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

Weidong Zhao; Jinlei Wang; Shige Peng

doi:10.3934/dcdsb.2009.12.905

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2009, Volume 12, Issue 4: 905-924. Doi: 10.3934/dcdsb.2009.12.905

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

1.
School of Mathematics, Shandong University, Jinan, Shandong

Received: March 31, 2008

Revised: February 28, 2009

Published: October 2009

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Abstract

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:

Backward stochastic differential equations,
$\theta$-scheme,
Error estimate.

Mathematics Subject Classification: Primary: 60H35, 65C20; Secondary: 65C30.

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