Made available in DSpace on 2016-06-02T20:28:24Z (GMT). No. of bitstreams: 1 3158.pdf: 931917 bytes, checksum: b087d03944cb71331eae19f40f0fe194 (MD5) Previous issue date: 2010-05-18 / Universidade Federal de Sao Carlos / This work is an approach to the famous "Product with a Line Problem". It investigates the class of topological spaces whose cartesian product with R is a topological manifold. Such spaces are called "Codimension One Manifold Factors". Based mainly on [5, 7, 14, 15, 24], we introduce the concept of generalized manifolds, which are separable ANR spaces with same local homological behavior that the topological manifolds, we define DAP, DADP, DDP, DHP, DCP general position properties and, through these concepts and a machinery topological-algebraic, we have got answers to the motivator problem. Even about the strategic importance of the DHP general position property, we studied a criterion to detect it into the generalized manifolds category, namely, the P2MP. / Este trabalho é uma abordagem do famoso "Problema do Produto com uma Reta", o qual investiga a classe dos espaços topológicos cujo produto cartesiano com R é uma variedade topológica. Tais espaços são chamados de "Fatores de Variedade de Codimensão Um". Com base principalmente em [5, 7, 14, 15, 24], introduzimos o conceito de variedades generalizadas, as quais são espaços separáveis ANR que têm mesmo comportamento homológico local que as variedades topológicas, definimos as propriedades de posição geral DAP, DADP, DDP, DHP e DCP e, através desses conceitos e um ferramentário topológico-algébrico, obtivemos respostas ao problema motivador. Dada ainda a importância estratégica da propriedade de posição geral DHP, estudamos um critério para detectá-la na categoria das variedades generalizadas, qual seja, a P2MP.

Submitted by Livia Mello (liviacmello@yahoo.com.br) on 2016-10-05T14:40:49Z No. of bitstreams: 1 DissCHFP.pdf: 1216039 bytes, checksum: e21e062b0283d2bfe6ec436442e824a5 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-10-20T19:23:38Z (GMT) No. of bitstreams: 1 DissCHFP.pdf: 1216039 bytes, checksum: e21e062b0283d2bfe6ec436442e824a5 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-10-20T19:23:43Z (GMT) No. of bitstreams: 1 DissCHFP.pdf: 1216039 bytes, checksum: e21e062b0283d2bfe6ec436442e824a5 (MD5) / Made available in DSpace on 2016-10-20T19:23:50Z (GMT). No. of bitstreams: 1 DissCHFP.pdf: 1216039 bytes, checksum: e21e062b0283d2bfe6ec436442e824a5 (MD5) Previous issue date: 2016-02-25 / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / In this work, we will use topological methods in combinatorics and geometry to present a proof of the topological Tverberg theorem and a result about many Tverberg partitions. / O objetivo principal desta dissertação consiste em desenvolver um estudo detalhado de métodos topológicos em combinatória e geometria visando apresentar uma prova da versão topológica do teorema de Tverberg e de um teorema sobre a quantidade de partições de Tverberg. / FAPESP: 2015/01264-7

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this dissertation, we study the Sturm Liouvile's theorems for the zeros of the solutions of linear differential equations of second order. These classical theorems are applied to analysis of the monotonicity of functions involving the zeros of classical orthogonal polynomials. in particular, Gegenbauer polynomials. / Neste trabalho estudamos os Teoremas de Sturm Liouville para zeros de soluções de equações diferenciais lineares de segunda ordem. Estes teoremas clássicos são aplicados para análise do crescimento e decrescimento de certas funções que envolvem os zeros de Polinômios Ortogonais Clássicos, como os Polinômios de Gegenbauer. / Mestre em Matemática

Vilela Mendes et al. (2015), based on the discovery of long-range dependence in the volatility of stock returns, proposed a stochastic volatility continuous mar- ket model where the volatility is given as a transform of the fractional Brownian motion (fBm) and studied its No-Arbitrage and completeness properties under va- rious assumptions. We investigate the possibility of generalization of their results from fBm to a wider class of Hermite processes. We have reworked and completed the proofs of the propositions in the cited article. Under the assumption of indepen- dence of the stock price and volatility driving processes the model is arbitrage-free. However, apart from a case of a special relation between the drift and the volatility, the model is proved to be incomplete. Under a different assumption that there is only one source of randomness in the model and the volatility driving process is bounded, the model is arbitrage-free and complete. All the above results apply to any Hermite process driving the volatility. 1

