Techn. University, Diss., 2007--Darmstadt.

Thesis (PhD.(Mathematics))-University of Pretoria, 2005. / Includes bibliographical references. Available on the Internet via the World Wide Web.

Universiẗat, Diss., 2002--Kiel.

Techn. University, Diss., 2005--Darmstadt.

Thesis (MSc (Mathematical Sciences. Applied Mathematics))--University of Stellenbosch, 2009. / In this thesis, we investigate two numerical methods to price nancial options. We look at two types of options, namely European options and Asian options. The numerical methods we use are the nite di erence method and numerical inversion of the Laplace transform. We apply nite di erence methods to partial di erential equations with both uniform and non-uniform spatial grids. The Laplace inversion method we use is due to Talbot. It is based on the midpoint-type approximation of the Bromwich integral on a deformed contour. When applied to Asian options, we have the problem of computing the hypergeometric function of the rst kind. We propose a new method for numerically calculating the hypergeometric function. This method too is based on using Talbot contours. Throughout the thesis, we use the Black-Scholes equation as our benchmark problem.

Thesis (MSc (Applied Mathematics))--University of Stellenbosch, 2009. / ENGLISH ABSTRACT: In recent years the Laplace inversion method has emerged as a viable alternative method for the numerical solution of PDEs. Effective methods for the numerical inversion are based on the approximation of the Bromwich integral. In this thesis, a numerical study is undertaken to compare the efficiency of the Laplace inversion method with more conventional time integrator methods. Particularly, we consider the method-of-lines based on MATLAB’s ODE15s and the Crank-Nicolson method. Our studies include an introductory chapter on the Laplace inversion method. Then we proceed with spectral methods for the space discretization where we introduce the interpolation polynomial and the concept of a differentiation matrix to approximate derivatives of a function. Next, formulas of the numerical differentiation formulas (NDFs) implemented in ODE15s, as well as the well-known second order Crank-Nicolson method, are derived. In the Laplace method, to compute the Bromwich integral, we use the trapezoidal rule over a hyperbolic contour. Enhancement to the computational efficiency of these methods include the LU as well as the Hessenberg decompositions. In order to compare the three methods, we consider two criteria: The number of linear system solves per unit of accuracy and the CPU time per unit of accuracy. The numerical results demonstrate that the new method, i.e., the Laplace inversion method, is accurate to an exponential order of convergence compared to the linear convergence rate of the ODE15s and the Crank-Nicolson methods. This exponential convergence leads to high accuracy with only a few linear system solves. Similarly, in terms of computational cost, the Laplace inversion method is more efficient than ODE15s and the Crank-Nicolson method as the results show. Finally, we apply with satisfactory results the inversion method to the axial dispersion model and the heat equation in two dimensions. / AFRIKAANSE OPSOMMING: In die afgelope paar jaar het die Laplace omkeringsmetode na vore getree as ’n lewensvatbare alternatiewe metode vir die numeriese oplossing van PDVs. Effektiewe metodes vir die numeriese omkering word gebasseer op die benadering van die Bromwich integraal. In hierdie tesis word ’n numeriese studie onderneem om die effektiwiteit van die Laplace omkeringsmetode te vergelyk met meer konvensionele tydintegrasie metodes. Ons ondersoek spesifiek die metode-van-lyne, gebasseer op MATLAB se ODE15s en die Crank-Nicolson metode. Ons studies sluit in ’n inleidende hoofstuk oor die Laplace omkeringsmetode. Dan gaan ons voort met spektraalmetodes vir die ruimtelike diskretisasie, waar ons die interpolasie polinoom invoer sowel as die konsep van ’n differensiasie-matriks waarmee afgeleides van ’n funksie benader kan word. Daarna word formules vir die numeriese differensiasie formules (NDFs) ingebou in ODE15s herlei, sowel as die welbekende tweede orde Crank-Nicolson metode. Om die Bromwich integraal te benader in die Laplace metode, gebruik ons die trapesiumreël oor ’n hiperboliese kontoer. Die berekeningskoste van al hierdie metodes word verbeter met die LU sowel as die Hessenberg ontbindings. Ten einde die drie metodes te vergelyk beskou ons twee kriteria: Die aantal lineêre stelsels wat moet opgelos word per eenheid van akkuraatheid, en die sentrale prosesseringstyd per eenheid van akkuraatheid. Die numeriese resultate demonstreer dat die nuwe metode, d.i. die Laplace omkeringsmetode, akkuraat is tot ’n eksponensiële orde van konvergensie in vergelyking tot die lineêre konvergensie van ODE15s en die Crank-Nicolson metodes. Die eksponensiële konvergensie lei na hoë akkuraatheid met slegs ’n klein aantal oplossings van die lineêre stelsel. Netso, in terme van berekeningskoste is die Laplace omkeringsmetode meer effektief as ODE15s en die Crank-Nicolson metode. Laastens pas ons die omkeringsmetode toe op die aksiale dispersiemodel sowel as die hittevergelyking in twee dimensies, met bevredigende resultate.

