About this Journal Submit a Manuscript Table of Contents 
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 643783, 16 pages
doi: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.

Linked References

  1. K. L. Cooke and J. Wiener, “Retarded differential equations with piecewise constant delays,” Journal of Mathematical Analysis and Applications, vol. 99, no. 1, pp. 265–297, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. J. Wiener, Generalized Solutions of Functional-Differential Equations, World Scientific Publishing, River Edge, NJ, USA, 1993.
  3. M. H. Song, Z. W. Yang, and M. Z. Liu, “Stability of θ-methods for advanced differential equations with piecewise continuous arguments,” Computers & Mathematics with Applications, vol. 49, no. 9-10, pp. 1295–1301, 2005. View at Publisher · View at Google Scholar
  4. A. Friedman, Stochastic Differential Equations and Applications, vol. 1, Academic Press, New York, NY, USA, 1975.
  5. X. Mao, Stochastic Differential Equations and Applications, Horwood Publishing, Chichester, UK, 2nd edition, 1997.
  6. X. Mao, “Numerical solutions of stochastic functional differential equations,” LMS Journal of Computation and Mathematics, vol. 6, pp. 141–161, 2003. View at Zentralblatt MATH
  7. X. Mao and S. Sabanis, “Numerical solutions of stochastic differential delay equations under local Lipschitz condition,” Journal of Computational and Applied Mathematics, vol. 151, no. 1, pp. 215–227, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, UK, 2006.
  9. H. Y. Dai and M. Z. Liu, “Mean square stability of stochastic differential equations with piecewise continuous arguments,” Journal of Natural Science of Heilongjiang University, vol. 25, no. 5, pp. 625–629, 2008.
  10. C. T. Ng and W. Zhang, “Invariant curves for planar mappings,” Journal of Difference Equations and Applications, vol. 3, no. 2, pp. 147–168, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH