Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2014 (2014), Article ID 678976, 6 pages
http://dx.doi.org/10.1155/2014/678976
Research Article

Impact of Correlated Noises on Additive Dynamical Systems

1School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China
2Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA

Received 14 April 2014; Accepted 4 August 2014; Published 3 September 2014

Academic Editor: Wuquan Li

Copyright © 2014 Chujin Li and Jinqiao Duan. 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. C. Bender, “An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter,” Stochastic Processes and their Applications, vol. 104, no. 1, pp. 81–106, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. J. Duan, C. Li, and X. Wang, “Modeling colored noise by fractional Brownian motion,” Interdisciplinary Mathematical Sciences, vol. 8, pp. 119–130, 2009. View at Google Scholar
  3. L. Arnold, Stochastic Differential Equations, John Wiley & Sons, New York, NY, USA, 1974.
  4. A. Du and J. Duan, “A stochastic approach for parameterizing unresolved scales in a system with memory,” Journal of Algorithms & Computational Technology, vol. 3, no. 3, pp. 393–405, 2009. View at Google Scholar
  5. J. Duan, “Stochastic modeling of unresolved scales in complex systems,” Frontiers of Mathematics in China, vol. 4, no. 3, pp. 425–436, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  6. B. Chen and J. Duan, “Stochastic quantification of missing mechanisms in dynamical systems,” Interdisciplinary Mathematical Sciences, vol. 8, pp. 67–76, 2009. View at Google Scholar
  7. D. Nualart, “Stochastic calculus with respect to the fractional Brownian motion and applications,” Contemporary Mathematics, vol. 336, pp. 3–39, 2003. View at Google Scholar
  8. E. Alòs and D. Nualart, “Stochastic integration with respect to the fractional Brownian motion,” Stochastics and Stochastics Reports, vol. 75, no. 3, pp. 129–152, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  9. A. Chronopoulou and F. Viens, “Hurst index estimation for self-similar processes with long-memory,” Interdisciplinary Mathematical Sciences, vol. 8, pp. 91–118, 2009. View at Google Scholar
  10. F. Biagini, Y. Hu, B. Øksendal, and T. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications, Springer, New York, NY, USA, 2008.
  11. Y. S. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, Springer, New York, NY, USA, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  12. C. A. Tudor and F. G. Viens, “Variations and estimators for self-similarity parameters via Malliavin calculus,” The Annals of Probability, vol. 37, no. 6, pp. 2093–2134, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  13. S. J. Lin, “Stochastic analysis of fractional Brownian motions,” Stochastics and Stochastics Reports, vol. 55, no. 1-2, pp. 121–140, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. T. E. Duncan, Y. Hu, and B. Pasik-Duncan, “Stochastic calculus for fractional Brownian motion. I. Theory,” SIAM Journal on Control and Optimization, vol. 38, no. 2, pp. 582–612, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. L. Decreusefond and A. S. Üstunel, “Stochastic analysis of the fractional Brownian motion,” Potential Analysis, vol. 10, no. 2, pp. 177–214, 1999. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. F. Baudoin and L. Coutin, “Operators associated with a stochastic differential equation driven by fractional Brownian motions,” Stochastic Processes and Their Applications, vol. 117, no. 5, pp. 550–574, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. M. A. Diop and Y. Ouknine, “A linear stochastic differential equation driven by a fractional Brownian motion with Hurst parameter >1/2,” Statistics & Probability Letters, vol. 81, no. 8, pp. 1013–1020, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. B. Saussereau, “A stability result for stochastic differential equations driven by fractional Brownian motions,” International Journal of Stochastic Analysis, vol. 2012, Article ID 281474, 13 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  19. L. Denis, M. Erraoui, and Y. Ouknine, “Existence and uniqueness for solutions of one dimensional SDE’s driven by an additive fractional noise,” Stochastics and Stochastics Reports, vol. 76, no. 5, pp. 409–427, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  20. Z. Huang and C. Li, “On fractional stable processes and sheets: white noise approach,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 624–635, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  21. G. Ünal, “Fokker-Planck-Kolmogorov equation for fBM: derivation and analytical solution,” in Proceedings of the 12th Regional Conference, pp. 53–60, Islamabad, Pakistan, 2006.
  22. D. Farrelly and J. E. Howard, “Double-well dynamics of two ions in the Paul and Penning traps,” Physical Review A, vol. 49, no. 2, pp. 1494–1497, 1994. View at Publisher · View at Google Scholar · View at Scopus
  23. E. Kierig, U. Schnorrberger, A. Schietinger, J. Tomkovic, and M. K. Oberthaler, “Single-particle tunneling in strongly driven double-well potentials,” Physical Review Letters, vol. 100, no. 19, Article ID 190405, 2008. View at Publisher · View at Google Scholar · View at Scopus