Thesis (MSc)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: We study Monte Carlo methods for pricing American options where the stock price dynamics follow exponential pure jump L évy models. Only stock price dynamics for a single underlying are considered. The thesis begins with a general introduction to American Monte Carlo methods. We then consider two classes of these methods. The fi rst class involves regression - we briefly consider the regression method of Tsitsiklis and Van Roy [2001] and analyse in detail the least squares Monte Carlo method of Longsta and Schwartz [2001]. The variance reduction techniques of Rasmussen [2005] applicable to the least squares Monte Carlo method, are also considered. The stochastic mesh method of Broadie and Glasserman [2004] falls into the second class we study. Furthermore, we consider the dual method, independently studied by Andersen and Broadie [2004], Rogers [2002] and Haugh and Kogan [March 2004] which generates a high bias estimate from a stopping rule. The rules we consider are estimates of the boundary between the continuation and exercise regions of the option. We analyse in detail how to obtain such an estimate in the least squares Monte Carlo and stochastic mesh methods. These models are implemented using both a pseudo-random number generator, and the preferred choice of a quasi-random number generator with bridge sampling. As a base case, these methods are implemented where the stock price process follows geometric Brownian motion. However the focus of the thesis is to implement the Monte Carlo methods for two pure jump L évy models, namely the variance gamma and the normal inverse Gaussian models. We first provide a broad discussion on some of the properties of L évy processes, followed by a study of the variance gamma model of Madan et al. [1998] and the normal inverse Gaussian model of Barndor -Nielsen [1995]. We also provide an implementation of a variation of the calibration procedure of Cont and Tankov [2004b] for these models. We conclude with an analysis of results obtained from pricing American options using these models. / AFRIKAANSE OPSOMMING: Ons bestudeer Monte Carlo metodes wat Amerikaanse opsies, waar die aandeleprys dinamika die patroon van die eksponensiële suiwer sprong L évy modelle volg, prys. Ons neem slegs aandeleprys dinamika vir 'n enkele aandeel in ag. Die tesis begin met 'n algemene inleiding tot Amerikaanse Monte Carlo metodes. Daarna bestudeer ons twee klasse metodes. Die eerste behels regressie - ons bestudeer die regressiemetode van Tsitsiklis and Van Roy [2001] vlugtig en analiseer die least squares Monte Carlo metode van Longsta and Schwartz [2001] in detail. Ons gee ook aandag aan die variansie reduksie tegnieke van Rasmussen [2005] wat van toepassing is op die least squares Monte Carlo metodes. Die stochastic mesh metode van Broadie and Glasserman [2004] val in die tweede klas wat ons onder oë neem. Ons sal ook aandag gee aan die dual metode, wat 'n hoë bias skatting van 'n stop reël skep, en afsonderlik deur Andersen and Broadie [2004], Rogers [2002] and Haugh and Kogan [March 2004] bestudeer is. Die reëls wat ons bestudeer is skattings van die grense tussen die voortsettings- en oefenareas van die opsie. Ons analiseer in detail hoe om so 'n benadering in die least squares Monte Carlo en stochastic mesh metodes te verkry. Hierdie modelle word geï mplementeer deur beide die pseudo kansgetalgenerator en die verkose beste quasi kansgetalgenerator met brug steekproefneming te gebruik. As 'n basisgeval word hierdie metodes geï mplimenteer wanneer die aandeleprysproses 'n geometriese Browniese beweging volg. Die fokus van die tesis is om die Monte Carlo metodes vir twee suiwer sprong L évy modelle, naamlik die variance gamma en die normal inverse Gaussian modelle, te implimenteer. Eers bespreek ons in breë trekke sommige van die eienskappe van L évy prossesse en vervolgens bestudeer ons die variance gamma model soos in Madan et al. [1998] en die normal inverse Gaussian model soos in Barndor -Nielsen [1995]. Ons gee ook 'n implimentering van 'n variasie van die kalibreringsprosedure deur Cont and Tankov [2004b] vir hierdie modelle. Ons sluit af met die resultate wat verkry is, deur Amerikaanse opsies met behulp van hierdie modelle te prys.

