Approximating the Value of Zero-Sum Differential Games with Linear Payoffs and Dynamics

We consider two-player zero-sum differential games of fixed duration, where the running payoff and the dynamics are both linear in the controls of the players. Such games have a value, which is determined by the unique viscosity solution of a Hamilton-Jacobi-type partial differential equation. Approximation schemes for computing the viscosity solution of Hamilton-Jacobi-type partial differential equations have been proposed that are valid in a more general setting, and such schemes can of course be applied to the problem at hand. However, such approximation schemes have a heavy computational burden. We introduce a discretized and probabilistic version of the differential game, which is straightforward to solve by backward induction, and prove that the solution of the discrete game converges to the viscosity solution of the partial differential equation, as the discretization becomes finer. The method removes part of the computational burden of existing approximation schemes.