Convergence rates for the full Gaussian rough paths

By Peter K. Friz. 2011

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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.

Subfields: Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/de1768a082ad382147496c965ee132b7d84b11fd

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Citations in the corpus

Rate of convergence and asymptotic error distribution of Euler approximation schemes for fractional diffusions (2014, result): “There are also a number of results on the rate of convergence of Euler-type numerical schemes in this case; see, for instance, the paper by Deya, Neuenkirch and Tindel [4] for a Milstein-type scheme without Lévy area in the case 13 < H < 1 2 , the paper by Friz and Riedel [5] for the N -step Euler s”
The Jain–Monrad criterion for rough paths and applications to random Fourier series and non-Markovian Hörmander theory (2013, methodology): “20Strictly speaking, [15], Theorem 5, assumes that ̃ is a certain approximation of .”