Path-dependent Martingale Problems and Additive Functionals

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.