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

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.

Automated summary

This article introduces an algorithm for solving the optimal stopping problem. By using Dynkin's characterization of the value function and Riesz's representation and inversion formula for alpha-excessive functions, we can obtain the density of the representing measure and determine its support. Through this algorithm, we can obtain the solution without verification and obtain the shape of the stopping region. The article also analyzes the generalizations to diffusions with atoms in the speed measure and non-smooth payoffs.