Weak Dirichlet processes and generalized martingale problems

TL;DR

This paper introduces weak Dirichlet processes as a generalization of semimartingales with jumps, providing new decompositions, characteristics, and a framework for martingale problems involving jumps and distributional drifts.

Contribution

It develops the theory of weak Dirichlet processes, including unique decompositions and characteristics, and extends martingale problem frameworks to include path-dependent jumps with distributional drifts.

Findings

Unique decomposition for weak Dirichlet processes

Introduction of characteristics for these processes

Framework for martingale problems with jumps and distributional drifts

Abstract

In this paper we explain how the notion of ''weak Dirichlet process'' is the suitable generalization of the one of semimartingale with jumps. For such a process we provide a unique decomposition which is new also for semimartingales: in particular we introduce ''characteristics'' for weak Dirichlet processes. We also introduce a weak concept (in law) of finite quadratic variation. We investigate a set of new useful chain rules and we discuss a general framework of (possibly path-dependent with jumps) martingale problems with a set of examples of SDEs with jumps driven by a distributional drift.