Abstract
This paper proposes consistent and asymptotically Gaussian estimators for the parameters \(\lambda , \sigma \) and \(H\) of the discretely observed fractional Ornstein–Uhlenbeck process solution of the stochastic differential equation \(d Y_t = -\lambda Y_t dt + \sigma d W_t^H\), where \((W_t^H, t\ge 0)\) is the fractional Brownian motion. For the estimation of the drift \(\lambda \), the results are obtained only in the case when \(\frac{1}{2} < H < \frac{3}{4}\). This paper also provides ready-to-use software for the R statistical environment based on the YUIMA package.
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References
Bercu B, Coutin L, Savy N (2010) Sharp large deviations for the fractional Ornstein-Uhlenbeck process. Teoriya Veroyatnostei i ee Primeneniya
Bertin K, Torres S, Tudor C (2011) Drift parameter estimation in fractional diffusions driven by perturbed random walk. Stat Probab Lett 81(2):243–249
Bertoin J (1998) Lévy processes. Cambridge University Press, Cambridge
Berzin C, Leon J (2008) Estimation in models driven by fractional Brownian motion. Annales de l’Institut Henri Poincaré 44(2):191–213
Bregni S, Erangoli W (2005) Fractional noise in experimental measurements of IP traffic in a metropolitan area network. Proc IEEE GlobeCom 2005(2):781–785
Brouste A, Kleptsyna M (2010) Asymptotic properties of MLE for partially observed fractional diffusion system. Stat Inference Stoch Process 13(1):1–13
Cheridito P, Kawaguchi H, Maejima M (2003) Fractional Ornstein–Uhlenbeck processes. Electron J Probab 8(3):1–14
Cialenco I, Lototsky S, Pospisil J (2009) Asymptotic properties of the maximum likelihood estimator for stochastic parabolic equations with additive fractional Brownian motion. Stoch Dyn 9(2):169–185
Cœurjolly JF (2001) Estimating the parameters of a fractional Brownian motion by discrete variations of its sample paths. Stat Inference Stoch Process 4(2):199–227
Daubechies I (1992) Ten lectures on wavelets. SIAM, Philadelphia
Florens-Zmirou D (1989) Approximate discrete time schemes for statistics of diffusion processes. Statistics 20:263–284
Genon-Catalot V (1990) Maximum constrast estimation for diffusion processes from discrete observation. Statistics 21:99–116
Gobet E (2002) LAN property for ergodic diffusions with discrete observations. Annales de l’Institut Henri Poincaré 38(5):711–737
Hu Y, Nualart D (2010) Parameter estimation for fractional Ornstein–Uhlenbeck processes. Stat Probab Lett 80(11–12):1030–1038
Hu Y, Nualart D, Xiao W, Zhang W (2011) Exact maximum likelihood estimator for drift fractional brownian motion at discrete time. Acta Mathematica Scientia 31(5):1851–1859
Istas J, Lang G (1997) Quadratic variations and estimation of the local Hölder index of a Gaussian process. Annales de l’Institut Henri Poincaré 23(4):407–436
Jacod J (2001) Inference for stochastic processes. Statistics 683
Kaarakka T, Salminem P (2011) On fractional Ornstein–Uhlenbeck processes. Commun Stoch Anal 5(1):121–133
Kessler M (1997) Estimation of an ergodic diffusion from discrete observations. Scand J Stat 24:211–229
Kleptsyna M, Breton AL (2002) Statistical analysis of the fractional Ornstein–Uhlenbeck type process. Stat Inference Stoch Process 5:229–241
Kolmogorov A (1940) Winersche Spiralen und einige andere interessante Kurven in Hilbertschen Raum. Acad Sci USSR 26:115–118
Ludena C (2004) Minimum contrast estimation for fractional diffusion. Scand J Stat 31:613–628
Mandelbrot B, Ness JV (1968) Fractional Brownian motions, fractional noises and application. SIAM Rev 10:422–437
Melichov D (2011) On estimation of the Hurst index of solutions of stochastic equations. Ph.D. thesis, Vilnius Gediminas Technical University
Neuenkirch A, Nourdin I (2007) Exact rate of convergence of some approximation schemes associated to SDEs driven by a fractional Brownian motion. J Theor Probl 20(4):871–899
Neuenkirch A, Tindel S (2011) A least square-type procedure for parameter estimation in stochastic differential equations with additive fractional noise. preprint. http://hal.inria.fr/hal-00639030/PS/zero-squares-nt.ps
R Development Core Team: R (2010) A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. http://www.R-project.org. ISBN 3-900051-07-0
Sato K (1999) Lévy processes and infinitely divisible distributions. Cambridge University Press, Cambridge
Willinger W, Taqqu M, Leland W, Wilson D (1995) Self-similarity in high-speed packet traffic: analysis and modeling of ethernet traffic measurements. Stat Sci 10:67–85
Wood A, Chan G (1994) Simulation of stationary Gaussian processes. J Comput Graph Stat 3(4):409–432
Xiao W, Zhang W, Xu W (2011) Parameter estimation for fractional Ornstein–Uhlenbeck processes at discrete observation. Appl Math Model 35:4196–4207
Yoshida N (1992) Estimation for diffusion processes from discrete observations. J Multivar Anal 41:220–242
Yuima Project Team: yuima (2011) The YUIMA project package (unstable version). http://R-Forge.R-project.org/projects/yuima/. R package version 0.1.1936
Acknowledgments
We would like to thank Marina Kleptsyna for the discussions and her interest for this work. Computing resources have been financed by Mostapad project in CNRS FR 2962. This work has been supported by the project PRIN 2009JW2STY, Ministero dell’Istruzione dell’Università e della Ricerca.
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Brouste, A., Iacus, S.M. Parameter estimation for the discretely observed fractional Ornstein–Uhlenbeck process and the Yuima R package. Comput Stat 28, 1529–1547 (2013). https://doi.org/10.1007/s00180-012-0365-6
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DOI: https://doi.org/10.1007/s00180-012-0365-6