Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2014, Article ID 523163, 11 pages
http://dx.doi.org/10.1155/2014/523163
Research Article

Numerical Implementation of Stochastic Operational Matrix Driven by a Fractional Brownian Motion for Solving a Stochastic Differential Equation

Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran

Received 16 November 2013; Revised 20 January 2014; Accepted 20 January 2014; Published 11 March 2014

Academic Editor: Hamid Reza Karimi

Copyright © 2014 R. Ezzati et al. 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. J. Bertoin, “Sur une intégrale pour les processus à α-variation bornée,” The Annals of Probability, vol. 17, no. 4, pp. 1277–1699, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  2. P. Cheng and M. Webster, “Stability analysis of impulsive stochastic functional differential equations with delayed impulses via comparison principle and impulsive delay differential inequality,” Abstract and Applied Analysis, vol. 2014, Article ID 710150, 2014. View at Publisher · View at Google Scholar
  3. W. Gao, F. Deng, R. Zhang, and W. Liu, “Finite-time H control for time-delayed stochastic systems with Markovian switching,” Abstract and Applied Analysis, vol. 2014, Article ID 809290, 2014. View at Publisher · View at Google Scholar
  4. J. Guerra and D. Nualart, “Stochastic differential equations driven by fractional Brownian motion and standard Brownian motion,” Stochastic Analysis and Applications, vol. 26, no. 5, pp. 1053–1075, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. D. Huang and S. K. Nguang, “Robust H static output feedback control of fuzzy systems: an ILMI approach,” IEEE Transactions on Systems, Man, and Cybernetics B, vol. 36, no. 1, pp. 216–222, 2006. View at Publisher · View at Google Scholar · View at Scopus
  6. A. Mark, F. Yao, and M. Hua, “Abstract functional stochastic evolution equations driven by fractional Brownian motion,” Abstract and Applied Analysis, vol. 2014, Article ID 516853, 2014. View at Publisher · View at Google Scholar
  7. K. Maleknejad, M. Khodabin, and M. Rostami, “A numerical method for solving m-dimensional stochastic Itô-Volterra integral equations by stochastic operational matrix,” Computers & Mathematics with Applications, vol. 63, no. 1, pp. 133–143, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  8. K. Maleknejad, M. Khodabin, and M. Rostami, “Numerical solution of stochastic Volterra integral equations by a stochastic operational matrix based on block pulse functions,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 791–800, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. J. Wang and K. Zhang, “Non-fragile H control for stochastic systems with Markovian jumping parameters and random packet losses,” Abstract and Applied Analysis, vol. 2014, Article ID 934134, 2014. View at Publisher · View at Google Scholar
  10. H. Zhang, Y. Shi, and A. Saadat Mehr, “Robust static output feedback control and remote PID design for networked motor systems,” IEEE Transactions on Industrial Electronics, vol. 58, no. 12, pp. 5396–5405, 2011. View at Publisher · View at Google Scholar · View at Scopus
  11. M. Zähle, “Integration with respect to fractal functions and stochastic calculus. I,” Probability Theory and Related Fields, vol. 111, no. 3, pp. 333–374, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. L. Coutin, “An introduction to (stochastic) calculus with respect to fractional Brownian motion,” in Séminaire de Probabilités XL, vol. 1899, pp. 3–65, Springer, Berlin, Germany, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. L. Decreusefond and A. S. Üstünel, “Fractional Brownian motion: theory and applications,” in Systèmes Différentiels Fractionnaires, vol. 5 of ESAIM Proceedings, pp. 75–86, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. D. Nualart and A. Răşcanu, “Differential equations driven by fractional Brownian motion,” Collectanea Mathematica, vol. 53, no. 1, pp. 55–81, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. H. Lisei and A. Soós, “Approximation of stochastic differential equations driven by fractional Brownian motion,” in Seminar on Stochastic Analysis, Random Fields and Applications Progress in Probability, vol. 59, pp. 227–241, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. Y. Mishura and G. Shevchenko, “The rate of convergence for Euler approximations of solutions of stochastic differential equations driven by fractional Brownian motion,” Stochastics, vol. 80, no. 5, pp. 489–511, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. F. Russo and P. Vallois, “Forward, backward and symmetric stochastic integration,” Probability Theory and Related Fields, vol. 97, no. 3, pp. 403–421, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. F. Biagini, Y. Hu, B. Øksendal, and T. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications, Springer, London, UK, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  19. T. Caraballo, M. J. Garrido-Atienza, and T. Taniguchi, “The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 11, pp. 3671–3684, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. L. Longjin, F.-Y. Ren, and W.-Y. Qiu, “The application of fractional derivatives in stochastic models driven by fractional Brownian motion,” Physica A, vol. 389, no. 21, pp. 4809–4818, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  21. B. Øksendal, Stochastic Differential Equations. An Introduction with Application, Springer, New York, NY, USA, 5th edition, 1998. View at MathSciNet