Time and Ratio Expected Average Cost Optimality for Semi-Markov Control Processes on Borel Spaces

We deal with semi-Markov control models with Borel state and control spaces, and unbounded cost functions under the ratio and the time expected average cost criteria. Under suitable growth conditions on the costs and the mean holding times together with stability conditions on the embedded Markov chains, we show the following facts: (i) the ratio and the time average costs coincide in the class of the stationary policies; (ii) there exists a stationary policy which is optimal for both criteria. Moreover, we provide a generalization of the classical Wald's Lemma to semi-Markov processes. These results are obtained combining the existence of solutions of the average cost optimality equation and the Optional Stopping Theorem.