Convergence rates for the full gaussian rough paths

Friz, Peter; Riedel, Sebastian

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014), p. 154-194 / Harvested from Numdam

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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.

Publié le : 2014-01-01

MR 3161527 | Zbl 1295.60045

DOI : https://doi.org/10.1214/12-AIHP507
Classification: 60H35, 60H10, 60G15, 65C30

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     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},
     volume = {50},
     year = {2014},
     pages = {154-194},
     doi = {10.1214/12-AIHP507},
     mrnumber = {3161527},
     zbl = {1295.60045},
     language = {en},
     url = {http://dml.mathdoc.fr/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, Tome 50 (2014) pp. 154-194. doi : 10.1214/12-AIHP507. http://gdmltest.u-ga.fr/item/AIHPB_2014__50_1_154_0/

Bibliographie

[1] T. Cass and P. Friz. Densities for rough differential equations under Hoermander's condition. Ann. of Math. (2) 171 (2010) 2115-2141. | MR 2680405 | Zbl 1205.60105

[2] L. Coutin and Z. Qian. Stochastic analysis, rough path analysis and fractional Brownian motions. Probab. Theory Related Fields 122 (2002) 108-140. | MR 1883719 | Zbl 1047.60029

[3] A. M. Davie. Differential equations driven by rough paths: an approach via discrete approximation. Appl. Math. Res. Express. AMRX (2007) Art. ID abm009, 40. | MR 2387018 | Zbl 1163.34005

[4] A. Deya, A. Neuenkirch and S. Tindel. A Milstein-type scheme without Lévy area terms for SDEs driven by fractional Brownian motion. Ann. Inst. Henri Poincaré Probab. Stat. 48 (2012) 518-550. | Numdam | MR 2954265 | Zbl 1260.60135

[5] P. Friz and S. Riedel. Convergence rates for the full Brownian rough paths with applications to limit theorems for stochastic flows. Bull. Sci. Math. 135 (2011) 613-628. | MR 2838093 | Zbl 1237.60044

[6] P. Friz and N. Victoir. Differential equations driven by Gaussian signals. Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010) 369-413. | Numdam | MR 2667703 | Zbl 1202.60058

[7] P. Friz and N. Victoir. Multidimensional Stochastic Processes as Rough Paths. Cambridge Univ. Press, Cambridge, 2010. | MR 2604669 | Zbl 1193.60053

[8] P. Friz and N. Victoir. A note on higher dimensional $p$ -variation. Electron. J. Probab. 16 (2011) 1880-1899. | MR 2842090 | Zbl 1244.60066

[9] I. Gyöngy and A. Shmatkov. Rate of convergence of Wong-Zakai approximations for stochastic partial differential equations. Appl. Math. Optim. 54 (2006) 315-341. | MR 2268661 | Zbl 1106.60050

[10] M. Hairer. Rough stochastic PDEs. Comm. Pure Appl. Math. 64 (2011) 1547-1585. | MR 2832168 | Zbl 1229.60079

[11] K. Hara and M. Hino. Fractional order Taylor's series and the neo-classical inequality. Bull. Lond. Math. Soc. 42 (2010) 467-477. | MR 2651942 | Zbl 1194.26027

[12] Y. Hu and D. Nualart. Rough path analysis via fractional calculus. Trans. Amer. Math. Soc. 361 (2009) 2689-2718. | MR 2471936 | Zbl 1175.60061

[13] S. Janson. Gaussian Hilbert Spaces. Cambridge Univ. Press, New York, 1997. | MR 1474726 | Zbl 1143.60005

[14] T. Lyons. Differential equations driven by rough signals. Rev. Mat. Iberoam. 14 (1998) 215-310. | MR 1654527 | Zbl 0923.34056

[15] T. Lyons and Z. Qian. System Control and Rough Paths. Oxford Univ. Press, New York, 2002. | MR 2036784 | Zbl 1029.93001

[16] A. Neuenkirch, S. Tindel and J. Unterberger. Discretizing the fractional Lévy area. Stochastic Process. Appl. 120 (2010) 223-254. | MR 2576888 | Zbl 1185.60076

[17] C. Reutenauer. Free Lie Algebras. Clarendon Press, New York, 1993. | MR 1231799 | Zbl 0798.17001

[18] N. Towghi. Multidimensional extension of L. C. Young's inequality. JIPAM J. Inequal. Pure Appl. Math. 3 (2002) 13 (electronic). | MR 1906391 | Zbl 0997.26007