Explicit solutions in one-sided optimal stopping problems for one-dimensional diffusions

TL;DR

This paper derives explicit solutions for one-sided optimal stopping problems involving one-dimensional diffusions, providing formulas for thresholds and value functions, and examining the smooth fit principle's validity.

Contribution

It introduces explicit equations and formulas for optimal stopping thresholds and value functions in one-sided diffusion problems, including cases where smooth fit fails.

Findings

Explicit equations for stopping thresholds

Formulas for value functions in diffusion problems

Analysis of smooth fit principle validity

Abstract

Consider the optimal stopping problem of a one-dimensional diffusion with positive discount. Based on Dynkin's characterization of the value as the minimal excessive majorant of the reward and considering its Riesz representation, we give an explicit equation to find the optimal stopping threshold for problems with one-sided stopping regions, and an explicit formula for the value function of the problem. This representation also gives light on the validity of the smooth fit principle. The results are illustrated by solving some classical problems, and also through the solution of: optimal stopping of the skew Brownian motion, and optimal stopping of the sticky Brownian motion, including cases in which the smooth fit principle fails.