Publications

2010

• Un ensemble de Julia dans l'ensemble des pseudo-quaternions (comme un 'MandelBulb' : un 'JuliaBulb') calculé pour A=(-0.5815147625160462,+0.6358885017421603,0,0)
  
  • Colonna Jean-François
  , 2010. A pseudo-quaternionic Julia set ('MandelBulb' like : a 'JuliaBulb') computed with A=(-0.5815147625160462,+0.6358885017421603,0,0) (Un ensemble de Julia dans l'ensemble des pseudo-quaternions (comme un 'MandelBulb' : un 'JuliaBulb') calculé pour A=(-0.5815147625160462,+0.6358885017421603,0,0))
• Planification de mouvements pour les systèmes non-holonomes et étude de la contrôlabilité spectrale pour les équations de Schrödinger linéarisées
  
  • Long Ruixing
  , 2010. L'objectif de cette thèse est, d'une part, de fournir des méthodes de planification de mouvements pour les systèmes non-holonomes, et d'autre part, d'étudier la contrôlabilité spectrale pour les équations de Schrödinger linéarisées. Nous avons apporté une double contribution au problème de la planification de mouvements pour les systèmes non-holonomes. Fondé sur la géométrie sous-riemannienne, nous avons conçu un nouvel algorithme qui résout complètement le problème dans un cadre général. Nous avons également proposé une implémentation numérique de la méthode de continuation qui fournit des solutions satisfaisantes au problème de la planification du roulement sur le plan, un exemple classique de systèmes non-holonomes à deux entrées. Nous avons donné des conditions nécessaires et suffisantes de contrôlabilité spectrale en temps fini des équations de Schrödinger linéarisées en dimension 2 et 3. Leur généricité par rapport au domaine a été étudiée par une technique originale basée sur les équations intégrales.
• La dynamique d'une texture fractale
  
  • Colonna Jean-François
  , 2010. Fractal texture dynamics (La dynamique d'une texture fractale)
• On a class of vector fields with discontinuity of divide-by-zero type and its applications
  
  • Ghezzi Roberta
  • Remizov Alexey
  , 2010. We study phase portraits and singular points of vector fields of a special type, that is, vector fields whose components are fractions with a common denominator vanishing on a smooth regular hypersurface in the phase space. We assume also some additional conditions, which are fulfilled, for instance, if the vector field is divergence-free. This problem is motivated by a large number of applications. In this paper, we consider three of them in the framework of differential geometry: singularities of geodesic flows in various singular metrics on surfaces.
• A Stochastic Algorithm for Probabilistic Independent Component Analysis
  
  • Allassonniere Stéphanie
  • Younes Laurent
  , 2010. The decomposition of a sample of images on a relevant subspace is a recurrent problem in many different fields from Computer Vision to medical image analysis. We propose in this paper a new learning principle and implementation of the generative decomposition model generally known as noisy ICA (for independent component analysis) based on the SAEM algorithm, which is a versatile stochastic approximation of the standard EM algorithm. We demonstrate the applicability of the method on a large range of decomposition models and illustrate the developments with experimental results on various data sets.
• New Trends in Model Coupling Theory, Numerics and Applications
  
  • Coquel Frédéric
  • Godlewski Edwige
  • Hérard Jean-Marc
  • Segré Jacques
  Networks and Heterogeneous Media, American Institute of Mathematical Sciences, 2010, 5. This special issue comprises selected papers from the workshop New Trends in Model Coupling, Theory, Numerics and Applications (NTMC'09) which took place in Paris, September 2 - 4, 2009. The research of optimal technological solutions in a large amount of industrial systems requires to perform numerical simulations of complex phenomena which are often characterized by the coupling of models related to various space and/or time scales. Thus, the so-called multi-scale modelling has been a thriving scientific activity which connects applied mathematics and other disciplines such as physics, chemistry, biology or even social sciences. To illustrate the variety of fields concerned by the natural occurrence of model coupling we may quote: meteorology where it is required to take into account several turbulence scales or the interaction between oceans and atmosphere, but also regional models in a global description, solid mechanics where a thorough understanding of complex phenomena such as propagation of cracks needs to couple various models from the atomistic level to the macroscopic level; plasma physics for fusion energy for instance where dense plasmas and collisionless plasma coexist; multiphase fluid dynamics when several types of flow corresponding to several types of models are present simultaneously in complex circuits; social behaviour analysis with interaction between individual actions and collective behaviour. (10.3934/NHM.2010.5.3I)
  DOI : 10.3934/NHM.2010.5.3I
• Le champ bidimensionnel définissant les trois coordonnées de la bouteille de Klein
  
