DYNAMIC PROGRAMMING PRINCIPLE FOR TUG-OF-WAR GAMES WITH NOISE

We consider a two-player zero-sum-game in a bounded open domain Omega described as follows: at a point x epsilon Omega, Players I and II play an epsilon-step tug-of-war game with probability alpha, and with probability beta (alpha + beta = 1), a random point in the ball of radius epsilon 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) -alpha/2 {sup u(y)(y is an element of(B) over bar epsilon(x)) + inf(y is an element of(B) over bar epsilon(x)) u(y)} + beta f(B epsilon(x)) u(y)dy, for x is an element of Omega with u( y) = F( y) when y is not an element of Omega. This principle implies the existence of quasioptimal Markovian strategies.