In this thesis, we study the existence of random periodic solutions of random dynamical systems (RDS) by geometric and topological approach. We employed an extension of ergodic theory to random setting to prove that a random invariant set with some kind of dissipative structure, can be expressed as union of random periodic curves. We extensively characterize the dissipative structure by random invariant measures and Lyapunov exponents. For stochastic flows induced by stochastic differential equations (SDEs), we studied the dissipative structure by two point motion of the SDE and prove the existence exponential stable random periodic solutions.

In this thesis the performances of different approximations are compared for a standard actuarial and financial problem: the estimation of quantiles and conditional tail expectations of the final value of a series of discrete cash flows. To calculate the risk measures such as quantiles and Conditional Tail Expectations, one needs the distribution function of the final wealth. The final value of a series of discrete payments in the considered model is the sum of dependent lognormal random variables. Unfortunately, its distribution function cannot be determined analytically. Thus usually one has to use time-consuming Monte Carlo simulations. Computational time still remains a serious drawback of Monte Carlo simulations, thus several analytical techniques for approximating the distribution function of final wealth are proposed in the frame of this thesis. These are the widely used moment-matching approximations and innovative comonotonic approximations. Moment-matching methods approximate the unknown distribution function by a given one in such a way that some characteristics (in the present case the first two moments) coincide. The ideas of two well-known approximations are described briefly. Analytical formulas for valuing quantiles and Conditional Tail Expectations are derived for both approximations. Recently, a large group of scientists from Catholic University Leuven in Belgium has derived comonotonic upper and comonotonic lower bounds for sums of dependent lognormal random variables. These bounds are bounds in the terms of "convex order". In order to provide the theoretical background for comonotonic approximations several fundamental ordering concepts such as stochastic dominance, stop-loss and convex order and some important relations between them are introduced. The last two concepts are closely related. Both stochastic orders express which of two random variables is the "less dangerous/more attractive" one. The central idea of comonotonic upper bound approximation is to replace the original sum, presenting final wealth, by a new sum, for which the components have the same marginal distributions as the components in the original sum, but with "more dangerous/less attractive" dependence structure. The upper bound, or saying mathematically, convex largest sum is obtained when the components of the sum are the components of comonotonic random vector. Therefore, fundamental concepts of comonotonicity theory which are important for the derivation of convex bounds are introduced. The most wide-spread examples of comonotonicity which emerge in financial context are described. In addition to the upper bound a lower bound can be derived as well. This provides one with a measure of the reliability of the upper bound. The lower bound approach is based on the technique of conditioning. It is obtained by applying Jensen's inequality for conditional expectations to the original sum of dependent random variables. Two slightly different version of conditioning random variable are considered in the context of this thesis. They give rise to two different approaches which are referred to as comonotonic lower bound and comonotonic "maximal variance" lower bound approaches. Special attention is given to the class of distortion risk measures. It is shown that the quantile risk measure as well as Conditional Tail Expectation (under some additional conditions) belong to this class. It is proved that both risk measures being under consideration are additive for a sum of comonotonic random variables, i.e. quantile and Conditional Tail Expectation for a comonotonic upper and lower bounds can easily be obtained by summing the corresponding risk measures of the marginals involved. A special subclass of distortion risk measures which is referred to as class of concave distortion risk measures is also under consideration. It is shown that quantile risk measure is not a concave distortion risk measure while Conditional Tail Expectation (under some additional conditions) is a concave distortion risk measure. A theoretical justification for the fact that "concave" Conditional Tail Expectation preserves convex order relation between random variables is given. It is shown that this property does not necessarily hold for the quantile risk measure, as it is not a concave risk measure. Finally, the accuracy and efficiency of two moment-matching, comonotonic upper bound, comonotonic lower bound and "maximal variance" lower bound approximations are examined for a wide range of parameters by comparing with the results obtained by Monte Carlo simulation. It is justified by numerical results that, generally, in the current situation lower bound approach outperforms other methods. Moreover, the preservation of convex order relation between the convex bounds for the final wealth by Conditional Tail Expectation is demonstrated by numerical results. It is justified numerically that this property does not necessarily hold true for the quantile.

This thesis contains theory of probability and statistical sets. Solved and unsolved problems of probability, random variable and distributions random variable, random vector, statistical sets, regression and correlation analysis. Unsolved problems contains solutions.

