Publications

2010

• Risques de taux et de longévité : Modélisation dynamique et Applications aux produits dérivés et à l'assurance-vie
  
  • Bensusan Harry
  , 2010. Cette thèse se divise en trois parties. La première partie est constituée des chapitres 2 et 3 dans laquelle nous considérons des modèles qui décrivent l'évolution d'un sous-jacent dans le monde des actions ainsi que l'évolution des taux d'intérêt. Ces modèles, qui utilisent les processus de Wishart, appartiennent à la classe affine et généralisent les modèles de Heston multi-dimensionnels. Nous étudions les propriétés intrinsèques de ces modèles et nous nous intéressons à l'évaluation des options vanilles. Après avoir rappelé certaines méthodes d'évaluation, nous introduisons des méthodes d'approximation fournissant des formules fermées du smile asymptotique. Ces méthodes facilitent la procédure de calibration et permettent une analyse intéressante des paramètres. La deuxième partie, du chapitre 4 au chapitre 6, étudie les risques de mortalité et de longévité. Nous rappelons tout d'abord les concepts généraux du risque de longévité et un ensemble de problématiques sous-jacentes à ce risque. Nous présentons ensuite un modèle de mortalité individuelle qui tient compte de l'âge et d'autres caractéristiques de l'individu qui sont explicatives de mortalité. Nous calibrons le modèle de mortalité et nous analysons l'influence des certaines caractéristiques individuelles. Enfin, nous introduisons un modèle microscopique de dynamique de population qui permet de modéliser l'évolution dans le temps d'une population structurée par âge et par traits. Chaque individu évolue dans le temps et est susceptible de donner naissance à un enfant, de changer de caractéristiques et de décéder. Ce modèle tient compte de l'évolution, éventuellement stochastique, des taux démographiques individuels dans le temps. Nous décrivons aussi un lien micro/macro qui fournit à ce modèle microscopique de bonnes propriétés macroscopiques. La troisième partie, concernant les chapitres 7 et 8, s'intéresse aux applications des modélisations précédentes. La première application est une application démographique puisque le modèle microscopique de dynamique de population permet d'effectuer des projections démographiques de la population française. Nous mettons aussi en place une étude démographique du problème des retraites en analysant les solutions d'une politique d'immigration et d'une réforme sur l'âge de départ à la retraite. La deuxième application concerne l'étude des produits d'assurance-vie associant les risques de longévité et de taux d'intérêt qui ont été étudiés en détails dans les deux premières parties de la thèse. Nous nous intéressons tout d'abord à l'étude du risque de base qui est généré par l'hétérogénéité des portefeuilles de rentes. De plus, nous introduisons la Life Nominal Chooser Swaption (LNCS) qui est un produit de transfert de risque des produits d'assurance-vie : ce produit a une structure très intéressante et permet à une assurance détenant un portefeuille de rente de transférer intégralement son risque de taux d'intérêt à une banque.
• Duality Between Invariant Spaces for Max-Plus Linear Discrete Event Systems
  
  • Di Loreto Michaël
  • Gaubert Stéphane
  • Katz Ricardo
  • Loiseau Jean-Jacques
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2010, 48 (8), pp.5606-5628. We extend the notions of conditioned and controlled invariant spaces to linear dynamical systems over the max-plus or tropical semiring. We establish a duality theorem relating both notions, which we use to construct dynamic observers. These are useful in situations in which some of the system coefficients may vary within certain intervals. The results are illustrated by an application to a manufacturing system. (10.1137/090747191)
  DOI : 10.1137/090747191
• Motion planning in quantum control via intersection of eigenvalues
  
  • Boscain Ugo
  • Chittaro Francesca
  • Mason Paolo
  • Pacqueau Rémi
  • Sigalotti Mario
  , 2010, pp.6 pages. In this paper we consider the problem of inducing a transition in a controlled quantum mechanical system whose spectrum loses simplicity for some values of the control. We study the situation in which the Hamiltonian of the system is real, and we are in presence of two controls. In this case, eigenvalue crossings are generically conical. Adiabatic approximation is used to decouple a finite dimensional sub-system from the original one (usually infinite dimensional). The main advantage of this method is that as a byproduct of the controllability result it permits to get an explicit expression of the controls. Moreover it may be used in the case in which the dependence of the Hamiltonian from the controls is non-linear, for which at the moment, no other method works. In this paper we study the basic block of this controllability method, that is a two by two system whose spectrum presents a conical intersection. We show how to control exactly this system with a control strategy that can be slowed down. The possibility of slowing down the control law is essential to obtain an adiabatic decoupling from the rest of the system with an arbitrary precision.
• Optimal design of experiments with application to the inference of traffic matrices in large networks: second order cone programming and submodularity
  
