Publications

2011

• Agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb')
  
  • Colonna Jean-François
  , 2011. Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') (Agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb'))
• Agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb') avec une transformation 'complexe' du module dans l'ensemble des pseudo-octonions
  
  • Colonna Jean-François
  , 2011. Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') with a 'complex' module transformation in the pseudo-octonionic space (Agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb') avec une transformation 'complexe' du module dans l'ensemble des pseudo-octonions)
• Agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb') avec une transformation 'complexe' du module dans l'ensemble des pseudo-octonions
  
  • Colonna Jean-François
  , 2011. Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') with a 'complex' module transformation in the pseudo-octonionic space (Agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb') avec une transformation 'complexe' du module dans l'ensemble des pseudo-octonions)
• Agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb')
  
  • Colonna Jean-François
  , 2011. Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') (Agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb'))
• Nuages
  
  • Colonna Jean-François
  , 2011. Clouds (Nuages)
• Normal forms and invariants for 2-dimensional almost-Riemannian structures
  
  • Boscain Ugo
  • Charlot Grégoire
  • Ghezzi Roberta
  , 2011. Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. Generically, there are three types of points: Riemannian points where the two vector fields are linearly independent, Grushin points where the two vector fields are collinear but their Lie bracket is not, and tangency points where the two vector fields and their Lie bracket are collinear and the missing direction is obtained with one more bracket. In this paper we consider the problem of finding normal forms and functional invariants at each type of point. We also require that functional invariants are ''complete'' in the sense that they permit to recognize locally isometric structures. The problem happens to be equivalent to the one of finding a smooth canonical parameterized curve passing through the point and being transversal to the distribution. For Riemannian points such that the gradient of the Gaussian curvature $K$ is different from zero, we use the level set of $K$ as support of the parameterized curve. For Riemannian points such that the gradient of the curvature vanishes (and under additional generic conditions), we use a curve which is found by looking for crests and valleys of the curvature. For Grushin points we use the set where the vector fields are parallel. Tangency points are the most complicated to deal with. The cut locus from the tangency point is not a good candidate as canonical parameterized curve since it is known to be non-smooth. Thus, we analyse the cut locus from the singular set and we prove that it is not smooth either. A good candidate appears to be a curve which is found by looking for crests and valleys of the Gaussian curvature. We prove that the support of such a curve is uniquely determined and has a canonical parametrization.
• Joyeux Noël 2011
  
  • Colonna Jean-François
  , 2011. Merry Christmas 2011 (Joyeux Noël 2011)
• Visualisation tridimensionnelle de la dynamique de Verhulst avec une transformation conforme QxQ dans l'ensemble des pseudo-quaternions
  
  • Colonna Jean-François
  , 2011. Tridimensional visualization of the Verhulst dynamics with a QxQ conformal transformation in the pseudo-quaternionic space (Visualisation tridimensionnelle de la dynamique de Verhulst avec une transformation conforme QxQ dans l'ensemble des pseudo-quaternions)
• Visualisation tridimensionnelle de la dynamique de Verhulst avec une transformation conforme QxQ dans l'ensemble des pseudo-quaternions
  
  • Colonna Jean-François
  , 2011. Tridimensional visualization of the Verhulst dynamics with a QxQ conformal transformation in the pseudo-quaternionic space (Visualisation tridimensionnelle de la dynamique de Verhulst avec une transformation conforme QxQ dans l'ensemble des pseudo-quaternions)
• Visualisation tridimensionnelle de la dynamique de Verhulst avec une transformation conforme 1/Q dans l'ensemble des pseudo-quaternions
  
  • Colonna Jean-François
  , 2011. Tridimensional visualization of the Verhulst dynamics with a 1/Q conformal transformation in the pseudo-quaternionic space (Visualisation tridimensionnelle de la dynamique de Verhulst avec une transformation conforme 1/Q dans l'ensemble des pseudo-quaternions)
• Visualisation tridimensionnelle de la dynamique de Verhulst avec une transformation conforme QxQ dans l'ensemble des pseudo-quaternions
  
