Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2014 (2014), Article ID 583930, 13 pages
http://dx.doi.org/10.1155/2014/583930
Research Article

The Stochastic -Method for Nonlinear Stochastic Volterra Integro-Differential Equations

1School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, China
2School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China

Received 6 May 2014; Accepted 28 August 2014; Published 27 October 2014

Academic Editor: Sanling Yuan

Copyright © 2014 Peng Hu and Chengming Huang. 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. T. A. Burton, Volterra Integral and Differential Equations, Elsevier, 2005. View at MathSciNet
  2. H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations, Cambridge University Press, 2004.
  3. A. Feldstein and J. R. Sopka, “Numerical methods for nonlinear Volterra integro-differential equations,” SIAM Journal on Numerical Analysis, vol. 11, pp. 826–846, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  4. P. Linz, “Linear multistep methods for Volterra integro-differential equations,” Journal of the Association for Computing Machinery, vol. 16, pp. 295–301, 1969. View at Publisher · View at Google Scholar · View at MathSciNet
  5. X. R. Mao, “Stability of stochastic integro differential equations,” Stochastic Analysis and Applications, vol. 18, no. 6, pp. 1005–1017, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. X. R. Mao and M. Riedle, “Mean square stability of stochastic Volterra integro-differential equations,” Systems and Control Letters, vol. 55, no. 6, pp. 459–465, 2006. View at Publisher · View at Google Scholar · View at Scopus
  7. J. A. D. Appleby, “pth mean integrability and almost sure asymptotic stability of solutions of It o^-volterra equations,” Journal of Integral Equations and Applications, vol. 15, no. 4, pp. 321–341, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. E. Buckwar and R. Winkler, “Multistep methods for SDEs and their application to problems with small noise,” SIAM Journal on Numerical Analysis, vol. 44, no. 2, pp. 779–803, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. E. Buckwar and R. Winkler, “Improved linear multi-step methods for stochastic ordinary differential equations,” Journal of Computational and Applied Mathematics, vol. 205, no. 2, pp. 912–922, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  10. K. Burrage, P. M. Burrage, and T. Tian, “Numerical methods for strong solutions of stochastic differential equations: an overview,” Proceedings of The Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 460, no. 2041, pp. 373–402, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. T. H. Tian and K. Burrage, “Implicit Taylor methods for stiff stochastic differential equations,” Applied Numerical Mathematics, vol. 38, no. 1-2, pp. 167–185, 2001. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. P. Burrage, Runge-Kutta methods for stochastic differential equations [Ph.D. thesis], University of Queensland, Brisbane, Australia, 1999.
  13. P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, Germany, 1992.
  14. J. Golec and S. Sathananthan, “Sample path approximation for stochastic integro-differential equations,” Stochastic Analysis and Applications, vol. 17, no. 4, pp. 579–588, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  15. J. Golec and S. Sathananthan, “Strong approximations of stochastic integro-differential equations,” Dynamics of Continuous, Discrete & Impulsive Systems B: Applications & Algorithms, vol. 8, no. 1, pp. 139–151, 2001. View at Google Scholar · View at MathSciNet · View at Scopus
  16. L. E. Shaikhet and J. A. Roberts, “Reliability of difference analogues to preserve stability properties of stochastic Volterra integro-differential equations,” Advances in Difference Equations, vol. 2006, Article ID 73897, 22 pages, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  17. D. J. Higham, “Mean-square and asymptotic stability of the stochastic theta method,” SIAM Journal on Numerical Analysis, vol. 38, no. 3, pp. 753–769, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. D. J. Higham, “A-stability and stochastic mean-square stability,” BIT Numerical Mathematics, vol. 40, no. 2, pp. 404–409, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. X. Mao, Stochastic Differential Equations and Applications, Horwood, Chichester, UK, 2nd edition, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  20. C. T. H. Baker and E. Buckwar, “Continuous θ-methods for the stochastic pantograph equation,” Electronic Transactions on Numerical Analysis, vol. 11, pp. 131–151, 2000. View at Google Scholar · View at MathSciNet · View at Scopus