Stochastic Maximum Principle for Mean-field Controls and Non-Zero Sum   Mean-field Game Problems for Forward-Backward Systems
  The objective of the present paper is to investigate the solution of fully coupled mean-field forward-backward stochastic differential equations (FBSDEs in short) and to study the stochastic control problems of mean-field type as well as the mean-field stochastic game problems both in which state processes are described as FBSDEs. By combining classical FBSDEs methods introduced by Hu and Peng [Y. Hu, S. Peng, Solution of forward-backward stochastic differential equations, Probab. Theory Relat. Fields 103 (1995)] with specific arguments for fully coupled mean-field FBSDEs, we prove the existence and uniqueness of the solution to this kind of fully coupled mean-field FBSDEs under a certain \textquotedblleft monotonicity" condition. Next, we are interested in optimal control problems for (fully coupled respectively) FBSDEs of mean-field type with a convex control domain. Note that the control problems are time inconsistent in the sense that the Bellman optimality principle does not hold. The stochastic maximum principle (SMP) in integral form for mean-field controls, which is different from the classical one, is derived, specifying the necessary conditions for optimality. Sufficient conditions for the optimality of a control is also obtained under additional assumptions. Then we are concerned the maximum principle for a new class of non-zero sum stochastic differential games. This game system differs from the existing literature in the sense that the game systems here are characterized by (fully coupled respectively) FBSDEs in the mean-field framework. Our paper deduces necessary conditions as well as sufficient conditions in the form of maximum principle for open equilibrium point of this class of games respectively. 
Ruimin Xu, Liangquan Zhang
math.OC