Na condução por saltos dispersiva, os portadores de carga são caracterizados por tempos de residência não exponenciais. Como uma conseqüência, efeitos hereditários aparecem e o problema deve ser conduzido no espaçõ de Laplace. Distribuições dielétricas conhecidas e algumas outras foram usadas como possíveis funções capazes de descrever o meio desordenado. Um estudo cuidadoso de seus espectros de freqüência foi realizado. O método de inversão de Widder truncado foi desenvolvido para possibilitar a volta do espaço de Laplace para o do tempo real. Dois tipos de problemas práticos foram abordados: condução dispersiva na qual o campo elétrico das cargas móveis pode ser desprezado e o caso de carga espacial para o qual essa aproximação não pode ser feita. Duas configurações de interesse foram escolhidas: decaimento do potencial de superfície de amostras carregadas por descarga corona e a subida do potencial no carregamento por corrente constante. / Dispersive conductive process are characterized by non-exponential residence times of the hopping carrier. As a consequence, hereditary effects appear and the problem must be conducted in the Laplace space. A bunch of known dielectric distributions functions and others were used as possible functions describing the disordered medium. A careful study of their frequency spectrum was carried out. The truncate Widder inversion was developed in order to allow returning from the Lapace to the real time space. Two kind of practical problems were studied: dispersive conduction in which the electric field of the mobile charge may be neglected and the space charge ones when that approximation cannot be made. Two configurations of interest were selected: the corona discharge potential decay and the constant current potentinal build-up.

Na condução por saltos dispersiva, os portadores de carga são caracterizados por tempos de residência não exponenciais. Como uma conseqüência, efeitos hereditários aparecem e o problema deve ser conduzido no espaçõ de Laplace. Distribuições dielétricas conhecidas e algumas outras foram usadas como possíveis funções capazes de descrever o meio desordenado. Um estudo cuidadoso de seus espectros de freqüência foi realizado. O método de inversão de Widder truncado foi desenvolvido para possibilitar a volta do espaço de Laplace para o do tempo real. Dois tipos de problemas práticos foram abordados: condução dispersiva na qual o campo elétrico das cargas móveis pode ser desprezado e o caso de carga espacial para o qual essa aproximação não pode ser feita. Duas configurações de interesse foram escolhidas: decaimento do potencial de superfície de amostras carregadas por descarga corona e a subida do potencial no carregamento por corrente constante. / Dispersive conductive process are characterized by non-exponential residence times of the hopping carrier. As a consequence, hereditary effects appear and the problem must be conducted in the Laplace space. A bunch of known dielectric distributions functions and others were used as possible functions describing the disordered medium. A careful study of their frequency spectrum was carried out. The truncate Widder inversion was developed in order to allow returning from the Lapace to the real time space. Two kind of practical problems were studied: dispersive conduction in which the electric field of the mobile charge may be neglected and the space charge ones when that approximation cannot be made. Two configurations of interest were selected: the corona discharge potential decay and the constant current potentinal build-up.

This thesis considers new methods that exploit recent developments in computer technology to address three extant problems in the area of Finance and Econometrics. The problem of Asian option pricing has endured for the last two decades in spite of many attempts to find a robust solution across all parameter values. All recently proposed methods are shown to fail when computations are conducted using standard machine precision because as more and more accuracy is forced upon the problem, round-off error begins to propagate. Using recent methods from numerical analysis based on multi-precision arithmetic, we show using the Mathematica platform that all extant methods have efficacy when computations use sufficient arithmetic precision. This creates the proper framework to compare and contrast the methods based on criteria such as computational speed for a given accuracy. Numerical methods based on a deformation of the Bromwich contour in the Geman-Yor Laplace transform are found to perform best provided the normalized strike price is above a given threshold; otherwise methods based on Euler approximation are preferred. The same methods are applied in two other contexts: the simulation of stable distributions and the computation of unit root densities in Econometrics. The stable densities are all nested in a general function called a Fox H function. The same computational difficulties as above apply when using only double-precision arithmetic but are again solved using higher arithmetic precision. We also consider simulating the densities of infinitely divisible distributions associated with hyperbolic functions. Finally, our methods are applied to unit root densities. Focusing on the two fundamental densities, we show our methods perform favorably against the extant methods of Monte Carlo simulation, the Imhof algorithm and some analytical expressions derived principally by Abadir. Using Mathematica, the main two-dimensional Laplace transform in this context is reduced to a one-dimensional problem.