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Continuous Gaussian Multifractional Processes with Random Pointwise Hölder Regularity

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  • Antoine Ayache

    (U.M.R. CNRS 8524)

Abstract

Let {X(t)} t∈ℝ be an arbitrary centered Gaussian process whose trajectories are, with probability 1, continuous nowhere differentiable functions. It follows from a classical result, derived from zero-one law, that, with probability 1, the trajectories of X have the same global Hölder regularity over any compact interval, i.e. the uniform Hölder exponent does not depend on the choice of a trajectory. A similar phenomenon occurs with their local Hölder regularity measured through the local Hölder exponent. Therefore, it seems natural to ask the following question: Does such a phenomenon also occur with their pointwise Hölder regularity measured through the pointwise Hölder exponent? In this article, using the framework of multifractional processes, we construct a family of counterexamples showing that the answer to this question is not always positive.

Suggested Citation

  • Antoine Ayache, 2013. "Continuous Gaussian Multifractional Processes with Random Pointwise Hölder Regularity," Journal of Theoretical Probability, Springer, vol. 26(1), pages 72-93, March.
    • Handle: RePEc:spr:jotpro:v:26:y:2013:i:1:d:10.1007_s10959-012-0418-3
    DOI: 10.1007/s10959-012-0418-3
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    References listed on IDEAS

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    1. Kenneth J. Falconer, 2002. "Tangent Fields and the Local Structure of Random Fields," Journal of Theoretical Probability, Springer, vol. 15(3), pages 731-750, July.
    2. Herbin, Erick & Lévy-Véhel, Jacques, 2009. "Stochastic 2-microlocal analysis," Stochastic Processes and their Applications, Elsevier, vol. 119(7), pages 2277-2311, July.
    3. Surgailis, Donatas, 2008. "Nonhomogeneous fractional integration and multifractional processes," Stochastic Processes and their Applications, Elsevier, vol. 118(2), pages 171-198, February.
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    Cited by:

    1. Daniele Angelini & Matthieu Garcin, 2024. "Market information of the fractional stochastic regularity model," Papers 2409.07159, arXiv.org.
    2. Angelini, Daniele & Bianchi, Sergio, 2023. "Nonlinear biases in the roughness of a Fractional Stochastic Regularity Model," Chaos, Solitons & Fractals, Elsevier, vol. 172(C).
    3. Loosveldt, L., 2023. "Multifractional Hermite processes: Definition and first properties," Stochastic Processes and their Applications, Elsevier, vol. 165(C), pages 465-500.

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