Abstract
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 A. S. Ustunel and M. Zakai Prob. Th. Rel. Fields 103 (1995), 409-429.
Highlights
Introduction and notationIn [3], sufficient conditions have been found for the Skorohod integral δ(Rh) to have a Gaussian law when h ∈ H = L2(IR+, IRd ) 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
For u ∈ ID2,1(H) we identify Du = (Dt us)s,t∈RI + to the random operator Du : H → H almost surely defined by
(D∗u)v(s) = (Ds†ut )vt d t, s ∈ IR+, v ∈ L2(W ; H), where Ds†ut denotes the transpose matrix of Dsut in IRd ⊗ IRd
Summary
Let (Bt )t∈RI + denote a standard IRd -valued Brownian motion on the Wiener space (W, μ) with W = 0(IR+, IRd ). Q > 1 such that p−1 + q−1 = 1 and k ≥ 1, let δ : IDp,k(X ⊗ H) → IDq,k−1(X ) denote the Skorohod integral operator adjoint of. For u ∈ ID2,1(H) we identify Du = (Dt us)s,t∈RI + to the random operator Du : H → H almost surely defined by (Du)v(s) = (Dt us)vt d t, s ∈ IR+, v ∈ L2(W ; H), and define its adjoint D∗u on H ⊗ H as (D∗u)v(s) = (Ds†ut )vt d t, s ∈ IR+, v ∈ L2(W ; H), where Ds†ut denotes the transpose matrix of Dsut in IRd ⊗ IRd. E[δ(u)2] = E[〈u, u〉H ] + E trace (Du) , u ∈ ID2,1(H), with trace (Du)2 = 〈Du, D∗u〉H⊗H.
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