  • Sagnol Guillaume
  , 2010. We approach the problem of optimizing the measurements in large IP networks, by using the theory of optimal experimental designs. This method gives raise to large scale optimization problems, for which we develop a resolution technique relying on Second Order Cone Programming (SOCP). The heart of our method is a rank reduction theorem in semidefinite programming. Some combinatorial problems --which arise when the goal is to find an optimal subset of the available experiments-- are also studied. The application to the inference of the traffic matrix in telecommunication networks is the object of the second part of this manuscript. We develop a method in which we optimize the estimation of several (randomly drawn) linear combinations of the traffic demands. This approach is compared to previous ones, and is fully evaluated by mean of simulations relying on real data. In particular, we handle some instances that were previously intractable.
• Identification par imagerie laser d'un objet dissimulé - Aspects mathématiques et numériques
  
  • Bellet Jean-Baptiste
  , 2010. Nous nous intéressons à l'imagerie d'un objet enfoui dans un milieu multi-couches inhomogène, avec des données ne contenant pas la phase. Nous résolvons un problème direct modèle de propagation des ondes dans un tel milieu, à l'aide de l'analyse asymptotique et des équations intégrales. Puis nous développons des algorithmes de reconstruction à base de dérivée topologique et des techniques de l'optimisation de forme.
• Un ensemble de 4x3 stéréogrammes d'une visualisation tridimensionnelle de la dynamique de Verhulst
  
  • Colonna Jean-François
  , 2010. A set of 4x3 stereograms of a tridimensional visualization of the Verhulst dynamics (Un ensemble de 4x3 stéréogrammes d'une visualisation tridimensionnelle de la dynamique de Verhulst)
• Modeling and simulation in photoacoustics
  
  • Jugnon Vincent
  , 2010. This thesis deals with the problem of photoacoustic imaging. In this imaging setting, one heats a medium up with an electromagnetic wave. The medium dilates and emits an ultrasonic wave. The aim is then to reconstruct inner properties of the medium from boundary measurements. It is an inverse problem on the initial condition for the wave equation. In an idealized frame, the reconstruction procedure has been thoroughly studied. The main goal of this thesis is to stray from the standard model by considering less restrictive assumptions. For each of them (boundary condition, partial view, attenuation, inhomogeneous sound speed) the thesis proposes a correction based on adapted mathematical tools (asymptotic analysis, dual approach , correlation...). Reconstruction of the initial condition of the wave equation is however not enough. It depends on the electromagnetique illumination. A second inverse problem has to be solved on the electromagnetic wave propagation to acces the physical coefficients of interest. The thesis presents algorithmic results in the frame of the diffusion equation and theoretic estimates in the frame of the transport equation. The thesis also presents an improving result for a topological derivative based imaging approach.
• Un ensemble de 4x3 stéréogrammes d'une visualisation tridimensionnelle de la dynamique de Verhulst
  
  • Colonna Jean-François
  , 2010. A set of 4x3 stereograms of a tridimensional visualization of the Verhulst dynamics (Un ensemble de 4x3 stéréogrammes d'une visualisation tridimensionnelle de la dynamique de Verhulst)
• Un ensemble de 4x3 stéréogrammes d'une visualisation tridimensionnelle de la dynamique de Verhulst
  
  • Colonna Jean-François
  , 2010. A set of 4x3 stereograms of a tridimensional visualization of the Verhulst dynamics (Un ensemble de 4x3 stéréogrammes d'une visualisation tridimensionnelle de la dynamique de Verhulst)
• Visualisation tridimensionnelle de la dynamique de Verhulst
  
  • Colonna Jean-François
  , 2010. Tridimensional visualization of the Verhulst dynamics (Visualisation tridimensionnelle de la dynamique de Verhulst)
• Un ensemble de 4x3 stéréogrammes d'une interpolation entre un ensemble de Mandelbrot dans l'ensemble des pseudo-quaternions (un 'Mandelbulb') et une dynamique de Verhulst tridimensionnelle
  
  • Colonna Jean-François
  , 2010. A set of 4x3 stereograms of an interpolation between a pseudo-quaternionic Mandelbrot set (a 'Mandelbulb') and a tridimensional Verhulst dynamics (Un ensemble de 4x3 stéréogrammes d'une interpolation entre un ensemble de Mandelbrot dans l'ensemble des pseudo-quaternions (un 'Mandelbulb') et une dynamique de Verhulst tridimensionnelle)
• Une interpolation entre un ensemble de Mandelbrot dans l'ensemble des pseudo-quaternions (un 'Mandelbulb') et une dynamique de Verhulst tridimensionnelle
  
