Cumulant operators and moments of the Itô and Skorohod integrals

where the sum runs over the partitions B1, . . . , Ba of {1, . . . , n} with cardinal |Bi| by the Faà di Bruno formula, cf. [5], [6] and references therein for background on combinatorial probability. When X is centered Gaussian, e.g. X is the Wiener integral of a deterministic function with respect to a standard Brownian motion (Bt)t∈R+ , we have κ X n = 0, n 6= 2, and (1.1) reads as Wick’s theorem for the computation of Gaussian moments of X counting the pair partitions of {1, . . . , n}, cf. [1].