Path-dependent Martingale Problems and Additive Functionals

TL;DR

This paper extends the classical Markov process framework to a path-dependent setting, introducing the concept of path-dependent canonical classes and generalizing semi-groups and additive functionals for path-dependent stochastic processes.

Contribution

It develops the theory of path-dependent canonical classes, generalizes semi-group and additive functional notions, and provides a foundation for analyzing path-dependent martingale problems and SDEs with jumps.

Findings

Introduces path-dependent canonical classes indexed by starting paths.

Generalizes semi-group and additive functional concepts to the path-dependent context.

Lays groundwork for future analysis of path-dependent SDEs and BSDEs.

Abstract

The paper introduces and investigates the natural extension to the path-dependent setup of the usual concept of canonical Markov class introduced by Dynkin and which is at the basis of the theory of Markov processes. That extension, indexed by starting paths rather than starting points will be called path-dependent canonical class. Associated with this is the generalization of the notions of semi-group and of additive functionals to the path-dependent framework. A typical example of such family is constituted by the laws $({\mathbb P}^{s,$\eta$})\_{(s,\eta) \in{\mathbb R} \times \Omega}$ , where for fixed time s and fixed path $\eta$ defined on [0, s], $({\mathbb P}^{s,$\eta$})\_{(s,\eta) \in {\mathbb R} \times \Omega}$ being the (unique) solution of a path-dependent martingale problem or more specifically a weak solution of a path-dependent SDE with jumps, with initial path $\eta$. In a companion paper we apply those results to study path-dependent analysis problems associated with BSDEs.