MOMENT IDENTITIES FOR SKOROHOD INTEGRALS ON THE WIENER SPACE AND APPLICATIONS

We prove a moment identity on the Wiener space that extends the Skorohod isometry to arbitrary powers of the Skorohod integral on the Wiener space. As simple consequences of this identity we obtain sufﬁcient conditions for the Gaussianity of the law of the Skorohod integral and a recurrence relation for the moments of second order Wiener integrals. We also recover and extend the sufﬁcient conditions for the invariance of the Wiener measure under random rotations given in [ 3 ]


Introduction and notation
In [3], sufficient conditions have been found for the Skorohod integral δ(Rh) to have a Gaussian law when h ∈ H = L 2 (I R + , I R d ) and R is a random isometry of H, using an induction argument.
In this paper we state a general identity for the moments of Skorohod integrals, which will allow us in particular to recover the result of [3] by a direct proof and to obtain a recurrence relation for the moments of second order Wiener integrals.
We refer to [1] and [4] for the notation recalled in this section. Let (B t ) t∈I R + denote a standard I R d -valued Brownian motion on the Wiener space (W, µ) with W = 0 (I R + , I R d ). For any separable Hilbert space X , consider the Malliavin derivative D with values in H = L 2 (I R + , X ⊗ I R d ), defined by . . , t n ∈ I R + , n ≥ 1. Let ID p,k (X ) denote the completion of the space of smooth X -valued random variables under the norm where X ⊗ H denotes the completed symmetric tensor product of X and H. For all p, q > 1 such that p −1 + q −1 = 1 and k ≥ 1, let denote the Skorohod integral operator adjoint of Recall that δ(u) coincides with the Itô integral of u ∈ L 2 (W ; H) with respect to Brownian motion, i.e.
when u is square-integrable and adapted with respect to the Brownian filtration.
Each element of X ⊗ H is naturally identified to a linear operator from H to X via For u ∈ ID 2,1 (H) we identify Du = (D t u s ) s,t∈I R + to the random operator Du : H → H almost surely defined by and define its adjoint D * u on H ⊗ H as and the commutation relation

Main results
First we state a moment identity for Skorohod integrals, which will be proved in Section 3.
In particular we obtain the following immediate consequence of Theorem 2.1. Recall that trace (Du) k = 0, k ≥ 1, when the process u is adapted with respect to the Brownian filtration.

Corollary 2.2.
Let n ≥ 1 and u ∈ ID n+1,2 (H) such that 〈u, u〉 H is deterministic and Then δ(u) has the same first n + 1 moments as the centered Gaussian distribution with variance 〈u, u〉 H .
Proof. The relation D〈u, u〉 = 2(D * u)u shows that We close this section with some applications.

Random rotations
As a consequence of Corollary 2.2 we recover Theorem 2.1-b) of [3], i.e. δ(Rh) has a centered Gaussian distribution with variance 〈h, h〉 H when u = Rh, h ∈ H, and R is a random mapping with values in the isometries of H, such that Rh ∈ ∩ p>1 ID p,2 (H) and trace (DRh) k+1 = 0, k ≥ 1. Note that in [3] the condition Rh ∈ ∩ p>1,k≥2 ID p,k (H) is assumed instead of Rh ∈ ∩ p>1 ID p,2 (H).

Second order Wiener integrals
Let d = 1. The second order Wiener integral I 2 ( f 2 ) of a symmetric function f 2 ∈ H ⊗ H = L 2 (I R 2 + ) can be written as I 2 ( f 2 ) = δ(u) with u t = δ( f 2 (·, t)), t ∈ I R + . Its law is infinitely divisible with Lévy measure when f 2 is decomposed as (s, t)dsd t, and using the relation hence Theorem 2.1 yields the recurrence relation , for the computation of the moments of second order Wiener integrals, by polarisation of (I 2 ( f 2 )) n−k I 2 (g (n−k+1) 2 ).

Proofs
In the sequel, all scalar products will be simply denoted by 〈·, ·〉.
We will need the following lemma.