Explicit Solutions for Optimal Stopping of Maximum Process with Absorbing Boundary that Varies with It

TL;DR

This paper develops explicit solutions for optimal stopping problems involving diffusions and their maximums, incorporating a flexible, state-dependent absorbing boundary, using excursion theory and measure change techniques.

Contribution

It introduces a novel approach to solve optimal stopping problems with variable absorbing boundaries by leveraging excursion theory and measure change, extending previous methods.

Findings

Explicit value functions derived for the stopping problem

Optimal strategies characterized for various boundary conditions

Method demonstrated through illustrative examples

Abstract

We provide, in a general setting, explicit solutions for optimal stopping problems that involve a diffusion process and its running maximum. Besides, a new feature includes absorbing boundaries that vary with the value of the running maximum. The existence of the absorbing boundary of this type makes the problem harder but more practical and flexible. 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.