Approximation of the Value Function for a Class of Differential Games with Target

We consider the approximation of a class of differential games with target by stochastic games. We use Kruzkov transformation to obtain discounted costs. The approximation is based on a space discretization of the state space and leads to consider the value function of the differential game as the limit of the value function of a sequence of stochastic games. To prove the convergence, we use the notion of viscosity solution for partial differential equations. This allows us to make assumptions only on the continuity of the value function and not on its differentiability. This technique of proof has been used before by M. Bardi, M. Falcone and P. Soravia for another kind of discretization. Under the additional hypothesis that the value function is Lipschitz continuous, we prove that the rate of convergence of this scheme is of order $\sqrt{h}$ where $h$ is the space parameter of discretization.\\ Some numerical experiments are presented in order to test the algorithm for a problem with discontinuous solution.