Near-optimality conditions in stochastic control of linear fully coupled FBSDEs

In this paper, we deal with near optimality for stochastic control problems where the controlled system is described by a linear fully coupled forward–backward stochastic differential equation. We assume that the forward diffusion coefficient depends explicitly on the control variable and the control domain is not necessarily convex. We establish necessary as well as sufficient conditions for near optimality, satisfied by all near optimal controls. The proof of the main result is based on Ekeland’s variational principle and some estimates on the state and the adjoint processes with respect to the control variable. Finally an example which illustrate our results is presented.