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

Understand

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.