A Hybrid ( Di erential - Stochastic ) Zero - SumGame with Fast Stochastic

We consider in this paper a continuous time stochastic hybrid system with a nite time horizon, controlled by two players with opposite objectives (zero-sum game). Player one wishes to maximize some linear function of the expected state trajectory, and player two wishes to minimize it. The state evolves according to a linear dynamics. The parameters of the state evolution equation may change at discrete times according to a MDP, i.e. a Markov chain that is directly controlled by both players, and has a countable state space. Each player has a nite action space. We use a procedure similar in form to the maximum principle; this determines a pair of stationary strategies for the players, which is asymptotically a saddle point, as the number of transitions during the nite time horizon grows to innnity.