Publications

2004

• Maturity randomization for stochastic control problems
  
  • Bouchard Bruno
  • El Karoui Nicole
  • Touzi Nizar
  , 2004. We study a maturity randomization technique for approximating optimal control problems. The algorithm is based on a sequence of control problems with random terminal horizon which converges to the original one. This is a generalization of the so-called {\it Canadization} procedure suggested by P. Carr in [2] for the fast computation of American put option prices. In addition to the original application of this technique to optimal stopping problems, we provide an application to another problem in finance, namely the super-replication problem under stochastic volatility, and we show that the approximating value functions can be computed explicitly.
• Mathematical Modeling in Optics
  
  • Soussi Sofiane
  , 2004. In the first part of the thesis, we study electromagnetic diffraction by bounded objects surrounded by nonlinear dielectric thin coating. We give a development of the fundamental and second-harmonic wave based on integral equation techniques. In the second part, we study the convergence of the (\em supercell) method, used by physicists for approaching modes introduced by compactly supported defects in a photonic crystal. We study the convergence of this method giving a sense to the spectrum convergence. The exponential convergence of defect eigenvalues is proved. The third part is dedicated to the study of wave propagation in photonic fibers. We derive a mathematical modelisation of these fibers which cladding is a 2D photonic crystal. Guided modes are characterized as eigenvalues of integral operators.
• Synthèse de textures tridimensionnelles grâce à une méthode fractale et a une dynamique du type Verhulst du quatrième degré
  
  • Colonna Jean-François
  , 2004. Synthesis of tridimensional textures by means of a fractal process and of a fourth degree Verhulst-like dynamics (Synthèse de textures tridimensionnelles grâce à une méthode fractale et a une dynamique du type Verhulst du quatrième degré)
• Synthèse de textures tridimensionnelles grâce à une méthode fractale et a une dynamique du type Verhulst du quatrième degré
  
  • Colonna Jean-François
  , 2004. Synthesis of tridimensional textures by means of a fractal process and of a fourth degree Verhulst-like dynamics (Synthese de textures tridimensionnelles grâce à une méthode fractale et a une dynamique du type Verhulst du quatrième degré)
• Lévy Processes in Finance: Inverse Problems and Dependence Modelling
  
  • Tankov Peter
  , 2004. This thesis deals with the modelling of stock prices by the exponentials of Lévy processes. In the first part we develop a non-parametric method allowing to calibrate exponential Lévy models, that is, to reconstruct such models from the prices of market-quoted options. We study the stability and convergence properties of this calibration method, describe its numerical implementation and give examples of its use. Our approach is first to reformulate the calibration problem as that of finding a risk-neutral exponential Lévy model that reproduces the option prices with the best possible precision and has the smallest relative entropy with respect to a given prior process, and then to solve this problem via the regularization methodology, used in the theory of ill-posed inverse problems. Applying this calibration method to the empirical data sets of index options allows us to study some properties of Lévy measures, implied by market prices. The second part of this thesis proposes a method allowing to characterize the dependence structures among the components of a multidimensional Lévy process and to construct multidimensional exponential Lévy models. This is done by introducing the notion of Lévy copula, which can be seen as an analog for Lévy processes of the notion of copula, used in statistics to model dependence between real-valued random variables. We give examples of parametric families of Lévy copulas and develop a method for simulating multidimensional Lévy processes with dependence given by a Lévy copula.
• Méthodes numériques pour la dynamique des structures non-linéaires incompressibles à deux échelles.
  
  • Hauret Patrice
  , 2004. Le travail exposé dans ce mémoire consiste en une étude mathématique et numérique d'outils permettant la simulation de la dynamique de structures complexes non-linéaires, quasi-incompressibles, et présentant deux échelles de longueurs caractéristiques. Pour être plus précis concernant ce dernier point, les structures considérées sont supposées comporter des détails géométriques fins sur leur bord. Cette étude, réalisée en partenariat avec la Manufacture Française des Pneumatiques Michelin, est largement motivée par l'importance de calculs dynamiques en roulage du pneumatique afin de prédire la valeur de différentes grandeurs physiques : contraintes dans les matériaux, pressions de contact au sol ou encore rayonnement acoustique. Dans ce cadre, les difficultés d'obtention de simulations complètes et réalistes pour des coûts de calcul raisonnables sont liées à la complexité de la géométrie, au comportement des matériaux, au mode de sollicitation par contact, ou encore à l'intervention de différentes échelles de longueur, de temps ou de rigidité caractéristiques de la structure. Après une description de l'anatomie du pneumatique, nous mentionnons quelques uns des enjeux de la simulation numérique lors de la phase de conception. Nous soulignons ensuite les propriétés intrinsèques de la structure qui rendent ces études délicates. Enfin, nous délimitons les problèmes qui occupent le reste de ce mémoire, et esquissons la démarche adoptée. Référence est faite au contenu des chapitres et aux contributions apportées.
• Vibrations 0001
  
