Kyoto Journal of Mathematics

A fractional calculus approach to rough integration

Yu Ito

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On the basis of fractional calculus, we introduce an integral of controlled paths against β-Hölder rough paths with β(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.

Article information

Kyoto J. Math., Volume 59, Number 3 (2019), 553-573.

Received: 18 December 2016
Revised: 31 March 2017
Accepted: 6 April 2017
First available in Project Euclid: 17 May 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 26A42: Integrals of Riemann, Stieltjes and Lebesgue type [See also 28-XX]
Secondary: 26A33: Fractional derivatives and integrals

Stieltjes integral fractional derivative rough path


Ito, Yu. A fractional calculus approach to rough integration. Kyoto J. Math. 59 (2019), no. 3, 553--573. doi:10.1215/21562261-2019-0017.

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