Multidimensional Backward Stochastic Riccati Equations and Applications

Multi-dimensional backward stochastic Riccati di erential equations (BSRDEs in short) are studied. A closed property for solutions of BSRDEs with respect to their coeÆcients 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 speci cally, the two classes of BSRDEs are (for the regular case N > 0) ( dK = [A K +KA+Q LD(N +D KD) D L] dt+ Ldw; K(T ) = M and (for the singular case) ><>: dK = [A K +KA+ C KC +Q+ C L+ LC (KB + C KD + LD)(D KD) (KB + C KD + LD) ] dt+ Ldw; K(T ) = M: This partially solves Bismut-Peng's problem which was initially proposed by Bismut (1978) in the Springer yellow book LNM 649. The arguments given in this paper are completely new, and they consist of some simple techniques of algebraic transformations and direct applications of the closed 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.