4.2.2):
o
Suppose all the conditions of Corollary (4.2.1) hold and
let g be constant i.e. g(t) = G E L(Rm,Rn) for all t E R.
Then the stochastic
FDE
dx(t) = H(xt)dt + Gdw(t)
t > 0
(XVI)
is (globally) asymptotically stochastically stable i.e. for every n E C lim
Po(nxt)-1 exists in Mp(C) and is an invariant Gaussian measure for the
t->W
stochastic FDE (XVI), (§(III.3)).
Proof:
As °°x is stationary, let Po("xt)-1 =
show that the transition probabilities
of (XVI) converge to y0 in Mp(C) as t
110 =
POO-1 for all t > 0.
We
Po(nxt)-1:t > 0, n E C} CD for every n E C.
Thus it is
sufficient to prove that l
J
mm
J
EEC
CEC
for every bounded uniformly continuous function 0 on C. But by Corollary 2 = 0 for every ri E C; so it follows from (4.2.1) above, lim EI1TI xt - xt1l
t-
C
217
the proof of Lemma (LLL.3.1)(i.i) that li.m [J
4(Xt(w))dP(w)] = 0,
q(nxt(w))dP(w) - J
in
t-
0 E Cb*
0
i.e.
(E)p(O,n,t,dE) -
lim CJ CEC
0(&)d j(&)] = 0, 0 E Cb. J&EC
Thus lim p(O,n,t,.) = u0, a Gaussian measure on C.
This implies that
t-,
Pt(u0) = vo for all t > 0, where
{Pt}.b0 is the adjoint semigroup on M(C)
associated with the stochastic FDE (XVL) (§(ILL.3)).
a
Remark (4.3)
Note that the invariant measure u0 of Corollary (4.2.2) is uniquely determined independently of all initial conditions n E C (or in 92(c,C)). In contrast with the asymptotically stable case, we finally look at the situation when a(AH) has some elements with positive real parts.
In this
case it turns out that the variance of every one-dimensional projection of nxt in U diverges to + - exponentially as t
for each n E C.
Indeed we
have
Theorem (4.3): (4.1.1).
Define T c U*, B E L(Rd), z:sl x e _ Rd as in Corollary (Rm,Rd).
For the stochastic FDE (XVI) (constant G), let C = '(0)G EL
Suppose {Ai = aj + ib:1 < j < p} is the set of all eigenvalues of B where a. > 0 and bj E R, j = 1,2,...,p.
Denote by
the (Euclidean) inner
product on Rd i.e. for u = (ul,...,ud), v = (v1,...,vd) E Rd, =
E uivi. i=1
(i)
Assume that the pair (B,C) is controllable viz.
(18)
rank (C, BC, B2C,...,Bd-1C) = d.
Then for every v E Rd, there exists 1 < j < p and t0 > 0 such that given e > 0 we can find D1, D2, D3 > 0 with the property
(2a+e)t
2a.t D1e
J
for all t > t0. 218
- D2 < EkI2 < D3e
i
(19)
( i i )
If the rank condition (18) is not satisfied, then either the inequal-
ities (19) hold or else EIl2 = 0 for all t > 0 and any v E Rd.
Proof:
The following argument is taken directly from (Mohammed, Scheutzow,
WeizsScker [61]). Fix t > 0.
Since z satisfies the stochastic ODE
dz.(t) = Bz(t)dt + Cdw(t)
it follows that z(t) =
etBz.(0)
+
t
(t-u)BC fe
dw(u)
0
and
Ez(t) = etBz(O) = etB(T,n),
t > 0.
Let v = (v1,v2,...,vd) E Rd, w = (w1,w2,.... wm).
Then
jt
e(t-u)B Cdw(u)>12
EI12 = EI
d
= E[ E
t
i=1 d = E[ E
0
(e(t-u)BC)ij
t
m
vi
i=1
Now the entry
e(t-u)B
v.(J
of the matrix
E
j=1
(e(t-u)BC)..dwj(u)]2
f
(20)
0 e(t-u)BC
is the i-th component of
the solution of the linear ODE x'(t) = Bx(t)
x(0) _ (C1j,...,Cdj) E Rd
where C = (Ciji = 1,...,d, j = 1,...,m; therefore using (Hirsch and Smale [33], §6.5, Theorem 1) we can write
(e(t-u)BC)1
=
p m a (t-u) E (t-u) k e k [aijksin bk(t-u) + sijkcos bk(t-u)] k=1
for some real constants aijk, Sijk and integers mk > 0, i = 1,2,...,m, k = 1,2,...,p.
i
= 1,2,...,d,
If we insert the above expression into (20) 219
e get 12
EI
mk
Jt p E E = E[ E i=1 j=1 k=1
ak(t-u)
(t-u)
[viaijksin bk(t-u)
e
0
+ vi aijk cos bk(t-u)]dwj(u)]2 mk ak(t-u) d m t p = E{ ,E1 JO kE1(t-u) [, E1 viaijk sin bk(t-u) e
d +
m =
j=1
ak(t-u)
mk
p
rt
E (t-u)
E J
O
a
viaijk cos bk(t-u)]dwj(u)}2
d
viaijk sin bk(t-u)
[ E
k=1
E i=1
i=1
d +
E viaijk cos bk(t-u)]}2du i=1
Now let the functions
because of the independence of w., j = 1,2,...,m.
gjkJ?" -R be defined by d
gjk(u) =
d
E viaijk i=1
sin bku +
E
viaijk cos bku ,
u > 0
i=1
Then m
E
ft
j=1
0
EII2 =
m { E
au gjk(u)}2du
u k e k
d If all
(21)
k=1
d
viaijk = 0 and if for all k with bk # 0 one has E i=1
then EI 0. [79]) this cannot happen if (B,C) is controllable. Now among all k with
E Igjk(u)I2 1 0 for some k. j=1
Z
i=1
viaijk = 0,
According to (Zabczyk
We therefore assume that E
Igjk(u)I2
0 pick
j=1
one for which ak is largest, and if this is not unique take one with the highest exponent mk. gjOkO(u) $ 0.
