Skip to main content
SpringerLink
Log in

Minimum contrast estimator for fractional Ornstein-Uhlenbeck processes

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

This paper proposes a minimum contrast methodology to estimate the drift parameter for the Ornstein-Uhlenbeck process driven by fractional Brownian motion of Hurst index, which is greater than one half. Both the strong consistency and the asymptotic normality of this minimum contrast estimator are studied based on the Laplace transform. The numerical simulation results confirm the theoretical analysis and show that the minimum contrast technique is effective and efficient.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aït-Sahalia Y. Maximum likelihood estimation of discretely sampled diffusions: A closed-form approximation approach. Econometrica, 2002, 70: 223–262

    Article  MathSciNet  MATH  Google Scholar 

  2. Anh V V, Leonenko N N, Sakhno L M. On a class of minimum contrast estimators for fractional stochastic processes and fields. J Stat Plan Infer, 2004, 123: 161–185

    Article  MathSciNet  MATH  Google Scholar 

  3. Bercu B, Coutin L, Savy N. Sharp large deviations for the fractional Ornstein-Uhlenbeck process. Theory Probab Appl, 2011, 55: 575–610

    Article  MathSciNet  MATH  Google Scholar 

  4. Bertin K, Torres S, Tudor C A. Drift parameter estimation in fractional diffusions driven by perturbed random walks. Statist Probab Lett, 2011, 81: 243–249

    Article  MathSciNet  MATH  Google Scholar 

  5. Bishwal J P N. Rates of weak convergence of approximate minimum contrast estimators for the discretely observed Ornstein-Uhlenbeck process. Statist Probab Lett, 2006, 76: 1397–1409

    Article  MathSciNet  MATH  Google Scholar 

  6. Bishwal J P N. Parameter estimation in stochastic differential equations. In: Lecture Notes in Math, 1923. Berlin: Springer, 2008

    Book  MATH  Google Scholar 

  7. Brouste A, Kleptsyna M. Asymptotic properties of MLE for partially observed fractional diffusion system. Statist Inference Stoch Process, 2010, 13: 1–13.

    Article  MathSciNet  MATH  Google Scholar 

  8. Brouste A. Asymptotic properties of MLE for partially observed fractional diffusion system with dependent noises. J Stat Plan Infer, 2010, 140: 551–558

    Article  MathSciNet  MATH  Google Scholar 

  9. Brouste A, Kleptsyna M, Popier A. Design for estimation of drift parameter in fractional diffusion system. Preprint, 2010

  10. Coeurjolly J F. Simulation and identification of the fractional Brownian motion: a bibliographical and comparative study. J Stat Softw, 2000, 5: 1–53

    Google Scholar 

  11. Davies R B, Harte D S. Tests for the Hurst effect. Biometrika, 1987, 74: 95–102

    Article  MathSciNet  MATH  Google Scholar 

  12. Dorogovcev A J. The consistency of an estimate of a parameter of a stochastic differential equation. Theory Probab Math Stat, 1976, 10: 73–82

    Google Scholar 

  13. Duncan T E, Hu Y, Pasik-Duncan B. Stochastic calculus for fractional Brownian motion I: Theory. SIAM J Control Optim, 2000, 38: 582–612

    Article  MathSciNet  MATH  Google Scholar 

  14. Hu Y, Long H. Least squares estimator for Ornstein-Uhlenbeck processes driven by α-stable motions. Stoch Process Appl, 2009, 119: 2465–2480

    Article  MathSciNet  MATH  Google Scholar 

  15. Hu Y, Nualart D. Parameter estimation for fractional Ornstein-Uhlenbeck processes. Statist Probab Lett, 2010, 80: 1030–1038

    Article  MathSciNet  MATH  Google Scholar 

  16. Hu Y, Nualart D, Xiao W, et al. Exact maximum likelihood estimator for drift fractional Brownian motion at discrete observation. Acta Math Sci, 2011, 31: 1851–1859

    MathSciNet  MATH  Google Scholar 

  17. Kasonga R A. The consistency of a nonlinear least squares estimator for diffusion processes. Stoch Process Appl, 1988, 30: 263–275

    Article  MathSciNet  MATH  Google Scholar 

  18. Kleptsyna M L, Le Breton A. Statistical analysis of the fractional Ornstein-Uhlenbeck type process. Statist Inference Stoch Process, 2002, 5: 229–248

    Article  MATH  Google Scholar 

  19. Kleptsyna M L, Le Breton A, Roubaud M C. Parameter estimation and optimal filtering for fractional type stochastic systems. Statist Inference Stoch Process, 2000, 3: 173–182

    Article  MATH  Google Scholar 

  20. Kleptsyna M L, Le Breton A, Viot M. Separation principle in the fractional Gaussian linear-quadratic regulator problem with partial observation. ESAIM Probab Stat, 2008, 12: 94–126

