Publications

2011

• Un ensemble de Julia brumeux dans l'ensemble des pseudo-octonions (comme un 'MandelBulb' : un 'JuliaBulb') calculé pour A=(-0.5815147625160462,+0.6358885017421603,0,0,0,0,0,0)
  
  • Colonna Jean-François
  , 2011. A foggy pseudo-octonionic Julia set ('MandelBulb' like : a 'JuliaBulb') computed with A=(-0.5815147625160462,+0.6358885017421603,0,0,0,0,0,0) (Un ensemble de Julia brumeux dans l'ensemble des pseudo-octonions (comme un 'MandelBulb' : un 'JuliaBulb') calculé pour A=(-0.5815147625160462,+0.6358885017421603,0,0,0,0,0,0))
• 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'))
• Un ensemble de Julia brumeux dans l'ensemble des pseudo-octonions (comme un 'MandelBulb' : un 'JuliaBulb') calculé pour A=(-0.5815147625160462,+0.6358885017421603,0,0,0,0,0,0)
  
  • Colonna Jean-François
  , 2011. A foggy pseudo-octonionic Julia set ('MandelBulb' like : a 'JuliaBulb') computed with A=(-0.5815147625160462,+0.6358885017421603,0,0,0,0,0,0) (Un ensemble de Julia brumeux dans l'ensemble des pseudo-octonions (comme un 'MandelBulb' : un 'JuliaBulb') calculé pour A=(-0.5815147625160462,+0.6358885017421603,0,0,0,0,0,0))
• Un ensemble de Julia brumeux dans l'ensemble des pseudo-octonions (comme un 'MandelBulb' : un 'JuliaBulb') calculé pour A=(-0.5815147625160462,+0.6358885017421603,0,0,0,0,0,0)
  
  • Colonna Jean-François
  , 2011. A foggy pseudo-octonionic Julia set ('MandelBulb' like : a 'JuliaBulb') computed with A=(-0.5815147625160462,+0.6358885017421603,0,0,0,0,0,0) (Un ensemble de Julia brumeux dans l'ensemble des pseudo-octonions (comme un 'MandelBulb' : un 'JuliaBulb') calculé pour A=(-0.5815147625160462,+0.6358885017421603,0,0,0,0,0,0))
• Un ensemble de Julia brumeux dans l'ensemble des pseudo-octonions (comme un 'MandelBulb' : un 'JuliaBulb') calculé pour A=(-0.5815147625160462,+0.6358885017421603,0,0,0,0,0,0)
  
  • Colonna Jean-François
  , 2011. A foggy pseudo-octonionic Julia set ('MandelBulb' like : a 'JuliaBulb') computed with A=(-0.5815147625160462,+0.6358885017421603,0,0,0,0,0,0) (Un ensemble de Julia brumeux dans l'ensemble des pseudo-octonions (comme un 'MandelBulb' : un 'JuliaBulb') calculé pour A=(-0.5815147625160462,+0.6358885017421603,0,0,0,0,0,0))
• Un ensemble de Julia brumeux dans l'ensemble des pseudo-octonions (comme un 'MandelBulb' : un 'JuliaBulb') calculé pour A=(-0.5815147625160462,+0.6358885017421603,0,0,0,0,0,0)
  
  • Colonna Jean-François
  , 2011. A foggy pseudo-octonionic Julia set ('MandelBulb' like : a 'JuliaBulb') computed with A=(-0.5815147625160462,+0.6358885017421603,0,0,0,0,0,0) (Un ensemble de Julia brumeux dans l'ensemble des pseudo-octonions (comme un 'MandelBulb' : un 'JuliaBulb') calculé pour A=(-0.5815147625160462,+0.6358885017421603,0,0,0,0,0,0))
• Un ensemble de Julia brumeux dans l'ensemble des pseudo-octonions (comme un 'MandelBulb' : un 'JuliaBulb') calculé pour A=(-0.5815147625160462,+0.6358885017421603,0,0,0,0,0,0)
  
