The Lent Particle Method, Application to Multiple Poisson Integrals

TL;DR

This paper introduces the lent particle method, a novel approach for applying Dirichlet form theory to Poisson measures, enabling the analysis of densities of jump process functionals with reduced regularity assumptions.

Contribution

It provides an explicit formula for the gradient on Poisson space and extends the method to multiple Poisson integrals, connecting with Fock space and second quantization.

Findings

Explicit lent particle formula for the gradient

Application to density existence of jump process functionals

Connections with Fock space and second quantization

Abstract

We give a extensive account of a recent new way of applying the Dirichlet form theory to random Poisson measures. The main application is to obtain existence of density for thelaws of random functionals of L\'evy processes or solutions of stochastic differential equations with jumps. As in the Wiener case the Dirichlet form approach weakens significantly theregularity assumptions. The main novelty is an explicit formula for the gradient or for the "carr\'e du champ' on the Poisson space called the lent particle formula because based on adding a new particle to the system, computing the derivative of the functional with respect to this new argument and taking back this particle before applying the Poisson measure. The article is expository in its first part and based on Bouleau-Denis [12] with several new examples, applications to multiple Poisson integrals are gathered in the last part which concerns the relation with the Fock space and some aspects of the second quantization.