Multistable processes, that is, processes which are, at each “time”, tangent to a stable
process, but where the index of stability varies along the path, have been recently
introduced as models for phenomena where the intensity of jumps is non constant. In this
work, we give further results on (multifractional) multistable processes related to their
local structure. We show that, under certain conditions, the incremental moments display a
scaling behaviour, and that the pointwise Hölder exponent is, as expected, related to the
local stability index. We compute the precise value of the almost sure Hölder exponent in
the case of the multistable Lévy motion, which turns out to reveal an interesting
phenomenon.