We introduce a class of multifractal processes, referred to as Multifractal Random Walks (MRWs). To our knowledge, it is the first multifractal processes with continuous dilation invariance properties and stationary increments. MRWs are very attractive alternative processes to classical cascade-like multifractal models since they do not involve any particular scale ratio. The MRWs are indexed by few parameters that are shown to control in a very direct way the multifractal spectrum and the correlation structure of the increments. We briefly explain how, in the same way, one can build stationary multifractal processes or positive random measures. Multifractal models have been used to account for scale invariance properties of various objects in very different domains ranging from the energy dissipation or the velocity field in turbulent flows to financial data. The scale invariance properties of a deterministic fractal function f (t) are generally characterized by the exponents ζ q which govern the power law scaling of the absolute moments of its fluctuations, i.e.,where, for instance, one can choose m(q, l) = t |f (t + l) − f (t)| q . When the exponents ζ q are linear in q, a single scaling exponent H is involved. One has ζ q = qH and f (t) is said to be monofractal. If the function ζ q is no longer linear in q, f (t) is said to be multifractal. In the case of a stochastic process X(t) with stationary increments, these definitions are naturally extended usingwhere E stands for the expectation. Some very popular monofractal stochastic processes are the so-called selfsimilar processes [1]. They are defined as processes X(t) which have stationary increments and which verify (in law)Widely used examples of such processes are fractional Brownian motions (fBm) and Levy walks. One reason for their success is that, as it is generally the case in experimental time-series, they do not involve any particular scale ratio (i.e., there is no constraint on l or λ in Eq. (3)). In the same spirit, one can try to build multifractal processes which do not involve any particular scale ratio. A common approach originally proposed by several authors in the field of fully developed turbulence [2,3,4,5,6], has been to describe such processes in terms of differential equations, in the scale domain, describing the cascading process that rules how the fluctuations evolves when going from coarse to fine scales. One can state that the fluctuations at scales l and λl (λ < 1) are related (for fixed t) through the infinitesimal (λ = 1 − η with η << 1) cascading rulewhere W λ is a stochastic variable which depends only on λ. Let us note that this latter equation can be simply seen as a generalization of Eq. (3) with H being stochastic. Since Eq. (4) can be iterated, it implicitely imposes the random variable W λ to have a log infinitely divisible law. However, according to our knowledge, nobody has succeeded in building effectively such processes yet, mainly because of the peculiar constraints in the timescale half-plane. The integral equation ...

In this paper, we provide a simple, "generic" interpretation of multifractal scaling laws and multiplicative cascade process paradigms in terms of volatility correlations. We show that in this context 1/f power spectra, as recently observed in Ref.[20], naturally emerge. We then propose a simple solvable "stochastic volatility" model for return fluctuations. This model is able to reproduce most of recent empirical findings concerning financial time series: no correlation between price variations, long-range volatility correlations and multifractal statistics. Moreover, its extension to a multivariate context, in order to model portfolio behavior, is very natural. Comparisons to real data and other models proposed elsewhere are provided.

We present an experimental study of velocity statistics for a partial layer of inelastic colliding beads driven by a vertically oscillating boundary. Over a wide range of parameters (accelerations 3-8 times the gravitational acceleration), the probability distribution P (v) deviates measurably from a Gaussian for the two horizontal velocity components. It can be described by P (v) ∼ exp(−|v/v c | 1.5 ), in agreement with a recent theory. The characteristic velocity v c is proportional to the peak velocity of the boundary. The granular temperature, defined as the mean square particle velocity, varies with particle density and exhibits a maximum at intermediate densities. On the other hand, for free cooling in the absence of excitation, we find an exponential velocity distribution. Finally, we examine the sharing of energy between particles of different mass. The more massive particles are found to have greater kinetic energy. PACS: 83.70.Fn, 05.20.Dd, 83.10.Pp Typeset using REVT E X 1 We determine the statistical properties of particles in a vibrated granular medium experimentally. While many similarities between ordinary gases and excited granular media have been found, a fundamental difference is that collisions between particles in granular matter are inelastic. As a consequence, the velocity distribution deviates measurably from a Gaussian, but can be described by P (v) ∼ exp(−|v/v c | 1.5 ) for a large range of parameters where external excitation is sufficiently frequent, in agreement with a recent theory.

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