2-microlocal analysis of martingales and stochastic integrals

Details 2-microlocal analysis of martingales and stochastic integrals 2-microlocal analysis of martingales and stochastic integrals

2011-07-29

arxiv.org/abs/1107.6016v2

Mathematics

Probability

40 pages, 3 figures

Scientific paper

Recently, a new approach in the fine analysis of stochastic processes sample paths 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 H\"older 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 behavior and square of Bessel processes.

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