We investigate the genericity of measure-preserving actions of the free group Fn, on possibly countably infinitely many generators, acting on a standard probability space. Specifically, we endow the space of all measure-preserving actions of Fn acting on a standard probability space with the weak topology and explore what properties may be verified on a comeager set in this topology. In this setting we show an analog of the classical Rokhlin Lemma. From this result we conclude that every action of Fn may be approximated by actions which factor through a finite group. Using this finite approximation we show the actions of Fn, which are rigid and hence fail to be mixing, are generic. Combined with a recent result of Kerr and Li, we obtain that a generic action of Fn is weak mixing but not mixing. We also show a generic action of Fn has sigma-entropy at most zero. With some additional work, we show the finite approximation result may be used to that show for any action of Fn, the crossed product embeds into the tracial ultraproduct of the hyperfinite II1 factor. We conclude by showing the finite approximation result may be transferred to a subspace of the space of all topological actions of Fn on the Cantor set. Within this class, we show the set of actions with sigma-entropy at most zero is generic.

The robust pricing and hedging approach in Mathematical Finance, pioneered by Hobson (1998), makes statements about non-traded derivative contracts by imposing very little assumptions about the underlying financial model but directly using information contained in traded options, typically call or put option prices. These prices are informative about marginal distributions of the asset. Mathematically, the theory of Skorokhod embeddings provides one possibility to approach robust problems. In this thesis we consider mostly robust pricing and hedging problems of Lookback options (options written on the terminal maximum of an asset) and Convex Vanilla Options (options written on the terminal value of an asset) and extend the analysis which is predominately found in the literature on robust problems by two features: Firstly, options with multiple maturities are available for trading (mathematically this corresponds to multiple marginal constraints) and secondly, restrictions on the total realized variance of asset trajectories are imposed. Probabilistically, in both cases, we develop new optimal solutions to the Skorokhod embedding problem. More precisely, in Part I we start by constructing an iterated Azema-Yor type embedding (a solution to the n-marginal Skorokhod embedding problem, see Chapter 2). Subsequently, its implications are presented in Chapter 3. From a Mathematical Finance perspective we obtain explicitly the optimal superhedging strategy for Barrier/Lookback options. From a probability theory perspective, we find the maximum maximum of a martingale which is constrained by finitely many intermediate marginal laws. Further, as a by-product, we discover a new class of martingale inequalities for the terminal maximum of a cadlag submartingale, see Chapter 4. These inequalities enable us to re-derive the sharp versions of Doob's inequalities. In Chapter 5 a different problem is solved. Motivated by the fact that in some markets both Vanilla and Barrier options with multiple maturities are traded, we characterize the set of market models in this case. In Part II we incorporate the restriction that the total realized variance of every asset trajectory is bounded by a constant. This has been previously suggested by Mykland (2000). We further assume that finitely many put options with one fixed maturity are traded. After introducing the general framework in Chapter 6, we analyse the associated robust pricing and hedging problem for convex Vanilla and Lookback options in Chapters 7 and 8. Robust pricing is achieved through construction of appropriate Root solutions to the Skorokhod embedding problem. Robust hedging and pathwise duality are obtained by a careful development of dynamic pathwise superhedging strategies. Further, we characterize existence of market models with a suitable notion of arbitrage.

