Necessary and sufficient conditions of optimality for optimal control problem with initial and terminal costs

We consider a stochastic control problem where the control domain need not be convex, the system is governed by a non linear forward-backward stochastic differential equation with nonconstant terminal condition. The criteria to be minimized is in the general form, with initial and terminal costs. We derive necessary as well as sufficient conditions of optimality by introducing three adjoint equations. This problem may have applications in the financial market and it can be adapted to the problem of the minimization of an initial investment and the maximization of a final wealth.\