Discrete approximations to the double-obstacle prtoblem, and optimal stopping of tug-of-war games

TL;DR

This paper connects the double-obstacle problem for the p-Laplace operator with tug-of-war games involving stopping times, providing a new discrete approximation method and numerical scheme.

Contribution

It introduces a novel link between double-obstacle PDE solutions and tug-of-war game strategies with stopping times, including a practical numerical scheme.

Findings

Viscosity solutions are unique and match variational solutions for Lipschitz data.

Solutions can be approximated by discrete min-max problems derived from tug-of-war games.

The proposed numerical scheme effectively solves examples with various obstacles and boundary conditions.

Abstract

We study the double-obstacle problem for the p-Laplace operator, p 2 [2;1). We prove that for Lipschitz boundary data and Lipschitz obstacles, viscosity solutions are unique and coincide with variational solutions. They are also uniform limits of solutions to discrete min-max problems that can be interpreted as the dynamic programming principle for appropriate tug-ofwar games with noise. In these games, both players in addition to choosing their strategies, are also allowed to choose stopping times. The solutions to the double-obstacle problems are limits of values of these games, when the step-size controlling the single shift in the token's position, converges to 0. We propose a numerical scheme based on this observation and show how it works for some examples of obstacles and boundary data.