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

University of Economics, The University of Danang, 71 Ngu Hanh Son street, 550000 Danang, Viet Nam.

^* Corresponding author: nhudtt@due.edu.vn

Received: 16 September 2018
Accepted: 19 June 2019

Abstract

In this paper we are interested in multifractional stable processes where the self-similarity index H becomes time-dependent, while the stability index α remains constant. Using β- negative power variations ( − 1∕2 < β < 0), we propose estimators for the value at a fixed time of the multifractional function H which satisfies an η-Hölder condition and for α in 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 together with the rate of convergence.