Infinite-time linear quadratic differential games for stochastic system with Markov jumps and multiplicative noise

This paper discusses the linear quadratic (LQ) differential games for stochastic systems with Markov jumps and multiplicative noise in infinite-time case. We introduce the definitions of exact detectability and stochastic detectability and the connection between them, which have close relation to Lyapunov equation. Based on Lyapunov equation, we obtain four-coupled algebraic Riccati equations (AREs), which are essential on finding the optimal strategies (Nash equilibrium strategies) and the optimal cost values for infinite stochastic differential games with Markov jumps. In addition, we also propose the PBH criterions of exact detectability for stochastic systems with Markov jumps.