Call this index k0.
ft
El
12
>
p { E
0
220
Let jO be any index with
Then from (21)
k=1
umk eaku g
(u)}2du.
Ok
(22)
Suppose Y is a positive constant such that I
{u:Ig.Ok0(u)I > y} # 0 and
let Mk, k = 1,2,...,p be constants so that IgjOk(u)I < Mk for all u > 0, all k.
Note that this is possible because each gj k is globally bounded. 0
Then there exists t0 > 0 such that
umk0 eak0u y > 2E Iumk eaku MkI
for all u > tot where the sum is taken over all k # k0 for which
10.
Igj0k(u)I2
Thus p {
k=1 umk eaku
[tO,t]nI
IumkOeakOu YI2du
1
>-
2
0
akOu
{u
k(u)} du >
E
ft 0
>
e
mk0
is periodic.
k
0 0
D1e2ak0t - D
[t0,t]nI
for some constants D1,D2 > 0,.since gj
(u)-EIu g j,
2 du
MkI} du
e
mk aku
2
Hence by (22) we
0 0
have
2ak0t
EI
12
> DIe
- D2
for all t > to.
On the other hand EII
<
p
p E
E
k=1 I=1
p
p
mk+mt (ak+at)u
E
a
E u = j=1 E 1 0 k=1 I=1
mk+mt
t
t
M
2
m
(ak+at)u e
fo u
gjk(u)gjt(u)du
Igjk(u)I I9jt(u)Idu, t> 0.
E
j=1
Defining k0, jO as before and letting
m MkI = sup
{ jE
I9jk(u)I I9jt(u)I
:
u > 01,
=1
we can find u1 > 0 such that 2mk
u
mk+mt
2ak u
0e
O Y>
2
E
u
(ak+at)u
e
MkI
(k,I)
for all u > u1, where summation is taken over all k,t
and Y = max {Mkt:1 < k,I < p}.
k0 with MkI
0,
Therefore 221
2mk0
t
Ell2
< D+
3
u
2ak0u e
Y du
fu l
for some D > 0 and all t > u1.
But for any e > 0, there exists a D > 0
such that
u20 e2ak0u
-
u2 Y du < 3 ft 1
3 It u1
0 e 2ak0u
(2ak0+c)t ydu
0
So
Ell 2 < D3 e
(2ak0+c)t
for all t > 0 and some constant D3 > 0. theorem.
This completes the-proof of the
a.
Remark (4.4)
In the hyperbotic case when no element of a(AH) lies on the imaginary axis, it is easy to see that there are constants M, A > 0 such that lEz(t)I> Melt 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 generalized Ornstein-Uhlenbeck process
(§2), stochastic FDE's with discontinuous initial data (0), stochastic integro-differential equations (§4) and stochastic DE's with infinite delays
(0).
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 following modification of the Ornstein-Uhlenbeck process
dE(t) = v(t)dt t
mdv(t) _ -m[
ft0
(I)
8(t-t')v(t')dt']dt + y(E(t),v(t))dw(t), t > t0.
The molecule is of mass m and moves under no external forces; a represents a frictional (viscosity) coefficient function having compact support, and Y:R3 x 1R3 , 1R is a function giving the random gas forces on the molecule.
The position and velocity of the molecule at time t > t0 are denoted by E(t), v(t) E I R 3 respectively.
Without loss of generality we may assume that
t0 = 0, supp a c [O,r] for some r > 0 and m _ 1.
It is clear that one needs
to specify vIJ, J = [-r,0], for equations (I) to make sense. stands for 3-dimensional Brownian motion.
As usual w
Now (I) is a stochastic FDE.
To
see this note that
223
t
to S(t-t')v(t')dt' =
Jo
J-t
S(-s)vt(s)ds,
t > 0,
ti and define the mappings H:J° x C(J,R3) . R3, H:R> x C(J,R6) --> p0 : C (J ,R3)
i R3 by
R6,
0
- f-t 0(-s)n(s)ds
0 < t < r
0
H(t.n) = I
s(-s)n(s)ds
t > r
pO(n) = n(0)
0 o
H
p0 H
J
(0) for n E C(J,R3).
Setting x(t) _ `v(t)) E R6, t > -r,
w(t) _
E
R6
w(t) it is easily seen that equation (I) is equivalent to the stochastic FDE
dx(t) = H(t,xt)dt + Y(x(t))dw(t), in R6.
t > 0
(II)
Note that this stochastic FDE is time-dependent for 0 < t < r but
becomes autonomous for all t > r.
If U E r1([0,r],R), it follows that H and
ti
H are continuous and Lipschitz in the second variable uniformly with respect ti
to t E R>O.
In fact H(t,.), H(t,.) are continuous linear maps with norms ti
uniformly bounded in t E R>O.
Now xJJ is specified by
v.1J and so from Theorem (11.2.1), (111.1.1) the stochastic FDE (II) has a
unique Markov trajectory {xt}b0 in C(J,R6) with given v!J. cess {xt}b0 is time-homogeneous for t > r.
The Markov pro-
In contrast to the classical
Ornstein-Uh.lenbeck process, observe here that the pair {(fi(t), v(t)):t > -r}
does not correspond to a Markov process on R6, yet the trajectory {(Et,vt): t > 0} 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) _ mYO E R for all x,y E R3.
Then v satisfies the autonomous stochastic FDE
dv(t) = H0(vt)dt + YO dw(t)
224
(III)
for t > r, where HO:C(J,R3) 0 H0(n) = - f-
.
R3 is given by
S(-s)n(s)ds,
n E C(J,R3).
r
By Theorem (VI..4.1), write t
vt = Tt-r(vr) + J
a.s., where {Tt)
r
Tt-u Dy0 dw(u),
t > 2r
is the semigroup of the deterministic linear drift FDE
t>O
t > r.
dy(t) = H0(yt)dt,
HO),
O
Now suppose
S(-s)ds < 7/2r.
f-r
We show that if A E a(A
the spectrum HO
of the generator of {Tt}, then Re A < 0.
