We study the asymptotic properties of an estimator of Hurst parameter of a stochastic differential equation driven by a fractional Brownian motion with $H > 1/2$. Utilizing the theory of asymptotic expansion of Skorohod integrals introduced by Nualart and Yoshida [NY19], we derive an asymptotic expansion formula of the distribution of the estimator. As an corollary, we also obtain a mixed central limit theorem for the statistic, indicating that the rate of convergence is $n^{-\frac12}$, which improves the results in the previous literature. To handle second-order quadratic variations appearing in the estimator, a theory of exponent has been developed based on weighted graphs to estimate asymptotic orders of norms of functionals involved.

翻译：我们研究了由分数布朗运动（$H > 1/2$）驱动的随机微分方程的Hurst参数估计量的渐近性质。利用Nualart和Yoshida [NY19]引入的Skorohod积分渐近展开理论，我们推导了该估计量分布的渐近展开公式。作为推论，我们还得到了该统计量的混合中心极限定理，表明其收敛速度为$n^{-\frac12}$，这改进了先前文献中的结果。为了处理估计量中出现的二阶二次变差，我们基于加权图发展了指数理论，以估计所涉及泛函范数的渐近阶。