Stochastic Optimal Linear Quadratic Controls with A Recursive Cost Functional

TL;DR

This paper studies stochastic linear quadratic control problems with recursive cost functionals, addressing the well-posedness of associated BSDEs and characterizing solvability through FBSDEs and Riccati equations.

Contribution

It introduces a suitable framework for the LQ problem with recursive costs and characterizes open-loop and closed-loop solvability via FBSDEs and Riccati equations.

Findings

Well-posedness of BSDEs in L^1 is established.

Solvability of the control problem is characterized by FBSDEs.

Riccati differential equations determine the control solutions.

Abstract

This paper is concerned with a stochastic linear quadratic (LQ, for short) control problem with a recursive cost functional. It involves BSDEs in $L^1$ whose well-posedness is a subtle issue. A suitable framework has been adopted so that the corresponding LQ problem is correctly formulated. Open-loop and closed-loop solvability of such an LQ problem have been investigated and characterized by the solvability of an FBSDE and that of Riccati differential equation.