Write A = Al + iA2 E a(A
Suppose if possible that Al > 0.
some A1,A2 E R.
), for
Using Hale [26] (Lemma
(2.1), p. 168), A satisfies the characteristic equation A + JO
S(-s)eAsds = 0
(1)
Hence
A1s
fo
+
S(-s)e
-
cos A2s ds = 0
2)
sin A s ds = 0
3)
r
and a1s 2 +
S(-s)e
fo
r
But from (3),
IA2sI < IA2Ir < r 1-r
Is
Isin A1sIds
O 0
< r J-
B(-s)ds < ir/2 r
for all s E J.
Therefore cos a2s > 0 for all s E J and is positive on some
open subinterval of J.
So from (2), Al = - t0r B(-s)exis cos A2s ds < 0
where S is assumed positive on a set of positive Lebesgue measure in J. is a contradiction and Rea
This
must be less than zero for all A E a(AHO). There-
fore, according to Corollary (VI.4.2.1), we obtain
225
In the system (I) assume that y is constant (-myO),S has compact
Theorem (2.1):
support in [O,r], a E f1([O,r],RO) and 0 < fr s(u)du < n/2r.
Then there
is a sample continuous Gaussian solution {( &(t),°°v(t)):t > r} of (I) and
positive real numbers K,a
such that
rt
v(u)du
0%(t) = fi(r) +
(i)
J
t
00
=
vt
J
00
r
Tt-u Ay0
dw(u)
t > r,
a.s.,
for every solution (E,v) of (1),
(ii)
E JI&t -t,2 <
Ke-at
E I I vt - -vt I 2 < Ke-at
for all t > 2r, -v is stationary and E has a.a. sample paths C1.
(iii)
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.
U.
Stochastic FDE's with Discontinuous Initial Data
This is a class of stochastic FDE's with initial process having a.a. sample paths of type t2 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) = H(t,x(t),xt)dt + G(t,x(t),xt)dz(t), t > 0 x(0) = v E r2(s,Rn) s
x(s) = 0(s)
for all s E [-r,0).
In (IV) the initial condition is a pair (v,0) where v E C2(S2,Rn) and 226
9 E C2(c,r2(J,Rn)).
Note that here we confuse t2 with L2, the Hilbert space
of all equivalence classes of (Lebesgue)-square integrable maps J -> Rn.
trajectory of (IV) is then defined as pairs {(x(t),xt):t > 01 in
The
Rnx.C2(J ,Rn).
We assume that the coefficients
H:R>0
x Rn x C2(J,Rn) -+ Rn, G:R0 x Rn x t2(J,Rn) - L(Rm,Rn)
are measurable with the maps H(t,.,.), G(t,.,.) globally Lipschitz on
Rn
x
92(J,Rn) having their Lipschitz constants independent of t E Rte.
The
is a sample continuous martingale on the filtered
noise process z:R x S2
probability space (SZ,F,(Ft)t>o,P) with z(t,.) E E2(S2,Rm;Ft) for all t E R>0
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,o') x 0 - Rn through
(v,o) E C2(S2
n;F0) x C2(S2,L2(J,Rn);FO)
with a continuous trajectory {(x(t),xt):t >01 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 FDE (IV) in the spirit of McShane ([53], Chapter V, §43,4, 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 FDE (IV), suppose the coefficients H,G satisfy the
conditions of existence mentioned above. motion adapted to (Ft)t>O.
Let z = w, m-dimensional Brownian
Then the trajectory {(x(t),xt):t > 01 corresponds
to a Feller process on Rn x C2(J,Rn).
If H, G are autonomous viz
dx(t) = H(x(t),xt)dt + G(x(t),xt)dw(t), t > 0, then the above process is time-homogeneous.
(V)
The transition probabilities
{p(t1,(v,n),t2,.):0 < t1 < t2, v E Rn, n E t2(J,Rn)} are given by
p(t1,(v,n),t2,B) = P{w :w E S2,
where B E Borel {Rn x
c2(J,Rn)}
((v,n)x(.)(t2), (v'n)xt2(w)) E B}
and (v.n)x is the unique solution of (IV)
through (v,n) E Rn x L2(J,Rn) at t = t1.
227
Let Cb = Cb(Rn x t2(J,Rn),R) be the Banach space of all uniformly
(ii)
continuous and bounded real functions on Rn x c2(J,Rn).
Define the semi-
group {Pt}y for the stochastic FDE (V) by
t > 0,
Pt(4)(v.n) = E4((v,n)x(t), (v,n)xt).
0 E C.
Define the shift semigroup St:Cb
Cb, t > 0, by setting
ti
St(4)(v,n) = 4(v,nt), for each 4 E Cb.
t > 0,
The semigroups {Pt}
t>O
and {St }will have the same domain
of strong continuity Cb c Cb (cf. Theorem (IV.2.1)), but it is not clear if Cb # 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 0 E V(S),$ C2, D@ globally bounded and D20 globally bounded and globally Lipschitz.
Then 0 E V(A) and
A(4)(v,n) = S(4)(v,n) + D14(v,n)(H(v,n)) n +
2
D, 4(v,n)(G(v,n)(eG(v,n)(ej))
E
j=1
1 where D0, D20 denote the partial derivatives of 0 with respect to the first
variable and
is any basis for Rn.
Remark
In contrast with the non-Hilbertable Banach space C(J,Rn), the state space
Rn x r2(J,Rn) carries a natural real Hilbert space structure and so Cb(Rn x L2(J,Rn),R) 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(J,Rn) with bounded support must be identically zero.
§4.
