A Maximum Principle For Controlled Time-symmetric Forward-backward Stochastic Differential Equations With Initial-terminal Sates Constraints And Its Applications

It is well known that the maximum principle is an important approach to study optimal control problems. When the controlled system under con-sideration is assumed to be with state constraints, especially with sample-wise constraints which means that the state be in a given set with Prob-ability 1, the corresponding stochastic optimal control problems are diffi-cult to solve. In order to deal with such optimal control problems, an ap-proach named "terminal perturbation method" was introduced and applied in financial optimization problems recently (see [10-13]). This method is based on the dual method or martingale method introduced by Bieleckiet in [3] and El Karoui, Peng and Quenez in [7]. It mainly applies Ekeland's variational principle to tackle the state constraints and derive a stochastic maximum principle which characterizes the optimal solution.In this paper, we study a stochastic optimal control problem with initial-terminal state constraints where the controlled system is described by the time-symmetric forward-backward stochastic differential equations. Applying the terminal perturbation method and Ekeland's variation prin-ciple, a necessary condition of the stochastic optimal control i.e. stochastic maximum principle is derived. Applications to backward doubly stochastic linear-quadratic control models as well as other specific models are investi-gated.