On the first part, we consider nonlinear operators I depending on a family of nonlocal linear operators [mathematical equations]. We study the solutions of the Dirichlet initial and boundary value problems [mathematical equations]. We do not assume even symmetry for the kernels. The odd part bring some sort of nonlocal drift term, which in principle competes against the regularization of the solution. Existence and uniqueness is established for viscosity solutions. Several Hölder estimates are established for u and its derivatives under special assumptions. Moreover, the estimates remain uniform as the order of the equation approaches the second order case. This allows to consider our results as an extension of the classical theory of second order fully nonlinear equations. On the second part, we study two phase problems posed over a two dimensional cone generated by a smooth curve [mathematical symbol] on the unit sphere. We show that when [mathematical equation] the free boundary avoids the vertex of the cone. When [mathematical equation]we provide examples of minimizers such that the vertex belongs to the free boundary. / text

Apresentamos um novo algoritmo, baseado no algoritmo de inversão de matrizes de Leverrier-Fadeev, para extrair os autovalores e os coeficientes do polinômio característico da matriz (si+ A), não-simétrica, que surge em conexão com o método LTSN - o qual utiliza a transformada de Laplace para a solução da equação de ordenadas discretas S N. O algoritmo baseia-se em propriedades exibidas pela matriz, cuja estrutura e valores dos elementos fazem com que todos os seus autovalores sejam reais e simétricos em relação a zero. Evidências experimentais demonstram que, os autovalores do bloco superior esquerdo da matriz, de dimensão N /2, entrelaçam os autovalores negativos de -A. O algoritmo foi implementado em FORTRAN 77, usando algumas rotinas do BLAS e do LAPACK, e estruturado de forma a explorar a estrutura da matriz, permitindo efetuar os cálculos necessários em um menor tempo e com um menor gasto de menória. No entanto, apesar de ganhos obtidos em comparação com o algoritmo usualmente utilizado, proposto por Barichello, nossos experimentos demonstram a instabilidade numérica do algoritmo de Leverrier-Fadeev. / We present a new algorithm to compute the eigenvalues and the coefficients o f the characteristic polynomial o f a nonsymmetric matrix o f the form (sI+ A), which arises in connection with the LTSN method for the solution of thc discrete ordinates equations S N. Our algorithm is a modifi.cation of the matrix inversion Leverrier-Fadeev algorithm, exploiting the pattern existent in the matrix -A and some properties exhibited by its eigenvalues, which have been determined experimentally. More specifi.cally, its eigenvalues alllie on the real axis and are symmetrically distributed around zero. Also, -A has a block structure and the eigenvalues of the left-hand superior block interleave the negative eigenvalues of the matrix. The algorithm was designed to exploit these characteristics, computing only the nega:tive eigenvalues of -A (due to their symmetrical distribution) by means of the well-know bisection method to obtain the zeros of thc characteristic polynomial. Since the eigenvalues of the left-hand superior block of A interleave those of the matrix, it is possible to use intervals made of pairs of those eigenvalues which contain just a single eigenvalue of - A. Also, the structure of -A was used to develop optimized sections of code of thc algorithm to reduce the number of operations required. The whole algorithm was implementcd in FORTRAN 77, making use of some of the BLAS and LAPACK routines. The results obtained although presenting a better performance than that used currently, due to Barichello, show that the algorithm is susceptible to the ill-conditioning of the matrix.

