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arXiv · 2407.02254

Asymptotic expansion of a Hurst index estimator for a stochastic differential equation driven by fBm

Hayate Yamagishi
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Abstract

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.

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Hayate Yamagishi. 2024-07-02. Asymptotic expansion of a Hurst index estimator for a stochastic differential equation driven by fBm. https://arxiv.org/abs/2407.02254

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