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Extension of a stochastic integral with respect to cylindrical martingales
I.rl~lt$ ELSEVIER
Statistics & Probability Letters 34 (1997) 103--III
Extension of a stochastic integral with respect to cylindrical martingales Leszek Gawarecki Deparmwnt qf Science and Mathematics, GMI Engineering and Management Institute, 1700 FE Third Ave., Flint, MI 48504, USA Rcccived Junc 1996; revised September 1996
Abstract
We extend stochastic integral of Metivier and Pellaumail with respect to cylindrical martingales which is necessary for constructing a Hilbert space-valued diffusion based on Nelson's kinematic theory of stochastic motion. Examples for inadequacy of existing stochastic integrals are provided.
Keyword~v Stochastic integral; Cylindrical martingales
I. Introduction
Given a real-valued stochastic process {Xt}t~l, I = [0, T] satisfying some regularity assumptions, Nelson (1967) constructed a brownian motion Wt and a stochastic differential equation, with the process Xt being its solution,
Xr =
f0' DX,. ds + f0t as d Ws.
The drift DXt c a n be interpreted as the mean forward velocity of Xt,
The process Yt =Xt - f o D X s d s
is a martingale and if the Ll(g2) limit
exists and satisfies some smoothness conditions, the process Wt = ./o a~-I dY~ is a brownian motion. In order to carry out the outlined program for Hilbert space-valued processes one can introduce a martingale in an analogous way as above, and needs to define a stochastic integral with respect to Yt for a sufficiently large class of processes a / 1 . The purpose of this paper is to provide examples explaining why the existing 0167-7152/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved PII $ 0 1 6 7 - 7 1 5 2 ( 9 6 ) 0 0 1 7 1 - X
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L. Gawarecki/ Statistics & Probability Letters 34 (1997) 103-111
stochastic integrals are not sufficient to support Nelson's technique and to introduce an extension of a stochastic integral with respect to cylindrical martingales of Metivier and Pellaumail. This extended integral will admit reasonable class of integrants.
2. Regularity assumptions and their consequences In what follows we assume that the filtration {~t}te; on (O,.~-,P) satisfies usual conditions. Denote the a-field of predictable sets of I2 by ~ and the collection of predictable rectangles by N. Let H be a separable Hilbert space and denote H ®2 H, and H ® H1 the Hilbert and Banach space of Hilbert-Schmidt and trace class operators on H. With an element x ® y E H ® H we associate an operator T : H ~ H by (Th, 9)n = (x, h)n(y, 9)H. Let us denote by (jt'2(H), (.,-)) the Hilbert space of H-valued square integrable martingales (i.e. E{ IIMr 112} < oG) and identify P-equivalent processes. We assume the following regularity assumptions on Y E,grZ(H) and call a process satisfying these assumptions reoular.
(A) a2(t)= lim E { (Y'+A - Yt)®2"~t exists in Ll(f2,~t,H ®t H) and t ~ y~2, t ~ o2(t) are continuous mappings from [0, T] to L1(f2,H®IH). Thus, a 2 takes values in positive, self-adjoined elements of H ®l H. (B- 1) a 2 is predictable. (B-2) The eigenvalues of a 2, {~.n(co,t)}~= l > 0. Now we will see how the regularity assumptions interfere with properties of the Dol~ans measure of the martingale Y. Theorem 1. Let {Yt}t~; be reoular. Then the following ham true: (1) There exists a jointly ,~ ® ~ ( I ) measurable version of a 2. Let us further consider this jointly measurable version and denote it by the same symbol. Also, let Ep<¢). denote the expectation with respect to P ® 2, with 2 denotin9 the Lebesgue measure on [0, T]. (2) The Dolbans measure ~llrl] of the process IIYl[2 is absolutely continuous with respect to P@ 2 with the density d~llrll/d(e ® 2) = tr a'. (3) The Dolkans measure ~v of the process y®2 is absolutely continuous with respect to P ® 2 with the density d~r/d(P ® 2 ) = a 2. It is also absolutely continuous with respect to all Eli with the density Qr = a2/tr o 2 a.e. P®)..
Note: It is the condition (A) that guarantees ( 1 ) - ( 3 ) with densities given by conditional expectations of the relevant expressions with respect to the predictable a-field ~ and the measure P ® 2. Proof. Statement (1) follows because the mapping t ~ a2(t) is continuous from [0, T] to L1(12,H ®1 H). To show (2) note that for predictable rectangles F x (s,t]EN we have OqlVll(F x (s,t]):E{1F([IYtII2H --IIY~I[~)} : E
= fF
×(s.t]
tr a2d(p ® 2),
{ft lF
tr a2(r)dr
}
L. Gawarecki I Statistics & Probability Letters 34 (1997) 103 111
105
(the proof of the second equality follows the same line as the proof of the corresponding statement in Nelson (1967)). The first and the last expression, both extend to measures on ~ and these extensions agree on predictable rectangles, hence they are identical. Finally, an analogous equality and extension argument as in the proof of (2) holds also for measures xr and P®)~, yielding d~r = a z d ( P ® 2 ) . Moreover, by Metivier and Pellaumail (1980), ~r ~ ~IIYI. Now, we have ay ,~ ~l',Vll '~ P®J- and (3) follows. U A process 0.2 satisfying conditions (B-l) and (B-2) has a form o.2(¢o,t)(h)=~n°°=12,(~o,t) (h,h,(oo, t))Hh.(ev, t),Vh EH with { h . } . ~ C H , an ONB, 2.(¢o,t)>0, n = 1,2 .... y~°°__~2 . ( o , t ) = No.2(co,t)[l~. Also, there exists the square root of 0.2, denoted by a, which is a Hilbert-Schmidt operator, o.(h) = ~=1 v/~. (h,h,)Hh., Vh E H (we will usually drop the dependence on (og, t)). The generalized inverse of a, denoted by 0.-, is defined as a composition P[Ker(,r)l-C o 0"-I OPcl(Ran(rr)), where P[Ker(a)]~ and Pcl(Ran(a)) are, respectively, projections on the orthogonal complement of the kernel space and on the closure of the range of a, and o.-l is the inverse relation to the operator a. Because a 2 is regular, cl(Ran(a)) = H. Then a - takes the form a-(h)= ~ - - l ( 1 / v / ~ - . ) ( h,h~)Hh~ Vh E Ran(o.). Both processes a and a - are predictable (see Mandrekar-Salehi, 1970/1971 for details).
