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

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 $α$ is a constant. Using $β$- negative power variations ($-1/2<β<0$), we propose estimators for the value of the multifractional function $H$ at a fixed time $t_0$ and for $α$ for two cases: multifractional Brownian motion ($α=2$) and linear multifractional stable motion ($0<α<2$). We get the consistency of our estimates for the underlying processes with the rate of convergence.