Infinite horizon linear quadratic differential games for discrete-time stochastic systems

This paper deals with the infinite horizon linear quadratic (LQ) differential games for discrete-time stochastic systems with both state and control dependent noise. The Popov-Belevitch-Hautus (PBH) criteria for exact observability and exact detectability of discrete-time stochastic systems are presented. By means of them, we give the optimal strategies (Nash equilibrium strategies) and the optimal cost values for infinite horizon stochastic differential games. It indicates that the infinite horizon LQ stochastic differential games are associated with four coupled matrix-valued equations. Furthermore, an iterative algorithm is proposed to solve the four coupled equations. Finally, an example is given to demonstrate our results.