Electronic Journal of Statistics

Consistency of the drift parameter estimator for the discretized fractional Ornstein–Uhlenbeck process with Hurst index $H\in(0,\frac{1}{2})$

Kęstutis Kubilius, Yuliya Mishura, Kostiantyn Ralchenko, and Oleg Seleznjev

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Abstract

We consider the Langevin equation which contains an unknown drift parameter $\theta$ and where the noise is modeled as fractional Brownian motion with Hurst index $H\in(0,\frac{1}{2})$. The solution corresponds to the fractional Ornstein–Uhlenbeck process. We construct an estimator, based on discrete observations in time, of the unknown drift parameter, that is similar in form to the maximum likelihood estimator for the drift parameter in Langevin equation with standard Brownian motion. It is assumed that the interval between observations is $n^{-1}$, i.e. tends to zero (high-frequency data) and the number of observations increases to infinity as $n^{m}$ with $m>1$. It is proved that for strictly positive $\theta$ the estimator is strongly consistent for any $m>1$, while for $\theta\leq0$ it is consistent when $m>\frac{1}{2H}$.

Article information

Source
Electron. J. Statist., Volume 9, Number 2 (2015), 1799-1825.

Dates
Received: January 2015
First available in Project Euclid: 25 August 2015

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1440507394

Digital Object Identifier
doi:10.1214/15-EJS1062

Mathematical Reviews number (MathSciNet)
MR3391120

Zentralblatt MATH identifier
1326.60048

Subjects
Primary: 60G22: Fractional processes, including fractional Brownian motion 60F15: Strong theorems 60F25: $L^p$-limit theorems 62F10: Point estimation 62F12: Asymptotic properties of estimators

Keywords
Fractional Brownian motion fractional Ornstein–Uhlenbeck process short-range dependence drift parameter estimator consistency strong consistency discretization high-frequency data

Citation

Kubilius, Kęstutis; Mishura, Yuliya; Ralchenko, Kostiantyn; Seleznjev, Oleg. Consistency of the drift parameter estimator for the discretized fractional Ornstein–Uhlenbeck process with Hurst index $H\in(0,\frac{1}{2})$. Electron. J. Statist. 9 (2015), no. 2, 1799--1825. doi:10.1214/15-EJS1062. https://projecteuclid.org/euclid.ejs/1440507394


