Anticipative Integrals with respect to a filtered Lévy Process and Lévy-Itô decomposition
Résumé
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 Lévy process, this is not possible. Alternatively, we can use the integrals defined by means of the so called S-transform or by means of the integral with respect to the process X and a linear operator K constructed from k. The usual fact that even for predictable Y , K * (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 Lévy process L, roughly speaking L = B + J, where Bis a Brownian motion and J is a pure jump Lévy process, behaves with respect to these integrals.