How to generate continuous cascade models with zero values: theory and simulations for a continuous beta-multifractal model

Abstract:

Multiplicative cascade models, when densified (continuous scale invariance) correspond to the exponential of a linear process. Hence this cannot generate zero values. Such framework is not complete and not purely multiplicative. We present here a stochastic framework which stays in the multiplicative realm and can be used to generate zero values.

The multiplicative continuous model for multifractal fields with zero values is built using infinitely multiplicative random variables, the multiplicative analog to infinitely divisible distributions for addition. It also needs stochastic multiplicative measures and multiplicative stochastic integrals. The model hence generates continuous multiplicative cascades.

The model produced possesses as special case a continuous generalization of the classical discrete beta-model. Applications are numerous in many fields of applied sciences, including smallscale rainfall, soil sciences. The theory is first proposed, then simulation algorithm is presented and simulations are shown in 1D and in 2D. Figure: a continuous lognormal multifractal with zero values (512x512).