One-sided solutions for optimal stopping problems with logconcave reward functions

TL;DR

This paper proves that optimal stopping problems with increasing, logconcave, right-continuous reward functions have one-sided solutions for general random walks and Lévy processes, extending known results and analyzing the smooth fit principle.

Contribution

It generalizes the class of reward functions for which one-sided solutions are guaranteed in optimal stopping problems involving Lévy processes and random walks.

Findings

All increasing, logconcave, right-continuous reward functions admit one-sided solutions.

The principle of smooth fit is examined for Lévy processes with such reward functions.

Explicit solutions are known for specific reward functions like $(x^+)^{ u}$ and $(e^x-K)^+$.

Abstract

In the literature on optimal stopping, the problem of maximizing the expected discounted reward over all stopping times has been explicitly solved for some special reward functions (including $(x^+)^{\nu}$, $(e^x-K)^+$, $(K-e^{-x})^+$, $x\in\mathbb{R}$, $\nu\in(0,\infty)$ and $K>0$) under general random walks in discrete time and L\'evy processes in continuous time (subject to mild integrability conditions). All of such reward functions are continuous, increasing and logconcave while the corresponding optimal stopping times are of threshold type (i.e. the solutions are one-sided). In this paper, we show that all optimal stopping problems with increasing, logconcave and right-continuous reward functions admit one-sided solutions for general random walks and L\'evy processes. We also investigate in detail the principle of smooth fit for L\'evy processes when the reward function is increasing and logconcave.