FBSDEs Approach for Solving Discrete-Time Rational Expectations Model With State Delays

In this article, we consider the discrete-time rational expectations model with state delays. In particular, the dynamic equation relies both on the history of states and the conditional expectation of the future states. The novelty of this article lies in transforming the solvability of the rational expectations model to that of the forward and backward stochastic difference equations (FBSDEs). The main contribution is to obtain the (unique) solvability and the explicit solutions for the FBSDEs in both finite horizon and infinite horizon. This accordingly gives the explicit solution of the rational expectations model. The key technology is to establish a nonhomogeneous relationship between forward and backward processes of the FBSDEs.