Multidimensional quadratic BSDEs with weak interactions and their applications in mean-field games of controls

Abstract

The well-posedness of multidimensional quadratic backward stochastic differential equations (qBSDEs) remains one of the central open problems in BSDE theory. Motivated by a mean-field utility maximization model with price impact, we introduce a new class of multidimensional qBSDEs that lies beyond the scope of existing well-posedness results (see (2.15) for the generator). In order to study the limit as the number of players tends to infinity, we establish existence and uniqueness for a large class of such qBSDEs under suitable smallness conditions imposed on each individual dynamics. A key feature of our approach is that the smallness condition is independent of the dimension of the BSDE system. In particular, the system itself is not confined to a small neighborhood, which allows us to analyze the mean-field limit of the underlying utility maximization problem. Such a condition is also natural in view of the well-known fact that general multidimensional qBSDEs may fail to admit solutions in the absence of suitable smallness assumptions. In addition, we derive a stability result for this class of equations based on the application of Picard iterations. Finally, using this stability result, we establish quantitative convergence rates toward the corresponding mean-field equilibria in two settings: Nash and Radner equilibria.