A General Method for Finding the Optimal Threshold in Discrete Time

TL;DR

This paper introduces a general method for determining optimal thresholds in discrete-time one-sided stopping problems involving Markov processes, simplifying solutions by transforming the problem into an auxiliary ladder height problem.

Contribution

The authors present a novel approach that reduces complex stopping problems to auxiliary problems, providing a unified framework and explicit threshold characterization for various models.

Findings

Applicable to prominent examples like Novikov-Shiryaev and Shepp-Shiryaev problems

Provides a sufficient condition for the method's validity

Characterizes reward functions leading to one-sided stopping as monotone and log-concave

Abstract

We develop an approach for solving one-sided optimal stopping problems in discrete time for general underlying Markov processes on the real line. The main idea is to transform the problem into an auxiliary problem for the ladder height variables. In case that the original problem has a one-sided solution and the auxiliary problem has a monotone structure, the corresponding myopic stopping time is optimal for the original problem as well. This elementary line of argument directly leads to a characterization of the optimal boundary in the original problem: The optimal threshold is given by the threshold of the myopic stopping time in the auxiliary problem. Supplying also a sufficient condition for our approach to work, we obtain solutions for many prominent examples in the literature, among others the problems of Novikov-Shiryaev, Shepp-Shiryaev, and the American put in option pricing under general conditions. As a further application we show that for underlying random walks (and L\'evy processes in continuous time), the reward functions $g$ leading to one-sided stopping problems are exactly the monotone and log-concave functions.