Linear Multistep Schemes for BSDEs

We study the convergence rate of a class of linear multistep methods for backward stochastic differential equations (BSDEs). We show that, under a sufficient condition on the coefficients, the schemes enjoy a fundamental stability property. Coupling this result to an analysis of the truncation error allows us to design approximation with arbitrary order of convergence. Contrary to the analysis performed in [W. Zhao, G. Zhang, and L. Ju, SIAM J. Numer. Anal., 48 (2010), pp. 1369--1394], we consider general diffusion models and BSDEs with driver depending on z. The class of methods we consider contains well-known methods from the ODE framework as Nystrom, Milne, or Adams methods. Finally, we provide a numerical illustration of the convergence of some methods.