Abstract and Applied Analysis
Volume 2012 (2012), Article ID 643783, 16 pages
http://dx.doi.org/10.1155/2012/643783
Research Article

Convergence of the Euler Method of Stochastic Differential Equations with Piecewise Continuous Arguments

Ling Zhang1,2 and Minghui Song1

1Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
2Institute of Mathematical Sciences, Daqing Normal University, Daqing 163712, China

Received 20 June 2012; Accepted 29 October 2012

Academic Editor: Roman Dwilewicz

Copyright © 2012 Ling Zhang and Minghui Song. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The main purpose of this paper is to investigate the strong convergence of the Euler method to stochastic differential equations with piecewise continuous arguments (SEPCAs). Firstly, it is proved that the Euler approximation solution converges to the analytic solution under local Lipschitz condition and the bounded th moment condition. Secondly, the Euler approximation solution converge to the analytic solution is given under local Lipschitz condition and the linear growth condition. Then an example is provided to show which is satisfied with the monotone condition without the linear growth condition. Finally, the convergence of numerical solutions to SEPCAs under local Lipschitz condition and the monotone condition is established.