BSDEs with no driving martingale, Markov processes and associated Pseudo Partial Differential Equations. Part II: Decoupled mild solutions and Examples

TL;DR

This paper introduces decoupled mild solutions for pseudo-PDEs associated with Markov processes, establishing their equivalence with martingale solutions and exploring their relation to BSDEs without driving martingales.

Contribution

It proposes the decoupled mild solution concept for pseudo-PDEs, demonstrating its equivalence to martingale solutions and analyzing well-posedness and connections to BSDEs without driving martingales.

Findings

Decoupled mild solutions are equivalent to martingale solutions.

The approach provides an alternative to viscosity solutions for non-PDE operators.

Well-posedness results are established for the proposed solutions.

Abstract

Let $(\mathbb{P}^{s,x})_{(s,x)\in[0,T]\times E}$ be a family of probability measures, where $E$ is a Polish space,defined on the canonical probability space ${\mathbb D}([0,T],E)$ of $E$-valued cadlag functions. We suppose that a martingale problem with respect to a time-inhomogeneous generator $a$ is well-posed. We consider also an associated semilinear {\it Pseudo-PDE} with generator $a$ for which we introduce a notion of so called {\it decoupled mild} solution and study the equivalence with the notion of martingale solution introduced in a companion paper. We also investigate well-posedness for decoupled mild solutions and their relations with a special class of BSDEs without driving martingale. The notion of decoupled mild solution is a good candidate to replace the notion of viscosity solution which is not always suitable when the map $a$ is not a PDE operator.