Thesis (MSc (Mathematics))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: In this thesis, we discuss and apply the Monte Carlo and integral transform methods in pricing options. These methods have proved to be very e ective in the valuation of options especially when acceleration techniques are introduced. By rst pricing European call options we have motivated the use of these methods in pricing arithmetic Asian options which have proved to be di cult to price and hedge under the Black􀀀Scholes framework. The arithmetic average of the prices in this framework, is a sum of correlated lognormal distributions whose distribution does not admit a simple analytic expression. However, many approaches have been reported in the academic literature for pricing these options. We provide a hedging strategy by manipulating the results by Geman and Yor [42] for continuous xed strike arithmetic Asian call options. We then derive a double Laplace transform formula for pricing continuous Asian call options following the approach by Fu et al. [39]. By applying the multi-Laguerre and iterated Talbot inversion techniques for Laplace transforms to the resulting pricing formula we obtain the option prices. Finally, we discuss the shortcomings of using the Laplace transform in pricing options. / AFRIKAANSE OPSOMMING: In hierdie tesis bespreek ons Monte Carlo- en integraaltransform metodes om die pryse van nansi ele opsies te bepaal. Hierdie metodes is baie e ektief, veral wanneer versnellingsmetodes ingevoer word. Ons bepaal eers die pryse van Europese opsies as motivering, voordat ons die bostaande metodes gebruik vir prysbepaling van Asiatiese opsies met rekenkundige gemiddeldes, wat baie moeiliker is om te hanteer in die Black􀀀Scholes raamwerk. Die rekenkundige gemiddelde van batepryse in hierdie raamwerk is 'n som van gekorreleerde lognormale distribusies wie se distribusie nie oor 'n eenvoudige analitiese vorm beskik nie. Daar is egter talle benaderings vir die prysbepaling van hierdie opsies in die akademiese literatuur. Ons bied 'n verskansingsstrategie vir Asiatiese opsies in kontinue tyd met 'n vaste trefprys aan deur die resultate van Geman en Yor [42] te manipuleer. Daarna volg ons Fu et al. [39] om 'n dubbele Laplace transform formule vir die pryse af te lei. Deur toepassing van multi-Laguerre en herhaalde Talbotinversie tegnieke vir Laplace transforms op hierdie formule, bepaal ons dan die opsiepryse. Ons sluit af met 'n bespreking van die tekortkominge van die gebruik van die Laplace transform vir prysbepaling.

In this thesis we study boundary value problems for the Laplace equation on do mains with smooth boundary. Central to our analysis is a relation, known as the global relation, that couples the boundary data for a given BVP. Previously, the global re lation has primarily been applied to elliptic PDEs defined on polygonal domains. In this thesis we extend the use of the global relation to domains with smooth boundary. This is done by introducing a new transform, denoted by F_p, that is an analogue of the Fourier transform on smooth convex curves. We show that the F_p-transform is a bounded and invertible integral operator. Following this, we show that the F_p-transform naturally arises in the global relation for the Laplace equation on domains with smooth boundary. Using properties of the F_p-transform, we show that the global relation defines a continuously invertible map between the Dirichlet and Neumann data for a given BVP for the Laplace equation. Following this, we construct a numerical method that uses the global relation to find the Neumann data, given the Dirichlet data, for a given BVP for the Laplace equation on a domain with smooth boundary.

