Eur. Phys. J. B 34, 85-98 (2003)
https://doi.org/10.1140/epjb/e2003-00199-x

A causal multifractal stochastic equation and its statistical properties

CNRS, UMR ELICO, Station Marine, Université de Lille 1, BP 80, 62930 Wimereux, France

Corresponding author: ^a francois.schmitt@univ-lille1.fr

Received: 16 December 2002
Revised: 15 April 2003
Published online: 23 July 2003

Abstract

Multiplicative cascades have been introduced in turbulence to generate random or deterministic fields having intermittent values and long-range power-law correlations. Generally this is done using discrete construction rules leading to discrete cascades. Here a causal log-normal stochastic process is introduced; its multifractal properties are demonstrated together with other properties such as the composition rule for scale dependence and stochastic differential equations for time and scale evolutions. This multifractal stochastic process is continuous in scale ratio and in time. It has a simple generating equation and can be used to generate sequentially time series of any length.