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

Luca Codenotti; Marta Lewicka; Juan Manfredi

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arXiv · 1511.01604

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

Luca Codenotti·Marta Lewicka·Juan Manfredi

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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.

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Luca Codenotti, Marta Lewicka, Juan Manfredi. 2015-11-05. Discrete approximations to the double-obstacle prtoblem, and optimal stopping of tug-of-war games. https://arxiv.org/abs/1511.01604

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