Multifractal stationary random measures and multifractal random walks with log-infinitely divisible scaling laws

TL;DR

This paper introduces a broad class of continuous-time multifractal measures and processes with log-infinitely divisible scaling laws, unifying and extending previous models like the log-normal MRW and log-Poisson processes.

Contribution

It develops a unified framework for multifractal measures using continuous stochastic multiplication, generalizing existing models and analyzing their statistical properties and convergence.

Findings

Proved stochastic convergence of the processes.

Established universality of limit multifractal processes.

Provided methods for numerical simulation and specific examples.

Abstract

We define a large class of continuous time 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 (MRW) [Bacry-Delour-Muzy] and the log-Poisson "product of cynlindrical pulses" [Barral-Mandelbrot]. Our construction is based on some ``continuous stochastic multiplication'' from coarse to fine scales that can be seen as a continuous interpolation of discrete multiplicative cascades. We prove the stochastic convergence of the defined processes and study their main statistical properties. The question of genericity (universality) of limit multifractal processes is addressed within this new framework. We finally provide some methods for numerical simulations and discuss some specific examples.