Local Malliavin Calculus for L\'evy Processes and Applications

TL;DR

This paper develops a Malliavin calculus framework for Le9vy processes, extending previous methods for Poisson and Gaussian processes, and applies it to analyze the absolute continuity of solutions to stochastic differential equations.

Contribution

It introduces a new family of true derivative operators for Le9vy processes, extending existing calculus methods and enabling analysis of the absolute continuity of their functionals.

Findings

Established a sufficient condition for absolute continuity of Le9vy process functionals.

Extended Malliavin calculus to include classical derivatives for Gaussian processes.

Applied the calculus to analyze solutions of stochastic differential equations.

Abstract

In this paper a Malliavin calculus for L\'evy processes based on a family of true derivative operators is developed. The starting point is an extension to L\'evy processes of the pioneering paper by Carlen and Pardoux [8] for the Poisson process, and our approach includes also the classical Malliavin derivative for Gaussian processes. We obtain a sufficient condition for the absolute continuity of functionals of the L\'evy process. As an application, we analyze the absolute continuity of the law of the solution of some stochastic differential equations.