Optimal stopping time on semi-Markov processes with finite horizon

TL;DR

This paper addresses the finite-horizon optimal stopping problem for semi-Markov processes by establishing theoretical foundations, proving the existence of optimal stopping times, and providing an algorithm for their computation.

Contribution

It extends finite horizon semi-Markov decision process results to include terminal costs and introduces an explicit construction linking SMPs and SMDPs for optimal stopping analysis.

Findings

Proves the existence of optimal stopping times for SMPs.

Provides an algorithm to compute optimal stopping times.

Characterizes optimal stopping times via hitting times of specific sets.

Abstract

In this paper, we consider the optimal stopping problem on semi-Markov processes (SMPs) with finite horizon, and aim to establish the existence and computation of optimal stopping times. To achieve the goal, we first develop the main results of finite horizon semi-Markov decision processes (SMDPs) to the case with additional terminal costs, introduce an explicit construction of SMDPs, and prove the equivalence between the optimal stopping problems on SMPs and SMDPs. Then, using the equivalence and the results on SMDPs developed here, we not only show the existence of optimal stopping time of SMPs, but also provide an algorithm for computing optimal stopping time on SMPs. Moreover, we show that the optimal and "-optimal stopping time can be characterized by the hitting time of some special sets, respectively.