Publications

2012

• Birth and death processes with neutral mutations
  
  • Champagnat Nicolas
  • Lambert Amaury
  • Richard Mathieu
  International Journal of Stochastic Analysis, Hindawi, 2012, 2012, pp.Article ID 569081, 20 pages. In this paper, we review recent results of ours concerning branching processes with general lifetimes and neutral mutations, under the infinitely many alleles model, where mutations can occur either at birth of individuals or at a constant rate during their lives. In both models, we study the allelic partition of the population at time t. We give closed formulae for the expected frequency spectrum at t and prove pathwise convergence to an explicit limit, as t goes to infinity, of the relative numbers of types younger than some given age and carried by a given number of individuals (small families). We also provide convergences in distribution of the sizes or ages of the largest families and of the oldest families. In the case of exponential lifetimes, population dynamics are given by linear birth and death processes, and we can most of the time provide general formulations of our results unifying both models. (10.1155/2012/569081)
  DOI : 10.1155/2012/569081
• Le modèle d'Ising tridimensionnel avec des spins à 2 états, température=0.2, des conditions initiales aléatoires et un nombre croissant d'itérations -de 100 à 4000
  
  • Colonna Jean-François
  , 2012. The tridimensional Ising Model with 2-state spins, temperature=0.2, random initial conditions and an increasing number of iterations -from 100 to 4000- (Le modèle d'Ising tridimensionnel avec des spins à 2 états, température=0.2, des conditions initiales aléatoires et un nombre croissant d'itérations -de 100 à 4000-)
• L'ensemble fractal dans les quaternions obtenu lors du calcul des racines de Q^3=1 grâce à la méthode de Newton avec translation le long du troisième axe de l'espace des quaternions -section tridimensionnelle
  
  • Colonna Jean-François
  , 2012. The quaternionic fractal set obtained when computing the roots of Q^3=1 using Newton's method with translation along the third axis of the quaternionic space -tridimensional cross-section- (L'ensemble fractal dans les quaternions obtenu lors du calcul des racines de Q^3=1 grâce à la méthode de Newton avec translation le long du troisième axe de l'espace des quaternions -section tridimensionnelle-)
• Seize tores entrelacés
  
  • Colonna Jean-François
  , 2012. Sixteen interlaced torus (Seize tores entrelacés)
• Représentation tridimensionnelle d'une variété quadridimensionnelle de Calabi-Yau fractale
  
  • Colonna Jean-François
  , 2012. Tridimensional representation of a fractal quadridimensional Calabi-Yau manifold (Représentation tridimensionnelle d'une variété quadridimensionnelle de Calabi-Yau fractale)
• Seize tores fractals entrelacés
  
  • Colonna Jean-François
  , 2012. Sixteen interlaced fractal torus (Seize tores fractals entrelacés)
• Analytical and numerical study of the apparent diffusion coefficient in diffusion MRI at long diffusion times and low b-values
  
  • Li Jing-Rebecca
  • Le Bihan Denis
  • Nguyen Thong Quoc
  • Grebenkov Denis S
  • Poupon Cyril
  • Haddar Houssem
  , 2012. Diffusion magnetic resonance imaging provides a measure of the average distance travelled by water molecules in a medium and can give useful information on cellular structure and structural change when the medium is biological tissue. In this paper, two approximate models for the apparent diffusion coefficient at low b-values and long diffusion times are formulated and validated. The first is a steady-state partial differential equation model that gives the steady-state (infinite time) effective diffusion tensor for general cellular geometries. For nearly isotropic diffusion where the intra-cellular compartment consists of non-elongated cells, a second approximate model is provided in the form of analytical formulae for the eigenvalue of the steady-state effective diffusion tensor. Both models are validated by numerical simulations on a variety of cells sizes and shapes.
• Méthodes numériques e fficaces pour la valorisation des GMWB
  
