Multifractional, multistable, and other processes with prescribed local form

Abstract

We present a general method for constructing stochastic processes with prescribed local form. Such processes include variable amplitude multifractional Brownian motion, multifractional $\alpha$-stable processes, and 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.