Integration by parts formula and applications to equations with jumps

TL;DR

This paper develops an integration by parts formula within an abstract framework to analyze the regularity of solutions to stochastic differential equations with jumps, especially those with discontinuous coefficients where traditional Malliavin calculus methods are ineffective.

Contribution

It introduces a novel integration by parts formula applicable to jump processes with discontinuous coefficients, extending the tools for studying regularity of such stochastic equations.

Findings

Established an integration by parts formula for jump processes

Analyzed regularity of solutions with discontinuous coefficients

Provided a new approach where Malliavin calculus fails

Abstract

We establish an integration by parts formula in an abstract framework in order to study the regularity of the law for processes solution of stochastic differential equations with jumps, including equations with discontinuous coefficients for which the Malliavin calculus developed by Bismut and Bichteler, Gravereaux and Jacod fails.