Academic literature on the topic 'Existence of equilibrium'

COORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIOR<br>Propomos um modelo de equilíbrio geral com securitização de ativos pro- tegidos por colateral físico e discutimos o papel do colateral para evitar, em equilíbrio, expectativas pessimistas em relação as taxas futuras de default. As promessas dos ativos garantidos por colateral são securitizadas em famílias de derivativos Pass-through, que dividem o default de forma prorata, ou de Collateralized Loans Obligations (CLO), que permitem diferentes níveis de prioridade entre os ativos dentro de uma mesma família. No caso de default, os tomadores de empréstimo podem sofrer penalidades extra-econômicas, que são internalizadas nas suas preferências não ordenadas. Nesse contexto, provamos a existência de um equilíbrio no qual os investidores não são muito pessimistas sobre os pagamentos dos derivativos.<br>We propose a specification of a general equilibrium model with securitization of collateral-backed promises and discuss the role of physical collateral to avoid, in equilibrium, pessimistic beliefs about the future rates of default. Promises are pooled in either pass-through securities or collateralized loans obligations (CLO), allowing the existence of different seniority levels among tranches in the same CLO. In case of default, borrowers may also suffer extra-economic penalties, which are internalized into a structure of nonordered preferences. In this context, we prove the existence of an equilibrium in that investors are not over-pessimistic about the payments of derivatives.

In this work the problematic issue of price determination in economic theory is re-examined. In the first chapter a state-of-the-art survey regarding the existence of equilibrium and the structure of the equilibrium set in price-setting games is provided. In chapter two a new core concept, the Bertrand core, is introduced and characterized. In chapter three a revealed preference perspective upon the Nash equilibria in price-setting games is provided. In chapter four, the issue of Bayesian equilibrium existence is addressed when traders have incomplete information regarding each others' types. Finally, a summary of possible future avenues for research in this area is provided.

Cette thèse est centrée sur l’étude d’équations issues de la théorie cinétique des gaz. Dans tous les problèmes qui y sont explorés, une analyse des problèmes linéaires ou linéarisés associés est réalisée d’un point de vue spectral et du point de vue des semi-groupes. A cela s’ajoute une analyse de la stabilité non linéaire lorsque le modèle est non linéaire. Plus précisément, dans une première partie, nous nous intéressons aux équations de Fokker-Planck fractionnaire et Boltzmann sans cut-off homogène en espace et nous prouvons un retour vers l’équilibre des solutions de ces équations avec un taux exponentiel dans des espaces de type L1 à poids polynomial. Concernant l’équation de Landau inhomogène en espace, nous développons une théorie de Cauchy de solutions perturbatives dans des espaces de type L2 avec différents poids (polynomiaux ou exponentiels) et nous prouvons également la stabilité exponentielle de ces solutions.Nous démontrons ensuite pour l’équation de Boltzmann inélastique inhomogène avec terme diffusif le même type de résultat dans des espaces L1 à poids polynomial dans un régime de faible inélasticité. Pour finir, nous étudions dans un cadre général et uniforme des modèles qui convergent vers l’équation de Fokker-Planck du point de vue de l’analyse spectrale et des semi-groupes<br>The topic of this thesis is the study of models coming from kinetic theory. In all the problems that are addressed, the associated linear or linearized problem is analyzed from a spectral point of view and from the point of view of semigroups. Tothat, we add the study of the nonlinear stability when the equation is nonlinear. More precisely, to begin with, we treat the problem of trend to equilibrium for the fractional Fokker-Planck and Boltzmann without cut-off equations, proving an exponential decay to equilibrium in spaces of type L1 with polynomial weights. Concerning the inhomogeneous Landau equation, we develop a Cauchy theory of perturbative solutions in spaces of type L2 with various weights such as polynomial and exponential weights and we also prove the exponential stability of these solutions. Then, we prove similar results for the inhomogeneous inelastic diffusively driven Boltzmann equation in a small inelasticity regime in L1 spaces with polynomial weights. Finally, we study in the same and uniform framework from the spectral analysis point of view with a semigroup approach several Fokker-Planck equations which converge towards the classical one

