Solutions for functional fully coupled forward–backward stochastic differential equations

In this paper, we study a functional fully coupled forward–backward stochastic differential equation (FBSDE). For this functional FBSDE, the classical Lipschitz and monotonicity conditions which guarantee the existence and uniqueness of the solution to FBSDE are no longer applicable. To overcome this difficulty, we propose a completely new type of Lipschitz and monotonicity condition in which an integral term with respect to the path of X(t)0≤t≤TX(t)0≤t≤T is involved. Under this integral Lipschitz and monotonicity conditions, the existence and uniqueness of solutions for functional fully coupled FBSDEs is proved.