Dynamic Programming Principle for tug-of-war games with noise

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\begin{equation*} u(x) = α{2} in ol Bε(x) u (y) + y in ol Bε(x) u (y) + β kint Bε(x) u(y) ud y, end equation* for x Ω with u(y) = F(y) when y Ω. This principle implies the existence of quasioptimal Markovian strategies.