Obtaining the distribution of the profit and loss (PL) of a portfolio is a key problem in market risk measurement. However, existing methods, such as those based on the Normal distribution, and historical simulation methods, which use empirical distribution of risk factors, face difficulties in dealing with at least one of the following three problems: describing the distributional properties of risk factors appropriately (description problem); deriving distributions of risk factors with time horizon longer than one day (time aggregation problem); and deriving the distribution of the PL given the distributional properties of the risk factors (risk aggregation problem). Here, we show that expansion methods can provide reasonable solutions to all three problems. Expansion methods approximate a probability density function by a sum of orthogonal polynomials multiplied by an associated weight function. One of the most important advantages of expansion methods is that they only require moments of the target distribution up to some order to obtain an approximation. Therefore they have the potential to be applied in a wide range of situations, including in attempts to solve the three problems listed above. On the other hand, it is also known that expansions lack robustness: they often exhibit unignorable negative density and their approximation quality can be extremely poor. This limits applications of expansion methods in existing studies. In this thesis, we firstly develop techniques to provide robustness, with which expansion methods result in a practical approximation quality in a wider range of examples than investigated to date. Specifically, we investigate three techniques: standardisation, use of Laguerre expansion and optimisation. Standardisation applies expansion methods to a variable which is transformed so that its first and second moments are the same as those of the weight function. Use of Laguerre expansions applies those expansions to a risk factor so that heavy tails can be captured better. Optimisation considers expansions with coefficients of polynomials optimised so that the difference between the approximation and the target distribution is minimised with respect to mean integrated squared error. We show, by numerical examples using data sets of stock index returns and log differences of implied volatility, and GARCH models, that expansions with our techniques are more robust than conventional expansion methods. As such, marginal distributions of risk factors can be approximated by expansion methods. This solves a part of the description problem: the information on the marginal distributions of risk factors can be summarised by their moments. Then we show that the dependence structure among risk factors can be summarised in terms of their cross-moments. This solves the other part of the description problem. We also use the fact that moments of risk factors can be aggregated using their moments and cross-moments, to show that expansion methods can be applied to both the time and risk aggregation problems. Furthermore, we introduce expansion methods for multivariate distributions, which can also be used to approximate conditional expectations and copula densities by rational functions.

This thesis descibes a theoretical study of nonlinear self-focusing as applied to the metrology of the nonlinear optical parameters of a medium. It also studies the phe- nomenon of optical power limiting which utilizes self-focusing e ects. As an analytical tool, a mode decomposition method which uses an orthogonal and complete set of Gaussian-Laguerre modes as a basis set is used to treat these problems. Nonlinear media both in the thin and thick limits are investigated. For thin media, a closed form expression is derived which describes the optical eld of an initally Gaussian beam that is perturbed by a thin nonlinear material which exhibits nonlinear absorption as well as nonlinear refraction. This result is valid for any regime of nonlinearity in the thin medium approximation. Thick media are treated using a numerical extension of the Gaussian-Laguerre Mode Decomposition technique. Spatial scanning techniques such as the Z-scan that rely on self-focusing e ects and that are used to measure the nonlinear optical parameters of a material are studied in detail. Optical limiting in both thick and thin media is also investigated.

This thesis descibes a theoretical study of nonlinear self-focusing as applied to the metrology of the nonlinear optical parameters of a medium. It also studies the phe- nomenon of optical power limiting which utilizes self-focusing e ects. As an analytical tool, a mode decomposition method which uses an orthogonal and complete set of Gaussian-Laguerre modes as a basis set is used to treat these problems. Nonlinear media both in the thin and thick limits are investigated. For thin media, a closed form expression is derived which describes the optical eld of an initally Gaussian beam that is perturbed by a thin nonlinear material which exhibits nonlinear absorption as well as nonlinear refraction. This result is valid for any regime of nonlinearity in the thin medium approximation. Thick media are treated using a numerical extension of the Gaussian-Laguerre Mode Decomposition technique. Spatial scanning techniques such as the Z-scan that rely on self-focusing e ects and that are used to measure the nonlinear optical parameters of a material are studied in detail. Optical limiting in both thick and thin media is also investigated.

Texte remanié de: Th. M.A.--Universidad de Puerto Rico, 1968. / Bibliogr. p. 171-184.

