Infinitely divisible distributions and processes have been the object of extensive research not only from the theoretical point of view but also for practical use, for example, in queueing theory or mathematical finance. In this thesis, we will study some of their subclasses with a view towards financial modeling. As generalizations of stable distributions, we study the tempered stable distributions and introduce the new classes of layered stable distributions as well as the mixed stable distributions, along with the corresponding Levy processes. As a further generalization of infinitely divisible processes, fractional tempered stable motions are defined. These theoretical studies will be complemented by some more practical ones, such as the simulation of sample paths, parameter estimations, financial portfolio hedging, and solving stochastic differential equations.

This thesis addresses some problems that arise in signal processing when the noise is impulsive and follows a heavy tailed distribution. After reviewing several of the more well known heavy- tailed distributions the common problem of which of these hest models the observations is considered. To this end, a test is proposed for the symmetric alpha stable distribution. The test threshold is found using both asymptotic theory and parametric bootstrap resampling. In doing so, some modifications are proposed for Koutrouvelis' estimator of the symmetric alpha stable distributions parameters that improve performance. In electrical systems impulsive noise is generated externally to the receiver while thermal Gaussian noise is generated internally by the receiver electronics, the resultant noise is an additive combination of these two independent sources. A characteristic function domain estimator for the parameters of the resultant distribution is developed for the case when the impulsive noise is modeled by a symmetric alpha stable distribution. Having concentrated on validation and parameter estimation for the noise model, some problems in signal detection and estimation are considered. Detection of the number of sources impinging on an array is an important first. step in many array processing problems for which the development of optimal methods can be complicated even in the Gaussian case. Here, a multiple hypothesis test for the equality of the eigenvalues of the sample array covariance is proposed. / The nonparametric bootstrap is used to estimate the distributions of the test statistics removing the assumption of Gaussianity and offering improved performance for heavy tailed observations. Finally, some robust estimators are proposed for estimating parametric signals in additive noise. These are based on M-estimators but implicitly incorporate an estimate of the noise distribution. enabling the estimator to adapt to the unknown noise distribution. Two estimators are developed, one uses a nonparametric kernel density estimator while the other models the score function of the noise distribution with a linear combination of basis functions.

The family of stable distributions represents an important generalization of the Gaussian family; stable random variables obey a generalized central limit theorem where the assumption of finite variance is replaced with one of power law decay in the tails. Possessing heavy tails, asymmetry, and infinite variance, non-Gaussian stable distributions can be suitable for inference in settings featuring impulsive, possibly skewed noise. A general lack of analytical form for the densities and distributions of stable laws has prompted research into computational methods of estimation. This report introduces stable distributions through a discussion of their basic properties and definitions in chapter 1. Chapter 2 surveys applications, and chapter 3 discusses a number of procedures for inference, with particular attention to time series models in the ARMA setting. Further details and an application can be found in the appendices. / text

This paper surveys various statistical methods that have been proposed for the examination of the efficiency of financial markets and proposes a novel procedure for testing the predictability of a time series. For illustration, this procedure is applied to Austrian stock return series.

L’objectif de cette thèse est d’étudier le comportement des codes LDPC dans un environnement où l’interférence générée par un réseau n’est pas de nature gaussienne mais présente un caractère impulsif. Un premier constat rapide montre que sans précaution, les performances de ces codes se dégradent très significativement. Nous étudions tout d’abord les différentes solutions possibles pour modéliser les bruits impulsifs. Dans le cas des interférences d’accès multiples qui apparaissent dans les réseaux ad hoc et les réseaux de capteurs, il nous semble approprié de choisir les distributions alpha-stables. Généralisation de la gaussienne, stables par convolution, elles peuvent être validées théoriquement dans plusieurs situations.Nous déterminons alors la capacité de l’environnement α-stable et montrons par une approche asymptotique que les codes LDPC dans cet environnement sont bons mais qu’une simple opération linéaire à l’entrée du décodeur ne permet pas d’obtenir de bonnes performances. Nous avons donc proposé différentes façons de calculer la vraisemblance en entrée du décodeur. L’approche optimale est très complexe à mettre en oeuvre. Nous avons étudié plusieurs approches différentes et en particulier le clipping dont nous avons cherché les paramètres optimaux. / The goal of this PhD is to study the performance of LDPC codes in an environment where interference, generated by the network, has not a Gaussian nature but presents an impulsive behavior.A rapid study shows that, if we do not take care, the codes’ performance significantly degrades.In a first step, we study different approaches for impulsive noise modeling. In the case of multiple access interference that disturb communications in ad hoc or sensor networks, the choice of alpha-stable distributions is appropriate. They generalize Gaussian distributions, are stable by convolution and can be theoretically justified in several contexts.We then determine the capacity if the α-stable environment and show using an asymptotic method that LDPC codes in such an environment are efficient but that a simple linear operation on the received samples at the decoder input does not allow to obtain the expected good performance. Consequently we propose several methods to obtain the likelihood ratio necessary at the decoder input. The optimal solution is highly complex to implement. We have studied several other approaches and especially the clipping for which we proposed several approaches to determine the optimal parameters.