Stochastic Integro-Differential Equations
In the stochastic integro-differential equation (SIDE)
228
0
r0
dx(t) = {I
h(s,x(t+r(s)))ds}dt + {J-
r
r
g(s,x(t+d(s)))ds}dz(t),t > 0 (Vt )
x(t) =
t E J = [-r,O]
z:52- C(I O,Rm) is a. continuous Rm-valued martingale on a filtered probability
space (S2,F,(Ft)t>,O,P), satisfying the usual conditions of McShane (Conditions The coefficients h:J x Rn -> R'
E(i) of Chapter II).
g:J x Rn
L(Rm,Rn) are
continuous maps which are globally Lipschi.tz 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 S2-> J are assumed to be (Borel J ® F0, Borel J)-
measurable and the initial condition e E 92(S2,C(J,Rn);FO).
To establish the
existence of a unique solution we shall first cast the stochastic IDE (VI) into the general format of Chapter II. §1.
h .j 2(S2,C)
L2(S2,Rn), g:r2(S2,C)
h(W)(w) = J
Indeed, let us define the maps
C2(S2,L(Rm,Rn))
as follows:
h(s,W(w)(r(s,w)))ds Or 0
ti
9(s,W(w)(d(s,w)))ds
9(W)(w) = J -r
for all T E 92(S2,C), a.a. W E Q.
Observe now that (VI) becomes the stochastic
FDE
dx(t) = h(xt)dt + g(xt)dz(t),
t > 0
x0 = 0.
Note also that the coefficients h, g are globally Lipschitz because if
y,1,W2 E
t2(S2,C), then IIh(T1)-h(W2)
112
2 n t (S2,R )
(0 f
= 1 /-r
{h(s,,U1(w)(r(s,w))) - h(s,W2(w)(r(s,w)))}dsl2dP
r
<
rL2
< rL2
IW1(w)(r(s,w)) - W2(w)(r(s,w))I2dP
II W1 - W2
112
2 t (S2,C)
229
A similar inequality holds for g.
To check that h, g satisfy the adaptability
condition E(iii) of Chapter LL §1, notice that the processes (s,w)l
-:>
h(s,W(w)(r(s,w)), (s,w) -a g(s,W(w)(d(s,w))) are(Borel J ® Ft)-measurable Thus by Theorem (11.2.1) the stochastic
whenever W E t2(St,C;Ft), for t > 0.
IDE (VI.) has a unique sample continuous trajectory {xt:t > 0} in C(J,Rn) through 0.
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
According to Theorem
just (deterministic) continuous functions r,d:J -> J.
(IV.3.2), the weak generator A of the associated semigroup {Pt}
t>O
is given
by the formula 0
A(O)(n) = S(O)(n) + DO(n)
h(s,n(r(s)))ds X{0}) J-r
1
+ -Z
n -2-
0
0
D 4(n)([
E
g(s,n(d(s)))(e.)dsX{0}, f-r g( s,n(d(s)))(ej)dsX{0})
J-r
j=t
where ¢ E Cb = Cb(C,R) satisfies the conditions of Theorem (IV.3.2), and the notation is the same as that of IN §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 (VI) corresponds to a random family of tiarkov processes on C(J,Rn) in very much the same spirit as that of Chapter VI. §3 (Theorem (VI.3.1) for stochastic DDS's). For deterministic delays r, d:J - J and Brownian noise z = w the coeffic-
ients H:C - Rn, G:C H(n) = J O
r
L(Rm)Rn), h(s,n(r(s)))ds, G(n) =
O f-r
g(s,n(d(s)))ds, n E C,
a re clearly globally Lipschitz and so all the regularity properties of
Section V §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.
§5.
Infinite Delays
The problem of determining sufficient conditions for the existence (and 230
uniqueness) of stationary solutions to stochastic FDE's was first considered by K. Ito and M. Nisio ([41], 1964) in the context of an infinite retardation time (r = °°).
We quote here some of their results wi.thout 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((-,O],R) of all real-valued continuous functions (-°°,0] -> R furnished with the metri.c
E 2'" n=1
Iln- Iln
1 + Iln-On
where IIn -EIIn = - supO In(s) - Vs)I,
n,
E C.
Consider the stochastic
FDE
dx(t) = H(xt)dt + G(xt)dw(t),
t > 0.
(VLI)
for a one-dimensional Brownian motion w on a filtered probability space
(S2,F,(Ft)t,.0,P) and continuous coefficients H,G:C -+R. Theorem (5.1) (Ito-Nisio):
Assume the following:
Equation (VLL) has a solution 0x:R x S2-> R such that °x(s) = 0 a.s. for
(i)
all s < 0 and there Is a constant a > 0 with EI°x(t)I4 < a for all t > 0;
(ii) There is a number M > 0 and a finite positive measure p on (--,0] so that J0
IH(n) I + IG(n) I < M +
- In(s) Idu(s)
for all n E C. Then the stochastic FDE (VII) has a stationary solution. In [41] several 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(0) + H1(n),
n E C,
231
for some continuous HO,H1:C + R; (ii)
G is continuous;
(iii) There are constants m, M, M1, M2 > 0 and finite positive measures 111, on (-oo,0] such that for all n E C 112
m < HO(n) < M 0
1H1(n)14 < Ml + J
In(s)14 duff (s) 00
j0
IG(n)14 < M2 +
1n(s)14 du2(s) Go
11111111/4 +3y
1111211 1/2 < m
where 11111 II
=
111(-°°,0],
1111211
= 112(--,oJ,
the stochastic FDE (VII) has a stationary solution.
Theorem (5.3) (Ito-Nisio):
Suppose H, G can be written in the form
H(n) = -H0(n)n(O) + H1(n), G(n) = -G0(n)n(0) + G1(n), Ti E C,
with HO, H1,GO,G1:C - R all continuous and bounded on C.
If there is a
positive constant m (> 0) such that 2H0(n) + IG0(n)12 < -m for all n E C, then the stochastic FDE (VII) has a stationary solution. Now in (VII) assume that the coefficients H, G are linear or rather affine of the form f0
H(n) = M1 + I-
n(s)d111(s)
1
G(n ) = M2 + fo
n(s)du2(s),
n E C,
for Ml, M2 E R, and 111,11 2 finite signed measures on (-°°,0].
denote the total variation measures of u,, 112.
