A Variational Formula of Forward-Backward Stochastic Differential System of Mean-Field Type with Observation Noise and Some Application

This paper examines an optimal control problem for mean-field systems under partial observation. The state system is described by a controlled mean-field forward-backward stochastic differential equation that features correlated noises between the system and the observation. Furthermore, the observation coefficients are allowed to depend not only on the control process but also on its probability distribution. Assuming a convex control domain and allowing all coefficients of the systems to be random, we derive directly a variational formula for the cost function in a given control process direction in terms of the Hamiltonian and the associated adjoint system without relying on variational systems under standard assumptions on the coefficients. As an application, we present the necessary and sufficient conditions for the optimality of our control problem using Pontryagin's maximum principle in a unified manner. The research provides insights into the optimization of mean-field systems under partial observation, which has practical implications for various applications.