Stochastic Linear Quadratic Optimal Control Problems with Regime-Switching Jumps in Infinite Horizon

This paper investigates a stochastic linear-quadratic (SLQ) control problem regulated by a time-invariant Markov chain in infinite horizon. Under the L²-stability framework, we study a class of linear backward stochastic differential equations (BSDE) in infinite horizon and discuss the open-loop and closed-loop solvabilities of the SLQ problem. The open-loop solvability is characterized by the solvability of a system of coupled forward-backward stochastic differential equations (FBSDEs) in infinite horizon and the convexity of the cost functional, and the closed-loop solvability is shown to be equivalent to the open-loop solvability, which, in turn, is equivalent to the existence of a static stabilizing solution to the associated constrained coupled algebra Riccati equations (CAREs). Under the uniform convexity assumption, we obtain the unique solvability of associated CAREs and construct the corresponding closed-loop optimal strategy. Finally, we also solve a class of discounted SLQ problems and give two concrete examples to illustrate the results developed in the earlier sections.