Convergence rates for the full Gaussian rough paths

Abstract

Under the key assumption of finite p-variation, 2 [1; 2), of the covariance of the underlying Gaussian process, sharp a.s. convergence rates for approximations of Gaussian rough paths are established. When applied to Brownian resp. fractional Brownian motion (fBM), p = 1 resp. p = 1 p= (2H), we recover and extend the respective results of [Hu{Nualart; Rough path analysis via fractional calculus; TAMS 361 (2009) 2689-2718] and [Deya{Neuenkirch{Tindel; A Milstein-type scheme without Lévy area terms for SDEs driven by fractional Brownian motion; AIHP (2011)]. In particular, we establish an a.s. rate k (1 1=2"), any > 0, for Wong- Zakai and Milstein-type approximations with mesh-size 1=k. When applied to fBM this answers a conjecture in the afore-mentioned references.