Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Fractional Calculus and Applied Analysis

Editor-in-Chief: Kiryakova, Virginia

6 Issues per year

IMPACT FACTOR 2017: 2.865
5-year IMPACT FACTOR: 3.323

CiteScore 2017: 3.06

SCImago Journal Rank (SJR) 2017: 1.967
Source Normalized Impact per Paper (SNIP) 2017: 1.954

Mathematical Citation Quotient (MCQ) 2017: 0.98

See all formats and pricing
More options …

The fBm-driven Ornstein-Uhlenbeck process: Probability density function and anomalous diffusion

Caibin Zeng
  • School of Sciences, South China University of Technology, Guangzhou, 510640, China
  • Center for Self-Organizing and Intelligent Systems Department of Electrical and Computer Engineering, Utah State University, Logan, Utah, 84322, USA
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ YangQuan Chen
  • Center for Self-Organizing and Intelligent Systems Department of Electrical and Computer Engineering, Utah State University, Logan, Utah, 84322, USA
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Qigui Yang
Published Online: 2012-06-30 | DOI: https://doi.org/10.2478/s13540-012-0034-z


This paper deals with the Ornstein-Uhlenbeck (O-U) process driven by the fractional Brownian motion (fBm). Based on the fractional Itô formula, we present the corresponding fBm-driven Fokker-Planck equation for the nonlinear stochastic differential equations driven by an fBm. We then apply it to establish the evolution of the probability density function (PDF) of the fBm-driven O-U process. We further obtain the closed form of such PDF by combining the Fourier transform and the method of characteristics. Interestingly, the obtained PDF has an infinite variance which is significantly different from the classical O-U process. We reveal that the fBm-driven O-U process can describe the heavy-tailedness or anomalous diffusion. Moreover, the speed of the sub-diffusion is inversely proportional to the viscosity coefficient, while is proportional to the Hurst parameter. Finally, we carry out numerical simulations to verify the above findings.

MSC: Primary 60G22; Secondary 26A33, 35R60, 60H15, 35Q84

Keywords: fractional Brownian motion; Ornstein-Uhlenbeck process; anomalous diffusion; Fokker-Planck equation

  • [1] E. Alòs, O. Mazet, D. Nualart, Stochastic calculus with respect to Gaussian processes. Ann. Probab. 29, No 2 (2001), 766–801. http://dx.doi.org/10.1214/aop/1008956692CrossrefGoogle Scholar

  • [2] B. Bercu, L. Coutin, N. Savy, Sharp large deviations for the fractional Ornstein-Uhlenbeck process. Theor. Probab. Appl. 55 (2011), 575–610. http://dx.doi.org/10.1137/S0040585X97985108CrossrefGoogle Scholar

  • [3] F. Biagini, Y. Hu, B. Oksendal, T. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications. Springer-Verlag (2008). Google Scholar

  • [4] F. Biagini, B. Øsendal, A. Sulem, N. Wallner, An introduction to white noise theory and Malliavin calculus for fractional Brownian motion. Proc. R. Soc. 460, No 2041 (2004), 347–372. http://dx.doi.org/10.1098/rspa.2003.1246CrossrefGoogle Scholar

  • [5] J. Bishwal, Minimum contrast estimation in fractional Ornstein-Uhlenbeck process: Continuous and discrete sampling. Fract. Calc. Appl. Anal. 14, No 3 (2011), 375–410; DOI:10.2478/s13540-011-0024-6; http://www.springerlink.com/content/1311-0454/14/3/ CrossrefWeb of ScienceGoogle Scholar

  • [6] P. Carmona, L. Coutin, G. Montseny, Stochastic integration with respect to fractional Brownian motion. Ann. I. H. Poincaré Probab. Stat. 39, No 1 (2003), 27–68. http://dx.doi.org/10.1016/S0246-0203(02)01111-1CrossrefGoogle Scholar

