We define a large class of continuous time multifractal random measures andprocesses with arbitrary log-infinitely divisible exact or asymptotic scalinglaw. These processes generalize within a unified framework both the recentlydefined log-normal Multifractal Random Walk (MRW) [Bacry-Delour-Muzy] and thelog-Poisson "product of cynlindrical pulses" [Barral-Mandelbrot]. Ourconstruction is based on some ``continuous stochastic multiplication'' fromcoarse to fine scales that can be seen as a continuous interpolation ofdiscrete multiplicative cascades. We prove the stochastic convergence of thedefined processes and study their main statistical properties. The question ofgenericity (universality) of limit multifractal processes is addressed withinthis new framework. We finally provide some methods for numerical simulationsand discuss some specific examples.
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