Nonlinear BSDEs in general filtration with drivers depending on the martingale part of a solution

Abstract

In the present paper, we consider multidimensional nonlinear backward stochastic differential equations (BSDEs) with a driver depending on the martingale part $M$ of a solution. We assume that the nonlinear term is merely monotone continuous with respect to the state variable. As to the regularity of the driver with respect to the martingale variable, we consider a very general condition which permits path-dependence on "the future" of the process $M$ as well as a dependence of its law (McKean-Vlasov-type equations). For such driver, we prove the existence and uniqueness of a global solution (i.e. for any maturity $T>0$) to BSDE with data satisfying natural integrability conditions.