Chaos expansion methods for stochastic differential equations involving the Malliavin derivative, Part I

Abstract

We consider Gaussian, Poissonian, fractional Gaussian and fractional Poissonian white noise spaces, all represented through the corresponding orthogonal basis of the Hilbert space of random variables with finite second moments, given by the Hermite and the Charlier polynomials. There exist unitary mappings between the Gaussian and Poissonian white noise spaces. We investigate the relationship of the Malliavin derivative, the Skorokhod integral, the Ornstein-Uhlenbeck operator and their fractional counterparts on a general white noise space.