Journées à la mémoire de Marc Yor
du 3 au 5 juin 2015
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Mercredi 3 juin
Amphi 15
Accueil dès 8h30
9h00 - 9h10
F. Comets
Introduction par le directeur du LPMA
9h10 - 9h50
J. Pitman
Martingale coupling of cumulative hazard and exponential variables by Azéma-Yor embedding in Brownian motion
Let $H(x) = \int_{- \infty}^x dF(y)/(1 - F(y-)$ be the cumulative hazard function derived from the cumulative distribution function $F$ of a real-valued random variable $X$. The Az\'ema-Yor embedding of $H(X)-1$ in Brownian motion couples $X$ with a standard exponential variable $T$ so that $H(X) = E(T|X)$ and $H(X) - H(X-) = E( T - H(X) )^2 | X )$. This construction extends to the compensator $(H(t), t \ge 0 )$ of an indicator process $1(X \le t)$ relative to a filtration with respect to which $X$ is a stopping time, and interprets some inequalities for compensators due to Meyer and Neveu.
Martingale coupling of cumulative hazard and exponential variables by Azéma-Yor embedding in Brownian motion
9h50 - 10h30
J. Bertoin
Markovian growth-fragmentation processes
Consider a Markov process ${X}$ on $[0,\infty)$ which has only negative jumps and converges at infinity a.s. We interpret $X(t)$ as the size of a typical cell at time $t$, and each jump as a birth event. More precisely, if $\Delta {X}(s)=-y<0$, then $s$ is the birth at time of a daughter cell with size $y$ which then evolves independently and according to the same dynamics. We provide a simple criterion on ${X}$ to ensure that the family of cell sizes at time $t$ can be ranked in the non-increasing order. The case when ${X}$ is self-similar is treated in details, and connexions with self-similar fragmentations and compensated fragmentations are emphasized.
Markovian growth-fragmentation processes
Pause
11h10 - 11h50
C. Donati-Martin
Spectrum of large deformed Wigner matrices
I will review recent results on the largest eigenvalues of large deformed Wigner matrices. Free probability plays a key rôle in the understanding of the limiting behaviour of the spectrum.
Spectrum of large deformed Wigner matrices
11h50 - 12h30
J.P. Kahane
L'équation de Langevin
Historique et actualité : la vitesse des particules browniennes
L'équation de Langevin
Déjeuner
14h00 - 14h40
T. Lyons
14h40 - 15h20
M. Emery
Geometric structure of multidimensional Azéma martingales
To each Azéma martingale is associated a family of orthogonal subspaces of the state space. They contain all jumps of the process, and in each of them the possible jumps are structured according to a pattern which obeys precise rules.
Geometric structure of multidimensional Azéma martingales
15h20 - 16h00
P. Salminen
Perpetual integral functionals of diffusions
This talk is a short survey of the work made mainly with Marc Yor on perpetual integral functionals of diffusions. Our main interest was first focused on deriving conditions for the finiteness of perpetual integral functionals of Brownian motion and, later on, of more general 1-dimensional diffusions. The approach is much based on the random time change techniques and was introduced by Marc Yor when analyzing perpetual exponential functionals of Brownian motion in his paper published year 1992 in JAP. On our agenda was also to provide new explicit examples as well as to study some known ones from this new perspective. In my talk I discuss, in particular,
an integral test for the finiteness of perpetual functionals,
explicit results for one-sided exponential functionals.
Perpetual integral functionals of diffusions
Pause
16h40 - 17h20
P. Vallois
Generalization of the Matsumoto-Yor independence properties. Relations between gamma, hypergeometric and Kummer's distributions
We define Letac–Wesolowski–Matsumoto–Yor (LWMY) functions as decreasing functions from (0,∞) onto (0,∞) with the following property: there exist independent, positive random variables X and Y such that the variables f (X + Y) and f (X) − f (X + Y) are independent. We prove that, under additional assumptions, there are essentially four such functions. The first one is f (x) = 1/x. In this case, referred to in the literature as the Matsumoto–Yor property, the law of X is generalized inverse Gaussian while Y is gamma distributed. In the three other cases, the associated densities are provided. As a consequence, we obtain new relations of convolution involving gamma, hypergeometric and Kummer’s of type 2 distributions.
Generalization of the Matsumoto-Yor independence properties. Relations between gamma, hypergeometric and Kummer's distributions
17h20 - 18h00
D. Dufresne
Yor’s contribution to Asian option pricing
Yor’s contributions to the problem of pricing Asian options go back to the early 1990s, with his first papers on the integral of geometric Brownian motion. This mathematical problem was not new but Yor made outstanding progress in its study right from the start, finding an integral expression for the density plus the fundamental expression for the law of the integral from 0 to an independent exponential variable. This result, together with the Bougerol identity, implies almost everything we know about the integral of geometric Brownian motion and hence about pricing options on the continuous average stock price. One consequence is the Geman-Yor Laplace transform, which may be used for numerical computations. More recent consequences will be given.