  • [7] P. Cheridito, H. Kawaguchi, M. Maejima, Fractional Ornstein-Uhlenbeck processes. Electron. J. Probab 8, No 3 (2003), 1–14. Google Scholar

  • [8] F. Debbasch, K. Mallick, J. Rivet, Relativistic Ornstein-Uhlenbeck process. J. Stat. Phys. 88, No 3 (1997), 945–966. http://dx.doi.org/10.1023/B:JOSS.0000015180.16261.53CrossrefGoogle Scholar

  • [9] L. Decreusefond, A. Üstünel, Stochastic analysis of the fractional Brownian motion. Potential Anal. 10, No 2 (1999), 177–214. http://dx.doi.org/10.1023/A:1008634027843CrossrefGoogle Scholar

  • [10] C. Dellacherie, P. Meyer, Probability and Potentials B. Theory of Martingales. North-Holland, Amsterdam (1982). Google Scholar

  • [11] S. Ditlevsen, P. Lansky, Estimation of the input parameters in the Ornstein-Uhlenbeck neuronal model. Phys. Rev. E 71, No 1 (2005), 011907. http://dx.doi.org/10.1103/PhysRevE.71.011907CrossrefGoogle Scholar

  • [12] T. Duncan, Y. Hu, B. Pasik-Duncan, Stochastic calculus for fractional Brownian motion I. theory. SIAM J. Control Optim. 38, No 2 (2000), 582–612. http://dx.doi.org/10.1137/S036301299834171XCrossrefGoogle Scholar

  • [13] R. Elliott, J. Van Der Hoek, A general fractional white noise theory and applications to finance. Math. Finan. 13, No 2 (2003), 301–330. http://dx.doi.org/10.1111/1467-9965.00018CrossrefGoogle Scholar

  • [14] P. Garbaczewski, R. Olkiewicz, Ornstein-Uhlenbeck-Cauchy process. J. Math. Phys. 41 (2000), 6843–6860. http://dx.doi.org/10.1063/1.1290054CrossrefGoogle Scholar

  • [15] C. Gardiner, Handbook of Stochastic Methods for Physics. Chemistry and the Natural Sciences Ser., Springer-Verlag, Berlin (1983). Google Scholar

  • [16] D. Gillespie, The mathematics of Brownian motion and Johnson noise. Am. J. Phys. 64, No 3 (1996), 225–239. http://dx.doi.org/10.1119/1.18210CrossrefGoogle Scholar

  • [17] D. Gillespie, Exact numerical simulation of the Ornstein-Uhlenbeck process and its integral. Phys. Rev. E 54, No 2 (1996), 2084–2091. http://dx.doi.org/10.1103/PhysRevE.54.2084CrossrefGoogle Scholar

  • [18] Y. Hu, D. Nualart, Parameter estimation for fractional Ornstein-Uhlenbeck processes. Stat. Probab. Lett. 80, No 11 (2010), 1030–1038. http://dx.doi.org/10.1016/j.spl.2010.02.018Web of ScienceCrossrefGoogle Scholar

  • [19] M. Jolis, On the wiener integral with respect to the fractional Brownian motion on an interval. J. Math. Anal. Appl. 330, No 2 (2007), 1115–1127. http://dx.doi.org/10.1016/j.jmaa.2006.07.100CrossrefGoogle Scholar

  • [20] C. Kahl, Modelling and Simulation of Stochastic Volatility in Finance. Universal-Publishers, Boca Raton (2008). Google Scholar

  • [21] M. Kleptsyna, A. L. Breton, Statistical analysis of the fractional Ornstein-Uhlenbeck type process. Stat. Inf. Stoch. Proces. 5 (2002), 229–248. http://dx.doi.org/10.1023/A:1021220818545CrossrefGoogle Scholar

  • [22] S. Lin, Stochastic analysis of fractional Brownian motions. Stoch. Stoch. Rep 55, No 1–2 (1995), 121–140. Google Scholar

