Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Convergence rates for the full Gaussian rough paths

Peter Friz and Sebastian Riedel

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Abstract

Under the key assumption of finite $\rho$-variation, $\rho\in\lbrack1,2)$, 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 $\varepsilon>0$, 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.

Résumé

Nous établissons des vitesses fines de convergence presque sûre pour les approximations des chemins rugueux Gaussiens, sous l’hypothèse que la fonction de covariance du processus Gaussien sous-jacent ait une $\rho$-variation finie, $\rho \in[1,2)$. Dans le cas du mouvement Brownien, respectivement du Brownien fractionnaire (fBM), pour lesquels $\rho=1$ resp. $\rho=1/(2H)$, ce résultat généralise les résultats respectifs de (Trans. Amer. Math. Soc. 361 (2009) 2689–2718) et (Ann. Inst. Henri Poincasé Probab. Stat. 48 (2012) 518–550).

Notamment, nous établissons le taux de convergence presque sure $k^{-(1/\rho-1/2-\varepsilon)}$, tout $\varepsilon>0$, pour les approximations de Wong–Zakai et de type Milstein avec pas de discrétisation $1/k$. Dans le cas du fBM, ce résultat résout une conjecture posée par les références ci-dessus.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 50, Number 1 (2014), 154-194.

Dates
First available in Project Euclid: 1 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1388545270

Digital Object Identifier
doi:10.1214/12-AIHP507

Mathematical Reviews number (MathSciNet)
MR3161527

Zentralblatt MATH identifier
1295.60045

Subjects
Primary: 60H35: Computational methods for stochastic equations [See also 65C30] 60H10: Stochastic ordinary differential equations [See also 34F05] 60G15: Gaussian processes 65C30: Stochastic differential and integral equations

Keywords
Gaussian processes Rough paths Numerical schemes Rates of convergence

Citation

Friz, Peter; Riedel, Sebastian. Convergence rates for the full Gaussian rough paths. Ann. Inst. H. Poincaré Probab. Statist. 50 (2014), no. 1, 154--194. doi:10.1214/12-AIHP507. https://projecteuclid.org/euclid.aihp/1388545270


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