Apresentamos um novo algoritmo, baseado no algoritmo de inversão de matrizes de Leverrier-Fadeev, para extrair os autovalores e os coeficientes do polinômio característico da matriz (si+ A), não-simétrica, que surge em conexão com o método LTSN - o qual utiliza a transformada de Laplace para a solução da equação de ordenadas discretas S N. O algoritmo baseia-se em propriedades exibidas pela matriz, cuja estrutura e valores dos elementos fazem com que todos os seus autovalores sejam reais e simétricos em relação a zero. Evidências experimentais demonstram que, os autovalores do bloco superior esquerdo da matriz, de dimensão N /2, entrelaçam os autovalores negativos de -A. O algoritmo foi implementado em FORTRAN 77, usando algumas rotinas do BLAS e do LAPACK, e estruturado de forma a explorar a estrutura da matriz, permitindo efetuar os cálculos necessários em um menor tempo e com um menor gasto de menória. No entanto, apesar de ganhos obtidos em comparação com o algoritmo usualmente utilizado, proposto por Barichello, nossos experimentos demonstram a instabilidade numérica do algoritmo de Leverrier-Fadeev. / We present a new algorithm to compute the eigenvalues and the coefficients o f the characteristic polynomial o f a nonsymmetric matrix o f the form (sI+ A), which arises in connection with the LTSN method for the solution of thc discrete ordinates equations S N. Our algorithm is a modifi.cation of the matrix inversion Leverrier-Fadeev algorithm, exploiting the pattern existent in the matrix -A and some properties exhibited by its eigenvalues, which have been determined experimentally. More specifi.cally, its eigenvalues alllie on the real axis and are symmetrically distributed around zero. Also, -A has a block structure and the eigenvalues of the left-hand superior block interleave the negative eigenvalues of the matrix. The algorithm was designed to exploit these characteristics, computing only the nega:tive eigenvalues of -A (due to their symmetrical distribution) by means of the well-know bisection method to obtain the zeros of thc characteristic polynomial. Since the eigenvalues of the left-hand superior block of A interleave those of the matrix, it is possible to use intervals made of pairs of those eigenvalues which contain just a single eigenvalue of - A. Also, the structure of -A was used to develop optimized sections of code of thc algorithm to reduce the number of operations required. The whole algorithm was implementcd in FORTRAN 77, making use of some of the BLAS and LAPACK routines. The results obtained although presenting a better performance than that used currently, due to Barichello, show that the algorithm is susceptible to the ill-conditioning of the matrix.

Apresentamos um novo algoritmo, baseado no algoritmo de inversão de matrizes de Leverrier-Fadeev, para extrair os autovalores e os coeficientes do polinômio característico da matriz (si+ A), não-simétrica, que surge em conexão com o método LTSN - o qual utiliza a transformada de Laplace para a solução da equação de ordenadas discretas S N. O algoritmo baseia-se em propriedades exibidas pela matriz, cuja estrutura e valores dos elementos fazem com que todos os seus autovalores sejam reais e simétricos em relação a zero. Evidências experimentais demonstram que, os autovalores do bloco superior esquerdo da matriz, de dimensão N /2, entrelaçam os autovalores negativos de -A. O algoritmo foi implementado em FORTRAN 77, usando algumas rotinas do BLAS e do LAPACK, e estruturado de forma a explorar a estrutura da matriz, permitindo efetuar os cálculos necessários em um menor tempo e com um menor gasto de menória. No entanto, apesar de ganhos obtidos em comparação com o algoritmo usualmente utilizado, proposto por Barichello, nossos experimentos demonstram a instabilidade numérica do algoritmo de Leverrier-Fadeev. / We present a new algorithm to compute the eigenvalues and the coefficients o f the characteristic polynomial o f a nonsymmetric matrix o f the form (sI+ A), which arises in connection with the LTSN method for the solution of thc discrete ordinates equations S N. Our algorithm is a modifi.cation of the matrix inversion Leverrier-Fadeev algorithm, exploiting the pattern existent in the matrix -A and some properties exhibited by its eigenvalues, which have been determined experimentally. More specifi.cally, its eigenvalues alllie on the real axis and are symmetrically distributed around zero. Also, -A has a block structure and the eigenvalues of the left-hand superior block interleave the negative eigenvalues of the matrix. The algorithm was designed to exploit these characteristics, computing only the nega:tive eigenvalues of -A (due to their symmetrical distribution) by means of the well-know bisection method to obtain the zeros of thc characteristic polynomial. Since the eigenvalues of the left-hand superior block of A interleave those of the matrix, it is possible to use intervals made of pairs of those eigenvalues which contain just a single eigenvalue of - A. Also, the structure of -A was used to develop optimized sections of code of thc algorithm to reduce the number of operations required. The whole algorithm was implementcd in FORTRAN 77, making use of some of the BLAS and LAPACK routines. The results obtained although presenting a better performance than that used currently, due to Barichello, show that the algorithm is susceptible to the ill-conditioning of the matrix.

