Stochastic Optimal Linear Quadratic Controls with A Recursive Cost Functional in Infinite Horizon

This paper is concerned with a stochastic linear quadratic (LQ, for short) control problem with a recursive cost functional in an infinite horizon. A main difficult is well-posedness of the BSDE in $L^1$ and in infinite horizon. A notion of weighted $L^2$-stabilizability is introduced and characterized, which will lead to an equivalence of the optimal control problem having recursive cost functional with a classical LQ problem. Then all the results of classical problems for open-loop and closed-loop solvability of such an LQ problem can be translated, in terms of the solvability of a forward-backward stochastic differential equation and that of algebraic Riccati equation. Finally, the nonhomogeneous is discussed.