On the Poisson equation for piecewise-deterministic Markov processes

In this paper we study the problem of the existence of a solution for the Poisson equation (PE) associated to a piecewise-deterministic Markov process (PDP). It is well known that the long run average cost of a stochastic process can be obtained through a solution of the PE associated with the process. Our first result will show that the existence of a solution for the PE of a PDP is equivalent to the existence of a solution for the PE of an embedded discrete-time Markov chain associated with the PDP. It is important to point out that, due to the possibility of jumps from the boundary, the differential formula for the PDPs has a special form, so that general results for the PE of continuous-time stochastic processes cannot be directly applied. Usually discrete-time conditions for the existence of a solution of a PE of a Markov chain are easier to apply than the continuous-time counterpart. We follow this approach to derive our second result, which establishes sufficient conditions for the existence of a PE to the embedded Markov chain, and consequently for the PE of the PDP. The condition is illustrated with an application to the capacity expansion model.