The necessary and sufficient conditions for stochastic differential systems with multi-time states cost functional

From economics point of view, we investigate a new optimal control problem driven by a stochastic differential equation with a multi-time states cost functional. By constructing a series of first-order adjoint equations, we establish the stochastic maximum principle and sufficient optimality conditions for this new optimal control problem. A constraints problem also be studied. In the end, we develop a near optimal control problem for a general cost functional.