Exact controllability of rational expectations model with multiplicative noise and input delay

This paper considers the rational expectations model with multiplicative noise and input delay, where the system dynamics rely on the conditional expectations of future states. The main contribution is to obtain a sufficient condition for the exact controllability of the rational expectations model. In particular, we derive a sufficient Gramian matrix condition and a rank condition for the delay-free case. The key is the solvability of the backward stochastic difference equations with input delay which is derived from the forward and backward stochastic system.