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Kinematics of stochastic motion in hilbert spaceLeszek Gawarecki aa Department of Science and Mathematics , GMI Engineering and ManagementInstitute , 1700 West Third Av, Flint, MI, 48504, U.S.APublished online: 03 Apr 2007.

To cite this article: Leszek Gawarecki (1997) Kinematics of stochastic motion in hilbert space, Stochastic Analysis andApplications, 15:4, 503-526, DOI: 10.1080/07362999708809492

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STOCHASTIC ANALYSIS AND APPLICATIONS, 15(4), 503-526 (1997)

KINEMATICS O F STOCHASTIC MOTION IN HILBERT SP.4CE

Leszek Gawaret ki

GMI Engineering and Management Institute,

Department of Science and 4Iathematlcs

1700 Kest Third Xv Flint. MI 18504, U S A

ABSTRACT

We formulate and prove results for development of Nelson's kinematic theory of stochastic motion in Hilbert space, extend stochastic integral of Metivier and Pellaumail with respect to cylindrical martingales and construct a diffusion from kinematical assumptions using the extended integration.

1. INTRODUCTIOK.

The Ornstein-Uhlenbeck theor?; of Brownian L,Iotion is based 011 New-

tonian mechanics of particle movement arid it employs physical quantities

like velocity. acceleration and force. This way, it offers strict derivation of

a Brownian Motion process, which is motivated on a very intuitive level.

Copyl-iphc O 1997 by Marcel D e k k e ~ . Inc

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504 GAWARECKI

Moreover it gives rise to kinematic theory of a stochastic motion which ,vas

considered by Kelson ([j]) and used for descript,ion of quantum phenomena.

111 this work we study two topics. First. we adopt Nelson's int,uitive ideas

on kinematics of stochastic irmtion t o Hilbert space valued stochastic motion.

T h e results show st,rorig relation between xelson t,ype regularity assumptions

for a diffusion and prope~.ties of DolPans measure of some martingale associ-

a ted with the diffusion Nest we obtain that the Brownian motion process

t h a t arises in this anal!.sis plays similar role as i ts finit,? dimensional coun-

terpart in t h e analysis of finite dimensional stochastic motion. For this, it is

necessary t o modify the stochastic integral of 2Ietivier arid Pellaunlail [4] with

respect to 2-cylindrical martingales. The properties of Dolkans measure are

used here extensiwl~. . which again emphasizes the role of Kelson's conditions.

2. OPERATORS ON HILBERT SPACE

For the reader's convenience we first introduce basic facts about tensor

product of Hilbert spaces and its identification with subclasses of continuous

linear operators on Hilbert space. We will always assume tha t H is a real

separable Hilbert space.

Let us think of H as a unitary space and take H @ H, a unitary space,

with the usual scalar product defined bj. ( h @ g, k @ 1 ) H B H = (h; k ) ( g , 1 ) W .

Now H S 2 , as a tensor product of Hilbert spaces is the ~ornple t~ ion of H @ H

in this usual scalar product and can be identified with the Hilbert space of

Hilbert-Schmidt operators T on H by (Th . g ) ~ = (T, h @ g ) H @ 2 .

For any continuous, linear operator T on H with a n N dimensional range

(N = 1 , 2 , ...), there exist orthonormal bases {e,,),M==,, {fn)El c H such t h a t

Vh E H, T h = xN X n ( h , e n ) ~ f , L , An > 0. n = 1, ..., N . Let us identify such n=l a n operator T with the element XI' Xn(e,, ~ l f , , ) E H @ H and define a norm n=l D

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KINEMATICS OF STOCHASTIC MOTION 505

N N in H 8 H by I / ,&(en f,,)ll~ = IlTlll = I,,=, A,,; where iiTill denotes

the trace class norm of the operator T. T h e Banach space II H is t h e

completion of the unitary space H @ H in t,he norm j / . / I I . Since tlic completion

in the trace class norm of the space of continuous; linear operators 011 H with

finite dimensional ranges is precisely the space of trace class operat,ors on H .

any element of If @, H can be uniqutl!. identified with a trace class operator.

Note that for g % h E H H we have

and in general ) / . I i L ( H ) < 1 1 . / I H % 2 _< 1 1 . , I l . For more details wi. refer to [4].

Identification of the spaces H FI and Ha* with subspaces of t h t space

of linear operators on a Hilbert space H a l l o m t o define symmetric a i d pos-

itive elements of H H and H X 2 . \ I-e say t h a t a n elenlent b E H B, H.

or b E H a 2 , is synmet r ic ( p o s i t i v ~ ) if the associated linear operator is self-

adjoint (positive), tha t is if (611. g)rI = ( 1 1 . 6,g)H. i.e. 6 = 6' wherc~ b* denotes

the adjoint operator.. ((bh? 1 1 ) ~ > 0) Vh. g E H.

