Optimal stopping and control near boundaries

TL;DR

This paper explores the characterization of optimal stopping and singular control problems near boundaries using fundamental ratios, revealing their local nature and illustrating findings with various examples.

Contribution

It introduces a ratio-based approach to characterize solutions of boundary-related control problems, emphasizing their local properties over global ones.

Findings

Ratios unambiguously characterize solutions near boundaries

The connection between problems is local, not global

Illustrated with multiple examples

Abstract

We will investigate the value and inactive region of optimal stopping and one-sided singular control problems by focusing on two fundamental ratios. We shall see that these ratios unambiguously characterize the solution, although usually only near boundaries. We will also study the well-known connection between these problems and find it to be a local property rather than a global one. The results are illustrated by a number of examples.