Abstract and Applied Analysis

An Averaging Principle for Stochastic Differential Delay Equations with Fractional Brownian Motion

Yong Xu, Bin Pei, and Yongge Li

Full-text: Open access

Abstract

An averaging principle for a class of stochastic differential delay equations (SDDEs) driven by fractional Brownian motion (fBm) with Hurst parameter in ( 1 / 2 , 1 ) is considered, where stochastic integration is convolved as the path integrals. The solutions to the original SDDEs can be approximated by solutions to the corresponding averaged SDDEs in the sense of both convergence in mean square and in probability, respectively. Two examples are carried out to illustrate the proposed averaging principle.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 479195, 10 pages.

Dates
First available in Project Euclid: 26 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1395858397

Digital Object Identifier
doi:10.1155/2014/479195

Mathematical Reviews number (MathSciNet)
MR3166618

Zentralblatt MATH identifier
07022457

Citation

Xu, Yong; Pei, Bin; Li, Yongge. An Averaging Principle for Stochastic Differential Delay Equations with Fractional Brownian Motion. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 479195, 10 pages. doi:10.1155/2014/479195. https://projecteuclid.org/euclid.aaa/1395858397


Export citation

References

  • J. B. Roberts and P. D. Spanos, “Stochastic averaging: an approximate method of solving random vibration problems,” International Journal of Non-Linear Mechanics, vol. 21, no. 2, pp. 111–134, 1986.
  • W. Q. Zhu, “Recent developments and applications of the stochastic averaging method in random vibration,” ASME Applied Mechanics Reviews, vol. 49, 10, pp. S72–S80, 1996.
  • N. Sri Namachchivaya and Y. K. Lin, “Application of stochastic averaging for nonlinear dynamical systems with high damping,” Probabilistic Engineering Mechanics, vol. 3, pp. 185–196, 1988.
  • W. Q. Zhu, “Stochastic averaging methods in random vibration,” ASME Applied Mechanics Reviews, vol. 41, no. 5, pp. 189–199, 1988.
  • R. L. Stratonovich, Topics in the Theory of Random Noise, Gordon and Breach, New York, NY, USA, 1967.
  • R. L. Stratonovich, Conditional Markov Processes and their Application to the Theory of Optimal Control, Elsevier, New York, NY, USA, 1967.
  • R. Z. Khasminskii, “A limit theorem for the solution of differential equations with random right-hand sides,” Theory of Probability & Its Applications, vol. 11, pp. 390–405, 1963.
  • R. Z. Khasminskii, “Principle of averaging of parabolic and elliptic differential equations for Markov process with small diffusion,” Theory of Probability & Its Applications, vol. 8, no. 1, pp. 1–21, 1963.
  • I. M. Stoyanov and D. D. Bainov, “The averaging method for a class of stochastic differential equations,” Ukrainian Mathematical Journal, vol. 26, no. 2, pp. 186–194, 1974.
  • V. G. Kolomiets and A. I. Mel'nikov, “Averaging of stochastic systems of integral-differential equations with Poisson noise,” Ukrainian Mathematical Journal, vol. 43, no. 2, pp. 242–246, 1991.
  • W. Q. Zhu, “Nonlinear stochastic dynamics and control in Hamiltonian formulation,” ASME Applied Mechanics Reviews, vol. 59, no. 4, pp. 230–248, 2006.
  • W. T. Jia, W. Q. Zhu, and Y. Xu, “Stochastic averaging of quasinon-integrable Hamiltonian systems under combined Gaussian and Poisson white noise excitations,” International Journal of Non-Linear Mechanics, vol. 51, pp. 45–53, 2012.
  • Y. Zeng and W. Q. Zhu, “Stochastic averaging of quasi-nonintegrable-Hamiltonian systems under Poisson white noise excitation,” ASME Journal of Applied Mechanics, vol. 78, no. 2, Article ID 021002, 11 pages, 2011.
  • Y. Xu, J. Duan, and W. Xu, “An averaging principle for stochastic dynamical systems with Lévy noise,” Physica D, vol. 240, no. 17, pp. 1395–1401, 2011.
  • W. E. Leland, M. S. Taqqu, W. Willinger, and D. V. Wilson, “On the self-similar nature of Ethernet traffic (extended version),” IEEE/ACM Transactions on Networking, vol. 2, no. 1, pp. 1–15, 1994.
  • Y. Hu and B. ${\text{\O}}$ksendal, “Fractional white noise calculus and applications to finance,” Infinite Dimensional Analysis, Quantum Probability and Related Topics, vol. 6, no. 1, pp. 1–32, 2003.
  • R. Scheffer and F. R. Maciel, “The fractional Brownian motion as a model for an industrial airlift reactor,” Chemical Engineering Science, vol. 56, no. 2, pp. 707–711, 2001.
  • N. Chakravarti and K. L. Sebastian, “Fractional Brownian motion models for polymers,” Chemical Physics Letters, vol. 267, no. 1-2, pp. 9–13, 1997.
  • Y. Xu and R. Guo, “Stochastic averaging principle for dynamical systems with fractional brownian motiončommentComment on ref. [19?]: Please update the information of this reference, if possible.,” AIMS Discrete and Continuous Dynamical Systems B. In press.
  • T. Taniguchi, K. Liu, and A. Truman, “Existence, uniqueness, and asymptotic behavior of mild solutions to stochastic functional differential equations in Hilbert spaces,” Journal of Differential Equations, vol. 181, no. 1, pp. 72–91, 2002.
  • X. Mao, “Numerical solutions of stochastic functional differential equations,” LMS Journal of Computation and Mathematics, vol. 6, pp. 141–161, 2003.
  • L. Tan and D. Lei, “The averaging method for stochastic differential delay equations under non-Lipschitz conditions,” Advances in Difference Equations, vol. 2013, article 38, 2013.
  • F. Biagini, Y. Hu, B. ${\text{\O}}$ksendal, and T. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications, Springer, London, UK, 2008.
  • 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.
  • Y. S. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, vol. 1929 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2008.
  • 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.
  • F. Russo and P. Vallois, “Forward, backward and symmetric stochastic integration,” Probability Theory and Related Fields, vol. 97, no. 3, pp. 403–421, 1993.
  • L. C. Young, “An inequality of the Hölder type, connected with Stieltjes integration,” Acta Mathematica, vol. 67, no. 1, pp. 251–282, 1936.
  • D. Nualart and G. Via, “Stochastic integration with respect to fractional Brownian motion and applications,” Contemporary Mathematics, vol. 336, pp. 3–39, 2003.
  • M. Ferrante and C. Rovira, “Convergence of delay differential equations driven by fractional Brownian motion,” Journal of Evolution Equations, vol. 10, no. 4, pp. 761–783, 2010. \endinput