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work we are treated in detail, three subjects of mathematics that relate to each other: Plane Geometry, Geometry and Spatial Revolution Solid. In this approach, we prioritized the calculation of the area of the lateral surfaces and full of Prism, Pyramid, Cylinder, Cone and Sphere, and the calculation of its volumes in the latter, using the principle of the deduction Cavalieri their formulas. In the study of Revolution Solids, we highlight the theorems of Pappus, used to derive the formulas of surface areas and volumes of cylinder, cone and revolution sphere. / Neste trabalho são tratados, de forma detalhada, três temas da Matemática que se relacionam entre si: Geometria Plana, Geometria Espacial e Sólidos de Revolução. Nessa abordagem, priorizou-se o cálculo da área das superfícies lateral e total do Prisma, da Pirâmide, do Cilindro, do Cone e da Esfera, bem como o cálculo de seus volumes, neste último, utilizando-se o princípio de Cavalieri na dedução de suas fórmulas. No estudo dos Sólidos de Revolução, destacam-se os Teoremas de Pappus, usados para deduzir as fórmulas das áreas das superfícies e dos volumes do Cilindro, do Cone e da Esfera de revolução.

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / O termo Teoria da Prova foi introduzido por Hilbert para identificar o estudo sobre provas formais. Pesquisas nessa Ãrea podem ser classificadas em: a) Teoria da Prova Redutiva ou Interpretacional, cujo objetivo à demonstrar, entre outras coisas, a consistÃncia da matemÃtica utilizando somente mÃtodos finitistas, e b) Teoria da Prova Estrutural, onde caracterÃsticas estruturais das provas formais sÃo investigadas por meio de sistemas dedutivos como DeduÃÃo Natural e CÃlculo de Sequentes. Prawitz, por meio da Teoria da Prova, definiu uma Teoria dos Significados para constantes logicas e propÃs regras esquemÃticas de introduÃÃo e de eliminaÃÃo para caracterizar os conectivos proposicionais. Schroeder-Heister estendeu as definiÃÃes de Prawitz e formalizou o uso de regras como hipÃteses, tornando possÃvel a utilizaÃÃo de cÃlculos para suposiÃÃes separados de cÃlculos para constantes lÃgicas. NÃo estamos interessados na investigaÃÃo de regras esquemÃticas para dar significado a constantes lÃgicas. Pretendemos, na verdade, definir procedimentos de normalizaÃÃo esquemÃticos, baseados em tais regras esquematicas, com objetivo de identificar condiÃÃes suficientes para um sistema ser normalizÃvel. Tais resultados sÃo pertinentes à Teoria Abstrata da Prova, termo usado para identificar o estudo das condiÃÃes abstratas e gerais para a anÃlise prova-teÃrica de sistemas formais. Teoria Abstrata da Prova nÃo estuda cÃlculos lÃgicos especÃficos, mas famÃlias de cÃlculos instÃncias de regras esquemÃticas. A nossa proposta, portanto, baseia-se em regras esquemÃticas que podem ser instanciadas por regras concretas, em particular, por regras que introduzem operadores modais. Provamos, tambÃm, Teoremas de NormalizaÃÃoo Fraca e Forte para sistemas esquemÃticos definidos em funÃÃoo de nossas regras esquemÃticas, obtemos condiÃÃes suficientes para que um sistema instÃncia destas regras seja normalizÃvel, definimos um procedimento que normaliza deduÃÃes concretas e comparamos nossas provas de normalizaÃÃo esquemÃtica com provas de normalizaÃÃo para sistemas definidos na literatura. / The term Theory Test was introduced by Hilbert to identify the study of formal proofs. Research in this area can be classified into: a) Proof Theory of reductive or interpretational, whose goal is to demonstrate, among other things, the consistency of mathematics using only methods finitistas, b) Structural Proof Theory, where the structural characteristics of the formal proofs are investigated by means of deductive systems as Natural Deduction and Sequent Calculus. Prawitz through Theory Proof set a Theory of Meaning for constants logics and proposed schematic introduction rules and elimination to characterize the propositional connectives. Schroeder-Heister settings Prawitz extended and formalized the use of rules as hypotheses, making possible the use of separate calculations for assumptions of calculations for logical constants. We are not interested in the investigation of schematic rules to give meaning to the logical constants. We intend to actually set schematic standardization procedures, based on such schematic rules? Attic, in order to identify sufficient conditions for a system to be normalizÃvel. These results are relevant to the Abstract Theory of Evidence, a term used to identify the study of the conditions abstract and general to the proof-theoretical analysis of formal systems. Abstract Theory of Evidence do not study specific logical calculations, but families of calculations instances of rules schematic. Our proposal is therefore based on rules schematic rules can be instantiated for concrete, in particular, by introducing rules modal operators. We prove also theorems NormalizaÃÃoo Weak and Strong systems defined in schematic funÃÃoo schematic of our rules, we obtain sufficient conditions for a system instance is normalizÃvel these rules, we define a procedure that normalizes deductions concrete evidence and compare our standards with evidence schematic standards for systems defined in the literature.

