On chaos representation and orthogonal polynomials for the doubly stochastic Poisson process

Abstract

In an L₂-framework, we study various aspects of stochastic calculus with respect to the centered doubly stochastic Poisson process. We introduce an orthogonal basis via multilinear forms of the value of the random measure and we analyze the chaos representation property. We revise the structure of non-anticipating integration for martingale random elds and in this framework we study non-anticipating differentiation.We present integral representation theorems where the integrand is explicitely given by the non-anticipating derivative. Stochastic derivatives of anticipative nature are also considered: The Malliavin type derivative is put in relationship with another anticipative derivative operator here introduced. This gives a new structural representation of the Malliavin derivative based on simple functions. Finally we exploit these results to provide a Clark-Ocone type formula for the computation of the non-anticipating derivative.