On optimal stopping of multidimensional diffusions

TL;DR

This paper introduces a novel approach for solving perpetual discounted optimal stopping problems for multidimensional diffusions, especially Wiener processes, using Green kernel and Martin boundary theories to characterize stopping regions.

Contribution

It develops verification theorems and integral equations that simplify the characterization of stopping regions in multidimensional diffusion problems.

Findings

Derived tractable integral equations for stopping regions.

Applied the method to Wiener processes with quadratic reward functions.

Provided a discretization scheme for approximate solutions.

Abstract

This paper develops an approach for solving perpetual discounted optimal stopping problems for multidimensional diffusions, with special emphasis on the $d$-dimensional Wiener process. We first obtain some verification theorems for diffusions, based on the Green kernel representation of the value function associated with the problem. Specializing to the multidimensional Wiener process, we apply the Martin boundary theory to obtain a set of tractable integral equations involving only harmonic functions that characterize the stopping region of the problem. These equations allow to formulate a discretization scheme to obtain an approximate solution. The approach is illustrated through the optimal stopping problem of a $d$-dimensional Wiener process with a positive definite quadratic form reward function.