Variational inequalities for the optimal stopping with continuous control of piecewise deterministic Markov processes

We consider the problem of optimal stopping and continuous control on some local parameters of a piecewise-deterministic Markov processes (PDPs). Optimality equations are obtained in terms of the infinitesimal operator as well as on the first jump time operator of the PDP. It is shown that if the final cost function is absolutely continuous along trajectories then so is the value function of the optimal stopping problem with continuous control. These results unify and generalize previous ones in the current literature