  • Colonna Jean-François
  , 2011. A foggy pseudo-octonionic Julia set ('MandelBulb' like : a 'JuliaBulb') computed with A=(-0.5815147625160462,+0.6358885017421603,0,0,0,0,0,0) (Un ensemble de Julia brumeux dans l'ensemble des pseudo-octonions (comme un 'MandelBulb' : un 'JuliaBulb') calculé pour A=(-0.5815147625160462,+0.6358885017421603,0,0,0,0,0,0))
• Un ensemble de Julia brumeux dans l'ensemble des pseudo-octonions (comme un 'MandelBulb' : un 'JuliaBulb') calculé pour A=(-0.5815147625160462,+0.6358885017421603,0,0,0,0,0,0)
  
  • Colonna Jean-François
  , 2011. A foggy pseudo-octonionic Julia set ('MandelBulb' like : a 'JuliaBulb') computed with A=(-0.5815147625160462,+0.6358885017421603,0,0,0,0,0,0) (Un ensemble de Julia brumeux dans l'ensemble des pseudo-octonions (comme un 'MandelBulb' : un 'JuliaBulb') calculé pour A=(-0.5815147625160462,+0.6358885017421603,0,0,0,0,0,0))
• A large time asymptotics for the solution of the Cauchy problem for the Novikov-Veselov equation at negative energy with non-singular scattering data
  
  • Kazeykina Anna
  , 2011. In the present paper we are concerned with the Novikov--Veselov equation at negative energy, i.e. with the $ ( 2 + 1 ) $--dimensional analog of the KdV equation integrable by the method of inverse scattering for the two--dimensional Schrödinger equation at negative energy. We show that the solution of the Cauchy problem for this equation with non--singular scattering data behaves asymptotically as $ \frac{ \const }{ t^{ 3/4 } } $ in the uniform norm at large times $ t $. We also present some arguments which indicate that this asymptotics is optimal.
• Magnetic Equations with FreeFem++: the Grad-Shafranov Equation & the Current Hole
  
  • Deriaz Erwan
  • Després Bruno
  • Faccanoni Gloria
  • Gostaf Kirill
  • Imbert-Gérard Lise-Marie
  • Sadaka Georges
  • Sart Remy
  ESAIM: Proceedings, EDP Sciences, 2011, 32, pp.76-94. FreeFem++ is a software for the numerical solution of partial differential equations. It is based on finite element method. The FreeFem++ platform aims at facilitating teaching and basic research through prototyping. For the moment this platform is restricted to the numerical simulations of problems which admit a variational formulation. Our goal in this work is to evaluate the FreeFem++ tool on basic magnetic equations arising in Fusion Plasma in the context of the ITER project. First we consider the Grad-Shafranov equation, which is derived from the static ideal MHD equations assuming axisymetry. Some of the properties of the equation and its analytical solutions are discussed. Second we discretize a reduced resistive MHD model which admits solutions of the Grad-Shafranov equation as stationary solutions. Then the physical stability of these stationary solutions is investigated through numerical experiments and the numerical stability of the algorithm is discussed. (10.1051/proc/2011013)
  DOI : 10.1051/proc/2011013
• Semimartingales and Contemporary Issues in Quantitative Finance
  