No description available.

Dans ce travail, nous nous intéressons aux estimateurs Multilevel Monte Carlo. Ces estimateurs vont apparaître sous leur forme standard, avec des poids et dans une forme randomisée. Nous allons rappeler leurs définitions et les résultats existants concernant ces estimateurs en termes de minimisation du coût de simulation. Nous allons ensuite montrer une loi forte des grands nombres et un théorème central limite. Après cela nous allons étudier deux cadres d'applications. Le premier est celui des diffusions avec schémas de discrétisation antithétiques, où nous allons étendre les estimateurs Multilevel aux estimateurs Multilevel avec poids. Le deuxième est le cadre nested, où nous allons nous concentrer sur les hypothèses d'erreur forte et faible. Nous allons conclure par l'implémentation de la forme randomisée des estimateurs Multilevel, en la comparant aux estimateurs Multilevel avec et sans poids. / In this work, we will focus on the Multilevel Monte Carlo estimators. These estimators will appear in their standard form, with weights and in their randomized form. We will recall the previous existing results concerning these estimators, in terms of minimization of the simulation cost. We will then show a strong law of large numbers and a central limit theorem.After that, we will focus on two application frameworks.The first one is the diffusions framework with antithetic discretization schemes, where we will extend the Multilevel estimators to Multilevel estimators with weights, and the second is the nested framework, where we will analyze the weak and the strong error assumptions. We will conclude by implementing the randomized form of the Multilevel estimators, comparing this to the Multilevel estimators with and without weights.