A virtualização de redes é um mecanismo que permite a coexistência de múltiplas redes virtuais sobre um mesmo substrato físico. Um dos desafios de pesquisa abordados na literatura é o mapeamento eficiente de recursos virtuais em infraestruturas físicas. Embora o referido desafio tenha recebido considerável atenção, as abordagens que constituem o estado-da-arte apresentam alta taxa de rejeição, i.e., a proporção de solicitações de redes virtuais negadas em relação ao total de solicitações efetuadas ao substrato é elevada. Nesta dissertação, caracteriza-se, inicialmente, a relação entre a qualidade dos mapeamentos de redes virtuais e as estruturas topológicas dos substratos subjacentes. Avalia-se as soluções exatas de um modelo de mapeamento online sob diferentes classes de topologias de rede. A partir do entendimento dos fatores topológicos que influenciam diretamente o processo de mapeamento de redes virtuais, propõe-se uma estratégia para planejar a expansão de redes de provedores de infraestrutura de forma a reduzir consistentemente a taxa de rejeição de requisições de redes virtuais e melhor aproveitar os recursos ociosos da mesma. Os resultados obtidos evidenciam que grande parte das rejeições de redes virtuais ocorre em situações em que há grande disponibilidade de recursos, mas alguns poucos já saturados acabam inviabilizando, em função de características de conectividade do substrato, o atendimento de novas requisições. Ademais, os resultados obtidos utilizando a estratégia proposta evidenciam que o fortalecimento de partes-chave da infraestrutura levam a uma ocupação muito mais satisfatória. Uma expansão de 10% a 20% dos recursos da infraestrutura contribui para um aumento sustentado de até 30% no número de redes virtuais aceitas e de até 45% no aproveitamento dos recursos em comparação com a rede original. / Network virtualization is a mechanism that allows the coexistence of multiple virtual networks on top of a single physical substrate. One of the research challenges addressed recently in the literature is the efficient mapping of virtual resources on physical infrastructures. Although this challenge has received considerable attention, state-of-the-art approaches present, in general, a high rejection rate, i.e., the ratio between the number of denied virtual network requests and the total amount of requests is considerably high. In this thesis, we characterize the relationship between the quality of virtual network mappings and the topological structures of the underlying substrates. Exact solutions of an online embedding model are evaluated under different classes of network topologies. From the understanding of the topological factors that directly influence the virtual network embedding process, we propose an expansion strategy of physical infrastructure in order to suggest adjustments that lead to higher virtual network acceptance and, in consequence, to improved physical resource utilization. The obtained results demonstrate that most of rejections occur in situations in which a significant amount of resource is available, but a few saturated devices and links, depending on connectivity features of the physical substrate, hinder the acceptance of new requests. Moreover, the obtained results using the proposed strategy evidence that an expansion of 10% to 20% of the infrastructure resources leads to a sustained increase of up to 30% in the number of accepted virtual networks and of up to 45% in resource usage compared to the original network.

A virtualização de redes é um mecanismo que permite a coexistência de múltiplas redes virtuais sobre um mesmo substrato físico. Um dos desafios de pesquisa abordados na literatura é o mapeamento eficiente de recursos virtuais em infraestruturas físicas. Embora o referido desafio tenha recebido considerável atenção, as abordagens que constituem o estado-da-arte apresentam alta taxa de rejeição, i.e., a proporção de solicitações de redes virtuais negadas em relação ao total de solicitações efetuadas ao substrato é elevada. Nesta dissertação, caracteriza-se, inicialmente, a relação entre a qualidade dos mapeamentos de redes virtuais e as estruturas topológicas dos substratos subjacentes. Avalia-se as soluções exatas de um modelo de mapeamento online sob diferentes classes de topologias de rede. A partir do entendimento dos fatores topológicos que influenciam diretamente o processo de mapeamento de redes virtuais, propõe-se uma estratégia para planejar a expansão de redes de provedores de infraestrutura de forma a reduzir consistentemente a taxa de rejeição de requisições de redes virtuais e melhor aproveitar os recursos ociosos da mesma. Os resultados obtidos evidenciam que grande parte das rejeições de redes virtuais ocorre em situações em que há grande disponibilidade de recursos, mas alguns poucos já saturados acabam inviabilizando, em função de características de conectividade do substrato, o atendimento de novas requisições. Ademais, os resultados obtidos utilizando a estratégia proposta evidenciam que o fortalecimento de partes-chave da infraestrutura levam a uma ocupação muito mais satisfatória. Uma expansão de 10% a 20% dos recursos da infraestrutura contribui para um aumento sustentado de até 30% no número de redes virtuais aceitas e de até 45% no aproveitamento dos recursos em comparação com a rede original. / Network virtualization is a mechanism that allows the coexistence of multiple virtual networks on top of a single physical substrate. One of the research challenges addressed recently in the literature is the efficient mapping of virtual resources on physical infrastructures. Although this challenge has received considerable attention, state-of-the-art approaches present, in general, a high rejection rate, i.e., the ratio between the number of denied virtual network requests and the total amount of requests is considerably high. In this thesis, we characterize the relationship between the quality of virtual network mappings and the topological structures of the underlying substrates. Exact solutions of an online embedding model are evaluated under different classes of network topologies. From the understanding of the topological factors that directly influence the virtual network embedding process, we propose an expansion strategy of physical infrastructure in order to suggest adjustments that lead to higher virtual network acceptance and, in consequence, to improved physical resource utilization. The obtained results demonstrate that most of rejections occur in situations in which a significant amount of resource is available, but a few saturated devices and links, depending on connectivity features of the physical substrate, hinder the acceptance of new requests. Moreover, the obtained results using the proposed strategy evidence that an expansion of 10% to 20% of the infrastructure resources leads to a sustained increase of up to 30% in the number of accepted virtual networks and of up to 45% in resource usage compared to the original network.

