Path-dependent SDEs with jumps and irregular drift: well-posedness and Dirichlet properties

TL;DR

This paper investigates path-dependent stochastic differential equations with jumps and irregular drift, establishing existence and uniqueness of solutions, and exploring their Dirichlet process properties with new theoretical insights.

Contribution

It introduces a framework for analyzing path-dependent SDEs with distributional drift and jumps, providing novel results on their well-posedness and Dirichlet process characteristics.

Findings

Existence and uniqueness of solutions via a martingale problem

Examples of solutions that are not Dirichlet processes

New theoretical results on Dirichlet process class

Abstract

We discuss a concept of path-dependent SDE with distributional drift with possible jumps. We interpret it via a suitable martingale problem, for which we provide existence and uniqueness. The corresponding solutions are expected to be Dirichlet processes, nevertheless we give examples of solutions which do not fulfill this property. In the second part of the paper we indeed state and prove significant new results on the class of Dirichlet processes.