An algorithm to solve optimal stopping problems for one-dimensional diffusions

TL;DR

This paper presents a comprehensive algorithm for solving optimal stopping problems in one-dimensional diffusions, leveraging Dynkin's characterization, Riesz's representation, and an inversion formula to determine the value function and stopping region.

Contribution

It introduces a novel, guaranteed-convergence algorithm that computes the optimal stopping rule and value function without requiring verification, applicable to generalized diffusion models.

Findings

Algorithm reliably finds the optimal stopping time and value function.

Method handles diffusions with atoms and non-smooth payoffs.

Provides explicit shape of the stopping region.

Abstract

Considering a real-valued diffusion, a real-valued reward function and a positive discount rate, we provide an algorithm to solve the optimal stopping problem consisting in finding the optimal expected discounted reward and the optimal stopping time at which it is attained. Our approach is based on Dynkin's characterization of the value function. The combination of Riesz's representation of $\alpha$-excessive functions and the inversion formula gives the density of the representing measure, being only necessary to determine its support. This last task is accomplished through an algorithm. The proposed method always arrives to the solution, thus no verification is needed, giving, in particular, the shape of the stopping region. Generalizations to diffusions with atoms in the speed measure and to non smooth payoffs are analyzed.