Estimation of the multifractional function and the stability index of linear multifractional stable processes

TL;DR

This paper develops estimators for the time-varying self-similarity function and the stability index of linear multifractional stable processes, demonstrating their consistency and convergence rates.

Contribution

It introduces novel estimation methods for the multifractional function and stability index in stable processes with time-changing self-similarity.

Findings

Estimates are consistent for multifractional processes.

Convergence rates of the estimators are established.

Applicable to both multifractional Brownian motion and stable motion.

Abstract

In this paper we are interested in multifractional stable processes where the self-similarity index $H$ is a function of time, in other words $H$ becomes time changing, and the stability index $\alpha$ is a constant. Using $\beta$- negative power variations ($-1/2<\beta<0$), we propose estimators for the value of the multifractional function $H$ at a fixed time $t_0$ and for $\alpha$ for two cases: multifractional Brownian motion ($\alpha=2$) and linear multifractional stable motion ($0<\alpha<2$). We get the consistency of our estimates for the underlying processes with the rate of convergence.