3. Inadequacy of existing stochastic integrals
3.1. The isometric integral With a martingale M E ./[2(H), we can uniquely associate a predictable, H fgl H-valued process QM satisfying Ctu(G)= Ja QM dCtllMIi for any predictable set G, with ~tM, ~tllMii denoting the Dolrans measure of M ~2 and IIMII~, respectively. The values of QM are trace class, self-adjoint, positive operators. A process X in the domain of the isometric integral of Metivier and Pellaumail has values in (possibly non-continuous) linear operators on H. The domain ~(X(og, t)), 'v'(og,t ) E O x [0, T], of X(~z~,t) must contain Q~2(H) and X(tn, t ) o Q~1/2 (~o,t) must be a Hilbert-Schmidt operator. By Theorem 1 we have Qy=o.2/tro. 2, however, 0 . - o /-) ~ ,1/ 2 = ( 1 / ~ ) o . - o o . = ( l / ~ ) P i K ~ ( o ) ] ~ = (l/t~ra2)IdH is not a Hilbert-Schmidt operator unless H is finite dimensional. Thus a - is not admissible for the isometric integral.
3.2. The cylindrical inteyral Failure of the isometric stochastic integral in Nelson's procedure is due to the non-existence of the standard H-valued brownian motion. In order to realize a brownian motion process with covariance associated with an identity operator on H one has to abandon H-valued processes and consider cylindrical processes. We call {/14t}t~/ a 2-cylindrical H-martin9ale, if each ?Qt is a continuous linear mapping from H to L2(f2, ~ ) and Vh E H the real-valued p r o c e s s {~/[t(h)}tEI is a martingale relative to {~tt}/el. The space of 2-cylindrical H-martingales can be identified with the space L(H,.//2(R)). For a 2-cylindrical H-martingale/k~t, the quadratic Dolkans function dpd is an additive, (H ®1 H)*-valued function on predictable rectangles ~ defined by (b,d~(F x ( s , t ] ) ) = E { IF(l~Tlt @ f / I t ( b ) - f 4 s ®/~ts(b))} where, for every t E [0, T], ftt®lC4t denotes the continuous linear mapping from H ® l H into Ll(t2,.~) which is the linear continuous extension of the mapping b=h®g~--~Mt(h)Mt(g). Also, above, bEH®1H, F E ,~, s, t E [0, T], s ~
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L. Gawarecki I Statistics & Probability Letters 34 (1997) 103-111
If for all h E H, M(h) has a cadlag version, then the Quadratic Dolrans measure exists (see Metivier and Pellaumail, 1980). This happens for a 2-cylindrical martingale associated with a martingale M E~g2(H) by ]t~t(h)=(Mt,h)H , ~/hEH.
For a 2-cylindrical H-martingale/14 with the quadratic Dol~ans measure 0tM of bounded variation [x~?[, there exists a process Q~ with values in the set of positive elements o f ( H ®1H)* (i.e. Q~(h ®h)>lO, Vh E H), such that for every b E H ®1 H the real-valued process (b, QR) is measurable for the [~t~[-completion of the a-field 2 , it is defined up to [~?[-equivalence and has the property that (b,~(A))=f~(b,Q~(to, t))[~[ (dco, dt)Vb E
H ®IH, A E ~ . An example of a 2-cylindrical martingale is a cylindrical brownian motion {Wt}t~, satisfying Vh E H, {Wt(h)}te~ is a brownian motion and Vh, g E H, t E I, E { ~ ( h ) ~ ( g ) } =tC(h,g), where C is a continuous bilinear form on H x H (in particular C can be the identity operator on H). Now we recall definition of a cylindrical stochastic integral with respect to a 2-cylindrical H-martingale with the quadratic Doi6ans measure of bounded variation. For an elementary process, X(w,t)= ~ = l uil~,(w,t), (here ui, i= 1..... n are continuous, linear operators on H and {Ai}n= 1 C ~ ) , define VhEH, (1) i=I
where u* denotes the adjoint operator. The integral ( f X d f 4 ) is a 2-cylindrical H-martingale and for every h E H the real-valued square integrable martingale (fXdl~)(h)E .¢¢~-(R) has norm given by (see Metivier and Pellaumail, 1980, 16.2.2)
(Sxd~)(h) 2~/(R) = f(X*(h)®X*(h),Q~)dl~l.
