  In this paper, we study the qualitative behaviour of approximation schemes
for Backward Stochastic Differential Equations (BSDEs) by introducing a new
notion of numerical stability. For the Euler scheme, we provide sufficient
conditions in the one-dimensional and multidimensional case to guarantee the
numerical stability. We then perform a classical Von Neumann stability analysis
in the case of a linear driver $f$ and exhibit necessary conditions to get
stability in this case. Finally, we illustrate our results with numerical
applications.
