Estimation in models driven by fractional Brownian motion
Résumé
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 $$X(t) = c + \int_{0}^{t} \sigma(X(u)) d b_{H}(u) + \int_{0}^{t} \mu(X(u)) d u \mbox{.}$$
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 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 $2^{nd}$-order increments of the fBm.
Brownian motion with parameter $0 < H <1$. When $1/2 < H$, we consider diffusion equations of the type $$X(t) = c + \int_{0}^{t} \sigma(X(u)) d b_{H}(u) + \int_{0}^{t} \mu(X(u)) d u \mbox{.}$$
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 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 $2^{nd}$-order increments of the fBm.
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