Optimal Stopping and the Sufficiency of Randomized Threshold Strategies

TL;DR

This paper extends classical optimal stopping theory to problems where the value depends on the law of the stopped process, showing that mixtures of threshold strategies are sufficient even when the objective is not quasi-convex.

Contribution

It demonstrates that in non-quasi-convex optimal stopping problems, mixtures of threshold strategies suffice, broadening the scope of optimal stopping solutions.

Findings

Mixtures of threshold strategies are sufficient for non-quasi-convex problems.

Classical threshold strategies are sufficient when the value function is quasi-convex.

The paper generalizes optimal stopping theory beyond traditional paradigms.

Abstract

In a classical optimal stopping problem the aim is to maximize the expected value of a functional of a diffusion evaluated at a stopping time. This note considers optimal stopping problems beyond this paradigm. We study problems in which the value associated to a stopping rule depends on the law of the stopped process. If this value is quasi-convex on the space of attainable laws then it is a well known result that it is sufficient to restrict attention to the class of threshold strategies. However, if the objective function is not quasi-convex, this may not be the case. We show that, nonetheless, it is sufficient to restrict attention to mixtures of threshold strategies.