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Multifractional Hermite processes: Definition and first properties

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  • Loosveldt, L.

Abstract

We define multifractional Hermite processes which generalize and extend both multifractional Brownian motion and Hermite processes. It is done by substituting the Hurst parameter in the definition of Hermite processes as a multiple Wiener–Itô integral by a Hurst function. Then, we study the pointwise regularity of these processes, their local asymptotic self-similarity and some fractal dimensions of their graph. Our results show that the fundamental properties of multifractional Hermite processes are, as desired, governed by the Hurst function. Complements are given in the second order Wiener chaos, using facts from Malliavin calculus.

Suggested Citation

  • Loosveldt, L., 2023. "Multifractional Hermite processes: Definition and first properties," Stochastic Processes and their Applications, Elsevier, vol. 165(C), pages 465-500.
    • Handle: RePEc:eee:spapps:v:165:y:2023:i:c:p:465-500
    DOI: 10.1016/j.spa.2023.09.003
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    References listed on IDEAS

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    1. Laurent Loosveldt & Samuel Nicolay, 2019. "Some equivalent definitions of Besov spaces of generalized smoothness," Mathematische Nachrichten, Wiley Blackwell, vol. 292(10), pages 2262-2282, October.
    2. Ayache, Antoine, 2020. "Lower bound for local oscillations of Hermite processes," Stochastic Processes and their Applications, Elsevier, vol. 130(8), pages 4593-4607.
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    4. Nourdin, Ivan & Poly, Guillaume, 2013. "Convergence in total variation on Wiener chaos," Stochastic Processes and their Applications, Elsevier, vol. 123(2), pages 651-674.
    5. Ayache, Antoine & Esser, Céline & Kleyntssens, Thomas, 2019. "Different possible behaviors of wavelet leaders of the Brownian motion," Statistics & Probability Letters, Elsevier, vol. 150(C), pages 54-60.
    6. Wu, Wei Biao, 2006. "Unit Root Testing For Functionals Of Linear Processes," Econometric Theory, Cambridge University Press, vol. 22(1), pages 1-14, February.
    7. Antoine Ayache, 2013. "Continuous Gaussian Multifractional Processes with Random Pointwise Hölder Regularity," Journal of Theoretical Probability, Springer, vol. 26(1), pages 72-93, March.
    8. Surgailis, Donatas, 2008. "Nonhomogeneous fractional integration and multifractional processes," Stochastic Processes and their Applications, Elsevier, vol. 118(2), pages 171-198, February.
    9. 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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