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June  2014, 19(4): 1197-1212. doi: 10.3934/dcdsb.2014.19.1197

## Stochastic averaging principle for dynamical systems with fractional Brownian motion

 1 Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an, 710072, China, China, China, China 2 Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616

Received  April 2013 Revised  September 2013 Published  April 2014

Stochastic averaging for a class of stochastic differential equations (SDEs) with fractional Brownian motion, of the Hurst parameter $H$ in the interval $(\frac{1}{2},1)$, is investigated. An averaged SDE for the original SDE is proposed, and their solutions are quantitatively compared. It is shown that the solution of the averaged SDE converges to that of the original SDE in the sense of mean square and also in probability. It is further demonstrated that a similar averaging principle holds for SDEs under stochastic integral of pathwise backward and forward types. Two examples are presented and numerical simulations are carried out to illustrate the averaging principle.
Citation: Yong Xu, Rong Guo, Di Liu, Huiqing Zhang, Jinqiao Duan. Stochastic averaging principle for dynamical systems with fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2014, 19 (4) : 1197-1212. doi: 10.3934/dcdsb.2014.19.1197
##### References:
 [1] E. Alos and D. Nualart, Stochastic integration with respect to the fractional Brownian motion, Stochastics and Stochastic Reports, 75 (2003), 129-152. doi: 10.1080/1045112031000078917. [2] R. T. Baillie, Long memory processes and fractional integration in econometrics, Journal of Econometrics, 73 (1996), 5-59. doi: 10.1016/0304-4076(95)01732-1. [3] F. Biagini, Y. Hu, B. Oksendal and T. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications, Springer-Verlag, London, 2008. doi: 10.1007/978-1-84628-797-8. [4] P. Carmona, L. Coutin and Gerard Montseny, Stochastic integration with respect to fractional Brownian motion, Ann. Inst. Henri Poincare, 39 (2003), 27-68. doi: 10.1016/S0246-0203(02)01111-1. [5] N. Chakravarti and K. L. Sebastian, Fractional Brownian motion models for ploymers, Chemical Physics Letter., 267 (1997), 9-13. [6] W. Dai and C. C. Heyde, Itô formula with respect to fractional Brownian motion and its application, Journal of Appl. Math. and Stoch. Anal., 9 (1996), 439-448. doi: 10.1155/S104895339600038X. [7] L. Decreusefond and A. S. Ustunel, Fractional Brownian motion: Theory and applications, ESAIM: Proceedings, 5 (1998), 75-86. doi: 10.1051/proc:1998014. [8] T. E. Duncan, Y. Hu and B. Pasik-Duncan, Stochastic calculus for fractional Brownian motion I: Theory, SIAM J. Control Optim., 38 (2000), 582-612. doi: 10.1137/S036301299834171X. [9] D. Feyel and A. de la Pradelle, Fractional integrals and Brownian processes, Potential Analysis., 10 (1996), 273-288. [10] H. Holden, B. Øksendal, J. Ubøe and T. Zhang, Stochastic Partial Differential Equations, Birkhäuser, Boston, 1996. [11] Y. Hu and B. Øksendal, Fractional white noise calculus and application to finance, Infin. Dimens. Anal. Quantum Probab. Relat. Topics, 6 (2003), 1-32. doi: 10.1142/S0219025703001110. [12] W. T. Jia, W. Q. Zhu and Yong Xu, Stochastic averaging of quasi-non-integrable Hamiltonian systems under combined Gaussian and Poisson white noise excitations, Int. J. Nonlin. Mech., 51 (2013), 45-53. doi: 10.1016/j.ijnonlinmec.2012.12.003. [13] G. Jumarie, Stochastic differential equations with fractional Brownian motion input, Int. J. Syst. Sci., 24 (1993), 1113-1132. doi: 10.1080/00207729308949547. [14] G. Jumarie, On the solution of the stochastic differential equation of exponential growth driven by fractional Brownian motion, Appl. Math. Lett., 18 (2005), 817-826. doi: 10.1016/j.aml.2004.09.012. [15] R. Z. Khasminskii, A