Solvability of general fully coupled forward–backward stochastic difference equations with delay and applications

A class of fully coupled forward–backward stochastic difference equations with delay (FBSDDEs) over infinite horizon are considered in this article. By establishing a non‐homogeneous explicit relation between the forward and backward equations in terms of Riccati‐like difference equations, we derive the unique solution to the FBSDDEs under certain conditions. Then, we deduce that the FBSDDEs are solvable if and only if the corresponding stochastic delayed system is β$$ \beta $$ ‐degree open‐loop mean‐square exponentially stabilizable. Finally, as an application, the FBSDDEs are employed to demonstrate the maximum principle of the stochastic LQ optimal control problem.