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2-microlocal analysis of martingales and stochastic integrals

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  • Balança, Paul
  • Herbin, Erick

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.

Suggested Citation

  • 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.
  • Handle: RePEc:eee:spapps:v:122:y:2012:i:6:p:2346-2382
    DOI: 10.1016/j.spa.2012.03.011
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    References listed on IDEAS

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    1. 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 327-343.
    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. 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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    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.

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