Convergence rates for the full Gaussian rough paths

TL;DR

This paper establishes sharp almost sure convergence rates for Gaussian rough path approximations under finite 9-variation, extending known results for Brownian and fractional Brownian motions and confirming a conjecture for fBM.

Contribution

It provides new sharp convergence rates for Gaussian rough path approximations, including Brownian and fractional Brownian motions, under finite 9-variation assumptions.

Findings

Sharp a.s. convergence rates for Gaussian rough paths are established.

Results extend previous work on Brownian and fractional Brownian motions.

Answers a conjecture regarding fBM approximation rates.

Abstract

Under the key assumption of finite {\rho}-variation, {\rho}\in[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), {\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\'evy area terms for SDEs driven by fractional Brownian motion; AIHP (2011)]. In particular, we establish an a.s. rate k^{-(1/{\rho}-1/2-{\epsilon})}, any {\epsilon}>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.