    Article  MathSciNet  MATH  Google Scholar 

  21. Kloeden P E, Neuenkirch A, Pavani R. Multilevel Monte Carlo for stochastic differential equations with additive fractional noise. Ann Oper Res, 2011, 189: 255–276

    Article  MathSciNet  MATH  Google Scholar 

  22. Lánska V. Minimum contrast estimation in diffusion processes. J Appl Probab, 1979, 16: 65–75

    Article  MathSciNet  MATH  Google Scholar 

  23. Le Breton A. Filtering and parameter estimation in a simple linear model driven by a fractional Brownian motion. Statist Probab Lett, 1998, 38: 263–274.

    Article  MathSciNet  MATH  Google Scholar 

  24. Liptser R S. Shiryaev A N. Statistics of Random Processes II: Applications, 2nd ed. Berlin: Springer, 2001

    Google Scholar 

  25. Ludeña C. Minimum contrast estimation for fractional diffusions. Scand J Stat, 2004, 31: 613–628.

    Article  MATH  Google Scholar 

  26. Mishra M N, Prakasa Rao B L S. On a Berry-Esséen type bound for the least squares estimator for diffusion processes based on discrete observations. Internat J Static Mana Syst, 2007, 2: 1–21

    MathSciNet  Google Scholar 

  27. Norros I, Valkeila E, Virtamo J. An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motion. Bernoulli, 1999, 5: 571–587

    Article  MathSciNet  MATH  Google Scholar 

  28. Phillips P B C, Yu J. Maximum likelihood and Gaussian estimation of continuous time models in finance. In: Andersen T G, Davis R A, Kreiss J-P, et al, Eds. Handbook of Financial Time Series. New York: Springer, 2008, 497–530

    Google Scholar 

  29. Prakasa Rao B L S. Asymptotic theory for nonlinear least squares estimator for diffusion processes. Math Oper Sch Stat Ser Stat, 1983, 14: 195–209

    MathSciNet  MATH  Google Scholar 

  30. Prakasa Rao B L S. Statistical Inference for Diffusion Type Processes. Arnold: Oxford University Press, 1999

    MATH  Google Scholar 

  31. Shimizu Y. M-estimation for discretely observed ergodic diffusion processes with infinite jumps. Stat Inference Stoch Process, 2006, 9: 179–225.

    Article  MathSciNet  MATH  Google Scholar 

  32. Shimizu Y, Yoshida N. Estimation of parameters for diffusion processes with jumps from discrete observations. Stat Inference Stoch Process, 2006, 9: 227–277

    Article  MathSciNet  MATH  Google Scholar 

  33. Tang C Y, Chen S X. Parameter estimation and bias correction for diffusion processes. J Econometrics, 2009, 149: 65–81

    Article  MathSciNet  Google Scholar 

  34. Taqqu M S. The modelling of Ethernet data and of signals that are heavy-tailed with infinite variance. Scand J Stat, 2002, 29: 273–295

    Article  MathSciNet  MATH  Google Scholar 

  35. Tsai H, Chan K S. Temporal aggregation of stationary and non-stationary continuous-time processes. Scand J Stat, 2005, 32: 583–597

    Article  MathSciNet  MATH  Google Scholar 

  36. Tudor C A, Viens F. Statistical aspects of the fractional stochastic calculus. Ann Statist, 2007, 25: 1183–1212

    Article  MathSciNet  Google Scholar 

  37. Valdivieso L, Schoutens W, Tuerlinckx F. Maximum likelihood estimation in processes of Ornstein-Uhlenbeck type. Stat Infer Stoch Process, 2009, 12: 1–19

    Article  MathSciNet  MATH  Google Scholar 

  38. Watson G N. A Treatise on the Theory of Bessel Functions, 2nd ed. New York: Cambridge University Press, 1944

    MATH  Google Scholar 

  39. Xiao W, Zhang W, Xu W. Parameter estimation for fractional Ornstein-Uhlenbeck processes at discrete observation. Appl Math Model, 2011, 35: 4196–4207

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Business and Administration, South China University of Technology, Guangzhou, 510640, China

    WeiLin Xiao & WeiGuo Zhang

  2. Department of Accounting and Finance, School of Management, Zhejiang University, Hangzhou, 310006, China

    XiLi Zhang

Authors
  1. WeiLin Xiao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. WeiGuo Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. XiLi Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to WeiGuo Zhang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Xiao, W., Zhang, W. & Zhang, X. Minimum contrast estimator for fractional Ornstein-Uhlenbeck processes. Sci. China Math. 55, 1497–1511 (2012). https://doi.org/10.1007/s11425-012-4386-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-012-4386-y

Keywords

MSC(2010)

Search

Navigation