Dynamic Programming Principle for tug-of-war games with noise
ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 1, pp. 81-90.

We consider a two-player zero-sum-game in a bounded open domain Ω described as follows: at a point x ∈ Ω, Players I and II play an ε-step tug-of-war game with probability α, and with probability β (α + β = 1), a random point in the ball of radius ε centered at x is chosen. Once the game position reaches the boundary, Player II pays Player I the amount given by a fixed payoff function F. We give a detailed proof of the fact that the value functions of this game satisfy the Dynamic Programming Principle

 $u\left(x\right)=\frac{\alpha }{2}\left\{\underset{y\in {\overline{B}}_{ϵ}\left(x\right)}{sup}u\left(y\right)+\underset{y\in {\overline{B}}_{ϵ}\left(x\right)}{inf}u\left(y\right)\right\}+{\beta }_{{B}_{}\left(x\right)}u\left(y\right)dy,$
for $x\in \Omega$ with $u\left(y\right)=F\left(y\right)$ when $y\notin \Omega$. This principle implies the existence of quasioptimal Markovian strategies.

DOI : https://doi.org/10.1051/cocv/2010046
Classification : 35J70,  49N70,  91A15,  91A24
Mots clés : Dirichlet boundary conditions, dynamic programming principle, p-laplacian, stochastic games, two-player zero-sum games
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author = {Manfredi, Juan J. and Parviainen, Mikko and Rossi, Julio D.},
title = {Dynamic {Programming} {Principle} for tug-of-war games with noise},
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pages = {81--90},
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Manfredi, Juan J.; Parviainen, Mikko; Rossi, Julio D. Dynamic Programming Principle for tug-of-war games with noise. ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 1, pp. 81-90. doi : 10.1051/cocv/2010046. http://www.numdam.org/articles/10.1051/cocv/2010046/

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