Asymptotic normality of least squares estimators to stochastic differential equations driven by fractional Brownian motions

TL;DR

This paper investigates the asymptotic normality of least squares estimators for parameters in stochastic differential equations driven by fractional Brownian motion with Hurst index greater than 1/2, providing theoretical insights into their statistical properties.

Contribution

It establishes the asymptotic normality of least squares estimators for SDEs driven by fractional Brownian motion, extending classical results to the fractional setting.

Findings

Proves asymptotic normality of estimators for Hurst index > 1/2

Provides conditions under which estimators are consistent and asymptotically normal

Extends classical SDE estimation theory to fractional Brownian motion context

Abstract

We will consider the following stochastic differential equation (SDE): \begin{equation} X_t=X_0+\int_0^tb(X_s,\theta_0)ds+\sigma B_t,~~~t\in(0,T], \end{equation} where $\{B_t\}_{t\ge 0}$ is a fractional Brownian motion with Hurst index $H\in(1/2,1)$, $\theta_0$ is a parameter that contains a bounded and open convex subset $\Theta\subset\mathbb{R}^d$, $\{b(\cdot,\theta),\theta\in\Theta\}$ is a family of drift coefficients with $b(\cdot,\theta):\mathbb{R}\rightarrow\mathbb{R}$, and $\sigma\in\mathbb{R}$ is assumed to be the known diffusion coefficient.