## Abstract and Applied Analysis

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

### 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 $(\mathrm{1}/\mathrm{2},\mathrm{1})$ 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

#### 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.Mathematical Reviews (MathSciNet): MR845160

Zentralblatt MATH: 0582.73077

Digital Object Identifier: doi:10.1016/0020-7462(86)90025-9 - 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.Mathematical Reviews (MathSciNet): MR0221860 - 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.Mathematical Reviews (MathSciNet): MR161044 - 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.Mathematical Reviews (MathSciNet): MR340746 - 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.Mathematical Reviews (MathSciNet): MR1106630

Zentralblatt MATH: 0725.60057

Digital Object Identifier: doi:10.1007/BF01060515 - 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.Mathematical Reviews (MathSciNet): MR2831776

Zentralblatt MATH: 1236.60060

Digital Object Identifier: doi:10.1016/j.physd.2011.06.001 - 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.Mathematical Reviews (MathSciNet): MR1976868

Zentralblatt MATH: 1045.60072

Digital Object Identifier: doi:10.1142/S0219025703001110 - 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.Mathematical Reviews (MathSciNet): MR1900461

Zentralblatt MATH: 1009.34074

Digital Object Identifier: doi:10.1006/jdeq.2001.4073 - X. Mao, “Numerical solutions of stochastic functional differential equations,”
*LMS Journal of Computation and Mathematics*, vol. 6, pp. 141–161, 2003.Mathematical Reviews (MathSciNet): MR1998145

Zentralblatt MATH: 1055.65011

Digital Object Identifier: doi:10.1112/S1461157000000425 - 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.Mathematical Reviews (MathSciNet): MR3032349

Digital Object Identifier: doi:10.1186/1687-1847-2013-38 - F. Biagini, Y. Hu, B. ${\text{\O}}$ksendal, and T. Zhang,
*Stochastic Calculus for Fractional Brownian Motion and Applications*, Springer, London, UK, 2008.Mathematical Reviews (MathSciNet): MR2387368 - 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.Mathematical Reviews (MathSciNet): MR1741154

Zentralblatt MATH: 0947.60061

Digital Object Identifier: doi:10.1137/S036301299834171X - Y. S. Mishura,
*Stochastic Calculus for Fractional Brownian Motion and Related Processes*, vol. 1929 of*Lecture Notes in Mathematics*, Springer, Berlin, Germany, 2008.Mathematical Reviews (MathSciNet): MR2378138 - 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.Mathematical Reviews (MathSciNet): MR1978896

Digital Object Identifier: doi:10.1080/1045112031000078917 - F. Russo and P. Vallois, “Forward, backward and symmetric stochastic integration,”
*Probability Theory and Related Fields*, vol. 97, no. 3, pp. 403–421, 1993.Mathematical Reviews (MathSciNet): MR1245252

Zentralblatt MATH: 0792.60046

Digital Object Identifier: doi:10.1007/BF01195073 - L. C. Young, “An inequality of the Hölder type, connected with Stieltjes integration,”
*Acta Mathematica*, vol. 67, no. 1, pp. 251–282, 1936.Mathematical Reviews (MathSciNet): MR1555421

Zentralblatt MATH: 0016.10404

Digital Object Identifier: doi:10.1007/BF02401743 - D. Nualart and G. Via, “Stochastic integration with respect to fractional Brownian motion and applications,”
*Contemporary Mathematics*, vol. 336, pp. 3–39, 2003.Mathematical Reviews (MathSciNet): MR2037156 - 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. \endinputMathematical Reviews (MathSciNet): MR2737158

Zentralblatt MATH: 1239.60040

Digital Object Identifier: doi:10.1007/s00028-010-0069-8

### More like this

- Stochastic delay equations with non-negativity constraints driven by fractional Brownian motion

Besalú, Mireia and Rovira, Carles, Bernoulli, 2012 - Numerical Implementation of Stochastic Operational Matrix Driven by a Fractional Brownian Motion for Solving a Stochastic Differential Equation

Ezzati, R., Khodabin, M., and Sadati, Z., Abstract and Applied Analysis, 2013 - Taylor expansion for the solution of a stochastic differential equation driven by fractional Brownian motions

Baudoin, Fabrice and Zhang, Xuejing, Electronic Journal of Probability, 2012

- Stochastic delay equations with non-negativity constraints driven by fractional Brownian motion

Besalú, Mireia and Rovira, Carles, Bernoulli, 2012 - Numerical Implementation of Stochastic Operational Matrix Driven by a Fractional Brownian Motion for Solving a Stochastic Differential Equation

Ezzati, R., Khodabin, M., and Sadati, Z., Abstract and Applied Analysis, 2013 - Taylor expansion for the solution of a stochastic differential equation driven by fractional Brownian motions

Baudoin, Fabrice and Zhang, Xuejing, Electronic Journal of Probability, 2012 - Short time kernel asymptotics for rough differential equation driven by fractional Brownian motion

Inahama, Yuzuru, Electronic Journal of Probability, 2016 - Delay equations driven by rough paths

Neuenkirch, Andreas, Nourdin, Ivan, and Tindel, Samy, Electronic Journal of Probability, 2008 - Integrability and tail estimates for Gaussian rough differential equations

Cass, Thomas, Litterer, Christian, and Lyons, Terry, The Annals of Probability, 2013 - Moment estimates for solutions of linear stochastic differential equations driven by analytic fractional Brownian motion

Unterberger, Jeremie, Electronic Communications in Probability, 2010 - Stochastic differential equations driven by fractional Brownian motions

Jien, Yu-Juan and Ma, Jin, Bernoulli, 2009 - Transportation inequalities for stochastic differential equations driven by a fractional Brownian motion

Saussereau, Bruno, Bernoulli, 2012 - Stochastic calculus for fractional Brownian motion with Hurst exponent H > ¼: A rough path method by analytic extension

Unterberger, Jérémie, The Annals of Probability, 2009