  • Colonna Jean-François
  , 2010. The bidimensional field defining the three coordinates of the Klein bottle (Le champ bidimensionnel définissant les trois coordonnées de la bouteille de Klein)
• La bouteille de Klein
  
  • Colonna Jean-François
  , 2010. The Klein bottle (La bouteille de Klein)
• Méthodes de Contrôle Stochastique pour la Gestion Optimale de Portefeuille
  
  • Espinosa Gilles-Edouard
  , 2010. Cette thèse présente trois sujets de recherche indépendants, le dernier étant décliné sous forme de deux problèmes distincts. Ces différents sujets ont en commun d'appliquer des méthodes de contrôle stochastique à des problèmes de gestion optimale de portefeuille. Dans une première partie, nous nous intéressons à un modèle de gestion d'actifs prenant en compte des taxes sur les plus-values. Dans une seconde partie, nous étudions un problème de détection du maximum d'un processus de retour à la moyenne. Dans les troisième et quatrième parties, nous regardons un problème d'investissement optimal lorsque les agents se regardent les uns les autres. Enfin dans une cinquième partie, nous étudions une variante de cette problématique incluant un terme de pénalisation au lieu de contraintes sur les portefeuilles admissibles.
• A first-order primal-dual algorithm for convex problems with applications to imaging
  
  • Chambolle Antonin
  • Pock Thomas
  , 2010. We study a first-order primal-dual algorithm for convex optimization problems with known saddle-point structure. We prove convergence to a saddle-point with rate O(1/N) in finite dimensions, which is optimal for the complete class of non-smooth problems we are considering in this paper. We further show accelerations of the proposed algorithm to yield optimal rates on easier problems. In particular we show that we can achieve O(1/N²) convergence on problems, where the primal or the dual objective is uniformly convex, and we can show linear convergence, i.e. O(1/e^N) on problems where both are uniformly convex. The wide applicability of the proposed algorithm is demonstrated on several imaging problems such as image denoising, image deconvolution, image inpainting, motion estimation and image segmentation.
• Transient Wave Imaging
  
  • Guadarrama Lili
  , 2010. Extensive work has been carried out in the past decade to image the elastic properties of human soft tissues by inducing motion. This broad field, called elasticity imaging or elastography, is based on the initial idea that shear elasticity can be correlated with the pathology of tissues. There are several techniques that can be classified according to the type of mechanical excitation chosen (static compression, monochromatic, or transient vibration) and the way these excitations are generated (externally or internally). Different imaging modalities can be used to estimate the resulting tissue displacements. A very interesting approach to assessing elasticity is to use the acoustic radiation force of an ultrasonic focused beam to remotely generate mechanical vibrations in organs. The acoustic force is generated by the momentum transfer from the acoustic wave to the medium. The radiation force essentially acts as a dipolar source. A spatio-temporal sequence of the propagation of the induced transient wave can be acquired, leading to a quantitative estimation of the viscoelastic parameters of the studied medium in a source-free region. Our aim in this thesis is to provide a solid mathematical foundation for this transient technique and to design accurate methods for anomaly detection using transient measurements. We consider both the acoustic and elastic cases. We develop efficient reconstruction techniques from not only complete measurements but also from limited-view transient data and adapt them in the case of viscous media, where the elastic waves are attenuated and/or dispersed. We begin with transient imaging in a non-dissipative medium. We develop anomaly reconstruction procedures that are based on rigorously established inner and outer time-domain asymptotic expansions of the perturbations in the transient measurements that are due to the presence of the anomaly. It is worth mentioning that in order to approximate the anomaly as a dipole with certain polarizability, one has to truncate the high-frequency component of the far-field measurements. Using the outer asymptotic expansion, we design a time-reversal imaging technique for locating the anomaly. Based on such expansions, we propose an optimization problem for recovering geometric properties as well as the physical parameters of the anomaly. We justify both theoretically and numerically that scale separation can be used to obtain local and precise reconstructions. We show the differences between the acoustic and the elastic cases, namely, the anisotropy of the focal spot and the birth of a near fieldlike effect by time reversing the perturbation due to an elastic anomaly. These interesting findings were experimentally observed before. Our asymptotic formalism clearly explains them. In the case of limited-view transient measurements, we construct Kirchhoff-, back-propagation-, MUSIC-, and arrival time-type algorithms for imaging small anomalies. Our approach is based on averaging of the limited-view data, using weights constructed by the geometrical control method. It is quite robust with respect to perturbations of the non-accessible part of the boundary. Our main finding is that if one can construct accurately the geometric control then one can perform imaging with the same resolution using partial data as using complete data. We also use our asymptotic formalism to explain how to reconstruct a small anomaly in a viscoelastic medium from wavefield measurements. The visco-elastic medium obeys a frequency power-law. For simplicity, we consider the Voigt model, which corresponds to a quadratic frequency loss. By using the stationary phase theorem, we express the ideal elastic field without any viscous effect in terms of the measured field in a viscous medium. We then generalize the imaging techniques developed for a purely quasi-incompressible elasticity model to recover the viscoelastic and geometric properties of an anomaly from wavefield measurements.
• Un détail d'un 'MandelBox' généralisé brumeux
  
