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In this paper, we consider the problem of estimating the drift parameter ofsolution to the stochastic differential equation driven by a fractionalBrownian motion with Hurst parameter less than $1/2$ under completeobservation. We derive a formula for the likelihood ratio and prove localasymptotic normality when $H \in (1/4,1/2)$. Our result shows that theconvergence rate is $T^{-1/2}$ for the parameters satisfying a certain equationand $T^{-(1-H)}$ for the others.

LAN property for stochastic differential equations driven by fractional Brownian motion of Hurst parameter $H\in(1/4,1/2)$