Piecewise Deterministic Markov Decision Processes

Abstract

In this chapter we deal with optimization problems where the state process is a Piecewise Deterministic Markov Process. These processes evolve through random jumps at random time points while the behavior between jumps is governed by an ordinary differential equation. They form a general and important class of non-diffusions. It is known that every strongMarkov process with continuous paths of bounded variation is necessarily deterministic.We assume that both the jump behavior as well as the drift behavior between jumps can be controlled. Hence this leads to a control problem in continuous-time which can be tackled for example via the Hamilton-Jacobi-Bellman equation. However, since the evolution between jumps is deterministic these problems can also be reduced to a discrete-time Markov Decision Process where however the action space is now a function space. We can treat these problems with the methods we have established in the previous chapters. More precisely we will restrict the presentation to problems with infinite horizon, thus we will use the results of Chapter 7. We show that under some continuity and compactness conditions the value function of the Piecewise Deterministic Markov Decision Process is a fixed point of the Bellman equation (Theorem 8.2.6) and the computational methods of Chapter 7 apply. In Section 8.3 the important special class of continuous-time Markov Decision Chains is investigated, in particular for problems with finite time horizon.