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)(X,Y,Z). The forward component XX is the solution of a Brownian stochastic differential equation and is approximated by a Euler scheme XNXN with NN time steps. The backward component is approximated by a backward scheme. Firstly, we prove that the errors (YN−Y,ZN−Z)(YN−Y,ZN−Z) measured in the strong LpLp-sense (p≥1p≥1) are of order N−1/2N−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 XN−XXN−X while residual terms are of order N−1N−1.