  • Colonna Jean-François
  , 2011. Tridimensional visualization of the Verhulst dynamics with a QxQ conformal transformation in the pseudo-quaternionic space (Visualisation tridimensionnelle de la dynamique de Verhulst avec une transformation conforme QxQ dans l'ensemble des pseudo-quaternions)
• Visualisation tridimensionnelle de la dynamique de Verhulst avec une transformation conforme 1/Q dans l'ensemble des pseudo-quaternions
  
  • Colonna Jean-François
  , 2011. Tridimensional visualization of the Verhulst dynamics with a 1/Q conformal transformation in the pseudo-quaternionic space (Visualisation tridimensionnelle de la dynamique de Verhulst avec une transformation conforme 1/Q dans l'ensemble des pseudo-quaternions)
• Second order analysis of optimal control problems with singular arcs. Optimality conditions and shooting algorithm.
  
  • Aronna Maria Soledad
  , 2011. This thesis deals with optimal control problems for systems that are affine in one part of the control variable. First, we state necessary and sufficient second order conditions when all control variables enter linearly. We have bound control constraints and a bang-singular solution. The sufficient condition is restricted to the scalar control case. We propose a shooting algorithm and provide a sufficient condition for its local quadratic convergence. This condition guarantees the stability of the optimal solution and the local quadratic convergence of the algorithm for the perturbed problem in some cases. We present numerical tests that validate our method. Afterwards, we investigate an optimal control problems with systems that are affine in one part of the control variable. We obtain second order necessary and sufficient conditions for optimality. We propose a shooting algorithm, and we show that the sufficient condition just mentioned is also sufficient for the local quadratic convergence. Finally, we study a model of optimal hydrothermal scheduling. We investigate, by means of necessary conditions due to Goh, the possible occurrence of a singular arc.
• Visualisation tridimensionnelle de la dynamique de Verhulst avec une transformation conforme 1/Q dans l'ensemble des pseudo-quaternions
  
  • Colonna Jean-François
  , 2011. Tridimensional visualization of the Verhulst dynamics with a 1/Q conformal transformation in the pseudo-quaternionic space (Visualisation tridimensionnelle de la dynamique de Verhulst avec une transformation conforme 1/Q dans l'ensemble des pseudo-quaternions)
• Stochastic control methods for optimal transportation and probabilistic numerical schemes for PDEs
  
  • Tan Xiaolu
  , 2011. This thesis deals with the numerical methods for a fully nonlinear degenerate parabolic partial differential equations (PDEs), and for a controlled nonlinear PDEs problem which results from a mass transportation problem. The manuscript is divided into four parts. In a first part of the thesis, we are interested in the necessary and sufficient condition of the monotonicity of finite difference $\theta$-scheme for a one-dimensional diffusion equations. An explicit formula is given in case of the heat equation, which is weaker than the classical Courant-Friedrichs-Lewy (CFL) condition. In a second part, we consider a fully nonlinear degenerate parabolic PDE and propose a splitting scheme for its numerical resolution. The splitting scheme combines a probabilistic scheme and the semi-Lagrangian scheme, and in total, it can be viewed as a Monte-Carlo scheme for PDEs. We provide a convergence result as well as a rate of convergence. In the third part of the thesis, we study an optimal mass transportation problem. The mass is transported by the controlled drift-diffusion dynamics, and the associated cost depends on the trajectories, the drift as well as the diffusion coefficient of the dynamics. We prove a strong duality result for the transportation problem, thus extending the Kantorovich duality to our context. The dual formulation maximizes a value function on the space of all bounded continuous functions, and every value function corresponding to a bounded continuous function is the solution to a stochastic control problem. In the Markovian cases, we prove the dynamic programming principle of the optimal control problems, and we propose a gradient-projection algorithm for the numerical resolution of the dual problem, and provide a convergence result. Finally, in a fourth part, we continue to develop the dual approach of mass transportation problem with its applications in the computation of the model-independent no-arbitrage price bound of the variance option in a vanilla-liquid market. After a first analytic approximation, we propose a gradient-projection algorithm to approximate the bound as well as the corresponding static strategy in vanilla options.
• A journey through second order BSDEs and other contemporary issues in mathematical finance.
  