(2)
Definition 1. (A) £(M,H) is the set of processes X with the following properties: (1) V(co,t)E f2 x I, X(to, t) is a linear operator on H with domain ~(X(co, t)) dense in H. (2) Denoting by X*(o~,t) the adjoint operator to X(og, t), the linear form (X*(og, t)(h)®X*(o~,t)(g), Q;t(og, t)) has [~,l-a.e. a unique continuous extension to H x H which results in a predictable process. (3) JV(X) = suPllhll< 1{f~×l (X*(h) ®X*(h), Q~?(og,t ) ) d l ~ [ } 1/2 < co. We define A(,Q,H), the closure in the seminorm ./V of the class of elementary processes in the space £(,~,H). (B) The unique extension of the isometric mapping X ~ ( f X d l Q ) given by (1), from the space of elementary processes into the space of 2-cylindrical H-martingales, to the isometric mapping from A(M,H) into the space of 2-cylindrical H-martingales is called the stochastic integral and is denoted again by X
(f X d~). Now we want to take advantage of the fact that the integrator in the cylindrical stochastic integral, which we consider, is actually a regular element of .g2(H). Note that again, this is a consequence of Nelson's regularity assumptions on a stochastic motion. Observe that any T E H ® I H extends uniquely to an element of (H@I H)* by T(h ® g)=(Th, g)H, with [ITIl(u®, H)* ~ IITII1. Therefore ~t~ =CtM as (H®I H)*-valued measures on 2 . Also the variation of ctM and the measure ~llgll are equivalent. Indeed, I~l~<[ctnl=~llnll, as well as 1~1(.4)=0 implies I~,Ml(Z)=~llml(A)=0. We can choose QM = (d~llMiI/d[ct;t[)QM to be a predictable process associated with/~t and this allows us to replace Q~ with QM and the measure I~1 with Cqlnll to get the condition (2) in the above definition hold for
L. Gawarecki / Statistics & Probability Letters 34 (1997) 103-111
107
(X*(co, t)(h)®X*(og, t)(g), QM(O, t)) and measure CqIM[I. This will not change the space L(/kC,H). Moreover, the seminorm •At(X) = sup { f t 2 ilhll~
3.3. Non-admissibility of aWe first verify that a - E £ ( I ~ , H ) . Condition (1) in Definition 1 holds, since ~ ( a - ) D R a n ( a ) . Equality (9,a-(h))n =Y]n~l (l/v~n)(h, hn)n (g, h. )n = ( a - (g), h)H, Vh, g E @(a- ) implies that ~ ( a - ) C ~ ( ( a - )* ). Now 'v'(g, h) C ~ ( a - ) x ~ ( a - ),
(g,h)H
H
((a-)*(g)®(a-)*(h),Qy) 1
=
(
:)
a - ( g ) ® a - ( h ) , t--~-fia2
1
-- tr a2(a2(a-(g)),a-(h))H = t-i : (g,h ). extends continuously to H x H. This extension is predictable in view of predictability of a 2. Finally, .4:(a-)2=
sup !lhll
IlhllHt--~--a~a2tra2d(P®2)
justifying (3). Define a ~ ( h ) = Y~=i ~j.(h~,h)Hhn. We have f~x, Ila'~' II~'H)d~llrll ~< f o x , (1 +tra2)d(P®2) < oo. Thus, aN EL2((f2 x I,~,~IIrlI);L(H))CA(Y,H) (see paragraph 16.3 in Metivier and Pellaumail, 1980). Now we are ready to prove that a - ~/I(17,H). First, note that .Ar(a~ - a - ) = suPllhll~l(fox I Y~i~N+l (hi, h)2d(p ® 2))b'2 = 2(I) 1/2. Thus aN does not converge to a - in the seminorm W. Assume that a - E/i(IT,H), so that there exists a sequence {X,,},= o0 l of elementary processes with lim,~o~ ,#" (X, - a - ) - - * 0. Denote by PN the orthogonal projection of H on the span{hi,h2 ..... hu}. We have the following: •.U(Xn o PN -- aN )
X.
= sup Ilhll
)*(h) ~
[~q(h.Xn*(h)) H -
= sup llhll~<]
o PN -- a N
(X n o
PN -- a;)*(h), ~
×1
xl i=l
,,,
® 2)
}
x tro'2d(P ® 2)
L. Gawarecki / Statistics & Probability Letters 34 (1997) 103-111
108
~< sup {foxl((Xn-~r-)*(h)®(Xn-cr-)*(h),Qy)dOtllYl.} Ilhll~l
=~V'(X~-cr-)~0
asn~.
We have proved that ~4r(X~ o PN - a7¢ ) ~ JV'(X~ - a - ), n, N = 1,2... Similarly, we verify that for any n = 1,2 . . . . . W'(X~oP,v-X~) ---, 0 as N ~ oo. Summarizing, X(a,~ - ¢ r - ) ~<2.A/'(Xn - or- ) + JV'(X~ o PN - X~), which gives a contradiction with JV'(a~ - or- ) 74 0.
4. Extension of the cylindrical stochastic integral and application to Nelson's construction of diffusion Motivation for further studies comes from the following lemma.
Lemma 1. For every hEH, fa×l((a~ - a- )(h) ® (a N - o'-)(h),Qy)dCqlVl I ~ 0, as N --+ co. Proof. V(og,t) E f2 x I, (tr O'2((O'N -- 0"- )(h) ~ bounded by Ilhll~ independently of (og, t). 5
(0" N --
0"- )(h), Qr))(og, t) = ~i°¢__N+t(h, hi(og, t))~ ~ 0 and is
Now we consider an extension of the cylindrical stochastic integral.
Definition 2. Let M be a 2-cylindrical H-martingale with the Dol6ans measure of finite variation. (A) Define Lw(~Q,H), the set of processes satisfying conditions (1) and (2) of Part (A) of Definition 1 and the following condition: (3) Vh EH, JV'~"(X) = [fa×z(X*(h) ®X*(h),Q;t)dl~;tl] 1/z < oc. For every h E H, "/~w h is a seminorm and we say that a sequence {X,},~ 1 c £ w ( , Q , H ) converges to X E £w(l(l,H) if VhEH, .W'~(X, -X)---. O. We will denote this convergence by Xn ~ X. (B) We denote by ./iw(&¢,H ) the closure in/[w(~, H ) of the class of elementary processes in the topology of convergence "=¢," defined in (A)-(3). For every XEflW(f4,H), h E H we define (fXdM)W(h) as the limit of ( f X , dF4)(h) in ~¢r2(R), where X, ~ X and X, are elementary processes. We call ( f x diQ)w E L(H, .At'r2(R)) the stochastic integral. oo is a sequence of Note. To justify correctness of Part (B) of the above definition assume that {X,~}~=l elementary processes, such that X~ =~ X. We have Vh E H,
=
= y~(x~
- x,.)
by equality (2). Therefore, whenever X, =~X, then Vh E H, {An }~=1 is a Cauchy sequence for ,/V~' and hence ( f x n d~4)(h) converges in Jg2(R) to a square integrable real-valued martingale. Now, the mappings h ~ (fXn d~t)(h) from H to Jt'2(R) are linear, continuous and for every h E H there exists a limit, which we denote by ( f X dffI)W(h). Therefore, by Banach-Steinhaus theorem (fXdA]t)w(.)E L(H,,g2(R)) - that means ( f X d l ( I ) w is a 2-cylindrical H-martingale. As we proved in Lemma 1, a~ =~ or- with a~ E/i(17,H)C/iW(I~,H). Therefore ¢r-E/iw(17,H).