Cette thèse a pour objectif la compréhension de plusieurs aspects du caractère rugueux de la volatilité observé de manière universelle sur les actifs financiers. Ceci est fait en six étapes. Dans une première partie, on explique cette propriété à partir des comportements typiques des agents sur le marché. Plus précisément, on construit un modèle de prix microscopique basé sur les processus de Hawkes reproduisant les faits stylisés importants de la microstructure des marchés. En étudiant le comportement du prix à long terme, on montre l’émergence d’une version rugueuse du modèle de Heston (appelé modèle rough Heston) avec effet de levier. En utilisant ce lien original entre les processus de Hawkes et les modèles de Heston, on calcule dans la deuxième partie de cette thèse la fonction caractéristique du log-prix du modèle rough Heston. Cette fonction caractéristique est donnée en terme d’une solution d’une équation de Riccati dans le cas du modèle de Heston classique. On montre la validité d’une formule similaire dans le cas du modèle rough Heston, où l’équation de Riccati est remplacée par sa version fractionnaire. Cette formule nous permet de surmonter les difficultés techniques dues au caractère non markovien du modèle afin de valoriser des produits dérivés. Dans la troisième partie, on aborde la question de la gestion des risques des produits dérivés dans le modèle rough Heston. On présente des stratégies de couverture utilisant comme instruments l’actif sous-jacent et la courbe variance forward. Ceci est fait en spécifiant la structure markovienne infini-dimensionnelle du modèle. Étant capable de valoriser et couvrir les produits dérivés dans le modèle rough Heston, nous confrontons ce modèle à la réalité des marchés financiers dans la quatrième partie. Plus précisément, on montre qu’il reproduit le comportement de la volatilité implicite et historique. On montre également qu’il génère l’effet Zumbach qui est une asymétrie par inversion du temps observée empiriquement sur les données financières. On étudie dans la cinquième partie le comportement limite de la volatilité implicite à la monnaie à faible maturité dans le cadre d’un modèle à volatilité stochastique général (incluant le modèle rough Bergomi), en appliquant un développement de la densité du prix de l’actif. Alors que l’approximation basée sur les processus de Hawkes a permis de traiter plusieurs questions relatives au modèle rough Heston, nous examinons dans la sixième partie une approximation markovienne s’appliquant sur une classe plus générale de modèles à volatilité rugueuse. En utilisant cette approximation dans le cas particulier du modèle rough Heston, on obtient une méthode numérique pour résoudre les équations de Riccati fractionnaires. Enfin, nous terminons cette thèse en étudiant un problème non lié à la littérature sur la volatilité rugueuse. Nous considérons le cas d’une plateforme cherchant le meilleur système de make-take fees pour attirer de la liquidité. En utilisant le cadre principal-agent, on décrit le meilleur contrat à proposer au market maker ainsi que les cotations optimales affichées par ce dernier. Nous montrons également que cette politique conduit à une meilleure liquidité et à une baisse des coûts de transaction pour les investisseurs. / The aim of this thesis is to study various aspects of the rough behavior of the volatility observed universally on financial assets. This is done in six steps. In the first part, we investigate how rough volatility can naturally emerge from typical behav- iors of market participants. To do so, we build a microscopic price model based on Hawkes processes in which we encode the main features of the market microstructure. By studying the asymptotic behavior of the price on the long run, we obtain a rough version of the Heston model exhibiting rough volatility and leverage effect. Using this original link between Hawkes processes and the Heston framework, we compute in the second part of the thesis the characteristic function of the log-price in the rough Heston model. In the classical Heston model, the characteristic function is expressed in terms of a solution of a Riccati equation. We show that rough Heston models enjoy a similar formula, the Riccati equation being replaced by its fractional version. This formula enables us to overcome the non-Markovian nature of the model in order to deal with derivatives pricing. In the third part, we tackle the issue of managing derivatives risks under the rough Heston model. We establish explicit hedging strategies using as instruments the underlying asset and the forward variance curve. This is done by specifying the infinite-dimensional Markovian structure of the rough Heston model. Being able to price and hedge derivatives in the rough Heston model, we challenge the model to practice in the fourth part. More precisely, we show the excellent fit of the model to historical and implied volatilities. We also show that the model reproduces the Zumbach’s effect, that is a time reversal asymmetry which is observed empirically on financial data. While the Hawkes approximation enabled us to solve the pricing and hedging issues under the rough Heston model, this approach cannot be extended to an arbitrary rough volatility model. We study in the fifth part the behavior of the at-the-money implied volatility for small maturity under general stochastic volatility models. In the same spirit as the Hawkes approximation, we look in the sixth part of this thesis for a tractable Markovian approximation that holds for a general class of rough volatility models. By applying this approximation on the specific case of the rough Heston model, we derive a numerical scheme for solving fractional Riccati equations. Finally, we end this thesis by studying a problem unrelated to rough volatility. We consider an exchange looking for the best make-take fees system to attract liquidity in its platform. Using a principal-agent framework, we describe the best contract that the exchange should propose to the market maker and provide the optimal quotes displayed by the latter. We also argue that this policy leads to higher quality of liquidity and lower trading costs for investors.

As the title already suggests the aim of the present work is to present Farkas - type results for inequality systems involving convex and/or non - convex functions. To be able to give the desired results, we treat optimization problems which involve convex and composed convex functions or non - convex functions like DC functions or fractions. To be able to use the fruitful Fenchel - Lagrange duality approach, to the primal problem we attach an equivalent problem which is a convex optimization problem. After giving a dual problem to the problem we initially treat, we provide weak necessary conditions which secure strong duality, i.e., the case when the optimal objective value of the primal problem coincides with the optimal objective value of the dual problem and, moreover, the dual problem has an optimal solution. Further, two ideas are followed. Firstly, using the weak and strong duality between the primal problem and the dual problem, we are able to give necessary and sufficient optimality conditions for the optimal solutions of the primal problem. Secondly, provided that no duality gap lies between the primal problem and its Fenchel - Lagrange - type dual we are able to demonstrate some Farkas - type results and thus to underline once more the connections between the theorems of the alternative and the theory of duality. One statement of the above mentioned Farkas - type results is characterized using only epigraphs of functions. We conclude our investigations by providing necessary and sufficient optimality conditions for a multiobjective programming problem involving composed convex functions. Using the well-known linear scalarization to the primal multiobjective program a family of scalar optimization problems is attached. Further to each of these scalar problems the Fenchel - Lagrange dual problem is determined. Making use of the weak and strong duality between the scalarized problem and its dual the desired optimality conditions are proved. Moreover, the way the dual problem of the scalarized problem looks like gives us an idea about how to construct a vector dual problem to the initial one. Further weak and strong vector duality assertions are provided.