  • Colonna Jean-François
  , 2010. Close-up on an extended foggy 'MandelBox' (Un détail d'un 'MandelBox' généralisé brumeux)
• The Hö̈lder infinite Laplacian and Hö̈lder extensions
  
  • Chambolle Antonin
  • Lindgren Erik
  • Monneau Régis
  , 2010. In this paper we study the limit as $p\to \infty$ of minimizers of the fractional $W^{s,p}$-norms. In particular, we prove that the limit satisfies a non-local and non-linear equation. We also prove the existence and uniqueness of solutions of the equation. Furthermore, we prove the existence of solutions in general for the corresponding inhomogeneous equation. By making strong use of the barriers in this construction, we obtain some regularity results.
• Characterizations of Lojasiewicz inequalities: Subgradient flows, talweg, convexity
  
  • Bolte Jerome
  • Daniilidis Aris
  • Ley Olivier
  • Mazet Laurent
  Transactions of the American Mathematical Society, American Mathematical Society, 2010, 362 (6), pp.3319-3363. The classical Lojasiewicz inequality and its extensions for partial differential equation problems (Simon) and to o-minimal structures (Kurdyka) have a considerable impact on the analysis of gradient-like methods and related problems: minimization methods, complexity theory, asymptotic analysis of dissipative partial differential equations, tame geometry. This paper provides alternative characterizations of this type of inequalities for nonsmooth lower semicontinuous functions defined on a metric or a real Hilbert space. In a metric context, we show that a generalized form of the Lojasiewicz inequality (hereby called the Kurdyka-Lojasiewicz inequality) relates to metric regularity and to the Lipschitz continuity of the sublevel mapping, yielding applications to discrete methods (strong convergence of the proximal algorithm). In a Hilbert setting we further establish that asymptotic properties of the semiflow generated by $-\partial f$ are strongly linked to this inequality. This is done by introducing the notion of a piecewise subgradient curve: such curves have uniformly bounded lengths if and only if the Kurdyka-Lojasiewicz inequality is satisfied. Further characterizations in terms of talweg lines -a concept linked to the location of the less steepest points at the level sets of $f$- and integrability conditions are given. In the convex case these results are significantly reinforced, allowing in particular to establish the asymptotic equivalence of discrete gradient methods and continuous gradient curves. On the other hand, a counterexample of a convex C^2 function in in the plane is constructed to illustrate the fact that, contrary to our intuition, and unless a specific growth condition is satisfied, convex functions may fail to fulfill the Kurdyka-Lojasiewicz inequality. (10.1090/S0002-9947-09-05048-X)
  DOI : 10.1090/S0002-9947-09-05048-X
• Transient Anomaly Imaging in Visco-Elastic Media Obeying a Frequency Power-Law
  
  • Bretin Elie
  • Guadarrama Bustos Lili
  • Wahab Abdul
  , 2010. In this work, we consider the problem of reconstructing a small anomaly in a viscoelastic medium from wave-field measurements. We choose Szabo's model to describe the viscoelastic properties of the medium. Expressing the ideal elastic field without any viscous effect in terms of the measured field in a viscous medium, we generalize the imaging procedures, such as time reversal, Kirchhoff Imaging and Back propagation, for an ideal medium to detect an anomaly in a visco-elastic medium from wave-field measurements.
• Multicomponent transport in weakly ionized mixtures
  
