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

TL;DR

This paper derives an asymptotic expansion and a mixed central limit theorem for a Hurst parameter estimator in stochastic differential equations driven by fractional Brownian motion with H > 1/2, improving convergence rate results.

Contribution

It introduces a new asymptotic expansion formula for the estimator's distribution and develops a graph-based theory to handle second-order variations.

Findings

Convergence rate of the estimator is $n^{-1/2}$.

Derived an asymptotic expansion formula for the estimator's distribution.

Established a mixed central limit theorem for the estimator.

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.