Estimation in models driven by fractional brownian motion
Annales de l'I.H.P. Probabilités et statistiques, Volume 44 (2008) no. 2, pp. 191-213.

Let b H (t),t 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 σ(x)=σ or σ(x)=σx and μ(x)=μ or μ(x)=μx, we propose a central limit theorem for estimators of H and of σ based on regression methods. Then we give tests of the hypothesis on σ for these models. We also consider functional estimation on σ(·) in the above more general models based in the asymptotic behavior of functionals of the 2nd-order increments of the fBm.

Soit b H (t),t 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, σ(x)=σ ou σ(x)=σx et μ(x)=μ ou μ(x)=μx, un théorème central limite pour des estimateurs de H et de σ, obtenus par une méthode de régression. Ensuite, pour ces modèles, nous proposons des tests d’hypothèses sur σ. Enfin, dans les modèles plus généraux ci-dessus nous proposons des estimateurs fonctionnels pour la fonction σ(·) dont les propriétés sont obtenues via la convergence de fonctionnelles des accroissements doubles du mBf.

DOI: 10.1214/07-AIHP105
Classification: 60F05,  60G15,  60G18,  60H10,  62F03,  62F12,  33C45
Keywords: central limit theorem, estimation, fractional brownian motion, gaussian processes, Hermite polynomials
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Berzin, Corinne; León, José R. Estimation in models driven by fractional brownian motion. Annales de l'I.H.P. Probabilités et statistiques, Volume 44 (2008) no. 2, pp. 191-213. doi : 10.1214/07-AIHP105. http://www.numdam.org/articles/10.1214/07-AIHP105/

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