Stochastic averaging principle for dynamical systems with fractional Brownian motion

Yong Xu; Rong Guo; Di Liu; Huiqing Zhang; Jinqiao Duan

doi:10.3934/dcdsb.2014.19.1197

Previous Article
Pullback attractors for three dimensional non-autonomous planetary geostrophic viscous equations of large-scale ocean circulation
DCDS-B Home
This Issue
Next Article
Asymptotic pattern of a migratory and nonmonotone population model

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

Stochastic averaging principle for dynamical systems with fractional Brownian motion

Yong Xu ^1, , Rong Guo ^1, , Di Liu ^1, , Huiqing Zhang ^1, and Jinqiao Duan ^2,

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

Abstract
Related Papers

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.

Keywords: stochastic differential equations, Correlated noise, averaging principle, stochastic calculus, fractional Brownian motion..

Mathematics Subject Classification: Primary: 34F05, 37H10, 60H10, 93E0.

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), 129-152. doi: 10.1080/1045112031000078917. Google Scholar
[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. Google Scholar
[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. Google Scholar
[4]	P. Carmona, L. Coutin and Gerard Montseny, Stochastic integration with respect to fractional Brownian motion, Ann. Inst. Henri Poincare, 39 (2003), 27-68. 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), 9-13. 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), 439-448. doi: 10.1155/S104895339600038X. Google Scholar
[7]	L. Decreusefond and A. S. Ustunel, Fractional Brownian motion: Theory and applications, ESAIM: Proceedings, 5 (1998), 75-86. 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), 582-612. doi: 10.1137/S036301299834171X. Google Scholar
[9]	D. Feyel and A. de la Pradelle, Fractional integrals and Brownian processes, Potential Analysis., 10 (1996), 273-288. Google Scholar
[10]	H. Holden, B. Øksendal, J. Ubøe and T. Zhang, Stochastic Partial Differential Equations, Birkhäuser, Boston, 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), 1-32. 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), 45-53. 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), 1113-1132. 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), 817-826. 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), 390-405. 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), 1-21. Google Scholar
[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. 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), 180603. 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), 1-15. 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), 225-249. doi: 10.1023/A:1009983802178. Google Scholar
[21]	T. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana, 14 (1998), 215-310. 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), 422-427. doi: 10.1137/1010093. Google Scholar
[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. Google Scholar
[24]	N. Sri. Namachchivaya and Y. K. Lin, Application of stochastic averaging for systems with high damping, Probab. Eng. Mech., 3 (1988), 185-196. 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), 571-587. doi: 10.2307/3318691. Google Scholar
[26]	D. Nualart and A. Rascanu, Differential equations driven by fractional Brownian motion, Collect. Math., 53 (2002), 55-81. 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), 111-134. 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), 403-421. 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), 707-711. doi: 10.1016/S0009-2509(00)00279-7. Google Scholar
[30]	A. N. Shiryaev, Essentials of Stochastic Finance: Facts, Models and Theory, World Scientific, New Jersey, 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), 041119. Google Scholar
[32]	R. L. Stratonovich, Topics in the Theory of Random Noise, New York, Gordon and Breach, 1, 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), 1395-1401. 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), 251-282. doi: 10.1007/BF02401743. Google Scholar
[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. 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), 021002-021011. 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), 230-248. 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), 129-152. doi: 10.1080/1045112031000078917. Google Scholar
[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. Google Scholar
[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. Google Scholar
[4]	P. Carmona, L. Coutin and Gerard Montseny, Stochastic integration with respect to fractional Brownian motion, Ann. Inst. Henri Poincare, 39 (2003), 27-68. 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), 9-13. 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), 439-448. doi: 10.1155/S104895339600038X. Google Scholar
[7]	L. Decreusefond and A. S. Ustunel, Fractional Brownian motion: Theory and applications, ESAIM: Proceedings, 5 (1998), 75-86. 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), 582-612. doi: 10.1137/S036301299834171X. Google Scholar
[9]	D. Feyel and A. de la Pradelle, Fractional integrals and Brownian processes, Potential Analysis., 10 (1996), 273-288. Google Scholar
[10]	H. Holden, B. Øksendal, J. Ubøe and T. Zhang, Stochastic Partial Differential Equations, Birkhäuser, Boston, 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), 1-32. 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), 45-53. 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), 1113-1132. 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), 817-826. 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), 390-405. 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), 1-21. Google Scholar
[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. 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), 180603. 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), 1-15. 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), 225-249. doi: 10.1023/A:1009983802178. Google Scholar
[21]	T. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana, 14 (1998), 215-310. 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), 422-427. doi: 10.1137/1010093. Google Scholar
[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. Google Scholar
[24]	N. Sri. Namachchivaya and Y. K. Lin, Application of stochastic averaging for systems with high damping, Probab. Eng. Mech., 3 (1988), 185-196. 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), 571-587. doi: 10.2307/3318691. Google Scholar
[26]	D. Nualart and A. Rascanu, Differential equations driven by fractional Brownian motion, Collect. Math., 53 (2002), 55-81. 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), 111-134. 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), 403-421. 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), 707-711. doi: 10.1016/S0009-2509(00)00279-7. Google Scholar
[30]	A. N. Shiryaev, Essentials of Stochastic Finance: Facts, Models and Theory, World Scientific, New Jersey, 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), 041119. Google Scholar
[32]	R. L. Stratonovich, Topics in the Theory of Random Noise, New York, Gordon and Breach, 1, 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), 1395-1401. 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), 251-282. doi: 10.1007/BF02401743. Google Scholar
[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. 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), 021002-021011. 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), 230-248. doi: 10.1115/1.2193137. Google Scholar

