The necessary conditions for the optimal control of partially observed,
fully coupled forward-backward mean-field stochastic differential
equations driven by Teugels martingales are discussed in this paper. In
this context, we make the assumption that the forward diffusion
coefficient and the martingale coefficient are independent of the
control variable, and the control domain may not necessarily be convex.
For this class of optimal control problems, we derive the stochastic
maximum principle based on the classical method of spike variations and
the filtering techniques. The adjoint processes that are related to the
variational equations are determined by the solutions of proposed
forward-backward stochastic differential equations in finite-dimensional
spaces. Further, the Hamiltonian function is used to obtain the maximum
principle for the optimality of the given control system. Our results
are then applied to the mean-field type problem of linear quadratic
stochastic optimization.