Convergence rates for the full gaussian rough paths

Peter Friz; Sebastian Riedel

Annales de l'I.H.P. Probabilités et statistiques (2014)

  • Volume: 50, Issue: 1, page 154-194
  • ISSN: 0246-0203

Abstract

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Under the key assumption of finite ρ -variation, ρ ∈ [ 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), ρ = 1 resp. ρ = 1 / ( 2 H ) , we recover and extend the respective results of (Trans. Amer. Math. Soc.361 (2009) 2689–2718) and (Ann. Inst. Henri Poincasé Probab. Stat.48(2012) 518–550). In particular, we establish an a.s. rate k - ( 1 / ρ - 1 / 2 - ε ) , any ε g t ; 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.

How to cite

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Friz, Peter, and Riedel, Sebastian. "Convergence rates for the full gaussian rough paths." Annales de l'I.H.P. Probabilités et statistiques 50.1 (2014): 154-194. <http://eudml.org/doc/272009>.

@article{Friz2014,
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 (Trans. Amer. Math. Soc.361 (2009) 2689–2718) and (Ann. Inst. Henri Poincasé Probab. Stat.48(2012) 518–550). In particular, we establish an a.s. rate $k^\{-(1/\rho -1/2-\varepsilon )\}$, any $\varepsilon &gt;0$, 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.},
author = {Friz, Peter, Riedel, Sebastian},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {gaussian processes; rough paths; numerical schemes; rates of convergence; Gaussian processes},
language = {eng},
number = {1},
pages = {154-194},
publisher = {Gauthier-Villars},
title = {Convergence rates for the full gaussian rough paths},
url = {http://eudml.org/doc/272009},
volume = {50},
year = {2014},
}

TY - JOUR
AU - Friz, Peter
AU - Riedel, Sebastian
TI - Convergence rates for the full gaussian rough paths
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2014
PB - Gauthier-Villars
VL - 50
IS - 1
SP - 154
EP - 194
AB - 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 (Trans. Amer. Math. Soc.361 (2009) 2689–2718) and (Ann. Inst. Henri Poincasé Probab. Stat.48(2012) 518–550). In particular, we establish an a.s. rate $k^{-(1/\rho -1/2-\varepsilon )}$, any $\varepsilon &gt;0$, 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.
LA - eng
KW - gaussian processes; rough paths; numerical schemes; rates of convergence; Gaussian processes
UR - http://eudml.org/doc/272009
ER -

References

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