Submitted by Viviane Lima da Cunha (viviane@biblioteca.ufpb.br) on 2015-11-09T13:29:03Z No. of bitstreams: 1 arquivototal.pdf: 3106591 bytes, checksum: 681468331a75115cdd412b8abaa0633f (MD5) / Approved for entry into archive by Maria Suzana Diniz (msuzanad@hotmail.com) on 2015-11-09T13:50:10Z (GMT) No. of bitstreams: 1 arquivototal.pdf: 3106591 bytes, checksum: 681468331a75115cdd412b8abaa0633f (MD5) / Made available in DSpace on 2015-11-09T13:50:10Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 3106591 bytes, checksum: 681468331a75115cdd412b8abaa0633f (MD5) Previous issue date: 2015-04-30 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work we study the graph theory, which although it is somewhat widespread content, including academia is extremely important for solving many mathematical problems and physical models. Moreover, this theme can be found in applications in several areas, including quote: computer, electrical, genetic. We adopt the bibliographic research and exploratory research to deal with the issue at hand, trying to de ne and clarify the aforesaid theory, but also contribute to its spread, which enables members of the basic and higher education have a contact with such an important and fruitful know, since elementary considerations graphs brings us closer to scienti c research. We alternate concepts and statements of lemmas and theorems to solve problems. We use simple language, so that a high school student can understand, without, however, distancing us from mathematical rigor. In time, we present the four color theorem the number of Ramsey, with detailed statements of the latter result. Finally, we use concepts and purely combinatorial results and probability, using the probabilistic method to prove the existence of graphs with certain properties that are di cult construction and, through this evidence, get other desired graph. / Neste trabalho, estudamos a teoria dos grafos, que embora seja um conteúdo pouco difundido, inclusive em âmbito acadêmico é de extrema importância para a resolução de inúmeros problemas matemáticos e modelos físicos. Além disso, essa temática pode ser veri cada em aplicações nas mais diversas áreas, entre as quais citamos: computacional, elétrica, genética. Adotamos a investigação bibliográ ca e a pesquisa exploratória para tratar o tema em questão, procurando de nir e esclarecer a teoria sobredita, como também contribuir em sua difusão, o que possibilita aos integrantes da educação básica e superior terem um contato com tão importante e fecundo saber, uma vez que considerações elementares sobre grafos nos aproxima da pesquisa cientí ca. Intercalamos os conceitos e as demonstrações de lemas e teoremas com a apresentação e resolução de problemas. Empregamos uma linguagem simples, de forma que um aluno do ensino médio possa compreender, sem, no entanto, nos distanciar do rigor matemático. Em tempo, apresentamos o teorema das quatro cores e o número de Ramsey, com demonstrações detalhadas deste último resultado. Por m, utilizamos conceitos e resultados puramente de combinatória e probabilidade, utilizando o método probabilístico para provar a existência de grafos com determinadas propriedades que são de difícil construção e, por meio desta comprovação, chegar a outro grafo desejado.

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / O presente trabalho tem por finalidades: demonstrar o problema da agulha de Buffon, fazer uma pequena generalizaÃÃo do resultado obtido e apresentar aplicaÃÃes baseadas nos fundamentos do referido problema. O problema da agulha de Buffon està inserido no estudo da Teoria das Probabilidades, particularmente na subÃrea de probabilidade geomÃtrica. Para chegarmos à soluÃÃo desta questÃo, alÃm dos conceitos e propriedades atinentes à Teoria das probabilidades à necessÃrio o conhecimento de noÃÃes bÃsicas do cÃlculo integral. Nos capÃtulos 2, 3 e 4 à apresentado um estudo preliminar sobre probabilidade, com os conceitos bÃsicos, propriedades e a formulaÃÃo de alguns modelos probabilÃsticos. Durante o desenvolvimento do trabalho, sempre que possÃvel, os conceitos e definiÃÃes sÃo inseridos com o auxÃlio de um problema motivador e para fixaÃÃo dos mesmos sÃo mostrados exemplos resolvidos. O Ãltimo capÃtulo evidencia a importÃncia do problema de Buffon como mÃtodo para realizar estimativas e como fundamento para o processo de captaÃÃo de imagens pelos aparelhos de tomografia computadorizada, um grande avanÃo para a Medicina no que diz respeito ao diagnÃstico por imagens. / This paper has the objective of showing Buffon's needle problem, doing a minor generalization of the results obtained hereby, and also presenting some applications based upon the fundamentals of such problem. Buffon's needle problem has been inserted into the study of Theory of Probability, particularly in its sub-area of geometrical probability. In order to attain the solution to this question, in addition to the concepts and the properties concerning the theory of probabilities, it is necessary that one should have some basic knowledge about integral calculus. In chapters 2, 3, and 4 there is a preliminary study of probability, with the basic concepts, properties and formulation of some probabilistic models being presented. During the development of this paper, whenever it was possible, the concepts and definitions were inserted with the aid of a motivational problem and they were solved by means of fixing the same examples as shown. The final chapter presents the importance of Buffon's needle problem as a method of making estimates and as a foundation for the process of capturing images in CT (computerized tomography) scanning machines, such a great breakthrough in what concerns the diagnosis by means of imaging.

