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The Annals of Probability 2010, Vol. 38, No. 2, 498531
 

Summary: The Annals of Probability
2010, Vol. 38, No. 2, 498531
DOI: 10.1214/09-AOP494
Institute of Mathematical Statistics, 2010
A STOCHASTIC DIFFERENTIAL GAME FOR THE
INHOMOGENEOUS -LAPLACE EQUATION
BY RAMI ATAR AND AMARJIT BUDHIRAJA1
Technion--Israel Institute of Technology and University of North Carolina
Given a bounded C2 domain G Rm, functions g C(G,R) and
h C(G,R \ {0}), let u denote the unique viscosity solution to the equation
-2 u = h in G with boundary data g. We provide a representation for u
as the value of a two-player zero-sum stochastic differential game.
1. Introduction.
1.1. Infinity-Laplacian and games. For an integer m 2, let a bounded C2 do-
main G Rm, functions g C(G,R) and h C(G,R \ {0}) be given. We study
a two-player zero-sum stochastic differential game (SDG), defined in terms of an
m-dimensional state process that is driven by a one-dimensional Brownian motion,
played until the state exits the domain. The functions g and h serve as terminal,
and, respectively, running payoffs. The players' controls enter in a diffusion coef-
ficient and in an unbounded drift coefficient of the state process. The dynamics are

  

Source: Atar, Rami - Department of Electrical Engineering, Technion, Israel Institute of Technology

 

Collections: Engineering