Optimal control of piecewise deterministic Markov processes with finite time horizon

In this paper we study controlled Piecewise Deterministic Markov Processes with finite time horizon and unbounded rewards. Using an embedding procedure we reduce these problems to discrete-time Markov Decision Processes. Under some continuity and compactness conditions we establish the existence of an optimal policy and show that the value function is the unique solution of the Bellman equation. It is remarkable that this statement is true for unbounded rewards and without any contraction assumptions. Further conditions imply the existence of optimal nonrelaxed controls. We highlight our findings by two examples from financial mathematics.