Thesis: S.M., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2015.<br>Cataloged from PDF version of thesis.<br>Includes bibliographical references (pages 63-65).<br>Competitive equilibrium provides a natural steady state for iterative combinatorial auctions that maximize social welfare, and therefore the first step in auction design is to establish its existence. Recent work by Baldwin and Klemperer (2012) has proved that the "demand type" of valuations being "unimodular" is a necessary and sufficient condition for the existence of a competitive equilibrium, but under the general setting where both buyers and sellers as well as multiple copies of items may exist, and the supply could be any combination of items available. In this work, we investigate the same condition under the more restrictive but standard setting for combinatorial auctions, where only buyers and a single copy of each distinct item are allowed and the supply is fixed to be the set of all available items. First, we provide an alternative proof of the sufficiency result for unimodular "complements" demand type, which defines a subclass of valuations for which a competitive equilibrium exists according to Baldwin and Klemperer (2012). While their original proof and analysis use tools from tropical geometry, our approach is based on linear programming. Relying on a result from Bikhchandani and Mamer (1999) that a competitive equilibrium exists if and only if a related linear program has an integral optimal solution, we provide a direct proof that the linear program has an integral optimal solution. Our analysis provides a fundamental understanding of the structure of the linear program and leads to various properties which may be helpful in auction design. Second, we provide an algorithm to determine the demand types of sign-consistent tree graphical valuations, for which competitive equilibria are known to exist due to Candogan et al. (2013). We then analyze the relationship between the set of the demand types of sign-consistent tree graphical valuations and the set of unimodular demand types. Our analysis implies that the unimodularity of demand type is not necessary for the existence of a competitive equilibrium in combinatorial auctions.<br>by Ji Young Lee.<br>S.M.

Ce mémoire de thèse est consacré à l’équation de Fokker-Planckpartial_ f=∆f+div(Ef).Il est subdivisé en deux parties :une partie linéaire et une partie non linéaire. Dans la partie linéaire on considère un champ de vecteur E(x) dépendant seulement de x. Cette partie est constituée des chapitres 3, 4 et 5. Dans le chapitre 3 on montre que l’opérateur linéaire Lf :=∆ f + div(E f ) est le générateur d’un semi-groupe fortement continu (SL(t))_{t≥0} dans tous les espaces L^p. On y établit également que le semi-groupe (SL(t))_{t≥0} est positif et ultracontractif. Dans le chapitre 4 nous montrons comment est qu’une décomposition adéquate de l’opérateur L permet d’établir certaines propriétés du semi-groupe (SL(t))_{t≥0} notamment sa bornitude. Le chapitre 5 est consacré à l’existence d’un état d’équilibre. De plus on y montre que cet état d’équi- libre est asymptotiquement stable. Dans la partie non linéaire on considère un champ de vecteur de la forme E(x,f) := x+nabla (a*f) ou a et f sont des fonctions assez régulières et * est l’opérateur de convolution. Cette parties est contituée des chapitre 6 et 7. Dans le chapitre 6 nous établissons que poura appartenant à W^{2,infini}_locl’équation de Fokker-Planck non linéaire admet une unique solution locale dans l’espace L^2_{K_alpha} (R^d). Dans le dernier chapitre nous montrons que le problème non linéaire admet une solution globale. De plus cette solution dépend continument des données<br>This thesis is devoted to the Fokker-Planck équation partial_t f =∆f + div(E f).It is divided into two parts. The rst part deals with the linear problem. In this part we consider a vector E(x) depending only on x. It is composed of chapters 3, 4 and 5. In chapter 3 we prove that the linear operator Lf :=∆f + div(Ef ) is an in nitesimal generator of a strong continuous semigroup (SL(t))_{t≥0}. We establish also that (SL(t))_{t≥0} is positive and ultracontractive. In chapter 4 we show how an adequate decomposition of the linear operator L allows us to deduce interesting properties for the semigroup (SL(t))_{t≥0}. Indeed using this decomposition we prove that (SL(t))_{t≥0} is a bounded semigroup. In the last chapter of this part we establish that the linear Fokker-Planck admits a unique steady state. Moreover this stationary solution is asymptotically stable.In the nonlinear part we consider a vector eld of the form E(x, f ) := x +nabla (a *f ), where a and f are regular functions. It is composed of two chapters. In chapter 6 we establish that fora in W^{2,infini}_locthe nonlinear problem has a unique local solution in L^2_{K_alpha}(R^d); . To end this part we prove in chapter 7 that the nonlinear problem has a unique global solution in L^2_k(R^d). This solution depends continuously on the data