In this thesis the performances of different approximations are compared for a standard actuarial and financial problem: the estimation of quantiles and conditional tail expectations of the final value of a series of discrete cash flows. To calculate the risk measures such as quantiles and Conditional Tail Expectations, one needs the distribution function of the final wealth. The final value of a series of discrete payments in the considered model is the sum of dependent lognormal random variables. Unfortunately, its distribution function cannot be determined analytically. Thus usually one has to use time-consuming Monte Carlo simulations. Computational time still remains a serious drawback of Monte Carlo simulations, thus several analytical techniques for approximating the distribution function of final wealth are proposed in the frame of this thesis. These are the widely used moment-matching approximations and innovative comonotonic approximations. Moment-matching methods approximate the unknown distribution function by a given one in such a way that some characteristics (in the present case the first two moments) coincide. The ideas of two well-known approximations are described briefly. Analytical formulas for valuing quantiles and Conditional Tail Expectations are derived for both approximations. Recently, a large group of scientists from Catholic University Leuven in Belgium has derived comonotonic upper and comonotonic lower bounds for sums of dependent lognormal random variables. These bounds are bounds in the terms of "convex order". In order to provide the theoretical background for comonotonic approximations several fundamental ordering concepts such as stochastic dominance, stop-loss and convex order and some important relations between them are introduced. The last two concepts are closely related. Both stochastic orders express which of two random variables is the "less dangerous/more attractive" one. The central idea of comonotonic upper bound approximation is to replace the original sum, presenting final wealth, by a new sum, for which the components have the same marginal distributions as the components in the original sum, but with "more dangerous/less attractive" dependence structure. The upper bound, or saying mathematically, convex largest sum is obtained when the components of the sum are the components of comonotonic random vector. Therefore, fundamental concepts of comonotonicity theory which are important for the derivation of convex bounds are introduced. The most wide-spread examples of comonotonicity which emerge in financial context are described. In addition to the upper bound a lower bound can be derived as well. This provides one with a measure of the reliability of the upper bound. The lower bound approach is based on the technique of conditioning. It is obtained by applying Jensen's inequality for conditional expectations to the original sum of dependent random variables. Two slightly different version of conditioning random variable are considered in the context of this thesis. They give rise to two different approaches which are referred to as comonotonic lower bound and comonotonic "maximal variance" lower bound approaches. Special attention is given to the class of distortion risk measures. It is shown that the quantile risk measure as well as Conditional Tail Expectation (under some additional conditions) belong to this class. It is proved that both risk measures being under consideration are additive for a sum of comonotonic random variables, i.e. quantile and Conditional Tail Expectation for a comonotonic upper and lower bounds can easily be obtained by summing the corresponding risk measures of the marginals involved. A special subclass of distortion risk measures which is referred to as class of concave distortion risk measures is also under consideration. It is shown that quantile risk measure is not a concave distortion risk measure while Conditional Tail Expectation (under some additional conditions) is a concave distortion risk measure. A theoretical justification for the fact that "concave" Conditional Tail Expectation preserves convex order relation between random variables is given. It is shown that this property does not necessarily hold for the quantile risk measure, as it is not a concave risk measure. Finally, the accuracy and efficiency of two moment-matching, comonotonic upper bound, comonotonic lower bound and "maximal variance" lower bound approximations are examined for a wide range of parameters by comparing with the results obtained by Monte Carlo simulation. It is justified by numerical results that, generally, in the current situation lower bound approach outperforms other methods. Moreover, the preservation of convex order relation between the convex bounds for the final wealth by Conditional Tail Expectation is demonstrated by numerical results. It is justified numerically that this property does not necessarily hold true for the quantile.

