Optimal control of Piecewise Deterministic Markov Processes: a BSDE representation of the value function

TL;DR

This paper develops a probabilistic representation of the value function for infinite horizon optimal control problems of piecewise deterministic Markov processes using constrained backward stochastic differential equations, linking it to an auxiliary control problem.

Contribution

It introduces a novel BSDE representation for the value function of PDMP control problems, extending the nonlinear Feynman-Kac formula to this setting.

Findings

Existence and uniqueness of the constrained BSDE solution.

Representation of the value function via the BSDE.

Connection between the constrained BSDE and an auxiliary control problem.

Abstract

We consider an infinite horizon discounted optimal control problem for piecewise deterministic Markov processes, where a piecewise open-loop control acts continuously on the jump dynamics and on the deterministic flow. For this class of control problems, the value function can in general be characterized as the unique viscosity solution to the corresponding Hamilton-Jacobi-Bellman equation. We prove that the value function can be represented by means of a backward stochastic differential equation (BSDE) on infinite horizon, driven by a random measure and with a sign constraint on its martingale part, for which we give existence and uniqueness results. This probabilistic representation is known as nonlinear Feynman-Kac formula. Finally we show that the constrained BSDE is related to an auxiliary dominated control problem, whose value function coincides with the value function of the original non-dominated control problem.