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

TL;DR

This paper establishes the existence and uniqueness of solutions for multidimensional nonlinear backward stochastic differential equations with drivers that depend on the martingale part, allowing for path and law dependence.

Contribution

It introduces a general framework for BSDEs with drivers depending on the martingale component, including path and law dependence, and proves global existence and uniqueness.

Findings

Proved existence and uniqueness of solutions for the class of BSDEs considered.

Allowed for drivers with very general regularity conditions, including path and law dependence.

Extended the theory of BSDEs to more general, possibly non-Markovian, settings.

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.