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A fractional Brownian field indexed by L2 and a varying Hurst parameter

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  • Richard, Alexandre

Abstract

Using structures of Abstract Wiener Spaces, we define a fractional Brownian field indexed by a product space (0,1/2]×L2(T,m), (T,m) a separable measure space, where the first coordinate corresponds to the Hurst parameter of fractional Brownian motion. This field encompasses a large class of existing fractional Brownian processes, such as Lévy fractional Brownian motions and multiparameter fractional Brownian motions, and provides a setup for new ones. We prove that it has satisfactory incremental variance in both coordinates and derive certain continuity and Hölder regularity properties in relation with metric entropy. Also, a sharp estimate of the small ball probabilities is provided, generalizing a result on Lévy fractional Brownian motion. Then, we apply these general results to multiparameter and set-indexed processes, proving the existence of processes with prescribed local Hölder regularity on general indexing collections.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:4:p:1394-1425
    DOI: 10.1016/j.spa.2014.11.003
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    References listed on IDEAS

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    1. Nualart, David & Tindel, Samy, 1995. "Quasilinear stochastic elliptic equations with reflection," Stochastic Processes and their Applications, Elsevier, vol. 57(1), pages 73-82, May.
    2. Jolis, Maria & Viles, Noèlia, 2010. "Continuity in the Hurst parameter of the law of the symmetric integral with respect to the fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 120(9), pages 1651-1679, August.
    3. Herbin, Erick & Lévy-Véhel, Jacques, 2009. "Stochastic 2-microlocal analysis," Stochastic Processes and their Applications, Elsevier, vol. 119(7), pages 2277-2311, July.
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    Cited by:

    1. 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.
    2. Zuopeng Fu & Yizao Wang, 2020. "Stable Processes with Stationary Increments Parameterized by Metric Spaces," Journal of Theoretical Probability, Springer, vol. 33(3), pages 1737-1754, September.

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