Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter November 11, 2016

Integration of controlled rough paths via fractional calculus

Yu Ito EMAIL logo
From the journal Forum Mathematicum


We develop a fractional calculus approach to rough path analysis, introduced by Y. Hu and D. Nualart [6], and show that our integration can be generalized so that it is consistent with the rough path integration introduced by M. Gubinelli [5].

MSC 2010: 26A42; 26A33; 60H05

Communicated by Jan Bruinier

Funding statement: This work was partially supported by JSPS Research Fellowships for Young Scientists.

A Appendix

In this appendix, we prove some statements used in Sections 2 and 3. Note that almost all statements follow from slight modifications of the proofs in the preceding studies [6, 8, 9, 14].

First, Proposition A.2 shows that 𝒟t-γRZ is β-Hölder continuous on the interval [0,t]. The proposition is proved similarly to [6, Lemma 6.3]. The proof makes use of the following lemma, which is a slight reformulation of [6, Lemmas 6.1 and 6.2].

Lemma A.1 ([8, Lemma 3.1]).

For 0<δ<ε1, there exists a positive constant C, depending only on δ and ε, such that

(A.1)yδ-xδCxδ-ε(y-x)εfor 0<x<y.

For δ,ε>0 with 0<ε-δ<1, there exists a positive constant C, depending only on δ and ε, such that

(A.2)01uε(u-δ-1-(u+z)-δ-1)𝑑uCzε-δfor 0z<.

Proposition A.2.

Let (Z,Z)QXβ(Rn), α(0,β), and b(0,T]. Then, Db-αRZ is β-Hölder continuous on the interval [0,b].


Take the real numbers s and t such that 0s<tb and estimate 𝒟b-αRZ(t)-𝒟b-αRZ(s). First, with regard to the first term of 𝒟b-αRZ(t)-𝒟b-αRZ(s), we estimate the following:


Then, by using (A.1), we have


and, from the equality Rs,bZ=Rs,tZ+Rt,bZ+δZs,tδXt,b, we have


Next, with regard to the second term of 𝒟b-αRZ(t)-𝒟b-αRZ(s), we estimate the following:






By using the change of variables u=(v-t)/(b-t) and (A.2) with z=(t-s)/(b-t), we have


and, from the equality Rs,vZ=Rs,tZ+Rt,vZ+δZs,tδXt,v for v[t,b], we have


Therefore, by combining these estimates, we obtain the statement of the proposition. ∎

Corollary A.3.

Let (X,X)Ωβ(Rd), α(0,β), and b(0,T]. Then, Db-αX is β-Hölder continuous on the interval [0,b].


We set (Z,Z)𝒬Xβ(dd) as Zt:=𝕏0,t and Zt(ξ):=(Xt-X0)ξ for t[0,T], where ξd. Then, the identity RZ=𝕏 holds from (2.1). Therefore, from Proposition A.2, we obtain the statement. ∎

We next prove the following proposition used in the proof of Theorem 2.3.

Proposition A.4.

In the setting of Definition 2.1, for each (s,t) with s<t,


where the limits are taken over all finite partitions P={t0,t1,,tN} of the interval [s,t] such that s=t0<t1<<tN=t and |P|=max0iN-1|ti+1-ti|.


We prove only that (A.5) holds, since (A.3) and (A.6) follow from [14, Theorem 4.1.1], and (A.4) follows from [9, Proposition 2.4]. The proof of (A.5) here is based on the proofs of [14, Theorem 4.1.1] and [9, Proposition 2.4]. For u(s,t], we set




for (v,u). It then suffices to show that


holds for k=1,2. First, by straightforward computation, we have


where Y denotes the supremum norm of Y on [s,t]. Thus, (A.7) holds for k=1. Next, by using the equalities


for u(ti,ti+1], we decompose the L1-norm of S𝒫2 on [s,t] as follows:


Then, from the equality Ψv,u𝒫=-YvδXv,uδZv,u for (v,u) with ti<vuti+1, we have


where we use that 2β+γ>1 holds from (1-β)/2<γ<β. Also, for (v,u) with tj<vtj+1ti<uti+1, we have


and so


Therefore, we have


where C is a positive constant that does not depend on 𝒫. The last inequality can be proved by straightforward computation, as in the proofs of [14, Theorem 4.1.1] and [9, Proposition 2.4]. Hence, it follows from the estimates of A1 and A2, and 2β+γ>1 that (A.7) holds for k=2. Thus, we obtain the statement of the proposition. ∎


[1] M. Besalú, D. Márquez-Carreras and C. Rovira, Delay equations with non-negativity constraints driven by a Hölder continuous function of order β(13,12), Potential Anal. 41 (2014), no. 1, 117–141. 10.1007/s11118-013-9365-6Search in Google Scholar

[2] M. Besalú and D. Nualart, Estimates for the solution to stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H(13,12), Stoch. Dyn. 11 (2011), no. 2–3, 243–263. 10.1142/S0219493711003267Search in Google Scholar

[3] P. K. Friz and M. Hairer, A Course on Rough Paths. With an Introduction to Regularity Structures, Universitext, Springer, Cham, 2014. 10.1007/978-3-319-08332-2Search in Google Scholar

[4] P. K. Friz and N. B. Victoir, Multidimensional Stochastic Processes as Rough Paths. Theory and Applications, Cambridge Stud. Adv. Math. 120, Cambridge University Press, Cambridge, 2010. 10.1017/CBO9780511845079Search in Google Scholar

[5] M. Gubinelli, Controlling rough paths, J. Funct. Anal. 216 (2004), no. 1, 86–140. 10.1016/j.jfa.2004.01.002Search in Google Scholar

[6] Y. Hu and D. Nualart, Rough path analysis via fractional calculus, Trans. Amer. Math. Soc. 361 (2009), no. 5, 2689–2718. 10.1090/S0002-9947-08-04631-XSearch in Google Scholar

[7] Y. Ito, Extension theorem for rough paths via fractional calculus, preprint (2014). 10.2969/jmsj/06930893Search in Google Scholar

[8] Y. Ito, Integrals along rough paths via fractional calculus, Potential Anal. 42 (2015), no. 1, 155–174. 10.1007/s11118-014-9428-3Search in Google Scholar

[9] Y. Ito, A fractional calculus approach to rough integration, preprint (2016). 10.1215/21562261-2019-0017Search in Google Scholar

[10] T. J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoam. 14 (1998), no. 2, 215–310. 10.4171/RMI/240Search in Google Scholar

[11] T. J. Lyons, M. J. Caruana and T. Lévy, Differential Equations Driven by Rough Paths, Lecture Notes in Math. 1908, Springer, Berlin, 2007. 10.1007/978-3-540-71285-5Search in Google Scholar

[12] T. J. Lyons and Z. Qian, System Control and Rough Paths, Oxford Math. Monogr., Clarendon Press, Oxford, 2002. 10.1093/acprof:oso/9780198506485.001.0001Search in Google Scholar

[13] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon & Breach, New York, 1993. Search in Google Scholar

[14] M. Zähle, Integration with respect to fractal functions and stochastic calculus. I, Probab. Theory Related Fields 111 (1998), no. 3, 333–374. 10.1007/s004400050171Search in Google Scholar

Received: 2016-3-20
Revised: 2016-9-18
Published Online: 2016-11-11
Published in Print: 2017-9-1

© 2017 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 28.1.2023 from
Scroll Up Arrow