Analysis and simulations of multifractal random walks

Multifractal time series, characterized by a scale invariance and large fluctuations at all scales, are found in many fields of natural and applied sciences. Here we consider a quite general type of multifractal time series, called multifractal random walk, as non stationary stochastic processes with intermittent stationary increments. We first quickly recall how such time series can be analyzed and characterized, using structure functions and arbitrary order Hilbert spectral analysis, and then we discuss the simulation approach. Here we review recent works on this topic. We provide an unification of the works published, and discuss how to choose parameters in stochastic simulations in order to simulate a multifractal series with desired properties. In the lognormal framework we provide a new h−μ plane expressing the scale invariant properties of these simulations.