Submitted by Paula Akari Fukushima (paula.fukushima@gmail.com) on 2018-06-07T13:24:40Z No. of bitstreams: 1 dissertacao_com_ficha_catalografica.pdf: 734238 bytes, checksum: 7e83c559085d2370c3126bb9787364f8 (MD5) / Rejected by Elza Mitiko Sato null (elzasato@ibilce.unesp.br), reason: Solicitamos que realize correções na submissão seguindo as orientações abaixo: Problema 01) Observamos que no seu arquivo consta o auxílio da Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), então na Folha de rosto e de aprovação deve constar a financiadora. Lembramos que o arquivo depositado no repositório deve ser igual ao impresso, o rigor com o padrão da Universidade se deve ao fato de que o seu trabalho passará a ser visível mundialmente. Agradecemos a compreensão on 2018-06-08T19:11:31Z (GMT) / Submitted by Paula Akari Fukushima (paula.fukushima@gmail.com) on 2018-06-12T13:25:21Z No. of bitstreams: 1 dissertacao_com_ficha_catalografica.pdf: 734279 bytes, checksum: 7a81403314e3b2da3076407e9672a090 (MD5) / Approved for entry into archive by Elza Mitiko Sato null (elzasato@ibilce.unesp.br) on 2018-06-12T19:15:16Z (GMT) No. of bitstreams: 1 fukushima_pa_me_sjrp.pdf: 734279 bytes, checksum: 7a81403314e3b2da3076407e9672a090 (MD5) / Made available in DSpace on 2018-06-12T19:15:16Z (GMT). No. of bitstreams: 1 fukushima_pa_me_sjrp.pdf: 734279 bytes, checksum: 7a81403314e3b2da3076407e9672a090 (MD5) Previous issue date: 2018-05-23 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Nesta dissertação estudamos sequências de polinômios ortogonais que surgem como auto-funções polinomiais do problema de Sturm-Liouville, sob a condição de que, nem todos os graus das auto-funções polinomiais estejam presentes na sequência de graus dos polinômios que formam o conjunto ortogonal completo. Estas sequências são chamadas de sequências de polinômios ortogonais excepcionais. Emparticular,realizamosumestudodospolinômiosortogonaisexcepcionais X1-Jacobi e X1-Laguerre. / In this dissertation we study sequences of orthogonal polynomials that arise as polynomial eigenfunctions of the Sturm-Liouville problem, with the condition that not all degrees of polynomial eigenfunctions are present in the sequence of degrees of the polynomials that form a complete orthogonal set. These sequences are called exceptional orthogonal polynomial sequences. In particular, we study the exceptional orthogonal polynomials X1-Jacobi and X1-Laguerre.

The motivation for this work comes from physics, when dealing with microstructures of polycrystalline materials. An adequate probabilistic model is a three-dimensional (3D) random tessellation. The original contribution of the author is dealing with the Gibbs-Voronoi and Gibbs- Laguerre tessellations in 3D, where the latter model is completely new. The energy function of the underlying Gibbs point process reflects interactions between geometrical characteristics of grains. The aim is the simulation, parameter estimation and degree-of-fit testing. Mathematical background for the methods is described and numerical results based on simulated data are presented in the form of tables and graphs. The interpretation of results confirms that the Gibbs-Laguerre model is promising for further investigation and applications.

Tato práce se zabývá použitím zobecněných Laguerrových funkcí pro modelování a identifikaci dynamických systémů. Po krátkém úvodu je uvedena definice zobecněných Laguerrových polynomů a funkcí a některé jejich vlastnosti. Je zavedena metoda pro identifikaci spojitých dynamických systémů z diskrétních nasamplovaných dat s použitím zobecněných Laguerrových funkcí a porovnána s metodou nejmenších čtverců. Je diskutována volba optimálních parametrů p a alpha pro konečné Fourierovy řady se zobecněnými Laguerrovými funkcemi. Na příkladech je porovnána navržená identifikační metoda s metodou nejmenších čtverců. Je zavedena diskrétní Laguerrova transformace a její rozšíření s použitím zobecněných Laguerrových funkcí. Na příkladech je předvedena aplikace diskrétní Laguerrovy transformace na kompresi dat. Je ukázáno, že použití diskrétní Laguerrovy transformace může vést k lepším výsledků při kompresi dat než tradiční postup s využitím diskrétní kosinové transformace.

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No / The causal and physically realizable Biot hysteretic model proves to be the simplest linear model able to describe the nearly rate-independent behaviour of engineering materials. In this paper, the performance of the Biot hysteretic model is analysed and compared with those of the ideal and causal hysteretic models. The Laguerre polynomial approximation (LPA) method, recently proposed for the time-domain analysis of linear viscoelastic systems, is then summarized and applied to the prediction of the dynamic response of linear hysteretic systems to deterministic and random excitations. The parameters of the LPA model generally need to be computed through numerical integrals; however, when this model is used to approximate the Biot hysteretic model, closed-form expressions can be found. Effective step-by-step procedures are also provided in the paper, which prove to be accurate also for high levels of damping. Finally, the method is applied to the dynamic analysis of a highway embankment excited by deterministic and random ground motions. The results show that in some cases the inaccuracy associated with the use of an equivalent viscous damping model is too large.