Characterizing the Structure of Optimal Stopping Policies

Abstract

This paper studies a stochastic model of optimal stopping processes that arise frequently in operational problems (e.g., when a manager needs to determine an optimal epoch to stop a process). For such problems, we propose an effective method that characterizes the structure of the optimal stopping policy for the class of discrete-time optimal stopping problems. Using the method, we also provide a set of metatheorems that characterize when a threshold or control-band type stopping policy is optimal. We show that our proposed method can characterize the structure of the optimal policy for some stopping problems for which conventional methods fail to do so. Our method also simplifies the analysis of some existing results. In addition, the metatheorems help identify sufficient conditions that yield simple optimal policies when such policies are not generally optimal. We show the aforementioned benefits of our method by applying it to several optimal stopping problems frequently encountered, for example, in operations, marketing, finance and economics literature. We remark that structural results make an optimal-stopping policy easier to follow, describe, compute and hence implement. They also help understand how a stopping policy should respond to changes in the operational environment. In addition, structural results are critical for the development of efficient algorithms to solve optimal stopping problems numerically.

Keywords: optimal stopping, structure of optimal policy, threshold policy, control-band policy