On extended stochastic integrals with respect to Lévy processes

Abstract

Let $L$ be a Levy process on $[0,+\infty)$. In particular cases, when $L$ is a Wiener or Poisson process, any square integrable random variable can be decomposed in a series of repeated stochastic integrals from nonrandom functions with respect to $L$. This property of $L$, known as the chaotic representation property (CRP), plays a very important role in the stochastic analysis. Unfortunately, for a general Levy process the CRP does not hold.

There are different generalizations of the CRP for Levy processes. In particular, under the Ito's approach one decomposes a Levy process $L$ in the sum of a Gaussian process and a stochastic integral with respect to a Poisson random measure, and then uses the CRP for both terms in order to obtain a generalized CRP for $L$. The Nualart-Schoutens's approach consists in decomposition of a square integrable random variable in a series of repeated stochastic integrals from nonrandom functions with respect to so-called orthogonalized centered power jump processes, these processes are constructed with using of a cadlag version of $L$. The Lytvynov's approach is based on orthogonalization of continuous monomials in the space of square integrable random variables.

In this paper we construct the extended stochastic integral with respect to a Levy process and the Hida stochastic derivative in terms of the Lytvynov's generalization of the CRP; establish some properties of these operators; and, what is most important, show that the extended stochastic integrals, constructed with use of the above-mentioned generalizations of the CRP, coincide.