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


Authors: Luca Codenotti, Marta Lewicka and Juan Manfredi
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 35J92
DOI: https://doi.org/10.1090/tran/6962
Published electronically: May 11, 2017
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Abstract: We study the double-obstacle problem for the $ p$-Laplace operator, $ p\in [2, \infty )$. 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-of-war 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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Additional Information

Luca Codenotti
Affiliation: Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, Pennsylvania 15260
Email: luc23@pitt.edu

Marta Lewicka
Affiliation: Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, Pennsylvania 15260
Email: lewicka@pitt.edu

Juan Manfredi
Affiliation: Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, Pennsylvania 15260
Email: manfredi@pitt.edu

DOI: https://doi.org/10.1090/tran/6962
Received by editor(s): January 20, 2016
Received by editor(s) in revised form: April 17, 2016, and April 22, 2016
Published electronically: May 11, 2017
Additional Notes: The first and second authors were partially supported by NSF award DMS-1406730
Article copyright: © Copyright 2017 American Mathematical Society