Well-posedness of mean-field forward–backward stochastic difference equations and applications to optimal control

This paper addresses the solvability of linear mean-field (MF) forward–backward stochastic difference equations (FBS$\Delta$Es) associated with discrete-time MF linear quadratic (LQ) optimal control problems. First of all, the relationships are investigated among the concerned equations and three different types of FBS$\Delta$Es arising from the available literature. It is found that the various formulations of FBS$\Delta$Es can be cast into a unified paradigm. Furthermore, through the solvability of two coupled difference Riccati equations, a necessary and sufficient condition is presented for the well-posedness of a general class of fully coupled linear MF-FBS$\Delta$Es in a finite horizon. Finally, based on stabilizability and detectability, a sufficient condition is proposed for an infinite-horizon MF-FBS$\Delta$E to admit an adapted solution, which can be explicitly characterized via the stabilizing solution of two coupled algebraic Riccati equations. As applications of the concerned MF-FBS$\Delta$Es, open-loop solvability is studied for finite- and infinite-horizon MF-LQ optimal control problems, respectively.