Estimation in models driven by fractional Brownian motion

Abstract

Let $\{b_H(t),t\in \mathbb{R}\}$ be the fractional Brownian motion with parameter $0<H<1$. When $1/2<H$, we consider diffusion equations of the type \[X(t)=c+\int_0^tσ\bigl(X(u)\bigr)\mathrm {d}b_H(u)+\int _0^tμ\bigl(X(u)\bigr)\mathrm {d}u.\] 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.

Published in at http://dx.doi.org/10.1214/07-AIHP105 the Annales de l'Institut Henri Poincaré - Probabilités et Statistiques (http://www.imstat.org/aihp/) by the Institute of Mathematical Statistics (http://www.imstat.org)