  • Kchia Younes
  , 2011. In this thesis, we study various contemporary issues in quantitative finance. The first chapter is dedicated to the stability of the semimartingale property under filtration expansion. We study first progressive filtration expansions with random times. We show how semimartingale decompositions in the expanded filtration can be obtained using a natural link between progressive and initial expansions. The link is, on an intuitive level, that the two coincide after the random time. We make this idea precise and use it to establish known and new results in the case of expansion with a single random time. The methods are then extended to the multiple time case, without any restrictions on the ordering of the individual times. We then look to the expanded filtrations from the point of view of filtration shrinkage. We turn then to studying progressive filtration expansions with processes. Using results from the weak convergence of sigma fields theory, we first establish a semimartingale convergence theorem, which we apply in a filtration expansion with a process setting and provide sufficient conditions for a semimartingale of the base filtration to remain a semimartingale in the expanded filtration. A first set of results is based on a Jacod's type criterion for the increments of the process we want to expand with. An application to the expansion of a Brownian filtration with a time reversed diffusion is given through a detailed study and some known examples in the litterature are recovered and generalized. Finally, we focus on filtration expansion with continuous processes and derive two new results. The first one is based on a Jacod's type criterion for the successive hitting times of some levels and the second one is based on honest times assumptions for these hitting times. We provide examples and see how those can be used as first steps toward harmful dynamic insider trading models. In the expanded filtration the finite variation term of the price process can become singular and arbitrage opportunities (in the sense of FLVR) can therefore arise in these models. In the second chapter, we reconcile structural models and reduced form models in credit risk from the perspective of the information induced credit contagion effect. That is, given multiple firms, we are interested on the behaviour of the default intensity of one firm at the default times of the other firms. We first study this effect within different specifications of structural models and different levels of information. Since almost all examples are non tractable and computationally very involved, we then work with the simplifying assumption that conditional densities of the default times exist. The classical reduced-form and filtration expansion framework is therefore extended to the case of multiple, non-ordered defaults times having conditional densities. Intensities and pricing formulas are derived, revealing how information-driven default contagion arises in these models. We then analyze the impact of ordering the default times before expanding the filtration. While not important for pricing, the effect is significant in the context of risk management, and becomes even more pronounced for highly correlated and asymmetrically distributed defaults. We provide a general scheme for constructing and simulating the default times, given that a model for the conditional densities has been chosen. Finally, we study particular conditional density models and the information induced credit contagion effect within them. In the third chapter, we provide a methodology for a real time detection of bubbles. After the 2007 credit crisis, financial bubbles have once again emerged as a topic of current concern. An open problem is to determine in real time whether or not a given asset's price process exhibits a bubble. Due to recent progress in the characterization of asset price bubbles using the arbitrage-free martingale pricing technology, we are able to propose a new methodology for answering this question based on the asset's price volatility. We limit ourselves to the special case of a risky asset's price being modeled by a Brownian driven stochastic differential equation. Such models are ubiquitous both in theory and in practice. Our methods use non parametric volatility estimation techniques combined with the extrapolation method of reproducing kernel Hilbert spaces. We illustrate these techniques using several stocks from the alleged internet dot-com episode of 1998 - 2001, where price bubbles were widely thought to have existed. Our results support these beliefs. During May 2011, there was speculation in the financial press concerning the existence of a price bubble in the aftermath of the recent IPO of LinkedIn. We analyzed stock price tick data from the short lifetime of this stock through May 24, 2011, and we found that LinkedIn has a price bubble. The last chapter is about discretely sampled variance swaps, which are volatility derivatives that trade actively in OTC markets. To price these swaps, the continuously sampled approximation is often used to simplify the computations. The purpose of this chapter is to study the conditions under which this approximation is valid. Our first set of theorems characterize the conditions under which the discretely sampled variance swap values are finite, given the values of the continuous approximations exist. Surprisingly, for some otherwise reasonable price processes, the discretely sampled variance swap prices do not exist, thereby invalidating the approximation. Examples are provided. Assuming further that both variance swap values exist, we study sufficient conditions under which the discretely sampled values converge to their continuous counterparts. Because of its popularity in the literature, we apply our theorems to the 3/2 stochastic volatility model. Although we can show finiteness of all swap values, we can prove convergence of the approximation only for some parameter values.
• Curse of dimensionality reduction in max-plus based approximation methods: theoretical estimates and improved pruning algorithms
  
  • Gaubert Stephane
  • Mceneaney William
  • Qu Zheng
  , 2011. Max-plus based methods have been recently developed to approximate the value function of possibly high dimensional optimal control problems. A critical step of these methods consists in approximating a function by a supremum of a small number of functions (max-plus "basis functions") taken from a prescribed dictionary. We study several variants of this approximation problem, which we show to be continuous versions of the facility location and $k$-center combinatorial optimization problems, in which the connection costs arise from a Bregman distance. We give theoretical error estimates, quantifying the number of basis functions needed to reach a prescribed accuracy. We derive from our approach a refinement of the curse of dimensionality free method introduced previously by McEneaney, with a higher accuracy for a comparable computational cost.
• Detecting Long Distance Conditional Correlations Between Anatomical Regions Using Gaussian Graphical Models
  