This thesis challenges several concepts in finance. Firstly, it is the Markowitz's solution to the portfolio problem. It introduces a new method which de- noises the covariance matrix - the cornerstone of the portfolio management. Random Matrix Theory originates in particle physics and was recently intro- duced to finance as the intersection between economics and natural sciences has widened over the past couple of years. Often discussed Efficient Market Hypothesis is opposed by adopting the assumption, that financial returns are driven by Paretian distributions, in- stead of Gaussian ones, as conjured by Mandelbrot some 50 years ago. The portfolio selection is set in a framework, where Expected Shortfall replaces the standard deviation as the risk measure. Therefore, direct optimi- sation of the portfolio is implemented to be compared with the performance of the classical solution and its denoised counterpart. The results are evalu- ated in a controlled environment of Monte Carlo simulation as well as using empirical data from S&P 500 constituents. 1

Made available in DSpace on 2016-03-15T19:38:05Z (GMT). No. of bitstreams: 1 Agostinho Benigno Monteiro Gutierrez.pdf: 1771214 bytes, checksum: b247e1232736a440c8348a9a5765749a (MD5) Previous issue date: 2007-09-19 / Fundo Mackenzie de Pesquisa / Evolutionary strategies normally use the Gaussian distributions in order to control the mutations over real values. Since there are other kinds of distributions in nature and in mathematics, such as those of Cauchy, Lévy and S-Lévy, in addition to several stable distributions, it seems a natural step to extend the standard approach, by using an algorithm that would be based upon other existing distributions, or that would even allow the choice of a stable distribution in a self-adaptive way. Such an idea is briefly sketched herein, in the context of populations of individuals that evolve towards the minimum of a test function (namely, the n-dimensional Rastrigin, Rosenberg, Griewangk and Schwefel functions) by means of evolutionary strategies, whose mutations are guided by eight types of specific types of distributions and by a self-adaptive scheme over a subset of the possible stable distributions. During the evolution of the experiment a remarkable influence on the right choice of the distribution family can be noted related to the search for the global minimum of a test function. This is due to the diversity used in the form of distribution: asymmetric and long tale (Lévy) and symmetric with various type of tale on the others. The choice of the type of distribution occurs determining four parameters properly: stability rate, asymmetric, scale and position. The choice of the type of distribution occurs determining if the four parameters above mentiones are part of the chromosome that also contains the possible coordinates of the global minimum that will be mutated according to the chosen distribution. Having applied this different mutation in the evolutionary process will lead to the global minimum of the chosen test function. The results indicate that the combined use of stable distribution controlling the mutations of the coordinates can result in a performance improvement regarding the convergence and consequent determination of the solution, when applied to spatially constrained benchmark functions. / Usualmente, as estratégias evolutivas utilizam as distribuições Gaussianas para governar as mutações sobre valores reais. Já que na natureza e na matemática existem outros tipos de distribuições, tais como de Cauchy, de Lévy e de S-Lévy, além de uma infinidade de distribuições estáveis, é razoável se pensar em expandir a abordagem tradicional, utilizando-se um algoritmo baseado em outras distribuições existentes, ou mesmo que possibilite a escolha de uma distribuição estável, de forma auto-adaptativa. Esta idéia é aqui ilustrada, no contexto de populações de indivíduos que evoluem em busca do mínimo de uma função de teste (no caso, a função de Rastrigin, vale de Rosenberg, Griewangk e Schwefel em n-dimensões) através de estratégias evolutivas cujas mutações são guiadas por oito tipos específicos de distribuições e de um esquema auto-adaptativo em um subconjunto das distribuições estáveis. Durante a evolução dos experimentos observa-se uma forte influência da escolha adequada da família de distribuição na correlação da busca do mínimo global na função de teste. Este fato se deve a diversidade utilizada na forma da distribuição: assimétrica e cauda longa (Lévy) e simétrica com vários tipos de cauda nas demais. A escolha do tipo de distribuição ocorre determinando-se adequadamente quatro parâmetros: Índice de estabilidade (α), assimétrico (β), escala (ϒ) e posição (δ). A escolha do tipo de distribuição ocorre determinando-se os quatro parâmetros acima que fazem parte do cromossomo que também contém as possíveis coordenadas do ponto de mínimo global que seram mutadas com base distribuição escolhida. Com aplicação desta mutação diferenciada no processo evolutivo chegasse ao mínimo global da função de teste escolhida. Os resultados indicaram que a utilização conjunta de distribuições estáveis governando as mutações das coordenadas podem acarretar uma melhora de desempenho com respeito à convergência e conseqüente determinação da solução, quando aplicadas sobre funções de teste delimitadas espacialmente.

