Dans cet article, nous considérons une équation différentielle stochastique multidimensionnelle dirigée par un mouvement brownien fractionnaire d'indice de Hurst H>1/3. Nous développons E[f(Xt)] par rapport à t, où on note X la solution de l'EDS et où f:ℝn→ℝ est une fonction régulière. Par rapport à F. Baudoin et L. Coutin, Stochastic Process. Appl. 117 (2007) 550-574, où le même problème est étudié, nous améliorons leur résultat dans trois directions différentes: nous traîtons le cas d'une équation avec dérive, nous paramétrons notre développement à l'aide d'arbres, ce qui le rend plus facile à utiliser, et nous proposons un contrôle plus fin du reste quand H>1/2.
In this article, we consider an n-dimensional stochastic differential equation driven by a fractional brownian motion with Hurst parameter H>1/3. We derive an expansion for E[f(Xt)] in terms of t, where X denotes the solution to the SDE and f:ℝn→ℝ is a regular function. Comparing to F. Baudoin and L. Coutin, Stochastic Process. Appl. 117 (2007) 550-574, where the same problem is studied, we provide an improvement in three different directions: we are able to consider equations with drift, we parametrize our expansion with trees, which makes it easier to use, and we obtain a sharp estimate of the remainder for the case H>1/2.
Mots-clés : fractional brownian motion, stochastic differential equations, trees expansions
@article{AIHPB_2009__45_1_157_0,
author = {Neuenkirch, A. and Nourdin, I. and R\"o{\ss}ler, A. and Tindel, S.},
title = {Trees and asymptotic expansions for fractional stochastic differential equations},
journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
pages = {157--174},
publisher = {Gauthier-Villars},
volume = {45},
number = {1},
year = {2009},
doi = {10.1214/07-AIHP159},
mrnumber = {2500233},
zbl = {1172.60017},
language = {en},
url = {http://www.numdam.org/articles/10.1214/07-AIHP159/}
}
TY - JOUR AU - Neuenkirch, A. AU - Nourdin, I. AU - Rößler, A. AU - Tindel, S. TI - Trees and asymptotic expansions for fractional stochastic differential equations JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2009 SP - 157 EP - 174 VL - 45 IS - 1 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/07-AIHP159/ DO - 10.1214/07-AIHP159 LA - en ID - AIHPB_2009__45_1_157_0 ER -
%0 Journal Article %A Neuenkirch, A. %A Nourdin, I. %A Rößler, A. %A Tindel, S. %T Trees and asymptotic expansions for fractional stochastic differential equations %J Annales de l'I.H.P. Probabilités et statistiques %D 2009 %P 157-174 %V 45 %N 1 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/07-AIHP159/ %R 10.1214/07-AIHP159 %G en %F AIHPB_2009__45_1_157_0
Neuenkirch, A.; Nourdin, I.; Rößler, A.; Tindel, S. Trees and asymptotic expansions for fractional stochastic differential equations. Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 1, pp. 157-174. doi : 10.1214/07-AIHP159. http://www.numdam.org/articles/10.1214/07-AIHP159/
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