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

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

2002-06-12

arxiv.org/abs/cond-mat/0206202v2

Physics

Condensed Matter

Statistical Mechanics

24 pages, 4 figures

Scientific paper

10.1103/PhysRevE.66.056121

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.

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