Yor’s contribution to Asian option pricing
Violin: Thierry Lévy, Piano: Joseph Najnudel
Frédéric Chopin, Barcarolle
Joseph Najnudel, Pièces pour piano
César Franck, Violin sonata
Allegretto moderato
Allegro
Recitativo-Fantasia
Allegretto poco mosso
18h30 Concert Amphi 25
Jeudi 4 juin
Amphi 25 puis 15
9h00 - 9h40
M. Barlow
Yor's conjectures on the structure of filtrations
I will talk about some conjectures Marc Yor made on the properties of filtrations at our first meeting in 1978. Some of these questions are still open.
Yor's conjectures on the structure of filtrations
9h40 - 10h20
P. Deheuvels
Extending Lévy's formula
A well-known formula of Paul Lévy gives the Laplace transform of the L2(0,1) norm of the Wiener process W(.) conditional on W(1)=x. We provide some extensions of this result to more general Gaussian processes.
Extending Lévy's formula
Pause
11h00 - 11h40
P. Biane
Polynomials associated with finite Markov chains
The Kirchoff matrix tree theorem, which relates the covering trees of a graph to the minors of its Laplacian matrix, can be interpreted probabilistcally by lifting a finite Markov chain to its set of covering trees. This leads us to associate a polynomial to any finite Markov chain, and we prove a remarkable factorization of this polynomial.
This is based on joint work with G. Chapuy.
Polynomials associated with finite Markov chains
11h40 - 12h20
A. Vershik
Standard filtrations and superstrong convergence of decreasing martingales
We define a class of filtrations (=decreasing sequences of sigma-fields) which called as "standard filtrations". The characteristic property of such filtrations is some kind of uniformicity of the convergence of the sequences of conditonal expectations for each random variable. For dyadic (or homogeneous) filtrations the standarness means that this is the tail filtration of independent random variables; in general case the standardness is the new kind of weak dependence. This notion is crucial for several areas (ergodic theory, invariant measures, Kolmogorov {0-1}-Law etc.
Standard filtrations and superstrong convergence of decreasing martingales
Déjeuner (changement d'amphi: 25 vers 15)
14h00 - 14h30
N. El Karoui
Les martingales browniennes sont-elles négociables ?
14h30 - 15h10
H. Föllmer
Weak Brownian Motions: An excursion guided by Marc Yor
A weak Brownian motion of order k has the same k-dimensional marginals as Brownian motion. Depending on the order, weak Brownian motions share many pathwise properties of Brownian motion, even though their law may be singular to Wiener measure. We review and comment on results obtained jointly with Marc Yor and Ching Tang Wu concerning the construction and the properties of weak Brownian motion of arbitrary order.
Weak Brownian Motions: An excursion guided by Marc Yor
15h10 - 15h50
M. Jeanblanc
Martingale representation property in progressively enlarged filtrations
We consider a filtration $\mathbb{G}$ which is the progressive enlargement of a filtration $\mathbb{F}$ with a random time $\tau$. Assuming that, in $\mathbb{F}$, the martingale representation property holds, we examine conditions under which the martingale representation property holds also in $\mathbb{G}$. This work is motivated by the development of models on the credit risk, which do not always satisfy the classical assumptions of enlargements of filtrations. A general methodology will be introduced in this paper, which extends the various classical results and covers the recent examples.
Martingale representation property in progressively enlarged filtrations
Pause
16h20 - 17h00
M. Rosenbaum
Testing the finiteness of the support of a distribution: a statistical look at Tsirelson’s equation
We consider the following statistical problem: based on an i.i.d. sample of size $n$ of integer valued random variables with common law $\nu$, is it possible to test whether or not the support of $\nu$ is finite as $n$ goes to infinity? This question is in particular connected to a simple case of Tsirelson’s equation, for which Marc Yor showed that it is natural to distinguish between two main configurations, the first one leading only to laws with finite support, and the second one including laws with infinite support. We show that it is in fact not possible to discriminate between the two situations, even using a very weak notion of statistical test. This is joint work with Sylvain Delattre.
Testing the finiteness of the support of a distribution: a statistical look at Tsirelson’s equation
17h00 - 17h40
D. Madan
Valuing stochastic perpetuities in booming and depressed markets
18h00 Cocktail Tour Zamansky, 24ème étage
Vendredi 5 juin
Amphi 25
9h00 - 9h40
D. Elworthy
Redundant Noise
I came to appreciate the relevance of redundant noise in conversations with Marc Yor. Here I will describe how it has been used and mention some recent developments.
