Variational View to Optimal Stopping Problems for Diffusion Processes and Threshold Strategies

TL;DR

This paper introduces a variational approach to solve optimal stopping problems for diffusion processes, focusing on threshold strategies and providing conditions for their optimality, with applications in investment and abandonment models.

Contribution

It presents a novel variational framework as an alternative to free-boundary methods and characterizes when threshold strategies are optimal.

Findings

Threshold strategies can simplify optimal stopping problems.

Necessary and sufficient conditions for threshold optimality are derived.

Applications include investment timing and abandonment models.

Abstract

We describe a variational approach to solving optimal stopping problems for diffusion processes, as an alternative to the traditional approach based on the solution of the free-boundary problem. We study smooth pasting conditions from a variational point of view, and give some examples when the solution to free-boundary problem is not the solution to optimal stopping problem. A special attention is paid to threshold strategies which allow reduce optimal stopping problem to more simple one-parametric optimization. Necessary and sufficient conditions for threshold structure of optimal stopping time are derived. We apply these results to both investment timing and optimal abandon models.