Linear-Quadratic Control of Backward Stochastic Differential Equations

This paper is concerned with optimal control of linear backward stochastic differential equations (BSDEs) with a quadratic cost criteria, or backward linear-quadratic (BLQ) control. The solution of this problem is obtained completely and explicitly by using an approach which is based primarily on the completion-of-squares technique. Two alternative, though equivalent, expressions for the optimal control are obtained. The first of these involves a pair of Riccati-type equations, an uncontrolled BSDE, and an uncontrolled forward stochastic differential equation (SDE), while the second is in terms of a Hamiltonian system. Contrary to the deterministic or stochastic forward case, the optimal control is no longer a feedback of the current state; rather, it is a feedback of the entire history of the state. A key step in our derivation is a proof of global solvability of the aforementioned Riccati equations. Although of independent interest, this issue has particular relevance to the BLQ problem since these Riccati equations play a central role in our solution. Last but not least, it is demonstrated that the optimal control obtained coincides with the solution of a certain forward linear-quadratic (LQ) problem. This, in turn, reveals the origin of the Riccati equations introduced.