In this thesis, we address two problems arising from the application of stochastic differential equations (SDEs). The first one pertains to the detection of asset bubbles, where the price process solves an SDE. We combine the strict local martingale model together with a statistical tool to instantaneously check the existence and severity of asset bubbles through the asset’s historical price process. Our approach assumes that the price process of interest is a CEV process. We relate the exponent parameter in the CEV process to an asset bubble by studying the future expectation and the running maximum of the CEV process. The detection of asset bubbles then boils down to the estimation of the exponent. With a dynamic linear regression model, inference on the exponent can be carried out using historical price data. Estimation of the volatility and calibration of the parameters in the dynamic linear regression model are also studied. When using SDEs in practice, for example, in the detection of asset bubbles, one often would like to simulate its paths using the Euler scheme to study the behavior of the solution. The second part of this thesis focuses on the convergence property of the Euler scheme under the assumption that the coefficients of the SDE are locally Lipschitz and that the solution has no finite explosion. We prove that if a numerical scheme converges uniformly on any compact time set (UCP) in probability with a certain rate under the globally Lipschitz condition, then when the globally Lipschitz condition is replaced with a locally Lipschitz one plus a no finite explosion condition, UCP convergence with the same rate holds. One contribution of this thesis is the proof of √n-weak convergence of the asymptotic normalized error process. The limit error process is also provided. We further study the boundedness for the second moment of the weak limit process and its running maximum under both the globally Lipschitz and the locally Lipschitz conditions. The convergence of the Euler scheme in the sense of approximating expectations of functionals is also studied under the locally Lipschitz condition

Neste trabalho, aplicamos o algoritmo Breadth-First Search para encontrar o tamanho de uma componente conectada no grafo aleatório de Erdos-Rényi. Uma cadeia de Markov é obtida deste procedimento. Apresentamos alguns resultados bem conhecidos sobre o comportamento dessa cadeia de Markov. Combinamos alguns destes resultados para obter uma proposição sobre a probabilidade da componente atingir um determinado tamanho e um resultado de convergência do estado da cadeia neste instante. Posteriormente, aplicamos o teorema de convergência de Darling (2002) a sequência de cadeias de Markov reescaladas e indexadas por N, o número de vértices do grafo, para mostrar que as trajetórias dessas cadeias convergem uniformemente em probabilidade para a solução de uma equação diferencial ordinária. Deste resultado segue a bem conhecida lei fraca dos grandes números para a componente gigante do grafo aleatório de Erdos-Rényi, no caso supercrítico. Além disso, obtemos o limite do fluído para um modelo epidêmico que é uma extensão daquele proposto em Kurtz et al. (2008). / In this work, we apply the Breadth-First Search algorithm to find the size of a connected component of the Erdos-Rényi random graph. A Markov chain is obtained of this procedure. We present some well-known results about the behavior of this Markov chain, and combine some of these results to obtain a proposition about the probability that the component reaches a certain size and a convergence result about the state of the chain at that time. Next, we apply the convergence theorem of Darling (2002) to the sequence of rescaled Markov chains indexed by N, the number of vertices of the graph, to show that the trajectories of these chains converge uniformly in probability to the solution of an ordinary dierential equation. From the latter result follows the well-known weak law of large numbers of the giant component of the Erdos-Renyi random graph, in the supercritical case. Moreover, we obtain the uid limit for an epidemic model which is an extension of that proposed in Kurtz et al. (2008).

Este trabalho tem por objetivo testar a hipótese da convergência de renda entre os estados e municípios brasileiros entre os anos de 1999 e 2013. Para avaliar os dados é empregado instrumental exposto em Tirado et al. (2015), no qual encontra-se análises de modalidade, mobilidade e agrupamento espacial. Os resultados apontam para a persistência das diferenças regionais tanto entre os estados, quanto entre os municípios, consequentemente não estaria ocorrendo um movimento de convergência. Foi constatado uma elevação da mobilidade de classes de renda entre 1999 e 2002, porém tal acréscimo não se manteve nos anos seguintes. Em relação ao agrupamento espacial, este mostrou-se relevante apenas para os estados. / This work aims to test the income convergence hypothesis among Brazilian states and counties from 1999 to 2013. For assessing the data was used the instrumental presented in Tirado et al. (2015), in which finds modality, mobility and spatial clustering analysis. The results indicate for persistence in regional differences among states as well as among counties, thereafter it would not be occurring a convergence movement. It was found a mobility elevation among income classes from 1999 to 2002, however this increase did not mantain to the following years. Concerning to spatial clustering, this was relevant only for the states.

