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Stochastic 2-microlocal analysis

Author

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  • Herbin, Erick
  • Lévy-Véhel, Jacques

Abstract

A lot is known about the Hölder regularity of stochastic processes, in particular in the case of Gaussian processes. Recently, a finer analysis of the local regularity of functions, termed 2-microlocal analysis, has been introduced in a deterministic frame: through the computation of the so-called 2-microlocal frontier, it allows us in particular to predict the evolution of regularity under the action of (pseudo-)differential operators. In this work, we develop a 2-microlocal analysis for the study of certain stochastic processes. We show that moments of the increments allow us, under fairly general conditions, to obtain almost sure lower bounds for the 2-microlocal frontier. In the case of Gaussian processes, more precise results may be obtained: the incremental covariance yields the almost sure value of the 2-microlocal frontier. As an application, we obtain new and refined regularity properties of fractional Brownian motion, multifractional Brownian motion, stochastic generalized Weierstrass functions, Wiener and stable integrals.

Suggested Citation

  • Herbin, Erick & Lévy-Véhel, Jacques, 2009. "Stochastic 2-microlocal analysis," Stochastic Processes and their Applications, Elsevier, vol. 119(7), pages 2277-2311, July.
    • Handle: RePEc:eee:spapps:v:119:y:2009:i:7:p:2277-2311
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    Citations

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    Cited by:

    1. Neuman, Eyal, 2014. "The multifractal nature of Volterra–Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 124(9), pages 3121-3145.
    2. Antoine Ayache, 2013. "Continuous Gaussian Multifractional Processes with Random Pointwise Hölder Regularity," Journal of Theoretical Probability, Springer, vol. 26(1), pages 72-93, March.
    3. Biermé, Hermine & Lacaux, Céline & Scheffler, Hans-Peter, 2011. "Multi-operator scaling random fields," Stochastic Processes and their Applications, Elsevier, vol. 121(11), pages 2642-2677, November.
    4. Alexandre Richard, 2017. "Some Singular Sample Path Properties of a Multiparameter Fractional Brownian Motion," Journal of Theoretical Probability, Springer, vol. 30(4), pages 1285-1309, December.
    5. Richard, Alexandre, 2015. "A fractional Brownian field indexed by L2 and a varying Hurst parameter," Stochastic Processes and their Applications, Elsevier, vol. 125(4), pages 1394-1425.
    6. Balança, Paul, 2015. "Some sample path properties of multifractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 125(10), pages 3823-3850.
    7. Hannebicque, Brice & Herbin, Érick, 2022. "Regularity of an abstract Wiener integral," Stochastic Processes and their Applications, Elsevier, vol. 154(C), pages 154-196.
    8. Peng, Qidi & Zhao, Ran, 2018. "A general class of multifractional processes and stock price informativeness," Chaos, Solitons & Fractals, Elsevier, vol. 115(C), pages 248-267.
    9. Balança, Paul & Herbin, Erick, 2012. "2-microlocal analysis of martingales and stochastic integrals," Stochastic Processes and their Applications, Elsevier, vol. 122(6), pages 2346-2382.

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