Magister Scientiae - MSc / Efficient numerical approaches for parameter dependent problems have been an inter- esting subject to numerical analysts and engineers over the past decades. This is due to the prominent role that these problems play in modeling many real life situations in applied sciences. Often, the choice and the e ciency of the approaches depend on the nature of the problem to solve. In this work, we consider the general linear first-order singularly perturbed Volterra integro-differential equations (SPVIDEs). These singularly perturbed problems (SPPs) are governed by integro-differential equations in which the derivative term is multiplied by a small parameter, known as "perturbation parameter". It is known that when this perturbation parameter approaches zero, the solution undergoes fast transitions across narrow regions of the domain (termed boundary or interior layer) thus affecting the convergence of the standard numerical methods. Therefore one often seeks for numerical approaches which preserve stability for all the values of the perturbation parameter, that is "numerical methods. This work seeks to investigate some "numerical methods that have been used to solve SPVIDEs. It also proposes alternative ones. The various numerical methods are composed of a fitted finite difference scheme used along with suitably chosen interpolating quadrature rules. For each method investigated or designed, we analyse its stability and convergence. Finally, numerical computations are carried out on some test examples to con rm the robustness and competitiveness of the proposed methods.

A Teoria das equações integrais, desde a segunda metade do século XX, tem assumido um papel cada vez maior no âmbito de problemas aplicados. Com isso, surge a necessidade do desenvolvimento de métodos numéricos cada vez mais eficazes para a resolução deste tipo de equação. Isso tem como consequência a possibilidade de resolução de uma gama cada vez maior de problemas. Nesse sentido, outros tipos de equações integrais estão sendo objeto de estudos, dentre elas as chamadas equações integro-diferenciais. O presente trabalho tem como objetivo o estudo das equações integro-diferenciais singulares lineares e não-lineares. Mais especificamente, no caso linear, apresentamos os principais resultados necessários para a obtenção de um método numérico e a formulação de suas propriedades de convergência. O caso não-linear é apresentado através de um modelo matemático para tubulações em um tipo específico de reator nuclear (LMFBR) no qual origina-se a equação integro-diferencial. A partir da equação integro-diferencial um modelo numérico é proposto com base nas condições físicas do problema / The theory of the integral equations, since the second half of the 20th century, has been assuming an ever more important role in the modelling of applied problems. Consequently, the development of new numerical methods for integral equations is called for and a larger range of problems has been possible to be solved by these new techniques. In this sense, many types of integral equations have been derived from applications and been the object of studies, among them the so called singular integro-differential equation. The present work has, as its main objective, the study of singular integrodifferential equations, both linear and non-linear. More specifically, in the linear case, we present our main results regarding the derivation of a numerical method and its uniform convergence properties. The non-linear case is introduced through the mathematical model of boiler tubes in a specific type of nuclear reactor (LMFBR) from which the integro-differential equation originates. For this integro-differential equation a numerical method is proposed based on the physical conditions of the problem

京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第21212号 / 情博第665号 / 新制||情||115(附属図書館) / 京都大学大学院情報学研究科先端数理科学専攻 / (主査)教授 磯 祐介, 教授 木上 淳, 助手 藤原 宏志 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

We study the application of Lie symmetry methods to solve some cases of the Boltzmann equation, a cornerstone of kinetic theory. The study explores hidden invariances and symmetry-based solutions that help to clarify the complexities inherent in the structure of the equation. Moreover, the study demonstrates a novel approach to solving the equation by rewriting it using the Fourier transform in the velocity variable, which resulted in a non-trivial solution to the Boltzmann equation. The findings not only clarify the mathematical underpinnings of the Boltzmann equation but also enhance our understanding of particle interactions in gases. Overall, this thesis not only enriches the theoretical understanding of integro-differential equations through its rigorous approach but also highlights the efficacy of Lie symmetry methods in unraveling the complexities of fundamental equations in physical sciences.

