  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.
