Multifractal theory up to date has been concerned mostly with random and deterministic singular measures, with the notable exception of fractional Brownian motion and Lévy motion. Real world problems involved with the estimation of the singularity structure of both, measures and processes, has revealed the need to broaden the known `multifractal formalism' to include more sophisticated tools such as wavelets. Moreover, the pool of models available at present shows a gap between `classical' multifractal measures, i.e.\ cascades in all variations with rich scaling properties, and stochastic processes with nice statistical properties such as stationarity of increments, Gaussian marginals, and long-range dependence but with degenerate scaling characteristics.
This paper has two objectives, then. For one it develops the multifractal formalism in a context suitable for functions and processes. Second, it introduces truly multifractal processes, building a bridge between multifractal cascades and self-similar processes.
Keywords: Multifractal analysis, self-similar processes, fractional Brownian motion, Lévy flights, stable motion, wavelets, long-range dependence, multifractal subordinator.