  • Allassonnière Stéphanie
  • Jolivet Pierre
  • Giraud Christophe
  , 2011, pp.111-122. The conditional correlation patterns of an anatomical shape may provide some important information on the structure of this shape. We propose to investigate these patterns by Gaussian Graphical Modelling. We design a model which takes into account both local and long-distance dependencies. We provide an algorithm which estimates sparse long-distance conditional correlations, highlighting the most significant ones. The selection procedure is based on a criterion which quantifies the quality of the conditional correlation graph in terms of prediction. The preliminary results on AD versus control population show noticeable differences.
• Homogenization of a coupled problem for sound propagation in porous media
  
  • Alouges François
  • Augier Adeline
  • Graille Benjamin
  • Merlet Benoit
  , 2011. In this paper we study the acoustic properties of a microstructured material such as glass wool or foam. In our model, the solid matrix is governed by linear elasticity and the surrounding fluid obeys Stokes equations. The microstructure is assumed to be periodic at some small scale $\eps$ and the viscosity coefficient of the fluid is assumed to be of order $\eps^2$. We consider the time-harmonic regime forced by vibrations applied on a part of the boundary. We use the two-scale convergence theory to prove the convergence of the displacements to the solution of a homogeneous problem as the size of the microstructure shrinks to 0.
• Self-adaptive congestion control for multi-class intermittent connections in a communication network
  
  • Graham Carl
  • Robert Philippe
  Queueing Systems, Springer Verlag, 2011, 69, pp.237–257. A Markovian model of the evolution of intermittent connections of various classes in a communication network is established and investigated. Any connection evolves in a way which depends only on its class and the state of the network, in particular as to the route it uses among a subset of the network nodes. It can be either active (ON) when it is transmitting data along its route, or idle (OFF). The congestion of a given node is defined as a functional of the transmission rates of all ON connections going through it, and causes losses and delays to these connections. In order to control this, the ON connections self-adaptively vary their transmission rate in TCP-like fashion. The connections interact through this feedback loop. A Markovian model is provided by the states (OFF, or ON with some transmission rate) of the connections. The number of connections in each class being potentially huge, a mean-field limit result is proved with an appropriate scaling so as to reduce the dimensionality. In the limit, the evolution of the states of the connections can be represented by a non-linear system of stochastic differential equations, of dimension the number of classes. Additionally, it is shown that the corresponding stationary distribution can be expressed by the solution of a fixed-point equation of finite dimension. (10.1007/s11134-011-9260-z)
  DOI : 10.1007/s11134-011-9260-z
• Montagnes octonioniques avec arithmétique étendue
  
  • Colonna Jean-François
  , 2011. Octonionic mountains with extended arithmetics (Montagnes octonioniques avec arithmétique étendue)
• Agrandissement d'un ensemble de Julia brumeux dans l'ensemble des pseudo-octonions (comme un 'MandelBulb' : un 'JuliaBulb') calculé pour A=(-0.5815147625160462,+0.6358885017421603,0,0,0,0,0,0)
  
  • Colonna Jean-François
  , 2011. Close-up on a foggy pseudo-octonionic Julia set ('MandelBulb' like : a 'JuliaBulb') computed with A=(-0.5815147625160462,+0.6358885017421603,0,0,0,0,0,0) (Agrandissement d'un ensemble de Julia brumeux dans l'ensemble des pseudo-octonions (comme un 'MandelBulb' : un 'JuliaBulb') calculé pour A=(-0.5815147625160462,+0.6358885017421603,0,0,0,0,0,0))
• Agrandissement d'un ensemble de Julia brumeux dans l'ensemble des pseudo-octonions (comme un 'MandelBulb' : un 'JuliaBulb') calculé pour A=(-0.5815147625160462,+0.6358885017421603,0,0,0,0,0,0)
  