232
Let 11111,
11121
Define the constants c E R,
:1,c2 > 0 by
c = - jump of
at 0, 111
c1 = 11111(-°°,0), C 2
= 11121(--,0].
Then the stochastic FDE (VII) will have a stationary solution in each of the following cases: (i)
c1 +
2 < c, provided 111,
have compact supports; 112
c2)1/2 < C. 2 + 1(c2 + 8c1 In this case the stationary solution tc2 IT is unique among those with sup EIx(t)12 < . ([41] §11 pp. 47-56). tEE (ii) c1 +
Taking the further special case G(n) = 1 for all n E C i.e. M2 =
1 and
112 = 0, Ito and Nisio ([41], §12, pp. 51-56) also proved the existence of
a unique stationary solution to dx(t) = H(xt)dt + dw(t),
t > 0, 0
under the conditions 11 1(-°°,0] < 0 and Jisi dl1111(s) < 1.
Note here that
in the case when 111 has compact support (i.e. a finite retardation time) the
last two conditions imply that the characteristic equation 0
n(A) = A -
eXs d111(s) = 0
J
-00
of H has all its roots to the left of the imaginary axis (Lemma (12.1), p. 54, in [41]).
Therefore this last result of [41] is indeed a special case
of our results in VI §4 viz.
Corollaries (VI.4.2.1), (VI.4.2.2).
Finally, in view of our analysis in Chapter III and the fact that C is a
Polish space, it is tempting to believe that for the general non-linear stochastic FDE (VII) the trajectory field {nxt:t > 0, n E C} also describes a time-homogeneous Markov process on C.
233
References
[1]
Stochastic Functional Differential Equations with Dis-
Ahmed, T.A.,
continuous Initial Data, M.Sc. Thesis, University of Khartoum, Khartoum, Sudan (1983). [2]
Stochastic Differential Equations: Theory and Applications,
Arnold, L.,
John Wi.ley and Sons, Inc., New York (1974). [3]
Banks, H.T.,
The Representation of Solutions of Linear Functional
Differential Equations, J. Differential Equations, 5 (1969), 399-410. [4]
Bellman, R. and Cooke, K.L.,
Differential Difference Equations,
Academic Press, New York-London (1963). [5]
Bismut, J- M.,
A Generalized Formula of It8 and Some Other Properties
of Stochastic Flows, Z. Wahr. verw. Geb., 55 (1981), 331-350. [6]
Bonic, R. and Frampton, J.,
Differentiable Functions on Certain Banach
Spaces, Bull. Amer. Math. Soc. ?1 (1965), 393-395. [7]
A Course in Probability Theory, Academic Press, New York-
Chung, K.L., London (1974).
[8]
Chung, K.L.,
Lectures from Markov Processes to Brownian Motion, Springer-
Verlag, New York-Heidelberg-Berlin (1982). [9]
Cohn, D.L.,
Measurable Choice of Limit Points and the Existence of
Separable and Measurable Processes, Z. Wahr. verw. Gob. 22 (1972), 161165
[10]
Courant, R. and Hilbert, D.,
Methods of Mathematical Physics Vol. 1,
I.nterscience Publishers, Inc., New York (1953). [11] [12]
Dieudonn6, J.A., Foundations of Modern Analysis, Academic Press (1960). Doss, H.,
Liens entre equations differentielles stochastiques et
ordi.naires, Ann. Inst. Henri Poinearee, Vol. XIII, no0 2 (1977), 99-125. [13]
Dudley, R.M., Sample Functions of the Gaussian Process, Ann. Prob. 1
[14]
Dudley, R.M.,
(1973), 66-103.
The Sizes of Compact Subsets of Hilbert Space and
Continuity of Gaussian Processes, J. Functional Analysis 1 (1967), 290330.
234
[15]
Dunford, N. and Schwartz, J.T.,
Linear Operators, Part I: General
Theory, Intersci.ence Publishers, New York (1958). [16]
Dynkin, E.B.
Markou Processes Vols. I, II, Springer-Verlag, Berlin
(1965). [17]
Investigations on the Theory of Brownian Movement,
Einstein, A.,
Methuen, London (1926). [18]
Introduction to the Theory of Differential Equations
El'sgol'tz., L.E.,
with Deviating Arguments, Holden-Day, Inc. (1966). [19]
Stochastic Differential Equations on Manifolds,
Elworthy, K.D.,
LMS Lecture Note Series 70, Cambridge University Press, Cambridge (1982). [20]
Feldman, J.,
Sets of Boundedness and Continuity for the Canonical
Normal Process, Proc. Sixth Berkeley Symp. Math. Statist. Prob. 2,
University of California Press (1971), 357-368. [21]
Fernique, X.,
R6gularitd de Processus Gaussiens, Invent. Math. 12
(1971), 304-320. [22]
Friedman, A.,
Stochastic Differential Equations and Applications
vols. 1,2, Academic Press, New York-San Francisco-London (1975). [23]
Garsia, A., Rodemich, E., and Ramsey (Jr.), H.,
A Real Variable Lemma
and the Continuity of Paths of Some Gaussian Processes, Indiana University Math. J., 20 (1970), 565-578. [24]
Gihman, I.I. and Skorohod, A.V.,
Stochastic Differential Equations,
Springer-Verlag, New York (1973). [25]
Halanay, A.,
Differential Equations, Stability, Oscillations, Time-
Lags, Academic Press (1966).