Thesis (MSc)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: Adolescents are regarded as a high risk group in South Africa with the highest human immunodeficiency virus (HIV) incidence occurring in this group. Prevention among adolescents is therefore a key in decreasing the HIV burden. This thesis aims to assist in the design of trials by simulating the potential outcomes of a combination prevention trial in adolescents. We develop a stochastic individual-based model stratified by sex and age. We then use this model to determine the impact of various prevention packages on HIV incidence among adolescents participating in a hypothetical trial over a three year period. The trial that is simulated involves an intervention arm, in which adolescents are offered a choice of a prevention methods (including medical male circumcision (MMC), oral pre-exposure prophylaxis (PrEP) and antiretroviral-based vaginal microbicides (ARV-VM)), and a control arm. We predict that the impact of a full prevention package on HIV incidence would be a 46% per personyear( PPY) (95% CI 45–47%) risk reduction. The combination of MMC and PrEP has a substantial impact on HIV incidence in males, with a 51% PPY (95% CI 49–53%) relative risk of HIV infection. Offering women the choice of PrEP, a microbicide gel or a microbicide in the form of a vaginal ring would be less effective, with a 57% PPY (95% CI 56–58%) relative risk of HIV acquisition. This is not substantially different from the relative risk estimated when the vaginal ring alone is offered, as the ring is assumed to be the most accept able of the three prevention methods. We determine a sample size requirement of approximately 1013 in each arm of a trial would achieve 80% power to detect a statistically significant reduction in HIV risk. We find that the relative risk is sensitive to the assumed degree of correlation between condom use and the acceptability of the prevention method. We also find that the most efficient trial design may be to offer both MMC and PrEP to males but to offer only a microbicide ring to females. Further work is required to better understand the processes by which adolescent prevention method choices are made. / AFRIKAANSE OPSOMMING: Adolessente word beskou as ‘n hoe risiko groep in Suid Afrika, met die hoogste menslike immuniteitsgebrekvirus (MIV) insidensie in hierdie groep. Voorkoming van MIV onder adolessente is daarom noodsaaklik om die MIV las te verminder. Die doel van hierdie tesis is om te help met die ontwerp van studies deur die moontlike uitkomste van ‘n kombinasie-voorkoming studie in adolessente te simuleer. Ons het ‘n stogastiese individu-gebaseerde model, gestratifiseer met betrekking tot seks en ouderdom, ontwikkel. Ons het toe die model gebruik om die impak van ‘n verskeinheid van voorkomingspakette op MIV insidensie onder adolessente wat deelneem aan ‘n hipotetiese proef oor ‘n drie jaar periode, te bepaal. Die proef wat gesimuleer word behels a intervensie groep, waarin die jong volwassenes ‘n keuse van voorbehoedings metodes (insluitende mediese manlike besnydenis (MMB), pre-blootstelling profilakse (PrBP) en anti-retrovirale vaginale mikrobisiedes (ARV-VM)) aangebied word, en ‘n kontrole groep. Ons voorspel dat die impak van ‘n volle voorkomingspaket op MIV insidensie ‘n 46% per persoon-jaar (PPJ) (95% VI 47–47%) risiko vermindering sal wees. Die kombinasie van MMB en PrBP het ‘n substansiele impak op MIV insidensie onder mans, met ‘n relatiewe risiko van MIV infeksie van 51% PPJ (95% VI 49–53%). Om die keuse van PrBP, ‘n mikrobisiede gel of ‘n mikrobisiede in die vorm van ‘n vaginale ring aan vrouens te bied, is minder effektief, met ‘n relatiewe risiko van MIV infeksie van 57% PPJ (95% VI 56%–58%). Hierdie verskil nie substansieel van die beraamde relatiewe risiko in die geval waar slegs die vaginale ring gebied word nie, aangesien daar aanvaar word dat die ring die mees aanvaarde van die drie voorkomingsmetodes is. Ons het bepaal dat ‘n steekproef van ongeveer 1013 individue in elke arm van die proef nodig is om ‘n 80% kans te he om ‘n statisties betekenisvolle afname in MIV-risiko te bespeur. Ons vind dat die relatiewe risiko sensitief is tot die aanvaarde graad van die korrelasies tussen kondoom-gebruik en die aanvaarding van die voorkomings metodes. Ons het ook gevind dat dit mag wees dat die mees doeltreffende proef ontwerp is om beide MMB en PrBP vir mans en slegs ‘n mikrobisiede ring vir vrouens te bied. Verdere werk word benodig om die prosesse waarby jong volwassenes keuses maak oor voorkomingsmetodes te verstaan.

