[Opérateurs cumulants et moments des intégrales dʼItô et de Skorohod]
Nous proposons une formule de calcul des moments dʼintégrales dʼItô et de Skorohod par rapport au mouvement brownien à lʼaide dʼopérateurs cumulants définis par le calcul de Malliavin. On retrouve ainsi certaines caractérisations de la loi gaussienne pour les intégrales stochastiques.
We propose a formula for the computation of the moments of all orders of Itô and Skorohod stochastic integrals with respect to Brownian motion, based on cumulant operators defined by the Malliavin calculus. Some characterizations of Gaussian distributions for stochastic integrals are recovered as a consequence.
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@article{CRMATH_2013__351_9-10_397_0,
author = {Privault, Nicolas},
title = {Cumulant operators and moments of the {It\^o} and {Skorohod} integrals},
journal = {Comptes Rendus. Math\'ematique},
pages = {397--400},
publisher = {Elsevier},
volume = {351},
number = {9-10},
year = {2013},
doi = {10.1016/j.crma.2013.05.014},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2013.05.014/}
}
TY - JOUR AU - Privault, Nicolas TI - Cumulant operators and moments of the Itô and Skorohod integrals JO - Comptes Rendus. Mathématique PY - 2013 SP - 397 EP - 400 VL - 351 IS - 9-10 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2013.05.014/ DO - 10.1016/j.crma.2013.05.014 LA - en ID - CRMATH_2013__351_9-10_397_0 ER -
%0 Journal Article %A Privault, Nicolas %T Cumulant operators and moments of the Itô and Skorohod integrals %J Comptes Rendus. Mathématique %D 2013 %P 397-400 %V 351 %N 9-10 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2013.05.014/ %R 10.1016/j.crma.2013.05.014 %G en %F CRMATH_2013__351_9-10_397_0
Privault, Nicolas. Cumulant operators and moments of the Itô and Skorohod integrals. Comptes Rendus. Mathématique, Tome 351 (2013) no. 9-10, pp. 397-400. doi : 10.1016/j.crma.2013.05.014. http://www.numdam.org/articles/10.1016/j.crma.2013.05.014/
[1] On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables, Biometrika, Volume 12 (1918) no. 1–2, pp. 134-139
[2] Applications of Faà di Brunoʼs formula in mathematical statistics, Am. Math. Mon., Volume 62 (1955), pp. 340-348
[3] Cumulants on the Wiener space, J. Funct. Anal., Volume 258 (2010) no. 11, pp. 3775-3791
[4] The Malliavin Calculus and Related Topics, Theory Probab. Appl., Springer-Verlag, Berlin, 2006
[5] Wiener Chaos: Moments, Cumulants and Diagrams: A Survey with Computer Implementation, Bocconi & Springer Series, Springer, 2011
[6] Combinatorial stochastic processes, Lectures from the 32nd Summer School on Probability Theory Held in Saint-Flour, July 7–24, 2002, Lect. Notes Math., vol. 1875, Springer-Verlag, Berlin, 2006, pp. 7-24
[7] Moment identities for Skorohod integrals on the Wiener space and applications, Electron. Commun. Probab., Volume 14 (2009), pp. 116-121 (electronic)
[8] Generalized Bell polynomials and the combinatorics of Poisson central moments, Electron. J. Comb., Volume 18 (2011) no. 1 (Research Paper 54, 10 pp)
[9] Laplace transform identities and measure-preserving transformations on the Lie–Wiener–Poisson spaces, J. Funct. Anal., Volume 263 (2012), pp. 2993-3023
[10] N. Privault, Cumulant operators for Lie–Wiener–Itô–Poisson stochastic integrals, preprint, 2013.
[11] On semi invariants in the theory of observations, Kjöbenhavn Overs (1899), pp. 135-141
[12] Random rotations of the Wiener path, Probab. Theory Relat. Fields, Volume 103 (1995) no. 3, pp. 409-429
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