Lye will study H-valued stochastic processrs {X,)t,I, I = [0, TI, T > 0.

defined on some probability spacc ( Q . 3. P) and adapted to an increasing

fami;y of a-fields { 3 t ) t E 1 . where Ft c 3. V t t I. For simplicit!. n . ~ a1naj.s

assume that .yo = 0. Let us recall that a stochastic process {St},,, is a n

H-valued martingale with respect to an iricreasi~ig farnily of' n-fit.]& { F t ) t E I

if V t E I, St t L1(R, ,Ft. H) and Vs 5 t . E(+YtlFs) = X , P-a.c~.\\ .c introduce.

as in Nelson (j:]), the following regularity assurriptions on the stocliastic mo-

tion ?it and. mostly using Nelson's techniques, we s tudy thcir constquenccs.

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506 GAWARECKI

( E l ) T h e napping t H S, is continuous from I to L1 (R. H) . D X , =

uous from I to L1 (0, H).

Note. DX, defined in condition (N1) can be interpreted as the mean for-

ward velocity

Il'ith a n ( E l ) process X we will associat,e the following process: Y, =

X t - J; D,Ysds. t E I .

We will introduce one more regularity condition for a process Y ; which

may or may not be associated with an (Ri) process S

- 5,;)"2 ( R 2 ) T h e following limit: u2 ( t ) = l i r n ~ ~ ~ E {

A 3,) exists in

L 1 ( R , Ft% H H ) and t H 1is2, t e c r 2 ( t ) are continuous mappings from I

M'e will now show t h a t the mean forward velocity has a simllar property

as its analogues: the velocity in a physical phenomenon of motion and the

mean forward velocity in stochastic motion in a finite dimensional space. T h e

latter was investigated by Nelson in [5]

Theorem 1 Let {,Yt)t,~ be an (RI) process. Then for any u 5 v with

Proof. Note tha t , by assumption, t ++ Xt and t ++ DdYt are continuous

mappings from I to L1 (S1, H) and so is t ++ DX,ds. Let E > 0 be arbitrary. i' We will prove tha t the following submterval of [u, v] c I is closed:

Denote T, = sup{t E 3). Let > 0 be arbitrary. Then, 36 > 0 such t h a t

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KINEMATICS OF STOCHASTIC MOTION 507

Tm E 1 VT,-6 5 s 5 T,, ~ l X T , , , - X s l / ~ , < 2. and 1 ) / D . L d i l l ~ , < - Hence, for

S Tm

2

s t J, by triangle inequality, l E { X r m - X u } - { D X , d r / F u } l ~ , 5 U

E~ + c(Tm - u ) Therefore T,,, E J To conlplete the proof assume that

T, < v , then 377 > 0. T,,,+q < 7 ' , such that for any 0 < A < rl. ~/E{.YT,+A - E T,+A

xTrn l3u) - E { A D x ~ F u } I I ~ l < and IIADxr. - kn DX,d7 ( / L , <

ii

First estlnlate follows from definition of irican forxia~d drii\atirrc and coil-

tractivity of conditional expectation (Theorern 4. Cliaptcr \ . (11) Second

estlmate IS a consequence of the fundamental theolein for Bochiicr Integral

(Theorem 9. Chapter 11, [ I]) . In 1 iew of contmuity of t h r rnapplng t e D X L

Therefore we obtain a contradiction

Theorem 2 Let { X t ) 2 E I has property ( R l ) , then 1; = X t - DX',ds I S an i' This is an immediate consequence of Theorem 1 Below wr ytate some

relations bettieen the processes I . and a2 Using Theorem 1 for the procesy

xB2 we obtaln

Theorem 3 Let 2' be as zn Tl~eorem 2 and has property ( R2). Let u < 1, .

Corollary 1 Let 2 - and a2 be u5 112 T / L P O T P ~ ~ L 3 Then { j / l ; / / : I } , E r zs ur2 (RI)

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508 G AWARECKI

Proof. M7ith a n element h 2 h E H @I H we associat,ed a trace class

operator b. so t>hat l t r (h @ h ) = ( t r6 / < Ilblll = l ~ h @ hill = i ~ h i / ~ , t,he

inequality bcing valid for anl. t r w c class operator. T h r r e f o r ~

T h e last equality follows froin t h ~ martingale prnpcrty of 1'. Since Y is

a n (R2) process the last expression converges t o zero as A \, 0. Clearly

the mapping i ct Dlll;((i = t r a 2 ( t ) is contiiluous from I to L I (R) . Finally,

the mapping t e j/Y;!($ from I to L1(R) is continuous because E{l ( (Yt / /$ -

lll<ll',!} = E{I l!q@2111 - /ll;M21'il} 5 ~{( j l / t ' ~ - I;X211i) and the mapping

t ++ 1,iB2 is assumed to b~ continuous from I t o L1( l l , H @I1 H).

The last assertion in the Corollary follows from Theorem 1.

Note. One can study processes with t h e property ( R 2 ) modified by replac-

ing the space H B 1 H with H X 2 . In this case Theorem 3 and Corollary 1 hold.

4. RECOVERING NOISE FRO11 STOCHASTIC MOTION. STOCHAS-

T I C INTEGRATION In' HILBERT SPACE

Nelson's idea t o recover the noise from a stochastic motion described by

a process A- was t o compute T i i ; u-'(s)dY,. Under some conditions, the

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KINEMATICS OF STOCHASTIC MOTION 509

process \.Ir turned out to he Brownian motion. ."\so the stochastic ~riotiori S

would satisf)~ the following stochastic integral equation (see Paragraph 11 in

151) X , = xa + / ' D X . ~ S + lt O ( S ) ~ M ; . 0

In our case, the stochastic integral a-'(s)dl: should be understood

as an integral of an operator-valued process with respect to an H--valued

martingale. If a-' existed and were an admissible process for either thc iso-

metric or cylindrical stochastic integral, then we would recover the noise I+\.

