Open Access
2013 A simple proof of distance bounds for Gaussian rough paths
Sebastian Riedel, Weijun Xu
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Electron. J. Probab. 18: 1-18 (2013). DOI: 10.1214/EJP.v18-2387


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.


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Sebastian Riedel. Weijun Xu. "A simple proof of distance bounds for Gaussian rough paths." Electron. J. Probab. 18 1 - 18, 2013.


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

zbMATH: 1290.60046
MathSciNet: MR3151728
Digital Object Identifier: 10.1214/EJP.v18-2387

Primary: 60G15
Secondary: 60H05 , 60H10 , 60H35

Keywords: Gaussian rough paths , iterated integrals , signatures

Vol.18 • 2013
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