Teoria Calitativa a Ecuatilior Diferent-
iale (Rumanian), Editura Acad. Rep. Populaire Romine (1963). [26]
Hale, J.K.,
Theory of Functional Differential Equations, Springer-
Verlag, New York-Heidelberg-Berlin (1977). [27]
Hale, J.K.,
Linear Functional Differential Equations with Constant
Coefficients, Cont. Diff. Eqns. 2 (1963), 291-319. [28]
Hale, J.K.,
Sufficient Conditions for Stability and Instability of
Autonomous Functional Differential Equations, J. Differential Equations 1 (1965), 452-482. [29]
Hale, J.K. and Meyer, K.R.,
A Class of Functional Differential Equat-
ions of Neutral Type, Mem. Amer. Math. Soc. 76 (1967). [30]
Halmos, P.R.,
Measure Theory, D. Van Nostrand Company, Inc., Toronto-
New York-London (1950). 235
[31]
Hida, T.,
Brownian Motion, Springer-1Lerlag, New York-Heidelberg-
Berlin (1980). [32]
Hirsch, M.W.,
Differential Topology, Graduate Texts in Mathematics
33, Springer (1976). [33]
Hirsch, M.W. and Smale, S.,
Differential Equations, Dynamical Systems
and Linear Algebra, Academic Press, New York-San Francisco-London (1974). [34]
Hoffmann-Jorgensen,
Existence of Measurable Modifications of Stochastic
Processes, Z.. Wahr. verve. Geb. 25, (1973), 205-207.
[35]
Ikeda, N. and Watanabe, S.,
Stochastic Differential Equations and
Diffusion Processes, North Holland-Kodansha, Amsterdam-Tokyo (1981). [36]
Ito, K.,
On Stochastic Differential Equations, Mem. Amer. Math. Soc.
4 (1951). [37]
Ito, K.,
On Stochastic Differential Equations on a Differentiable
Manifold, Nagoya Math. J. 1 (1950), 35-47. [38]
Ito, K.,
On a Formula Concerning Stochastic Differentials, Nagoya
Math. J. 3 (1951), 55-65. [39]
Ito, K.,
Stochastic Integral, Proc. Imp. Acad. Tokyo, 20 (1944),
519-524. [40]
I.to, K. and McKean, H.P.,
Diffusion Processes and Their Sample Paths,
Springer-V.erlag, Berlin (1965). [41]
Ito, K. and Nisio, M.,
On Stationary Solutions of a Stochastic
Differential Equation, J. Math. Kyoto University, 4-1 (1964), 1-75. [42]
Jones, G.S.
Asymptotic Fixed Point Theorems and Periodic Solutions of
Functional Differential Equations, Cont. Diff. Eqns. 2 (1963), 385405 [43]
Krasovskii, N.
Stability of Motion, Moscow (1959).
Translated by
Stanford University Press (1963). [44]
Kubo, R.,
The Fluctuation-Dissipation Theorem and Brownian Motion,
in Many-Body Theory, Edited by R. Kubo, Syokabo and Benjamin (1966), 1-16. [45]
Kunita, H.,
On the Decomposition of Solutions of Stochastic Differen-
tial Equations, Proc. Durham LMS Symposium on Stochastic Integrals (1980), Lecture Notes in Mathematics 851, Springer-Verlag, BerlinHeidelberg-New York (1981), 213-255.
236
[46]
Kunita, H.,
[47]
Lang, S.,
On Backward Stochastic Differential Equations, to appear
in Stochastics (1981).
Differential Manifolds, Addison-Wesley Publishing Company,
I.nc. (1.972).
[48]
Lidskii, E.A.,
Stability of Motions of a System with Random Retarda-
tions, Differentsial'nye Uraveneniya, Vol. 1, No. [49]
Mallet-Paret, J.,
1
(1965), 96-101.
Generic and Qualitative Properties of Retarded
Functional Differential Equations, Meeting Func. Diff. Eqns. Braz. Math. Soc., Sao Carlos, July, 1975. [50]
Mallet-Paret, J.,
Generic Properties of Retarded Functional Differen-
tial Equations, Bull. Amer. Math. Soc. 81 (1975), 750-752. [51]
Malliavin, P.,
Stochastic Calculus of Variation and Hypoelliptic
Operators, Proc. Intern. Synrp. Stoch. Diff. Eqns., Kyoto 1976, Edited
by K. Ito, Wiley, Tokyo-New York (1978), 195-263. [52]
McKean, H.P.
Stochastic Integrals, Academic Press, New York (1969).
[53]
McShane, E.J.
Stochastic Calculus and Stochastic Models, Academic
Press, New York (1974). [54]
Meti.vier, M. and Pellaumail, J.,
Stochastic Integration, Academic
Press, London-New York (1980). [55]
Meyer, P.A.
Un Cours sur Les Integrates Stochastiques, Seminaire de
Probabilitds X, Proceedings 1974-75, edited by P.A. Meyer, Lecture
Notes in Mathematics No. 511, Springer-Verlag, Berlin-Heidelberg-New York (1976). [56]
Mishkis, A.D.,
General Theory of Differential Equations with a
Retarded Argument, Amer. Math. Soc. TransZ. No. 55 (1951). [57]
Mohammed, S.E.A.,
Retarded Functional Differential Equations: A Global
Point of View, Research Notes in Mathematics, 21, Pitman Books Limited, London-San Francisco-Melbourne (1978). [58]
Mohammed, S.E.A.,
Stochastic Functional Differential Equations and
Markov Processes I, II, School of Mathematical Sciences, University of Khartoum, Khartoum, Sudan (1978) (Preprints). [59]
Mohammed, S.E.A.,
Generators of Stochastic Functional Differential
Equations, School of Mathematical Sciences, University of Khartoum, Khartoum, Sudan (1980) (Preprint).
237
[60]
Mohammed, S.E.A.,
The Infinitesimal Generator of a Stochastic Funct-
ional Differential Equation, Proceedings of the Seventh Conference on Ordinary and Partial Differential Equations, Dundee, Scotland, March 29
April 2, 1982; Lecture Notes in Mathematics 964, Springer-Verlag,
Berlin-Heidelberg-New York (1982). [61]
Mohammed, S.E.A.,
Scheutzow, M. and Weizskcker, H.v., Growth of the
Solutions of Stochastic Delay Equations on Certain Subspaces of the State Space (Preprint). [62]
Some Asymptotic Fixed-Point Theorems, Trans. Amer.
