A Simple Proof of Indefinite Linear-Quadratic Stochastic Optimal Control With Random Coefficients

We consider the indefinite linear-quadratic optimal control problem for stochastic systems with random coefficients, where the corresponding cost parameters need not be definite matrices. Although the solution to this problem was obtained previously via the stochastic maximum principle or by solving the stochastic Hamilton–Jacobi–Bellman (HJB) equation, our approach is simple and direct. Specifically, we develop a direct approach, also known as the completion of squares method, to characterize the explicit optimal solution and the optimal cost. The corresponding optimal solution is linear characterized in terms of the stochastic Riccati differential equation (SRDE) and the linear backward stochastic differential equation (BSDE). In our approach, the completion of squares method, together with the SRDE and the BSDE, allows us to construct an equivalent cost functional that is quadratic in the control $u$, provided that an additional positive definiteness condition in terms of the SRDE holds. Then, the optimal control and the associated cost can be obtained by eliminating the quadratic term of $u$ in the equivalent cost functional. The additional positive definiteness condition is induced due to the indefiniteness of the cost parameters and the dependence of the control on the diffusion term. We verify the optimal solution by solving the stochastic HJB equation.