  We introduce a wide family of stochastic processes that are obtained as sums
of self-similar localized "waveforms" with multiplicative intensity in the
spirit of the Richardson cascade picture of turbulence. We establish the
convergence and the minimum regularity of our construction. We show that its
continuous wavelet transform is characterized by stochastic self-similarity and
multifractal scaling properties. This model constitutes a stationary, "grid
free", extension of $\cal W$-cascades introduced in the past by Arneodo, Bacry
and Muzy using wavelet orthogonal basis. Moreover our approach generically
provides multifractal random functions that are not invariant by time reversal
and therefore is able to account for skewed multifractal models and for the
so-called "leverage effect". In that respect, it can be well suited to
providing synthetic turbulence models or to reproducing the main observed
features of asset price fluctuations in financial markets.
