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

TL;DR

This paper studies stochastic linear quadratic control problems with recursive costs over an infinite horizon, addressing well-posedness and establishing conditions for classical solutions.

Contribution

Introduces a weighted L^2-stabilizability concept, linking recursive LQ problems to classical ones via FBSDEs and algebraic Riccati equations.

Findings

Characterizes well-posedness of BSDEs in L^1 and infinite horizon.

Establishes equivalence between recursive and classical LQ problems.

Discusses nonhomogeneous cases in the control framework.

Abstract

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.