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Parameter estimation for the discretely observed fractional Ornstein–Uhlenbeck process and the Yuima R package

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  • Alexandre Brouste
  • Stefano Iacus

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. Copyright Springer-Verlag 2013

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  • Alexandre Brouste & Stefano Iacus, 2013. "Parameter estimation for the discretely observed fractional Ornstein–Uhlenbeck process and the Yuima R package," Computational Statistics, Springer, vol. 28(4), pages 1529-1547, August.
    • Handle: RePEc:spr:compst:v:28:y:2013:i:4:p:1529-1547
    DOI: 10.1007/s00180-012-0365-6
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    References listed on IDEAS

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    1. Bertin, Karine & Torres, Soledad & Tudor, Ciprian A., 2011. "Drift parameter estimation in fractional diffusions driven by perturbed random walks," Statistics & Probability Letters, Elsevier, vol. 81(2), pages 243-249, February.
    2. Alexandre Brouste & Marina Kleptsyna, 2010. "Asymptotic properties of MLE for partially observed fractional diffusion system," Statistical Inference for Stochastic Processes, Springer, vol. 13(1), pages 1-13, April.
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    Cited by:

    1. Liu, Yanghui & Nualart, Eulalia & Tindel, Samy, 2019. "LAN property for stochastic differential equations with additive fractional noise and continuous time observation," Stochastic Processes and their Applications, Elsevier, vol. 129(8), pages 2880-2902.
    2. El Mehdi Haress & Yaozhong Hu, 2021. "Estimation of all parameters in the fractional Ornstein–Uhlenbeck model under discrete observations," Statistical Inference for Stochastic Processes, Springer, vol. 24(2), pages 327-351, July.
    3. Pavel Kříž & Leszek Szała, 2020. "The Combined Estimator for Stochastic Equations on Graphs with Fractional Noise," Mathematics, MDPI, vol. 8(10), pages 1-21, October.
    4. Guangjun Shen & Qian Yu, 2019. "Least squares estimator for Ornstein–Uhlenbeck processes driven by fractional Lévy processes from discrete observations," Statistical Papers, Springer, vol. 60(6), pages 2253-2271, December.
    5. Stefano Iacus & Lorenzo Mercuri, 2015. "Implementation of Lévy CARMA model in Yuima package," Computational Statistics, Springer, vol. 30(4), pages 1111-1141, December.
    6. Brouste, Alexandre & Fukasawa, Masaaki & Hino, Hideitsu & Iacus, Stefano & Kamatani, Kengo & Koike, Yuta & Masuda, Hiroki & Nomura, Ryosuke & Ogihara, Teppei & Shimuzu, Yasutaka & Uchida, Masayuki & Y, 2014. "The YUIMA Project: A Computational Framework for Simulation and Inference of Stochastic Differential Equations," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 57(i04).
    7. Qian Yu, 2021. "Least squares estimator of fractional Ornstein–Uhlenbeck processes with periodic mean for general Hurst parameter," Statistical Papers, Springer, vol. 62(2), pages 795-815, April.
    8. Marko Voutilainen & Lauri Viitasaari & Pauliina Ilmonen & Soledad Torres & Ciprian Tudor, 2022. "Vector‐valued generalized Ornstein–Uhlenbeck processes: Properties and parameter estimation," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(3), pages 992-1022, September.
    9. Pavel Kříž & Leszek Szała, 2020. "Least-Squares Estimators of Drift Parameter for Discretely Observed Fractional Ornstein–Uhlenbeck Processes," Mathematics, MDPI, vol. 8(5), pages 1-20, May.

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