Stopping with expectation constraints: 3 points suffice

Stefan Ankirchner,Maike Klein,Thomas Kruse,Nabil Kazi-Tani

doi:10.1214/19-ejp309

Abstract

We consider the problem of optimally stopping a one-dimensional regular continuous strong Markov process with a stopping time satisfying an expectation constraint. We show that it is sufficient to consider only stopping times such that the law of the process at the stopping time is a weighted sum of 3 Dirac measures. The proof uses recent results on Skorokhod embeddings in order to reduce the stopping problem to a linear optimization problem over a convex set of probability measures.

Highlights

Let (Yt)t∈R≥0 be a one-dimensional regular continuous strong Markov process with respect to a right-continuous filtration (Ft)
The problem (1.1) arises whenever an average time constraint applies for any stopping rule
In this article we show that for the stopping problem (1.1) it is sufficient to consider only stopping times τ such that the law of Yτ is a weighted sum of at most 3 Dirac measures

Summary

Introduction

Let (Yt)t∈R≥0 be a one-dimensional regular continuous strong Markov process with respect to a right-continuous filtration (Ft). In this article we show that for the stopping problem (1.1) it is sufficient to consider only stopping times τ such that the law of Yτ is a weighted sum of at most 3 Dirac measures. Our idea for proving a reduction to 3 Dirac measures is to rewrite the stopping problem (1.1) as a linear optimization problem over a set of probability measures To this end we use recent results on the Skorokhod embedding problem characterizing the set A(T ) of probability distributions that can be embedded into Y with stopping times having expectation smaller than or equal to T (see [1] and [13]). Different constraints have been recently studied: Bayraktar and Miller [4] consider the problem of optimally stopping the Brownian motion with a stopping time whose distribution is atomic with finitely many points of mass.

Stopping after consecutive exit times

Optimal stopping as a measure optimization

Existence of an optimizer