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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
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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
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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
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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.
- JACQUES ISTAS, UPMF
Autour de l'auto-similarité locale
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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
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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
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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
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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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