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A least squares estimator for discretely observed Ornstein–Uhlenbeck processes driven by symmetric α-stable motions

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  • Shibin Zhang
  • Xinsheng Zhang

Abstract

We study the problem of parameter estimation for Ornstein–Uhlenbeck processes driven by symmetric α-stable motions, based on discrete observations. A least squares estimator is obtained by minimizing a contrast function based on the integral form of the process. Let h be the length of time interval between two consecutive observations. For both the case of fixed h and that of h → 0, consistencies and asymptotic distributions of the estimator are derived. Moreover, for both of the cases of h, the estimator has a higher order of convergence for the Ornstein–Uhlenbeck process driven by non-Gaussian α-stable motions (0 > α > 2) than for the process driven by the classical Gaussian case (α = 2). Copyright The Institute of Statistical Mathematics, Tokyo 2013

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  • Shibin Zhang & Xinsheng Zhang, 2013. "A least squares estimator for discretely observed Ornstein–Uhlenbeck processes driven by symmetric α-stable motions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 65(1), pages 89-103, February.
    • Handle: RePEc:spr:aistmt:v:65:y:2013:i:1:p:89-103
    DOI: 10.1007/s10463-012-0362-0
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    1. Fred Espen Benth & Martin Groth & Rodwell Kufakunesu, 2007. "Valuing Volatility and Variance Swaps for a Non-Gaussian Ornstein-Uhlenbeck Stochastic Volatility Model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 14(4), pages 347-363.
    2. Geurt Jongbloed & Frank H. Van Der Meulen, 2006. "Parametric Estimation for Subordinators and Induced OU Processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 33(4), pages 825-847, December.
    3. Emanuele Taufer, 2008. "Characteristic function estimation of non-Gaussian Ornstein-Uhlenbeck processes," DISA Working Papers 0805, Department of Computer and Management Sciences, University of Trento, Italy, revised 07 Jul 2008.
    4. Ole E. Barndorff‐Nielsen & Neil Shephard, 2001. "Non‐Gaussian Ornstein–Uhlenbeck‐based models and some of their uses in financial economics," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(2), pages 167-241.
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    Cited by:

    1. Yurong Pan & Litan Yan, 2019. "The Least Squares Estimation for the α-Stable Ornstein-Uhlenbeck Process with Constant Drift," Methodology and Computing in Applied Probability, Springer, vol. 21(4), pages 1165-1182, December.
    2. 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.
    3. Mathias Mørck Ljungdahl & Mark Podolskij, 2022. "Multidimensional parameter estimation of heavy‐tailed moving averages," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(2), pages 593-624, June.
    4. Yanfeng Wu & Jianqiang Hu & Xiangyu Yang, 2022. "Moment estimators for parameters of Lévy‐driven Ornstein–Uhlenbeck processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 43(4), pages 610-639, July.
    5. Yiying Cheng & Yaozhong Hu & Hongwei Long, 2020. "Generalized moment estimators for $$\alpha $$α-stable Ornstein–Uhlenbeck motions from discrete observations," Statistical Inference for Stochastic Processes, Springer, vol. 23(1), pages 53-81, April.
    6. Xuekang Zhang & Huisheng Shu & Haoran Yi, 2023. "Parameter Estimation for Ornstein–Uhlenbeck Driven by Ornstein–Uhlenbeck Processes with Small Lévy Noises," Journal of Theoretical Probability, Springer, vol. 36(1), pages 78-98, March.
    7. Szarek, Dawid & Bielak, Łukasz & Wyłomańska, Agnieszka, 2020. "Long-term prediction of the metals’ prices using non-Gaussian time-inhomogeneous stochastic process," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 555(C).
    8. Alexander Gushchin & Ilya Pavlyukevich & Marian Ritsch, 2020. "Drift estimation for a Lévy-driven Ornstein–Uhlenbeck process with heavy tails," Statistical Inference for Stochastic Processes, Springer, vol. 23(3), pages 553-570, October.
    9. Kevin W. Lu, 2022. "Calibration for multivariate Lévy-driven Ornstein-Uhlenbeck processes with applications to weak subordination," Statistical Inference for Stochastic Processes, Springer, vol. 25(2), pages 365-396, July.

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