Solutions to Forward–Backward Stochastic Differential Equations with Volterra Delayed and Anticipated Terms and Applications to Optimal Control

This paper studies a class of fully coupled forward–backward stochastic differential equations (FBSDEs in short) with Volterra delayed and anticipated terms. Such equations naturally arise in stochastic control problems involving memory effects and future-dependent terms. We establish the existence and uniqueness of adapted solutions by using the continuation method. We also discuss a linear quadratic optimal control problem with Volterra delayed terms and provide an explicit representation of the unique optimal control in terms of the solution to the associated FBSDE.