Error expansion for the discretization of backward stochastic differential equations

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Abstract

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 XN with N time steps. The backward component is approximated by a backward scheme. Firstly, we prove that the errors (YNY,ZNZ) measured in the strong Lp-sense (p1) are of order N1/2 (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 XNX while residual terms are of order N1.

MSC

60H07
60F05
60H10
65G99

Keywords

Backward stochastic differential equation
Discretization scheme
Malliavin calculus
Semi-linear parabolic PDE

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