Optimal Stopping Time on Semi-Markov Processes with Finite Horizon

Abstract

In this paper, we consider the optimal stopping problems on semi-Markov processes (sMPs) with finite horizon and aim to establish the existence and algorithm of optimal stopping times. The key method is the equivalence between optimal stopping problems on sMPs and a special class of semi-Markov decision processes (sMDPs). We first introduce the optimality equation and show the existence of the optimal policies of finite-horizon sMDPs with additional terminal costs. Based on the optimal stopping problems on sMPs, we give an explicit construction of sMDPs such that the optimal stopping times of sMPs are equivalent to the optimal policies of the constructed sMDPs. Then, using the results of sMDPs developed here, we not only prove the existence of the optimal stopping times of sMPs, but also provide an algorithm for computing the optimal stopping times of sMPs. Moreover, we show that the optimal and \(\varepsilon \)-optimal stopping time can be characterized by the hitting time of some special sets. Finally, we give an example to illustrate the effectiveness of our conclusions.