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L. Gawarecki / Statistics & Probability Letters 34 (1997) 103-111
Theorem 2. Assume that Y E JI~(H) is regular and the associated process a 2 is also regular. Then if'= (fa-dl~) w is a 2-cylindrical H-martin#ale. such that E{(~(h) - ~ ( h ) ) ( ~ ( g ) - ~ ( g ) ) [ ~ } = (t - s)(h,g)H
Vh, gEH.
Proof. Let us first prove that for X E J W ( ) ' , H ) we have
=E
(X*(h)@X*(O),~2(r))dro~.
,
VhEH.
Recall that by condition (2) of Definition 1 and because Qr =a2/tra 2 and dallrll =tr~2d(P ® ,~), the following expression: (X*(h)®,X*(g),a 2) = ( X * ( h ) ® X * ( g ) , Q r ) t r a 2 is well defined on H x H. It is easy to verify that equality (3) holds for elementary processes. Next, E
[(aoX*(h),~oX*(g))n -(aoX*(h),aoX*(9))nld2
<~ (E { / ' liaro ( X . - X)* (h)ll~d2 }),/2 (E { f ' ],a o X* (9)111 d2 })1/2 + (E { / / I l a o (X.-X)*(g)ll2nd2}) '/2 (E { / ' .,~.~"(x,
-
IIGoX*(h)l,,~d,~})~/2
x),Gw(x) + ~ w ( x . - x ) ~ w ( x . ).
Therefore convergence Xn :::vX implies that, for every h E H, E{fts(X*(h)®X*(g), (r2)d2l~s } -* E{fts (X*(h) ® X*(g),a:)d2[.~} in Ll(12) by contractivity of conditional expectation. Convergence X, ~ X implies also that, Vh EH, ( f X , dlT)W(h)--* ( f X d)~)W(h) in ~¢~-(R) which, in turn, implies
,{ -.
,.} {[((/..,);-(f.d');)(.,] [((f-.');- (f-.');)(0,] ,}
in Ll(12), VhEH. This concludes the proof of (3). Using (3) we obtain that
=E
{f t(h,g)n dA ~s} =(t -
s)(h,g)H.
[]
The last Theorem states in particular that I~ is a 2-cylindrical standard brownian motion, provided ~(h) has continuous sample paths in t for every h E H. Now we are ready to carry out Nelson's program in Hilbert space. This was the motivation for the new stochastic integral.
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L. Gawarecki / Statistics & Probability Letters 34 (1997) 103-111
Assume that {Xt)tE I is an H-valued stochastic process satisfying the following conditions: t~--~Xt is continuous from I to LI(f2,H) and D X t = lim,~\oE{(Xt, A - S t ) / A l ~ } exists in Ll(f2,H) with t ~ DXt continuous from I to LI(f2,H). The process Yt =Xt - J o D X s d s is an H-valued martingale (the proof in Nelson (1967) applies if adjusted to a Hilbert space case).
Theorem 3. Let a process {Xt}tE 1 satisfy the above conditions and the associated martingale {Yt}t~l together with the process {at}t~1 be as in Theorem 2. Then Vh E H, ( X t - foDXsds, h )H = ( f t r d f f ' ) t ( h ) in .~/g2(R), with W = ( f tr-dY) w. In particular, f crdW is a 2-cylindrical H-martingale associated with an ordinary H-valued martingale. Furthermore, if X • I --, H is continuous, then Vh E H the real-valued martingale /~(h) = ( f tr-dl?)"'(h) has P-a.e. continuous paths.
Proof (sketch). By noting that Ilallz,
E{IlaII~,H}d). < re,
xl
we obtain that a E , 4 ( f f ' , H ) (see Metivier and Pellaumail, 1980, 16.3). Next we can show that for elementary processes X, of the form (1), X, o a - E/iw(I~,H), and
This is achieved by approximating a - with a sequence of elementary processes in/]w(17,H). By selecting X~ ---, tr i n / ] ( f f ' , H ) and noting that X~ o a - ~ Id,~ in flW(ff',H) we obtain that
(/)
Xndff"
(h)---*
(/)
adW
(h),
(/)w Xnoa-d~"
(h)--'(Y,h)H
in .g~-(R). Hence (Xt - fo DX.~ ds, h )H = ( Yt, h ) =( f a d lTV),( h ) in ~¢t'2(R). The final statement of the theorem is a consequence of a lemma in Metivier and Pellaumail (1980, 10.1). We note that we have obtained a stochastic integral representation of the martingale { Yt}t~r. Such representation result was established in Yor (1974) under different assumptions involving the existence of a solution to a martingale problem. While providing a detailed construction of a new stochastic integral we omitted details not directly related to the presented topic. Extended discussion and additional results will be the subject of a future publication.
Acknowledgements The author is indebted to Professor V. Mandrekar for introducing him to this problem and for his continuous support and encouragement. The author would like to thank an unknown referee for valuable suggestions, which led to an improvement of the original presentation.
L. Gawarecki / Statistics & Probability Letters 34 (1997) 103-I11
III
References Mandrekar, V. and H. Salehi (1970/71), The square-integrability of operator-valued functions with respect to a non-negative operator-valued measure and the Kolmogorov isomorphism theorem, Indiana Unit. Math. J. 20, 545-563. Metivier, M. and J. Pellaumail (1980), Stochastic Integration (Academic Press, New York). Nelson, E. (1967), Dynamical Theories of Brownian Motion, Mathematical Notes (University Press, Princeton, N J). Yor, M (1974), Existence et unicit6 de diffusions ~i valeurs dans un espace Hilbert, Ann. Inst. H. Poincar~ Ser. B 10, 55-88.
