Well-posedness of McKean-Vlasov generalized multivalued BSDEs

TL;DR

This work addresses well-posedness of McKean–Vlasov backward stochastic variational inequalities, where the generator depends on the joint law of the solution. It first proves existence and uniqueness under globally Lipschitz and linear growth via Yosida approximation and contraction arguments, complemented by robust a priori estimates. It then relaxes to locally Lipschitz and nonlinear growth by constructing globally Lipschitz approximations $F_n$ and passing to the limit, establishing a comprehensive well-posedness theory for mean-field BSVIs. The results connect MFBSDEs and RBSDEs within a unified variational framework, offering a solid foundation for constrained mean-field stochastic systems.

Abstract

This paper investigates McKean-Vlasov backward stochastic variational inequalities (BSVIs) whose generator depends on the joint law of the solution. We first establish the existence and uniqueness of the solution under globally Lipschitz and linear growth conditions. The analysis is then extended to the more general case of locally Lipschitz and non-linear growth conditions.