Eur. Phys. J. B 58, 149-158 (2007)
https://doi.org/10.1140/epjb/e2007-00205-5

On causal stochastic equations for log-stable multiplicative cascades

¹ CNRS FRE 2816 ELICO, Wimereux Marine Station, Université des Sciences et Technologies de Lille - Lille 1, 28 av. Foch, 62930 Wimereux, France
² LIMOS UMR CNRS 6158-Université Blaise Pascal, 63173 Aubière, France

Corresponding authors: ^a francois.schmitt@univ-lille1.fr - ^b pchainai@isima.fr

Received: 23 March 2007
Published online: 10 August 2007

Abstract

We reformulate various versions of infinitely divisible cascades proposed in the literature using stochastic equations. This approach sheds a new light on the differences and common points of several formulations that have been recently provided by several teams. In particular, we focus on the simplification occurring when the infinitely divisible noise at the heart of such model is stable: an independently scattered random measure becomes a stable stochastic integral. In the last section we discuss the D-dimensional generalization.