THE HU-MEYER FORMULA FOR NON DETERMINISTIC KERNELS

In this paper we prove the analogue of the HuMeyer formula for random kernels. More precisely, using a suitable notion of trace we give the relation between the multiple Stratonovich integral of a non adapted process and the multiple Skorohod integral.


l. INTRODUCTION
This paper has been motivated by the problern of finding the analogue of the Hu-Meyer formula for randorn kernels. In [1] the authors present a relation between the multiple Stratonovich integral of deterrninistic kernels and the rnultiple Ito-Wiener integral of traces, giving an intuitive explanation for its validity. Recently, different authors have given rigorous proofs of this formula, see for instance [2], [3], [9], [10] and [11].
Considera non adapted stochastic process X = {Xt, t E [O, 1]}. Under sorne srnoothness requirernents, the stochastic integrals of X with respect to a Brownian rnotion 1]} in the Skorohod and in the Stratonovich sense can be defined. They usually have been denoted by <5(X) and P(X) respectively. These two notions of integrals can be related by rneans of a trace type terrn (see Theorern 7.3 in [6] and Theorem 1.9 in [9]. Formally, Iª(X) = 6(X) + TX. The multiple k-th Skorohod integral bk(X) can be defined as an extension of the onedimensional parameter case (see for instance [5] and [4]), as well as the k-Stratonovich integral, IZ(X). For k = 2 the relation between I~(X) and <5 2 (X) has been studied in Section 2.B of [9]. To this end different notions of traces are introduced, say T 1 X, TX and T 1 , 2 X, and it is proved that (0.2) Both formulae (0.1) and (0.2) are of the Hu-Meyer type for non deterministic kernels.
Our purpose here has been to find an appropiate notion of trace which unifies all the notions explained before, allowing us to relate the multiple Stratonovich and Skorohod integrals of a multiparameter process X. This notion is given in Definition 2.4. The basic result is presented in Theorem 3.1, where the formula is proved. Here Tj,r(X) denote the traces, and it should be pointed out that for X deterministic this formula reduces to the Hu-Meyer formula [½] '°"' k' . IZ(X) = L.,; (k _ 2 j)! j! 2 j Ik-2j (Tr' f) , j=O (see Remark 3.2).
Consider a standard one-dimensional Brownian motion {Wt, t E T} defined on the canonical probability space (n, :F, P), that means n = C(T), :F = B(n) and P is the Wiener measure. Consider also a measurable stochastic process X= {Xt, t E Tk} defined on (n, :F, P), such that E (fr1c Xf dt) < +oo. For any partition 1r of T we set The process X is said to be Stratonovich integrable if the family {S1r(X), 1r partition of T} converges in L 2 (n) as lnl --+ O. We will call this limit the k-Stratonovich integral of the process X and it will be denoted by IZ(X).

Remarks
(2.2) In the previous definition the value of IZ(X) can be obtained, equivalently, as the limit of { S1r(n): .¡ 2:: 1} for any increasing sequence of partitions { n( n ), n 2:: 1} of T such that ln(n)I--+ O as n tends to infinity. Then it is obvious that the process X is k-Stratonovich integrable if and only if X possess the same property, and in this case It(X) = It(X). Hence, when dealing with the Stratonovich integral, we will only consideras integrands processes {X 1 , ' í. = (t 1 , .•. , tk) E Tk} which are symmetric in the variables ti, ... , tk.
For any integer k ~ 1, Dk denotes the Malliavin k-th derivative operator. Given any real p > l we call ok,p the set of Wiener functionals F in Dom Dk such that k IIFllk,p := IIFIIP + L 11 IIDi FIIP(Ti) llp i=l is finite.
The adjoint of the operator Dk is the multiple k-Skorohod integral. It will be denoted by bk.
By definition LZ' 2 is the space L 2 (Tk, Dk, 2 ). That means, LZ' 2 is the class of processes X E L 2 (Tk x n) such that X 1 E ok, 2 for any í. E Tk, and there exists a measurable version We refer the reader to [5] and [4] for an extensive treatment of questions concerning the multiple Skorohod integral.
In the next definition we introduce the notion of trace for processes of LZ• 2 • As it will be shown in the next section, this is the suitable concept to compare the k-th Stratonovich and Skorohod integrals, and unifi.es different definitions given in (9], [6J. This limit defines a stochastic process with a ( k -2jr )-dimensional parameter which will be denoted by T;,r(X).

Remarks
(2.5) By convention T 0 , 0 (X) =X.  (2.8) As in Proposition 1.8 [9) (see also Proposition 2.1 [11)) the existence of the (j, r)-trace of X, T;,r(X) can be given in terms of the existence of traces for the kernels of the Wiener-chaos decomposition of the random variable Xi. The precise statement is as Denote by ri+r f m the L 2 (Tm+k-2 i-2 r)-limit, if it exists, of the sums t . ,·.
LE{l r Then, if X has (j,r)-trace, ,-i+r fm exists and oo m' .

HU-MEYER FORMULA FOR RANDOM KERNELS
The aim of this section is to prove the following result. where Im(·) denotes the multiple Ito-Wiener integral.
Before giving the proof of Theorem 3.1 we quote sorne _known results that will be used in the sequel.  The proof is straightforward and will be omitted.
ICY Proof of Theorem 9.1. It has to be shown that, for any increasing sequence of partitions {7í(n), n ~ 1} of T such that 17l'(n)I--+ O as n tends to infinity, the sequence of randorn variables {S1r(n) (X), n ~ l} defined by (2.1) converges in L 2 (S1) to the right hand side of (3.1 ). The proof will be done in two steps, following along the ideas of the proof of theorem 3.4 [9]. However the random character of X adds sorne difficulties.
(1) Let 71' be a partition of T, and 91r the cr-field on T generated by the intervals .6.i , i = 1, ... , r1r, whe:·e we use the notation introduced in Section 2.
Proof. The conditional expectation Ek-2 ;-r [trl,r(f)] can be developed as follows Therefore the lemma is proved. •