However this does not happen and we will provide examples explaining why

neither of the stochastic integrals is a sufficient tool for Nelson's technique.

Stochastic integration in Hilbert and Banach spaces is a subject of the

monograph [4]. iVe recall here the isometric and cylindrical integrals and

introduce a stochastic integral with respect to a ky l ind r i ca l H-rriartingale.

which admits a wider class of processcs as integrauds than the integrals in

[4]. Eventually, wc use our results to give a partial answer to the question of

the role of Brownian motion in stochastic motion in Hilbert space.

4 .1 General .\ssumptions and their Consequences

In what foliows we always assume that the filtration {F~}lEI=[O,~~ satisfies

usual conditions. This means that the filtration is right-continuous (i.e. V t E

I , Ft = n F S and that the probability space (R,.FT. P) is complete and s>t

V t E I, .Fi contains all sets of P-measure zero. which belong to &)

Two processes X and Y are said to be P-equivalent if P ( { w : 3. St(&) #

.4 stochastic process X 1s called cadlag ~f V d i, E the sample path t t+

X t ( w ) is right continuous and has left limits.

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510 GAWARECKI

\17c define M:, the spacc of H-valued, cadlag. square integrable mar-

tingales (i.e. E{1112dTl/2,) < OC) and ident,ify P-equivalent processes. In

the case of H = R we will write M:(R) to avoid possible confusion. The

space M : is a Hilhert space with the scalar product defined by (:I[, lY)M; =

E { ( J I T . A'T)lf}.

Every martingale I ' ~vhich is L , ( R , H) continuous has a cadlag version

by Proposition C-G. Chaptcx 11; in !GI, therefore we have the following.

R e m a r k 1 I/ 1.; = X, - L' DX,ds wzth X an ( R l ) process then there eizsti

a verszon Y' of the process I' (i. e. V t E I , P(x = I.;') = 1 ) which zs cadlag.

Moreover, zf E{ljYT(ji} < x then Y ' E M ; and Y and I." a?? P-equzvalrnt.

In rielv of the last Remark. from now on, we assume that 1; = X 1 -

d ' D l , d s is a cadlag nrartmgale and i f I' is an (R2) process, then 1' E M :

follows from equality (1).

4 2 Doleans Measure of (R2) Elements of M :

First we recall basic definitions and properties of Dolkans measure as in

[A] . Sections 1.15, 2.6 and 14.3.

-4 set A = F x ( s , t ] C 'd x I, where F E 7, is called a p red ic t ab le

rec tangle and the collection of predictable rectangles is denoted here by

R. The a-field generakd by R is called the a-field of p red ic t ab le sets

and denoted by P. A ~t~ochastic process is called p red ic t ab le if it is P

measurable.

Assume tha t { X t ) t E l . X t E LI (R , H), V t E I. For each A = F x ( s , t ] E !I?

define a ( A ) = E{lF(X, - X,)). If cu extends to a a-additive, H-valued mea-

sure on P, then it is called Doldans measu re of process X. The following

results are proved in [4], Sections 2.6 and 14.3. Dow

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KINEMATICS OF STOCHASTIC MOTION 51 1

Let M E M ; , then {llMt//L),,~ has Dolkans measure, which will be

denoted here by cr l l ,q~. .41so, {MF2}t ,I has Dolhans measure, denoted by c u ~ .

with values in the set of positive, symmetric e lemrnt ,~ of H B1 H. Moreover,

a l l ~ l l = t r a M = jaAW1, where 1 . / is the variatioli of a measure.

There exists a unique, up t o al i~ l i equivaknce, predictable H @ , H-valued

process Qni, such that oAi(G) = / Q n ~ d ~ , , ~ ~ . VG' E E . T h e process Q,v G

takes its values in the set of positive, spmrnrtric elements of H H and

~ T Q M ( w , t ) = IIQ.v(~J, ~ ) / ( H @ ~ H = 1 a.e . all.v I .

Now we will see how Nelson's regularity assuniptions interfere with prop-

erties of Dolkans measure

Theorem 4 Let A' be an ( R 1 ) process and I ; = ,I7, - DX,ds be an (R2) it element of M:. Then. with the previous notatzon

(1) There exist jointly F 8 B(1) measurable ~ierszons of D/JYJj$ and v2

Let us further conszder these jointly measurable versions and denote them

b y the same symbols. Also, let us denote by EPaX the expectation wzth respect

t o P @ A, where X zs the Lebesgue measure on I .

