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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 & 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. doi: 10.1080/1045112031000078917. Google Scholar

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F. Biagini, Y. Hu, B. Oksendal and T. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications,, Springer-Verlag, (2008). doi: 10.1007/978-1-84628-797-8. Google Scholar

[4]

P. Carmona, L. Coutin and Gerard Montseny, Stochastic integration with respect to fractional Brownian motion,, Ann. Inst. Henri Poincare, 39 (2003), 27. doi: 10.1016/S0246-0203(02)01111-1. Google Scholar

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N. Chakravarti and K. L. Sebastian, Fractional Brownian motion models for ploymers,, Chemical Physics Letter., 267 (1997), 9. Google Scholar

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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. doi: 10.1155/S104895339600038X. Google Scholar

[7]

L. Decreusefond and A. S. Ustunel, Fractional Brownian motion: Theory and applications,, ESAIM: Proceedings, 5 (1998), 75. doi: 10.1051/proc:1998014. Google Scholar

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T. E. Duncan, Y. Hu and B. Pasik-Duncan, Stochastic calculus for fractional Brownian motion I: Theory,, SIAM J. Control Optim., 38 (2000), 582. doi: 10.1137/S036301299834171X. Google Scholar

[9]

D. Feyel and A. de la Pradelle, Fractional integrals and Brownian processes,, Potential Analysis., 10 (1996), 273. Google Scholar

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H. Holden, B. Øksendal, J. Ubøe and T. Zhang, Stochastic Partial Differential Equations,, Birkhäuser, (1996). Google Scholar

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Y. Hu and B. Øksendal, Fractional white noise calculus and application to finance,, Infin. Dimens. Anal. Quantum Probab. Relat. Topics, 6 (2003), 1. doi: 10.1142/S0219025703001110. Google Scholar

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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. doi: 10.1016/j.ijnonlinmec.2012.12.003. Google Scholar

[13]

G. Jumarie, Stochastic differential equations with fractional Brownian motion input,, Int. J. Syst. Sci., 24 (1993), 1113. doi: 10.1080/00207729308949547. Google Scholar

[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. doi: 10.1016/j.aml.2004.09.012. Google Scholar

[15]

R. Z. Khasminskii, A limit theorem for the solution of differential equations with random right-hand sides,, Theory Probab. Appl., 11 (1963), 390. Google Scholar

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R. Z. Khasminskii, Principle of averaging of parabolic and elliptic differential equations for Markov process with small diffusion,, Theory Probab. Appl., 8 (1963), 1. Google Scholar

[17]

A. N. Kolmogorov, Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen,, Raum, 26 (1940), 115. Google Scholar

[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). doi: 10.1103/PhysRevLett.93.180603. Google Scholar

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T. Lyons, Differential equations driven by rough signals,, Rev. Mat. Iberoamericana, 14 (1998), 215. doi: 10.4171/RMI/240. Google Scholar

[22]

B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications,, SIAM Review, 10 (1968), 422. doi: 10.1137/1010093. Google Scholar

[23]

Y. S. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes,, Springer-Verlag, (2008). doi: 10.1007/978-3-540-75873-0. Google Scholar

[24]

N. Sri. Namachchivaya and Y. K. Lin, Application of stochastic averaging for systems with high damping,, Probab. Eng. Mech., 3 (1988), 185. doi: 10.1007/978-3-642-83254-3_15. Google Scholar

[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. doi: 10.2307/3318691. Google Scholar

[26]

D. Nualart and A. Rascanu, Differential equations driven by fractional Brownian motion,, Collect. Math., 53 (2002), 55. Google Scholar

[27]

J. Roberts and P. Spanos, Stochastic averaging: an approximate method of solving random vibration problems,, Int. J. Non-linear Mech., 21 (1986), 111. doi: 10.1016/0020-7462(86)90025-9. Google Scholar

[28]

F. Russo and P. Vallois, Forward, backward and symmetric stochastic integration,, Probab. Theory Rel. Fields, 97 (1993), 403. doi: 10.1007/BF01195073. Google Scholar

[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. doi: 10.1016/S0009-2509(00)00279-7. Google Scholar

[30]

A. N. Shiryaev, Essentials of Stochastic Finance: Facts, Models and Theory,, World Scientific, (1999). doi: 10.1142/9789812385192. Google Scholar

[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). Google Scholar

[32]

R. L. Stratonovich, Topics in the Theory of Random Noise,, New York, (1963). Google Scholar

[33]

R. L. Stratonovich, Conditional Markov Processes and Their Application to the Theory of Optimal Control,, American Elsevier, (1967). Google Scholar

[34]

Y. Xu, J. Duan and W. Xu, An averaging principle for stochastic dynamical systems with Levy noise,, Physica D., 240 (2011), 1395. doi: 10.1016/j.physd.2011.06.001. Google Scholar

[35]

L. C. Young, An inequality of the Holder type connected with Stieltjes integratin,, Acta. Math., 67 (1936), 251. doi: 10.1007/BF02401743. Google Scholar

[36]

M. Zahle, Integration with respect to fractal functions and stochastic calculus II,, Math. Nachr., 225 (2001), 145. doi: 10.1002/1522-2616(200105)225:1<145::AID-MANA145>3.0.CO;2-0. Google Scholar

[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. doi: 10.1115/1.4002528. Google Scholar

[38]

W. Q. Zhu, Nonlinear stochastic dynamics and control in Hamiltonian formulation,, ASME Appl. Mech. Rev., 59 (2006), 230. doi: 10.1115/1.2193137. Google Scholar