Les systèmes à canaux non fiables ont été introduits à l'origine comme un modèle de communication. Ils ont donné naissance à une classe de complexité restée mal comprise pendant longtemps. Dans cette thèse, nous étudions et comblons certaines des plus importantes lacunes dans la connaissance de cette classe. Nous fournissons entre autres des bornes inférieure et supérieure qui se rejoignent pour la complexité en temps. Puis nous proposons un nouvel outil de preuve : le Problème de Sous Mot de Post (PEP). C'est un problème simple, inspiré du Problème de Correspondance de Post, et complet pour cette classe de complexité. Nous étudions ensuite PEP et ses variantes, ainsi que les langages de solutions de PEP sur lesquels nous avons fourni des résultats de complexité et des outils de preuve tels que des lemmes de pompage. / Lossy channel systems were originally introduced to model communication protocols. It gave birth to a complexity class wich remained scarcely undersood for a long time. In this thesis we study some of the most important gaps. In particular, we bring matching upper and lower bounds for the time complexity. Then we describe a new proof tool : the Post Embedding Problem (PEP) which is a simple problem, closely related to the Post Correspondence Problem, and complete for this complexity class. Finally, we study PEP, its variants and the languages of solutions of PEP on which we provide complexity results and proof tools like pumping lemmas.

A virtualização de redes é um mecanismo que permite a coexistência de múltiplas redes virtuais sobre um mesmo substrato físico. Um dos desafios de pesquisa abordados na literatura é o mapeamento eficiente de recursos virtuais em infraestruturas físicas. Embora o referido desafio tenha recebido considerável atenção, as abordagens que constituem o estado-da-arte apresentam alta taxa de rejeição, i.e., a proporção de solicitações de redes virtuais negadas em relação ao total de solicitações efetuadas ao substrato é elevada. Nesta dissertação, caracteriza-se, inicialmente, a relação entre a qualidade dos mapeamentos de redes virtuais e as estruturas topológicas dos substratos subjacentes. Avalia-se as soluções exatas de um modelo de mapeamento online sob diferentes classes de topologias de rede. A partir do entendimento dos fatores topológicos que influenciam diretamente o processo de mapeamento de redes virtuais, propõe-se uma estratégia para planejar a expansão de redes de provedores de infraestrutura de forma a reduzir consistentemente a taxa de rejeição de requisições de redes virtuais e melhor aproveitar os recursos ociosos da mesma. Os resultados obtidos evidenciam que grande parte das rejeições de redes virtuais ocorre em situações em que há grande disponibilidade de recursos, mas alguns poucos já saturados acabam inviabilizando, em função de características de conectividade do substrato, o atendimento de novas requisições. Ademais, os resultados obtidos utilizando a estratégia proposta evidenciam que o fortalecimento de partes-chave da infraestrutura levam a uma ocupação muito mais satisfatória. Uma expansão de 10% a 20% dos recursos da infraestrutura contribui para um aumento sustentado de até 30% no número de redes virtuais aceitas e de até 45% no aproveitamento dos recursos em comparação com a rede original. / Network virtualization is a mechanism that allows the coexistence of multiple virtual networks on top of a single physical substrate. One of the research challenges addressed recently in the literature is the efficient mapping of virtual resources on physical infrastructures. Although this challenge has received considerable attention, state-of-the-art approaches present, in general, a high rejection rate, i.e., the ratio between the number of denied virtual network requests and the total amount of requests is considerably high. In this thesis, we characterize the relationship between the quality of virtual network mappings and the topological structures of the underlying substrates. Exact solutions of an online embedding model are evaluated under different classes of network topologies. From the understanding of the topological factors that directly influence the virtual network embedding process, we propose an expansion strategy of physical infrastructure in order to suggest adjustments that lead to higher virtual network acceptance and, in consequence, to improved physical resource utilization. The obtained results demonstrate that most of rejections occur in situations in which a significant amount of resource is available, but a few saturated devices and links, depending on connectivity features of the physical substrate, hinder the acceptance of new requests. Moreover, the obtained results using the proposed strategy evidence that an expansion of 10% to 20% of the infrastructure resources leads to a sustained increase of up to 30% in the number of accepted virtual networks and of up to 45% in resource usage compared to the original network.

