A game-theoretic approach to the parabolic normalized p-Laplacian obstacle problem

TL;DR

This paper introduces a stochastic game model for the parabolic normalized p-Laplacian obstacle problem, establishing a probabilistic representation and proving convergence of the game value to the unique viscosity solution.

Contribution

It develops a novel zero-sum tug-of-war game with noise that characterizes the solution of the parabolic obstacle problem for the normalized p-Laplacian.

Findings

Existence of the game value functions.

Dynamic programming principle established.

Uniform convergence to the viscosity solution.

Abstract

This paper establishes a probabilistic representation for the solution of the parabolic obstacle problem associated with the normalized $p$-Laplacian. We introduce a zero-sum stochastic tug-of-war game with noise in a space-time cylinder, where one player has the option to stop the game at any time to collect a payoff given by an obstacle function. We prove that the value functions of this game exist, satisfy a dynamic programming principle, and converge uniformly to the unique viscosity solution of the continuous obstacle problem as the step size $\varepsilon$ tends to zero.