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Multifractional, Multistable, and Other Processes with Prescribed Local Form

Author

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  • K. J. Falconer

    (University of St Andrews)

  • J. Lévy Véhel

    (INRIA Rocquencourt)

Abstract

We present a general method for constructing stochastic processes with prescribed local form, encompassing examples such as variable amplitude multifractional Brownian and multifractional α-stable processes. We apply the method to Poisson sums to construct multistable processes, that is, processes that are locally α(t)-stable but where the stability index α(t) varies with t. In particular we construct multifractional multistable processes, where both the local self-similarity and stability indices vary.

Suggested Citation

  • K. J. Falconer & J. Lévy Véhel, 2009. "Multifractional, Multistable, and Other Processes with Prescribed Local Form," Journal of Theoretical Probability, Springer, vol. 22(2), pages 375-401, June.
    • Handle: RePEc:spr:jotpro:v:22:y:2009:i:2:d:10.1007_s10959-008-0147-9
    DOI: 10.1007/s10959-008-0147-9
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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. Antoine Ayache & Jacques Vehel, 2000. "The Generalized Multifractional Brownian Motion," Statistical Inference for Stochastic Processes, Springer, vol. 3(1), pages 7-18, January.
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    Cited by:

    1. Ronan Le Guével & Jacques Lévy Véhel & Lining Liu, 2015. "On Two Multistable Extensions of Stable Lévy Motion and Their Semi-martingale Representations," Journal of Theoretical Probability, Springer, vol. 28(3), pages 1125-1144, September.
    2. Olivier Le Courtois, 2018. "Some Further Results on the Tempered Multistable Approach," Post-Print hal-02312142, HAL.
    3. K. J. Falconer & J. Lévy Véhel, 2020. "Self-Stabilizing Processes Based on Random Signs," Journal of Theoretical Probability, Springer, vol. 33(1), pages 134-152, March.
    4. R. Guével, 2019. "The Hausdorff dimension of the range of the Lévy multistable processes," Journal of Theoretical Probability, Springer, vol. 32(2), pages 765-780, June.

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