Lossy channel systems were originally introduced to model communication protocols. It gave birth to a complexity class wich remained scarcely undersood for a long time. In this thesis we study some of the most important gaps. In particular, we bring matching upper and lower bounds for the time complexity. Then we describe a new proof tool : the Post Embedding Problem (PEP) which is a simple problem, closely related to the Post Correspondence Problem, and complete for this complexity class. Finally, we study PEP, its variants and the languages of solutions of PEP on which we provide complexity results and proof tools like pumping lemmas.

Proximity problems and point set embeddability problems are fundamental and well-studied in computational geometry and graph drawing. Examples of such problems that are of particular interest to us in this dissertation include: finding the closest pair among a set P of points, finding the k-nearest neighbors to each point p in P, answering reverse k-nearest neighbor queries, computing the Yao graph, the Semi-Yao graph and the Euclidean minimum spanning tree of P, and mapping the vertices of a planar graph to a set P of points without inducing edge crossings. In this dissertation, we consider so-called kinetic version of these problems, that is, the points are allowed to move continuously along known trajectories, which are subject to change. We design a set of data structures and a mechanism to efficiently update the data structures. These updates occur at critical, discrete times. Also, a query may arrive at any time. We want to answer queries quickly without solving problems from scratch, so we maintain solutions continuously. We present new techniques for giving kinetic solutions with better performance for some these problems, and we provide the first kinetic results for others. In particular, we provide: • A simple kinetic data structure (KDS) to maintain all the nearest neighbors and the closest pair. Our deterministic kinetic approach for maintenance of all the nearest neighbors improves the previous randomized kinetic algorithm. • An exact KDS for maintenance of the Euclidean minimum spanning tree, which improves the previous KDS. • The first KDS's for maintenance of the Yao graph and the Semi-Yao graph. • The first KDS to consider maintaining plane graphs on moving points. • The first KDS for maintenance of all the k-nearest neighbors, for any k ≥ 1. • The first KDS to answer the reverse k-nearest neighbor queries, for any k ≥ 1 in any fixed dimension, on a set of moving points. / Graduate

Cette thèse porte sur les processus croissants pour l'ordre convexe que nous désignons sous le nom de peacocks. Un résultat remarquable dû à Kellerer stipule qu'un processus stochastique à valeurs réelles est un peacock si et seulement s'il possède les mêmes marginales unidimensionnelles qu'une martingale. Une telle martingale est dite associée à ce processus. Mais dans son article, Kellerer ne donne ni d'exemple de peacock, ni d'idée précise sur la construction d'une martingale associée pour un peacock donné. Ainsi, comme d'autres travaux sur les peacocks, notre étude vise deux objectifs. Il s'agit d'exhiber de nouvelles familles de peacocks et de construire des martingales associées pour certains peacocks. Dans les trois premiers chapitres, nous exhibons diverses classes de peacocks en utilisant successivement les notions de monotonie conditionnelle, de peacock très fort et de positivité totale d'ordre 2. En particulier, nous fournissons plusieurs extensions du résultat de Carr-Ewald-Xiao selon lequel la moyenne arithmétique du mouvement brownien géométrique, encore appelée "option asiatique" est un peacock. L'objet du dernier chapitre est de construire des martingales associées pour une classe de peacocks. Pour cela, nous utilisons les plongements d'Azéma-Yor et de Bertoin-Le Jan. L'originalité de ce chapitre est l'utilisation de la positivité totale d'ordre 2 dans l'étude du plongement d'Azéma-Yor / This thesis deals with real valued stochastic processes which increase in the convex order. We call them peacocks. A remarkable result due to Kellerer states that a real valued process is a peacock if and only if it has the same one-dimensional marginals as a martingale. Such a martingale is said to be associated to this process. But in his article, Kellerer provides neither an example of peacock nor a concrete idea to construct an associated martingale to a given peacock. Hence, as other investigations on peacocks, our study has two purposes. We first exhibit new families of peacocks and then, we contruct associated martingales to certain of them. In the first three chapters, we exhibit several classes of peacocks using successively the notions of conditional monotonicity, very strong peacock and total positivity of order 2. In particular, we provide many extensions of Carr-Ewald-Xiao result which states that the arithmetic mean of geometric Brownian motion, also called "Asian option" is a peacock. The purpose of the last chapter is to construct associated martingales to certain peacocks. To this end, we use Azéma-Yor and Bertoin-Le Jan embedding algorithms. The originality of this chapter is the use of total positivity of order 2 in the study of Azéma-Yor embedding algorithm

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