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We introduce a wide family of stochastic processes that are obtained as sumsof self-similar localized "waveforms" with multiplicative intensity in thespirit of the Richardson cascade picture of turbulence. We establish theconvergence and the minimum regularity of our construction. We show that itscontinuous wavelet transform is characterized by stochastic self-similarity andmultifractal scaling properties. This model constitutes a stationary, "gridfree", extension of $\cal W$-cascades introduced in the past by Arneodo, Bacryand Muzy using wavelet orthogonal basis. Moreover our approach genericallyprovides multifractal random functions that are not invariant by time reversaland therefore is able to account for skewed multifractal models and for theso-called "leverage effect". In that respect, it can be well suited toproviding synthetic turbulence models or to reproducing the main observedfeatures of asset price fluctuations in financial markets.

Continuous cascades in the wavelet space as models for synthetic turbulence