  In this paper, we study the stability of the solutions of Backward Stochastic
Differential Equations (BSDE for short) with an almost surely finite random
terminal time. More precisely, we are going to show that if $(W^n)$ is a
sequence of scaled random walks or a sequence of martingales that converges to
a Brownian motion $W$ and if $(\tau^n)$ is a sequence of stopping times that
converges to a stopping time $\tau$, then the solution of the BSDE driven by
$W^n$ with random terminal time $\tau^n$ converges to the solution of the BSDE
driven by $W$ with random terminal time $\tau$.
