2-microlocal analysis of martingales and stochastic integrals

TL;DR

This paper develops a 2-microlocal analysis framework to understand the local regularity of martingales and stochastic integrals, linking their path regularity to their quadratic variation and coefficients.

Contribution

It introduces a novel 2-microlocal approach to analyze the fine regularity of martingales and stochastic integrals, connecting path regularity to quadratic variation and coefficients.

Findings

The 2-microlocal frontier of a martingale is determined by its quadratic variation.

Regularity of stochastic integrals is linked to the regularity of integrand and integrator.

Methodology enables prediction of diffusion regularity from coefficients.

Abstract

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.