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 ( x ) = α 2 sup y B ¯ ϵ ( x ) u ( y ) + inf y B ¯ ϵ ( x ) u ( y ) + β B ( x ) u ( y ) d y ,
for x Ω with u ( y ) = F ( y ) when y Ω . This principle implies the existence of quasioptimal Markovian strategies.

DOI : 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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     title = {Dynamic {Programming} {Principle} for tug-of-war games with noise},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {81--90},
     publisher = {EDP-Sciences},
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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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