  We consider a class of multi-dimensional BSDEs on a finite time horizon
(containing in particular 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 a given time horizon.
In the second result, via an ordinary differential equation, we compute a
minimum horizon up to which any BSDE of this class has a solution. The
combination of these two results leads to a new scheme to solve quadratic
BSDEs.
