LAN property for stochastic differential equations with additive fractional noise and continuous time observation

TL;DR

This paper establishes the LAN property for a class of stochastic differential equations driven by fractional noise with Hurst parameter greater than 1/2, demonstrating asymptotic normality and efficiency of estimators with continuous observations.

Contribution

It proves the LAN property for SDEs with fractional noise and non-linear drift, and analyzes the efficiency of the MLE in the fractional Ornstein-Uhlenbeck case.

Findings

LAN property holds with rate √τ as τ→∞

MLE is asymptotically efficient in the fractional Ornstein-Uhlenbeck process

Ergodic properties and Girsanov transform are key to the proof

Abstract

We consider a stochastic differential equation with additive fractional noise with Hurst parameter $H>1/2$, and a non-linear drift depending on an unknown parameter. We show the Local Asymptotic Normality property (LAN) of this parametric model with rate $\sqrt{\tau}$ as $\tau\rightarrow \infty$, when the solution is observed continuously on the time interval $[0,\tau]$. The proof uses ergodic properties of the equation and a Girsanov-type transform. We analyse the particular case of the fractional Ornstein-Uhlenbeck process and show that the Maximum Likelihood Estimator is asymptotically efficient in the sense of the Minimax Theorem.