Séminaire Probabilités et Statistique

The De Vylder and Goovaerts conjecture is an open problem in risk theory, stating that the finite time ruin probability in a standard risk model is greater or equal to the corresponding ruin probability evaluated in the associated model with equalized claim amounts. Equalized means here that the jump sizes of the associated model are equal to the average jump in the initial model between 0 and a terminal time T.
In this talk, we will consider the diffusion approximations of both the standard risk model and the associated risk model. We will prove that the associated model, when conveniently renormalized, converges in distribution to a gaussian process satisfying a simple SDE with explicit coefficients. We will then compute the probability that this diffusion hits the level 0 before time T and compare it with the same probability for the diffusion approximation for the standard risk model, which is well known. We will then conclude that the De Vylder and Goovaerts conjecture holds true for these diffusion limits.
This is a joint work with Stefan Ankirchner (University of Jena) and Christophette Blanchet-Scalliet (Ecole Centrale de Lyon and ICJ).

In a quasicrystal, the arrangement of the atoms is highly ordered (as
in an ordinary crystal) but non-periodic (unlike in a crystal). There
are various mathematical challenges in connection with quasicrystals.
From the point of view of statistical mechanics, the major open
problem is to provide a mathematical explanation of the formation and
stability of quasicrystals in presence of thermal fluctuations. In
this talk, I will present a (toy) lattice gas model with finite-range
interactions that has stable quasicrystal phases at positive
temperature (i.e., Gibbs measures supported at perturbations of
non-periodic tilings). The construction is based on old results on
cellular automata and tilings, in particular, a method of simulating
one cellular automaton with another that is resilient against noise,
and the existence of aperiodic sets of Wang tiles that are
deterministic in one direction.

The distribution of a Markov process with killing, conditioned to be
still alive at a given time, can be approximated by a Fleming-Viot
particle system. In such a system, each particle is simulated
independently according to the law of the underlying Markov process, and
branches onto another particle at each killing time. The purpose of this
talk is to present a central limit theorem for the law of the
Fleming-Viot particle system at a given time in the large population
limit. We will illustrate this result on an application in molecular
dynamics. This is a joint work with Frédéric Cérou, Bernard Delyon and
Mathias Rousset.

In the majority of solid cancers, secondary tumors (metastases) and associated complications are the main cause of death. In order to define the optimal therapeutic strategy for a given patient, one of the major current challenges is to estimate, at diagnosis, the burden of invisible metastases and how they will respond to treatments. In this talk, I will present research efforts towards the establishment of a predictive computational tool of metastatic development, with a particular emphasis on the assessment of mathematical models to empirical data (both experimental and clinical). I will first present the model’s framework, which is based on a physiologically-structured partial differential equation for the time dynamics of a population of metastases, combined to a nonlinear mixed-effects model for statistical representation of the distribution of the parameters in the population. Then, I will show results about the descriptive power of the model on data from clinically relevant ortho-surgical animal models of metastasis (breast and kidney tumors), with recent findings about differential effect of therapies between primary and secondary tumors. The talk will further be devoted to the translation of this modeling approach toward the clinical reality. Using clinical imaging data of brain metastasis from non-small cell lung cancer, several biological processes will be investigated to establish a minimal and biologically realistic model able to describe the data. Integration of this model into a biostatistical approach for individualized prediction of the model’s parameters from data only available at diagnosis will also be discussed. Together, these results represent a step forward towards the integration of mathematical modeling as a predictive tool for personalized medicine in oncology.

Nous présentons plusieurs algorithmes d’exploration de graphes aléatoires, markoviens dans le sens o๠leur implémentation est simultanée à  la construction-même du graphe par le modèle de configuration. Pour différents modèles, par des approximations fluides des processus markoviens sous-jacents, nous obtenons des estimations en grand graphe de (i) la taille de la famille indépendante maximale, avec des applications au protocole de télécommunication CSMA; (ii) la dynamique d’une épidémie de type SIR sur un réseau hétérogène et (iii) la taille d’un couplage maximal sur un grand graphe aléatoire, éventuellement orienté.

I will talk about the problem of detection of a change in mean in a Gaussian vector. A bump is a stepwise change within an interval of a given length but unknown location. We consider the problem of heterogeneous bump detection when the change occurs in mean and in variance of the observed vector and the detection of a bump for dependent observations. Minimax detectability conditions will be presented.
Joint work with A. Munk, M. Pohlmann and F. Werner.