  • Colonna Jean-François
  , 2004. Vibrations 0001 (Vibrations 0001)
• Le cri
  
  • Colonna Jean-François
  , 2004. The scream (Le cri)
• Coupling Harmonic Functions-Finite Elements for Solving the Stream Function-Vorticity Stokes Problem
  
  • Abboud Toufic
  • Salaün Michel
  • Salmon Stéphanie
  Numerical Methods for Partial Differential Equations, Wiley, 2004, vol. 20, pp.pp. 765-788. We consider the bidimensional Stokes problem for incompressible fluids in stream function-vorticity form. The classical finite element method of degree one usually used does not allow the vorticity on the boundary of the domain to be computed satisfactorily when the meshes are unstructured and does not converge optimally. To better approach the vorticity along the boundary, we propose that harmonic functions obtained by integral representation be used. Numerical results are very satisfactory, and we prove that this new numerical scheme leads to an optimal convergence rate of order 1 for the natural norm of the vorticity and, under higher regularity assumptions, from 3/2 to 2 for the quadratic norm of the vorticity. (10.1002/num.2005)
  DOI : 10.1002/num.2005
• Financial modelling with jump processes
  
  • Cont Rama
  • Tankov Peter
  , 2004. During the last decade, financial models based on jump processes have acquired increasing popularity in risk management and option pricing. Much has been published on the subject, but the technical nature of most papers makes them difficult for nonspecialists to understand, and the mathematical tools required for applications can be intimidating. Potential users often get the impression that jump and Lévy processes are beyond their reach. Financial Modelling with Jump Processes shows that this is not so. It provides a self-contained overview of the theoretical, numerical, and empirical aspects involved in using jump processes in financial modelling, and does so in terms within the grasp of nonspecialists. The introduction of new mathematical tools is motivated by its use in the modelling process, and precise mathematical statements of results are accompanied by intuitive explanations. Topics covered in this book include: jump-diffusion models, Lévy processes, stochastic calculus for jump processes, pricing and hedging in incomplete markets, implied volatility smiles, time-inhomogeneous jump processes and stochastic volatility models with jumps. The authors illustrate the mathematical concepts with many numerical and empirical examples and provide the details of numerical implementation of pricing and calibration algorithms. This book demonstrates that the concepts and tools necessary for understanding and implementing models with jumps can be more intuitive that those involved in the Black Scholes and diffusion models. If you have even a basic familiarity with quantitative methods in finance, Financial Modelling with Jump Processes will give you a valuable set of tools for modelling market fluctuations.
• Recovering exponential Lévy models from option prices: regularization of an ill-posed inverse problem.
  
  • Cont Rama
  • Tankov Peter
  , 2004. We propose a stable nonparametric method for constructing an option pricing model of exponential Lévy type, consistent with a given data set of option prices. After demonstrating the ill-posedness of the usual and least squares version of this inverse problem, we suggest to regularize the calibration problem by reformulating it as the problem of finding an exponential Lévy model that minimizes the sum of the pricing error and the relative entropy with respect to a prior exponential Lévy model. We prove the existence of solutions for the regularized problem and show that it yields solutions which are continuous with respect to the data, stable with respect to the choice of prior and converge to the minimum-entropy least square solution of the calibration problem.
• Diagrammes de Voronoi
  
  • Colonna Jean-François
  , 2004. Voronoi diagrams (Diagrammes de Voronoi)
• Diagrammes de Voronoi
  
  • Colonna Jean-François
  , 2004. Voronoi diagrams (Diagrammes de Voronoi)
• A PBP2x from a clinical isolate of Streptococcus pneumoniae exhibits an alternative mechanism for reduction of susceptibility to beta-lactam antibiotics.
  
