Convergence rates for the full Gaussian rough paths

Details Convergence rates for the full Gaussian rough paths Convergence rates for the full Gaussian rough paths

2011-08-04

arxiv.org/abs/1108.1099v1

Mathematics

Probability

53 pages

Scientific paper

Under the key assumption of finite $\rho $-variation, $\rho \in \lbrack 1,2)$, 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, $\rho =1$ resp. $\rho =1/(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 L\'{e}vy area terms for SDEs driven by fractional Brownian motion; AIHP (2011)]. In particular, we establish an a.s. rate $n^{-(1/\rho -1/2-\epsilon)}$, any $\epsilon >0 $, for Wong-Zakai and Milstein-type approximations with mesh-size $1/n$. When applied to fBM this answers a conjecture in the afore-mentioned references.

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