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Probability and Math Finance Seminar

Amarjit Budhiraja
University North Carolina, Chapel Hill

Elliott-Kalton Stochastic Differential Games Associated with the Infinity Laplacian

Abstract: In a recent work, Peres, Schramm, Sheffield, Wilson [PSSW] have considered a two player, zero sum, discrete time stochastic game, called Tug of War. In this game two competing players drive the state dynamics in a bounded domain as follows. At each time instant one of the two players is selected by toss of a fair coin who is then allowed to move the state by an amount that is bounded by c. The game ends at the first time instant when the boundary is reached with a payoff given in terms of a terminal cost function and a suitably scaled running cost. Player 1 seeks to maximize the expected payoff while Player 2 aims to minimize it. It is shown in [PSSW] that if the running cost is bounded away from zero then the game has a value u(c) and as c approaches 0, u(c) converges uniformly to the ``continuum value'' u which is the unique viscosity solution of an inhomogeneous infinite Laplace equation with a Dirichlet boundary data. In this work we consider a continuous time two player zero sum stochastic differential game that is motivated by the Tug of War game of [PSSW]. The state dynamics are driven by a one dimensional Brownian motion and each player can control both drift and diffusion coefficients. Controls can be unbounded and possibly lead to degenerate dynamics. We show that, under certain conditions, the game has a value in the usual Elliott-Kalton sense which is characterized as the unique viscosity solution of the equation in [PSSW]. Thus the result provides a game theoretic interpretation for the ``continuum value" in the [PSSW] analysis and a stochastic representation for the solution of an inhomogeneous infinite Laplace equation. This is a joint work with Rami Atar.

MONDAY, April 6, 2009
Time: 5:00 P.M.
Location: WeH 6423