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.

中文翻译：

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时滞正向和后向随机差分方程的解及其应用

在本文中，我们将研究一类具有时滞的前向和后向随机差分方程（离散时间FBSDE）。通过建立前向和后向随机过程之间的非均匀关系，我们根据一类类似里卡蒂方程，给出了具有时间延迟的离散时间FBSDE的显式解。显式解对解决随机控制问题具有重要意义。为了说明这一点，我们研究了具有输入延迟的随机领导者－跟随者博弈问题。通过导出具有时滞的离散时间FBSDE的显式解决方案，我们以预测变量的反馈形式给出了领导者和跟随者的最优控制器。