Convergence rates for the full gaussian rough paths
Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 1, p. 154-194

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/(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/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.

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[1,2). Dans le cas du mouvement Brownien, respectivement du Brownien fractionnaire (fBM), pour lesquels ρ=1 resp. ρ=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/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.

DOI : https://doi.org/10.1214/12-AIHP507
Classification:  60H35,  60H10,  60G15,  65C30
Keywords: gaussian processes, rough paths, numerical schemes, rates of convergence
@article{AIHPB_2014__50_1_154_0,
     author = {Friz, Peter and Riedel, Sebastian},
     title = {Convergence rates for the full gaussian rough paths},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {1},
     year = {2014},
     pages = {154-194},
     doi = {10.1214/12-AIHP507},
     zbl = {1295.60045},
     mrnumber = {3161527},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2014__50_1_154_0}
}
Friz, Peter; Riedel, Sebastian. Convergence rates for the full gaussian rough paths. Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 1, pp. 154-194. doi : 10.1214/12-AIHP507. http://www.numdam.org/item/AIHPB_2014__50_1_154_0/

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