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Convergence rates for the full Gaussian rough paths. (English. French summary) Zbl 1295.60045
The authors consider multi-dimensional Gaussian processes in the rough-paths framework and their results are established under the assumption that its covariance is of finite \(\rho\)-variation for some \(\rho\in[1,2)\). The main results are sharp a.s. convergence rates for approximations of Gaussian rough paths. As special cases prior results regarding Brownian motion and fractional Brownian motion are recovered. Even more specifically, for Wong-Zakai- (including piecewise linear and mollifier approximations) and Milstein-type approximations an a.s. convergence rate of \(k^{-(1/\rho-1/2-\epsilon)}\) for any \(\epsilon>0\) is obtained. The main tool for establishing the result is a multi-dimensional Young integration. Additionally, the article contains a brief introduction into the basic notions of Gaussian rough paths and shuffle algebras and interated integrals.

60G15 Gaussian processes
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
65C30 Numerical solutions to stochastic differential and integral equations
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