Integration with respect to Hölder rough paths of order greater than 1/4: an approach via fractional calculus

On the basis of fractional calculus, we introduce an integral of controlled paths with respect to Hölder rough paths of order β∈(1/4,1/3]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \in (1/4,1/3]$$\end{document}. Our definition of the integral is given explicitly in terms of Lebesgue integrals for fractional derivatives, without using any arguments from discrete approximation. We demonstrate that for suitable classes of β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta$$\end{document}-Hölder rough paths and controlled paths, our definition of the integral is consistent with the usual definition given by the limit of the compensated Riemann–Stieltjes sum. The results of this paper also provide an approach to the integral of 1-forms against geometric β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta$$\end{document}-Hölder rough paths.