Publications

2013

• Stochastic expansion for the diffusion processes and applications to option pricing
  
  • Bompis Romain
  , 2013. This thesis deals with the approximation of the expectation of a functional (possibly depending on the whole path) applied to a diffusion process (possibly multidimensional). The motivation for this work comes from financial mathematics where the pricing of options is reduced to the calculation of such expectations. The rapidity for price computations and calibration procedures is a very strong operational constraint and we provide real-time tools (or at least more competitive than Monte Carlo simulations in the case of multidimensional diffusions) to meet these needs. In order to derive approximation formulas, we choose a proxy model in which analytical calculus are possible and then we use stochastic expansions around the proxy model and Malliavin calculus to approach the quantities of interest. In situation where Malliavin calculus can not be applied, we develop an alternative methodology combining Itô calculus and PDE arguments. All the approaches (from PDEs to stochastic analysis) allow to obtain explicit formulas and tight error estimates in terms of the model parameters. Although the final result is generally the same, the derivation can be quite different and we compare the approaches, first regarding the way in which the corrective terms are made explicit, second regarding the error estimates and the assumptions used for that. We consider various classes of models and functionals throughout the four Parts of the thesis. In the Part I, we focus on local volatility models and provide new price, sensitivity (delta) and implied volatility approximation formulas for vanilla products showing an improving accuracy in comparison to previous known formulas. We also introduce new results concerning the pricing of forward start options. The Part II deals with the analytical approximation of vanilla prices in models combining both local and stochastic volatility (Heston type). This model is very difficult to analyze because its moments can explode and because it is not regular in the Malliavin sense. The error analysis is original and the idea is to work on an appropriate regularization of the payoff and a suitably perturbed model, regular in the Malliavin sense and from which the distance with the initial model can be controlled. The Part III covers the pricing of regular barrier options in the framework of local volatility models. This is a difficult issue due to the indicator function on the exit times which is not considered in the literature. We use an approach mixing Itô calculus, PDE arguments, martingale properties and temporal convolutions of densities to decompose the approximation error and to compute correction terms. We obtain explicit and accurate approximation formulas under a martingale hypothesis. The Part IV introduces a new methodology (denoted by SAFE) for the efficient weak analytical approximation of multidimensional diffusions in a quite general framework. We combine the use of a Gaussian proxy to approximate the law of the multidimensional diffusion and a local interpolation of the terminal function using Finite Elements. We give estimates of the complexity of our methodology. We show an improved efficiency in comparison to Monte Carlo simulations in small and medium dimensions (up to 10).
• Random walk with heavy tail and negative drift conditioned by its minimum and final values
  
  • Bansaye Vincent
  • Vatutin Vladimir
  , 2013. We consider random walks with finite second moment which drifts to $-\infty$ and have heavy tail. We focus on the events when the minimum and the final value of this walk belong to some compact set. We first specify the associated probability. Then, conditionally on such an event, we finely describe the trajectory of the random walk. It yields a decomposition theorem with respect to a random time giving a big jump whose distribution can be described explicitly.
• Controllability and Optimal Strokes for N-link Micro-swimmer
  
  • Giraldi Laetitia
  • Martinon Pierre
  • Zoppello Marta
  , 2013. In this paper we focus on the N-link swimmer, a generalization of the classical 3-link Purcell swimmer. We use the Resistive Force Theory to express the equation of motion in a fluid with a low Reynolds number, and prove that the swimmer is controllable in the whole plane for N greater or equal than 3 and for almost every set of stick lengths. As a direct result, there exists an optimal swimming strategy to reach a desired configuration in minimum time. Numerical experiments for N=3 (Purcell swimmer) suggest that the optimal strategy is periodic, namely a sequence of identical strokes. Our results indicate that this candidate for an optimal stroke indeed gives a better displacement speed than the classical Purcell stroke. (10.1109/CDC.2013.6760480)
  DOI : 10.1109/CDC.2013.6760480
• Solving multichain stochastic games with mean payoff by policy iteration
  
