Enhanced gaussian processes and applications
ESAIM: Probability and Statistics, Volume 13 (2009), p. 247-260

We propose some construction of enhanced gaussian processes using Karhunen-Loeve expansion. We obtain a characterization and some criterion of existence and uniqueness. Using rough-path theory, we derive some Wong-Zakai Theorem.

DOI : https://doi.org/10.1051/ps:2008007
Classification:  60G15,  60G17
Keywords: gaussian processes, Volterra processes, rough path theory
@article{PS_2009__13__247_0,
     author = {Coutin, Laure and Victoir, Nicolas},
     title = {Enhanced gaussian processes and applications},
     journal = {ESAIM: Probability and Statistics},
     publisher = {EDP-Sciences},
     volume = {13},
     year = {2009},
     pages = {247-260},
     doi = {10.1051/ps:2008007},
     zbl = {pre05660767},
     mrnumber = {2528082},
     language = {en},
     url = {http://www.numdam.org/item/PS_2009__13__247_0}
}
Coutin, Laure; Victoir, Nicolas. Enhanced gaussian processes and applications. ESAIM: Probability and Statistics, Volume 13 (2009) pp. 247-260. doi : 10.1051/ps:2008007. http://www.numdam.org/item/PS_2009__13__247_0/

[1] Ph. Biane and M. Yor, Variation sur une formule de Paul Lévy. Ann. Inst. H. Poincaré 23 (1987) 359-377. | Numdam | MR 898500 | Zbl 0623.60099

[2] C. Borell, On polynomial chaos and integrability. Probab. Math. Statist. 3 (1984) 191-203. | MR 764146 | Zbl 0555.60008

[3] P. Cheridito and Nualart, D. Stochastic integral of divergence type with respect to fractional Brownian motion with Hurst parameter H(0,1 2). Ann. Inst. H. Poincaré Probab. Statist. 41 (2005) 1049-1081. | Numdam | MR 2172209 | Zbl 1083.60027

[4] L. Coutin, An introduction to (stochastic) calculus with respect to fractional Brownian motion, Séminaire de Probabilités XL, Lect. Notes Math. 1899 (2007) 3-65. Springer, Berlin. | MR 2408998 | Zbl 1126.60042

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

[6] L. Coutin, P. Friz and N. Victoir, Good rough path sequences and applications to anticipating calculus. Ann. Probab. 35 (2007) 1172-1193. | MR 2319719 | Zbl 1132.60053

[7] L. Decreusefond, Stochastic Integration with respect to Volterra processes. Ann. Inst. H. Poincaré 41 (2005) 123-149. | Numdam | MR 2124078 | Zbl 1071.60040

[8] L. Decreusefond and A.S. Üstünel, Stochastic Analysis of the Fractional Brownian Motion. Potential Anal. 10 (1997) 177-214. | MR 1677455 | Zbl 0924.60034

[9] X.M. Fernique, Régularité des trajectoires des fonctions aléatoires gaussiennes, École d'été de probabilités de Saint-Flour, 1974. Lect. Notes Math. 480 (1974) 1-96. | Zbl 0331.60025

[10] P. Friz and N. Victoir, Approximations of the Brownian rough path with applications to stochastic analysis. Ann. Inst. H. Poincaré 41 (2005) 703-724. | Numdam | MR 2144230 | Zbl 1080.60021

[11] A. Lejay, Introduction to Rough Paths, Séminaire de probabilités XXXVII. Lect. Notes Math. 1832 (2003) 1-59. | MR 2053040 | Zbl 1041.60051

[12] P. Levy, Wiener's random function and other Laplacian random function, Proc. 2 Berkeley Symp. Math. Proba. (1950) 171-186, Univ. of California. | MR 44774 | Zbl 0044.13802

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

[14] T. Lyons and Z. Qian, System Control and Rough Paths, Oxford University Press (2002). | MR 2036784 | Zbl 1029.93001

[15] A. Millet and M. Sanz-Sole, Approximation of rough path of fractional Brownian motion, Seminar on Stochastic Analysis, Random Fields and Application V, Ascona 2005, Progr. Probab. 59. Birkhäuser Verlag (to appear) and arXiv math. PR/0509353. | MR 2401962 | Zbl 1142.60027

[16] V. Pipiras and M.S. Taqqu, Are classes of deterministic integrands for fractional Brownian motion on interval complete? Bernoulli 7 (2001) 873-897. | MR 1873833 | Zbl 1003.60055