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November  2009, 12(4): 905-924. doi: 10.3934/dcdsb.2009.12.905

Error estimates of the $\theta$-scheme for backward stochastic differential equations

Weidong Zhao 1, , Jinlei Wang 1, and  Shige Peng 1,

1. 

School of Mathematics, Shandong University, Jinan, Shandong, China, China, China

Received  April 2008 Revised  March 2009 Published  August 2009

In this paper, we study the error estimate of the $\theta$-scheme for the backward stochastic differential equation $y_t=\varphi(W_T)+\int_t^Tf(s,y_s)ds-\int_t^Tz_sdW_s$. We show that this scheme is of first-order convergence in $y$ for general $\theta$. In particular, for the case of $\theta=\frac{1}{2}$ (i.e., the Crank-Nicolson scheme), we prove that this scheme is of second-order convergence in $y$ and first-order in $z$. Some numerical examples are also given to validate our theoretical results.
Keywords: $\theta$-scheme, Backward stochastic differential equations, Error estimate..
Mathematics Subject Classification: Primary: 60H35, 65C20; Secondary: 65C3.
Citation: Weidong Zhao, Jinlei Wang, Shige Peng. Error estimates of the $\theta$-scheme for backward stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2009, 12 (4) : 905-924. doi: 10.3934/dcdsb.2009.12.905
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