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 .