Statistics & Probability Letters 34 (1997) 103--III
Extension of a stochastic integral with respect to cylindrical martingales Leszek Gawarecki Deparmwnt qf Science and Mathematics, GMI Engineering and Management Institute, 1700 FE Third Ave., Flint, MI 48504, USA Rcccived Junc 1996; revised September 1996
Abstract
We extend stochastic integral of Metivier and Pellaumail with respect to cylindrical martingales which is necessary for constructing a Hilbert space-valued diffusion based on Nelson's kinematic theory of stochastic motion. Examples for inadequacy of existing stochastic integrals are provided.
Keyword~v Stochastic integral; Cylindrical martingales
I. Introduction
Given a real-valued stochastic process {Xt}t~l, I = [0, T] satisfying some regularity assumptions, Nelson (1967) constructed a brownian motion Wt and a stochastic differential equation, with the process Xt being its solution,
Xr =
f0' DX,. ds + f0t as d Ws.
The drift DXt c a n be interpreted as the mean forward velocity of Xt,
The process Yt =Xt - f o D X s d s
is a martingale and if the Ll(g2) limit
exists and satisfies some smoothness conditions, the process Wt = ./o a~-I dY~ is a brownian motion. In order to carry out the outlined program for Hilbert space-valued processes one can introduce a martingale in an analogous way as above, and needs to define a stochastic integral with respect to Yt for a sufficiently large class of processes a / 1 . The purpose of this paper is to provide examples explaining why the existing 0167-7152/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved PII $ 0 1 6 7 - 7 1 5 2 ( 9 6 ) 0 0 1 7 1 - X
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L. Gawarecki/ Statistics & Probability Letters 34 (1997) 103-111
stochastic integrals are not sufficient to support Nelson's technique and to introduce an extension of a stochastic integral with respect to cylindrical martingales of Metivier and Pellaumail. This extended integral will admit reasonable class of integrants.
2. Regularity assumptions and their consequences In what follows we assume that the filtration {~t}te; on (O,.~-,P) satisfies usual conditions. Denote the a-field of predictable sets of I2 by ~ and the collection of predictable rectangles by N. Let H be a separable Hilbert space and denote H ®2 H, and H ® H1 the Hilbert and Banach space of Hilbert-Schmidt and trace class operators on H. With an element x ® y E H ® H we associate an operator T : H ~ H by (Th, 9)n = (x, h)n(y, 9)H. Let us denote by (jt'2(H), (.,-)) the Hilbert space of H-valued square integrable martingales (i.e. E{ IIMr 112} < oG) and identify P-equivalent processes. We assume the following regularity assumptions on Y E,grZ(H) and call a process satisfying these assumptions reoular.
(A) a2(t)= lim E { (Y'+A - Yt)®2"~t exists in Ll(f2,~t,H ®t H) and t ~ y~2, t ~ o2(t) are continuous mappings from [0, T] to L1(f2,H®IH). Thus, a 2 takes values in positive, self-adjoined elements of H ®l H. (B- 1) a 2 is predictable. (B-2) The eigenvalues of a 2, {~.n(co,t)}~= l > 0. Now we will see how the regularity assumptions interfere with properties of the Dol~ans measure of the martingale Y. Theorem 1. Let {Yt}t~; be reoular. Then the following ham true: (1) There exists a jointly ,~ ® ~ ( I ) measurable version of a 2. Let us further consider this jointly measurable version and denote it by the same symbol. Also, let Ep<¢). denote the expectation with respect to P ® 2, with 2 denotin9 the Lebesgue measure on [0, T]. (2) The Dolbans measure ~llrl] of the process IIYl[2 is absolutely continuous with respect to P@ 2 with the density d~llrll/d(e ® 2) = tr a'. (3) The Dolkans measure ~v of the process y®2 is absolutely continuous with respect to P ® 2 with the density d~r/d(P ® 2 ) = a 2. It is also absolutely continuous with respect to all Eli with the density Qr = a2/tr o 2 a.e. P®)..
Note: It is the condition (A) that guarantees ( 1 ) - ( 3 ) with densities given by conditional expectations of the relevant expressions with respect to the predictable a-field ~ and the measure P ® 2. Proof. Statement (1) follows because the mapping t ~ a2(t) is continuous from [0, T] to L1(12,H ®1 H). To show (2) note that for predictable rectangles F x (s,t]EN we have OqlVll(F x (s,t]):E{1F([IYtII2H --IIY~I[~)} : E
= fF
×(s.t]
tr a2d(p ® 2),
{ft lF
tr a2(r)dr
}
L. Gawarecki I Statistics & Probability Letters 34 (1997) 103 111
105
(the proof of the second equality follows the same line as the proof of the corresponding statement in Nelson (1967)). The first and the last expression, both extend to measures on ~ and these extensions agree on predictable rectangles, hence they are identical. Finally, an analogous equality and extension argument as in the proof of (2) holds also for measures xr and P®)~, yielding d~r = a z d ( P ® 2 ) . Moreover, by Metivier and Pellaumail (1980), ~r ~ ~IIYI. Now, we have ay ,~ ~l',Vll '~ P®J- and (3) follows. U A process 0.2 satisfying conditions (B-l) and (B-2) has a form o.2(¢o,t)(h)=~n°°=12,(~o,t) (h,h,(oo, t))Hh.(ev, t),Vh EH with { h . } . ~ C H , an ONB, 2.(¢o,t)>0, n = 1,2 .... y~°°__~2 . ( o , t ) = No.2(co,t)[l~. Also, there exists the square root of 0.2, denoted by a, which is a Hilbert-Schmidt operator, o.(h) = ~=1 v/~. (h,h,)Hh., Vh E H (we will usually drop the dependence on (og, t)). The generalized inverse of a, denoted by 0.-, is defined as a composition P[Ker(,r)l-C o 0"-I OPcl(Ran(rr)), where P[Ker(a)]~ and Pcl(Ran(a)) are, respectively, projections on the orthogonal complement of the kernel space and on the closure of the range of a, and o.-l is the inverse relation to the operator a. Because a 2 is regular, cl(Ran(a)) = H. Then a - takes the form a-(h)= ~ - - l ( 1 / v / ~ - . ) ( h,h~)Hh~ Vh E Ran(o.). Both processes a and a - are predictable (see Mandrekar-Salehi, 1970/1971 for details).
