Multifractal stationary random measures and multifractal random walks with log infinitely divisible scaling laws.
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 [J.F. Muzy, J. Delour, and E. Bacry, Eur. J. Phys. B 17, 537 (2000), E. Bacry, J. Delour, and J.F. Muzy, Phys. Rev. E 64, 026103 (2001)] and the log-Poisson "product of cylindrical pulses" [J. Barral and B.B. Mandelbrot, Cowles Foundation Discussion Paper No. 1287, 2001 (unpublished)]. Our construction is based on some "continuous stochastic multiplication" [as introduced in F. Schmitt and D. Marsan, Eur. J. Phys. B. 20, 3 (2001)] 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 a method for numerical simulations and discuss some specific examples.
Muzy
Jean-François
JF
CNRS UMR 6134, Université de Corse, Quartier Grossetti, 20250 Corte, France. muzy@univ-corse.fr
Bacry
Emmanuel
E