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 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.