  We consider an optimal control problem for piecewise deterministic Markov
processes (PDMPs) on a bounded state space. The control problem under study is
very general: a pair of controls acts continuously on the deterministic flow
and on the two transition measures (in the interior and from the boundary of
the domain) describing the jump dynamics of the process. For this class of
control problems, the value function can be characterized as the unique
viscosity solution to the corresponding fully-nonlinear Hamilton-Jacobi-Bellman
equation with a non-local type boundary condition. By means of the recent
control randomization method, we are able to provide a probabilistic
representation for the value function in terms of a constrained backward
stochastic differential equation (BSDE), known as nonlinear Feynman-Kac
formula. This result considerably extends the existing literature, where only
the case with no jumps from the boundary is considered. The additional boundary
jump mechanism is described in terms of a non quasi-left-continuous random
measure and induces predictable jumps in the PDMP's dynamics. The existence and
uniqueness results for BSDEs driven by such a random measure are non trivial,
even in the unconstrained case, as emphasized in the recent work [2].
