Pareto optimality in cooperative differential game of nonlinear mean-field backward stochastic system

This paper is concerned with a necessary condition and a sufficient condition for the existence of Pareto optimal strategy in a cooperative game driven by a nonlinear mean-field backward stochastic system. First, the game problem is equivalently converted into a set of constrained single objective optimal control problems. Next, we utilise Ekeland's variational principle to dispose constraints, and propose a necessary condition in the form of stochastic maximum principle. Meanwhile, we point out that under certain convex assumptions, the necessary condition we proposed is also sufficient. Finally, we give an example to explain the above theoretical results.