  • Colonna Jean-François
  , 2010. An interpolation between a pseudo-quaternionic Mandelbrot set (a 'Mandelbulb') and a tridimensional Verhulst dynamics (Une interpolation entre un ensemble de Mandelbrot dans l'ensemble des pseudo-quaternions (un 'Mandelbulb') et une dynamique de Verhulst tridimensionnelle)
• Visualisation tridimensionnelle de la dynamique de Verhulst
  
  • Colonna Jean-François
  , 2010. Tridimensional visualization of the Verhulst dynamics (Visualisation tridimensionnelle de la dynamique de Verhulst)
• Visualisation tridimensionnelle de la dynamique de Verhulst
  
  • Colonna Jean-François
  , 2010. Tridimensional visualization of the Verhulst dynamics (Visualisation tridimensionnelle de la dynamique de Verhulst)
• Visualisation tridimensionnelle de la dynamique de Verhulst
  
  • Colonna Jean-François
  , 2010. Tridimensional visualization of the Verhulst dynamics (Visualisation tridimensionnelle de la dynamique de Verhulst)
• Application of the linear sampling method to retrieve cracks with impedance boundary conditions
  
  • Ben Hassen Fahmi
  • Boukari Yosra
  • Haddar Houssem
  , 2010. We use the Linear Sampling Method (LSM) to detect a crack with impedance boundary conditions. This paper extends the work of Cakoni and Colton [1] that uses the LSM to reconstruct a crack with mixed boundary conditions from measurements of the far field patterns associated with different incident plane waves. The performance of our method is illustrated through some numerical examples.
• Visualisation tridimensionnelle de la dynamique de Verhulst
  
  • Colonna Jean-François
  , 2010. Tridimensional visualization of the Verhulst dynamics (Visualisation tridimensionnelle de la dynamique de Verhulst)
• Statistical analysis of the multifractal random walk processes
  
  • Duvernet Laurent
  , 2010. We study some properties of a class of real-valued, continuous-time random processes, namely multifractal random walks. A striking feature of these processes lies in their scaling property: the distribution of the process at small scales is the same as the distribution at large scales, given some random factor independent of the process. The first part of the dissertation concerns the convergence of the empirical moment of the increment of the process, in a rather general asymptotic setting where the step of the increment may go to zero while the observation horizon may also go to infinity. In the second part, we propose a family of nonparametric tests that separate multifractal random walks from Itô semi-martingales. After showing the consistency of these tests, we study their behavior on simulations. In the third part, we build a skewed multifractal random walk process, such that the past increment is negatively correlated with the future squared increment. Such a "leverage effect" is notably seen on financial stock and index prices. We compare the empirical properties of the obtained process with real data. The fourth part deals with the parametric estimation of the process in a Gaussian case. We first show that under certain conditions, one can not estimate two of the three parameters, even if the sample path is continuously observed on some interval. We next study the theoretical and empirical performances of some estimators of the third parameter, the intermittency coefficient.
• DIRECT AND INVERSE MEDIUM SCATTERING IN A 3D HOMOGENEOUS PLANAR WAVEGUIDE
  
  • Arens Tilo
  • Gintides Drossos
  • Lechleiter Armin
  , 2010. Time-harmonic acoustic waves in an ocean of finite height are modeled by the Helmholtz equation inside a layer with suitable boundary conditions. Scattering in this geometry features phenomena unknown in free space: resonances might occur at special frequencies and wave fields consist of partly evanescent modes. Inverse scattering in waveguides hence needs to cope with energy loss and limited aperture data due to the planar geometry. In this paper, we analyze direct wave scattering in a 3D planar waveguide and show that resonance frequencies do not exist for a certain class of bounded penetrable scatterers. More important, we propose the Factorization method for solving inverse scattering problems in the 3D waveguide. This fast inversion method requires near-field data for special incident fields and we rigorously show how to generate this data from standard point sources. Finally, we discuss our theoretical results in the light of numerical examples.
• Multifractal analysis in a mixed asymptotic framework
  
  • Bacry Emmanuel
  • Gloter Arnaud
  • Hoffmann Marc
  • Muzy J.-F.
  The Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2010, pp.1729-1760. Multifractal analysis of multiplicative random cascades is revisited within the framework of mixed asymptotics. In this new framework, the observed process can be modeled by a concatenation of independent binary cascades and statistics are estimated over a sample whose size increases as the resolution scale (or the sampling period) becomes finer. This allows one to continuously interpolate between the situation where one studies a single cascade sample at arbitrary fine scales and where, at fixed scale, the sample length (number of cascades realizations) becomes infinite. We show that scaling exponents of "mixed" partitions functions, that is, the estimator of the cumulant generating function of the cascade generator distribution depends on some "mixed asymptotic" exponent χ, respectively, above and below two critical value pχ− and pχ+. We study the convergence properties of partition functions in mixed asymtotics regime and establish a central limit theorem. Moreover, within the mixed asymptotic framework, we establish a "box-counting" multifractal formalism that can be seen as a rigorous formulation of Mandelbrot's negative dimension theory. Numerical illustrations of our results on specific examples are also provided. A possible application of these results is to distinguish data generated by log-Normal or log-Poisson models. (10.1214/09-AAP670)
  DOI : 10.1214/09-AAP670
• EXISTENCE RESULTS FOR NONSMOOTH SECOND-ORDER DIFFERENTIAL INCLUSIONS, CONVERGENCE RESULT FOR A NUMERICAL SCHEME AND APPLICATION TO THE MODELING OF INELASTIC COLLISIONS
  