  • [23] B. Mandelbrot, J. Van Ness, Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10, No 4 (1968), 422–437. http://dx.doi.org/10.1137/1010093CrossrefGoogle Scholar

  • [24] B. Mandelbrot, On the geometry of homogeneous turbulence, with stress on the fractal dimension of the iso-surfaces of scalars. J. Fluid Mech. 72, No 3 (1975), 401–416. http://dx.doi.org/10.1017/S0022112075003047CrossrefGoogle Scholar

  • [25] Y. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes. Springer (2008). Google Scholar

  • [26] F. Reif, Fundamentals of Statistical and Thermal Physics. McGraw-Hill, New York (1965). Google Scholar

  • [27] L. Rogers, Arbitrage with fractional Brownian motion. Math. Finan. 7, No 1 (1997), 95–105. http://dx.doi.org/10.1111/1467-9965.00025CrossrefGoogle Scholar

  • [28] Y. Shao, The fractional Ornstein-Uhlenbeck process as a representation of homogeneous eulerian velocity turbulence. Physica D 83, No 4 (1995), 461–477. http://dx.doi.org/10.1016/0167-2789(95)00051-5CrossrefGoogle Scholar

  • [29] G. Uhlenbeck, L. Ornstein, On the theory of the Brownian motion. Phys. Rev. 36, No 5 (1930), 823–841. http://dx.doi.org/10.1103/PhysRev.36.823CrossrefGoogle Scholar

  • [30] O. Vasicek, An equilibrium characterization of the term structure. J. Finan. Econ. 5, No 2 (1977), 177–188. http://dx.doi.org/10.1016/0304-405X(77)90016-2CrossrefGoogle Scholar

  • [31] W. Xiao, W. Zhang, W. Xu, Parameter estimation for fractional Ornstein-Uhlenbeck processes at discrete observation. Appl. Math. Model. 35 (2011), 4196–4207. http://dx.doi.org/10.1016/j.apm.2011.02.047CrossrefWeb of ScienceGoogle Scholar

  • [32] L. Yan, M. Tian, On the local times of fractional Ornstein-Uhlenbeck process. Lett. Math. Phys. 73, No 3 (2005), 209–220. http://dx.doi.org/10.1007/s11005-005-0018-6CrossrefGoogle Scholar

  • [33] C. Zeng, Q. Yang, Y. Q. Chen, Solving nonlinear stochastic differential equations with fractional Brownian motion using reducibility approach. Nonlinear Dyn. 67, No 4 (2012), 2719–2726. http://dx.doi.org/10.1007/s11071-011-0183-3Web of ScienceCrossrefGoogle Scholar

About the article

Published Online: 2012-06-30

Published in Print: 2012-09-01

Citation Information: Fractional Calculus and Applied Analysis, Volume 15, Issue 3, Pages 479–492, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.2478/s13540-012-0034-z.

Export Citation

© 2012 Diogenes Co., Sofia. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Jiro Akahori, Xiaoming Song, and Tai-Ho Wang
Stochastic Processes and their Applications, 2018
Caibin Zeng, YangQuan Chen, and Qigui Yang
Fractional Calculus and Applied Analysis, 2013, Volume 16, Number 2
Caibin Zeng and YangQuan Chen
Fractional Calculus and Applied Analysis, 2014, Volume 17, Number 2
Ciprian Tudor
Fractional Calculus and Applied Analysis, 2014, Volume 17, Number 1
Caibin Zeng, Qigui Yang, and YangQuan Chen
Abstract and Applied Analysis, 2014, Volume 2014, Page 1
Qigui Yang, Caibin Zeng, and Cong Wang
Chaos: An Interdisciplinary Journal of Nonlinear Science, 2013, Volume 23, Number 4, Page 043120
Xiao-Li Ding and Yao-Lin Jiang
Fractional Calculus and Applied Analysis, 2013, Volume 16, Number 3

Comments (0)

Please log in or register to comment.
Log in