Publications

2012

• MODEL UNCERTAINTY IN FINANCE AND SECOND ORDER BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS
  
  • Zhou Chao
  , 2012. The main objective of this PhD thesis is to study some financial mathematics problems in an incomplete market with model uncertainty. In recent years, the theory of second order backward stochastic differential equations (2BSDEs for short) has been developed by Soner, Touzi and Zhang on this topic. In this thesis, we adopt their point of view. This thesis contains of four key parts related to 2BSDEs. In the first part, we generalize the 2BSDEs theory initially introduced in the case of Lipschitz continuous generators to quadratic growth generators. This new class of 2BSDEs will then allow us to consider the robust utility maximization problem in non-dominated models. In the second part, we study this problem for exponential utility, power utility and logarithmic utility. In each case, we give a characterization of the value function and an optimal investment strategy via the solution to a 2BSDE. In the third part, we provide an existence and uniqueness result for second order reflected BSDEs with lower obstacles and Lipschitz generators, and then we apply this result to study the problem of American contingent claims pricing with uncertain volatility. In the fourth part, we define a notion of 2BSDEs with jumps, for which we prove the existence and uniqueness of solutions in appropriate spaces. We can interpret these equations as standard BSDEs with jumps, under both volatility and jump measure uncertainty. As an application of these results, we shall study a robust exponential utility maximization problem under model uncertainty, where the uncertainty affects both the volatility process and the jump measure.
• Anaglyphe d'une interpolation tridimensionnelle entre une structure fractale et un réseau cubique
  
  • Colonna Jean-François
  , 2012. Anaglyph of a tridimensional interpolation between a fractal structure and a cubic mesh (Anaglyphe d'une interpolation tridimensionnelle entre une structure fractale et un réseau cubique)
• Anaglyphe d'un agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb')
  
  • Colonna Jean-François
  , 2012. Anaglyph of a close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') (Anaglyphe d'un agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb'))
• Anaglyphe d'une structure fractale tridimensionnelle -la 'mousse' de l'espace-temps ?
  
  • Colonna Jean-François
  , 2012. Anaglyph of a tridimensional fractal structure -the space-time foam ?- (Anaglyphe d'une structure fractale tridimensionnelle -la 'mousse' de l'espace-temps ?-)
• Anaglyphe d'une sphère fractale
  
  • Colonna Jean-François
  , 2012. Anaglyph of a fractal sphere (Anaglyphe d'une sphère fractale)
• Anaglyphe d'noeud '7-trèfle' torique fractal
  
  • Colonna Jean-François
  , 2012. Anaglyph of a fractal 7-foil torus knot (Anaglyphe d'noeud '7-trèfle' torique fractal)
• Anaglyphe d'une structure fractale filamenteuse tridimensionnelle
  
  • Colonna Jean-François
  , 2012. Anaglyph of a tridimensional filamentous fractal structure (Anaglyphe d'une structure fractale filamenteuse tridimensionnelle)
• Anaglyphe d'un ruban de Möbius fractal
  
  • Colonna Jean-François
  , 2012. Anaglyph of a fractal Möbius strip (Anaglyphe d'un ruban de Möbius fractal)
• Anaglyphe d'un ensemble de Mandelbrot dans l'ensemble des pseudo-quaternions (un 'MandelBulb') -'la ronde des enfants' ou 'la conscience émergeant des Mathématiques'
  
  • Colonna Jean-François
  , 2012. Anaglyph of a pseudo-quaternionic Mandelbrot set (a 'MandelBulb') -'the children round' or 'the consciousness emerging from Mathematics'- (Anaglyphe d'un ensemble de Mandelbrot dans l'ensemble des pseudo-quaternions (un 'MandelBulb') -'la ronde des enfants' ou 'la conscience émergeant des Mathématiques'-)
• Anaglyphe d'une structure fractale tridimensionnelle
  
  • Colonna Jean-François
  , 2012. Anaglyph of a tridimensional fractal structure (Anaglyphe d'une structure fractale tridimensionnelle)
• Global stability for the multi-channel Gel'fand-Calderón inverse problem in two dimensions
  
  • Santacesaria Matteo
  Bulletin des Sciences Mathématiques, Elsevier, 2012, 136 (7), pp.731-744. We prove a global logarithmic stability estimate for the multi-channel Gel'fand-Calderón inverse problem on a two-dimensional bounded domain, i.e. the inverse boundary value problem for the equation $-\Delta \psi + v\, \psi = 0$ on $D$, where $v$ is a smooth matrix-valued potential defined on a bounded planar domain $D$. (10.1016/j.bulsci.2012.02.004)
  DOI : 10.1016/j.bulsci.2012.02.004
• Anaglyphe d'noeud '3-trèfle' torique fractal
  
  • Colonna Jean-François
  , 2012. Anaglyph of a fractal 3-foil torus knot (Anaglyphe d'noeud '3-trèfle' torique fractal)
• Anaglyphe d'un agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-quaternions (un 'Mandelbulb')
  
  • Colonna Jean-François
  , 2012. Anaglyph of a close-up on a pseudo-quaternionic Mandelbrot set (a 'Mandelbulb') (Anaglyphe d'un agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-quaternions (un 'Mandelbulb'))
• Local stability and robustness of sparse dictionary learning in the presence of noise
  