limit theorem for the solution of differential equations with random right-hand sides, Theory Probab. Appl., 11 (1963), 390-405. [16] R. Z. Khasminskii, Principle of averaging of parabolic and elliptic differential equations for Markov process with small diffusion, Theory Probab. Appl., 8 (1963), 1-21. [17] A. N. Kolmogorov, Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen, Raum, C. R. (Dokaldy) Acad. Sci. URSS (N.S.), 26 (1940), 115-118. [18] S. C. Kou and X. S. Xie, Generalized Langevin Equation with Fractional Gaussian Noise: Subdiffusion within a Single Protein Molecule, Phys. Rev. Lett., 93 (2004), 180603. doi: 10.1103/PhysRevLett.93.180603. [19] W. E. Leland, M. S. Taqqu,W. Willinger and D. V. Wilson, On the self-similar nature of ethernet traffic, IEEE/ACM Trans. Networking, 2 (1994), 1-15. [20] R. Liptser and V. Spokoiny, On estimating a dynamic function of a stochastic system with averaging, Statistical Inference for Stochastic Processes, 3 (2000), 225-249. doi: 10.1023/A:1009983802178. [21] T. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana, 14 (1998), 215-310. doi: 10.4171/RMI/240. [22] B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Review, 10 (1968), 422-427. doi: 10.1137/1010093. [23] Y. S. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-75873-0. [24] N. Sri. Namachchivaya and Y. K. Lin, Application of stochastic averaging for systems with high damping, Probab. Eng. Mech., 3 (1988), 185-196. doi: 10.1007/978-3-642-83254-3_15. [25] I. Norros, E. Valkeila and J. Virtamo, An elementary approach to a Girsanov formula and other analytivcal resuls on fractional Brownian motion, Bernoulli., 5 (1999), 571-587. doi: 10.2307/3318691. [26] D. Nualart and A. Rascanu, Differential equations driven by fractional Brownian motion, Collect. Math., 53 (2002), 55-81. [27] J. Roberts and P. Spanos, Stochastic averaging: an approximate method of solving random vibration problems, Int. J. Non-linear Mech., 21 (1986), 111-134. doi: 10.1016/0020-7462(86)90025-9. [28] F. Russo and P. Vallois, Forward, backward and symmetric stochastic integration, Probab. Theory Rel. Fields, 97 (1993), 403-421. doi: 10.1007/BF01195073. [29] R. Scheffer and F. R. Maciel, The fractional Brownian motion as a model for an industrial airlift reactor, Chemical Engineering Science, 56 (2001), 707-711. doi: 10.1016/S0009-2509(00)00279-7. [30] A. N. Shiryaev, Essentials of Stochastic Finance: Facts, Models and Theory, World Scientific, New Jersey, 1999. doi: 10.1142/9789812385192. [31] O. Y. Sliusarenko, V. Y. Gonchar, A. V. Chechkin, I. M. Sokolov and R. Metaler, Kramers-like escape driven by fractional Brownian noise, Phys. Rev. E., 81 (2010), 041119. [32] R. L. Stratonovich, Topics in the Theory of Random Noise, New York, Gordon and Breach, 1, 1963. [33] R. L. Stratonovich, Conditional Markov Processes and Their Application to the Theory of Optimal Control, American Elsevier, 1967. [34] Y. Xu, J. Duan and W. Xu, An averaging principle for stochastic dynamical systems with Levy noise, Physica D., 240 (2011), 1395-1401. doi: 10.1016/j.physd.2011.06.001. [35] L. C. Young, An inequality of the Holder type connected with Stieltjes integratin, Acta. Math., 67 (1936), 251-282. doi: 10.1007/BF02401743. [36] M. Zahle, Integration with respect to fractal functions and stochastic calculus II, Math. Nachr., 225 (2001), 145-183. doi: 10.1002/1522-2616(200105)225:1<145::AID-MANA145>3.0.CO;2-0. [37] Y. Zeng and W. Q. Zhu, Stochastic averageing of quasi-nonintegrable-hamiltonian systems under poisson white noise excitation, J. Appl. Mech.-Trans. ASME, 78 (2011), 021002-021011. doi: 10.1115/1.4002528. [38] W. Q. Zhu, Nonlinear stochastic dynamics and control in Hamiltonian formulation, ASME Appl. Mech. Rev., 59 (2006), 230-248. doi: 10.1115/1.2193137.