This dissertation investigates uncertainty quantification in dynamic problems. The Advanced Mean Value (AMV) method is used to calculate probabilistic sound power and the sensitivity of elastically supported panels with small uncertainty (coefficient of variation). Sound power calculations are done using Finite Element Method (FEM) and Boundary Element Method (BEM). The sensitivities of the sound power are calculated through direct differentiation of the FEM/BEM/AMV equations. The results are compared with Monte Carlo simulation (MCS). An improved method is developed using AMV, metamodel, and MCS. This new technique is applied to calculate sound power of a composite panel using FEM and Rayleigh Integral. The proposed methodology shows considerable improvement both in terms of accuracy and computational efficiency. In systems with large uncertainties, the above approach does not work. Two Spectral Stochastic Finite Element Method (SSFEM) algorithms are developed to solve stochastic eigenvalue problems using Polynomial chaos. Presently, the approaches are restricted to problems with real and distinct eigenvalues. In both the approaches, the system uncertainties are modeled by Wiener-Askey orthogonal polynomial functions. Galerkin projection is applied in the probability space to minimize the weighted residual of the error of the governing equation. First algorithm is based on inverse iteration method. A modification is suggested to calculate higher eigenvalues and eigenvectors. The above algorithm is applied to both discrete and continuous systems. In continuous systems, the uncertainties are modeled as Gaussian processes using Karhunen-Loeve (KL) expansion. Second algorithm is based on implicit polynomial iteration method. This algorithm is found to be more efficient when applied to discrete systems. However, the application of the algorithm to continuous systems results in ill-conditioned system matrices, which seriously limit its application. Lastly, an algorithm to find the basis random variables of KL expansion for non-Gaussian processes, is developed. The basis random variables are obtained via nonlinear transformation of marginal cumulative distribution function using standard deviation. Results are obtained for three known skewed distributions, Log-Normal, Beta, and Exponential. In all the cases, it is found that the proposed algorithm matches very well with the known solutions and can be applied to solve non-Gaussian process using SSFEM. / Ph. D.

Es werden verschiedene Stetigkeitskonzepte, die in der statistischen Theorie und Methodik eine Rolle spielen, erläutert.

Das vorliegende Skript ist aus einer Lehrveranstaltung hervorgegangen, die von mir mehrere Jahre an der Fakultät Wirtschaftswissenschaften der TU Dresden gehalten wurde. Diese Lehrveranstaltung hatte erst die Bezeichnung "Monetäre Risikomaße" und später "Risikomaße". Mehrere frühere Fassungen dieses Skripts, das häufig überarbeitet und erweitert wurde, trugen den Namen Monetäre Risikomaße (Auflagen 1 bis 7). Die einzelnen Kapitel enthalten in der Regel die drei abschließenden Abschnitte "Übungsaufgaben", "Beweise" und "Ergänzung und Vertiefung" mit Material zum jeweiligen Kapitel, das nicht in der Vorlesung vorgetragen wurde.

Electrodialysis performance strongly depends on the boundary layer near ion exchange membranes. The thickness of the boundary layer has not been clearly evaluated due to its substantial fluctuation around the spacer geometry. In this study, the boundary layer thickness was defined with three statistical parameters: the mean, standard deviation, and correlation coefficient between the two boundary layers facing across the spacer. The relationship between the current and potential under conditions of the competitive transport between mono- and di-valent cations was used to estimate the statistical parameters. An uncertainty model was developed for the steady-state ionic transport in a two-dimensional cell pair. Faster ionic separations were achieved with smaller means, greater standard deviations, and more positive correlation coefficients. With the increasing flow velocity from 1.06 to 4.24 cm/s in the bench-scale electrodialyzer, the best fit values for the mean thickness reduced from 40 to less than 10 μm, and the standard deviation was in the same order of magnitude as the mean. For the partitioning of mono- and di-valent cations, a CMV membrane was examined in various KCl and CaCl₂ mixtures. The equivalent fraction correlation and separation factor responded sensitively to the composition of the mixture; however, the selectivity coefficient was consistent over the range of aqueous-phase ionic contents between 5 and 100 mN and the range of equivalent fractions of each cation between 0.2 and 0.8. It was shown that small analytic errors in measuring the concentration of the mono-valent cation are amplified when estimating the selectivity coefficient. To minimize the effects of such error propagation, a novel method employing the least square fitting was proposed to determine the selectivity coefficient. Each of thermodynamic factors, such as the aqueous- and membrane-phase activity coefficients, water activity, and standard state, was found to affect the magnitude of the selectivity coefficient. The overlimiting current, occurring beyond the electroneutral limit, has not been clearly explained because of the difficulty in solving the singularly perturbed Nernst-Planck-Poisson equations. The steady-state Nernst-Planck-Poisson equations were converted into the Painlevé equation of the second kind (P[subscript II] equation). The converted model domain is explicitly divided into the space charge and electroneutral regions. Given this property, two mathematical formulae were proposed for the limiting current and the width of the space charge region. The Airy solution of the P[subscript II] equation described the ionic transport in the space charge region. By using a hybrid numerical scheme including the fixed point iteration and Newton Raphson methods, the P[subscript II] equation was successfully solved for the ionic transport in the space charge and electroneutral regions as well as their transition zone. Above the limiting current, the sum of the ionic charge in the aqueous-phase electric double layer and in the space charge region remains stationary. Thus, growth of the space charge region involves shrinkage of the aqueous-phase electric double layer. Based on this observation, a repetitive mechanism of expansion and shrinkage of the aqueous-phase electric double layer was suggested to explain additional current above the limiting current. / text