Estimation of the Hurst and the stability indices of a $H$-self-similar stable process

TL;DR

This paper develops consistent estimators for the Hurst and stability indices of H-self-similar stable processes using $eta$-variations, providing asymptotic normality results for certain processes.

Contribution

It introduces a novel estimation method based on $eta$-variations for H-self-similar stable processes, with proven consistency and asymptotic normality for key cases.

Findings

Consistent estimators with convergence rates for H and alpha.

Asymptotic normality established for fractional Brownian motion and Lévy motion.

Applicable to various classical H-sssi alpha-stable processes.

Abstract

In this paper we estimate both the Hurst and the stable indices of a H-self-similar stable process. More precisely, let $X$ be a $H$-sssi (self-similar stationary increments) symmetric $\alpha$-stable process. The process $X$ is observed at points $\frac{k}{n}$, $k=0,\ldots,n$. Our estimate is based on $\beta$-variations with $-\frac{1}{2}<\beta<0$. We obtain consistent estimators, with rate of convergence, for several classical $H$-sssi $\alpha$-stable processes (fractional Brownian motion, well-balanced linear fractional stable motion, Takenaka's processes, L\'evy motion). Moreover, we obtain asymptotic normality of our estimators for fractional Brownian motion and L\'evy motion. Keywords: H-sssi processes; stable processes; self-similarity parameter estimator; stability parameter estimator.