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Regularity of multifractional moving average processes with random Hurst exponent

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  • Loboda, Dennis
  • Mies, Fabian
  • Steland, Ansgar

Abstract

A recently proposed alternative to multifractional Brownian motion (mBm) with random Hurst exponent is studied, which we refer to as Itô-mBm. It is shown that Itô-mBm is locally self-similar. In contrast to mBm, its pathwise regularity is almost unaffected by the roughness of the functional Hurst parameter. The pathwise properties are established via a new polynomial moment condition similar to the Kolmogorov–Centsov theorem, allowing for random local Hölder exponents. Our results are applicable to a broad class of moving average processes where pathwise regularity and long memory properties may be decoupled, e.g. to a multifractional generalization of the Matérn process.

Suggested Citation

  • Loboda, Dennis & Mies, Fabian & Steland, Ansgar, 2021. "Regularity of multifractional moving average processes with random Hurst exponent," Stochastic Processes and their Applications, Elsevier, vol. 140(C), pages 21-48.
    • Handle: RePEc:eee:spapps:v:140:y:2021:i:c:p:21-48
    DOI: 10.1016/j.spa.2021.05.008
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    References listed on IDEAS

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    1. Bardet, Jean-Marc & Surgailis, Donatas, 2013. "Nonparametric estimation of the local Hurst function of multifractional Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 123(3), pages 1004-1045.
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    3. Mark Podolskij & Mathias Vetter, 2010. "Understanding limit theorems for semimartingales: a short survey," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 64(s1), pages 329-351.
    4. Stoev, Stilian A. & Taqqu, Murad S., 2006. "How rich is the class of multifractional Brownian motions?," Stochastic Processes and their Applications, Elsevier, vol. 116(2), pages 200-221, February.
    5. Surgailis, Donatas, 2008. "Nonhomogeneous fractional integration and multifractional processes," Stochastic Processes and their Applications, Elsevier, vol. 118(2), pages 171-198, February.
    6. Lebovits, Joachim & Lévy Véhel, Jacques & Herbin, Erick, 2014. "Stochastic integration with respect to multifractional Brownian motion via tangent fractional Brownian motions," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 678-708.
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    Cited by:

    1. Ayache, Antoine & Bouly, Florent, 2022. "Moving average Multifractional Processes with Random Exponent: Lower bounds for local oscillations," Stochastic Processes and their Applications, Elsevier, vol. 146(C), pages 143-163.
    2. Angelini, Daniele & Bianchi, Sergio, 2023. "Nonlinear biases in the roughness of a Fractional Stochastic Regularity Model," Chaos, Solitons & Fractals, Elsevier, vol. 172(C).

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