Redundant Noise
9h40 - 10h20
J.F. Le Gall
Géométrie aléatoire sur la sphère
Considérons une triangulation aléatoire de la sphère choisie uniformément au hasard dans l'ensemble des triangulations avec un nombre fixé n de faces. L'ensemble des sommets de cette triangulation, vu comme un espace métrique pour la distance de graphe convenablement renormalisée, converge en loi quand n tend vers l'infini, au sens de la distance de Gromov-Hausdorff, vers un espace métrique compact aléatoire appelé la carte brownienne. Nous décrirons divers résultats suggérant que la carte brownienne est un modèle universel de géométrie aléatoire. Si le temps le permet, nous présenterons aussi des résultats récents obtenus en collaboration avec N. Curien, montrant que la carte brownienne apparaît encore si on affecte aux arêtes de la triangulation des longueurs aléatoires (modèle de percolation de premier passage).
Géométrie aléatoire sur la sphère
Pause
11h00 - 11h40
P. Bourgade
Ergodicite quantique pour des matrices aleatoires
Pour divers modeles de matrices aleatoires, j'expliquerai comment la distribution des vecteurs propres peut etre etudiee via l'etude de marches aleatoires en environnement aleatoire dynamique. L'ergodicite quantique ainsi obtenue (distribution uniforme sur la sphere de ces etats propres) est un outil pour comprendre la distribution des valeurs propres.
Ergodicite quantique pour des matrices aleatoires
11h40 - 12h20
B. Bru
Marc Yor et Wolfgang Doeblin
On rappellera la rencontre de Marc et de Wolfgang, une rencontre qui ne doit rien au hasard, mais qui reste énigmatique à bien des égards. On tentera une approche indirecte, en évoquant, en particulier, la figure de Pierre de Jean Olivi, l’un des grands théoriciens du juste prix, qui permet peut-être d’aller plus loin, sans qu’on prétende en rien atteindre ni même approcher le fond des choses.
Marc Yor et Wolfgang Doeblin
Déjeuner
14h00 - 14h40
G. Peccati
A Weak Wick Version of the Gaussian Product Conjecture
We show how to establish a new class of moment inequalities, involving Hermite-type transformations of arbitrary Gaussian vectors in any dimension. This result represents an extension of estimates by Frenkel (2007), and is tightly connected to the so-called 'Real Linear Polarization Problem' arising in linear algebra. Our approach is based on semigroup interpolations. Based on joint work with D. Malicet, I. Nourdin and G. Poly.
A Weak Wick Version of the Gaussian Product Conjecture
14h40 - 15h20
N. O'Connell
Exponential functionals and Toda
In the late 90’s, while exploring process-level extensions of Dufresne’s identity, Matsumoto and Yor discovered a version of Pitman’s 2M-X theorem for exponential functions of Brownian motion. I will explain how their result is related to an integrable Hamiltonian system which was introduced by Toda in the mid 60’s, and briefly describe some higher-dimensional generalisations with applications to random polymers.
Exponential functionals and Toda
15h20 - 16h00
A. Nikeghbali
A new approach to ratios and related conjectures for the Riemann zeta function
We study the limiting behavior of ratios of characteristic polynomials of random unitary matrices at the microscopic scale and we deduce some new and/or simplified conjectures for the value distribution of the Riemann zeta function on the critical line.
A new approach to ratios and related conjectures for the Riemann zeta function
Pause
16h40 - 17h20
Y. Hu
Randomly biased random walks on trees
This is based on a joint work with Zhan Shi. We are interested in the randomly biased random walk on the supercritical Galton--Watson tree. Our attention is focused on a slow regime when the biased random walk $(X_n)$ is null recurrent, making a maximal displacement of order of magnitude $(\log n)^3$ in the first $n$ steps. We study the localization problem of $X_n$ and prove that the quenched law of $X_n$ can be approximated by a certain invariant probability depending on $n$ and the random environment. As a consequence, we establish that upon the survival of the system, $\frac{|X_n|}{(\log n)^2}$ converges in law to some non-degenerate limit on $(0, \infty)$ whose law is explicitly computed.
Randomly biased random walks on trees
17h20 - 18h00
J. Najnudel
Limiting random operators for the circular unitary ensemble
In this talk, we provide a construction of a flow of infinite dimensional random operators constructed from the space of random virtual isometries which are stochastic processes on the infinite unitary group.
Limiting random operators for the circular unitary ensemble