  • Colonna Jean-François
  , 2011. Close-up on a foggy pseudo-octonionic Julia set ('MandelBulb' like : a 'JuliaBulb') computed with A=(-0.5815147625160462,+0.6358885017421603,0,0,0,0,0,0) (Agrandissement d'un ensemble de Julia brumeux dans l'ensemble des pseudo-octonions (comme un 'MandelBulb' : un 'JuliaBulb') calculé pour A=(-0.5815147625160462,+0.6358885017421603,0,0,0,0,0,0))
• Agrandissement d'un ensemble de Julia brumeux dans l'ensemble des pseudo-octonions (comme un 'MandelBulb' : un 'JuliaBulb') calculé pour A=(-0.5815147625160462,+0.6358885017421603,0,0,0,0,0,0)
  
  • Colonna Jean-François
  , 2011. Close-up on a foggy pseudo-octonionic Julia set ('MandelBulb' like : a 'JuliaBulb') computed with A=(-0.5815147625160462,+0.6358885017421603,0,0,0,0,0,0) (Agrandissement d'un ensemble de Julia brumeux dans l'ensemble des pseudo-octonions (comme un 'MandelBulb' : un 'JuliaBulb') calculé pour A=(-0.5815147625160462,+0.6358885017421603,0,0,0,0,0,0))
• Computing estimates on material properties from transmission eigenvalues
  
  • Giorgi Giovanni
  • Haddar Houssem
  , 2011. This work is motivated by inverse scattering problems, those problems where one is interested in reconstructing the shape and the material properties of an inclusion from electromagnetic farfields measurements. More precisely we are interested in complementing the so called sampling methods, (those methods that enables one to reconstruct just the geometry of the scatterer), by providing estimates on the material properties. We shall use for that purpose the so-called transmission eigenvalues. Our method is based on reformulating the so-called interior transmission eigenvalue problem into an eigenvalue problem for the material coefficients. We shall restrict ourselves to the two dimensional setting of the problem and treat the cases of both TE and TM polarizations. We present a number of numerical experiments that validate our methodology for homogeneous and inhomogeneous inclusions and backgrounds. We also treat the case of a background with absorption and the case of scatterers with multiple connected components of different refractive indexes.
• L'ensemble de Julia dans l'ensemble des octonions calculé pour A=(0,1,0,0,0,0,0,0)
  
  • Colonna Jean-François
  , 2011. The octonionic Julia set computed with A=(0,1,0,0,0,0,0,0) (L'ensemble de Julia dans l'ensemble des octonions calculé pour A=(0,1,0,0,0,0,0,0))
• Un ensemble de Julia brumeux dans l'ensemble des pseudo-octonions (comme un 'MandelBulb' : un 'JuliaBulb') calculé pour A=(-0.5815147625160462,+0.6358885017421603,0,0,0,0,0,0)
  
  • Colonna Jean-François
  , 2011. A foggy pseudo-octonionic Julia set ('MandelBulb' like : a 'JuliaBulb') computed with A=(-0.5815147625160462,+0.6358885017421603,0,0,0,0,0,0) (Un ensemble de Julia brumeux dans l'ensemble des pseudo-octonions (comme un 'MandelBulb' : un 'JuliaBulb') calculé pour A=(-0.5815147625160462,+0.6358885017421603,0,0,0,0,0,0))
• L'ensemble de Julia dans l'ensemble des octonions calculé pour A=(-0.5815147625160462,+0.6358885017421603,0,0,0,0,0,0)
  
  • Colonna Jean-François
  , 2011. The octonionic Julia set computed with A=(-0.5815147625160462,+0.6358885017421603,0,0,0,0,0,0) (L'ensemble de Julia dans l'ensemble des octonions calculé pour A=(-0.5815147625160462,+0.6358885017421603,0,0,0,0,0,0))
• Un ensemble de Julia brumeux dans l'ensemble des pseudo-octonions (comme un 'MandelBulb' : un 'JuliaBulb') calculé pour A=(0,1,0,0,0,0,0,0)
  
  • Colonna Jean-François
  , 2011. A foggy pseudo-octonionic Julia set ('MandelBulb' like : a 'JuliaBulb') computed with A=(0,1,0,0,0,0,0,0) (Un ensemble de Julia brumeux dans l'ensemble des pseudo-octonions (comme un 'MandelBulb' : un 'JuliaBulb') calculé pour A=(0,1,0,0,0,0,0,0))
• 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'))