Variability and the existence of rough integrals with irregular coefficients

TL;DR

This paper demonstrates how variability can be employed within rough path analysis to establish the existence of integrals with irregular coefficients, especially for certain Gaussian processes like fractional Brownian motion.

Contribution

It introduces a novel approach using variability to prove the existence of rough integrals with irregular coefficients in the context of fractional calculus.

Findings

Existence of rough integrals with irregular coefficients established.

Application to fractional Brownian motions with Hurst index between 1/3 and 1/2.

Method extends rough path analysis to broader classes of stochastic processes.

Abstract

Within the context of rough path analysis via fractional calculus, we show how variability can be used to prove the existence of integrals with respect to H\"older continuous multiplicative functionals in the case of Lipschitz coefficients with first order partial derivatives of bounded variation. We discuss applications to certain Gaussian processes, in particular, fractional Brownian motions with Hurst index $\frac13<H\leq \frac12$.