  • Ben Zineb Tarik
  , 2012. Cette thèse traite du problème de valorisation par des méthodes numériques efficaces de contrats GMWB dans une optique de calcul par formules fermées ou par méthode de Monte Carlo sous contrainte de faible nombre de simulations. Les produits GMWB sont des produits très complexes qui ont connu ces dernières années un grand succès de par la garantie dont bénéficie l'assuré sur les retraits futurs avec un effet upside dépendant de la performance du fond sous-jacent au contrat. En outre, le souscripteur dispose de nombreuses options attrayantes qu'il peut exercer à tout moment dont l'option de racheter partiellement ou totalement son contrat, la possibilité de modifier l'allocation de la prime payée (fund switching) pendant la durée du contrat et enfin l'option d'avancer ou de reporter la date de début des paiements. Cependant, de telles options cumulées avec la complexité du produit et les risques de marché et de mortalité exposent l'assureur qui doit gérer des dizaines de milliers de contrats sous plusieurs contraintes opérationnelles (temps de calcul,faible nombre de simulation, etc.) à une difficulté majeure en terme de valorisation et de couverture. Une grande partie de cette thèse (chapitres de 2, 4, 5 et 6) est consacrée à l'étude de l'option de rachat partiel ou total dans les contrats GMWB selon deux angles : le point de vue assuré rationnel et le point de vue couvreur appréhendant le pire cas. A ce propos, dans notre cadre général en temps discret avec une volatilité locale et taux d'intérêt à la Hull-White 1 facteur, la stratégie optimale déterminant le coût du contrat dans les deux cas est la solution d'un problème de contrôle stochastique optimal en temps discret. Néanmoins, grâce à une propriété d'homogénéité partielle sur le prix et les flux, on démontre qu'elle est explicite et de type Bang-Bang. Le problème de valorisation étant ainsi ramené à celui d'arrêt optimal , nous avons proposé une méthode de Monte Carlo de type Longstaff Schwartz dont l'étape de régression empirique a été traitée par la méthode de moindres carrés habituels et par une nouvelle méthode appelée VCP (Variables de contrôle préliminaires). Cette dernière consiste dans un premier temps à réduire la variance empirique des flux à régresser à travers une projection L2 sur des variables de contrôle adaptées et centrées, et puis à faire la régression par moindres carrées habituels sur les nouveaux flux à variance réduite. Une étude numérique sur un cas test ainsi qu'une quantification théorique de l'erreur par les techniques de régression non paramétriques ont conclu à son efficacité dans un contexte de faible nombre de simulation (contrainte d'Axa). Quant au chapitre 3, il est consacré à justifier numériquement et théoriquement l'hypothèse de mutualisation du risque de mortalité souvent supposée par les praticiens dans le cas américain sur un produit simple sensible à ce risque. Enfin, la dernière partie de la thèse (chapitre 7) est consacrée à la valorisation par formules fermées approchées pour des contrats GMWB simplifiés dans un modèle Black Scholes avec taux d'intérêt de dynamique Hull-White à 1 facteur. En effectuant un développement asymptotique sur le montant des retraits, on obtient des formules approchées du prix du contrat GMWB par un prix Black-Scholes corrigé par une somme explicite de Grecques, le tout étant plus rapide à évaluer. Des estimations d'erreur sont établies lorsque la fonction payoff est régulière. La précision des formules asymptotiques est testée numériquement et montre un excellent comportement de ces approximations, même pour des contrats à longue maturité (20 ans).
• Entrelacs fractal Entrelacs fractal
  
  • Colonna Jean-François
  , 2012. Fractal intertwining (Entrelacs fractal)
• Structure paradoxale fractale Structure paradoxale fractale
  
  • Colonna Jean-François
  , 2012. Fractal paradoxal structure (Structure paradoxale fractale)
• Using Salomé to reproduce the structure and to observe the diffusion of water molecules in biological tissue
  
  • Nguyen Dang Van
  • Grebenkov Denis S
  • Li Jing-Rebecca
  , 2012. Diffusion magnetic resonance imaging (DMRI) can give useful information on cellular structure and its structural changes. Salomé is used to reproduce some complicated shapes in d-dimensions (d=2,3) that are used to represent the natural structures of various biological tissue. The meshes representing these shapes are used as inputs to a finite element code that we built upon FENICS C++. Results were obtained for a model of globlastoma (cerebral tumor) as a Voronoi diagram which was used to observe the convergence of the apparent diffusion tensor in long-time limit to the effective diffusion tensor computed by homogenization theory.
• Structure paradoxale périodique fractale
  
  • Colonna Jean-François
  , 2012. Fractal periodical impossible structure (Structure paradoxale périodique fractale)
• A fast random walk algorithm for computing diffusion-weighted NMR signals in multiscale porous media: a feasibility study for a Menger sponge
  
