Estimation in models driven by fractional brownian motion

Berzin, Corinne ; León, José R.

Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008), p. 191-213 / Harvested from Numdam

Résumé

Soit $\left\lbrace b_H(t),t\in \mathbb {R} \right\rbrace $ le mouvement Brownien fractionnaire de paramètre $0<H<1$. Lorsque $1/2<H$, nous considérons des équations de diffusion de la forme X(t)=c+∫ 0 t σ(X(u))db H (u)+∫ 0 t μ(X(u))du. Nous proposons dans des modèles particuliers où, $\sigma (x)=\sigma $ ou $\sigma (x)=\sigma x$ et $\mu (x)=\mu $ ou $\mu (x)=\mu x$, un théorème central limite pour des estimateurs de $H$ et de $\sigma $, obtenus par une méthode de régression. Ensuite, pour ces modèles, nous proposons des tests d’hypothèses sur $\sigma $. Enfin, dans les modèles plus généraux ci-dessus nous proposons des estimateurs fonctionnels pour la fonction $\sigma (\cdot )$ dont les propriétés sont obtenues via la convergence de fonctionnelles des accroissements doubles du mBf.

Abstract

Let $\left\lbrace b_H(t),t\in \mathbb {R} \right\rbrace $ be the fractional brownian motion with parameter $0<H<1$. When $1/2<H$, we consider diffusion equations of the type X(t)=c+∫ 0 t σ(X(u))db H (u)+∫ 0 t μ(X(u))du. In different particular models where $\sigma (x)=\sigma $ or $\sigma (x)=\sigma x$ and $\mu (x)=\mu $ or $\mu (x)=\mu x$, we propose a central limit theorem for estimators of $H$ and of $\sigma $ based on regression methods. Then we give tests of the hypothesis on $\sigma $ for these models. We also consider functional estimation on $\sigma (\cdot )$ in the above more general models based in the asymptotic behavior of functionals of the 2nd-order increments of the fBm.

Détail

Publié le : 2008-01-01

MR 2446320 | Zbl pre05611436

DOI : https://doi.org/10.1214/07-AIHP105
Classification:  60F05,  60G15,  60G18,  60H10,  62F03,  62F12,  33C45

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