Special weak Dirichlet processes and BSDEs driven by a random measure

TL;DR

This paper studies backward stochastic differential equations driven by a random measure, with the forward process being a special weak Dirichlet process, and identifies key solution components using stochastic calculus tailored for such processes.

Contribution

It extends the theory of BSDEs to cases where the forward process is a special weak Dirichlet process, providing explicit characterizations of solution components.

Findings

Identification of Z and U in terms of the deterministic function u

Application of stochastic calculus for weak Dirichlet processes

Extension of BSDE theory to more general semimartingale processes

Abstract

This paper considers a forward BSDE driven by a random measure, when the underlying forward process X is special semimartingale, or even more generally, a special weak Dirichlet process. Given a solution (Y, Z, U), generally Y appears to be of the type u(t, X\_t) where u is a deterministic function. In this paper we identify Z and U in terms of u applying stochastic calculus with respect to weak Dirichlet processes.