O mundo globalizado e o estado-da-arte dos desenvolvimentos nos remetem a um ambiente tecnologicamente complexo. Nos dias de hoje a convergência tecnológica é indiscutivelmente realidade. O usuário tem a necessidade de movimentar-se fisicamente dentro do meio em que vive e interagir com o aplicativo ou serviço que lhe é prestado. As novas gerações de sistemas de comunicação, que não mais utilizam fios em suas conexões, prevêem a integração de vários sistemas heterogêneos já existentes, unidos em uma infraestrutura capaz de, transparentemente, entregar a usuários móveis uma ampla gama de serviços com a finalidade de permitir sua comunicabilidade e acesso à informação. Com base nessas premissas, este trabalho apresenta um método para descrever um modelo de negócio para um usuário em movimento dentro de um ambiente convergente heterogêneo. O trânsito do usuário dentro das várias tecnologias de acesso para que esteja conectado a um determinado serviço é mostrado através de um framework de mobilidade e interatividade e, fundamentado nesse framework, é construído um modelo de negócio para a consecução da entrega desse serviço ao usuário. Este trabalho conceitua um modelo para um framework de mobilidade e interatividade aplicáveis a sistemas abertos; e um modelo de negócio voltado aos serviços num cenário com soluções tecnológicas interoperáveis. / The global world today and the state of the art of the technology send us to a very complex environment. Nowadays, the technological convergence a reality, where the users have the necessity of moving and where they need to interact with services and applications while moving on a wireless base. The new generations of wireless networks can integrate some existing heterogeneous systems providing a new infrastructure capable of delivering several services to moving users with transparency, allowing the communicability and the access to the users’ information. This work presents a method for the description of a generic business model for a moving user inside of a convergent heterogeneous environment. The user’s movement in this environment is modelled on a mobility and interactivity framework, where there has been built a business model to deliver services to the moving user. This thesis also presents the basic concepts of a business model for mobility and interactivity applicable to open systems.

Tettigoniidae (katydids) has more than 7200 species and is the largest family within the insect order Orthoptera. Their unique biology including leaf-like crypsis, acoustic signaling, and courtship rituals garners much of their academic attention. However, the taxonomy of katydids is chaotic and previous to these studies, little work had been done to decipher the phylogenetic relationships within this family. Without a robust phylogenetic framework, questions regarding the evolution of katydid disguises including the leaf-like crypsis cannot be addressed. This dissertation contains three chapters. Chapter 1 provides the first phylogenetic hypothesis focusing on Tettigoniidae. In this chapter we show a character thought to be taxonomically informative, the thoracic auditory spiracle, is homoplasious within Tettigoniidae. We provide evidence that the leaf-like wings of katydids have been derived independently in multiple lineages. Additionally, in Chapter 1 the problematic taxonomy within Tettigoniidae, particularly the lack of monophyly in many of the larger and widespread subfamilies, is addressed. Chapter 2 contains a more in depth look into the evolution of crypsis. Leaf-like wings are common throughout Tettigoniidae, but the definition of leaf-like has varied by author. In this second chapter we provide a ratio method for differentiating between leaf-like and non leaf-like wings. Our ratio method was then verified using geometric morphometics. We found at least 15 independent derivations of leaf-like wings in Tettigoniidae. Furthermore we found that throughout Tettigoniidae the leaf-like wings are not a driver of speciation and selection may favor a shift away from the leaf-like wings. Within the cosmopolitan Phaneropterinae, the trend differs, as there is no significant difference between the speciation and transition rates of the leaf-like and non leaf-like lineages. Chapter 3 presents the largest and most comprehensive phylogeny for Tettigoniidae to date and provides a hypothesis for origins and biogeographic dispersal of katydids. Characters that define subfamilies are similar due to similar selective pressures and are not taxonomically informative. As a result, many of the larger and widespread subfamilies, particularly those with species in similar but geographically distant habitats, are paraphyletic. In this chapter we also provide temporary names to define the two large clades containing the bulk of Tettigoniidae diversity (tettigonioid clade and phaneropteroid clade) in addition to smaller subfamily groups to simplify discussion of katydid relationships until a higher-level taxonomic revision is completed.

I give an overview of the basic theory of Hilbert spaces necessary to understand the convergence of the Fourier series for square integrable functions. I state the necessary theorems and definitions to understand the formulations of the problem in a Hilbert space framework, and then I give some applications of the theory along the way.