show all references

##### References:
 [1] E. Alos and D. Nualart, Stochastic integration with respect to the fractional Brownian motion, Stochastics and Stochastic Reports, 75 (2003), 129-152. doi: 10.1080/1045112031000078917. [2] R. T. Baillie, Long memory processes and fractional integration in econometrics, Journal of Econometrics, 73 (1996), 5-59. doi: 10.1016/0304-4076(95)01732-1. [3] F. Biagini, Y. Hu, B. Oksendal and T. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications, Springer-Verlag, London, 2008. doi: 10.1007/978-1-84628-797-8. [4] P. Carmona, L. Coutin and Gerard Montseny, Stochastic integration with respect to fractional Brownian motion, Ann. Inst. Henri Poincare, 39 (2003), 27-68. doi: 10.1016/S0246-0203(02)01111-1. [5] N. Chakravarti and K. L. Sebastian, Fractional Brownian motion models for ploymers, Chemical Physics Letter., 267 (1997), 9-13. [6] W. Dai and C. C. Heyde, Itô formula with respect to fractional Brownian motion and its application, Journal of Appl. Math. and Stoch. Anal., 9 (1996), 439-448. doi: 10.1155/S104895339600038X. [7] L. Decreusefond and A. S. Ustunel, Fractional Brownian motion: Theory and applications, ESAIM: Proceedings, 5 (1998), 75-86. doi: 10.1051/proc:1998014. [8] T. E. Duncan, Y. Hu and B. Pasik-Duncan, Stochastic calculus for fractional Brownian motion I: Theory, SIAM J. Control Optim., 38 (2000), 582-612. doi: 10.1137/S036301299834171X. [9] D. Feyel and A. de la Pradelle, Fractional integrals and Brownian processes, Potential Analysis., 10 (1996), 273-288. [10] H. Holden, B. Øksendal, J. Ubøe and T. Zhang, Stochastic Partial Differential Equations, Birkhäuser, Boston, 1996. [11] Y. Hu and B. Øksendal, Fractional white noise calculus and application to finance, Infin. Dimens. Anal. Quantum Probab. Relat. Topics, 6 (2003), 1-32. doi: 10.1142/S0219025703001110. [12] W. T. Jia, W. Q. Zhu and Yong Xu, Stochastic averaging of quasi-non-integrable Hamiltonian systems under combined Gaussian and Poisson white noise excitations, Int. J. Nonlin. Mech., 51 (2013), 45-53. doi: 10.1016/j.ijnonlinmec.2012.12.003. [13] G. Jumarie, Stochastic differential equations with fractional Brownian motion input, Int. J. Syst. Sci., 24 (1993), 1113-1132. doi: 10.1080/00207729308949547. [14] G. Jumarie, On the solution of the stochastic differential equation of exponential growth driven by fractional Brownian motion, Appl. Math. Lett., 18 (2005), 817-826. doi: 10.1016/j.aml.2004.09.012. [15] R. Z. Khasminskii, A limit theorem for the solution of differential equations with random right-hand sides, Theory Probab. Appl., 11 (1963), 390-405. [16] R. Z. Khasminskii, Principle of averaging of parabolic and elliptic differential equations for Markov process with small diffusion, Theory Probab. Appl., 8 (1963), 1-21. [17] A. N. Kolmogorov, Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen, Raum, C. R. (Dokaldy) Acad. Sci. URSS (N.S.), 26 (1940), 115-118. [18] S. C. Kou and X. S. Xie, Generalized Langevin Equation with Fractional Gaussian Noise: Subdiffusion within a Single Protein Molecule, Phys. Rev. Lett., 93 (2004), 180603. doi: 10.1103/PhysRevLett.93.180603. [19] W. E. Leland, M. S. Taqqu,W. Willinger and D. V. Wilson, On the self-similar nature of ethernet traffic, IEEE/ACM Trans. Networking, 2 (1994), 1-15. [20] R. Liptser and V. Spokoiny, On estimating a dynamic function of a stochastic system with averaging, Statistical Inference for Stochastic Processes, 3 (2000), 225-249. doi: 10.1023/A:1009983802178. [21] T. