On piecewise deterministic Markov control processes: Control of jumps and of risk processes in insurance

Dynamic programming for piecewise deterministic Markov processes is studied where only the jumps but not the deterministic flow can be controlled. Then one can dispense with relaxed controls. There exists an optimal stationary policy of feedback form. Further, a piecewise deterministic Markov model for the control of dividend pay-out and reinsurance is introduced. This model can be transformed to a model with uncontrolled flow. It is shown that a classical solution to the Bellman equation exists and that a non-relaxed optimal policy of feedback form can be obtained via the Bellman equation. Lipschitz continuity of the one-dimensional vector field defining the controlled flow will be replaced by strict positivity.