Tug-of-war games associated with boundary value problems involving derivatives

In this paper, we study a certain type of noisy tug-of-war game which can be regarded as an interpretation of a certain type of boundary value problem for the normalized $p$-Laplace equation, where $1<p<2$. More precisely, we will investigate the boundary regularity of the value function to the game and its convergence to a viscosity solution of the model problem.