Stochastic Linear-Quadratic Differential Game with Regime-Switching in an Infinite Horizon

Abstract. This paper investigates a two-person nonhomogeneous linear-quadratic stochastic differential game (LQ-SDG) in an infinite horizon for a system regulated by a time-invariant Markov chain. Both nonzero-sum and zero-sum LQ-SDG problems are studied, where the latter can be viewed as a special case of the former. It is shown that the open-loop Nash equilibrium point of the nonzero-sum (respectively, zero-sum) LQ-SDG problem is characterized by the solvability of a system of constrained forward-backward stochastic differential equations in an infinite horizon and the convexity (respectively, convexity-concavity) of the performance functional, and their corresponding closed-loop Nash equilibrium strategies are equivalent to the solvability of a system of constrained coupled algebraic Riccati equations (CAREs) with certain stabilizing conditions. In addition, the closed-loop representation of the open-loop Nash equilibrium point for a nonzero-sum (respectively, zero-sum) LQ-SDG is provided by the nonsymmetric (respectively, symmetric) solution to a system of CAREs. At the end of this paper, we provide three concrete examples and solve their open-loop/closed-loop Nash equilibrium strategy based on the obtained results.