Naussbaum, R.,
Math. Soc. 171 (1972), 349-375. [63]
Nussbaum, R.,
Periodic Solutions of Some Non-Linear Autonomous
Functional Differential Equations, Ann. Mat. Pura AppZ. 10 (1974), 263306.
[64]
Oliva, W.M.,
Functional Differential Equations on Compact Manifolds
and an Approximation Theorem, J. Differential Equations 5 (1969), 483496. [65]
Oliva, W.M.,
Functional Differential Equations - Generic Theory, Proc.
Int. Symp. Diff. Eqns. Dyn. Syst.,
Brown University, August 1974;
Dynamical Systems - An International Symposium, vol. 1, Academic Press (1976), 195-209. [66]
Parthasarathy, K.R.,
Probability Measures on Metric Spaces, Academic
Press, New York-London (1967). [67]
Rao, M.M.,
Foundations of Stochastic Analysis, Academic Press, New
York (1981). [68]
Riesz., F. and Sz-Nagy, B.,
Functional Analysis, translated by L.F.
Boron, Frederick Ungar Publishing Co., New York (1978). [69]
Scheutzow, M.,
Qualitative Behaviour of Stochastic Delay Equations
with a Bounded Memory (To appear in Stochastics) (1982). [70]
Scheutzow, M.,
Qualitatives Verhalten der L8sungen von eindimensionalen
michtlinearen stochastischen DifferentialgZeichungen mit Gedlichtnis,
Ph.D. Thesis, Kaiserslautern (1982). [71]
Schwartz, L.,
Radon Measures on Arbitrary Topological Spaces and
Cylindrical Measures, Tata Institute of Fundamental Research, Oxford University Press (1973). [72]
Stricker, C. and Yor, M.,
Calcul Stochastique ddpendant d'un param6tre,
Z. Wahr. verve. Geb., 45 (1978), 109-133. 238
[73]
Stroock, D.N. and Varadhan, S.R.S.,
Multidimensional Diffusion Pro-
cesses, Springer-Verlag, Berlin-Hei.delberg-New York (1979). [74]
On the Gap between Deterministic and Stochastic
Sussman, H.J.,
Ordinary Differential Equations, Ann. Prob., 6, [75]
Treves, F.,
1
(1978), 19-41.
Topological Vector Spaces, Distributions and Kernels,
Academic Press, New York (1967). [76]
Uhlenbeck, G.E. and Ornstein, L.S.,
On the Theory of Brownian Motion,
Physical Review 36 (1930), 823-841. [77]
Yli.nen, K.,
On Vector Bimeasures, Ann. Mat. Pura Appi. (4), 117 (1978),
11.5-1.38.
[78]
Yosida, K.,
Functional Analysis, Springer-Verlag, Berlin-Heidelberg-
New York (1971). [79]
Zabczyk, J.,
Linear Stochastic Systems in Hilbert Spaces: Structural
Properties and Limit Behaviour, Polish Academy of Sciences, Reprint No. 236.
239
Index
Absorbing state
Adapted
Cauchy-Maruyama scheme
111
14, 30
Cp-bounded function
Adjoint semi-group
69
a-Hdlder continuous
norm
98
Characteristic equation
(iii)
function
(i)
49
Chapman-Kolmogorov identity
1.14
Algebraic tensor product
8
Almost sure convergence linearity
Chebyshev's inequality
6
Compact Hausdorff space
190
map
Asymptotic behaviour
1
121
Compactifying version
172
113
Compactness in probability
165
19
(iii), 31
Coefficient process
5
Approximation of stochastic DDE's
227
155
fixed point theorems
236
Complete filtered probability space
stochastic stability
184
Completion of a probability space Condition (D)
Backward stochastic DE's
237
(DA)
(ii)
Banach space
independence
Bilinear pairing
integrable random variable Borel-Cantelli. lemma
m-dimensional
Brownian a-algebra
240
(i), 22
bilinear maps
dependence
99
148
12
(iii)
Feller process function
122
40
Continuous bilinear form
46
169
(ii)
Continuity of the semi-group in probability
22
31
Continuation property of trajectories
(iii)
(iv), 2
motion, one-dimensional
time
4
6
Borel measurable version
Browni.an filtration
7
Configuration space
4
7
7
probability
2
Bochner Integration
Bump function
32
Conditional expectation
4
a-algebra
94
Condition (E)
Banach-space valued random variable
41
(ii)
(ii)
linear functional
2
12
71
28 1
Continuous path
Drift coefficient
(iii)
20, 67
Contraction semi-group 218 rk
Controllable Convergence in
in probability Convolution
RFDE
46
Dunford-Schwartz (D-S)
8, 10
integral
6
(D-S)-u-integrable
5
21
Eigenfunctions
function
92
Eigenvalues
Cylinder sets
92
92
Eigenvalue problem
23
Erratic behaviour Decomposition of solutions
moments
processes
229
a-algebra
170
norm
150
(i)
(i)
Evaluation map Existence
31
Deviating argument
235
30
(iii), 33
Expectation
4
(iii), 117
Differential system
Feller process
(i)
234
difference equations
Filtration
(ii), 20
14
Filtered probability space (iii), 14
70
Differentiation of stochastic
Finite-dimensional distributions
integral with respect to
jump discontinuity
parameters
memory (v)
196
Diffusion coefficient RFDE
First hitting time
(iv)
Dirac measure
151
Forced linear system
77
Formal adjoint
79
236
191
206
Discontinuous initial data (v), 226
Frdchet differentiability
Discrete-time Gaussian system
Frictional coefficient function
Distribution
21
Frobenius condition (Fr)
59
Distributional regularity Domain of strong continuity Dominated convergence Doob's inequality
17
6
(iv)
15
79
Fluctuation-dissipation
46
Direct sum
218
23
space
(iii)
Deterministic delay equation
Diffeomorphism
144
Euclidean inner product
Delayed diffusions
memory map
13
Estimates on higher order
236
(i)
notation
9
99
Covariance