(2) The Dole'ans measure a,,ll of the process /1l'/I2 zs absolutely contznuous

wzth respect to P 8 X ~uath the denszty = = d ( P @ A)

EPBX ( t r a 2 /PI (3) Thr Dole'ans measure 01 of the proceas 1.'' zh absolutely contznuous

zuzth respect to P @ X uzth the denszty ' = E ~ , ~ { ~ ~ I P } ~t zs also d ( P 8 A)

absolutely contznuous wzth respect to qj~ j j a n d the denszty Qy satzsJes 0' =

Ql t r g 2 a e P@ X

Proof. ( I ) Note that the mapprng t e uL( t ) IS ~ o n t i n u o u s from I to

L1(!2, H H) and hence, the mapping t H tlaL(t) is continuous from I to

L1 (0) Therefore (1) fcilloas b\ Theorem 1 2 in [2]

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512 GAWARECKI

(2) For predictable rectangles F x (s, t] E T? we have

T h e evprcssions at thc beginning and at the ?rid of the e q u a l ~ t ~ , both

t o nicasurcs oli P and these exteiisio~is ~tgree on R, hence t h ~ are

identical

(3) An analogous equnlit> as in the proof of (2) holds also here and the same

extension drgurlient jieida da) = Epg~{a21P}dlP 8 A)

Yo\\, we have a , << all, 1 1 << P @ X glving the result

For the forthcoming analysis let us make some regularity assumptioris

about t h e process a2

Definition 1 A process a2 z.s regular zf

(1) a2 zs pl-edictable u!zth values zn positzve, self-adjoznt elements of H @ l H.

(2) V(W. t) E R x I a11 ezgenvalues X,(ij, t), n = 1.2 ..., of a2(u. t) U T r strzctly

posztzve.

Thus for a regular process a2. Q(w,t) E R x I, 3{hn)F=1 c H, a n ONB,

such t h a t a2(w,t)(h) = Xn(w,t)(h,hn(u.t))~hn(u,t), Vh E H with n = l

m A,(w, t) > 0, n = 1; 2 ..., In=, Xn(w, t) = jlo2(u, t ) l l l . Also, there exists

the square-root of a2. denoted by a, which is a Hilbert-Schmidt operator,

ojh) = x y = l &(h, hn)Hhnr Vh E H (we will usually drop the dependence

on (w, t)).

T h e generalized inverse of a ([3]), denoted by a- , is defined by a compo-

sition p [ ~ e , ( o ) ] ~ O a-' Pcl(~an(o)), where P [ h . e r ( o ) ] ~ and P c l ( ~ a n ( o ) ) are respec-

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KINEMATICS OF STOCHASTIC MOTION 513

tlvely projections on the orthogonal coniplement of the kernel spate and on

the closure of the range of a and a-' 1s the inverse relatlon to the operator a

Note that because a2 1s regular, c l (Ran(a)) = H Then a- takes the form

~ ( h ) = xy=l &(h, h , ) ~ h , V h E Rnnia) .

4 . 3 Inadequacy of Existing Stochastic Integrals

We will restrict oursel\res to processes with values only in linear operators

on H as it is enough for our purposes. \\'ith a martingale .\I E M : , we can

uniquely associate predictable, H B1 H-valued process Qhf.Csing the usual

identification of Section 2. we obtain that the values of Q.21 are trace class.

self-adjoint, positive operators. Thus there exists the square-root. denoted 1

by QL! which is a Hilbert-Schmidt operator.

The domain of the isometric integral of Metivier and Pellaurnail is con-

tained in the class of processes X with values in (possibly non--continuous)

linear operators on H. with the follo\ving properties:

(1) For every (;. t ) E n x I, the domain D ( X ( d , t ) ) of X ( d . t ) contains

I

(2) For every h E H the H-valued process S o Qi l (h ) 1s predictable I

(3) For every (LL t ) E 0 x I, X ( u . t ) oQ;,(d. t ) is a Hilbert-Schmldt operator

and

It follows from Theorem 2.10: Corollary 2.13 in [3] arid regular-it!. of o2

that a and a- are predictable processes. By Theorem 4 wc h a ~ r (2,. = a2 I 1 ---- Hence. D(a-(* , t ) ) > R a n ( n ) = & I . ( H ) . Moreover, a- o I):- is a tra2 predictable process. so that requirenlents (1) and (2) above are satisfied.

1 1 _ 1 1 However, u- o Q; = --- &pff O a = - P I K e T l a r j ~ = 1s not m

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514 GAWARECKI

a Hilbert-Scliniidt opcrator unless H is finite dimcnsional. Thus D- is not

admissible for the isometric i n ~ g r a l .

Failurr of the isomrtric stochastic integral in Nelson's procedure is due t o

non-esistcricc of the s tandard H - ~ d u e d Dronnian mot,ion. In order t o realize

a Brownian motion process with covariance associated with an identit!. op-

erator on H one has to abandon H-valued processes and consider cylindrical

processes. It is enough for our purposes ro study 2-cylindrical H-martingales,

with H - a separable Hilbert space.

A 2 - c y l i n d r i c a l LY(!! 3 ) - v a l u e d H - r a n d o m e l e m e n t C is a continu-

ous linear mapping from H to L2(R. 3). Ii'e call {i??t)t,I a 2 -cy l indr ica l

H - m a r t i n g a l e , if ~ a c h .ift is a 2-cylindrical, L2(R, 3,)-valurd H-random ele-

ment and V h E H the real valued process {Al?t(h)),,, is a martingale relative

to {Ft}tEf. T h e space of '-cylindrical H-martingales can he identified with

the space L ( H . M:(R)).

