Maximum Principle For Differential Games Of Mean-field Forward-backward Stochastic Systems

In2009, Buckdahn, Djehiche, Li and Peng [1] firstly introduced the mean-field theory to backward stochastic differential equations (BSDEs), and obtain-ed a new type of backward stochastic differential equations--Mean-field backw-ard stochastic differential equations (Mean-field BSDEs). We will study the di-fferential games of mean-field forward-backward stochastic systems:where λv1,v2(t)=(xv1,v2(t),yv1,v2(t),zv1,v2(t)).Under appropriate assumptions, this paper mainly works on the maximum principle for both zero-sum and nonzero-sum games. We give a necessary con--dition and a sufficient condition in the form of maximum principle for the games. In the end, we give an example of a nonzero-sum game of mean-field FBSDE to explain our main results.Since the general stochastic control problems could be regarded as the zero-sum differential games with only one player, and in reality, the observers can only observe the partial information which is a sub-filtration in probability lang--uage, we will firstly focus on the optimization problems of mean-field FBSDEs with partial information. With the help of the classical convex variational techn--ique, we establish a necessary maximum principle for the optimization proble--ms, where the stochastic system is described as follows:...