Threshold Strategies in Optimal Stopping Problem for Diffusion Processes and Free-Boundary Problem

TL;DR

This paper characterizes when optimal stopping times for diffusion processes can be represented as threshold strategies, providing conditions to identify valid solutions and exclude non-optimal free-boundary solutions.

Contribution

It establishes necessary and sufficient conditions for threshold strategies in diffusion optimal stopping problems, including second-order conditions to validate solutions.

Findings

Threshold strategies are optimal under specific conditions.

Second-order conditions help distinguish true solutions from non-optimal free-boundary solutions.

The continuation set is characterized as a semi-interval.

Abstract

We study a problem when a solution to optimal stopping problem for one-dimensional diffusion will generate by threshold strategy. Namely, we give necessary and sufficient conditions under which an optimal stopping time can be specified as the first time when the process exceeds some level (threshold), and a continuation set is a semi-interval. We give also second-order conditions, which allow to discard such solutions to free-boundary problem that are not the solutions to optimal stopping problem.