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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 |
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. |
[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. |
[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. |
[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. |
[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. |
[7] |
L. Decreusefond and A. S. Ustunel, Fractional Brownian motion: Theory and applications,, ESAIM: Proceedings, 5 (1998), 75.
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.
doi: 10.1137/S036301299834171X. |
[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).
|
[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. |
[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. |
[13] |
G. Jumarie, Stochastic differential equations with fractional Brownian motion input,, Int. J. Syst. Sci., 24 (1993), 1113.
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.
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. 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.
|
[17] |
A. N. Kolmogorov, Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen,, Raum, 26 (1940), 115.
|
[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. |
[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. |
[21] |
T. Lyons, Differential equations driven by rough signals,, Rev. Mat. Iberoamericana, 14 (1998), 215.
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.
doi: 10.1137/1010093. |
[23] |
Y. S. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes,, Springer-Verlag, (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.
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.
doi: 10.2307/3318691. |
[26] |
D. Nualart and A. Rascanu, Differential equations driven by fractional Brownian motion,, Collect. Math., 53 (2002), 55.
|
[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. |
[28] |
F. Russo and P. Vallois, Forward, backward and symmetric stochastic integration,, Probab. Theory Rel. Fields, 97 (1993), 403.
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.
doi: 10.1016/S0009-2509(00)00279-7. |
[30] |
A. N. Shiryaev, Essentials of Stochastic Finance: Facts, Models and Theory,, World Scientific, (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). Google Scholar |
[32] |
R. L. Stratonovich, Topics in the Theory of Random Noise,, New York, (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.
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.
doi: 10.1007/BF02401743. |
[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. |
[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. |
[38] |
W. Q. Zhu, Nonlinear stochastic dynamics and control in Hamiltonian formulation,, ASME Appl. Mech. Rev., 59 (2006), 230.
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.
doi: 10.1080/1045112031000078917. |
[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. |
[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. |
[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. |
[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. |
[7] |
L. Decreusefond and A. S. Ustunel, Fractional Brownian motion: Theory and applications,, ESAIM: Proceedings, 5 (1998), 75.
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.
doi: 10.1137/S036301299834171X. |
[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).
|
[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. |
[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. |
[13] |
G. Jumarie, Stochastic differential equations with fractional Brownian motion input,, Int. J. Syst. Sci., 24 (1993), 1113.
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.
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. 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.
|
[17] |
A. N. Kolmogorov, Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen,, Raum, 26 (1940), 115.
|
[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. |
[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. |
[21] |
T. Lyons, Differential equations driven by rough signals,, Rev. Mat. Iberoamericana, 14 (1998), 215.
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.
doi: 10.1137/1010093. |
[23] |
Y. S. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes,, Springer-Verlag, (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.
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.
doi: 10.2307/3318691. |
[26] |
D. Nualart and A. Rascanu, Differential equations driven by fractional Brownian motion,, Collect. Math., 53 (2002), 55.
|
[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. |
[28] |
F. Russo and P. Vallois, Forward, backward and symmetric stochastic integration,, Probab. Theory Rel. Fields, 97 (1993), 403.
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.
doi: 10.1016/S0009-2509(00)00279-7. |
[30] |
A. N. Shiryaev, Essentials of Stochastic Finance: Facts, Models and Theory,, World Scientific, (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). Google Scholar |
[32] |
R. L. Stratonovich, Topics in the Theory of Random Noise,, New York, (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.
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.
doi: 10.1007/BF02401743. |
[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. |
[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. |
[38] |
W. Q. Zhu, Nonlinear stochastic dynamics and control in Hamiltonian formulation,, ASME Appl. Mech. Rev., 59 (2006), 230.
doi: 10.1115/1.2193137. |
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