Thesis (MSc)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: In recent years, HIV-associated cancers have proven to be the bane of our time, since HIV is decimating humanity across the globe, even in the twilight of the last century. Cancer rates continue to rise in developing countries, where 95% of the world’s HIV-infected population lives, yet less than 1% have access to antiretroviral therapy. HIV-infected individuals have a higher proclivity to develop cancers, mainly from immunosuppression. An understanding of the immunopathogenesis of HIV-related cancers (HRC) is therefore a major prerequisite for rationally developing and/or improving therapeutic strategies, developing immunotherapeutics and proplylatic vaccines. In this study, we explore the pathology of HIV-related cancer malignancies, taking into account the pathogenic mechanisms and their potential for improving the treatment of management of these malignancies especially in developing countries. We mathematically model the dynamics of malignant tumors in an HIV-free environment, investigate the impact of cancer malignancies on HIV-positive patients and explore the benefits of various therapeutic intervention strategies in the management of HIV-related cancers. We present two deterministic models of infectious diseases to implement these, and they were analysed. We use HIV-related lymphomas in the Western Cape of South Africa as a case study. We validated the proposed models using lymphoma incidence data from the Tygerberg Lymphoma Study Group (TLSG), Tygerberg Hospital, Western Cape, South Africa. We show that the increasing prevalence of HIV increases lymphoma cases, and thus, other HIV-related cancers. Our models also suggests that an increase in the roll-out of the HAART program can reduce the number of lymphoma cases in the nearest future, while it averts many deaths. Furthermore, the results indicate that a highly crucial factor to consider in the prognosis of the incidence of lymphoma (and other cancer types) in HIV-infected patients is their CD4 cell count, irrespective of whether the patient has developed an HRC or not.

Thesis (MSc)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: In this thesis we develop a self-dual categorical approach to some topics in non-abelian algebra, which is based on replacing the framework of a category with that of a category equipped with a functor to it. We also make some first steps towards a possible link between this theory and the theory of databases in computer science. Both of these theories are based around the study of Grothendieck bifibrations and their generalisations. The main results in this thesis concern correspondences between certain structures on a category which are relevant to the study of categories of non-abelian group-like structures, and functors over that category. An investigation of these correspondences leads to a system of dual axioms on a functor, which can be considered as a solution to the proposal of Mac Lane in his 1950 paper "Duality for Groups" that a self-dual setting for formulating and proving results for groups be found. The part of the thesis concerned with the theory of databases is based on a recent approach by Johnson and Rosebrugh to views of databases and the view update problem. / AFRIKAANSE OPSOMMING: In hierdie tesis word ’n self-duale kategoriese benadering tot verskeie onderwerpe in nie-abelse algebra ontwikkel, wat gebaseer is op die vervanging van die raamwerk van ’n kategorie met dié van ’n kategorie saam met ’n funktor tot die kategorie. Ons neem ook enkele eerste stappe in die rigting van ’n skakel tussen hierdie teorie and die teorie van databasisse in rekenaarwetenskap. Beide hierdie teorieë is gebaseer op die studie van Grothendieck bifibrasies en hul veralgemenings. Die hoof resultate in hierdie tesis het betrekking tot ooreenkomste tussen sekere strukture op ’n kategorie wat relevant tot die studie van nie-abelse groep-agtige strukture is, en funktore oor daardie kategorie. ’n Verdere ondersoek van hierdie ooreemkomste lei tot ’n sisteem van duale aksiomas op ’n funktor, wat beskou kan word as ’n oplossing tot die voorstel van Mac Lane in sy 1950 artikel “Duality for Groups” dat ’n self-duale konteks gevind word waarin resultate vir groepe geformuleer en bewys kan word. Die deel van hierdie tesis wat met die teorie van databasisse te doen het is gebaseer op ’n onlangse benadering deur Johnson en Rosebrugh tot aansigte van databasisse en die opdatering van hierdie aansigte.

Thesis (MSc (Mathematical Sciences))--Stellenbosch University, 2008. / Class field theory describes the abelian extensions of a given field K in terms of various class groups of K, and can be viewed as one of the great successes of 20th century number theory. However, the main results in class field theory are pure existence results, and do not give explicit constructions of these abelian extensions. Such explicit constructions are possible for a variety of special cases, such as for the field Q of rational numbers, or for quadratic imaginary fields. When K is a global function field, however, there is a completely explicit description of the abelian extensions of K, utilising the theory of sign-normalised Drinfeld modules of rank one. In this thesis we give detailed survey of explicit class field theory for rational function fields over finite fields, and of the fundamental results needed to master this topic.