  • Possamaï Dylan
  , 2011. This PhD dissertation presents two independent research topics dealing with contemporary issues in mathematical finance, the second one being divided into into two distinct problems. Throughout the first part of the dissertation, we study the notion of second order backward stochastic differential equations (2BSDE in the following), first introduced by Cheredito, Soner, Touzi and Victoir, then reformulated by Soner, Touzi and Zhang. We start by proving an extension of their existence and uniqueness results to the case of a continuous generator with linear growth. Then, we pursue our study with another extension to the case of a quadratic generator. The theoretical results obtained in that chapter allow us to solve a problem of utility maximization for an investor in an incomplete market, the source of incompleteness being on one hand the restrictions on the class of admissible trading strategies, and on the other hand the fact that the volatility of the market is uncertain. We prove the existence of optimal strategies, we characterize the value function of the problem thanks to a 2BSDE and solve explicetely several examples which give further insight into the main modifications introduced by the uncertain volatility framework. We conclude the first part of the dissertation by introducing the notion of 2BSDEs reflected on an obstacle. We prove existence and uniqueness of the solutions of those equations and propose an application to the pricing problem of American options under volatility uncertainty. The first chapter of the second part of the dissertation deals with a problem of option pricing in an illiquidity model. We provide asymptotic expansions of those prices in the infinite liquidity limit and highlight a transition phase effect depending on the regularity of the payoff considered. We also give numerical results. Finally, the last chapter of this thesis is devoted to a Principal/Agent problem with moral hazard. A bank (the agent) has a certain number of defaultable loans and is ready to exchange their interests with the promess of payments. The bank can influence the default probabilities by choosing whether it monitors the loans or not, this monitoring being costly for the bank. Those choices are only known by the bank itself. Investors (the principal) want to design contracts which maximize their utility while implicitely giving incentives to the bank to monitor all the loans at all times. We solve explicitely this optimal control problem, we describe the associated optimal contract and its economic implications and provide some numerical simulations.
• A Stochastic Dynamic Principle for Hybrid Systems with Execution Delay and Decision Lags
  
  • Aouchiche K.
  • Bonnans J. Frederic
  • Granato Giovanni
  • Zidani Hasnaa
  , 2011, pp.6788-6793. This work presents a stochastic dynamic programming (SDP) algorithm that aims at minimizing an economic criteria based on the total energy consumption of a range extender electric vehicle (REEV). This algorithm integrates information from the REEV's navigation system in order to obtain some information about future expected vehicle speed. The model of the vehicle's energetic system, which consists of a high-voltage (HV) battery, the main energy source, and an internal combustion engine (ICE), working as an auxiliary energy source), is written as a hybrid dynamical system and the associated optimization problem in the hybrid optimal control framework. The hybrid optimal control problem includes two important physical constraints on the ICE, namely, an activation delay and a decision lag. Three methods for the inclusion of such physical constraints are studied. After introducing the SDP algorithm formulation we comment on numerical results of the stochastic algorithm and its deterministic counterpart.
• Matrices aléatoires et probabilités libres
  
  • Benaych-Georges Florent
  , 2011. Dans ce texte est présentée une sélection des travaux de l'auteur, portant, par exemple, sur la convolution libre rectangulaire, la transition de phase BBP, l'infinie divisibilité libre, les vecteurs propres de matrices de Wigner, etc...
• La pseudo-sphère -un hommage à Jacques Cartier
  
  • Colonna Jean-François
  , 2011. The pseudo-sphere -a Tribute to Jacques Cartier- (La pseudo-sphère -un hommage à Jacques Cartier-)
• On some applications of functions of bounded variation in finite and infinite dimension
  