  • Akian Marianne
  • Cochet-Terrasson Jean
  • Detournay Sylvie
  • Gaubert Stéphane
  , 2013, pp.1834-1841. Zero-sum stochastic games with finite state and action spaces, perfect information, and mean payoff criteria arise in particular from the monotone discretization of mean-payoff pursuit-evasion deterministic differential games. In that case no irreducibility assumption on the Markov chains associated to strategies are satisfied (multichain games). The value of such a game can be characterized by a system of nonlinear equations, involving the mean payoff vector and an auxiliary vector (relative value or bias). Cochet-Terrasson and Gaubert proposed in (C. R. Math. Acad. Sci. Paris, 2006) a policy iteration algorithm relying on a notion of nonlinear spectral projection (Akian and Gaubert, Nonlinear Analysis TMA, 2003), which allows one to avoid cycling in degenerate iterations. We give here a complete presentation of the algorithm, with details of implementation in particular of the nonlinear projection. This has led to the software PIGAMES and allowed us to present numerical results on pursuit-evasion games. (10.1109/CDC.2013.6760149)
  DOI : 10.1109/CDC.2013.6760149
• Sampling of singularly perturbed switched linear systems
  
  • El Hachemi Fouad
  • Sigalotti Mario
  • Daafouz Jamal
  , 2013, pp.CDROM. We consider several time-discretization algorithms for singularly perturbed switched systems. The algorithms correspond to different sampling times and the discretization procedure respects the splitting of each mode in fast and slow dynamics. We study whether such algorithms preserve the asymptotic or quadratic stability of the original continuous-time singularly perturbed switched system.
• Invariant sets of defocused switched systems
  
  • Nilsson Petter
  • Boscain Ugo
  • Sigalotti Mario
  • Newling James
  , 2013, pp.5987-5992. Abstract: We consider affine switched systems as perturbations of linear ones, the equilibria playing the role of perturbation parameters. We study the stability properties of an affine switched system under arbitrary switching, assuming that the corresponding linear system is uniformly exponentially stable. It turns out that the affine system admits a minimal invariant set Ω, whose properties we investigate. In the two-dimensional bi-switched case when both subsystems have non-real eigenvalues we are able to characterize Ω completely and to prove that all trajectories of the system converge to Ω. We also explore the behavior of minimal-time trajectories in Ω by constructing optimal syntheses. (10.1109/CDC.2013.6760834)
  DOI : 10.1109/CDC.2013.6760834
• La conjecture de Goldbach -la comète de Goldbach- pour les entiers pairs de 6 à 411678
  
  • Colonna Jean-François
  , 2013. The Goldbach conjecture -the Goldbach comet- for the even numbers from 6 to 411678 (La conjecture de Goldbach -la comète de Goldbach- pour les entiers pairs de 6 à 411678)
• La conjecture de Goldbach -la comète de Goldbach ou l'arc-en ciel de Goldbach- pour les entiers pairs de 6 à 411678
  
  • Colonna Jean-François
  , 2013. The Goldbach conjecture -the Goldbach comet or the Goldbach rainbow- for the even numbers from 6 to 411678 (La conjecture de Goldbach -la comète de Goldbach ou l'arc-en ciel de Goldbach- pour les entiers pairs de 6 à 411678)
• Geometric modeling of the movement based on an inverse optimal control approach
  
  • Jean Frédéric
  • Mason Paolo
  • Chittaro Francesca
  , 2013, pp.1816-1821. The present paper analyses a class of optimal control problems on geometric paths of the euclidean space, that is, curves parametrized by arc length. In the first part we deal with existence and robustness issues for such problems and we define the associated inverse optimal control problem. In the second part we discuss the inverse optimal control problem in the special case of planar trajectories and under additional assumptions. More precisely we define a criterion to restrict the study to a convenient class of costs based on the analysis of experimentally recorded trajectories. This method applies in particular to the case of human locomotion trajectories. (10.1109/cdc.2013.6760146)
  DOI : 10.1109/cdc.2013.6760146
• Formal Proofs for Global Optimization -- Templates and Sums of Squares
  
