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Identification of filtered white noises

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  • Benassi, Albert
  • Cohen, Serge
  • Istas, Jacques
  • Jaffard, Stéphane

Abstract

In this paper, a class of Gaussian processes, having locally the same fractal properties as fractional Brownian motion, is studied. Our aim is to give estimators of the relevant parameters of these processes from one sample path. A time dependency of the integrand of the classical Wiener integral, associated with the fractional Brownian motion, is introduced. We show how to identify the asymptotic expansion for high frequencies of these integrands on one sample path. Then, the identification of the first terms of this expansion is used to solve some filtering problems. Furthermore, rates of convergence of the estimators are then given.

Suggested Citation

  • Benassi, Albert & Cohen, Serge & Istas, Jacques & Jaffard, Stéphane, 1998. "Identification of filtered white noises," Stochastic Processes and their Applications, Elsevier, vol. 75(1), pages 31-49, June.
    • Handle: RePEc:eee:spapps:v:75:y:1998:i:1:p:31-49
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    References listed on IDEAS

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    1. Adler, Robert J. & Pyke, Ron, 1993. "Uniform quadratic variation for Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 48(2), pages 191-209, November.
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    Cited by:

    1. Kęstutis Kubilius & Dmitrij Melichov, 2016. "Exact Confidence Intervals of the Extended Orey Index for Gaussian Processes," Methodology and Computing in Applied Probability, Springer, vol. 18(3), pages 785-804, September.
    2. Bégyn, Arnaud, 2007. "Functional limit theorems for generalized quadratic variations of Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 117(12), pages 1848-1869, December.
    3. Ayache, Antoine & Lévy Véhel, Jacques, 2004. "On the identification of the pointwise Hölder exponent of the generalized multifractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 111(1), pages 119-156, May.
    4. Vu, Huong T.L. & Richard, Frédéric J.P., 2020. "Statistical tests of heterogeneity for anisotropic multifractional Brownian fields," Stochastic Processes and their Applications, Elsevier, vol. 130(8), pages 4667-4692.
    5. Biermé, Hermine & Lacaux, Céline & Scheffler, Hans-Peter, 2011. "Multi-operator scaling random fields," Stochastic Processes and their Applications, Elsevier, vol. 121(11), pages 2642-2677, November.
    6. Kubilius, K. & Mishura, Y., 2012. "The rate of convergence of Hurst index estimate for the stochastic differential equation," Stochastic Processes and their Applications, Elsevier, vol. 122(11), pages 3718-3739.
    7. Kubilius, K., 2020. "CLT for quadratic variation of Gaussian processes and its application to the estimation of the Orey index," Statistics & Probability Letters, Elsevier, vol. 165(C).
    8. Andreas Neuenkirch & Ivan Nourdin, 2007. "Exact Rate of Convergence of Some Approximation Schemes Associated to SDEs Driven by a Fractional Brownian Motion," Journal of Theoretical Probability, Springer, vol. 20(4), pages 871-899, December.
    9. Peng, Qidi, 2011. "Uniform Hölder exponent of a stationary increments Gaussian process: Estimation starting from average values," Statistics & Probability Letters, Elsevier, vol. 81(8), pages 1326-1335, August.
    10. Frezza, Massimiliano, 2012. "Modeling the time-changing dependence in stock markets," Chaos, Solitons & Fractals, Elsevier, vol. 45(12), pages 1510-1520.
    11. Kubilius, K. & Skorniakov, V., 2016. "On some estimators of the Hurst index of the solution of SDE driven by a fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 109(C), pages 159-167.

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    Keywords

    Gaussian processes Identification;

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