Journal of Applied Mathematics
Volume 2012 (2012), Article ID 696849, 21 pages
Research Article

Numerical Solutions of Stochastic Differential Equations with Piecewise Continuous Arguments under Khasminskii-Type Conditions

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

Received 9 April 2012; Accepted 13 June 2012

Academic Editor: F. Marcellán

Copyright © 2012 Minghui Song and Ling Zhang. 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.


The main purpose of this paper is to investigate the convergence of the Euler method to stochastic differential equations with piecewise continuous arguments (SEPCAs). The classical Khasminskii-type theorem gives a powerful tool to examine the global existence of solutions for stochastic differential equations (SDEs) without the linear growth condition by the use of the Lyapunov functions. However, there is no such result for SEPCAs. Firstly, this paper shows SEPCAs which have nonexplosion global solutions under local Lipschitz condition without the linear growth condition. Then the convergence in probability of numerical solutions to SEPCAs under the same conditions is established. Finally, an example is provided to illustrate our theory.