Random Cascade Processes Research Articles

Model of Continuous Random Cascade Processes in Financial Markets

This article presents a continuous cascade model of volatility formulated as a stochastic differential equation. Two independent Brownian motions are introduced as random sources triggering the volatility cascade: one multiplicatively combines with volatility; the other does so additively. Assuming that the latter acts perturbatively on the system, the model parameters are estimated by the application to an actual stock price time series. Numerical calculation of the Fokker–Planck equation derived from the stochastic differential equation is conducted using the estimated values of parameters. The results reproduce the probability density function of the empirical volatility, the multifractality of the time series, and other empirical facts.

Complete multiplier statistics explained by stochastic cascade processes

Abstract The prevalent understanding of fully developed turbulence is that of a cascade in which vortices successively break up into smaller ones. Based on this idea, theoretical developments were often concerned with multiplicative random cascade processes. We focus on the velocity fluctuations and derive a closed as well as complete statistical description of the underlying velocity multipliers using a Fokker–Planck equation, which is estimated directly from experimental data. This shed new light on the statistics of multipliers and their often assumed independence. For the heavy-tailed statistical features of the multipliers, close to a Cauchy distribution, no intermittency of turbulence is needed.

Analysis and modeling of solute transport dynamics by breakdown coefficients and random cascades

The small‐scale variability and scaling properties of solute transport dynamics were investigated by laboratory experiments. Dye‐stained water was applied at a constant flux in a Plexiglas Hele‐Shaw cell filled with soil material, and the transport process was registered by digital photographs of the cell front, producing a very high resolution in both time (minutes) and space (millimeters). The experiments comprised different cell materials (uniform and natural sand) and different water fluxes. Scaling properties of vertically integrated dye mass distributions were analyzed and modeled using so‐called breakdown coefficients (BDCs), which represent the multiplicative weights in a microcanonical random cascade process. The pdfs of BDCs varied with scale, indicating self‐affine scaling, and were accurately approximated by beta distributions with a scaling parameter. Two versions of BDC‐based random cascade models were used to simulate mass distributions at different time steps. The results support the applicability of random cascade models to subsurface transport processes.

Multiscale Probability Density Function Analysis: Non-Gaussian and Scale-Invariant Fluctuations of Healthy Human Heart Rate

For a detailed characterization of intermittency and non-Gaussianity of human heart rate, we introduce an analysis method to investigate the deformation process of the probability density function (PDF) of detrended increments when going from fine to coarse scales. To characterize the scale dependence of the multiscale PDF, we use two methods: 1) calculation of Kullback-Leibler relative entropy; 2) parameter estimation based on Castaing's equation (B. Castaing et al, 1990). We compare scale-dependence of the increment PDFs between actual heart rate fluctuations and artificially generated Gaussian and non-Gaussian noise, including a widely used autoregressive model and a recently proposed multifractal model based on a random cascade process. Our analysis highlights an essential difference between heart rate fluctuations and those generated by other models. The outstanding feature of human heart rate is the robust scale-invariance of the non-Gaussian PDF, which is preserved not only in a quiescent condition, but also in a dynamic state during waking hours, in which the mean level of heart rate is dramatically changing. Our results strongly suggest the need for revising existing models of heart rate variability to incorporate the scale-invariance in the PDF.

Multiscaling analysis and random cascade modeling of dye infiltration

We aimed at investigating whether the spatial variability of infiltration in field soils, as visualized through dye infiltration experiments, is characterized by a multiscaling behavior. Digitized high‐resolution dye images from three sites in an experimental catchment in Tunisia were analyzed using three indicators of scaling: empirical probability distribution functions, power spectra, and raw statistical moments. The two former indicators suggested a general scaling behavior of the data, which through the moments' analysis was found to be of multiscaling type. Random cascade processes are frequently used to model multiscaling processes, and we fitted the “universal multifractal” (UM) model of Schertzer and Lovejoy [1987] to our data. The UM model closely reproduced the empirical K(q) functions, and simulated fields reproduced key features in the observed ones. The results indicate that multiscaling random cascade modeling is useful for statistically describing flow processes and solute transport under field conditions.

Left-sided multifractality in a binary random multiplicative cascade.

In this paper we study a binary random multiplicative cascade. Specifically, the cascade is used to produce and study left-sided multifractal random measures. Extensive numerical simulations of the random cascade process were undertaken and f(alpha) spectra obtained and compared with the analytical results. We believe that this model and approach can serve as a simple and fundamental tool in the analysis and understanding of physical systems possessing an underlying multiplicative structure.

Tests of a space‐time model of daily rainfall in southwestern Australia based on nonhomogeneous random cascades

A space‐time rainfall model is constructed to generate synthetic fields of space‐time rainfall, with a daily time step. The model has two components: a temporal model based on a first‐order, four‐state discrete Markov chain which generates a daily time series of the regionally averaged rainfall and a spatial model based on a nonhomogeneous random cascade process which disaggregates the above regionally averaged rainfall amounts to produce spatial patterns of daily rainfall across the region. The cascade generators used to disaggregate the rainfall spatially are a product of stochastic and deterministic factors; the latter enable the model to capture systematic spatial gradients exhibited by measured data. The model is applied to a 400 km × 400 km region encompassing the Swan‐Avon River Basin in the southwest of Western Australia. The model parameters are estimated on the basis of 11 years of observed daily rainfall data from 490 rain gauges located in the region. A detailed testing of the model is undertaken on the basis of a comparison of the statistical characteristics of the spatial and temporal variability of rainfall between the rainfall fields obtained from the rain gauge network and those generated by the simulation model. It was found that the spatial and temporal characteristics of the simulated rainfall field are in very good agreement with those of observed rainfall. In particular, the model preserves the observed systematic spatial variations of daily, monthly, and annual rainfall across the region and the observed seasonal variations. Statistical characteristics of storm durations and interstorm periods at individual gauge locations are not reproduced that well, which suggests generalizations need to be made to include space‐time correlations.

Analysis of Random Cascades Using Space-Scale Correlation Functions

We introduce a formalism that allows us to study space-scale correlations in multiscale processes. This method, based on the wavelet transform, is particularly well suited to study multiplicative random cascade processes for which the correlation functions take a very simple expression. This two-point space-scale statistical analysis is illustrated on some pedagogical examples and then applied to fully developed turbulence data.

Experimental Analysis of Self-Similarity and Random Cascade Processes: Application to Fully Developed Turbulence Data

In the context of fully developed turbulence, Castaing et al. [10] have recently advocated a description of a random cascade process in terms of a kernel G aa , (x) that charac-terizes the nature of the cascade when going from a scale a' to a finer scale a. We elaborate on a method to estimating, directly from experimental data, the shape of G aa , for all scales a and a'. We apply this method to turbulent velocity data and we show that it provides very instructive informations about the soundness of various phenomenological models for the intermittency character of turbulent flows.

The importance of phase space dimension in the intermittency analysis of multi hadron production

It is shown for a simple model (“α-model”) realizing a self-similar random cascade process in d=2 (or d=3) dimensions of momentum space that intermittency of the multiplicity distribution occurs in the d-dimensional analysis but disappears in the one-dimensional projection. This demonstrates the importance of multidimensional analysis for the interpretation of data in terms of intermittency. However, in a variety of two- and three-dimensional cascade models a power-type relation between moments remains for the one dimensional projection into rapidity space and these relations are supported by data from all collision processes in the full rapidity range. The difference to the negative binomial description is pointed out.