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2-microlocal analysis of martingales and stochastic integrals

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  • Balança, Paul
  • Herbin, Erick
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    Abstract

    Recently, a new approach in the fine analysis of sample paths of stochastic processes has been developed to predict the evolution of the local regularity under (pseudo-)differential operators. In this paper, we study the sample paths of continuous martingales and stochastic integrals. We proved that the almost sure 2-microlocal frontier of a martingale can be obtained through the local regularity of its quadratic variation. It allows to link the Hölder regularity of a stochastic integral to the regularity of the integrand and integrator processes. These results provide a methodology to predict the local regularity of diffusions from the fine analysis of its coefficients. We illustrate our work with examples of martingales with unusual complex regularity behaviour and square of Bessel processes.

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    Bibliographic Info

    Article provided by Elsevier in its journal Stochastic Processes and their Applications.

    Volume (Year): 122 (2012)
    Issue (Month): 6 ()
    Pages: 2346-2382

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    Handle: RePEc:eee:spapps:v:122:y:2012:i:6:p:2346-2382

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    Keywords: 2-microlocal analysis; Bessel processes; Hölder regularity; Multifractional Brownian motion; Stochastic differential equations; Stochastic integral;

    References

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    1. S. Bianchi & A. Pianese, 2008. "Multifractional Properties Of Stock Indices Decomposed By Filtering Their Pointwise Hölder Regularity," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 11(06), pages 567-595.
    2. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-43.
    3. Herbin, Erick & Lévy-Véhel, Jacques, 2009. "Stochastic 2-microlocal analysis," Stochastic Processes and their Applications, Elsevier, vol. 119(7), pages 2277-2311, July.
    4. Stoev, Stilian A. & Taqqu, Murad S., 2006. "How rich is the class of multifractional Brownian motions?," Stochastic Processes and their Applications, Elsevier, vol. 116(2), pages 200-221, February.
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