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana, 14 (1998), 215-310. doi: 10.4171/RMI/240. [22] B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Review, 10 (1968), 422-427. doi: 10.1137/1010093. [23] Y. S. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-75873-0. [24] N. Sri. Namachchivaya and Y. K. Lin, Application of stochastic averaging for systems with high damping, Probab. Eng. Mech., 3 (1988), 185-196. doi: 10.1007/978-3-642-83254-3_15. [25] I. Norros, E. Valkeila and J. Virtamo, An elementary approach to a Girsanov formula and other analytivcal resuls on fractional Brownian motion, Bernoulli., 5 (1999), 571-587. doi: 10.2307/3318691. [26] D. Nualart and A. Rascanu, Differential equations driven by fractional Brownian motion, Collect. Math., 53 (2002), 55-81. [27] J. Roberts and P. Spanos, Stochastic averaging: an approximate method of solving random vibration problems, Int. J. Non-linear Mech., 21 (1986), 111-134. doi: 10.1016/0020-7462(86)90025-9. [28] F. Russo and P. Vallois, Forward, backward and symmetric stochastic integration, Probab. Theory Rel. Fields, 97 (1993), 403-421. doi: 10.1007/BF01195073. [29] R. Scheffer and F. R. Maciel, The fractional Brownian motion as a model for an industrial airlift reactor, Chemical Engineering Science, 56 (2001), 707-711. doi: 10.1016/S0009-2509(00)00279-7. [30] A. N. Shiryaev, Essentials of Stochastic Finance: Facts, Models and Theory, World Scientific, New Jersey, 1999. doi: 10.1142/9789812385192. [31] O. Y. Sliusarenko, V. Y. Gonchar, A. V. Chechkin, I. M. Sokolov and R. Metaler, Kramers-like escape driven by fractional Brownian noise, Phys. Rev. E., 81 (2010), 041119. [32] R. L. Stratonovich, Topics in the Theory of Random Noise, New York, Gordon and Breach, 1, 1963. [33] R. L. Stratonovich, Conditional Markov Processes and Their Application to the Theory of Optimal Control, American Elsevier, 1967. [34] Y. Xu, J. Duan and W. Xu, An averaging principle for stochastic dynamical systems with Levy noise, Physica D., 240 (2011), 1395-1401. doi: 10.1016/j.physd.2011.06.001. [35] L. C. Young, An inequality of the Holder type connected with Stieltjes integratin, Acta. Math., 67 (1936), 251-282. doi: 10.1007/BF02401743. [36] M. Zahle, Integration with respect to fractal functions and stochastic calculus II, Math. Nachr., 225 (2001), 145-183. doi: 10.1002/1522-2616(200105)225:1<145::AID-MANA145>3.0.CO;2-0. [37] Y. Zeng and W. Q. Zhu, Stochastic averageing of quasi-nonintegrable-hamiltonian systems under poisson white noise excitation, J. Appl. Mech.-Trans. ASME, 78 (2011), 021002-021011. doi: 10.1115/1.4002528. [38] W. Q. Zhu, Nonlinear stochastic dynamics and control in Hamiltonian formulation, ASME Appl. Mech. Rev., 59 (2006), 230-248. doi: 10.1115/1.2193137.
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