Linear-quadratic control of backward stochastic differential equations

Authors
Citation
Aeb. Lim et Xy. Zhou, Linear-quadratic control of backward stochastic differential equations, SIAM J CON, 40(2), 2001, pp. 450-474
Citations number
20
Language
INGLESE
art.tipo
Article
Categorie Soggetti
Mathematics,"Engineering Mathematics
Journal title
SIAM JOURNAL ON CONTROL AND OPTIMIZATION
ISSN journal
0363-0129 → ACNP
Volume
40
Issue
2
Year of publication
2001
Pages
450 - 474
Database
ISI
SICI code
0363-0129(20010830)40:2<450:LCOBSD>2.0.ZU;2-1
Abstract
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 co mpletely and explicitly by using an approach which is based primarily on th e completion-of-squares technique. Two alternative, though equivalent, expr essions for the optimal control are obtained. The rst of these involves a p air of Riccati-type equations, an uncontrolled BSDE, and an uncontrolled fo rward stochastic differential equation (SDE), while the second is in terms of a Hamiltonian system. Contrary to the deterministic or stochastic forwar d case, the optimal control is no longer a feedback of the current state; r ather, it is a feedback of the entire history of the state. A key step in o ur derivation is a proof of global solvability of the aforementioned Riccat i equations. Although of independent interest, this issue has particular re levance to the BLQ problem since these Riccati equations play a central rol e in our solution. Last but not least, it is demonstrated that the optimal control obtained coincides with the solution of a certain forward linear-qu adratic (LQ) problem. This, in turn, reveals the origin of the Riccati equa tions introduced.