Delay
(iv)
Functional calculus
41
223
(iv), 114
185
(iv), 76
Gateaux differentiable Gaussian distribution
42 21
241
Gaussian process random field
(1)
Its formula
21
random variable system
Ito calculus
21
Ito integral
21
126
27
21
Generalized. eigenspace
Kolmogorov-Bochner theorem
191
223
Ornstei.n-Uhlenbeck process
Generic conditions
Kronecker delta
stochastically stable Globally Lipschitz
146
24
Laplacian
46
Law of the iterated logarithm 117
t2-continuous process
119
Linear drift
Growth of solutions
238
Hausdorff topological space
222
123
144
Linearity in probability
58
Lipschitz coefficients
46
Local uniqueness
Independence of events random variables sub-6-algebras
23
6
(v), 239 24
I.nfi.ni.tesimal generator, strong
r1-semi-norm
5
r2-semi-norm
(iii)
Markov behaviour
31
property
I.nitial path
(ii)
trajectories Martingale
(i)
I.nvariant Di.rac measures
242
69
type noise
Mean
15
26
(ii), 19, 51
46
16
35
McShane belated integral
165
Invariant probability measure
Ito'belated integral
inequality
112
(iii)
19
Initial data
Isonomous
(iv)
p-almost everywhere convergence
process
77
Gaussian measure
143
49
Infinite memory (or delay)
process
(iv)
(iii)
compactifying version
6
6
Indicator function
weak
Locally bounded
28, 190
150
Localization technique Increments of Brownian motion
191
(iv)
stochastic DDE
Hypotheses (A) (M)
maps
223
Hyperbolic case
34
FDE's with white noise
2
23
165
growth condition
Heat bath
147
216
165
Group of diffeomorphisms property
Kolmogorov-Totoki theorem
69
Globally asymptotically stable
147
21
Measure
1
(iii)
31
9
Measure, vector-valued finite
Physical Brownian motion
1
positive signed
path
1
(v), 223
15, 148
Mercer's theorem
2
223
Position
Probability measure
13
Probability space
24
p-essentially bounded Minimal representation
1
(i),
1
Product probability measure
10
98
Progressively measurable Prohorov's theorem
9
Noise
148
16
155
Projective tensor product Neutral FDE's
187
100
Polish space
1
Measurable version
u-measurable
209
Piece-wise linear approximations
1
Measurable space
Mesh
Periodicity in distribution
1
8
235
Pseudo-metric
224
Quasi-tame function
(iv), 105
Random coefficients
33
5
(ii.i)
coefficient Noise process
31
Non-existence of continuous versions
delays
144
locally bounded versions
family of Feller processes 169,
145
measurable linear versions
148
Non-negative definite kernel
167
92
171
field
15
gas forces One-dimensional polynomial
delay equation
infinitesimal generators
189
RFDE
quadratic delay equation One-parameter semi-group
semi-groups
(iv) (iv)
237 185
transition probabilities Regular measure
235
22
in probability
Partition
24
Resolvent
belated
24
Retardation
Cauchy of unity
53
Periodic family of Gaussian measures
165
113
113, 149
78 (v)
Retarded FDE's
24
(v)
Riemann-Stieltjes integral sums
171
2
Regularity properties Parameter space
185
122
retardations
186
Ordinary diffusion coefficient Oscillations
223
35
35
Riemann sum
24
243
Riesz. representation theorem.
Right continuous
It
square-integrable vector-valued
16
14
14
with continuous sample functions. a-additive a-algebra
Stochastic indefinite integral
t
Sample function regularity
(iv)
Semi-martingale
228
68
32 163
71
15
(iv), 20
Strong continuity dual
2
infinitesimal generator
semi-group
78
Submartingale
Signed bimeasure Simple delay
Symmetric kernel
8
continuous
(ii)
Solution process Splitting of C
31,
208
Stationary transition
(1), 167
dynamical system
(i)
(v), 49, 165
Stochastic process a.s. continuous continuous
rk 14
183
Total variation measure norm
3
77 (ii)
Trajectory field
(ii), 111
Transition probabilities
19
14
Uniform Lipschitz constant
14
locally bounded of class
(i)
(ii), 20, 64
235
Tonelli's theorem
Trajectories
3.1
95
2
Time-homogeneous -lags
23
probabilities
Stationary solutions (v), 231
Stochastic DDE
98
Tame function
Tight measure
15
memory map
13 13
Taylor's theorem
165
State space
13
191
Stable projection subspace
positive
36
(ii)
Supremum norm
169
166
17
Successive approximation
12
Simple function
244
194
151
Stricker-Yor lemma
9
measurable version
ODE's
(i), 31
Stopping time
Separable martingale
Slice
RFDE
variation of parameters
143
(ii.)
Semi-variation
Shift
integro-differential equations
(iii)
Semi-group property Semi-flow
24
integral (McShane)
1
14
Uniformly continuous Lipschitz
32
14
26
(iv)
33
(v),
Uniformly tight Uniqueness
155
Wiener process
(iii.), 33
measure
Unstable stochastic ODE subspace
23
206
165
Vague topology
3
Vector fields
118
Velocity process Version
22
224
(iii)
Borel measurable discontinuous smooth
15
145
(111)
non-linear measurable
148
Weak continuity of linear operators
192
property (w1) property (w2)
79
82
Weak convergence in Banach space in Cb
11
77
Weak Derivative
generator
78
(iv)
sequential compactness
Weak * topology
11
3
Weak (or narrow) topology
3
Weak topology on Banach space
11
on Cb
77
Weakly closed
78
Weakly compact map complete
11
12
continuous extensions
79
continuous semi-group
20
dense
(iv.)
measurable White noise
4 (v)
245
