Convergence rates for the full gaussian rough paths

Friz, Peter; Riedel, Sebastian

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

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

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

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