For a 2-cylindrical H-mart ingale ,<f. the q u a d r a t i c D o l k a n s f u n c t i o n

d, is a n addi t ive . (H~81H)*-~~slued funct,iori on R defined by (b, dci(F x ( s , t ] ) )

= ~ { l ~ ( : V f , @ ;Gt(b) - .<fs @ A?,(b))} where. for every t t I. ,?4, 9 denotes

the continuous linear mapping from H H into L I ( n , F,) which is the lin-

ear continuous extension of the mapping b = h @ g ++ i a ( h ) ~ x ( ~ ) . Also,

above, b E H Qi H . F E F,, s , t E I , s 5 t . If d, extends to a D-additive

measure on P then the extension is called q u a d r a t i c D o l k a n s measure of

the 2-cylindrical H-mart ingale 6l and will be denoted by a,<,.

A simple condition for the existence of quadratic DolBans measure for a 2-

cylindrical H-mart ingale A? is t h a t for all h E H, ~ ( h ) had a cadlag version

(141). Note t h a t it assures existence of Doleans measure for a 2-cylindrical

martingale associated with a martingale AP E M$ by ~ , ( h ) = (M,, h)H,

Vh E H.

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KINEMATICS OF STOCHASTIC MOTION 515

For a 2-cylindrical H-martingale ill nit11 the quadratic DolC-ans mcasure

C Y ~ of bounded variation (a,/, there exists a process Qi4 wit11 values in the

set of positive elements of (H B1 H ) * (i.e. Q j i f ( h €3 h) > 0, Qh E H ) , such

that for every b E H g1 H the real-valued process (b. Qhi) is measurable

for the JcuMj-completion of the 0-field P, it is defined up to l~ ,~~l-equivalence

and has the property that ( 6 a,,(.l)) = / ( b Qni(yl. t)) I a r f ( d i ; . d f ) Qb t 4

H @ H , A E P

Here is ail important example of a 2 c> lindrical mdrtingale Let us recall

(see Proposition 4.11 in [4]) that the H-valued Brownian inotion LZ' has

The covariance C. being a trace class operator. cannot be an identity on

infinlte dlmenslonal Hllbert space H \ l e saj that a 2-cvlindr~c~l inartingale

{Ct/,ltE1 1s a cylindrical Brownian motion if Q h E H, { 1 i ; ( / 2 ) ) t c l 15 a

Brownian motion and Vh,g E H, t E I , E { I S , ( ~ L ) I ~ / , ( ~ ) } = tC (h y ) where C

is a continuous bilinear form on H x H

Given any continuous bilinear form C on H x H, there exists cylindrical

Browman motion with C as its covariance In the case of C(11 g) = ( h , g ) H .

C is associated with an identity operator I d H and we call the cylindrical

Brownian motion standard

Now we recall definition of cylindrical stochastic integral wi th respect to a

2-cylindrical H-martingale with the quadratic Dolkans measule of bounded

varisiiotl Let S be an elementan process, i e S ( w , 1) = u,l (w. t ) .

where u,, z = 1 , , 7 ~ are continuous, linear operators on H arid { A , } : = , C !f?

We will always assume that l f .4, = F, x ( s f , t ,] . A, = F, x (s, t,] and 2 # j

then (s,, t,] n (s,, t,] = 0 by taking more refined partition of I i f necessary.

We define Q h E H,

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516 GAWARECKI

where u* denotes the adjolnt operator. The integral, ( / X d h j ) , is r 2-

cylindrical H-martingale and for every h E H the real valued square in-

tegrable martingale ( ~ X d f i ) ( h ) € M:(R) has norm given by (see 16.2.2 in

141)

Definition 2 (A) L(k, H ) is the set of processes X wzth the following ,,rap-

erties:

(1) V ( w , t ) E Q x I , X ( w , t ) is a h e a r operator on H with domazn D ( X ( w , t ) )

dense i n H .

(2 ) Denoting b y X * ( w , t ) the adjoznt o f X ( w , t ) , the lznear form < X * ( w , t ) (h )@

X 8 ( w , t ) ( g ) , Q ~ ( w , t ) > has la.c,l-a.e. a unique continuous extension to

H x H ,which results i n a predzctable process.

W e define .%(?li, H ) - the closure i n the sernznorm JV of the class of ele-

mentary processes zn the space t(:G, H ) .

( B ) The unique extension of the zsometric mapping X i (1 X ~ M ) given

b y (Z), from the space of elementary processes into the space of 2-cylindrical

H-rnartzngales, to the isometrzc mappang from A ( M , H ) into the space of 2-

cylindrical H-martingales is called the stochastic integral and is denoted

again b y X i (/ x ~ M ) .

Now we want to take advantage of the fact that the integrator in the

cylindrical stochastic integral, which we consider, is actually a square inte-

grable martingale and even an (R2) process. Our results from Section 4.2 now

become handy. Note that again, this is a consequence of Nelson's regularity

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KINEMATICS OF STOCHASTIC MOTION 5 17

Lemma 1 The Dole'ans measure of a martingale A4 E M : and quadratic

Dole'ans measure of ii4 coinczde as ( H H)*-valued measures o n P.

