Issue |
ESAIM: COCV
Volume 18, Number 1, January-March 2012
|
|
---|---|---|
Page(s) | 81 - 90 | |
DOI | https://doi.org/10.1051/cocv/2010046 | |
Published online | 02 December 2010 |
Dynamic Programming Principle for tug-of-war games with noise
1
Department of Mathematics, University of Pittsburgh,
Pittsburgh, PA
15260,
USA
manfredi@pitt.edu
2
Institute of Mathematics, Helsinki University of
Technology, P.O. Box
1100, 02015
TKK,
Finland
Mikko.Parviainen@tkk.fi
3
Departamento de Matemática, FCEyN UBA (1428),
Buenos Aires,
Argentina
jrossi@dm.uba.ar
Received:
1
December
2009
Revised:
13
August
2010
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 for x ∈ Ω with
u(y) = F(y) when
y ∉ Ω. This principle implies the existence of
quasioptimal Markovian strategies.
Mathematics Subject Classification: 35J70 / 49N70 / 91A15 / 91A24
Key words: Dirichlet boundary conditions / Dynamic Programming Principle / p-Laplacian / stochastic games / two-player zero-sum games
© EDP Sciences, SMAI, 2010
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