  • Grebenkov Denis S.
  • Nguyen Hang Tuan
  • Li Jing-Rebecca
  , 2012. A fast random walk (FRW) algorithm is adapted to compute diffusion-weighted NMR signals in a Menger sponge which is formed by multiple channels of broadly distributed sizes and often considered as a model for soils and porous materials. The self-similar structure of a Menger sponge allows for rapid simulations that were not feasible by other numerical techniques. The role of multiple length scales onto diffusion-weighted NMR signals is investigated.
• Coquillage quaternionique fractal Coquillage quaternionique fractal
  
  • Colonna Jean-François
  , 2012. Fractal quaternionic shell (Coquillage quaternionique fractal)
• Optimisation de Lois de Gestion Énergétiques des Véhicules Hybrides
  
  • Granato Giovanni
  , 2012. The purpose of the this work is to apply optimal control techniques to enhance the performance of the power management of hybrid vehicles. More precisely, the techniques concerned are viscosity solutions of Hamilton-Jacobi equations, level set methods in reachability analysis, stochastic dynamic programming, stochastic dual dynamic programming and chance constrained optimal control. This document starts by presenting the necessary technical background and models for the study of optimal power management of hybrid vehicles. The synthesis of efficient power management strategies for hybrid vehicles accounting for uncertainty in the vehicle speed is studied next. This is done via a stochastic dynamic algorithm, at a first time, and then by a stochastic dual dynamic programming algorithm. In addition, we introduce a chance constrained optimal control problem that can be used to synthesize more flexible optimal control strategies. We detail a dynamic programming principle in a form that can be readily used for the numerical synthesis of optimal feedback using a dynamic programming algorithm. Later, theoretical results regarding the reachability analysis of hybrid systems are obtained. The reachability set of a continuous-time hybrid system is characterized by a value function via a level set approach. Furthermore, we show that the value function of a hybrid optimal control problem is the unique solution of a system of quasi-variational inequalities in the viscosity sense. Then, we prove the convergence of a class of numerical schemes for the computation of the value function. As a further step in the reachability analysis, we study of the discrete-time dynamical system and the discrete-time optimal control problem for the reachability analysis of hybrid systems. Here, the focus is on a discrete-time modeling of the hybrid system, which leads to dynamic programming principle, which can be used to characterize the value function. Lastly, we describe the construction of a stochastic model of the speed profile for electric vehicles.
• Analysis of Backward SDEs with Jumps and Risk Management Issues
  
  • Kazi-Tani Mohamed Nabil
  , 2012. This PhD dissertation deals with issues in management, measure and transfer of risk on the one hand and with problems of stochastic analysis with jumps under model uncertainty on the other hand. The first chapter is dedicated to the analysis of Choquet integrals, as non necessarily law invariant monetary risk measures. We first establish a new representation result of convex comonotone risk measures, then a representation result of Choquet integrals by introducing the notion of local distortion. This allows us then to compute in an explicit manner the inf-convolution of two Choquet integrals, with examples illustrating the impact of the absence of the law invariance property. Then we focus on a non-proportional reinsurance pricing problem, for a contract with reinstatements. After defining the indifference price with respect to both a utility function and a risk measure, we prove that is is contained in some interval whose bounds are easily calculable. Then we pursue our study in a time dynamic setting. We prove the existence of bounded solutions of quadratic backward stochastic differential equations (BSDEs for short) with jumps, using a direct fixed point approach. Under an additional standard assumption, or under a convexity assumption of the generator, we prove a uniqueness result, thanks to a comparison theorem. Then we study the properties of the corresponding non-linear expectations, we obtain in particular a non linear Doob-Meyer decomposition for g-submartingales and their regularity in time. As a consequence of this results, we obtain a converse comparison theorem for our class of BSDEs. We give applications for dynamic risk measures and their dual representation, and compute their inf-convolution, with some explicit examples, when the filtration is generated by both a Brownian motion and an integer valued random measure. The second part of this PhD dissertation is concerned with the analysis of model uncertainty, in the particular case of second order BSDEs with jumps. These equations hold P-almost surely, where P lies in a wide family of probability measures, corresponding to solutions of some martingale problems on the Skorohod space of càdlàg paths. We first extend the definition given by Soner, Touzi and Zhang of second order BSDEs to the case with jumps. For this purpose, we prove an aggregation result, in the sense of Soner, Touzi and Zhang, on the Skorohod space D. This allows us to use a quasi-sure version of the canonical process jump measure compensator. Then we prove a wellposedness result for our class of second order BSDEs. These equations include model uncertainty, affecting both the volatility and the jump measure of the canonical process.
• Controllability of the bilinear Schrödinger equation with several controls and application to a 3D molecule
  
