Cumulant operators and moments of the Itô and Skorohod integrals

We propose a formula for the computation of the moments of all orders of Itô and Skorohod stochastic integrals with respect to Brownian motion, based on cumulant operators defined by the Malliavin calculus. Some characterizations of Gaussian distributions for stochastic integrals are recovered as a consequence.     

On the Existence of Weak Local in Time Solutions in the Form of a Cumulant Expansion for a Chain of Bogolyubov’s Equations of a One-Dimensional Symmetric Particle System

We consider a Cauchy problem for a chain of Bogolyubov equations of an infinite onedimensional symmetric particle system, where the particles interact with each other by a finite-range pair potential with a hard core. We consider it in the space of sequences of bounded measurable functions. Based on the proposed method of a joint interval for estimates of the volume of the interaction domain and on the derived estimate itself we find a representation of a weak local with respect to time solution in the form of a cumulant expansion. We prove that the considered weak local with respect to time solution is an equilibrium solution if the initial data are equilibrium distribution functions.