Thesis (PhD (Mathematical Sciences))--University of Stellenbosch, 2011. / Includes bibliography. / ENGLISH ABSTRACT: Latimer & MacDuffee showed in 1933 that there is a one-to-one correspondence between equivalence classes of matrices with a given minimum polynomial and equivalence classes of ideals of a certain ring. In the case where the matrices are taken over the integers, Behn and Van der Merwe developed an algorithm in 2002 to produce a representative in each equivalence class. We extend this algorithm to matrices taken over the ring Fq[T] of polynomials over a finite field and prove a modified version of the Latimer-MacDuffee theorem which holds for proper equivalence classes of matrices. / AFRIKAANSE OPSOMMING: Latimer & MacDuffee het in 1933 bewys dat daar 'n een-tot-een korrespondensie is tussen ekwivalensieklasse van matrikse met 'n gegewe minimumpolinoom en ekwivalensieklasse van ideale van 'n sekere ring. In die geval waar die matrikse heeltallige inskrywings het, het Behn en Van der Merwe in 2002 'n algoritme ontwikkel om verteenwoordigers in elke ekwivalensieklas voort te bring. Ons brei hierdie algoritme uit na die geval van matrikse met inskrywings in die ring Fq[T] van polinome oor 'n eindige liggaam en ons bewys 'n gewysigde weergawe van die Latimer-MacDuffee stelling wat geld vir klasse van streng ekwivalente matrikse.

Thesis (PhD)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: Water-borne infections have been a menace in many countries around the globe, claiming millions of lives. Cholera in particular has spread to all continents and now on its seventh epidemic. Although control measures have been continually developed through sanitation, vaccination and rehydration, the infection still devastates populations whenever there is an outbreak. In this research work, mathematical models for cholera transmission dynamics with focus on the impact of sanitation and hygiene, metapopulation spread, optimal control and biological control using a bacteriophage specific for pathogenic Vibrio cholerae are constructed and analysed. Vital analyses for the models are precisely given as well as numerical results depicting long term behaviour and the evolution of populations over time. The results of our analysis indicate that; improved sanitation and hand-hygiene are vital in reducing cholera infections; the spread of disease across metapopulations characterised by exchange of individuals and no cross community infection is associated with synchronous fluctuation of populations in both adjacent communities; during control of cholera, the control measures/efforts ought to be optimal especially at the beginning of the epidemic where the outbreak is often explosive in nature; and biological control if well implemented would avert many potential infections by lowering the concentration of pathogenic vibrios in the aquatic environment to values lower than the infectious dose. / AFRIKAANSE OPSOMMING: Water-infeksies is ’n bedreiging in baie lande regoor die wêreld en eis miljoene lewens. Cholera in die besonder, het op sy sewende epidemie na alle kontinente versprei. Hoewel beheermaatreëls voortdurend ontwikkel word deur middel van higiëne, inentings en rehidrasie, vernietig die infeksie steeds bevolkings wanneer daar ’n uitbraak voorkom. In hierdie navorsingswerk, word wiskundige modelle vir cholera-oordrag dinamika met die fokus op die impak van higiëne, metabevolking verspreiding, optimale beheer en biologiese beheer met behulp van ’n bakteriofaag spesifiek vir patogene Vibrio cholerae gebou en ontleed. Noodsaaklike ontledings vir die modelle is gegee sowel as numeriese resultate wat die langtermyn gedrag uitbeeld en die ontwikkeling van die bevolking oor tyd. Die resultate van ons ontleding dui daarop dat; verbeterde higiëne is noodsaaklik in die vermindering van cholera infeksies; die verspreiding van die siekte oor metapopulaties gekenmerk deur die uitruil van individue en geen kruis gemeenskap infeksie wat verband houmet sinchrone skommeling van bevolkings in beide aangrensende gemeenskappe; tydens die beheer van cholera,behoort die beheermaatreëls/pogings optimaal te wees veral aan die begin van die epidemie waar die uitbreking dikwels plofbaar in die natuur is; en biologiese beheer, indien dit goed geïmplementeer word, kan baie potensiële infeksies voorkom deur ’n vermindering in die konsentrasie van patogene vibrio in die water tot waardes laer as die aansteeklike dosis.

