Explicit solution to forward and backward stochastic differential equations with state delay and its application to optimal control

This paper is concerned with linear forward–backward stochastic differential equations (FBSDEs) with state delay, the solvability which is much more complex than the case of no delay or input delay caused by the prediction of the backward processes of the future time. To overcome this difficulty, we innovatively establish the non-homogeneous relationship between the backward and forward processes with the help of the corresponding discrete-time system. The main contribution is to give the explicit solution to the FBSDEs with state delay in terms of partial Riccati equations for the first time. The presented results form the basis to solve the challenging problem of linear quadratic optimal control for multiplicative-noise stochastic systems with state delay.