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

Abstract

We will consider the following stochastic differential equation (SDE): \begin{equation} X_t=X_0+\int_0^tb(X_s,θ_0)ds+σB_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 $b(\cdot ,\theta ):\mathbb{R}\to \mathbb{R}$, and $\sigma \in \mathbb{R}$ is assumed to be the known diffusion coefficient.