  • Magron Victor
  , 2013. The aim of this work is to certify lower bounds for real-valued multivariate functions, defined by semialgebraic or transcendental expressions and to prove their correctness by checking the certificates in the Coq proof system. The application range for such a tool is widespread; for instance Hales' proof of Kepler's conjecture involves thousands of nonlinear inequalities. The functions we are dealing with are nonlinear and involve semialgebraic operations as well as some transcendental functions like cos, arctan, exp, etc. Our general framework is to use different approximation methods to relax the original problem into a semialgebraic optimization problem. It leads to polynomial optimization problems which we solve by sparse sums of squares relaxations. First, we implement a classical scheme in global optimization. Namely, we approximate univariate transcendental functions with best uniform degree-d polynomial estimators. Then, we present an alternative method, which consists in bounding some of the constituents of the nonlinear function by suprema or infima of quadratic polynomials (max-plus approximation method, initially introduced in optimal control) with a carefully chosen curvature. Finally, we improve this approximation algorithm, by combining the ideas of the maxplus estimators and of the linear template method developed by Manna et al. (in static analysis). The nonlinear templates control the complexity of the semialgebraic estimators at the price of coarsening the maxplus approximations. In that way, we arrive at a new - template based - global optimization method, which exploits both the precision of sums of squares/SDP relaxations and the scalability of abstraction methods. We successfully implemented these approximation methods in a software package named NLCertify. This tool interleaves semialgebraic approximations with sums of squares witnesses to form certificates. It is interfaced with Coq and thus benefits from the trusted arithmetic available inside the proof assistant. This feature is used to produce, from the certificates, both valid underestimators and lower bounds for each approximated constituent. We illustrate the efficiency of NLCertify with various examples from the global optimization literature, as well as tight inequalities issued from the Flyspeck project.
• Les 12.723 premières décimales -base 6- de 'pi' considérées comme un grand nombre entier et visualisées comme une marche aléatoire tridimensionnelle 'absolue
  
  • Colonna Jean-Francois
  , 2013. The 12.723 first digits -base 6- of 'pi' viewed as a huge integer number and displayed as an 'absolute' tridimensional random walk (Les 12.723 premières décimales -base 6- de 'pi' considérées comme un grand nombre entier et visualisées comme une marche aléatoire tridimensionnelle 'absolue')
• Les 12.723 premières décimales -base 6- de 'pi' visualisées comme une marche aléatoire tridimensionnelle 'absolue
  
  • Colonna Jean-François
  , 2013. The 12.723 first digits -base 6- of 'pi' displayed as an 'absolute' tridimensional random walk (Les 12.723 premières décimales -base 6- de 'pi' visualisées comme une marche aléatoire tridimensionnelle 'absolue')
• Les 12.723 premières décimales -base 6- de 'pi' considérées comme un grand nombre entier et visualisées comme une marche aléatoire tridimensionnelle 'absolue
  
  • Colonna Jean-Francois
  , 2013. The 12.723 first digits -base 6- of 'pi' viewed as a huge integer number and displayed as an 'absolute' tridimensional random walk (Les 12.723 premières décimales -base 6- de 'pi' considérées comme un grand nombre entier et visualisées comme une marche aléatoire tridimensionnelle 'absolue')
• Les 12.723 premières décimales -base 6- de 'pi' considérées comme un grand nombre entier et visualisées comme une marche aléatoire tridimensionnelle 'absolue
  
  • Colonna Jean-François
  , 2013. The 12.723 first digits -base 6- of 'pi' viewed as a huge integer number and displayed as an 'absolute' tridimensional random walk (Les 12.723 premières décimales -base 6- de 'pi' considérées comme un grand nombre entier et visualisées comme une marche aléatoire tridimensionnelle 'absolue')
• Les 127.237 premières décimales -base 6- de 'pi' considérées comme un grand nombre entier et visualisées comme une marche aléatoire tridimensionnelle 'absolue
  
  • Colonna Jean-François
  , 2013. The 127.237 first digits -base 6- of 'pi' viewed as a huge integer number and displayed as an 'absolute' tridimensional random walk (Les 127.237 premières décimales -base 6- de 'pi' considérées comme un grand nombre entier et visualisées comme une marche aléatoire tridimensionnelle 'absolue')
• Modélisation probabiliste et éco-évolution d'une population diploïde
  
