Equations Involving Malliavin Derivative: A Chaos Expansion Approach

Abstract

We study equations involving the Malliavin derivative operator and the Wick product with a Gaussian process. In particular, we solve an equation with first-order Malliavin derivative operator by the chaos expansion method in white noise spaces. We prove necessary and sufficient conditions for existence and uniqueness of the solution and represent it in explicit way. We characterize the domains of the Malliavin operators in spaces of Kondratiev distributions in general form. In addition, as an illustration we apply stochastic Galerkin method for solving numerically a stationary version of the equation we considered.