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

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