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 triangle 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 triangle Es arising from the available literature. It is found that the various formulations of FBS triangle 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 triangle Es in a finite horizon. Finally, based on stabilizability and detectability, a sufficient condition is proposed for an infinite-horizon MF-FBS triangle 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 triangle Es, open-loop solvability is studied for finiteand infinite-horizon MF-LQ optimal control problems, respectively. (c) 2025 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.