Multidimensional backward stochastic Riccati equations and applications

Backward stochastic Riccati differential equations (BSRDEs for short) arise from the solution of general linear quadratic optimal stochastic control problems with random coefficients. The existence and uniqueness question of the global adapted solutions has been open since Bismut's pioneering research publication in 1978 [Seminaire de Probabilites XII, Lecture Notes in Math. 649, C. Dellacherie, P. A. Meyer, and M. Weil, eds., Springer-Verlag, Berlin, 1978, pp. 180-264]. One distinguishing difficulty lies in the quadratic nonlinearity of the drift term in the second unknown component. In a previous article [ Stochastic Process. Appl., 97 ( 2002), pp. 255-288], the authors solved the one-dimensional case driven by Brownian motions. In this paper the multidimensional case driven by Brownian motions is studied. A closeness property for solutions of BSRDEs with respect to their coefficients is stated and is proved for general BSRDEs, which is used to obtain the existence of a global adapted solution to some BSRDEs. The global existence and uniqueness results are obtained for two classes of BSRDEs, whose generators contain a quadratic term of L ( the second unknown component). More specifically, the two classes of BSRDEs are ( for the regular case N > 0) [GRAPHICS] under the condition d = 1, and (for the singular case) [GRAPHICS] under the condition d = 1 and m = n. The arguments given in this paper are completely new, and they consist of some simple techniques of algebraic matrix transformations and direct applications of the closeness property mentioned above. We make full use of the special structure ( the nonnegativity of the quadratic term, for example) of the underlying Riccati equation. Applications in optimal stochastic control are exposed.