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) no. 2, pp. 191-213.

Soit $\left\{b_H\left(t\right),t\in ℝ\right\}$ le mouvement Brownien fractionnaire de paramètre $0<H<1$. Lorsque $1/2<H$, nous considérons des équations de diffusion de la forme

$$X\left(t\right)=c+{\int }_0^t\sigma \left(X\left(u\right)\right)\mathrm{d}{b}_H\left(u\right)+{\int }_0^t\mu \left(X\left(u\right)\right)\mathrm{d}u.$$

Nous proposons dans des modèles particuliers où, $\sigma \left(x\right)=\sigma$ ou $\sigma \left(x\right)=\sigma x$ et $\mu \left(x\right)=\mu$ ou $\mu \left(x\right)=\mu x$, 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 $\sigma (·)$ dont les propriétés sont obtenues via la convergence de fonctionnelles des accroissements doubles du mBf.

Let $\left\{b_H\left(t\right),t\in ℝ\right\}$ be the fractional brownian motion with parameter $0<H<1$. When $1/2<H$, we consider diffusion equations of the type

$$X\left(t\right)=c+{\int }_0^t\sigma \left(X\left(u\right)\right)\mathrm{d}{b}_H\left(u\right)+{\int }_0^t\mu \left(X\left(u\right)\right)\mathrm{d}u.$$

In different particular models where $\sigma \left(x\right)=\sigma$ or $\sigma \left(x\right)=\sigma x$ and $\mu \left(x\right)=\mu$ or $\mu \left(x\right)=\mu x$, 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 $\sigma (·)$ in the above more general models based in the asymptotic behavior of functionals of the 2nd-order increments of the fBm.

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