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SS11 - Processus multifractals et à longue mémoire / SS11 - Multifractals and Long Memory Processes
Org: J-M Azaïs (Toulouse) et/and B. Remillard (HEC, Montréal)

ANTOINE AYACHE, CMLA, ENS de Cachan, 61, Avenue du President Wilson, 94235 Cachan
Analyse par ondelettes du Drap Brownien Fractionnaire

Il existe deux extensions possibles à RN du Mouvement Brownien Fractionnaire (MBF) sur R. L'une d'elles est le Champ Brownien Fractionnaire isotrope de Lévy et l'autre est le Drap Brownien Fractionnaire (DBF) anisotrope. La covariance du DBF est un produit tensoriel de covariances de MBF. Ce champ Gaussien suscite de plus en plus d'intérêt depuis plusieurs années. Il intervient de façon naturelle dans de multiples domaines comme par exemples les équations aux dérivées partielles stochastiques et l'étude des sites les plus visités des processus de Markov symétriques.

Les décompositions en ondelettes du MBF se sont déjà avérées très utiles pour son étude. Il semble donc important d'introduire des décompositions en ondelettes du DBF. Ce problème sera traité dans la première partie de notre exposé. Sa principale difficulté provient de l'anisotropie du DBF. Dans la seconde partie de notre exposé nous donnons de nouveaux résultats concernants ce champ : module de continuité, irrégularité uniforme et estimation fine du comportement à l'infini. Ces nouveaux résultats sont obtenus au moyen de méthodes d'ondelettes.

JULIEN BARRAL, INRIA Rocquencourt, France
Multifractal Analysis of a Class of Additive Processes with Correlated Non-Stationary Increments

We consider a family of stochastic processes built from infinite sums of independent positive random functions on R+. Each of these functions increases linearly between two consecutive negative jumps, with the jump points following a Poisson point process on R+. The motivation for studying these processes stems from the fact that they constitute simplified models for TCP traffic on the Internet. Such processes bear some analogy with Lévy processes, but they are more complex in the sense that their increments are neither stationary nor independent. Nevertheless, we show that their multifractal behavior is very much the same as that of certain Lévy processes. More precisely, we compute the Hausdorff multifractal spectrum of our processes, and find that it shares the shape of the spectrum of a typical Lévy process. This result yields a theoretical basis to the empirical discovery of the multifractal nature of TCP traffic.

Joint work with J. Lévy Véhel.

HERMINE BIERMÉ, MAPMO, Université d'Orléans, Rue de Chartres BP 6759, 45067 Orléans Cedex 2
X-ray Transform of Anisotropic Models for Bones

The aim of this study is to find a parameter, easily computable, to detect osteoporosis from radiographic images. We consider two types of anisotropic models for bones: a Gaussian random field characterized by its spectral density on one side, a microball model characterized by the intensity of a Poisson measure on another side. This last one is obtained by throwing balls whose center and radius are given by a point Poisson process.

These models are anisotropic generalizations of the Fractional Brownian Motion (resp. the isotropic microball model). They are obtained through an anisotropic deformation of the spectral density (resp. intensity) of these two isotropic models. Moreover, classes of such fields are stable through X-ray transform.

In each case the anisotropy is given by a function of the direction, which one would like to recover from radiographs. Self-similarity properties seem appropriate tools for this, once one has performed an X-ray transform.

Autour de l'auto-similarité locale

La propriété d'auto-similarité locale d'un processus stochastique est une propriété qui généralise la notion classique d'auto-similarité (globale). Nous étudierons les propriétés de processus localement auto-similaires: caractérisation des processus tangents qui sont (p.p.) auto-similaires à accroissements stationnaires; dimension de Hausdorff des graphes des trajectoires; estimation du paramètre d'auto-similarité locale à partir d'une observation locale du processus.

STEPHANE JAFFARD, Université Paris 12-Val de Marne, 61 avenue du General de Gaulle, 94010 Creteil Cedex, France
Analysis of multifractal random processes: The wavelet leaders method

The purpose of multifractal analysis is to determine the dimensions of the sets of points where a signal f(t) has a given Hölder regularity. In practice, this is performed through the application of a multifractal formalism which is expected to derive these dimensions from global, numerically computable quantities. Several such quantities have been introduced in the past: In the seminal paper of Parisi and Frisch, they were based on increments of f(t); afterwards, Arneodo and his collaborators proposed to base them on the continuous wavelet transform of f; new formulas are now based on the wavelet leaders, which are local suprema of the coefficients of f on an orthonormal wavelet basis. We will compare such formulas from three points of view:

  • the general mathematical properties of each multifractal formalism;

  • theoretical and numerical results for several classes of stochastic processes;

  • examples in signal processing.

ERIC MOULINES, Ecole Nationale Supérieure des Télécommunications
Long Range Dependent Markov Chains

We present a new drift condition which implies rates of convergence to the stationary distribution of the iterates of a y-irreducible aperiodic and positive recurrent transition kernel. This condition, extending a condition introduced by Jarner and Roberts (2001) for polynomial convergence rates, turns out to be very convenient to establish subgeometric rates of convergence.

This condition allows in particular to construct nontrivial examples of Markov Chains (including nonlinear autoregressive models, stochastic unit root models etc.) showing long-range dependence p. We will in particular discuss the connection of these LRD Markovian processes with long-memory renewal processes.

DAVID NUALART, Université de Barcelona, Gran Via 585, 08007 Barcelona, Espagne
Stochastic calculus with respect to fractional Brownian motion and applications

Fractional Brownian motion (fBm) is a centered self-similar Gaussian process with stationary increments, which depends on a parameter 0 < H < 1 called the Hurst index. In this conference we will survey some recent advances in the stochastic calculus with respect to fBm. In the particular case H=1/2, the process is an ordinary Brownian motion, but otherwise it is not a semimartingale and Itô calculus cannot be used.

Different approaches have been introduced to construct stochastic integrals with respect to fBm: pathwise techniques, Malliavin calculus, approximation by Riemann sums. We will describe these methods and present the corresponding change of variable formulas. Some applications will be discussed.


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