Cumulant operators and moments of the Itô and Skorohod integrals (Q357427)

Consider a variable \(F\) and a process \(u_t\) defined on the Wiener space. Denote by \(\delta(u)\) the Skorohod integral of \(u\). The classical duality formula of Malliavin's calculus states that the expectation \(E[F\,\delta(u)]\) is equal to \(E[\langle DF,u\rangle_H]\) for the gradient operator \(D\). The aim of this work is to obtain more generally a formula for \(E[F\,\delta(u)^n]\). This formula involves some cumulant operators which are defined in terms of \(D\).