Convergence rates for the full gaussian rough paths

Friz, Peter; Riedel, Sebastian

doi:10.1214/12-AIHP507

Friz, Peter ; Riedel, Sebastian

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 1, pp. 154-194.

Résumé
Abstract

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 $ρ$ -variation finie, $ρ \in [1, 2)$ . Dans le cas du mouvement Brownien, respectivement du Brownien fractionnaire (fBM), pour lesquels $ρ = 1$ resp. $ρ = 1 / (2 H)$ , 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 $k^{- (1 / ρ - 1 / 2 - ε)}$ , tout $ε > 0$ , 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.

Under the key assumption of finite $ρ$ -variation, $ρ \in [1, 2)$ , 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), $ρ = 1$ 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 $ε > 0$ , 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.

MR Zbl

DOI : 10.1214/12-AIHP507

Classification : 60H35, 60H10, 60G15, 65C30
Mots clés : gaussian processes, rough paths, numerical schemes, rates of convergence

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     title = {Convergence rates for the full gaussian rough paths},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {154--194},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {1},
     year = {2014},
     doi = {10.1214/12-AIHP507},
     mrnumber = {3161527},
     zbl = {1295.60045},
     language = {en},
     url = {http://www.numdam.org/articles/10.1214/12-AIHP507/}
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Friz, Peter; Riedel, Sebastian. Convergence rates for the full gaussian rough paths. Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 1, pp. 154-194. doi : 10.1214/12-AIHP507. http://www.numdam.org/articles/10.1214/12-AIHP507/

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