Optimal stopping involving a diffusion and its running maximum: a generalisation of the maximality principle

TL;DR

This paper generalizes the maximality principle for optimal stopping problems involving diffusions and their running maximum, linking the optimal boundary to solutions of a nonlinear ODE derived from the variational inequality.

Contribution

It extends the maximality principle to a broader class of optimal stopping problems by connecting the boundary function to solutions of a nonlinear ODE related to the variational inequality.

Findings

Derived a generalized maximality principle for optimal stopping boundaries.

Connected the boundary function to solutions of a nonlinear ODE.

Provided a framework for solving more complex stopping problems.

Abstract

The maximality principle has been a valuable tool in identifying the free-boundary functions that are associated with the solutions to several optimal stopping problems involving one-dimensional time-homogeneous diffusions and their running maximum processes. In its original form, the maximality principle identifies an optimal stopping boundary function as the maximal solution to a specific first-order nonlinear ODE that stays strictly below the diagonal in $\mathbb{R}^2$. In the context of a suitably tailored optimal stopping problem, we derive a substantial generalisation of the maximality principle: the optimal stopping boundary function is the maximal solution to a specific first-order nonlinear ODE that is associated with a solution to the optimal stopping problem's variational inequality.