Multifractional, multistable, and other processes with prescribed local form

TL;DR

This paper introduces a general method for constructing stochastic processes with specific local behaviors, including multifractional and multistable processes with variable local properties.

Contribution

It provides a unified framework for creating processes with prescribed local forms, encompassing multifractional Brownian motion and multistable processes with variable stability and self-similarity indices.

Findings

Constructed processes with desired local stability and self-similarity properties.

Unified approach applicable to a wide class of stochastic processes.

Demonstrated the flexibility in modeling complex local behaviors.

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.