From Self-Similarity to Local Self-Similarity: the Estimation Problem

In this article we review some methods used to identify the order H of a fractional Brownian motion. This discussion is introduced to see how such techniques can be extended to locally self-similar processes. Moreover the model of the multifractional Brownian motion which is a locally self-similar process is further studied. In particular it is shown that its presentation given by J. Levy Vehel and R. Peltier and the one given by A. Benassi, S. Jaffard and D. Roux are in some sense Fourier transformed of each other. Last some results for the estimation of the multifractionnal Brownian motion are recalled.