  • Goldman Michael
  , 2011. The aim of this thesis is to investigate some applications of the functions of bounded variation and sets of finite perimeter. We mainly focus on applications in image processing, geometry and infinite dimensional analysis. We study first a Primal-Dual method proposed by Appleton and Talbot for solving some imaging problems. We give a new interpretation of this method which leads to a better mathematical understanding. This enables us for example to prove the convergence of the method and give new a posteriori estimates which are very important for numerical use. We then consider the problem of prescribed mean curvature surfaces in periodic environment. Using the theory of sets of finite perimeter, we prove existence of compact approximated solutions to this problem. We also study the asymptotic behavior of these solutions when their volume goes to infinity. The last two parts of the thesis are devoted to the study of some geometric problems in Wiener spaces. Studying on the one hand, the relationship between symmetrization, semi-continuity and isoperimetric inequalities, we compute the relaxation of the perimeter in this infinite dimensional setting and give an elliptic approximation of this lower semicontinuous envelope. On the other hand, we show convexity of the minimizers for some variational problems in Wiener spaces. One of the main ingredients in this study is the generalization of representations formulas for integral functionals in this setting.
• Entrelacs
  
  • Colonna Jean-François
  , 2011. Intertwining (Entrelacs)
• Entrelacs
  
  • Colonna Jean-François
  , 2011. Intertwining (Entrelacs)
• Asymptotic expansions for interior solutions of semilinear elliptic problems
  
  • Bonnans J. Frederic
  • Silva Francisco J.
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2011, 49 (6), pp.2494-2517. In this work we consider 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 nonnegative. The approach is to consider an associated parametrized family of penalized problems, whose solutions define a central path converging to the solution of the original problem. Our aim is to obtain an asymptotic expansion for the solutions of the penalized problems around the solution of the original problem. This approach allows us to obtain some specific error bounds in various norms and for a general class of barrier functions. In this manner, we generalize the results of the previous work which were obtained in the ODE framework.
• Limit theorems for Markov processes indexed by continuous time Galton-Watson trees
  
  • Bansaye Vincent
  • Delmas Jean-François
  • Marsalle Laurence
  • Tran Viet Chi
  The Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2011, 21 (6), pp.2263-2314. We study the evolution of a particle system whose genealogy is given by a supercritical continuous time Galton-Watson tree. The particles move independently according to a Markov process and when a branching event occurs, the offspring locations depend on the position of the mother and the number of offspring. We prove a law of large numbers for the empirical measure of individuals alive at time $t$. This relies on a probabilistic interpretation of its intensity by mean of an auxiliary process. This latter has the same generator as the Markov process along the branches plus additional branching events, associated with jumps of accelerated rate and biased distribution. This comes from the fact that choosing an individual uniformly at time $t$ favors lineages with more branching events and larger offspring number. The central limit theorem is considered on a special case. Several examples are developed, including applications to splitting diffusions, cellular aging, branching Lévy processes and ancestral lineages. (10.1214/10-AAP757)
  DOI : 10.1214/10-AAP757
• Distributions quasi-stationnaires et méthodes particulaires pour l'approximation de processus conditionnés
  
  • Villemonais Denis
  , 2011. Ma thèse porte sur l'étude de la distribution de processus stochastiques avec absorption et leur approximation. Ces processus trouvent des applications dans de nombreux domaines, tels que l'écologie, la finance ou les études de fiabilité. Nous étudions en particulier l'évolution en temps long de la distribution de processus de Markov avec absorption. La distribution limite d'un processus conditionné à ne pas être éteint au moment où on l'observe permet de décrire et d'expliquer des comportements non-triviaux, comme les plateaux de mortalité. Lorsqu'une telle distribution existe, elle est appelée distribution quasi-stationnaire. Dans le premier chapitre, nous rappelons et démontrons en toute généralités des propriétés propres à ces distributions. Dans les chapitres suivants, nous démontrons dans une grande généralité une méthode particulaire d'approximation des distributions de processus de Markov conditionnés à ne pas être absorbés et de leur limite distribution quasi-stationnaire. Des programmes en C++ ont été écrits afin d'implémenter numériquement l'approximation particulaire de distribution quasi-stationnaires de processus provenant de modèles biologiques, tels que les diffusions de Wright-Fisher et les diffusions de Lotka-Volterra. La méthode d'approximation démontrée dans cette thèse associée à des méthodes de couplage nous permet également d'obtenir des nouveaux résultats d'existence et d'unicité de distributions quasi-stationnaires, ainsi que de démontrer des propriétés de mélanges nouvelles pour les diffusions conditionnées à ne pas sortir d'un ouvert borné.