  • Giovangigli Vincent
  • Graille Benjamin
  • Magin Thierry
  • Massot Marc
  Plasma Sources Science and Technology, IOP Publishing, 2010, 19 (3), pp.1-6. We discuss transport coefficients in weakly ionized mixtures. We investigate the situations of weak and strong magnetic fields as well as electron temperature nonequilibrium. We present in each regime the Boltzmann equations, examples of transport fluxes, the structure of transport linear systems and discuss their solution by efficient iterative techniques. Numerical simulations are presented for partially ionized high-temperature air. (10.1088/0963-0252/19/3/034002)
  DOI : 10.1088/0963-0252/19/3/034002
• Green's functions and integral equations for the Laplace and Helmholtz operators in impedance half-spaces
  
  • Hein Hoernig Ricardo Oliver
  , 2010. Dans cette thèse on calcule la fonction de Green des équations de Laplace et Helmholtz en deux et trois dimensions dans un demi-espace avec une condition à la limite d'impédance. Pour les calculs on utilise une transformée de Fourier partielle, le principe d'absorption limite, et quelques fonctions spéciales de la physique mathématique. La fonction de Green est après utilisée pour résoudre numériquement un problème de propagation des ondes dans un demi-espace qui est perturbé de manière compacte, avec impédance, en employant des techniques des équations intégrales et la méthode d'éléments de frontière. La connaissance de son champ lointain permet d'énoncer convenablement la condition de radiation dont on a besoin. Des expressions pour le champ proche et lointain de la solution sont données, dont l'existence et l'unicité sont discutées brièvement. Pour chaque cas un problème benchmark est résolu numériquement. On expose étendument le fond physique et mathématique et on inclut aussi la théorie des problèmes de propagation des ondes dans l'espace plein qui est perturbé de manière compacte, avec impédance. Les techniques mathématiques développées ici sont appliquées ensuite au calcul de résonances dans un port maritime. De la même façon, ils sont appliqués au calcul de la fonction de Green pour l'équation de Laplace dans un demi-plan bidimensionnel avec une condition à la limite de dérivée oblique.
• Modélisation mathématique et simulation numérique avancée des phénomènes de propagation d'ondes dans les médias élastiques sans limite.
  
  • Godoy Eduardo
  , 2010. Motivée par des applications en géophysique et ingénierie sismique, cette thèse cherche à contribuer à l'étude de phénomènes de propagation d'ondes en milieux élastiques non bornés. Nous développons des techniques mathématiques et numériques pour résoudre des problèmes de diffraction en régime harmonique, dans des domaines infinis extérieurs et demi-infinis localement perturbés. En plus, nous introduisons une nouvelle condition aux limites du type impédance en élasticité, laquelle généralise la condition de frontière libre utilisée d'habitude pour décrire la surface de la terre en problèmes géophysiques. Les ondes de surface qui apparaissent avec cette condition aux limites sont étudiées. Nous montrons l'existence de l'onde de Rayleigh et comment elle dépend de l'impédance. En plus, nous prouvons qu'il apparaît une onde de surface additionnelle dans un cas particulière. Pour traiter numériquement les domaines non bornés, nous considérons des approches basées sur des conditions aux limites exactes et des méthodes d'équations intégrales de frontière. Les premières s'appliquent à des domaines extérieurs, pendant que les deuxièmes s'emploient pour les deux types de domaine. Un accent particulier est mis sur les équations intégrales et les méthodes d'éléments de frontière pour résoudre des problèmes de diffraction dans des demi-plans localement perturbés. Nous calculons de manière efficace et précise la fonction de Green d'un demi-plan élastique avec des conditions aux limites d'impédance, à l'aide d'une méthode de calcul qui combine de façon appropriée des techniques analytiques et numériques. Nous proposons aussi une méthode d'équations intégrales de frontière basée sur la fonction de Green calculée. Finalement, les procédures numériques sont validées en utilisant des problèmes benchmark appropriés.
• Multifractal models for asset prices
  
