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, ω), (t, x, ω) ∈ [0, T ] × Rn × Ω, 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 semi-martingale of the form