  Two discretizations of a class of locally Lipschitz Markovian backward
stochastic differential equations (BSDEs) are studied. The first is the
classical Euler scheme which approximates a projection of the processes Z, and
the second a novel scheme based on Malliavin weights which approximates the
mariginals of the process Z directly. Extending the representation theorem of
Ma and Zhang leads to advanced a priori estimates and stability results for
this class of BSDEs. These estimates are then used to obtain competitive
convergence rates for both schemes with respect to the number of points in the
time-grid. The class of BSDEs considered includes Lipschitz BSDEs with
fractionally smooth terminal condition as well as quadratic BSDEs with bounded,
H\"older continuous terminal condition (for bounded, differentiable
volatility), and BSDEs related to proxy methods in numerical analysis.
