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Linear Multifractional Stable Motion: Wavelet Estimation of H(·) and α Parameters*

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Abstract

Stoev and Taqqu introduced a linear multifractional stable motion (LMSM), an extension of a linear fractional stable motion (LFSM) such that the Hurst parameter H becomes a function H(t). The stability parameter α determines tail heaviness of marginal distributions of LMSM. Under some conditions, Stoev and Taqqu showed that H(t 0) is its self-similarity exponent at a t 0 ≠ 0; also, recently, Ayache and Hamonier established that H(t 0)−1/α and min t∈I H(t)−1/α are its local Hölder exponent at t 0 and uniform Hölder exponent on a compact interval I.

We construct, strongly consistent wavelet estimators of min t∈I H(t), H(t 0), and α when α ∈ (1, 2) and H(·) is smooth with values in \( \left[\underset{\bar{\mkern6mu}}{H},\overline{H}\right]\subset \left(1/\alpha, 1\right) \).

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Authors and Affiliations

  1. UMR CNRS 8524, Laboratoire Paul Painlevé, Bât. M2, Université Lille 1, 59655, Villeneuve d’Ascq Cedex, France

    Antoine Ayache

  2. EA 2694, Laboratoire de Biomathématiques, Université Lille 2, 3, Rue du Professeur Laguesse, BP 83, 59006, Lille Cedex, France

    Julien Hamonier

Authors
  1. Antoine Ayache
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  2. Julien Hamonier
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Correspondence to Antoine Ayache.

Additional information

*This work has been partially supported by ANR-11-BS01-0011 (AMATIS), GDR 3475 (Analyse Multifractale), and ANR-11-LABX-0007-01 (CEMPI).

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Ayache, A., Hamonier, J. Linear Multifractional Stable Motion: Wavelet Estimation of H(·) and α Parameters* . Lith Math J 55, 159–192 (2015). https://doi.org/10.1007/s10986-015-9272-1

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  • DOI: https://doi.org/10.1007/s10986-015-9272-1

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