Made available in DSpace on 2016-03-15T19:38:05Z (GMT). No. of bitstreams: 1 Leopoldo Bulgarelli de Carvalho.pdf: 696477 bytes, checksum: f90764d3c257bf63305bda69583c731e (MD5) Previous issue date: 2007-09-12 / Fundo Mackenzie de Pesquisa / Recent applications in evolutionary programming have suggested the use of different stable probability distributions, such as Cauchy and Lévy, in the random process associated with the mutations, as an alternative to the traditional (and also stable) Normal distribution. The motivation for this is the attempt to improve the results in some classes of optimisation problems, over those obtained with Normal distribution. Based upon an algorithm proposed in the literature, mostly its version in [Lee and Yao, 2004], that use non Normal stable distributions, we study herein the effect of turning it adaptive in respect to the determination of the more adequate stable distribution parameters for each problem. The evaluations relied upon standard benchmarking functions of the literature, and the comparative performance tests were carried out in respect to the baseline defined by a standard algorithm using Normal distribution. The results suggest numerical and statistical superiority of the stable distribution based approach, when compared with the baseline. However, they showed no improvement over the adaptive method of [Lee and Yao, 2004], possibly due to a consequence of implementation decisions that had to be made in the present implementation, that were not made explicit therein. / Aplicações recentes em programação evolutiva tem sugerido a utilização de diferentes distribuições estáveis de probabilidade, tais como de Cauchy e de Lévy, no processo aleatório associado às mutações, como alternativa à tradicional (e também estável) distribuição Normal. A motivação para tanto é melhorar os resultados em algumas classes de problemas de otimização, com relação aos obtidos através da distribuição Normal. Esse trabalho propõe uma nova classe de algoritmos auto-adaptativos com respeito à determinação dos parâmetros da distribuição estável mais adequada para cada problema de otimização. Tais algoritmos foram derivados de um existente na literatura, especialmente sua versão apresentada em [Lee e Yao, 2004]. Em um primeiro momento foram estudadas as principais características das distribuições estáveis que são, nesse trabalho, o foco dos processos aleatórios associados às mutações. Posteriormente, foram apresentadas as diferentes abordagens descritas pela literatura e as sugestões de algoritmos com características auto-adaptativas. As avaliações dos algoritmos propostos utilizaram funções de teste padrão da literatura, e os resultados comparativos de desempenho foram realizados com relação a um algoritmo tradicional baseado na distribuição Normal. Posteriormente, foram aplicados novos comparativos entre as diversas abordagens auto-adaptativas definidas no presente estudo, e feito um comparativo do melhor algoritmo auto-adaptativo aqui proposto com o melhor algoritmo adaptativo obtido de [Lee e Yao, 2004]. Os resultados evidenciaram superioridade numérica e estatística da abordagem baseada em distribuições estáveis, sobre o método tradicional baseado na distribuição Normal. No entanto, o método proposto não se mostrou mais eficaz que o método adaptativo sugerido em [Lee e Yao, 2004], o que pode ter sido decorrente de decisões de implementação não explícitas naquele trabalho, que tiveram de ser tomadas no presente contexto.

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Modelos GARCH tendo a normal e a t-Student como distribuições condicionais são amplamente utilizados para modelagem da volatilidade de dados financeiros. No entanto, tais distribuições podem não ser apropriadas para algumas séries com caudas pesadas e comportamento leptocúrtico. As chamadas distribuições estáveis podem ser mais adequadas para sua modelagem, como já explorado na literatura. Por outro lado, os modelos GAS (Generalized Autoregressive Score), com desenvolvimento recente, tratam-se de modelos dinâmicos que possuem em sua estrutura a função score (derivada do logaritmo da verossimilhança). Tal abordagem oferece uma direção natural para a evolução dos parâmetros da distribuição dos dados. Neste trabalho, é proposto um novo modelo GAS em conjunção com distribuições estáveis simétricas para a modelagem da volatilidade - de fato, é uma generalização do GARCH, pois, para uma particular escolha de distribuição estável e de estrutura do modelo, tem-se o clássico modelo GARCH gaussiano. Como em geral a função densidade das distribuições estáveis não possui forma analítica fechada, é apresentado seu procedimento de cálculo, bem como de suas derivadas, para o completo desenvolvimento do método de estimação dos parâmetros. Também são analisadas as condições de estacionariedade e a estrutura de dependência do modelo. Estudos de simulação são conduzidos, bem como uma aplicação a dados reais, para comparação entre modelos usuais, que utilizam distribuições normal e t-Student, e o modelo proposto, demonstrando a eficácia deste. / GARCH models with normal and t-Student conditional distributions are widely used for volatility modeling in financial data. However, such distributions may not be suitable for some heavy-tailed and leptokurtic series. The stable distributions may be more adequate to fit such characteristics, as already exploited in the literature. On the other hand, the recently developed GAS (Generalized Autoregressive Score) models are dynamic models in which the updating mechanism of the time-varying parameters is based on the score function (first derivative of the log-likelihood function). This provides the natural direction for updating the parameters, based on the complete density. We propose a new GAS model with symmetric stable distribution for volatility modeling. The model can be interpreted as a generalization of the GARCH models, since the classic gaussian GARCH model is derived from it by using particular choices of the stable distribution and the model structure. There are no closed analytical expressions for general stable densities in most cases, hence its numeric computation and derivatives are detailed for the sake of complete development of the estimation process. The stationarity conditions and the dependence structure of the model are analysed. Simulation studies, as well as an application to real data, are presented for comparisons between the usual models and the proposed model, illustrating the effectiveness of the latter.