Under the key assumption of finite -variation, , 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), resp. , 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 , any , for Wong-Zakai and Milstein-type approximations with mesh-size . 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 -variation finie, . Dans le cas du mouvement Brownien, respectivement du Brownien fractionnaire (fBM), pour lesquels resp. , 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 , tout , pour les approximations de Wong-Zakai et de type Milstein avec pas de discrétisation . Dans le cas du fBM, ce résultat résout une conjecture posée par les références ci-dessus.
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}, 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://archive.numdam.org/articles/10.1214/12-AIHP507/} }
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 SP - 154 EP - 194 VL - 50 IS - 1 PB - Gauthier-Villars UR - http://archive.numdam.org/articles/10.1214/12-AIHP507/ DO - 10.1214/12-AIHP507 LA - en ID - AIHPB_2014__50_1_154_0 ER -
%0 Journal Article %A Friz, Peter %A Riedel, Sebastian %T Convergence rates for the full gaussian rough paths %J Annales de l'I.H.P. Probabilités et statistiques %D 2014 %P 154-194 %V 50 %N 1 %I Gauthier-Villars %U http://archive.numdam.org/articles/10.1214/12-AIHP507/ %R 10.1214/12-AIHP507 %G en %F 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://archive.numdam.org/articles/10.1214/12-AIHP507/
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