Several existing pricing models of financial derivatives as well as the effects of volatility risk are analyzed. A new option pricing model is proposed which assumes that stock price follows a diffusion process with square-root stochastic volatility. The volatility itself is mean-reverting and driven by both diffusion and compound Poisson process. These assumptions better reflect the randomness and the jumps that are readily apparent when the historical volatility data of any risky asset is graphed. The European option price is modeled by a homogeneous linear second-order partial differential equation with variable coefficients. The case of underlying assets that pay continuous dividends is considered and implemented in the model, which gives the capability of extending the results to American options. An American option price model is derived and given by a non-homogeneous linear second order partial integro-differential equation. Using Fourier and Laplace transforms an exact closed-form solution for the price formula for European call/put options is obtained.

Dans cette thèse, je me suis interessé a deux aspects de la gestion de portefeuille : la maximisation de l'utilité e d'un portefeuille financier lorsque on impose une contrainte sur l'exposition au risque, et la couverture quadratique en marché incomplet. Part I. Dans la première partie, j' étudie un problème d'assurance de portefeuille du point de vue du manager d'un fond d'investissement, qui veut structurer un produit financier pour les investisseurs du fond avec une garantie sur la valeur du portefeuille a la maturité . Si, a la maturité, la valeur du portefeuille est au dessous d'un seuil x e, l'investisseur sera remboursé a la hauteur de ce seuil par une troisième partie, qui joue le rôle d'assureur du fond (on peut imaginer que le fond appartient à une banque et que donc c'est la banque elle même qui joue le rôle d'assureur). En échange de cette assurance, la troisième partie impose une contrainte sur l'exposition au risque que le manager du fond peut tolérer, mesurée avec une mesure de risque monétaire convexe. Je donne la solution complet e de ce problème de maximisation non convexe en marché complet et je prouve que le choix de la mesure de risque est un point crucial pour avoir existence d'un portefeuille optimal. J'applique donc mes résultats lorsque on utilise la mesure de risque entropique (pour laquelle le portefeuille optimal existe toujours), les mesures de risque spectrales (pour lesquelles le portefeuille optimal peut ne pas exister dans certains cas) et la G-divergence. Mots-cl es : Assurance de portefeuille ; maximisation d'utilité ; mesure de risque convexe ; VaR, CVaR et mesure de risque spectrale ; entropie et G-divergence. Part II. Dans la deuxième partie, je m'intéresse au problème de couverture quadratique en marché incomplet. J'assume que le marché est d écrit par un processus Markovien tridimensionnel avec sauts. La premi ère variable d' état décrit l'actif - financier, échangeable sur le marché, qui sert comme instrument de couverture ; la deuxième variable d' état modélise un actif financier que intervient dans la dynamique de l'instrument de couverture mais qui n'est pas échangeable sur le march é : il peut donc être vu comme un facteur de volatilité de l'instrument de couverture, ou comme un actif financier que l'on ne peut pas acheter (pour de raisons légales par exemple) ; la troisième et dernière variable d' état représente une source externe de risque qui affecte l'option Européenne qu'on veut couvrir, et qui, elle aussi, n'est pas échangeable sur le marché. Pour résoudre le problème j'utilise l'approche de la programmation dynamique, qui me permet d' écrire l' équation de Hamilton-Jacobi- Bellman associé e au problème de couverture quadratique, qui est non locale en non linéaire. Je prouve que la fonction valeur associée au problème de couverture quadratique peut être caractérisée par un système de trois équations integro- différentielles aux dérivées partielles, dont l'une est semilinéaire et ne dépends pas du choix de l'option a couvrir, et les deux autres sont simplement linéaires , et que ce système a une unique solution r régulière dans un espace de Hölder approprié, qui me permet donc de caractériser la stratégie de couverture optimale . Ce résultat est démontré lorsque le processus est non dégénéré (c'est a dire que la composante Brownienne est strictement elliptique) et lorsque le processus est a sauts purs. Je conclus avec une application de mes résultats dans le cadre du marché de l' électricité. Mots-cl es : Couverture quadratique ; modèle a sauts ; programmation dynamique ; équation de Hamilton-Jacobi-Bellman ; équations aux dérivées partielles integro-différentielles.