  • Pernot Lucile
  • Chesnel Laurent
  • Le Gouellec Audrey
  • Croizé Jacques
  • Vernet Thierry
  • Dideberg Otto
  • Dessen Andréa
  Journal of Biological Chemistry, American Society for Biochemistry and Molecular Biology, 2004, pp.16463-70. The human pathogen Streptococcus pneumoniae is one of the main causative agents of respiratory tract infections. At present, clinical isolates of S. pneumoniae often exhibit decreased susceptibility toward beta-lactams, a phenomenon linked to multiple mutations within the penicillin-binding proteins (PBPs). PBP2x, one of the six PBPs of S. pneumoniae, is the first target to be modified under antibiotic pressure. By comparing 89 S. pneumoniae PBP2x sequences from clinical and public data bases, we have identified one major group of sequences from drug-sensitive strains as well as two distinct groups from drug-resistant strains. The first group includes proteins that display high similarity to PBP2x from the well characterized resistant strain Sp328. The second group includes sequences in which a signature mutation, Q552E, is found adjacent to the third catalytic motif. In this work, a PBP2x from a representative strain from the latter group (S. pneumoniae 5259) was biochemically and structurally characterized. Phenotypical analyses of transformed pneumococci show that the Q552E substitution is responsible for most of the reduction of strain susceptibility toward beta-lactams. The crystal structure of 5259-PBP2x reveals a change in polarity and charge distribution around the active site cavity, as well as rearrangement of strand beta3, emulating structural changes observed for other PBPs that confer drug resistance to Gram-positive pathogens. Interestingly, the active site of 5259-PBP2x is in closed conformation, whereas that of Sp328-PBP2x is open. Consequently, S. pneumoniae has evolved to employ the same protein in two distinct mechanisms of antibiotic resistance. (10.1074/jbc.M313492200)
  DOI : 10.1074/jbc.M313492200
• A Steady Performance Stopping Criterion for Pareto-based Evolutionary Algorithms
  
  • Roudenko Olga
  • Schoenauer Marc
  , 2004. The most commonly used stopping criterion in Evolutionary Multi-objective Algorithms is an a priori fixed number of generations (or evaluations). But it is rather difficult to speak about achieving a particular compromise between the quality of the final solutions and the computation time when stopping an algorithm this way. Unfortunately, whereas single-objective Evolutionary Algorithms can stop when the fitness does not improve during a given number of generations, such "steady-fitness" stopping criterion does not easily extend to the multi-objective framework. This paper introduces a stability measure based on the density of the non-dominated solutions and proposes to use it to stop the optimization process when no significant improvement is likely to take place on further iterations. This approach is validated by the empirical results obtained applying NSGA-II to the well-known bi-objective ZDT-benchmarks. In particular, the problem ZDT4 best illustrates the ability of the proposed criterion to avoid useless continuation of a wedged optimization process when a local Pareto-optimal set is reached.
• A functional central limit theorem in equilibrium for a large network in which customers join the shortest of several queues
  
  • Graham Carl
  Probability Theory and Related Fields, Springer Verlag, 2005, 131 (1), pp.97–120. We consider N single server infinite buffer queues with service rate beta. Customers arrive at rate N times alpha,choose L queues uniformly, and join the shortest one. The stability condition is alpha strictly less than beta. We study in equilibrium the sequence of the fraction of queues of length at least k, in the large N limit. We prove a functional central limit theorem on an infinite-dimensional Hilbert space with its weak topology, with limit a stationary Ornstein-Uhlenbeck process. We use ergodicity and justify the inversion of limits of long times and large sizes N by a compactness-uniqueness method. The main tool for proving tightness of the ill-known invariant laws and ergodicity of the limit is a global exponential stability result for the nonlinear dynamical system obtained in the functional law of large numbers limit.
• Out of equilibrium functional central limit theorems for a large network where customers join the shortest of several queues
  
  • Graham Carl
  , 2004. Customers arrive at rate N times alpha on a network of N single server infinite buffer queues, choose L queues uniformly, join the shortest one, and are served there in turn at rate beta. We let N go to infinity.We prove a functional central limit theorem (CLT) for the tails of the empirical measures of the queue occupations,in a Hilbert space with the weak topology, with limit given by an Ornstein-Uhlenbeck process. The a priori assumption is that the initial data converge.This completes a recent functional CLT in equilibrium result for which convergence for the initial data was not known in advance, but was deduced a posteriori from the functional CLT.
• Synthèse de textures tridimensionnelles grâce à une méthode fractale
  
  • Colonna Jean-François
  , 2004. Synthesis of tridimensional textures by means of a fractal process (Synthèse de textures tridimensionnelles grâce à une méthode fractale)
• Application des algorithmes évolutionnaires aux problèmes d'optimisation multi-objectif avec contraintes.
  