Population dynamics of pathogens invading new territories continues to be of primary concern for both biologists and mathematicians. Extensive researches are mainly carried out throughout mathematical modeling to reconstruct the past dynamics of the alien species.We present a mechanistic-statistical approach that allows us to date and localize the invasion of an alien species and describe other epidemiological parameters as per example, the diffusion, the reproduction and the mortality parameter. The used approach is based on (i) a coupled reaction-diffusion-absorption sub-model that describes the dynamics of the epidemics in a heterogeneous domain and (ii) a stochastic sub-model that represents the observation process. Then, we will jointly estimate the initial conditions (date and site) and the epidemiological parameters using a Bayesian framework through an adaptive multiple importance sampling algorithm. We will show the results obtained in this framework on the basis of abundant post-introduction data gathered to draw up a surveillance plan on the expansion of Xylella fastidiosa, a phytopathogenic bacterium detected in South Corsica in 2015. Nevertheless, this approach could be applied to other post-emergent species in order to endorse a fast reaction.

The ovarian follicles are the basic anatomical and functional units of the ovaries, which are renewed from a
quiescent pool all along reproductive life. Follicular development involves a finely tuned sequence of growth and
maturation processes, involving complex cell dynamics. In their early stages of development, ovarian follicles are made up of a germ cell (oocyte), whose diameter increases steadily, and of surrounding proliferating somatic
cells, which are layered in a globally spherical and compact structure.
Here, we present two complementary modeling approaches dedicated to the first stages of a follicle develop-
ment, starting with the exit from the pool of quiescent (primordial) follicles leading to growth initiation, and
ending up just before the breaking of the spherical symmetry induced by the follicle cavitation (formation of
the antrum cavity).
The initiation phase is described by joint stochastic dynamics accounting for cell shape transitions (from
a flattened to a cuboidal shape) and proliferation of reshaped cells. We can derive the mean time elapsed before all cells have changed shapes and the corresponding increment in the total cell number, which is fitted
to experimental data retrieved from primordial follicles (single layered follicle with only flattened cells) and primary follicles (single layered follicles with only cuboidal cells).
The next stages, characterized by the accumulation of cell layers around the oocyte, are described by
multi-type structured models in either a stochastic or deterministic framework. We have designed a linear age-structured stochastic (Bellman-Harris branching) process ruling the changes in the number of follicular cells and their distribution into successive layers, which is inspired from the nonlinear model initially introduced in [1], as well as is deterministic counterpart (multi-dimensional Mc Kendrick Von Foerster). We have studied the large-time behavior of the models and derived explicit analytical formulas characterizing an exponential growth
of the population (Malthus parameter, asymptotic cell number moments and stable age distribution). We have
compared the theoretical and numerical outputs of the models with experimental biological data informing on follicle morphology in the ovine species (follicle and oocyte diameters, layer number and total cell number) from the primary to the pre-antral stage. In addition, in the case of age independent division rates, we have established the structural identifiability of the parameters, and estimated the parameter values fitting the cell numbers in each layer during the early stages of follicle development.

[1 ] Clément F., Michel P., Monniaux D., Stiehl T., Coupled somatic cell kinetics and germ cell growth:
mutliscale model-based insight on ovarian follicular development,
Multiscale et Modeling & Simulation
, 11(3), 719-746, 2013.

[2 ] Clément F., Robin F., Yvinec R., Analysis and calibration of a linear model for structured cell populations with unidirectional motion : Application to the morphogenesis of ovarian follicles,
Submitted. https://arxiv.or/abs/1712.05372

Nous considérons des systèmes de processus de Hawkes structurés en un nombre fini de populations, avec des interactions du type champ moyen. Ces processus décrivent les instants de décharge électrique de neurones en interactions. Nous décrivons la limite lorsque le nombre de neurones par population tend vers l’infini et nous montrons que dans certains cas, le processus limite possède des solutions périodiques qui correspondent aux orbites périodiques d’un système dynamique associé au processus limite. Nous établissons aussi des théorèmes limites qui décrivent la convergence du système de neurones à  taille finie vers ces orbites périodiques.

Ces résultats sont basés sur un travail en collaboration avec S. Ditlevsen.

In this talk, we present some inference methods in some multivariate models with
multiple unknown change-points when the target parameter is suspected to satisfy an uncertain constraint. We waive the assumptions on the error terms and establish the joint asymptotic normality of the unrestricted estimator and the restricted estimator. Further, we propose a class of shrinkage estimators that includes as a special case the unrestricted estimator, the estimator restricted as well as James-Stein type estimators. To study the performance of the proposed estimators, we generalize some classical identities underlying the multivariate Gaussian random samples or, more generally, the multivariate elliptically contoured random samples. Finally, we prove that shrinkage estimators dominate the unrestricted estimator.