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

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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 t₀ 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.