Proof. Indeed, first note that .14p2 E L1((R, Ft), ( H @1 H) ' ) Thls IS

because ~f T E H H then T ( k @ g) = (Th.g)H extends unlquel! to an

element of ( H g l H) ' 1~1th I(TII(HR,~)- 5 /jT//I -41~0 I I E ~ N A ! 1 f L ( ~ 1 8 g ) =

( ~ l , , h ) , , ( ~ i ~ , g ) ~ = ii, @ X f t ( r l E q ) Hence 31f2 = .<I, 8 !\jt as elements of

L I ( ( R , F i ) , (H H)') Therefore V b E H H , F E 3<, s. t E I . s < t x\e

have.

Kote that a n d IS an H @ , H-lalucd measure and can be treated as an ( H E ,

H)*-valued measure Because ,<1 is a 2-cyllndr~cal martingale associated w t h

iZI E M:, d,,, on the LHS of the above expression extends to aZf, therefole

a , = cii1 as ( H H)*-valued mcasures on P

Now n e explaln how the c \lmdrlcal integral n i th respect to a square

lntegrahle martmgale can be computed uslng onlx the Doleans measure and

the a<soc~ated process Qw In thls case we knon that (I,, = a{, as ( H @ , H)*-

valued measures Thus the opcrntlon of extension to an element of ( H @ , H ) '

and ~ntegratlon are mterchangrable

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518 GAWARECKI

(we denote a n element of H@, H and ~ t s extension t o ( H H ) * by thc szme

symbol)

.Use l a n ; l l ( ~ i ~ ~ ~ ~ . = J C Y A I I ( I ~ B ~ H ) . 5 I N M I = C Y I I M : ~ . where, to avoid confu-

sion, we denoted by 1 , ] ( H ~ , H ) . the variation of an ( H H)* valued mea-

sure. On the other hand if I J T I I ( H ~ , H ) ~ = 0 then, by uniqueness of the

extension, also //TI/1 = 0. This gives tha t if laMI(A) = 0 then lanf l (A) =

- o l ! , ~ f I I ( A ) = 0 and we arrived a t the conclusion that I c L ~ I ( ~ ~ , ~ ) . = r y l l n f l i .

we can choose Q, = --- d r r i ' l Q , b f to be a predictable process dla,,,'

\Z'e summarize, t h a t if in Definition 2 we replace the process Q i , with

Q,,, and the measure I Q , ~ ~ , , ~ with aii~ll t o get the Definition's condition (2)

hold for (X"(w, t ) ( h ) @ X * ( J . t ) (g) , Qb,(w, t ) ) and measure crll~11, i t will not

change the space L(:2.i', H ) . the seminorni

the space of integrable processes .?(,G. H ) together with the stochastic inte-

gral all remain unchanged Thus we can integrate processes from the space

.i(.if, H ) wth respect to an element M E M : in the sense of cylindrical

stochastic integration

4.4 .4n Example Motivating Modification of the Cylindrical Stochastic Inte-

gral

T h e problem of non-admissibility of a- extends t o the cylindrical case

Lemma 2 For an (R2) process I' E M: we h w e a- E L(Y, H).

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KINEMATICS OF STOCHASTIC MOTION 519

Proof. We need to verify conditions (1)-(3) in par t (A) of Definition 2.

Condition (1) is satisfied easily since D(a-) > Runja). For condition (2) . co 1

note t h a t Vh. g E V ( o - ) we have, ( g , a - ( h ) ) ~ = En=, --(h, h , ) ~ ( y : h , , ) ~ = ""XI ( u - ( ~ ) , h ) H . Therefore D(a-) C V ( ( 6 ) ' ) . Xow V(g, h) E V ( o - ) x D(o- )

we obtain tha t

As a consequence of regularity of o2 we obtain.

\ 1 Corollary 2 Let a, (12) = XI,=, ----(hn, l l )Hh, , Then, 0 , E L2( (R x

V'X I . 'P. 0 ~ ~ 1 , I ) , L ( H ) ) c .i(? H)

Proof. /Ve have

since the Doleans measlirp of a square integrable rnartirigale has b o u n d ~ d vari-

ation. T h e assertion fo l low because L2( (Q x I, P. q~-~~); L ( B ) ) C .i(?. H)

in view of Proposition in Paragraph 1G.3 of [4].

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520 GAWARECKI

Now M.P are ready to prove t h a t o- @ .I(?, H). First let us note tha t m

N o - 0 ) = s u p ( (h, fl)'d(~ @ A)); = ~ ( 1 ) ; Thus o, does " I ~=h '+ l . . -

not converge to a- in the seminorm H. . issume that 0- E ij?, H ) , so t h a t therc twsts ri sequenw {Xn):==, of

e le~nentarv plocessPs ~11th &'(I, - a-) + 0 a5 71 + oc Denote by PA,

thc orthogonal projectloll In H on the 5pnn{hl h2? % h h } \Ve h a ~ e the

follonmg

N(X, o Px - a;) =

= Sup {llx, ((x,, 0 p\ - u<.)'(~L) @ (-yn 0 P~v - LJi , )*(h) , I I ~ I I < I tra

x tra2d(p @ A ) }

= sup { / Jh,(h, . ~,(h))ii - Ih,. L)xl2d(p @ A ) } l l l l ! ! ~ l n x 1

Note that xw proved that Ac(.Y, 0 P,v - 0,;) 5 h r ( X n - o-), n. &Y = 1 . 2 . . .