3. Inadequacy of existing stochastic integrals
3.1. The isometric integral With a martingale M E ./[2(H), we can uniquely associate a predictable, H fgl H-valued process QM satisfying Ctu(G)= Ja QM dCtllMIi for any predictable set G, with ~tM, ~tllMii denoting the Dolrans measure of M ~2 and IIMII~, respectively. The values of QM are trace class, self-adjoint, positive operators. A process X in the domain of the isometric integral of Metivier and Pellaumail has values in (possibly non-continuous) linear operators on H. The domain ~(X(og, t)), 'v'(og,t ) E O x [0, T], of X(~z~,t) must contain Q~2(H) and X(tn, t ) o Q~1/2 (~o,t) must be a Hilbert-Schmidt operator. By Theorem 1 we have Qy=o.2/tro. 2, however, 0 . - o /-) ~ ,1/ 2 = ( 1 / ~ ) o . - o o . = ( l / ~ ) P i K ~ ( o ) ] ~ = (l/t~ra2)IdH is not a Hilbert-Schmidt operator unless H is finite dimensional. Thus a - is not admissible for the isometric integral.
3.2. The cylindrical inteyral Failure of the isometric stochastic integral in Nelson's procedure is due to the non-existence of the standard H-valued brownian motion. In order to realize a brownian motion process with covariance associated with an identity operator on H one has to abandon H-valued processes and consider cylindrical processes. We call {/14t}t~/ a 2-cylindrical H-martin9ale, if each ?Qt is a continuous linear mapping from H to L2(f2, ~ ) and Vh E H the real-valued p r o c e s s {~/[t(h)}tEI is a martingale relative to {~tt}/el. The space of 2-cylindrical H-martingales can be identified with the space L(H,.//2(R)). For a 2-cylindrical H-martingale/k~t, the quadratic Dolkans function dpd is an additive, (H ®1 H)*-valued function on predictable rectangles ~ defined by (b,d~(F x ( s , t ] ) ) = E { IF(l~Tlt @ f / I t ( b ) - f 4 s ®/~ts(b))} where, for every t E [0, T], ftt®lC4t denotes the continuous linear mapping from H ® l H into Ll(t2,.~) which is the linear continuous extension of the mapping b=h®g~--~Mt(h)Mt(g). Also, above, bEH®1H, F E ,~, s, t E [0, T], s ~
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L. Gawarecki I Statistics & Probability Letters 34 (1997) 103-111
If for all h E H, M(h) has a cadlag version, then the Quadratic Dolrans measure exists (see Metivier and Pellaumail, 1980). This happens for a 2-cylindrical martingale associated with a martingale M E~g2(H) by ]t~t(h)=(Mt,h)H , ~/hEH.
For a 2-cylindrical H-martingale/14 with the quadratic Dol~ans measure 0tM of bounded variation [x~?[, there exists a process Q~ with values in the set of positive elements o f ( H ®1H)* (i.e. Q~(h ®h)>lO, Vh E H), such that for every b E H ®1 H the real-valued process (b, QR) is measurable for the [~t~[-completion of the a-field 2 , it is defined up to [~?[-equivalence and has the property that (b,~(A))=f~(b,Q~(to, t))[~[ (dco, dt)Vb E
H ®IH, A E ~ . An example of a 2-cylindrical martingale is a cylindrical brownian motion {Wt}t~, satisfying Vh E H, {Wt(h)}te~ is a brownian motion and Vh, g E H, t E I, E { ~ ( h ) ~ ( g ) } =tC(h,g), where C is a continuous bilinear form on H x H (in particular C can be the identity operator on H). Now we recall definition of a cylindrical stochastic integral with respect to a 2-cylindrical H-martingale with the quadratic Doi6ans measure of bounded variation. For an elementary process, X(w,t)= ~ = l uil~,(w,t), (here ui, i= 1..... n are continuous, linear operators on H and {Ai}n= 1 C ~ ) , define VhEH, (1) i=I
where u* denotes the adjoint operator. The integral ( f X d f 4 ) is a 2-cylindrical H-martingale and for every h E H the real-valued square integrable martingale (fXdl~)(h)E .¢¢~-(R) has norm given by (see Metivier and Pellaumail, 1980, 16.2.2)
(Sxd~)(h) 2~/(R) = f(X*(h)®X*(h),Q~)dl~l.
