Tohoku Mathematical Journal
2020
March
SECOND SERIES VOL. 72, NO. 1

Tohoku Math. J.
72 (2020), 39-62

Title ROUGH INTEGRATION VIA FRACTIONAL CALCULUS

Author Yu Ito

(Received April 2, 2018, revised July 5, 2018)
Abstract. On the basis of fractional calculus, the author's previous study [9] introduced an approach to the integral of controlled paths against Hölder rough paths. The integral in [9] 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 [9] and show that, for a suitable class of Hölder rough paths including geometric Hölder rough paths, the integral in [9] is consistent with that obtained by the usual integration theory of rough path analysis, given by the limit of the compensated Riemann--Stieltjes sums.

Mathematics Subject Classification. Primary 26A42; Secondary 26A33, 60H05.
Key words and phrases. Stieltjes integral, fractional derivative, rough path.

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