Transformations of index set for Skorokhod integral with respect to Gaussian processes

We consider a Gaussian process {Xt, t E T} with an arbitrary index set T and study consequences of transformations of the index set on the Skorokhod integral and Skorokhod derivative with respect to X. The results applied to Skorokhod SDEs of diffusion type provide uniqueness of the solution for the time-reversed equation and, to Ogawa line integral, give an analogue of the fundamental theorem of calculus.


Introduction
The purpose of this article is to prove that, in a general case of Gaussian processes and under mild assumptions, transformations of a parameter set do not change the Skorokhod integral and Skorokhod derivative, and to indicate some applications of this fact.Let T be any set, C a covariance on T and H(C) H the reproducing kernel Hil- bert space (RKHS) on C (note that H may not be separable).With covariance C, we associate a Gaussian process {Xt, t e T} defined on (ft,,P), where r{Xt, T}.For the details of the constructions above, see [3].Let H (R) p be the p-fold tensor product of H.The p-Multiple Wiener Integral (MWI) Ip'H (R) P--.L2(f,,P was defined in [6] (see also [5]) as a linear mapping satisfying the following properties.Here f is the symmetrization of f. a) EIp(f)-O, { 0 ifpyq for f G H (R)p gG H (R)q b) EIp(f)Iq(g) pI(7 , )H (R) p if p q, p c) Ip + l(gh) Ip(g)Ii(h )-Iv_ l(g ?h), for g H (R) P, h E H. k=l Above, (g ?h) (tl,...,tk_ 1, tlc + 1," tp) (g(tl,"-,tlc-1, tlc + 1," ", tp), h( ))H" We note that Ip(f)-Ip(']) and hence Ip(H (R) P)-Ip(H (R) P) where H (R) p is the p-fold symmetric tensor product.
Let u.:f-H be a Bochner measurable function with I I u ll H E L2(f,,P).

Skorokhod Integral Under Transformation of a Parameter Set
For a Gaussian process {Xt, t T}, let H(X) cl(svan{Xt, t T}), the closure being taken in L2(ft,5,P).With a transformation R:S--,T we associate a Gaussian process X/-{X s S} and we call R nondegenerate if it is onto and if H(Xt:t) H(X).R))r main result on transformations of the Skorokhod derivative and integral is the following: Theorem 1: Let {Xt} E T be a Gaussian process and R:S---,T be a nondegener- ate transformation.Denote by ISx and I s the Skorokhod integrals with respect to X R X and XR, respectively.Then: R--f(R(Sl),.R(sp)) is an isometry from H(Cx) (R)p onto 1) fp-fp H(CxR)(R)P. 2) If u e (ISx) then u l-{uR(s),s e S} e (IsXR and lSx(U) I s (uR) X R 3)

4)
Moreover, denote by D X and D Xt the Skorokhod derivatives with respect to X and X R respectively If for t G T u G 2(DX), then uRs G (DXR) for sGR-l{t} and Df u =D ,u P-ae for s () ()..
Hence, for s S, v s p OIp (f p (', s)) p According to 1), u p=olXp(fp(.,t))L2(f,H(Cx) and equality of norms claimed in 4) is satisfied.The last part of assertion 4) follows from 1), 2) and 3) since failure to satisfy any stated condition by u implies violation of this condition by V.

VI
Example 2: Transformations of parameter set and Skorokhod integral.
B .Ogawa Line Integral.We recall the definition of the Ogawa integral ([4, 9]) with respect to a Gaussian process {Xt, [0, 1]} with the RKHS H. Let .'Hbe an H-valued Bochner measurable function.Then, on a set of P-measure one, .()takes values in a separable subspace of H. Let {en, n N} be an ONB of this subspace.The (universal) Ogawa integral of is defined as follows" ()-(u, en)HI(en) (limit in probability) n=l if it exists with respect to all ONBs and is independent of the choice of basis.
The relation between Skorokhod and Ogawa integrals is explained in [4].
Thus, 5x(U) 5y(V) for v s u(s) provided either of the integrals exists.
Consider Brownian sheet {W(x,t), (x,t) [0,112}.Assume that F C [0, 1] 2 is a curve parametrized by a function 7:[a,b]F, 0 a b 1.We define the Ogawa line integral, F-5, over F with respect to {Wtx t,(x,t) F} using F as the parameter set.
Example 3: Skorokhod-type stochastic differential equations.The following class of Skorokhod SDEs was considered by Buckdahn in [1], where, under smoothness assumptions, the author proved existence and uniqueness results Z r -f b(Z(s))ds -F Ii(a(Z(s))l[o, t](s)), 0 _< t _< 1. (2)   0 The initial condition q needs to be bounded.However, this restriction vanishes if equation ( 2) is reversed.
Lemma 1: Let {us} s [0,1] be such that usl[0,tl(s) e (I) t e [0, 1] Then for the time reversed, process -s Ul s, we have slio, t](s) G (I) Vt G [;, 1] and if we denote X I*B(I[o t](s)us), then X1 -t-Xl I (1[0 t](S)s).B Using time reversal and Lemma 1, Buckdahn's result can be extended to time reversed SDEs with the initial condition being a terminal value of the solution of the original equation.
Theorem 2: Assume that coefficients b and r of a Skorokhod SDE (2) satisfy assumptions for existence and uniqueness of the solution._If {ZtitE[O, 1 iS the solution of Equation ( 2), then the time reversed process Z -ZI_ is thd unique solution in LI([0 1] x f) of the time reversed equation X 2 0 + / --(Xs)ds + Ii (-l[o,t](s) (X(s))) B 0 where b (Xt) b(X 1 t), (Xt) if(X1 t), and B B 1 B 1 t" The above theorem gives a partial answer to a question in [8], Proposition 5.2.The technique of time reversal has been used in [10] to solve a problem regarding anticipative stochastic models in finance.
) and I I v I I L 2 I I I I L 2, 8 oov, v e (Ix) impis e (I)) e v e (vx) impis u(s) e (Dx) wiCh Dv D(,)R()for s,s' e S. g D R 2 s' S), Ih n Dt,utcH(CX) 2 v scH(cxR (s, C e (t, t' G T), and the H-S norms of those derivatives are equal.
) w Uow o the inductive relation (c) for MWI to complete the proof.For fp H(Cx) arbitrary,