Thesis (MSc (Mathematical Sciences))--University of Stellenbosch, 2008. / Context-free grammars are well-studied and well-behaved in terms of decidability, but many real-world problems cannot be described with context-free grammars. Grammars with regulated rewriting are grammars with mechanisms to regulate the applications of rules, so that certain derivations are avoided. Thus, with context-free rules and regulated rewriting mechanisms, one can often generate languages that are not context-free. In this thesis we study grammars with regulated rewriting mechanisms. We consider problems in which context-free grammars are insufficient and in which more descriptive grammars are required. We compare bag context grammars with other well-known classes of grammars with regulated rewriting mechanisms. We also discuss the relation between bag context grammars and recognizing devices such as counter automata and Petri net automata. We show that regular bag context grammars can generate any recursively enumerable language. We reformulate the pumping lemma for random permitting context languages with context-free rules, as introduced by Ewert and Van der Walt, by using the concept of a string homomorphism. We conclude the thesis with decidability and complexity properties of grammars with regulated rewriting.

Thesis (MSc (Mathematical Sciences))--University of Stellenbosch, 2007. / The ever-increasing reliance of society on computer systems has led to a need for highly reliable systems. There are a number of areas where computer systems perform critical functions and the development of such systems requires a higher level of attention than any other type of system. The appropriate approach in this situation is known as formal methods. Formal methods refer to the use of mathematical techniques for the specification, development and verification of software and hardware systems. The two main goals of this thesis are: 1. The design of mathematical models as a basis for the implementation of error-free software for the safety interlock system at iThemba LABS (http://www.tlabs.ac.za/). 2. The comparison of formal method techniques that addresses the lack of much-needed empirical studies in the field of formal methods. Mathematical models are developed using model checkers: Spin, Uppaal, Smv and a theorem prover Pvs. The criteria used for the selection of the tools was based on the popularity of the tools, support of the tools, representation of properties, representativeness of verification techniques, and ease of use. The procedure for comparing these methods is divided into two phases. Phase one involves the time logging of activities followed by a novice modeler to model check and theorem prove software systems. The results show that it takes more time to learn and use a theorem prover than a model checker. Phase two involves the performance of the tools in relation to the time taken to verify a property, memory used, number of states and transitions generated. In spite of the differences between models, the results are in favor of Smv and this maybe attributed to the nature of the safety interlock system, as it involves a lot of hard-wired lines.

Thesis (MScEng (Mathematical Sciences)--University of Stellenbosch, 2009. / ENGLISH ABSTRACT: To reliably classify parts of an image sequence as foreground or background is an important part of many computer vision systems, such as video surveillance, tracking and robotics. It can also be important in applications where bandwidth is the limiting factor, such as video conferencing. Independent foreground motion is an attractive source of information for this task, and with static cameras, background subtraction is a particularly popular type of approach. The idea behind background subtraction is to compare the current image with a reference image of the background, and from there decide on a pixel by pixel basis, what is foreground and what is background by observing the changes in the pixel sequence. The problem is to get the useful reference image, especially when large parts of the background are occluded by moving/stationary foreground objects; i.e. some parts of the background are never seen. In this thesis four algorithms are reviewed that segment an image sequence into foreground and background components with varying degrees of success that can be measured on speed, comparative accuracy and/or memory requirements. These measures can be then effectively used to decide the application scope of the individual algorithms. / AFRIKAANSE OPSOMMING: Om betroubaar dele van ’n beeld reeks te klassifiseer as voorgrond of agtergrond is ’n belangrike deel van baie rekenaarvisie sisteme, byvoorbeeld video bewaking, volging en robotika. Dit kan ook belangrik wees in toepassings waar bandwydte die beperkende faktor is, byvoorbeeld video konferensie gesprekke. Onafhanklik voorgrond beweging is ’n aantreklike bron van informasie vir hierdie taak, en met statiese kameras, is agtergrond aftrekking ’n populêre benadering. Die idee agter agtergrond aftrekking is om die huidige beeld met ’n naslaan beeld van die agtergrond te vergelyk, en daarvandaan besluit op ’n piksel-na-piksel basis, wat is voorgrond en wat is agtergrond deur die observasies van die veranderinge in die piksel-reeks. Die probleem is om die naslaan beeld te kry om mee te werk, veral wanneer groot dele van die agtergrond onsigbaar bly as gevolg van bewegende of stilstaande voorgrond objekte en sommige dele van die agtergrond word dalk nooit gesien nie. In hierdie tesis word vier algorithms ondersoek wat ’n beeld reeks segmenteer in respektiewe voorgrond en agtergrond komponente met wisselende grade van sukses wat gemeet kan word deur spoed, vergelykbare akkuraatheid en/of geheu gebruik. Hierdie metings kan dan effektief gebruik word om die applikasie veld van die individuele algoritmes the bepaal.