Ce travail consiste en une tentative de ré-examiner les propriétés électroniques du Fe, Cr et Au au sein de Ge, qui ont déjà été étudiées classiquement par DLTS (Deep Level Transient Spectroscopy). L'image générale qui en découle est que les métaux de transition dans Ge forment de manière prépondérante des centres accepteurs multiples, introduisant plusieurs niveaux profonds dans la bande de gap. A partir d'un modèle de liaison de valence simple, cette conclusion est en accord avec une survenue des impuretés sur des sites de substitution. Cependant, plusieurs questions demeurent ouvertes, comme le rôle de l'hydrogène en tant que contaminant dans l'élargissement des spectres DLTS. Notre contribution se base sur l'utilisation d'une approche plus performante nommée Laplace DLTS, en ce sens qu'elle autorise une meilleure résolution du signal. Nous présentons une analyse extensive par DLTS, MCTS et Laplace DLTS, afin d'étudier les propriétés électroniques des états accepteurs multiples, induits par les 4 métaux de transition sus-nommés. On distingue, parmis les paramètres étudiés, les barrières de capture des porteurs, les vraies sections efficaces de capture de sporteurs majoritaires (déterminées directement par la méthode de variation du pulse de remplissage), L'effet Pool-Frenkel (en lien avec la détermination de l'état de charge du niveau concerné). Ceci permet d'indiquer avec précision la position exacte des niveaux dans la bande interdite. Nous confirmons la plupart des résultats mis en évidence précédemment, tout en ajoutant quelques précisions sir le rôle de l'hydrogène dans la formation de nouveaux complexes. Une mise en parallèle avec le silicium. Dans le cas de Au, de nouveaux niveaux attribués aux complexes Au-Hn et Au-Sb sont observés. De manière générale, l'analyse des porteurs majoritaires et minoritaires par MCYS est toujours sujette à étude. En ce qui concerne le cas du Fe, la faible différence d'énergie entre ses deux niveaux soulève la possibilité d'un caractère de type U-négatif. L'ensemble de ses points devraient faire l'objet d'un travail approfondis dans un avenir proche. / The present work is an attempt to re-examine the electronic properties of Fe, Ni Cr and Au in Ge which have been already studied by conventional DLTS (Deep Level Transient Spectroscopy). A general picture, that emerged, is that transition metals in Ge predominantly form multiple-acceptor centres, introducing several deep levels in the band gap, which according to a simple valence bond model is in agreement with a preferential occurrence of the impurities on substitutional sites. However, some questions remained open such as the role of hydrogen as a natural contaminant in the broadening of the DLTS spectra. Our contribution is based on the use of a more powerfull approach called Laplace DLTS in the sense that it allows a better resolution of the signal. We present extensive DLTS, MCTS and Laplace DLTS results to investigate the electronic properties of the multi-acceptor states, induced by mentioned four transition metals. Among the studied parameters we may cite the barrier for carrier capture, the true majority carrier capture cross section directly measured by the variable pulse length method, the Poole-Frenkel effect related to the assignment of the charge states - all these parameters are important to locate the level positions in the band gap. We confirm in the present work most of the results obatained in the past, adding some insight into the role of hydrogen in the formation of new complexes and in this respect a parallel is made with silicon. In case of Au new levels attributed to conjectural Au-Hn and Au-Sb complexes are observed. In addition development of both majority and minority carriers in MCTS analysis still is under consideration. For the Fe case, the small difference in energy of its two levels raises the question as to the possibility of negative-U character. These mentioned points should be treated more thoroughly in a future work.

Neste trabalho são desenvolvidos métodos numéricos para inversão da transformada de Laplace, fazendo-se uso de polinômios trigonométricos e de Laguerre. Sua utilização é ilustrada num problema de fronteira móvel da área de engenharia nuclear, através do algoritmo computacional ALG-619. Uma revisão dos aspectos analíticos básicos da transformada de Laplace e sua utilização na resolução de equações diferenciais parciais é apresentada de maneira suscinta.