Multi-Fractal Spectral Analysis of the 1987 Stock Market Crash

The multifractal model of asset returns captures the volatility persistence of many financial time series. Its multifractal spectrum computed from wavelet modulus maxima lines provides the spectrum of irregularities in the distribution of market returns over time and thereby of the kind of uncertainty or randomness in a particular market. Changes in this multifractal spectrum display distinctive patterns around substantial market crashes or drawdowns. In other words, the kinds of singularities and the kinds of irregularity change in a distinct fashion in the periods immediately preceding and following major market drawdowns. This paper focuses on these identifiable multifractal spectral patterns surrounding the stock market crash of 1987. Although we are not able to find a uniquely identifiable irregularity pattern within the same market preceding different crashes at different times, we do find the same uniquely identifiable pattern in various stock markets experiencing the same crash at the same time. Moreover, our results suggest that all such crashes are preceded by a gradual increase in the weighted average of the values of the Lipschitz regularity exponents, under low dispersion of the multifractal spectrum. At a crash, this weighted average irregularity value drops to a much lower value, while the dispersion of the spectrum of Lipschitz exponents jumps up to a much higher level after the crash. Our most striking result, therefore, is that the multifractal spectra of stock market returns are not stationary. Also, while the stock market returns show a global Hurst exponent of slight persistence 0.5Financial Markets, Persistence, Multi-Fractal Spectral Analysis, Wavelets

Multifractal analysis of complex random cascades

We achieve the multifractal analysis of a class of complex valued statistically self-similar continuous functions. For we use multifractal formalisms associated with pointwise oscillation exponents of all orders. Our study exhibits new phenomena in multifractal analysis of continuous functions. In particular, we find examples of statistically self-similar such functions obeying the multifractal formalism and for which the support of the singularity spectrum is the whole interval $[0,\infty]$.Comment: 37 pages, 8 figure

A survey on prescription of multifractal behavior

International audienceMultifractal behavior has been identified and mathematically established for large classes of functions, stochastic processes and measures. Multifractality has also been observed on many data coming from Geophysics, turbulence, Physics, Biology, to name a few. Developing mathematical models whose scaling and multifractal properties fit those measured on data is thus an important issue. This raises several still unsolved theoretical questions about the prescription of multifractality (i.e. how to build mathematical models with a singularity spectrum known in advance), typical behavior in function spaces, and existence of solutions to PDEs or SPDEs with possible multifractal behavior. In this survey, we gather some of the latest results in this area. Dedicated to Alain Arnéodo, pioneer in the development of wavelet tools for data analysis

Function spaces vs. Scaling functions: Some issues in image classification

Criteria based on the computation of fractal dimensions have been used in order to perform image analysis and classification; we show that such criteria often amount to deter- mine the regularity of the image in some classes of function spaces, and that looking for richer criteria naturally leads to the introduction of new classes of function spaces. We will investigate the properties of some of these classes, and show which type of additional information they yield for the initial image

Experimental Evidence for Statistical Scaling and Intermittency in Sediment Transport Rates

Understanding bed load transport fluctuations in rivers is crucial for development of a transport theory and for choosing a sampling interval for “mean” transport rates. Field-scale studies lack sufficient resolution to statistically characterize these fluctuations, while laboratory experiments are limited in scale and hence cannot be directly compared to field cases. Here we use a natural-scale laboratory channel to examine bed load transport fluctuations in a heterogeneous gravel substrate under normal flow conditions. The novelty of our approach is the application of a geometrical/statistical formalism (called the multifractal formalism), which allows characterization of the “roughness” of the series (depicting the average strength of local abrupt fluctuations in the signal) and the “intermittency” (depicting the temporal heterogeneity of fluctuations of different strength). We document a rougher and more intermittent behavior in bed load sediment transport series at low-discharge conditions, transitioning to a smoother and less intermittent behavior at high-discharge conditions. We derive an expression for the dependence of the probability distribution of bed load sediment transport rates on sampling interval. Our findings are consistent with field observations demonstrating that mean bed load sediment transport rate decreases with sampling time at low-transport conditions and increases with sampling time at high-transport conditions. Simultaneous measurement of bed elevation suggests that the statistics of sediment transport fluctuations are related to the statistics of bed topography

Common turbulent signature in sea surface temperature and chlorophyll maps

5 pages, 3 figuresOceans and 2D turbulence present similar characteristics, as for instance the dominant role played by eddies in energy and matter transport. However, providing a complete justification of this analogy is difficult, as it requires knowledge of the ocean's dynamic state at different instants and over large scales. Recently, new techniques coming from the Microcanonical Multifractal Formalism have made it possible to infer the streamlines from the analysis of satellite images of some scalar variables. In this paper, we will show that this information is enough to characterize the scaling properties of the energy cascade, which is manifested as a multifractal signature; further, the multifractal signature is obtained at each location in a local basis. Different scalars obtained from satellite measurements such as Sea Surface Temperature or Surface Chlorophyll Concentration present essentially the same multifractal structure, which is interpreted as a consequence of the pervading character of the turbulent advection at the scales of observationA. Turiel is contracted under the Ramón y Cajal program by the Spanish Ministry of Education. V. Nieves is supported by a Ph.D. grant funded by MERSEA. C. Llebot is supported by a Ph.D. grant funded by CSIC, J. Solé is supported by a post-doc grant funded by EU Streps Project SEEDS. This work is a contribution to the MERSEA project. Partial support from the European Commission under contract SIP3-CT-2003-502885 is gratefully acknowledged. This work is also a contribution to the Spanish projects ESEOO (VEM2003- 20577-C14-10), MIDAS-4 (ESP2005-06823-C05-1) and OCEANTECH (PIF-2006)Peer reviewe