Multi-dimensional BSDEs whose terminal values are bounded and have bounded Malliavin derivatives

We consider a class of multi-dimensional BSDEs on a finite time horizon (containing the Lipschitzian-quadratic BSDEs), whose terminal values are bounded as well as their corresponding Malliavin derivatives. We prove two results. The first one is an exponential integrability condition which determines when a BSDE in this class has a solution up to the given time horizon. In the second result, via a (deterministic) differential equation, we compute a minimum horizon up to which any BSDE of this class has a solution.