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. doi: 10.1080/1045112031000078917. Google Scholar

[2]

R. T. Baillie, Long memory processes and fractional integration in econometrics,, Journal of Econometrics, 73 (1996), 5. doi: 10.1016/0304-4076(95)01732-1. Google Scholar

[3]

F. Biagini, Y. Hu, B. Oksendal and T. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications,, Springer-Verlag, (2008). doi: 10.1007/978-1-84628-797-8. Google Scholar

[4]

P. Carmona, L. Coutin and Gerard Montseny, Stochastic integration with respect to fractional Brownian motion,, Ann. Inst. Henri Poincare, 39 (2003), 27. doi: 10.1016/S0246-0203(02)01111-1. Google Scholar

[5]

N. Chakravarti and K. L. Sebastian, Fractional Brownian motion models for ploymers,, Chemical Physics Letter., 267 (1997), 9. Google Scholar

[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. doi: 10.1155/S104895339600038X. Google Scholar

[7]

L. Decreusefond and A. S. Ustunel, Fractional Brownian motion: Theory and applications,, ESAIM: Proceedings, 5 (1998), 75. doi: 10.1051/proc:1998014. Google Scholar

[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. doi: 10.1137/S036301299834171X. Google Scholar

[9]

D. Feyel and A. de la Pradelle, Fractional integrals and Brownian processes,, Potential Analysis., 10 (1996), 273. Google Scholar

[10]

H. Holden, B. Øksendal, J. Ubøe and T. Zhang, Stochastic Partial Differential Equations,, Birkhäuser, (1996). Google Scholar

[11]

Y. Hu and B. Øksendal, Fractional white noise calculus and application to finance,, Infin. Dimens. Anal. Quantum Probab. Relat. Topics, 6 (2003), 1. doi: 10.1142/S0219025703001110. Google Scholar

[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. doi: 10.1016/j.ijnonlinmec.2012.12.003. Google Scholar

[13]

G. Jumarie, Stochastic differential equations with fractional Brownian motion input,, Int. J. Syst. Sci., 24 (1993), 1113. doi: 10.1080/00207729308949547. Google Scholar

[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. doi: 10.1016/j.aml.2004.09.012. Google Scholar

[15]

R. Z. Khasminskii, A limit theorem for the solution of differential equations with random right-hand sides,, Theory Probab. Appl., 11 (1963), 390. Google Scholar

[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. Google Scholar

[17]

A. N. Kolmogorov, Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen,, Raum, 26 (1940), 115. Google Scholar

[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). doi: 10.1103/PhysRevLett.93.180603. Google Scholar

[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. Google Scholar

[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. doi: 10.1023/A:1009983802178. Google Scholar

[21]

T. Lyons, Differential equations driven by rough signals,, Rev. Mat. Iberoamericana, 14 (1998), 215. doi: 10.4171/RMI/240. Google Scholar

[22]

B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications,, SIAM Review, 10 (1968), 422. doi: 10.1137/1010093. Google Scholar

[23]

Y. S. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes,, Springer-Verlag, (2008). doi: 10.1007/978-3-540-75873-0. Google Scholar

[24]

N. Sri. Namachchivaya and Y. K. Lin, Application of stochastic averaging for systems with high damping,, Probab. Eng. Mech., 3 (1988), 185. doi: 10.1007/978-3-642-83254-3_15. Google Scholar

[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. doi: 10.2307/3318691. Google Scholar

[26]

D. Nualart and A. Rascanu, Differential equations driven by fractional Brownian motion,, Collect. Math., 53 (2002), 55. Google Scholar

[27]

J. Roberts and P. Spanos, Stochastic averaging: an approximate method of solving random vibration problems,, Int. J. Non-linear Mech., 21 (1986), 111. doi: 10.1016/0020-7462(86)90025-9. Google Scholar

[28]

F. Russo and P. Vallois, Forward, backward and symmetric stochastic integration,, Probab. Theory Rel. Fields, 97 (1993), 403. doi: 10.1007/BF01195073. Google Scholar

[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. doi: 10.1016/S0009-2509(00)00279-7. Google Scholar

[30]

A. N. Shiryaev, Essentials of Stochastic Finance: Facts, Models and Theory,, World Scientific, (1999). doi: 10.1142/9789812385192. Google Scholar

[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). Google Scholar

[32]

R. L. Stratonovich, Topics in the Theory of Random Noise,, New York, (1963). Google Scholar

[33]

R. L. Stratonovich, Conditional Markov Processes and Their Application to the Theory of Optimal Control,, American Elsevier, (1967). Google Scholar

[34]

Y. Xu, J. Duan and W. Xu, An averaging principle for stochastic dynamical systems with Levy noise,, Physica D., 240 (2011), 1395. doi: 10.1016/j.physd.2011.06.001. Google Scholar

[35]

L. C. Young, An inequality of the Holder type connected with Stieltjes integratin,, Acta. Math., 67 (1936), 251. doi: 10.1007/BF02401743. Google Scholar

[36]

M. Zahle, Integration with respect to fractal functions and stochastic calculus II,, Math. Nachr., 225 (2001), 145. doi: 10.1002/1522-2616(200105)225:1<145::AID-MANA145>3.0.CO;2-0. Google Scholar

[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. doi: 10.1115/1.4002528. Google Scholar

[38]

W. Q. Zhu, Nonlinear stochastic dynamics and control in Hamiltonian formulation,, ASME Appl. Mech. Rev., 59 (2006), 230. doi: 10.1115/1.2193137. Google Scholar

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