(2)
Definition 1. (A) £(M,H) is the set of processes X with the following properties: (1) V(co,t)E f2 x I, X(to, t) is a linear operator on H with domain ~(X(co, t)) dense in H. (2) Denoting by X*(o~,t) the adjoint operator to X(og, t), the linear form (X*(og, t)(h)®X*(o~,t)(g), Q;t(og, t)) has [~,l-a.e. a unique continuous extension to H x H which results in a predictable process. (3) JV(X) = suPllhll< 1{f~×l (X*(h) ®X*(h), Q~?(og,t ) ) d l ~ [ } 1/2 < co. We define A(,Q,H), the closure in the seminorm ./V of the class of elementary processes in the space £(,~,H). (B) The unique extension of the isometric mapping X ~ ( f X d l Q ) given by (1), from the space of elementary processes into the space of 2-cylindrical H-martingales, to the isometric mapping from A(M,H) into the space of 2-cylindrical H-martingales is called the stochastic integral and is denoted again by X
(f X d~). Now we want to take advantage of the fact that the integrator in the cylindrical stochastic integral, which we consider, is actually a regular element of .g2(H). Note that again, this is a consequence of Nelson's regularity assumptions on a stochastic motion. Observe that any T E H ® I H extends uniquely to an element of (H@I H)* by T(h ® g)=(Th, g)H, with [ITIl(u®, H)* ~ IITII1. Therefore ~t~ =CtM as (H®I H)*-valued measures on 2 . Also the variation of ctM and the measure ~llgll are equivalent. Indeed, I~l~<[ctnl=~llnll, as well as 1~1(.4)=0 implies I~,Ml(Z)=~llml(A)=0. We can choose QM = (d~llMiI/d[ct;t[)QM to be a predictable process associated with/~t and this allows us to replace Q~ with QM and the measure I~1 with Cqlnll to get the condition (2) in the above definition hold for
L. Gawarecki / Statistics & Probability Letters 34 (1997) 103-111
107
(X*(co, t)(h)®X*(og, t)(g), QM(O, t)) and measure CqIM[I. This will not change the space L(/kC,H). Moreover, the seminorm •At(X) = sup { f t 2 ilhll~
3.3. Non-admissibility of aWe first verify that a - E £ ( I ~ , H ) . Condition (1) in Definition 1 holds, since ~ ( a - ) D R a n ( a ) . Equality (9,a-(h))n =Y]n~l (l/v~n)(h, hn)n (g, h. )n = ( a - (g), h)H, Vh, g E @(a- ) implies that ~ ( a - ) C ~ ( ( a - )* ). Now 'v'(g, h) C ~ ( a - ) x ~ ( a - ),
(g,h)H
H
((a-)*(g)®(a-)*(h),Qy) 1
=
(
:)
a - ( g ) ® a - ( h ) , t--~-fia2
1
-- tr a2(a2(a-(g)),a-(h))H = t-i : (g,h ). extends continuously to H x H. This extension is predictable in view of predictability of a 2. Finally, .4:(a-)2=
sup !lhll
IlhllHt--~--a~a2tra2d(P®2)
justifying (3). Define a ~ ( h ) = Y~=i ~j.(h~,h)Hhn. We have f~x, Ila'~' II~'H)d~llrll ~< f o x , (1 +tra2)d(P®2) < oo. Thus, aN EL2((f2 x I,~,~IIrlI);L(H))CA(Y,H) (see paragraph 16.3 in Metivier and Pellaumail, 1980). Now we are ready to prove that a - ~/I(17,H). First, note that .Ar(a~ - a - ) = suPllhll~l(fox I Y~i~N+l (hi, h)2d(p ® 2))b'2 = 2(I) 1/2. Thus aN does not converge to a - in the seminorm W. Assume that a - E/i(IT,H), so that there exists a sequence {X,,},= o0 l of elementary processes with lim,~o~ ,#" (X, - a - ) - - * 0. Denote by PN the orthogonal projection of H on the span{hi,h2 ..... hu}. We have the following: •.U(Xn o PN -- aN )
X.
= sup Ilhll
)*(h) ~
[~q(h.Xn*(h)) H -
= sup llhll~<]
o PN -- a N
(X n o
PN -- a;)*(h), ~
×1
xl i=l
,,,
® 2)
}
x tro'2d(P ® 2)
L. Gawarecki / Statistics & Probability Letters 34 (1997) 103-111
108
~< sup {foxl((Xn-~r-)*(h)®(Xn-cr-)*(h),Qy)dOtllYl.} Ilhll~l
=~V'(X~-cr-)~0
asn~.
We have proved that ~4r(X~ o PN - a7¢ ) ~ JV'(X~ - a - ), n, N = 1,2... Similarly, we verify that for any n = 1,2 . . . . . W'(X~oP,v-X~) ---, 0 as N ~ oo. Summarizing, X(a,~ - ¢ r - ) ~<2.A/'(Xn - or- ) + JV'(X~ o PN - X~), which gives a contradiction with JV'(a~ - or- ) 74 0.
4. Extension of the cylindrical stochastic integral and application to Nelson's construction of diffusion Motivation for further studies comes from the following lemma.
Lemma 1. For every hEH, fa×l((a~ - a- )(h) ® (a N - o'-)(h),Qy)dCqlVl I ~ 0, as N --+ co. Proof. V(og,t) E f2 x I, (tr O'2((O'N -- 0"- )(h) ~ bounded by Ilhll~ independently of (og, t). 5
(0" N --
0"- )(h), Qr))(og, t) = ~i°¢__N+t(h, hi(og, t))~ ~ 0 and is
Now we consider an extension of the cylindrical stochastic integral.
Definition 2. Let M be a 2-cylindrical H-martingale with the Dol6ans measure of finite variation. (A) Define Lw(~Q,H), the set of processes satisfying conditions (1) and (2) of Part (A) of Definition 1 and the following condition: (3) Vh EH, JV'~"(X) = [fa×z(X*(h) ®X*(h),Q;t)dl~;tl] 1/z < oc. For every h E H, "/~w h is a seminorm and we say that a sequence {X,},~ 1 c £ w ( , Q , H ) converges to X E £w(l(l,H) if VhEH, .W'~(X, -X)---. O. We will denote this convergence by Xn ~ X. (B) We denote by ./iw(&¢,H ) the closure in/[w(~, H ) of the class of elementary processes in the topology of convergence "=¢," defined in (A)-(3). For every XEflW(f4,H), h E H we define (fXdM)W(h) as the limit of ( f X , dF4)(h) in ~¢r2(R), where X, ~ X and X, are elementary processes. We call ( f x diQ)w E L(H, .At'r2(R)) the stochastic integral. oo is a sequence of Note. To justify correctness of Part (B) of the above definition assume that {X,~}~=l elementary processes, such that X~ =~ X. We have Vh E H,
=
= y~(x~
- x,.)
by equality (2). Therefore, whenever X, =~X, then Vh E H, {An }~=1 is a Cauchy sequence for ,/V~' and hence ( f x n d~4)(h) converges in Jg2(R) to a square integrable real-valued martingale. Now, the mappings h ~ (fXn d~t)(h) from H to Jt'2(R) are linear, continuous and for every h E H there exists a limit, which we denote by ( f X dffI)W(h). Therefore, by Banach-Steinhaus theorem (fXdA]t)w(.)E L(H,,g2(R)) - that means ( f X d l ( I ) w is a 2-cylindrical H-martingale. As we proved in Lemma 1, a~ =~ or- with a~ E/i(17,H)C/iW(I~,H). Therefore ¢r-E/iw(17,H).
