2-microlocal analysis of martingales and stochastic integrals

Paul Balança and Erick Herbin

Stochastic Processes and their Applications, 2012, vol. 122, issue 6, 2346-2382

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.

Keywords: 2-microlocal analysis; Bessel processes; Hölder regularity; Multifractional Brownian motion; Stochastic differential equations; Stochastic integral (search for similar items in EconPapers)
Date: 2012
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Citations: View citations in EconPapers (2)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:122:y:2012:i:6:p:2346-2382

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DOI: 10.1016/j.spa.2012.03.011

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