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

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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).

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References

  • Barndorff-Nielsen O.E., Shephard N. (2001) Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics. Journal of the Royal Statistical Society B-Statistical Methodology 63: 167–241

    Article  MathSciNet  MATH  Google Scholar 

  • Benth F.E., Gorth M., Kufakunesu R. (2007) Valuing volatility and variance swaps for a non-Gaussian Ornstein–Uhlenbeck stochastic volatility model. Applied Mathematical Finance 14: 347–363

    Article  MathSciNet  MATH  Google Scholar 

  • Brockwell P.J., Davis R.A., Yang Y. (2007) Estimation for nonnegative Lévy-driven Ornstein–Uhlenbeck processes. Journal of Applied Probability 44: 977–989

    Article  MathSciNet  MATH  Google Scholar 

  • Cariboni C., Schoutens W. (2009) Jumps in intensity models: Investigating the performance of Ornstein–Uhlenbeck processes in credit risk modeling. Metrika 69: 173–198

    Article  MathSciNet  Google Scholar 

  • Chambers J.M., Mallows C.L., Stuck B.W. (1976) A method for simulating stable random variables. Journal of the American Statistical Association 71: 340–344

    Article  MathSciNet  MATH  Google Scholar 

  • Davis R., Resnick S. (1986) Limit theory for the sample covariance and correlation functions of moving averages. Annals of Statistics 14: 533–558

    Article  MathSciNet  MATH  Google Scholar 

  • Durrett R. (2004) Probability: Theory and examples (3rd ed). Belmont, Thomson

    Google Scholar 

  • Feller W. (1946) A limit theorem for random variables with infinite moments. American Journal of Mathematics 68: 257–262

    Article  MathSciNet  MATH  Google Scholar 

  • Hall P. (1981) A comedy of errors: The canonical form for a stable characteristic function. Bulletin of the London Mathematical Society 13: 23–27

    Article  MathSciNet  MATH  Google Scholar 

  • Hu Y., Long H. (2009) Least squares estimator for Ornstein–Uhlenbeck processes driven by α-stable motions. Stochastic Processes and Their Applications 119: 2465–2480

    Article  MathSciNet  MATH  Google Scholar 

  • Jongbloed G., VanDer Meulen F.H. (2006) Parametric estimation for subordinators and induced OU processes, Scandinavian Journal of Statistics 33: 825–847

    MathSciNet  MATH  Google Scholar 

  • Jongbloed G., VanDer Meulen F.H., VanDer Vaart A.W. (2005) Non-parametric inference for Lévy driven Ornstein-Uhlenbeck processes. Bernoulli 11: 759–791

    Article  MathSciNet  MATH  Google Scholar 

  • Lévy P. (1934) Sur les intégrales dont les éléments sont des variables aléatoires indépendantes. Annali della Scuola Normale Superiore di Pisa 3: 337–366

    Google Scholar 

  • Lin Z., Bai Z. (2010) Probability inequalities. Science Press, Beijing

    MATH  Google Scholar 

  • Masuda H. (2004) On multidimensional Ornstein–Uhlenbeck processes driven by a general Lévy process. Bernoulli 10: 97–120

    Article  MathSciNet  MATH  Google Scholar 

  • Mikosch T., Gadrich T., Klüppelberg C., Adler R. (1995) Parameter estimation for ARMA models with infinite variance innovations. Annals of Statistics 23: 305–326

    Article  MathSciNet  MATH  Google Scholar 

  • Nolan J.P. (2010) Stable distributions—models for heavy tailed data. Birkhäuser, Boston

    Google Scholar 

  • Rosinski J., Woyczynski W.A. (1986) On Itô stochastic integration with respect to p-stable motion: Inner clock, integrability of sample paths, double and multiple integrals. Annals of Probability 14: 271–286

    Article  MathSciNet  MATH  Google Scholar 

  • Sato K. (1999) Lévy processes and infinitely divisible distributions. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Sato K., Yamazato M. (1984) Operator-self-decomposable distributions as limit distributions of processes of Ornstein–Uhlenbeck type. Stochastic Processes and Their Applications 17: 73–100

    Article  MathSciNet  MATH  Google Scholar 

  • Sun S., Zhang X. (2009) Empirical likelihood estimation of discretely sampled processes of OU type. Science China Mathematics 52: 908–931

    Article  MATH  Google Scholar 

  • Taufer E., Leonenko N. (2009) Characteristic function estimation of non-Gaussian Ornstein–Uhlenbeck processes. Journal of Statistical Planning and Inference 139: 3050–3063

    Article  MathSciNet  MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Mathematics, Shanghai Maritime University, 1550 Haigang Avenue In New Harbor City, Shanghai, 201306, China

    Shibin Zhang

  2. Department of Statistics, Fudan University, Shanghai, 200433, China

    Xinsheng Zhang

Authors
  1. Shibin Zhang
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  2. Xinsheng Zhang
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Correspondence to Shibin Zhang.

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Zhang, S., Zhang, X. A least squares estimator for discretely observed Ornstein–Uhlenbeck processes driven by symmetric α-stable motions. Ann Inst Stat Math 65, 89–103 (2013). https://doi.org/10.1007/s10463-012-0362-0

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  • DOI: https://doi.org/10.1007/s10463-012-0362-0

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