Estimation in models driven by fractional Brownian motion

TL;DR

This paper develops statistical estimation methods for parameters in models driven by fractional Brownian motion with Hurst parameter H>1/2, including CLTs, hypothesis tests, and functional estimation techniques.

Contribution

It introduces new central limit theorems and hypothesis tests for estimating H and  in fractional Brownian motion-driven models, extending to functional estimation of .

Findings

CLTs for estimators of H and  in specific models

Hypothesis tests for  in these models

Asymptotic behavior of functionals of 2nd-order increments

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 \[X(t)=c+\int_0^t\sigma\bigl(X(u)\bigr)\mathrm {d}b_H(u)+\int _0^t\mu\bigl(X(u)\bigr)\mathrm {d}u.\] 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.