On causal stochastic equations for log-stable multiplicative cascades

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. Copyright EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2007