Fully-coupled mean-field FBSDE and maximum principle for related optimal control problem

Abstract

In this paper, motivated by the study of Stackelberg differential game of McKean–Vlasov type, we first investigate the solvability for the mean-field forward–backward stochastic differential equation (MF-FBSDE), which involves one McKean–Vlasov stochastic differential equation (SDE) and one mean-field type backward stochastic differential equation (BSDE). Under some weakly coupled conditions, we use a purely probabilistic approach to prove the well-posedness of such MF-FBSDE with arbitrary fixed time horizon. Then we consider a mean-field control problem, where the state dynamic is given by controlled MF-FBSDE. We establish the stochastic maximum principle and verification theorem for this optimal control problem. We mention that our weakly coupled conditions introduced in studying the solvability for the MF-FBSDE can be applied to simultaneously guarantee the well-posedness of variational equation and adjoint equation which are both fully-coupled MF-FBSDE. As an application, we consider a Stackelberg differential game whose system is described by a McKean–Vlasov SDE. We give a necessary condition for the existence of Stackelberg solutions.