  • Bacry Emmanuel
  • Muzy J.-F.
  Encyclopedia of quantitative finance, 2010, pp.1-10. We present an overview of multifractal models of asset returns. All the proposed models rely upon the notion of random multiplicative cascades. We focus in more details on the simplest of such models namely the log-normal multifractal random walk. This model can be seen as a stochastic volatility model where the (log-) volatility has a peculiar long-range correlated memory. We briefly address calibration issues of such models and their applications to volatility and Value at Risk (VaR) forecasting. (10.1002/9780470061602.eqf20004)
  DOI : 10.1002/9780470061602.eqf20004
• Paris-Princeton Lectures on Mathematical Finance
  
  • Scheinkman Jose
  • Carmona René
  • Cinlare Erhan
  • Ekeland Ivar
  • Jouini Elyès
  • Touzi Nizar
  , 2010, pp.376. This is the fourth volume of the Paris-Princeton Lectures in Mathematical Finance. The goal of this series is to publish cutting edge research in self contained articles prepared by established academics or promising young researchers invited by the editors. Contributions are refereed and particular attention is paid to the quality of the exposition, the goal being to publish articles that can serve as introductory references for research. The series is a result of frequent exchanges between researchers in finance and financial mathematics in Paris and Princeton. Many of us felt that the field would benefit from timely exposés of topics in which there is important progress. René Carmona, Erhan Cinlar, Ivar Ekeland, Elyes Jouini, José Scheinkman and Nizar Touzi serve in the first editorial board of the Paris-Princeton Lectures in Financial Mathematics. Although many of the chapters involve lectures given in Paris orPrinceton, we also invite other contributions. Springer Verlag kindly offered to hostthe initiative under the umbrella of the Lecture Notes in Mathematics series, and weare thankful to Catriona Byrne for her encouragement and her help. This fourth volume contains five chapters. In the first chapter, Areski Cousin, Monique Jeanblanc, and Jean -Paul Laurent discuss risk management and hedging of credit derivatives. The latter are over-the-counter (OTC) financial instruments designed to transfer credit risk associated to are ference entity from one counter party to another. The agreement involves a seller and a buyer of protection, the sellerbeing committed to cover the losses induced by the default. The popularity of theseinstruments lead a runaway market of complex derivatives whose risk management did not developas fast. This first chapter fills the gap by providing rigorous tools for quantifying and hedging counterparty risk in some of these markets. In the second chapter, Stéphane Crépey reviews the general theory of for-ward backward stochastic differential equations and their associated systems of partial integro-differential obstacle problems and applies it to pricing and hedging financial derivatives. Motivated by the optimal stopping and optimal stopping game formulations of American option and convertible bond pricing, he discussesthe well-posedness and sensitivities of reflected and doubly reflected Markovian Backward Stochastic Differential Equations. The third part of the paper is devotedto the variational inequality formulation of these problems and to a detailed discussion of viscosity solutions. Finally he also considers discrete path-dependenceissues such as dividend payments. The third chapter written by Olivier Guéant Jean-Michel Lasry and Pierre-Louis Lions presents an original and unified account of the theory and the applications of the mean field games as introduced and developed by Lasry and Lions in a seriesof lectures and scattered papers. This chapter provides systematic studies illustrating the application of the theory to domains as diverse as population behavior (theso-called Mexican wave), or economics (management of exhaustible resources). Some of the applications concern optimization of individual behavior when inter-acting with a large population of individuals with similar and possibly competing objectives. The analysis is also shown to apply to growth models and for example, to their application to salary distributions. The fourth chapter is contributed by David Hobson. It is concerned with the applications of the famous Skorohod embedding theorem to the proofs of model in dependent bounds on the prices of options. Beyond the obvious importance of thefinancial application, the value of this chapter lies in the insightful and extremely pedagogical presentation of the Skorohodem bedding problem and its application to the analysis of martingales with given one-dimensional marginals, providing a one-to-one correspondence between candidate price processes which are consistent with observed call option prices and solutions of the Skorokhod embedding problem, extremal solutions leading to robust model in dependent prices and hedges for exoticoptions. The final chapter is concerned with pricing and hedging in exponential Lévy models. Peter Tankov discusses three aspects of exponential Lévy models: absenceof arbitrage, including more recent results on the absence of arbitrage in multi dimensional models, properties of implied volatility, and modern approaches tohedging in these models. It is a self contained introduction surveying all the results and techniques that need to be known to be able to handle exponential Lévy models in finance.
• Un ordinateur est-il une parfaite machine à calculer ?
  