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References

  • [1] Belfadli, R., Es-Sebaiy, K. and Ouknine, Y. (2011). Parameter estimation for fractional Ornstein–Uhlenbeck processes: Non-ergodic case., Frontiers in Science and Engineering 1 1–16.
  • [2] Bertin, K., Torres, S. and Tudor, C. A. (2011). Drift parameter estimation in fractional diffusions driven by perturbed random walks., Stat. Probab. Lett. 81 243–249.
  • [3] Bianchi, S., Pantanella, A. and Pianese, A. (2013). Modeling stock prices by multifractional Brownian motion: An improved estimation of the pointwise regularity., Quant. Finance 13 1317–1330.
  • [4] Bishwal, J. P. N. (2011). Minimum contrast estimation in fractional Ornstein-Uhlenbeck process: Continuous and discrete sampling., Fract. Calc. Appl. Anal. 14 375–410.
  • [5] Le Breton, A. (1976). On continuous and discrete sampling for parameter estimation in diffusion type processes., Stoch. Syst.: Model., Identif., Optim. I; Math. Program. Study 5 124–144.
  • [6] Brouste, A. and Iacus, S. M. (2013). Parameter estimation for the discretely observed fractional Ornstein–Uhlenbeck process and the Yuima R package., Computational Statistics 28 1529–1547.
  • [7] Buchmann, B. and Chan, N. H. (2009). Integrated functionals of normal and fractional processes., Ann. Appl. Probab. 19 49–70.
  • [8] Cénac, P. and Es-Sebaiy, K. (2012). Almost sure central limit theorems for random ratios and applications to LSE for fractional Ornstein-Uhlenbeck processes., Preprint. arXiv:1209.0137 [math.PR].
  • [9] Cheridito, P., Kawaguchi, H. and Maejima, M. (2003). Fractional Ornstein-Uhlenbeck processes., Electron. J. Probab. 8.
  • [10] Clarke De la Cerda, J. and Tudor, C. A. (2012). Least squares estimator for the parameter of the fractional Ornstein–Uhlenbeck sheet., J. Korean Stat. Soc. 41 341–350.
  • [11] Comte, F., Coutin, L. and Renault, E. (2012). Affine fractional stochastic volatility models., Ann. Finance 8 337–378.
  • [12] Diedhiou, A., Manga, C. and Mendy, I. (2011). Parametric estimation for SDEs with additive sub-fractional Brownian motion., Journal of Numerical Mathematics and Stochastics 3 37–45.
  • [13] Es-Sebaiy, K. (2013). Berry-Esséen bounds for the least squares estimator for discretely observed fractional Ornstein–Uhlenbeck processes., Stat. Probab. Lett. 83 2372–2385.
  • [14] Es-sebaiy, K. and Ndiaye, D. (2014). On drift estimation for non-ergodic fractional Ornstein–Uhlenbeck process with discrete observations., Afr. Stat. 9 615–625.
  • [15] Gradinaru, M. and Nourdi, I. (2007). Convergence of weighted power variations of fractional Brownian motion., Preprint.
  • [16] Hu, Y. and Nualart, D. (2010). Parameter estimation for fractional Ornstein-Uhlenbeck processes., Stat. Probab. Lett. 80 1030–1038.
  • [17] Hu, Y., Nualart, D., Xiao, W. and Zhang, W. (2011). Exact maximum likelihood estimator for drift fractional Brownian motion at discrete observation., Acta Mathematica Scientia 31 1851–1859.
  • [18] Hu, Y. and Song, J. (2013). Parameter estimation for fractional Ornstein-Uhlenbeck processes with discrete observations. In, Malliavin Calculus and Stochastic Analysis. A Festschrift in Honor of David Nualart. New York, NY: Springer. 427–442.
  • [19] Jacod, J. (2006). Parametric inference for discretely observed non-ergodic diffusions., Bernoulli 12 383–401.
  • [20] Kasonga, R. A. (1988). The consistency of a nonlinear least squares estimator from diffusion processes., Stochastic Processes Appl. 30 263–275.
  • [21] Kleptsyna, M. L. and Le Breton, A. (2002). Statistical analysis of the fractional Ornstein–Uhlenbeck type process., Stat. Inference Stoch. Process. 5 229–248.
  • [22] Kozachenko, Y., Melnikov, A. and Mishura, Y. (2014). On drift parameter estimation in models with fractional Brownian motion., Statistics. Advance online publication. doi: 10.1080/02331888.2014.907294.
  • [23] Mendy, I. (2013). Parametric estimation for sub-fractional Ornstein-Uhlenbeck process., J. Stat. Plann. Inference 143 663–674.
  • [24] Mishura, Y. (2014). Standard maximum likelihood drift parameter estimator in the homogeneous diffusion model is always strongly consistent., Stat. Probab. Lett. 86 24–29.
  • [25] Mishura, Y., Ralchenko, K., Seleznev, O. and Shevchenko, G. (2014). Asymptotic properties of drift parameter estimator based on discrete observations of stochastic differential equation driven by fractional Brownian motion. In, Modern Stochastics and Applications (V. Korolyuk, N. Limnios, Y. Mishura, L. Sakhno and G. Shevchenko, eds.). Springer Optimization and Its Applications 90 303–318.
  • [26] Nourdin, I. (2008). Asymptotic behavior of weighted quadratic and cubic variations of fractional Brownian motion., Ann. Probab. 36 2159–2175.
  • [27] Prakasa Rao, B. L. S. (1999)., Statistical Inference for Diffusion Type Processes. London: Arnold.
  • [28] Prakasa Rao, B. L. S. (2004). Sequential estimation for fractional Ornstein–Uhlenbeck type process., Sequential Anal. 23 33–44.
  • [29] Shimizu, Y. (2009). Notes on drift estimation for certain non-recurrent diffusion processes from sampled data., Stat. Probab. Lett. 79 2200–2207.
  • [30] Tanaka, K. (2013). Distributions of the maximum likelihood and minimum contrast estimators associated with the fractional Ornstein–Uhlenbeck process., Stat. Inference Stoch. Process. 16 173–192.
  • [31] Tudor, C. A. and Viens, F. G. (2007). Statistical aspects of the fractional stochastic calculus., Ann. Stat. 35 1183–1212.
  • [32] Xiao, W., Zhang, W. and Xu, W. (2011). Parameter estimation for fractional Ornstein–Uhlenbeck processes at discrete observation., Appl. Math. Modelling 35 4196–4207.
  • [33] Zhang, P., Xiao, W.-l., Zhang, X.-l. and Niu, P.-q. (2014). Parameter identification for fractional Ornstein–Uhlenbeck processes based on discrete observation., Economic Modelling 36 198–203.