Solution to Delayed Forward and Backward Stochastic Difference Equations and Its Applications

In this article, we will study a class of forward and backward stochastic difference equations (discrete-time FBSDEs) with time delay. By establishing a nonhomogeneous relationship between the forward and backward stochastic processes, we give the explicit solution of the discrete-time FBSDEs with time delay in terms of a class of Riccati-like equations. The explicit solution is of great significance in solving the stochastic control problem. To show this point, we study a stochastic leader–follower game problem with input delay. With the derived explicit solution of the discrete-time FBSDEs with time delay, we present the optimal controllers of the leader and the follower in the feedback form of the predictors.