General Linear Quadratic Optimal Stochastic Control Problems with Random Coefficients: Linear Stochastic Hamilton Systems and Backward Stochastic Riccati Equations

Abstract

Consider the minimization of the following quadratic cost functional: $$J(u):=E\langle Mx_T,x_T\rangle +E\int_0^T(\langle Q_sx_s,x_s\rangle +\langle N_su_s,u_s\rangle )\, ds,$$ where x is the solution of the following stochastic control system: $$ \eq{dx_t=&(A_tx_t+B_tu_t)\, dt +\sum_{i=1}^d(C_t^ix_t+D_t^iu_t)\, dW_t^i,\cr x_0=h \cr} $$ u is a square integrable adapted process. The problem is conventionally 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. We prove the existence and uniqueness result for the associated Riccati equation, which in our general case is a backward stochastic differential equation with the generator (the drift term) being highly nonlinear in the two unknown variables. This solves Bismut and Peng's long-standing open problem (for the case of a Brownian filtration), which was initially proposed by the French mathematician 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 interrelationship 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 arguments. The first one is to connect the existence of the solution of the Riccati equation to the homomorphism of the stochastic flows derived from the optimally controlled system. Actually, we establish their equivalence. As a consequence, we can construct solutions to a sequence of suitably modified Riccati equations in terms of the associated stochastic Hamilton systems (and the optimal controls). The second key point is to establish a new type of a priori estimate for solutions of Riccati equations, with which we show that the sequence of constructed solutions has a limit which is a solution to the original Riccati equation.