On the robustness of backward stochastic differential equations

In this paper, we study the robustness of backward stochastic differential equations (BSDEs for short) w.r.t. the Brownian motion; more precisely, we will show that if Wn is a martingale approximation of a Brownian motion W then the solution to the BSDE driven by the martingale Wn converges to the solution of the classical BSDE, namely the BSDE driven by W. The particular case of the scaled random walks has been studied in Briand et al. (Electron. Comm. Probab. 6 (2001) 1). Here, we deal with a more general situation and we will not assume that the Wn has the predictable representation property: this yields an orthogonal martingale in the BSDE driven by Wn. As a byproduct of our result, we obtain the convergence of the “Euler scheme” for BSDEs corresponding to the case where Wn is a time discretization of W.