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L. Gawarecki / Statistics & Probability Letters 34 (1997) 103-111
Theorem 2. Assume that Y E JI~(H) is regular and the associated process a 2 is also regular. Then if'= (fa-dl~) w is a 2-cylindrical H-martin#ale. such that E{(~(h) - ~ ( h ) ) ( ~ ( g ) - ~ ( g ) ) [ ~ } = (t - s)(h,g)H
Vh, gEH.
Proof. Let us first prove that for X E J W ( ) ' , H ) we have
=E
(X*(h)@X*(O),~2(r))dro~.
,
VhEH.
Recall that by condition (2) of Definition 1 and because Qr =a2/tra 2 and dallrll =tr~2d(P ® ,~), the following expression: (X*(h)®,X*(g),a 2) = ( X * ( h ) ® X * ( g ) , Q r ) t r a 2 is well defined on H x H. It is easy to verify that equality (3) holds for elementary processes. Next, E
[(aoX*(h),~oX*(g))n -(aoX*(h),aoX*(9))nld2
<~ (E { / ' liaro ( X . - X)* (h)ll~d2 }),/2 (E { f ' ],a o X* (9)111 d2 })1/2 + (E { / / I l a o (X.-X)*(g)ll2nd2}) '/2 (E { / ' .,~.~"(x,
-
IIGoX*(h)l,,~d,~})~/2
x),Gw(x) + ~ w ( x . - x ) ~ w ( x . ).
Therefore convergence Xn :::vX implies that, for every h E H, E{fts(X*(h)®X*(g), (r2)d2l~s } -* E{fts (X*(h) ® X*(g),a:)d2[.~} in Ll(12) by contractivity of conditional expectation. Convergence X, ~ X implies also that, Vh EH, ( f X , dlT)W(h)--* ( f X d)~)W(h) in ~¢~-(R) which, in turn, implies
,{ -.
,.} {[((/..,);-(f.d');)(.,] [((f-.');- (f-.');)(0,] ,}
in Ll(12), VhEH. This concludes the proof of (3). Using (3) we obtain that
=E
{f t(h,g)n dA ~s} =(t -
s)(h,g)H.
[]
The last Theorem states in particular that I~ is a 2-cylindrical standard brownian motion, provided ~(h) has continuous sample paths in t for every h E H. Now we are ready to carry out Nelson's program in Hilbert space. This was the motivation for the new stochastic integral.
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L. Gawarecki / Statistics & Probability Letters 34 (1997) 103-111
Assume that {Xt)tE I is an H-valued stochastic process satisfying the following conditions: t~--~Xt is continuous from I to LI(f2,H) and D X t = lim,~\oE{(Xt, A - S t ) / A l ~ } exists in Ll(f2,H) with t ~ DXt continuous from I to LI(f2,H). The process Yt =Xt - J o D X s d s is an H-valued martingale (the proof in Nelson (1967) applies if adjusted to a Hilbert space case).
Theorem 3. Let a process {Xt}tE 1 satisfy the above conditions and the associated martingale {Yt}t~l together with the process {at}t~1 be as in Theorem 2. Then Vh E H, ( X t - foDXsds, h )H = ( f t r d f f ' ) t ( h ) in .~/g2(R), with W = ( f tr-dY) w. In particular, f crdW is a 2-cylindrical H-martingale associated with an ordinary H-valued martingale. Furthermore, if X • I --, H is continuous, then Vh E H the real-valued martingale /~(h) = ( f tr-dl?)"'(h) has P-a.e. continuous paths.
Proof (sketch). By noting that Ilallz,
E{IlaII~,H}d). < re,
xl
we obtain that a E , 4 ( f f ' , H ) (see Metivier and Pellaumail, 1980, 16.3). Next we can show that for elementary processes X, of the form (1), X, o a - E/iw(I~,H), and
This is achieved by approximating a - with a sequence of elementary processes in/]w(17,H). By selecting X~ ---, tr i n / ] ( f f ' , H ) and noting that X~ o a - ~ Id,~ in flW(ff',H) we obtain that
(/)
Xndff"
(h)---*
(/)
adW
(h),
(/)w Xnoa-d~"
(h)--'(Y,h)H
in .g~-(R). Hence (Xt - fo DX.~ ds, h )H = ( Yt, h ) =( f a d lTV),( h ) in ~¢t'2(R). The final statement of the theorem is a consequence of a lemma in Metivier and Pellaumail (1980, 10.1). We note that we have obtained a stochastic integral representation of the martingale { Yt}t~r. Such representation result was established in Yor (1974) under different assumptions involving the existence of a solution to a martingale problem. While providing a detailed construction of a new stochastic integral we omitted details not directly related to the presented topic. Extended discussion and additional results will be the subject of a future publication.
Acknowledgements The author is indebted to Professor V. Mandrekar for introducing him to this problem and for his continuous support and encouragement. The author would like to thank an unknown referee for valuable suggestions, which led to an improvement of the original presentation.
L. Gawarecki / Statistics & Probability Letters 34 (1997) 103-I11
III
References Mandrekar, V. and H. Salehi (1970/71), The square-integrability of operator-valued functions with respect to a non-negative operator-valued measure and the Kolmogorov isomorphism theorem, Indiana Unit. Math. J. 20, 545-563. Metivier, M. and J. Pellaumail (1980), Stochastic Integration (Academic Press, New York). Nelson, E. (1967), Dynamical Theories of Brownian Motion, Mathematical Notes (University Press, Princeton, N J). Yor, M (1974), Existence et unicit6 de diffusions ~i valeurs dans un espace Hilbert, Ann. Inst. H. Poincar~ Ser. B 10, 55-88.