A Feynman-Kac result via Markov BSDEs with generalized driver

TL;DR

This paper extends the Feynman-Kac formula to BSDEs with drivers involving distributional terms, introducing an integral operator and establishing existence of solutions through PDE techniques, even when the solutions are weak Dirichlet processes.

Contribution

It introduces a novel integral operator to handle distributional drivers in BSDEs and proves the existence of solutions via PDE methods, broadening the scope of Feynman-Kac representations.

Findings

Established a Feynman-Kac formula for BSDEs with generalized drivers.

Proved existence of strong solutions using PDE techniques.

Demonstrated solutions can be weak Dirichlet processes due to driver irregularity.

Abstract

In this paper we investigate BSDEs where the driver contains a distributional term (in the sense of generalised functions) and derive general Feynman-Kac formulae related to these BSDEs. We introduce an integral operator to give sense to the equation and then we show the existence of a strong solution employing results on a related PDE.Due to the irregularity of the driver, the $Y$-component of a couple $(Y,Z)$ solving the BSDE is not necessarily a semimartingale but a weak Dirichlet process.