  • Bernicot Frederic
  • Lefebvre-Lepot Aline
  Confluentes Mathematici, Université de Lyon, 2010, 02 (04), pp.445-471. We are interested in the existence results for second-order differential inclusions, involving finite number of unilateral constraints in an abstract framework. These constraints are described by a set-valued operator, more precisely a proximal normal cone to a timedependent set. In order to prove these existence results, we study an extension of the numerical scheme introduced in [10] and prove a convergence result for this scheme. (10.1142/S1793744210000247)
  DOI : 10.1142/S1793744210000247
• Interior penalty approximation for optimal control problems. Optimality conditions in stochastic optimal control theory.
  
  • Silva Francisco J.
  , 2010. This thesis is divided in two parts. In the first one we consider deterministic optimal control problems and we study interior approximations for two model problems with non-negativity constraints. The first model is a quadratic optimal control problem governed by a nonautonomous affine ordinary differential equation. We provide a first-order expansion for the penalized state an adjoint state (around the corresponding state and adjoint state of the original problem), for a general class of penalty functions. Our main argument relies on the following fact: if the optimal control satisfies strict complementarity conditions for its Hamiltonian, except for a set of times with null Lebesgue measure, the functional estimates of the penalized optimal control problem can be derived from the estimates of a related finite dimensional problem. Our results provide three types of measure to analyze the penalization technique: error estimates of the control, error estimates of the state and the adjoint state and also error estimates for the value function. The second model we study is the optimal control problem of a semilinear elliptic PDE with a Dirichlet boundary condition, where the control variable is distributed over the domain and is constrained to be non-negative. Following the same approach as in the first model, we consider an associated family of penalized problems, whose solutions define a central path converging to the solution of the original one. In this fashion, we are able to extend the results obtained in the ODE framework to the case of semilinear elliptic PDE constraints. In the second part of the thesis we consider stochastic optimal control problems. We begin withthe study of a stochastic linear quadratic problem with non-negativity control constraints and we extend the error estimates for the approximation by logarithmic penalization. The proof is based is the stochastic Pontryagin's principle and a duality argument. Next, we deal with a general stochastic optimal control problem with convex control constraints. Using the variational approach, we are able to obtain first and second-order expansions for the state and cost function, around a local minimum. This analysis allows us to prove general first order necessary condition and, under a geometrical assumption over the constraint set, second-order necessary conditions are also established.
• Interacting particle systems and Yaglom limit approximation of diffusions with unbounded drift
  
  • Villemonais Denis
  , 2010. We study the existence and the exponential ergodicity of a general interacting particle system, whose components are driven by independent diffusion processes with values in an open subset of $\mathds{R}^d$, $d\geq 1$. The interaction occurs when a particle hits the boundary: it jumps to a position chosen with respect to a probability measure depending on the position of the whole system. Then we study the behavior of such a system when the number of particles goes to infinity. This leads us to an approximation method for the Yaglom limit of multi-dimensional diffusion processes with unbounded drift defined on an unbounded open set. While most of known results on such limits are obtained by spectral theory arguments and are concerned with existence and uniqueness problems, our approximation method allows us to get numerical values of quasi-stationary distributions, which find applications to many disciplines. We end the paper with numerical illustrations of our approximation method for stochastic processes related to biological population models.
• Visualisation tridimensionnelle de la dynamique de Verhulst
  
  • Colonna Jean-François
  , 2010. Tridimensional visualization of the Verhulst dynamics (Visualisation tridimensionnelle de la dynamique de Verhulst)
• Visualisation tridimensionnelle de la dynamique de Verhulst -'Les Vaisseaux du Temps', un hommage à Stephen Baxter
  
  • Colonna Jean-François
  , 2010. Tridimensional visualization of the Verhulst dynamics -'Time Ships', a tribute to Stephen Baxter- (Visualisation tridimensionnelle de la dynamique de Verhulst -'Les Vaisseaux du Temps', un hommage à Stephen Baxter-)