[1]	Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213
[2]	Yong Xu, Bin Pei, Rong Guo. Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2257-2267. doi: 10.3934/dcdsb.2015.20.2257
[3]	María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 473-493. doi: 10.3934/dcdsb.2010.14.473
[4]	Ahmed Boudaoui, Tomás Caraballo, Abdelghani Ouahab. Stochastic differential equations with non-instantaneous impulses driven by a fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2521-2541. doi: 10.3934/dcdsb.2017084
[5]	David Cheban, Zhenxin Liu. Averaging principle on infinite intervals for stochastic ordinary differential equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2021014
[6]	Litan Yan, Xiuwei Yin. Optimal error estimates for fractional stochastic partial differential equation with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 615-635. doi: 10.3934/dcdsb.2018199
[7]	Yong Ren, Xuejuan Jia, Lanying Hu. Exponential stability of solutions to impulsive stochastic differential equations driven by $G$-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2157-2169. doi: 10.3934/dcdsb.2015.20.2157
[8]	Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control & Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401
[9]	Guolian Wang, Boling Guo. Stochastic Korteweg-de Vries equation driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5255-5272. doi: 10.3934/dcds.2015.35.5255
[10]	Brahim Boufoussi, Soufiane Mouchtabih. Controllability of neutral stochastic functional integro-differential equations driven by fractional brownian motion with Hurst parameter lesser than $ 1/2 $. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020096
[11]	Defei Zhang, Ping He. Functional solution about stochastic differential equation driven by $G$-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 281-293. doi: 10.3934/dcdsb.2015.20.281
[12]	Phuong Nguyen, Roger Temam. The stampacchia maximum principle for stochastic partial differential equations forced by lévy noise. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2289-2331. doi: 10.3934/cpaa.2020100
[13]	Tyrone E. Duncan. Some linear-quadratic stochastic differential games for equations in Hilbert spaces with fractional Brownian motions. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5435-5445. doi: 10.3934/dcds.2015.35.5435
[14]	María J. Garrido-Atienza, Bohdan Maslowski, Jana Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088
[15]	Jin Li, Jianhua Huang. Dynamics of a 2D Stochastic non-Newtonian fluid driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2483-2508. doi: 10.3934/dcdsb.2012.17.2483
[16]	Henryk Leszczyński, Monika Wrzosek. Newton's method for nonlinear stochastic wave equations driven by one-dimensional Brownian motion. Mathematical Biosciences & Engineering, 2017, 14 (1) : 237-248. doi: 10.3934/mbe.2017015
[17]	Haiyan Zhang. A necessary condition for mean-field type stochastic differential equations with correlated state and observation noises. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1287-1301. doi: 10.3934/jimo.2016.12.1287
[18]	Kexue Li. Effects of the noise level on nonlinear stochastic fractional heat equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5437-5460. doi: 10.3934/dcdsb.2019065
[19]	Peng Gao. Averaging principle for stochastic Kuramoto-Sivashinsky equation with a fast oscillation. Discrete & Continuous Dynamical Systems, 2018, 38 (11) : 5649-5684. doi: 10.3934/dcds.2018247
[20]	Zhen Li, Jicheng Liu. Synchronization for stochastic differential equations with nonlinear multiplicative noise in the mean square sense. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5709-5736. doi: 10.3934/dcdsb.2019103

2019 Impact Factor: 1.27

American Institute of Mathematical Sciences

Stochastic averaging principle for dynamical systems with fractional Brownian motion

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Stochastic averaging principle for dynamical systems with fractional Brownian motion

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors