Multifractal random walk

Abstract

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 ...

Coupled uncertainty provided by a multifractal random walker

“…More complementary explanations can be found in [13,42,43]. Here we assume that time is a dynamical parameter, therefore x(t) is only taken to be time dependant.…”

“…For a multiplicative cascading process which starts from a large scale, L, tending to small scales, ℓ, implementation of the multifractal random walk approach enables us to rewrite the increment of fluctuation as ∆ ℓ x(t) ≡ ξ ℓ (t)e ω ℓ (t) , in which ξ ℓ (t) and ω ℓ (t) are independent of each other and have Gaussian distributions with zero means. The corresponding variances are denoted by σ 2 (ℓ) and λ 2 (ℓ) for ξ ℓ (t) and ω ℓ (t), respectively [13]. In this approach, a non-Gaussian probability density function (PDF) with fat tails is expressed by [15] …”

“…The multifractal formalism provides almost adequate tools for studying the scaling relations and properties of objects possessing a fractal geometry and/or generalized multifractal exponents [10][11][12][13][14]. Once, the infinitely divisible cascades [15][16][17] in hydrodynamic turbulence was confirmed to be statistically scale-invariant, one important motivation for multifractal formulation to become prominent was established [18,19].…”

“…This factor only depends on the scale ratio of the processes. In a further study, by developing this theory, it has been shown that a continuous random walk model can posses multifractal properties due to the existence of a correlation in the logarithm of the stochastic variances [13,17]. It is worth noting that a strong log-normal deviation from the normal case leads to a robust state of multifractality or in other words a critical state in the underlying system.…”

Properties of a simple bilinear stochastic model: Estimation and predictability

“…(FI)-GARCH (Baillie et al, 1996), α-ARCH (Diebolt and Guegan, 1991), multifractal 1 (Barral and Mandelbrot, 2002;Mandelbrot et al, 1997;Lux, 2004) or many other stochastic volatility models (Heston, 1993;Taylor, 1994) have been used to describe the nonlinear behavior associated with volatility clustering and long-term memory. Many of the stylized fact of monovariate financial returns can be captured with multiplicative nonlinear processes, of which multifractal stochastic volatility models constitute a prominent example, for instance in the form of the so-called the "multifractal random walk" (Bacry et al, 2001) and its generalizations (Pochard and Bouchaud, 2002;Bacry and Muzy, 2003). See also Muzy et al (2001) for a general multifractal multivariate generalization.…”