LAN Property for the Drift and Hurst Paramters in The Mixed Fractional O-U Process with Continuous Observations
Chunhao Cai, Cong Zhang
TL;DR
This work derives local asymptotic normality (LAN) for the joint estimation of the drift parameter $\alpha$ and the Hurst parameter $H$ in the mixed fractional Ornstein–Uhlenbeck process under the regime $H>\tfrac{3}{4}$. By leveraging the semimartingale structure of the mixed fractional Brownian motion via its optimal filtering, the authors apply a Girsanov-based innovation approach to obtain a LAN expansion with rate $\phi(T)=T^{-1/2}I(\theta_0)^{-1/2}$ and an explicit Fisher-information functional $I(\theta)$. They establish the Hájek lower bound for local minimax risk in this continuous-observation setting and provide a detailed roadmap and supporting lemmas (including Laplace-transform properties of the kernel $g(t,s;H)$) to prove LAN. A simulation discussion indicates that discretized (Whittle-type) estimators can achieve the asymptotic bound under sufficiently fine sampling, highlighting practical implications for efficient inference on long-range dependent diffusions driven by mixed fractional noise.
Abstract
This paper deals with the Local Asymptotical normality for the joint drift parameter and Hurst parameter $H>3/4$ in the mixed fractional Ornstein-Uhlenbeck process. Different from the only estimation of the drift parameter when $H$ is known, we will use the fact that the mixed fractional Brownian motion is a semimartingale with its own filtering when $H>3/4$.