Multifractal stationary random measures and multifractal random walks with log-infinitely divisible scaling laws
arXiv:cond-mat/0206202 · doi:10.1103/PhysRevE.66.056121
Abstract
We define a large class of continuous time multifractal random measures and processes with arbitrary log-infinitely divisible exact or asymptotic scaling law. These processes generalize within a unified framework both the recently defined log-normal Multifractal Random Walk (MRW) [Bacry-Delour-Muzy] and the log-Poisson "product of cynlindrical pulses" [Barral-Mandelbrot]. Our construction is based on some ``continuous stochastic multiplication'' from coarse to fine scales that can be seen as a continuous interpolation of discrete multiplicative cascades. We prove the stochastic convergence of the defined processes and study their main statistical properties. The question of genericity (universality) of limit multifractal processes is addressed within this new framework. We finally provide some methods for numerical simulations and discuss some specific examples.
24 pages, 4 figures
References in corpus (4)
Cited by in corpus (49)
- Multifractal analysis of financial markets
- Wavelet versus Detrended Fluctuation Analysis of multifractal structures
- Arbitrary-order Hilbert spectral analysis for time series possessing scaling statistics: a comparison study with detrended fluctuation analysis and wavelet leaders
- Multifractal Scaling of Thermally-Activated Rupture Processes
- Magnitude-Dependent Omori Law: Empirical Study and Theory
- Intermittent process analysis with scattering moments
- Multiple time scales and the exponential Ornstein-Uhlenbeck stochastic volatility model
- Extreme values and fat tails of multifractal fluctuations
- Consistency of detrended fluctuation analysis
- Generic Multifractality in Exponentials of Long Memory Processes
- Intermittency of surface layer wind velocity series in the mesoscale range
- Best attainable rates of convergence for the estimation of the memory parameter
- Self-Excited Multifractal Dynamics
- Information theory for non-stationary processes with stationary increments
- Stochastic energy-cascade model for 1+1 dimensional fully developed turbulence
- One-dimensional Langevin models of fluid particle acceleration in developed turbulence
- On Barnes Beta Distributions and Applications to the Maximum Distribution of the 2D Gaussian Free Field
- Approximated maximum likelihood estimation in multifractal random walks
- A class of spatio-temporal and causal stochastic processes, with application to multiscaling and multifractality
- The Dynamics of Financial Markets -- Mandelbrot's multifractal cascades, and beyond
- Inside singularity sets of random Gibbs measures
- Continuous cascades in the wavelet space as models for synthetic turbulence
- A Review of Conjectured Laws of Total Mass of Bacry-Muzy GMC Measures on the Interval and Circle and Their Applications
- From Rough to Multifractal volatility: the log S-fBM model
- Localization in fractal and multifractal media
- Random cascade model in the limit of infinite integral scale as the exponential of a non-stationary noise. Application to volatility fluctuations in stock markets
- Multifractal processes: Definition, properties and new examples
- On a multi-timescale statistical feedback model for volatility fluctuations
- Renewal of singularity sets of statistically self-similar measures
- Self-similar continuous cascades supported by random Cantor sets. Application to rainfall data
- Stochastic models of Lagrangian acceleration of fluid particle in developed turbulence
- On Riemann zeroes, Lognormal Multiplicative Chaos, and Selberg Integral
- Interdisciplinarity in Socio-economics, mathematical analysis and predictability of complex systems
- Random cascade models of multifractality : real-space renormalization and travelling-waves
- Quantifying and containing the curse of high resolution coronal imaging
- Estimating the scaling function of multifractal measures and multifractal random walks using ratios
- Revisiting the framework for intermittency in Lagrangian stochastic models for turbulent flows: a way to an original and versatile numerical approach
- Scale Symmetry in the Universe
- Detrended Structure-Function in Fully Developed Turbulence
- Apparent multifractality of self-similar Lévy processes
- Market memory and fat tail consequences in option pricing on the expOU stochastic volatility model
- Extremal-point density of scaling processes from fractal Brownian motion to turbulence in one dimension
- Shot noise multifractal model for turbulent pseudo-dissipation
- The Markov Switching Multi-fractal models as a new class of REM-like models in 1-dimensional space
- A Note on Moments of Limit Log Infinitely Divisible Stochastic Measures of Bacry and Muzy
- A Theory of Intermittency Differentiation of 1D Infinitely Divisible Multiplicative Chaos Measures
- Integer-valued multifractal processes
- The spatio-temporal statistical structure of the turbulent dissipation field and its stochastic representation as a Gaussian Multiplicative Chaos
- Intermittency in the small-time behavior of Lévy processes