Electronic Journal of Probability
- Electron. J. Probab.
- Volume 18 (2013), paper no. 108, 18 pp.
A simple proof of distance bounds for Gaussian rough paths
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.
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]
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