Backward stochastic differential equations associated to jump Markov processes and applications

TL;DR

This paper develops a framework for backward stochastic differential equations driven by jump Markov processes, applying it to nonlinear PDEs and optimal control problems, thereby linking probabilistic methods with classical equations.

Contribution

It introduces a novel approach to BSDEs associated with jump Markov processes and applies this to establish well-posedness of nonlinear PDEs and solve optimal control problems.

Findings

Proved well-posedness of a class of nonlinear parabolic equations.

Established a probabilistic representation for the Hamilton-Jacobi-Bellman equation.

Linked the value function and optimal control law to BSDEs.

Abstract

In this paper we study backward stochastic differential equations (BSDEs) driven by the compensated random measure associated to a given pure jump Markov process X on a general state space K. We apply these results to prove well-posedness of a class of nonlinear parabolic differential equations on K, that generalize the Kolmogorov equation of X. Finally we formulate and solve optimal control problems for Markov jump processes, relating the value function and the optimal control law to an appropriate BSDE that also allows to construct probabilistically the unique solution to the Hamilton-Jacobi-Bellman equation and to identify it with the value function.