On the basis of fractional calculus, the author's previous study  introduced an approach to the integral of controlled paths against Hölder rough paths. The integral in  is defined by the Lebesgue integrals for fractional derivatives without using any arguments based on discrete approximation. In this paper, we revisit the approach of  and show that, for a suitable class of Hölder rough paths including geometric Hölder rough paths, the integral in  is consistent with that obtained by the usual integration theory of rough path analysis, given by the limit of the compensated Riemann–Stieltjes sums.
"Rough integration via fractional calculus." Tohoku Math. J. (2) 72 (1) 39 - 62, 2020. https://doi.org/10.2748/tmj/1585101620