On the basis of fractional calculus, we introduce an integral of controlled paths with respect to Hölder rough paths of order \(\beta \in (1/4,1/3]\). 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 \(\beta\)-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 \(\beta\)-Hölder rough paths.

在分数阶微积分的基础上，相对于阶阶\（\ beta \ in（1 / 4,1 / 3] \）的Hölder粗糙路径，我们引入了受控路径的积分，我们对积分的定义明确表示为分数阶导数的Lebesgue积分，而没有使用离散逼近的任何参数。我们证明，对于\（\ beta \）- Hölder粗糙路径和受控路径的合适类别，我们对积分的定义与由本文的结果还提供了一种针对几何\（\ beta \）- Hölder粗糙路径的1形式积分的方法。