A fractional calculus approach to rough integration

Abstract

On the basis of fractional calculus, we introduce an integral of controlled paths against $\beta$-Hölder rough paths with $\beta \in (1/3,1/2]$. The integral is defined by the Lebesgue integrals for fractional derivative operators, without using any argument based on discrete approximation. We show in this article that the integral is consistent with that obtained by the usual integration in rough path analysis, given by the limit of the compensated Riemann–Stieltjes sums.