  • Coron Camille
  , 2013. 0n s'intéresse à la modélisation probabiliste pour l'évolution génétique de populations diploïdes, dans un contexte d'éco-évolution. La population considérée est modélisée par un processus de naissance et mort multi-types, avec interaction, et dont les taux de naissance modélisent la reproduction mendélienne. En particulier, la taille de la population considérée n'est pas constante et peut être petite. Une première partie du travail est consacrée à l'étude probabiliste du vortex d'extinction démo-génétique, un phénomène au cours duquel la taille d'une petite population décroît de plus en plus rapidement suite à des fixations de plus en plus fréquentes de mutations délétères. Nous donnons notamment une formule pour la probabilité de fixation d'un allèle légèrement délétère en fonction de la composition génétique de la population et nous prouvons l'existence d'un vortex d'extinction sous une hypothèse de mutations rares. Nous donnons par ailleurs des résultats numériques et une analyse biologique détaillée des comportements obtenus. Nous étudions en particulier l'impact du vortex sur la dynamique de la taille moyenne de population, et nous quantifions ce phénomène en fonction des paramètres écologiques. Dans une deuxième partie, sous une asymptotique de grande taille de population et événements de naissance et mort fréquents, nous étudions d'abord la convergence vers une dynamique lente-rapide et le comportement quasi-stationnaire d'une population diploïde caractérisée par sa composition génétique à un locus bi-allélique. Nous étudions en particulier la possibilité de coexistence en temps long de deux allèles dans la population conditionnée à ne pas être éteinte. Ensuite nous généralisons cette dynamique lente-rapide à une population présentant un nombre fini quelconque d'allèles. La population est alors modélisée par un processus à valeurs mesures dont nous prouvons la convergence lorsque le nombre d'allèles tend vers l'infini vers un superprocessus de Fleming-Viot généralisé, avec une taille de population variable et une sélection diploïde additive.
• Convex Relaxations for Permutation Problems
  
  • Fogel Fajwel
  • Jenatton Rodolphe
  • Bach Francis
  • d'Aspremont Alexandre
  , 2013, pp.http://nips.cc/Conferences/2013/Program/speaker-info.php?ID=12863. Seriation seeks to reconstruct a linear order between variables using unsorted similarity information. It has direct applications in archeology and shotgun gene sequencing for example. We prove the equivalence between the seriation and the combinatorial 2-sum problem (a quadratic minimization problem over permutations) over a class of similarity matrices. The seriation problem can be solved exactly by a spectral algorithm in the noiseless case and we produce a convex relaxation for the 2-sum problem to improve the robustness of solutions in a noisy setting. This relaxation also allows us to impose additional structural constraints on the solution, to solve semi-supervised seriation problems. We present numerical experiments on archeological data, Markov chains and gene sequences.
• La conjecture de Goldbach -la comète de Goldbach- pour les entiers pairs de 6 à 6244
  
  • Colonna Jean-Francois
  , 2013. The Goldbach conjecture -the Goldbach comet- for the even numbers from 6 to 6244 (La conjecture de Goldbach -la comète de Goldbach- pour les entiers pairs de 6 à 6244)
• La conjecture de Goldbach -la comète de Goldbach ou l'arc-en ciel de Goldbach- pour les entiers pairs de 6 à 41518
  
  • Colonna Jean-Francois
  , 2013. The Goldbach conjecture -the Goldbach comet or the Goldbach rainbow- for the even numbers from 6 to 41518 (La conjecture de Goldbach -la comète de Goldbach ou l'arc-en ciel de Goldbach- pour les entiers pairs de 6 à 41518)
• La distance entre les nombres premiers consécutifs, lorsqu'elle est égale à 2 -nombres premiers jumeaux- la couleur blanc est utilisée
  
  • Colonna Jean-Francois
  , 2013. The distance between consecutive prime numbers, when it equals 2 -twin prime numbers- the white color is used (La distance entre les nombres premiers consécutifs, lorsqu'elle est égale à 2 -nombres premiers jumeaux- la couleur blanc est utilisée)
• Numerical simulation of diffusion MRI signals using an adaptive time-stepping method
  
