Maximum Principle for Near-Optimality of Mean-Field FBSDEs

The present paper concerns with a near-optimal control problem for systems governed by mean-field forward-backward stochastic differential equations (FBSDEs) with mixed initial-terminal conditions. Utilizing Ekeland’s variational principle as well as the reduction method, the necessary and sufficient near-optimality conditions are established in the form of Pontryagin’s type. The results are obtained under restriction on the convexity of the control domain. As an application, a linear-quadratic stochastic control problem is solved explicitly.