The lent particle method for marked point processes

TL;DR

This paper extends the lent particle method to marked point processes, utilizing Dirichlet forms for Malliavin calculus on marks, simplifying proofs and broadening applicability beyond Poisson measures.

Contribution

It introduces a generalized lent particle method for marked point processes using Dirichlet forms, applicable to non-Poisson processes and simplifying existing proofs.

Findings

Applicable to isotropic processes

Handles processes with jump modifications by diffusions

Simplifies the proof structure of the method

Abstract

Although introduced in the case of Poisson random measures, the lent particle method applies as well in other situations. We study here the case of marked point processes. In this case the Malliavin calculus (here in the sense of Dirichlet forms) operates on the marks and the point process doesn't need to be Poisson. The proof of the method is even much simpler than in the case of Poisson random measures. We give applications to isotropic processes and to processes whose jumps are modified by independent diffusions.