In a reinforcement learning task an agent must learn a policy for performing actions so as to perform well in a given environment. Policy gradient methods consider a parameterized class of policies, and using a policy from the class, and a trajectory through the environment taken by the agent using this policy, estimate the performance of the policy with respect to the parameters. Policy gradient methods avoid some of the problems of value function methods, such as policy degradation, where inaccuracy in the value function leads to the choice of a poor policy. However, the estimates produced by policy gradient methods can have high variance.¶ In Part I of this thesis we study the estimation variance of policy gradient algorithms, in particular, when augmenting the estimate with a baseline, a common method for reducing estimation variance, and when using actor-critic methods. A baseline adjusts the reward signal supplied by the environment, and can be used to reduce the variance of a policy gradient estimate without adding any bias. We find the baseline that minimizes the variance. We also consider the class of constant baselines, and find the constant baseline that minimizes the variance. We compare this to the common technique of adjusting the rewards by an estimate of the performance measure. Actor-critic methods usually attempt to learn a value function accurate enough to be used in a gradient estimate without adding much bias. In this thesis we propose that in learning the value function we should also consider the variance. We show how considering the variance of the gradient estimate when learning a value function can be beneficial, and we introduce a new optimization criterion for selecting a value function.¶ In Part II of this thesis we consider online versions of policy gradient algorithms, where we update our policy for selecting actions at each step in time, and study the convergence of the these online algorithms. For such online gradient-based algorithms, convergence results aim to show that the gradient of the performance measure approaches zero. Such a result has been shown for an algorithm which is based on observing trajectories between visits to a special state of the environment. However, the algorithm is not suitable in a partially observable setting, where we are unable to access the full state of the environment, and its variance depends on the time between visits to the special state, which may be large even when only few samples are needed to estimate the gradient. To date, convergence results for algorithms that do not rely on a special state are weaker. We show that, for a certain algorithm that does not rely on a special state, the gradient of the performance measure approaches zero. We show that this continues to hold when using certain baseline algorithms suggested by the results of Part I.

Cette thèse se concentre sur le modèle de classification binaire. Etant donné $n$ couples de variables aléatoires indépendantes et identiquement distribuées (i.i.d.) $(X_i,Y_i)$, $i=1,\ldots ,n$ de loi $P$, on cherche à prédire la classe $Y\in\{-1,+1\}$ d'une nouvelle entrée $X$ où $(X,Y)$ est de loi $P$. La règle de Bayes, notée $f^*$, minimise l'erreur de généralisation $R(f)=P(f(X)\not=Y)$. Un algorithme de classification doit s'approcher de la règle de Bayes. Cette thèse suit deux axes : établir des vitesses de convergence vers la règle de Bayes et proposer des procédures adaptatives.<br /><br />Les méthodes de régularisation ont montrées leurs intérêts pour résoudre des problèmes de classification. L'algorithme des Machines à Vecteurs de Support (SVM) est aujourd'hui le représentant le plus populaire. Dans un premier temps, cette thèse étudie les performances statistiques de cet algorithme, et considère le problème d'adaptation à la marge et à la complexité. On étend ces résultats à une nouvelle procédure de minimisation de risque empirique pénalisée sur les espaces de Besov. Enfin la dernière partie se concentre sur une nouvelle procédure de sélection de modèles : la minimisation de l'enveloppe du risque (RHM). Introduite par L.Cavalier et Y.Golubev dans le cadre des problèmes inverses, on cherche à l'appliquer au contexte de la classification.

Dans ce travail de thèse sont étudiés des problèmes mathématiques (théorie et approximation) issus de la théorie de la dynamique des gaz compressibles et modélisés par les équations d'Euler. L'approximation numérique de ces problèmes physiques a nécessite une étude détaillée de quelques problèmes de conditions aux limites. Les approximations numériques obtenues sont basées sur la methode des volumes finis, qui nous a semble la mieux adaptée pour la discrétisation des systèmes hyperboliques de lois de conservation en général. La convergence de la methode des volumes finis est obtenue pour les problèmes de lois de conservation scalaires avec des conditions aux limites, a l'aide d'unicité dans l'espace des solutions mesures

La solution projection itérée pour l'équation intégrale de Fredholm de seconde espèce. Résultats de superconvergence pour la methode de projection itérée appliquée à une équation intégrale de Fredholm de 2ème espèce et problème aux valeurs propres. Résultats de superconvergence pour des problèmes aux valeurs propres différentiels : une methode de Galerkin sur la formulation intégrale. Superconvergence des vecteurs propres généralisés d'opérateurs différentiels et intégraux aux nœuds.