Application of the lent particle method to Poisson driven SDE's

TL;DR

This paper introduces the lent particle method, leveraging Dirichlet forms and Malliavin calculus, to simplify the analysis of Poisson-driven SDEs with jumps, reducing assumptions and computational complexity.

Contribution

It presents a novel explicit formula for the Malliavin matrix and introduces the lent particle method for more efficient analysis of jump SDEs.

Findings

Explicit formula for the Malliavin matrix derived

Simplified computations for Poisson-driven SDEs demonstrated

Reduced assumptions on coefficients in jump SDE analysis

Abstract

We apply the Dirichlet forms version of Malliavin calculus to stochastic differential equations with jumps. As in the continuous case this weakens significantly the assumptions on the coefficients of the SDE. In spite of the use of the Dirichlet forms theory, this approach brings also an important simplification which was not available nor visible previously : an explicit formula giving the carr\'e du champ matrix, i.e. the Malliavin matrix. Following this formula a new procedure appears, called the lent particle method which shortens the computations both theoretically and in concrete examples.