  • Boscain Ugo
  • Caponigro Marco
  • Sigalotti Mario
  , 2013, pp.3038-3043. We show the approximate rotational controllability of a polar linear molecule by means of three nonresonant linear polarized laser fields. The result is based on a general approximate controllability result for the bilinear Schrödinger equation, with wavefunction varying in the unit sphere of an infinite-dimensional Hilbert space and with several control potentials, under the assumption that the internal Hamiltonian has discrete spectrum. (10.1109/CDC.2012.6426289)
  DOI : 10.1109/CDC.2012.6426289
• Controllability of the Schroedinger equation via adiabatic methods and conical intersections of the eigenvalues
  
  • Chittaro Francesca
  • Mason Paolo
  • Boscain Ugo
  • Sigalotti Mario
  , 2012, pp.3044-3049. We present a constructive method to control the bilinear Schroedinger equation by means of two or three controlled external fields. The method is based on adiabatic techniques and works if the spectrum of the Hamiltonian admits eigenvalue intersections, with respect to variations of the controls, and if the latter are conical. We provide sharp estimates of the relation between the error and the controllability time.
• Factorization Method for Electromagnetic Inverse Scattering from Biperiodic Structures
  
  • Lecheiter Armin
  • Nguyen Dinh Liem
  , 2012. This paper is concerned with the inverse scattering problem of electromagnetic waves from penetrable biperiodic structures in three dimensions. We study the Factorization method as a tool for reconstructing the periodic media from measured data consisting of scattered electromagnetic waves for incident plane electromagnetic waves. We propose a rigorous analysis for the method. A simple criterion is provided to reconstruct the biperiodic structures. We also provide three-dimensional numerical experiments to indicate the performance of the method.
• Un 'mélange' entre une structure fractale tridimensionnelle et un agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb') -section tridimensionnelle
  
  • Colonna Jean-François
  , 2012. A 'mixing' between a tridimensional fractal structure and a close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') -tridimensional cross-section- (Un 'mélange' entre une structure fractale tridimensionnelle et un agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb') -section tridimensionnelle-)
• Spectral Methods for Direct and Inverse Scattering from Periodic Structures
  
  • Nguyen Dinh Liem
  , 2012. The main topic of the thesis are inverse scattering problems of electromagnetic waves from periodic structures. We study first the direct problem and its numerical resolution using volume integral equation methods with a focus on the case of strongly singular integral operators and discontinuous coefficients. In a second investigation of the direct problem we study conditions on the material parameters under which well-posedness is ensured for all positive wave numbers. Such conditions exclude the existence of guided waves. The considered inverse scattering problem is related to shape identification. To treat this class of inverse problems, we investigate the so-called Factorization method as a tool to identify periodic patterns from measured scattered waves. In this thesis, these measurements are always related to plane incident waves. The outline of the thesis is the following: The first chapter is the introduction where we give the state of the art and new results of the topics studied in the thesis. The main content consists of five chapters, divided into two parts. The first part deals with the scalar case where the TM electromagnetic polarization is considered. In the second chapter we present the volume integral equation method with new results on Garding inequalities, convergence theory and numerical validation. The third chapter is devoted to the analysis of the Factorization method for the inverse scalar problem as well as some numerical experiments. The second part is dedicated to the study of 3-D Maxwell's equations. The fourth and fifth chapters are respectively generalizations of the results of the second and third ones to the case of Maxwell's equations. The sixth chapter contains the analysis of uniqueness conditions for the direct scattering problem, that is, absence of guided modes.
• Structure fractale tridimensionnelle Structure fractale tridimensionnelle
  
  • Colonna Jean-François
  , 2012. Tridimensional fractal structure (Structure fractale tridimensionnelle)
• Seize tores fractals entrelacés
  
  • Colonna Jean-François
  , 2012. Sixteen interlaced fractal torus (Seize tores fractals entrelacés)
• Représentation tridimensionnelle d'une variété quadridimensionnelle de Calabi-Yau fractale
  
  • Colonna Jean-François
  , 2012. Tridimensional representation of a fractal quadridimensional Calabi-Yau manifold (Représentation tridimensionnelle d'une variété quadridimensionnelle de Calabi-Yau fractale)
• Soixante-quatre tores entrelacés
  
  • Colonna Jean-François
  , 2012. Sixty-four interlaced torus (Soixante-quatre tores entrelacés)