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

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.