Dynamic Programming for General Linear Quadratic Optimal Stochastic Control with Random Coefficients

We are concerned with the linear-quadratic optimal stochastic control problem where all the coefficients of the control system and the running weighting matrices in the cost functional are allowed to be predictable (but essentially bounded) processes and the terminal state-weighting matrix in the cost functional is allowed to be random. Under suitable conditions, we prove that the value field V (t, x, ), (t, x, ) 0, T Rⁿ, is quadratic in x and has the following form: V (t, x) = Kₜx, x, where K is an essentially bounded nonnegative symmetric matrix-valued adapted processes. Using the dynamic programming principle (DPP), we prove that K is a continuous semimartingale of the form Kₜ=K₀-₀ᵗ \, dkₛ+₀ᵗ₈=₁ᵈ Lₛⁱ\, dWₛⁱ, \ t 0, T, with k being a continuous process of bounded variation and the stochastic integral ₀^₈=₁ᵈ Lⁱₛ\, dWⁱₛ being a bounded mean oscillation martingale, and that (K, L) with L: = (L¹, , Lᵈ) is a solution to the associated backward stochastic Riccati equation (BSRE), whose generator is highly nonlinear in the unknown pair of processes. The uniqueness is also proved via a localized completion of squares in a self-contained manner for a general BSRE. The existence and uniqueness of an adapted solution to a general BSRE was initially proposed by the French mathematician J. M. Bismut SIAM J. Control Optim. , 14 (1976), pp. 419--444; Lecture Notes in Math. 649, C. Dellacherie, P. A. Meyer, and M. Weil, eds. , Springer-Verlag, Berlin, 1978, pp. 180--264 and subsequently listed by S. Peng Control of Distributed Parameter and Stochastic Systems (Hangzhou, 1998), S. Chen et al. , eds. , Kluwer, Boston, 1999, pp. 265--273 as an open problem for backward stochastic differential equations. It had remained open until a general solution by the author SIAM J. Control Optim. , 42 (2003), pp. 53--75 via the stochastic maximum principle with a viewpoint of stochastic flow for the associated stochastic Hamiltonian system. The present paper is its companion and gives the second but more comprehensive (seemingly much simpler, but appealing to the advanced tool of Doob--Meyer's decomposition theorem and the general theory of stochastic processes, in addition to the DPP) adapted solution to a general BSRE via the DPP. Further extensions to the jump-diffusion control system and to the general nonlinear control system are possible.