Similarly, we can verify tha t for any 11 = 1. Z.... N(X, o P,\ - S,) + 0

a s Y -t m. Summarizing. ,2i(oi. - 0 - ) 5 ?.ju(S, - a-) + . u ( X n o Pv - X,).

which gives a contradiction wit,h ,U(a,; - a - ) jii 0.

5 EXTENSIOK O F THE CYLINDICAL STOCHASTIC INTEGR-4L AND

.WPLIC.;ZTIOK T O KELSON'S CONSTRUCTION O F DIFFUSION

R l o t ~ v a t ~ o n for further s t u d ~ e s comes from the following Lemrna

Lemma 3 For eurry 11 t H . / ( ( 0 - o-)(h) 8 ( o ~ - o-)(h). Q), ) dov l l n x c

-+ 0, as N -t oo.

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KINEMATICS OF STOCHASTIC MOTION 52 1

Proof. V(w, t ) E n x I . ( t r o 2 ( (ok - a - ) ( h i C ( a & - o - ) (h ) , Q Y ) ) ( ~ . 1 ) =

x z N + l ( h , h , ( d , t )); , i 0 and is bounded by Ihlli independently of (d. t ) .

Now we consider an extensiorl of the cylindrical stochastic integral

Definition 3 Let .I1 be a 2-cylzndrz~al H-~nnrtzrigalc wzth the Dolic17is rnen-

sure of finzte uartatzorl

( A ) Define Lw(?;I, H ) . the set o f p ~ o c e s s r i >(~ t z~ fYzng rondztzons ( 1 ) and (2)

of Part ( A ) of Dcfinttzon 2 and the f o l h znq condztzon

(3) ~h E H , N;(\.) = [/ ( x * ( / L ) .\*(/L) Q,,) rinnill i < 5: R x l

For every h E H I S a s e m z n o r ~ n and 70e say that a sequence {.Y,,)r=, c

L" (M, H ) converqes to S E it' (10 H ) ~f b'h E H . N : ( X , - S ) + O W e

wzll denote thzs convergence b y S,, + .\

( B ) W e denote b y .iujAI?, H ) the closure t n i w ( h ; l . H ) of the class of ele-

mentary processes zn the topology of convergence " =+ " defined zn ( A ) - ( 3 )

For every X E iu(i"?, H ) , h E H ule define ( xdh-I)"(h) as a h z t of / (1 X,dh?)(h) i n M : ( R ) , whpre S, + .Y and .Y, are elementar y processes

W e call ( X d h l ) " E L ( H . M ~ ( R ) ) t h ~ stochastic integral J Note. To justify correctness of Part (B) of the above Definition assume that

{X,)r==, is a sequence of elementary processes, such that X, + .Y. \\'e have

V h E H,

by equality (3). Therefore whenever S, .Y, then V h E H. {.\-,}:=;"=, 1s a

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522 GAWARECKI

Cauchy sequencc for Nl and hcnce. (/ ~ . , d l ? ) ( h ) converges in ,lrl:(R) LO a

square int,egral)le real valued martingale.

Now, the mappings lz -i ( . ~ , d ~ f f ) ( h ) from H to ..2/l:(R) are linear, J continuous and for every h E H there exists a limit, which we denote by

( d ) ( h ) Therefore, by Banach-St,einliaos theorem ( ~ d A ? ) ~ ' ( h ) E / L ( H , M:(R)) - that means ( ~ d f i ) ' " is a 2-cylindrical H-martingale. J

\5'r conclude our considerat,ions on S r l s o n ' s ideas with an analog%ie of

Theorem 11.6 in [5]. . I s we proved in Lemma 3, a; + a- n.it,h n.; E

.I(?', H ) c .iW(?':'. H ) (see Corollary 2 ) . Therefore a - E .iw(f', H ) .

Theorem 5 Let .Y be on (Ri) process and let Yt = X, - lo DX,ds be an

(R2) process. Assunze that o2 is regular. Then there ex& a 2-cylzizdrzcnl

H-martzngale 11.) such that

The above equalzty zs zn the follou~ng smse . Vh E H , (X- D.Y,ds, h ) H = l (/ o d ~ ) , ( h ) zn M;(R) In paiticuhr, od1i7 is a 2-cylzndrzcol H-martzngole i assocaated wzth an o7dznary H-valued martzngale

Proof. We definr I%- = ( a-dp)" Let us first prove that for X E

.i"(?, H) we have

a 2 Recall that by condition (2) of Definition 2 and because Q y = - and

t r d daljvii = tra2d(P 8 A), the following expression: ( X * ( h ) 8 X" ( g ) , u2) =

( X * ( h ) 8 X * ( y ) , Q,) t ra2 is well defined on H x H.

