Stochastic Differential Games With Multiple Modes

We have studied two person stochastic differential games with multiple modes. For the zero-sum game we have established the existence of optimal strategies for both players. For the nonzero sum case we have proved the existence of a Nash equilibrium. Key Words. Controlled diffusions, Markov chains, Markov strategy, Optimal strategy, Isaacs equation, Nash equilibrium. INTRODUCTION We study a two-person stochastic differential game with multiple modes. The state of the system at time t is given by a pair (X(t); `(t)) 2 R d \Theta S; S = f1; 2; \Delta \Delta \Delta ; Ng. The discrete component `(t) describes the various modes of the system. The continuous component X(t) is governed by a "controlled diffusion process" with a drift vector which depends on the discrete component `(t). Thus X(t) switches from one diffusion path to another at random times as the mode `(t) changes. On the other hand, the discrete component `(t) is a "controlled Markov chain" with a transition rate matri..