Error expansion for the discretization of Backward Stochastic Differential Equations

Emmanuel Gobet (1, 2) , Céline Labart (3)

1 LMC - IMAG - Laboratoire de Modélisation et Calcul
2 MATHFI - Mathématiques financières
3 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique

We study the error induced by the time discretization of a 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 ($p \geq 1$) are of order $N^{-1/2}$ (this generalizes the results by Zhang 2004). Secondly, an error expansion is derived: surprisingly, the first term is proportional to $X^N-X$ while residual terms are of order $N^{-1}$.

Mots clés: semi-linear parabolic PDE; backward stochastic differential equation; discretization scheme; Malliavin calculus

Domaines: Probabilités [math.PR]

Emmanuel Gobet, Céline Labart. Error expansion for the discretization of Backward Stochastic Differential Equations. Stochastic Processes and their Applications, 2007, 117 (7), pp.803-829. ⟨10.1016/j.spa.2006.10.007⟩. ⟨hal-00019463⟩