Electronic Journal of Probability

A simple proof of distance bounds for Gaussian rough paths

Sebastian Riedel and Weijun Xu

Full-text: Open access


We derive explicit distance bounds for Stratonovich iterated integrals along two Gaussian processes (also known as signatures of Gaussian rough paths) based on the regularity assumption of their covariance functions. Similar estimates have been obtained recently in [Friz-Riedel, AIHP, to appear]. One advantage of our argument is that we obtain the bound for the third level iterated integrals merely based on the first two levels, and this reflects the intrinsic nature of rough paths. Our estimates are sharp when both covariance functions have finite $1$-variation, which includes a large class of Gaussian processes. Two applications of our estimates are discussed. The first one gives the a.s. convergence rates for approximated solutions to rough differential equations driven by Gaussian processes. In the second example, we show how to recover the optimal time regularity for solutions of some rough SPDEs.

Article information

Electron. J. Probab., Volume 18 (2013), paper no. 108, 18 pp.

Accepted: 30 December 2013
First available in Project Euclid: 4 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G15: Gaussian processes
Secondary: 60H05: Stochastic integrals 60H10: Stochastic ordinary differential equations [See also 34F05] 60H35: Computational methods for stochastic equations [See also 65C30]

Gaussian rough paths iterated integrals signatures

This work is licensed under a Creative Commons Attribution 3.0 License.


Riedel, Sebastian; Xu, Weijun. A simple proof of distance bounds for Gaussian rough paths. Electron. J. Probab. 18 (2013), paper no. 108, 18 pp. doi:10.1214/EJP.v18-2387. https://projecteuclid.org/euclid.ejp/1465064333

Export citation


  • Coutin, Laure; Qian, Zhongmin. Stochastic analysis, rough path analysis and fractional Brownian motions. Probab. Theory Related Fields 122 (2002), no. 1, 108–140. MR1883719
  • Friz, Peter; Hairer, Martin. A Short Course on Rough Paths. Preprint.
  • Friz, Peter; Riedel, Sebastian. Convergence rates for the full Brownian rough paths with applications to limit theorems for stochastic flows. Bull. Sci. Math. 135 (2011), no. 6-7, 613–628. MR2838093
  • Friz, Peter; Riedel, Sebastian. Convergence rates for the full Gaussian rough paths. To appear in Annales de l'Institut Henri Poincaré (B) Probability and Statistics.
  • Friz, Peter; Victoir, Nicolas. Differential equations driven by Gaussian signals. Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010), no. 2, 369–413. MR2667703
  • Friz, Peter K.; Victoir, Nicolas B. Multidimensional stochastic processes as rough paths. Theory and applications. Cambridge Studies in Advanced Mathematics, 120. Cambridge University Press, Cambridge, 2010. xiv+656 pp. ISBN: 978-0-521-87607-0 MR2604669
  • Friz, Peter; Victoir, Nicolas. A note on higher dimensional $p$-variation. Electron. J. Probab. 16 (2011), no. 68, 1880–1899. MR2842090
  • Gubinelli, M. Controlling rough paths. J. Funct. Anal. 216 (2004), no. 1, 86–140. MR2091358
  • Hairer, Martin. An introduction to Stochastic PDEs. Preprint, available at http://arxiv.org/abs/0907.4178 (2009).
  • Hairer, M. Rough stochastic PDEs. Comm. Pure Appl. Math. 64 (2011), no. 11, 1547–1585. MR2832168
  • Lyons, Terry J. Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 (1998), no. 2, 215–310. MR1654527
  • Lyons, Terry; Caruana, Michael; Lévy Thierry. Differential Equations Driven by Rough Paths. Lecture Notes in Mathematics Volume 1908 (2006), Springer Berlin.
  • Lyons, Terry; Qian, Zhongmin. System control and rough paths. Oxford Mathematical Monographs. Oxford Science Publications. Oxford University Press, Oxford, 2002. x+216 pp. ISBN: 0-19-850648-1 MR2036784
  • Lyons, Terry J.; Xu, Weijun. A uniform estimate for rough paths. Bull. Sci. Math. 137 (2013), no. 7, 867–879. MR3116217
  • Lyons, Terry; Zeitouni, Ofer. Conditional exponential moments for iterated Wiener integrals. Ann. Probab. 27 (1999), no. 4, 1738–1749. MR1742886
  • Towghi, Nasser. Multidimensional extension of L. C. Young's inequality. JIPAM. J. Inequal. Pure Appl. Math. 3 (2002), no. 2, Article 22, 13 pp. (electronic). MR1906391