Discrete-Time Stochastic LQ Optimal Control Problem with Random Coefficients

In this paper, the discrete-time linear quadratic (LQ) optimal control problem for a stochastic system with random coefficients is studied. Unlike the classical LQ optimal control problem, there exists a great difficulty in the LQ optimal control problem when the coefficient matrices of the stochastic system and weighting matrices in the cost functional are not assumed to be deterministic. Therefore, two innovative points are mentioned. First, we mainly consider structural changes and innovations in discrete-time LQ optimal control problems once the coefficients are randomized. Second, the stochastic system of this article includes nonhomogeneous terms. Interestingly, we show that the maximum principle leads to a Riccati equation. Specifically speaking, the fully coupled forward-backward stochastic difference equations (FBSDEs) are used to characterize the optimal control. Through decoupling the FBSDEs, we derive the expression corresponding to the Riccati equation with nonhomogeneous terms and get a state feedback representation of the optimal control. Finally, we construct the expression of the value function.