  • Colonna Jean-François
  Images des mathématiques, CNRS, 2010, pp.http://images.math.cnrs.fr/Un-ordinateur-est-il-une-parfaite.html. Un ordinateur est une machine à calculer programmable à la fois finie et discrète. La plupart des nombres et en particulier les nombres réels ne peuvent donc pas y être mémorisés et manipulés exactement. Dans la plupart des calculs, cela introduit des erreurs d'arrondi qui, principalement dans les problèmes sensibles aux conditions initiales, peuvent se propager et s'amplifier. Les propriétés mathématiques usuelles, telle l'associativité, sont perdues et ainsi, deux ordinateurs différents aux niveaux matériel et/ou logiciel pourront donner à partir d'un même programme des résultats différents.
• Optimal control of state constrained integral equations
  
  • Bonnans J. Frederic
  • de La Vega Constanza
  Set-Valued and Variational Analysis, Springer, 2010, 18 (3), pp.307-326. We consider the optimal control problem of a class of integral equations with initial and final state constraints, as well as running state coinstraints. We prove Pontryagin's principle, and study the continuity of the optimal control and of the measure associated with first order state constraints. We also establish the Lipschitz continuity of these two functions of time for problems with only first order state constraints.
• Upper large deviations for Branching Processes in Random Environment with heavy tails
  
  • Bansaye Vincent
  • Boeinghoff Christian
  , 2010. Branching Processes in a Random Environment (BPREs) $(Z_n:n\geq0)$ are a generalization of Galton Watson processes where in each generation the reproduction law is picked randomly in an i.i.d. manner. We determine here the upper large deviation of the process when the reproduction law may have heavy tails. The behavior of BPREs is related to the associated random walk of the environment, whose increments are distributed like the logarithmic mean of the offspring distributions. We obtain an expression of the upper rate function of $(Z_n:n\geq0)$, that is the limit of $-\log \mathbb{P}(Z_n\geq e^{\theta n})/n$ when $n\rightarrow \infty$. It depends on the rate function of the associated random walk of the environment, the logarithmic cost of survival $\gamma:=-\lim_{n\rightarrow\infty} \log \mathbb{P}(Z_n>0)/n$ and the polynomial decay $\beta$ of the tail distribution of $Z_1$. We give interpretations of this rate function in terms of the least costly ways for the process $(Z_n: n \geq 0)$ of attaining extraordinarily large values and describe the phase transitions. We derive then the rate function when the reproduction law does not have heavy tails, which generalizes the results of Böinghoff and Kersting (2009) and Bansaye and Berestycki (2008) for upper large deviations. Finally, we specify the upper large deviations for the Galton Watson processes with heavy tails.
• A Probabilistic Numerical Method for Fully Non-linear Parabolic Partial Differential Equations
  
  • Fahim Arash
  , 2010. The second part of the thesis deals with the optimal production policy under the carbon emission allowance market. The carbon emission allowance market is a market approach to implement Kyoto protocol. We calculated the optimal production in 3 cases: when there is such a market but without any large carbon producer, when there is a large producer who is not market maker, and when there is a large producer market maker. We showed that in second cases, the optimal production is always less than the first case and in the third case it is even less than the second case. On the other hand, we showed that the market maker (if there exist any) can benefit from the market by changing the risk premium of the carbon allowance due to her extra production. The model we used here for the price of carbon allowance is a BSDE. Then we introduce a stochastic optimization problem. The carbon producer wants to maximaze her utility from her wealth. Her wealth consists of two parts: a self--financing portfolio over the carbon emission allowance papers and the benefit from her production. As expected, the optimal production does not depend on the utility. One could pass to a new optimization problem which gives the optimal production. We choose to solve the stochastic optimization problem by the means of HJB equations. We obtained the verification and uniqueness result for the HJB equation. This part is closed by some numerical experiments which shows cases which the large producer can benefit from extra production.
• Stochastic Utilities With a Given Optimal Portfolio : Approach by Stochastic Flows
  
  • El Karoui Nicole
  • M'Rad Mohamed
  , 2010. The paper generalizes the construction by stochastic flows of consistent utility processes introduced by M. Mrad and N. El Karoui in (2010). The utilities random fields are defined from a general class of processes denoted by $\GX$. Making minimal assumptions and convex constraints on test-processes, we construct by composing two stochastic flows of homeomorphisms, all the consistent stochastic utilities whose the optimal-benchmark process is given, strictly increasing in its initial condition. Proofs are essentially based on stochastic change of variables techniques.