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

In this paper we are interested in multifractional stable processes where the self-similarity index H becomes time-dependent, while the stability index alpha remains constant. Using beta- negative power variations ( - 1/2 < beta < 0), we propose estimators for the value at a fixed time of the multifractional function H which satisfies an eta-Holder condition and for alpha in two cases: multifractional Brownian motion (alpha = 2) and linear multifractional stable motion (0 < alpha < 2). We get the consistency of our estimates for the underlying processes together with the rate of convergence.