  • Roudenko Olga
  , 2004. Ce travail est une contribution au développement des Algorithmes Evolutionnaires Multi-objectif. La hausse remarquable d'intérêt pour ces méthodes récentes constatée depuis la dernière décénie s'explique notamment par leur capacité de trouver une (bonne) approximation de l'ensemble des compromis de Pareto en un seul essai de l'algorithme, à la différence des approches traditionnelles pour l'optimisation multi-critère, qui ne trouvent qu'une solution-compromis à la fois (d'autant que cette solution dépend fortement du choix subjectif de certains paramètres). En effet, lors de la résolution des problèmes réels d'optimisation multi-critère, et en particulier, des problèmes de conception, il est souvent préférable de prendre la décision finale à partir des informations les plus complètes possibles, même si cela nécessite un effort de calcul supplémentaire. Dans cette thèse, deux problèmes de l'industrie automobile sont étudiés. Le premier concerne l'optimisation paramétrique de la forme d'un pare-choc de voiture, un problème a 10 objectifs issus de 3 domaines mécaniques: crash, acoustique et statique. Le second problème qui se pose lors du calibrage du moteur diesel Common Rail (rampe commune) consiste à minimiser la consommation spécifique du carburant ainsi que le bruit de la combustion tout en respectant les normes européennes de fonctionnement en terme de nuisances à l'environnement. Une tendance remarquable des Algorithmes Evolutionnaires est que ces méthodes pénètrent" aujourd'hui dans de nombreux nouveaux domaines d'application malgré l'absence de bases théoriques (notamment, de preuves de convergence) aussi solides que celles qu'on peut trouver pour des approches alternatives. Inspirée par cette observation, la motivation principale de ce travail était de contribuer au développement des Algorithmes Evolutionnaires Multi-objectif de façon à rendre leur application aux problèmes réels la plus efficace possible. Ainsi, une contribution originale de cette thèse consiste à répondre à un manque criant dans ce domaine, le manque de critère d'arrêt plus fin qu'une simple borne sur le nombre d'itérations. Le critère d'arrêt proposé dans ce travail est destiné à optimiser le rapport entre la qualité des solutions et le coût de calcul: dans la pratique c'est ce compromis qui est le plus souvent recherché. De même, un nouvel opérateur de croisement basé sur la relation de la dominance de Pareto est proposé et nous montrons l'accélération de la progression vers la surface des compromis optimaux qu'il apporte.
• Synthèse de textures tridimensionnelles grâce à une méthode fractale
  
  • Colonna Jean-François
  , 2004. Synthesis of tridimensional textures by means of a fractal process (Synthèse de textures tridimensionnelles grâce à une méthode fractale)
• Entropic structure of multicomponent reactive flows with partial equilibrium chemistry
  
  • Giovangigli Vincent
  • Massot Marc
  Mathematical Methods in the Applied Sciences, Wiley, 2004, 27 (7), pp.739--768. In this paper, we investigate the system of equations modelling multicomponent reactive ows with detailed transport and complex chemistry in the limit of partial equilibrium. The reduced system is obtained using a projection step compatible with the chemical entropy production. The reduced multicomponent transport and convection uxes are shown to be compatible with the mathematical entropy thus providing a symmetric form as well as normal forms for the reduced system. This yields global existence and asymptotic stability around constant equilibrium states for the Cauchy problem on the partial equilibrium manifold in all space dimensions. (10.1002/mma.429)
  DOI : 10.1002/mma.429
• Synthese de textures bidimensionnelles
  
  • Colonna Jean-François
  , 2004. Synthesis of bidimensional textures (Synthese de textures bidimensionnelles)
• Montagnes multifractales sous la neige
  
  • Colonna Jean-François
  , 2004. Snowy multifractal mountains (Montagnes multifractales sous la neige)
• Montagnes multifractales sous la neige
  
  • Colonna Jean-François
  , 2004. Snowy multifractal mountains (Montagnes multifractales sous la neige)
• Montagnes multifractales sous la neige
  
  • Colonna Jean-François
  , 2004. Snowy multifractal mountains (Montagnes multifractales sous la neige)