Multifractional, multistable, and other processes with prescribed local form

We present a general method for constructing stochastic processes with prescribed local form, encompassing examples such as variable amplitude multifractional Brownian and multifractional alpha-stable processes. We apply the method to Poisson sums to construct multistable processes, that is, processes that are locally alpha(t)-stable but where the stability index alpha(t) varies with t. In particular we construct multifractional multistable processes, where both the local self-similarity and stability indices vary.