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Statistical Papers

, Volume 60, Issue 6, pp 2253–2271 | Cite as

Least squares estimator for Ornstein–Uhlenbeck processes driven by fractional Lévy processes from discrete observations

  • Guangjun ShenEmail author
  • Qian Yu
Regular Article

Abstract

In this paper, we consider the problem of parameter estimation for Ornstein–Uhlenbeck processes with small fractional Lévy noises, based on discrete observations at n regularly spaced time points \(t_i=i/n,\)\(i=1,\ldots ,n\) on [0, 1]. Least squares method is used to obtain an estimator of the drift parameter. The consistency and the asymptotic distribution of the estimator have been established.

Keywords

Ornstein–Uhlenbeck processes Fractional Lévy processes Least squares estimator Asymptotic distribution 

Mathematics Subject Classification

60G18 65C30 93E24 

Notes

Acknowledgements

The authors are very grateful to the anonymous referee and the editor for their insightful and valuable comments, which have improved the presentation of the paper. Research supported by the Distinguished Young Scholars Foundation of Anhui Province (1608085J06), the Top Talent Project of University Discipline (Speciality) (gxbjZD03), the National Natural Science Foundation of China (11271020).

References

  1. Benassi A, Cohen S, Istas J (2002) Identification and properties of real harmonizable fractional Lévy motions. Bernoulli 8:97–115MathSciNetzbMATHGoogle Scholar
  2. Bender C, Lindner A, Schicks M (2012) Finite variation of fractional Lévy processes. J Theor Probab 25:594–612CrossRefGoogle Scholar
  3. Brockwell PJ, Davis RA, Yang Y (2007) Estimation for non-negative Lévy-driven Ornstein–Uhlenbeck processes. J Appl Probab 44:977–989MathSciNetCrossRefGoogle Scholar
  4. Brouste A, Iacus SM (2013) Parameter estimation for the discretely observed fractional Ornstein–Uhlenbeck process and the Yuima R package. Comput Stat 28:1529–1547MathSciNetCrossRefGoogle Scholar
  5. Engelke S (2013) A unifying approach to fractional Lévy processes. Stoch Dyn 13:1250017MathSciNetCrossRefGoogle Scholar
  6. Dietz HM, Kutoyants YA (2003) Parameter estimation for some non-recurrent solutions of SDE. Stat Decis 21:29–46MathSciNetCrossRefGoogle Scholar
  7. Fink H, Klüppelberg C (2011) Fractional Lévy-driven Ornstein–Uhlenbeck processes and stochastic differential equations. Bernoulli 17:484–506MathSciNetCrossRefGoogle Scholar
  8. Gao F, Jiang H (2009) Deviation inequalities and moderate deviations for estimators of parameters in an Ornstein–Uhlenbeck process with linear drift. Electron Commun Probab 14:210–223MathSciNetCrossRefGoogle Scholar
  9. Genon-Catalot V (1990) Maximum contrast estimation for diffusion processes from discrete observations. Statistics 21:99–116MathSciNetCrossRefGoogle Scholar
  10. Gloter A, Sørensen M (2009) Estimation for stochastic differential equations with a small diffusion coefficient. Stoch Process Appl 119:679–699MathSciNetCrossRefGoogle Scholar
  11. Hu Y, Long H (2009) Least squares estimator for Ornstein–Uhlenbeck processes driven by \(\alpha \)-stable motions. Stoch Process Appl 119:2465–2480MathSciNetCrossRefGoogle Scholar
  12. Jiang H, Dong X (2015) Parameter estimation for the non-stationary Ornstein–Uhlenbeck process with linear drift. Stat Papers 56:257–268MathSciNetCrossRefGoogle Scholar
  13. Kuang N, Liu B (2016) Least squares estimator for \(\alpha \)-sub-fractional bridges. Stat Papers. doi: 10.1007/s00362-016-0795-2 MathSciNetCrossRefGoogle Scholar
  14. Kutoyants YA (2004) Statistical inference for Ergodic diffusion processes. Springer, BerlinCrossRefGoogle Scholar
  15. Lacaux C (2004) Real harmonizable multifractional Lévy motions. Ann I H Poincaré 40:259–277MathSciNetCrossRefGoogle Scholar
  16. Laredo CF (1990) A sufficient condition for asymptotic sufficiency of incomplete observations of a diffusion process. Ann Stat 18:1158–1171MathSciNetCrossRefGoogle Scholar
  17. Lin Z, Cheng Z (2009) Existence and joint continuity of local time of multiparameter fractional Lévy processes. Appl Math Mech 30:381–390MathSciNetCrossRefGoogle Scholar
  18. Long H (2009) Least squares estimator for discretely observed Ornstein–Uhlenbeck processes with small Lévy noises. Stat Probab Lett 79:2076–2085CrossRefGoogle Scholar
  19. Long H, Shimizu Y, Sun W (2013) Least squares estimators for discretely observed stochastic processes driven by small Lévy noises. J Multivar Anal 116:422–439CrossRefGoogle Scholar
  20. Ma C (2010) A note on “Least squares estimator for discretely observed Ornstein–Uhlenbeck processes with small Lévy noises”. Stat Probab Lett 80:1528–1531CrossRefGoogle Scholar
  21. Ma C, Yang X (2014) Small noise fluctuations of the CIR model driven by \(\alpha \)-stable noises. Stat Probab Lett 94:1–11MathSciNetCrossRefGoogle Scholar
  22. Mandelbrot BB, Van Ness JW (1968) Fractional Brownian motions, fractional noises and applications. SIAM Rev 10:422–437MathSciNetCrossRefGoogle Scholar
  23. Marquardt T (2006) Fractional Lévy processes with an application to long memory moving average processes. Bernoulli 12:1009–1126CrossRefGoogle Scholar
  24. Masuda H (2005) Simple estimators for parametric Markovian trend of ergodic processes based on sampled data. J Jpn Stat Soc 35:147–170MathSciNetCrossRefGoogle Scholar
  25. Revuz D, Yor M (1999) Continuous martingales and Brownian motion. Springer, New YorkCrossRefGoogle Scholar
  26. Samorodnitsky G, Taqqu MS (1994) Stable non-Gaussian random processes: stochastic models with infinite variance. Chapman and Hall, New YorkzbMATHGoogle Scholar
  27. Shimizu Y, Yoshida N (2006) Estimation of parameters for diffusion processes with jumps from discrete observations. Stat Inference Stoch Process 9:227–277MathSciNetCrossRefGoogle Scholar
  28. Sørensen M (2000) Small dispersion asymptotics for diffusion martingale estimating functions. Preprint No. 2000–2002, Department of Statistics and Operation Research, University of Copenhagen, CopenhagenGoogle Scholar
  29. Sørensen M, Uchida M (2003) Small diffusion asymptotics for discretely sampled stochastic differential equations. Bernoulli 9:1051–1069MathSciNetCrossRefGoogle Scholar
  30. Tikanmäki H, Mishura Y (2011) Fractional Lévy processes as a result of compact interval integral transformation. Stoch Anal Appl 29:1081–1101MathSciNetCrossRefGoogle Scholar
  31. Xiao W, Zhang W, Xu W (2011) Parameter estimation for fractional Ornstein–Uhlenbeck processes at discrete observation. Appl Math Model 35:4196–4207MathSciNetCrossRefGoogle Scholar
  32. Zhang S, Zhang X (2013) A least squares estimator for discretely observed Ornstein–Uhlenbeck processes driven by symmetric \(\alpha \)-stable motions. Ann Inst Stat Math 65:89–103MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of MathematicsAnhui Normal UniversityWuhuChina

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