  • Li Jing-Rebecca
  • Calhoun Donna
  • Poupon Cyril
  • Le Bihan Denis
  Physics in Medicine and Biology, IOP Publishing, 2013. The effect on the MRI signal of water diffusion in biological tissues in the presence of applied magnetic field gradient pulses can be modelled by a multiple compartment Bloch-Torrey partial differential equation. We present a method for the numerical solution of this equation by coupling a standard Cartesian spatial discretization with an adaptive time discretization. The time discretization is done using the explicit Runge-Kutta-Chebyshev method, which is more efficient than the forward Euler time discretization for diffusive-type problems. We use this approach to simulate the diffusion MRI signal from the extra-cylindrical compartment in a tissue model of the brain gray matter consisting of cylindrical and spherical cells and illustrate the effect of cell membrane permeability. (10.1088/0031-9155/59/2/441)
  DOI : 10.1088/0031-9155/59/2/441
• Second order reflected backward stochastic differential equations
  
  • Matoussi Anis
  • Possamaï Dylan
  • Zhou Chao
  The Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2013, 23 (6), pp.2420-2457. In this article, we build upon the work of Soner, Touzi and Zhang [Probab. Theory Related Fields 153 (2012) 149-190] to define a notion of a second order backward stochastic differential equation reflected on a lower c\'{a}dl\'{a}g obstacle. We prove existence and uniqueness of the solution under a Lipschitz-type assumption on the generator, and we investigate some links between our reflected 2BSDEs and nonclassical optimal stopping problems. Finally, we show that reflected 2BSDEs provide a super-hedging price for American options in a market with volatility uncertainty. (10.1214/12-AAP906)
  DOI : 10.1214/12-AAP906
• Comparison of Numerical Methods in the Contrast Imaging Problem in NMR
  
  • Bonnard Bernard
  • Claeys Mathieu
  • Cots Olivier
  • Martinon Pierre
  , 2013. In this article, the contrast imaging problem in nuclear magnetic resonance is modeled as a Mayer problem in optimal control. A first synthesis of locally optimal solutions is given in the single-input case using geometric methods based on Pontryagin's maximum principle. We then compare these results using direct methods and a moment-based approach, and make a first step towards global optimality. Finally, some preliminary results are given in the bi-input case.
• On the inverse optimal control problems of the human locomotion: stability and robustness of the minimizers
  
  • Chittaro Francesca
  • Jean Frédéric
  • Mason Paolo
  Journal of Mathematical Sciences, Springer Verlag (Germany), 2013, 195 (3), pp.269-287. In recent papers models of the human locomotion by means of an optimal control problem have been proposed. In this paradigm, the trajectories are assumed to be solutions of an optimal control problem whose cost has to be determined. The purpose of the present paper is to analyze the class of optimal control problems defined in this way. We prove strong convergence result for their solutions on the one hand for perturbations of the initial and final points (stability), and on the other hand for perturbations of the cost (robustness). (10.1007/s10958-013-1579-z)
  DOI : 10.1007/s10958-013-1579-z
• Stabilité et instabilité dans les problèmes inverses
  
  • Isaev Mikhail
  , 2013. Dans cette thèse nous nous intéressons aux questions de stabilité et d'instabilité dans certains problèmes inverses classiques pour l'équation de Schrödinger et l'équation acoustique en dimension d>=2. Les problèmes considérés sont le problème inverse de Gel'fand de valeurs au bord et les problèmes inverses de diffusion en champ proche et en champ lointain. Les résultats de stabilité et d'instabilité présentés dans cette thèse se complètent mutuellement et contribuent à une meilleure compréhension de la nature des problèmes précités. En particulier, nous démontrons des nouvelles estimations de stabilité globale qui dépendent explicitement de la régularité du coefficient et de l'énergie. En outre, nous considérons le problème inverse de valeurs au bord pour l'équation de Schrödinger à l'énergie fixée avec des mesures frontières représentées comme l'opérateur frontière d'impédance (ou l'opérateur Robin-Robin). Nous démontrons des estimations de stabilité globale pour détermination du potentiel à partir de mesures frontières dans cette représentation d'impédance. De plus, des techniques similaires donnent aussi une procédure de reconstruction globale pour ce problème.