Anticipative integrals with respect to a filtered Lévy process and Lévy-Itô decomposition

Abstract

A filtered process $X^k$ is defined as an integral of a deterministic kernel $k$ with respect to a stochastic process $X$. One of the main problems to deal with such processes is to define a stochastic integral with respect to them. When $X$ is a Brownian motion one can use the Gaussian properties of $X^k$ to define an integral intrinsically. When $X$ is a jump process or a Levy process, this is not possible. Alternatively, we can use the integrals defined by means of the so called $\mathcal{S}$-transform or by means of the integral with respect to the process $X$ and a linear operator $\mathcal{K}$ constructed from $k$. The usual fact that even for predictable $Y$, $K^{\ast}(Y)$ may not be predictable forces us to consider only anticipative integrals. The aim of this paper is, on the one hand, to clarify the links between these integrals for a given $X$ and on the other hand, to investigate how the Lévy-Itô decomposition of a Levy process $L$, roughly speaking $L=B+J$, where $B$ is a Brownian motion and $J$ is a pure jump Lévy process, behaves with respect to these integrals.