Iteration of the lent particle method for existence of smooth densities of Poisson functionals

TL;DR

This paper extends the lent particle method to prove the smoothness of densities for Poisson functionals, constructing Sobolev spaces of any order and applying the results to Poisson-driven SDEs.

Contribution

It introduces an iterative approach to the lent particle method, enabling the proof of smoothness of densities for Poisson functionals and related SDEs.

Findings

Established Sobolev spaces of arbitrary order for Poisson functionals

Proved a Malliavin-type criterion for smooth density existence

Applied the method to non-trivial Poisson-driven SDE examples

Abstract

In previous works we have introduced a new method called the lent particle method which is an efficient tool to establish existence of densities for Poisson functionals. We now go further and iterate this method in order to prove smoothness of densities. More precisely, we construct Sobolev spaces of any order and prove a Malliavin-type criterion of existence of smooth density. We apply this approach to SDE's driven by Poisson random measures and also present some non-trivial examples to which our method applies.