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KINEMATICS OF STOCHASTIC MOTION 523

It IS easy to ~ m f y that equalit\ (4) holds for elementary processes Next,

Therefore con\ergencc .Y, + .Y lrnpl~ci tha t , for c\ci \ ir E N,

E{/; ' ( ~ : ( h ) @ Xd(g) n2) d h ( ~ ~ } i E{/' ( ~ ~ ( h ) @ .Ya(q) n2) d i ~ ~ } in E

L1(R) by cont rx t l \ i t l of corldltio~ial expectation Con~crgent r S, =+ ,Y

m p l i e ~ also that Vh t H (/ I,,& )u \h I i (/ \d?)" (ti) ln l.l;(R) r h c h .

In turn, implies

in Ll(R), V h E H This concludes the proof of (4 ) . Using (1) and the proof

of Lemma 2 we obtain that

To shon th t last assertion of the thrwrem, let us note that a E .%(IT , H)

(for this see the proof of Corollan 2) \ov n e \nil1 prove that for an plen~en-

tar) proces\ .Y,, X,, o a- E iu (y7 H) and

The domain 'D(S,, o u - ) is denst i n H and

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extends uniquely to a continuous billneai form L(S:(~), S: ( g ) ) H Fur- froL

tller. V h t H

Therefore S,, o o E i(i Mj c I,"(?', H) Smcr a E iu(f', H), there

exists a sequence {Z,}z=, of clrmentdri proccsses such that Z,,, a - as

m + c c 30x1 V h e I f ,

But {X, o Z,},"=, is a sequence of elementar) processes, hence Xn o a- E

.Iw ( ? H) Next. n r n.111 prove that ~f A-, + CT 111 .<(\I7, H) then .Yn 0 a- + I d H in

.Iw(?'. H) as n i ~x, C l e a r l ~ Id,y E ,i(J?, H) C 2 iw(p , H) Observe that

1 extends uniquely to a continuous bllinear form on H x H, namely> t o - -((Xf: -

tra2 a ) ( h ) . (X; - a ) ( g ) ) ~ . For every h E H, and for this extension, we have

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KINEMATICS OF STOCHASTIC MOTION 525

Since convergence in 1" implies convergence of stochastic integrals iri

M$(R), we conclude t h a t d h E H. (/ X , o Z,dP)(h) i (/ in oo-df')'(h)

as ni i m and ( X, o ~ - d i ' ) ~ ( h ) i (1 i d ~ d i ' ) ( h ) = (1: h ) ~ as n i x s in M:(R). Equality (5) can be proved as follows:

Because, dh t H, ( / ~ , d l ( - ) ( h ) i ( / o d < l ) ( h ) in Mf(R) , we obtain

( X , - D&ds. h ) x = (I;, i l) , = (1 ~ d f i ' ) ~ ( h ) , in M:(R) This concludes

the proof

0

Last Theorein s tates in particular tha t li7 1s a 2-cylindrical s tandard

Brownian motion, provided l?/,(h) has contmuous sample paths in t for every

h E H The followng Proposition glves some regularltv of the process c?..

Proposition 1 Under the assumptzons of Theorem 5, if S : I t H zs

continuous; thin V h E H the real valued martzngale f i7(h) = ( l o - d ~ ) " ( h )

has P-a.e. co~~tznuous paths.

Proof. Since t H D X , is continuous from I to L , ((1. H) we can choose its

jointly measurable version in ( t . d) (see [2], Theorem 1.2) . For this version,

t i 1' DXSds is continuous from I t o H This, ragrther with continuity of

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526 GAWARECKI

A' : I -+ H gives continuit,y of I' : I + H. Since for an elementary process

the stochastic procpss (/ \ , d ~ ) , ( i i ) = 2 l ~ , [ l i , , t ( u ; ( h ) ) - l : t A t ( u : ( l ~ ) ) ] has r = l

continuous sample paths, then, by choosing S,, * a - . we gr t , t f h E H

( ) ( h ) -+ ( d ) ) in M R ) Refercnip to Lemma 111 P a n -

graph 10.1, [4] completes the proof

This work was supported by ONR Grant. N0001.2-91 -,I 1087

The author would like to thank Professor 1'. hlandrekar for introducing hini

to this problem and for his cont,inuous support and encouragement

REFERENCES

[l] Diestel J . , Uhl J .J . Jr . . 1'ector measures. Mathematical Surveys l5, -4mer- ican Mathematical Society. Providence, Rhodc Island. 1977.

[2] Fcrnlque X , RCgularlte de fonctions aleatoires Gaussiennes statlorindms a valeurs vectorielle Probability Theory on bector Spaces I\ , tanclit 1987, Lecture Kotes in hIath 1391, 1989, 66-73

[3] Mandrekar V. , Salehi H.. The square-integrability of operator-valued functions with respect to a non-negative operator-valued measure and the Kolmogorov isomorphism theorem, Indiana Univ. Math. J . 20. 1970/71, 545-563.

[4] Metivier M., Pellaumail J . , Stochastic integration. Academic Press. New York, 1980.

[5] Nelson E., Dynamical theories of Brownian motion , Mathematical Not,es. University Press, Princeton, New Jersey, 1967.

[6] Pellaumail J., Sur l'integrale stochastique et la dCcomposition de Doob- Meyer, Astirisque 9. SocietC Mathkmatique de France, 1973.

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