Multifractal stationary random measures and multifractal random walks with log infinitely divisible scaling laws (original) (raw)

On the scaling function of multifractal processes

2018

Benôıt Mandelbrot, the father of Fractal Geometry, developed a multifractal model for describing price changes. Despite the commonly used models such as the Brownian motion, the Mutifractal Model of Asset Return (MMAR) takes into account scale-consistency, longrange dependence and heavy tails, thus having a great flexibility in depicting the real-market peculiarities. In section 2 a review of the mathematics involved into multifractals is presented; Section 3 shows how to extend multifractality to stochastic processes. Contributions in Section 4 are new in the literature and extend Mandelbrot’s results to canonical multifractal measures. Proof of Theorem 5 is unpublished and highlights which are the drivers of heavy tails in the process, thus possibly creating a bridge between multifractal formalism and jump processes.

Estimating the scaling function of multifractal measures and multifractal random walks using ratios

Bernoulli, 2014

In this paper, we prove central limit theorems for bias reduced estimators of the structure function of several multifractal processes, namely mutiplicative cascades, multifractal random measures, multifractal random walk and multifractal fractional random walk as defined by Ludeña [Ann. Appl. Probab. 18 (2008) 1138-1163]. Previous estimators of the structure functions considered in the literature were severely biased with a logarithmic rate of convergence, whereas the estimators considered here have a polynomial rate of convergence.

Limit theorems for multifractal products of geometric stationary processes

Bernoulli, 2016

We investigate the properties of multifractal products of geometric Gaussian processes with possible long-range dependence and geometric Ornstein-Uhlenbeck processes driven by Lévy motion and their finite and infinite superpositions. We present the general conditions for the L q convergence of cumulative processes to the limiting processes and investigate their q-th order moments and Rényi functions, which are nonlinear, hence displaying the multifractality of the processes as constructed. We also establish the corresponding scenarios for the limiting processes, such as log-normal, loggamma, log-tempered stable or log-normal tempered stable scenarios.

Confidence intervals for the scaling function of multifractal random walks

Statistics & Probability Letters, 2009

Multiscaling properties are typically described by multifractal random measures or multifractal random walks which are characterized by a nonlinear scaling function. In this paper we discuss statistical issues related to the estimation of this function based on the empirical structure function and provide approximate confidence intervals in certain cases.

Scaling exponents and multifractal dimensions for independent random cascades

Communications in Mathematical Physics, 1996

This paper is concerned with Mandelbrot's stochastic cascade measures. The problems of (i) scaling exponents of structure functions of the measure, τ(#), and (ii) multifractal dimensions are considered for cascades with a generator vector (wι w c ) of the general type. These problems were previously studied for independent strongly bounded variables w l : 0 < a < w/ ^ c. Consequently, a broad class of models used in applications, including Kolmogorov's log-normal model in turbulence, log-stable "universal" cascades in atmospheric dynamics, has not been covered. Roughly speaking, problems (i), (ii) are here solved under the condition that the scaling exists; the τ-function is calculated for all arguments (previously this was done for positive q) and a new effect emerges: the τ-function can generally involve discontinuities in the first derivative as well as in the second.

L p -variations for multifractal fractional random walks

The Annals of Applied Probability, 2008

A multifractal random walk (MRW) is defined by a Brownian motion subordinated by a class of continuous multifractal random measures M[0, t], 0 ≤ t ≤ 1. In this paper we obtain an extension of this process, referred to as multifractal fractional random walk (MFRW), by considering the limit in distribution of a sequence of conditionally Gaussian processes. These conditional processes are defined as integrals with respect to fractional Brownian motion and convergence is seen to hold under certain conditions relating the self-similarity (Hurst) exponent of the fBm to the parameters defining the multifractal random measure M. As a result, a larger class of models is obtained, whose fine scale (scaling) structure is then analyzed in terms of the empirical structure functions. Implications for the analysis and inference of multifractal exponents from data, namely, confidence intervals, are also provided.

Hard and soft multifractal processes

Physica A-statistical Mechanics and Its Applications, 1992

We show that multifractal notions encompass a wider variety of phenomena than often believed. Ranked by increasing highest order of singularities we have geometric, then microcanonical and finally canonical multifractals. They are respectively localized and "calm", delocalized and "calm", and delocalized and "wild". Canonical multifractals may also involve rare violent ("hard") singularities which cause high order statistical moments to diverge. We demonstrate this classification in the framework of universal multifractals characterized by three fundamental exponents.

Apparent scale correlations in a random multifractal process

The European Physical Journal B, 2008

We discuss various properties of a homogeneous random multifractal process, which are related to the issue of scale correlations. By design, the process has no built-in scale correlations. However, when it comes to observables like breakdown coefficients, which are based on a coarse-graining of the multifractal field, scale correlations do appear. In the log-normal limit of the model process, the conditional distributions and moments of breakdown coefficients reproduce the observations made in fully developed small-scale turbulence. These findings help to understand several puzzling empirical details, which have been extracted from turbulent data already some time ago.

To be and not to be : scale correlations in random multifractal processes

2006

We discuss various properties of a random multifractal process, which are related to the issue of scale correlations. By design, the process is homogeneous, non-conservative and has no built-in scale correlations. However, when it comes to observables like breakdown coefficients, which are based on a coarse-graining of the multifractal field, scale correlations do appear. In the log-normal limit of the model process, the conditional distributions and moments of breakdown coefficients reproduce the observations made in fully developed small-scale turbulence. These findings help to understand several puzzling empirical details, which have been extracted from turbulent data already some time ago.