  • Jenatton Rodolphe
  • Gribonval Rémi
  • Bach Francis
  , 2012, pp.41. A popular approach within the signal processing and machine learning communities consists in modelling signals as sparse linear combinations of atoms selected from a learned dictionary. While this paradigm has led to numerous empirical successes in various fields ranging from image to audio processing, there have only been a few theoretical arguments supporting these evidences. In particular, sparse coding, or sparse dictionary learning, relies on a non-convex procedure whose local minima have not been fully analyzed yet. In this paper, we consider a probabilistic model of sparse signals, and show that, with high probability, sparse coding admits a local minimum around the reference dictionary generating the signals. Our study takes into account the case of over-complete dictionaries and noisy signals, thus extending previous work limited to noiseless settings and/or under-complete dictionaries. The analysis we conduct is non-asymptotic and makes it possible to understand how the key quantities of the problem, such as the coherence or the level of noise, can scale with respect to the dimension of the signals, the number of atoms, the sparsity and the number of observations.
• Anaglyphe d'une interpolation tridimensionnelle entre une structure fractale et un réseau cubique
  
  • Colonna Jean-François
  , 2012. Anaglyph of a tridimensional interpolation between a fractal structure and a cubic mesh (Anaglyphe d'une interpolation tridimensionnelle entre une structure fractale et un réseau cubique)
• Modified Suliciu relaxation system and exact resolution of isolated shock waves
  
  • Chalons Christophe
  • Coquel Frédéric
  , 2012. We present a new Approximate Riemann Solver (ARS) for the gas dynamics equations in Lagrangian coordinates and with general non linear pressure laws. The design of this new ARS relies on a generalized Suliciu pressure relaxation approach. It gives by construction the exact solutions for isolated entropic shocks and we prove that it is Lipschitz-continuous and satisfies an entropy inequality. Finally, the ARS is used to develop either a classical entropy conservative Godunov-type method, or a Glimm-type (random sampling based Godunov-type) method able to generate infinitely sharp discrete shock profiles. Numerical experiments are proposed to prove the validity of these approaches.
• Asymptotic expansions for interior penalty solutions of control constrained linear-quadratic problems
  
  • Alvarez Felipe
  • Bolte Jérôme
  • Bonnans J. Frederic
  • Silva Francisco
  Mathematical Programming, Series A, Springer, 2012, 135 (1-2), pp.473-507. We consider a quadratic optimal control problem governed by a nonautonomous affine differential equation subject to nonnegativity control constraints. For a general class of interior penalty functions, we show how to compute the principal term of the pointwise expansion of the state and the adjoint state. Our main argument relies on the following fact: If the control of the initial problem satisfies strict complementarity conditions for its Hamiltonian except for a finite number of times, the estimates for the penalized optimal control problem can be derived from the estimations of a related stationary problem. Our results provide several types of efficiency measures of the penalization technique: error estimations of the control for $L^s$ norms ($s$ in $[1,+\infty]$), error estimations of the state and the adjoint state in Sobolev spaces $W^{1,s}$ ($s$ in $[1,+\infty)$) and also error estimates for the value function. For the $L^1$ norm and the logarithmic penalty, the optimal results are given. In this case we indeed establish that the penalized control and the value function errors are of order $O(\eps|\log\eps|)$.
• Enhanced controllability of low Reynolds number swimmers in the presence of a wall
  
  • Alouges François
  • Giraldi Laetitia
  , 2012. Swimming, i.e., being able to advance in the absence of external forces by performing cyclic shape changes, is particularly demanding at low Reynolds numbers which is the regime of interest for micro-organisms and micro-robots. We focus on self-propelled stokesian robots composed of assemblies of balls and we prove that the presence of a wall has an effect on their motility. More precisely, we demonstrate that a controllable swimmer remains controllable in a half space whereas the reachable set of a non fully controllable one is affected by the presence of a wall.
• Visualisation tridimensionnelle de la conjecture ABC
  
  • Colonna Jean-François
  , 2012. Tridimensional visualization of the ABC conjecture (Visualisation tridimensionnelle de la conjecture ABC)
• Variation artistique sur des noeuds '3/5/7-trèfle' toriques fractals
  
  • Colonna Jean-François
  , 2012. Artistic variation on fractal 3/5/7-foil torus knots (Variation artistique sur des noeuds '3/5/7-trèfle' toriques fractals)
• La conjecture ABC
  
  • Colonna Jean-François
  , 2012. The ABC conjecture (La conjecture ABC)
• Variation artistique sur des noeuds '3/5/7-trèfle' toriques fractals
  
  • Colonna Jean-François
  , 2012. Artistic variation on fractal 3/5/7-foil torus knots (Variation artistique sur des noeuds '3/5/7-trèfle' toriques fractals)
• Un ensemble de 4x3 stéréogrammes d'un noeud '5-trèfle' torique fractal
  
  • Colonna Jean-François
  , 2012. A set of 4x3 stereograms of a fractal 5-foil torus knot (Un ensemble de 4x3 stéréogrammes d'un noeud '5-trèfle' torique fractal)
• Un ensemble de 4x3 stéréogrammes d'un noeud '3-trèfle' torique fractal
  
  • Colonna Jean-François
  , 2012. A set of 4x3 stereograms of a fractal 3-foil torus knot (Un ensemble de 4x3 stéréogrammes d'un noeud '3-trèfle' torique fractal)
• Un ensemble de 4x3 stéréogrammes d'un noeud '7-trèfle' torique fractal
  
  • Colonna Jean-François
  , 2012. A set of 4x3 stereograms of a fractal 7-foil torus knot (Un ensemble de 4x3 stéréogrammes d'un noeud '7-trèfle' torique fractal)