Numerical solution of certain minimax problems of stochastic control

We study the discretized version of a system of quasi-variational inequalities that arise in a problem of stochastic game theory, where the two players take discrete actions with both continuous and discrete costs for each player. We prove convergence of an algorithm for the iterative solution of the discrete quasi-variational inequalities. For certain particular cases, we prove the contraction property of the iterative methods we consider here. Our results imply stability of the corresponding finite-difference schemes. The iterative methods we develop are ideally suitable for parallel computing implementation.