On the robustness of backward stochastic differential equations

• Published in: Stochastic processes and their applications 2002-02, Vol.97 (2), p.229-253
• Main Authors: Briand, Philippe, Delyon, Bernard, Mémin, Jean
• Format: Article
• Language: English
• Subjects: Backward stochastic differential equation (BSDE); Backward stochastic differential equation (BSDE) Stability of BSDEs Weak convergence of filtrations Discretization; Discretization; Exact sciences and technology; Limit theorems; Mathematics; Probability; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; Stability of BSDEs; Stochastic analysis; Weak convergence of filtrations
• ISSN: 0304-4149; 1879-209X
• DOI: 10.1016/S0304-4149(01)00131-4

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 W n is a martingale approximation of a Brownian motion W then the solution to the BSDE driven by the martingale W n 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 W n has the predictable representation property: this yields an orthogonal martingale in the BSDE driven by W n . As a byproduct of our result, we obtain the convergence of the “Euler scheme” for BSDEs corresponding to the case where W n is a time discretization of W.