Log-infinitely divisible multifractal processes

TL;DR

This paper introduces a broad class of multifractal processes with log-infinitely divisible scaling laws, unifying previous models like MRW and log-Poisson processes, and studies their statistical properties and limits.

Contribution

It defines a new, unified framework for multifractal processes with arbitrary scaling laws, extending existing models and analyzing their statistical behavior and convergence.

Findings

Established existence of limit processes as scale approaches zero

Proved convergence of moments and multifractal scaling properties

Unified framework encompassing log-normal and log-Poisson models

Abstract

We define a large class of multifractal random measures and processes with arbitrary log-infinitely divisible exact or asymptotic scaling law. These processes generalize within a unified framework both the recently defined log-normal "Multifractal Random Walk" processes (MRW) and the log-Poisson "product of cynlindrical pulses". Their construction involves some ``continuous stochastic multiplication'' from coarse to fine scales. They are obtained as limit processes when the finest scale goes to zero. We prove the existence of these limits and we study their main statistical properties including non degeneracy, convergence of the moments and multifractal scaling.