Explicit Solutions for Optimal Stopping of Linear Diffusion and its Maximum

TL;DR

This paper develops explicit solutions for optimal stopping problems involving diffusion processes and their maxima by employing excursion theory and measure change techniques, simplifying complex two-dimensional problems into manageable one-dimensional solutions.

Contribution

It introduces a novel approach combining excursion theory and measure change to solve complex optimal stopping problems for diffusions and their maxima.

Findings

Explicit value functions derived for diffusion and maximum problems

Method reduces two-dimensional problems to one-dimensional solutions

Examples illustrating the application of the solution methods

Abstract

We provide, in a general setting, explicit solutions for optimal stopping problems that involve diffusion process and its running maximum. Our approach is to use the excursion theory for Levy processes. Since general diffusions are, in particular, not of independent increments, we use an appropriate measure change to make the process have that property. Then we rewrite the original two-dimensional problem as an infinite number of one-dimensional ones and complete the solution. We show general solution methods with explicit value functions and corresponding optimal strategies, illustrating them by some examples.