Error expansion for the discretization of backward stochastic differential equations

We study the error induced by the time discretization of decoupled forward-backward stochastic differential equations (X, Y, Z). The forward component X is the solution of a Brownian stochastic differential equation and is approximated by a Euler scheme X^N with N time steps. The backward component is approximated by a backward scheme. Firstly, we prove that the errors (Y^N - Y, Z^N - Z) measured in the strong L_p-sense (p ≥ 1) are of order N^{- 1 / 2} (this generalizes the results by Zhang [J. Zhang, A numerical scheme for BSDEs, The Annals of Applied Probability 14 (1) (2004) 459-488]). Secondly, an error expansion is derived: surprisingly, the first term is proportional to X^N - X while residual terms are of order N^{- 1}.