This article is concerned with the optimal control problem for the linear stochastic system Xt = x +∫t0(AsXs + Bsus +∫s) ds +∫t0∫di=1[Ci(s)Xs + Di(s)us + gi(s)] dwi(s) with the convex risk functional J(u) = EM(XT) + E∫T 0G(t,Xt, ut) dt. In order to guarantee the existence of an optimal control without any(w eak) compactness assumption on the admissible control set, we assume that the risk function M is coercive and that∫di=1 D?i Di is uniformlyp ositive, rather than to assume like in the control literature that the running risk function G is coercive with respect to the control variable. In this new setting, the running risk function G mayb e independent of the control variable, and therefore the so-called singular linear-quadratic (LQ) stochastic control problem is included. A rigorous theoryis developed for the general stochastic LQ problem with random coefficients, and the bounded mean oscillation–martingale theoryis used to account for the concerned integrability. It plays a crucial role in the following exposition: (a) to connect the stochastic LQ problem to two associated backward stochastic differential equations (BSDEs)-one is an n× n symmetric matrix-valued nonlinear Riccati BSDE and the other is an n-dimensional linear BSDE with unbounded coefficients; (b) to show that the latter BSDE has an adapted solution pair of the suitablynecessaryregularit y. This seems to be the first application in a stochastic LQ theory of the BMO-martingale theory, which roots in harmonic analysis. Furthermore, with the help of an a priori estimate on the risk functional, existence and uniqueness of the solutions of backward stochastic Riccati differential equations (BSRDEs) in the singular case is reduced to the regular case via a perturbation method, and then a new existence and uniqueness result on BSRDEs is obtained for the singular case.

This paper studies, under some natural monotonicity conditions, the theory (existence and uniqueness, a priori estimate, contmuous dependence on a parameter) of forwald-backward stochastic differential equations and their connection with quaslinear parabolic partial differential equations. We use a purely probabilistic approach, and allow the forward equation to be degenerate.

Consider the minimization of the following quadratic cost functional: J (u):=E<Mx/T, x/T>+E ∫T (<QsTs, xs>+<Nsus, us>) ds, where x is the solution of the following linear stochastic control system: dxt=(Atxt+btut)dt +∑d (cixt+diut) dWi, u is a square integrable adapted proce;s. The problem :s convenally called the stochastic LQ (the abbreviation of "linear quadratic") problem. We are concerned with the following general case: the coefficients A, B, Ci, Di, Q, N, and M are allowed to be adapted processes or random matrices or random matrices We prove the existence and uniqueness result for the associated Riccati equation, which in our general case is a backward stochaatic differential equation with the generator (the drift term) being highly nonlinear in the two unknown variables. This solves solves Bismut and Peng's long-standing open problem (for the case of a Brownian filtration), which was initially proposed by the French mathemation J. M. Bismut [in 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]. We also provide a rigorous derivation of the Riccati equation from the stochastic Hamilton system. This completes the interrellationship between the Riccati equation and the stochastic Hamilton system as two different but equivalent tools for the stochastic LQ problem. There are, two key points in our argumentsl. The first one is to connect the existence of the solution of the Riccati equation to the homomorphism of the stochastic flows derived form theoptimally contrrolled system. Actually, we establish their equivalence. As a consequence, we canconstruct solutions to a sequence of suitably modified Riccati equations in terms of the associated stochasic Hamilton systems (and the optimal controls). The soco:nd koy point is to establish a new type of a priori estimate for solutions of Riccati equations, with which we show that the sequence of construoted solutions has a Limit which is a solution to the original Riccati equation.

Brockett在1982年国际数学家大会上提出了对非线性滤波器的所有的有限维的估计代数进行分类。最近，Chen，Yau和Leung【SIAM J Control Optim.，35（1997），1132-1141】宣称：当状态空间的维数小于或者等于4时，他们对所有的具极大秩的有限维的估计代数进行了分类。在该文里，我们对估计代数引入一系列全新的计算，找到了两组关于阵的新的方程。由此我们可以证明在任意维数的状态空间里，阵都是常阵，从而在任意维数的状态空间里，我们成功地对所有的具极大秩的有限维的估计代数进行了分类。