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LAN property for stochastic differential equations with additive fractional noise and continuous time observation
Yanghui Liu, Eulalia Nualart, Samy Tindel · 2015-09-01 · via math.PR updates on arXiv.org

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 $\sqrtτ$ as $τ\rightarrow \infty$, when the solution is observed continuously on the time interval $[0,τ]$. 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.

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