Existence, Uniqueness and Stability of $$L^1$$ L 1 Solutions for Multidimensional Backward Stochastic Differential Equations with Generators of One-Sided Osgood Type

Abstract

We establish a general existence and uniqueness result of $$L^1$$ solution for a multidimensional backward stochastic differential equation (BSDE for short) with generator g satisfying a one-sided Osgood condition as well as a general growth condition in y, and a Lipschitz condition together with a sublinear growth condition in z, which improves some existing results. In particular, we put forward and prove a stability theorem of the $$L^1$$ solutions for the first time. A new type of $$L^1$$ solution is also investigated. Some delicate techniques involved in the relationship between convergence in $$L^1$$ and in probability and dividing appropriately the time interval play crucial roles in our proofs.