  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.
