Thesis (MSc)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: Recently, new strains of influenza such as bird flu and swine flu have emerged. These strains have the capacity to infect people on a quite large scale and are characterized by their resistance to existing influenza treatment and their high mortality rates. In this thesis, we consider two models for influenza transmission dynamics that include both sensitive and resistant strains and accounts for disease induced mortality. The first model allows for immigration/migration and does not include any control measure. The second one explores the effects of vaccination and treatment of the sensitive strain but ignores immigration/migration. We studied the two models mathematically and numerically. We started with the model without any control measures; we calculated the basic reproductive numbers, determined the equilibrium points and investigated their stability. Our analysis showed that when the basic reproduction numbers of both strains are less than one then the two strains will die out. When at least one of the basic reproduction numbers is greater than one, then the strain with the higher basic reproduction number is the one that will persist. Numerical simulations were carried out to confirm the stability results and a bifurcation diagram was given. We also studied numerically the impact of the mortality rate of influenza on the dynamics of the disease. Especially, we investigated the effect of the mortality rate on the time needed for the pandemic to reach its peak, the value at the peak for each strain and, when eradication is possible, the time it takes for the disease to be eradicated. For the model with control, we also calculated the control reproductive number and the equilibrium points. The stability analysis was carried out numerically and bifurcation diagrams with vaccination and treatment parameters were given to determine the regions where eradication of the disease is possible. Our results suggest that in the presence of a resistant strain, treating more infected individuals will not eradicate the disease as the resistant strain will always persist. In such a case vaccination and antiviral treatment should be implemented simultaneously. / AFRIKAANSE OPSOMMING: Geen opsomming

Thesis (MSc)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: Truth preservation is an important topic in model theory. However a brief examination of the models for a logic often show that isomorphism is needlessly restrictive as a truth preserving construction. In the case of modal logics with Kripke semantics the notions of simulation and bisimulation prove far more practical and interesting than isomorphism. We present and study these various notions, followed by a discussion of Shehtman’s frame product as semantics for certain many-dimensional modal logics. We show how simulations and bisimulations can be interpreted inside models over frame products. This is followed by a discussion on a category-theoretic setting for frame products, where the arrows may run between frames with different types. / AFRIKAANSE OPSOMMING: Die behou van waarheid is ’n prominente onderwerp in modelteorie. ’n Vlugtige ondersoek van die modelle vir ’n besondere logika wys egter dat isomorfisme onnodig beperkend as waarheid-behoudende konstruksie is. In die geval van modale logika met Kripke se semantiek is simulasie en bisimulasie heelwat meer prakties en interessant as isomorfisme. Na die bekendstel en studie van hierdie onderskeie begrippe bespreek ons Shehtman se raamproduk as semantiek vir sekere meer-dimensionele modale logikas. Ons wys ons hoe simulasies en bisimulasies binne modelle oor sulke raamprodukte geïnterpreteer kan word. Daarna bespreek ons ’n kategorie-teoretiese konteks vir raamprodukte, waar die pyle tussen rame met verskillende tipes mag loop.

Thesis (MSc)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: Bacterial chemotaxis, as observed for Escherichia coli, in a field of chemoattractant molecules is characterised by a run-and-tumble motion. The motion is effected by the clockwise (CW) or counter-clockwise (CCW) rotation of flagella; filamentous appendages attached to molecular motors on the cell body. Runs appear when all flagella turn in the CCW-direction and are used to maintain a favourable direction. Tumbles emerge as soon as one flagellum starts to turn CW and are used for reorientation. Because of the variation observed between individual bacteria displaying run-and-tumble motion, we choose to model this behaviour within a probabilistic framework. An important feature of the chemotactic ability of E.coli is that the cell increases run while moving in the right direction and shortens it in the opposite case. This underlines that tumbles are used for reorientation. It has been found from experiments that there can be significant variation in the tumble fashion depending on the fraction of CW-rotating motors (Turner et al., 2000). The change in angle produced when fewer flagella are rotating CW was found to be smaller when compared to the case for many CW-rotating flagella. In addition, the change of direction contributed by a small portion of CW-rotating flagella is rarely significant for bacteria with many flagella. Based on these observations, we have distinguished between models for the one-flagellated and the multi-flagellated cases. Furthermore, since the tumbling angle change increases with the fraction of CW-rotating motors, it would not be impossible to have some cases where the amount of turn produced by the CW-rotating motors induces the bacterium to have a change of direction greater than 2π. But, this feature could not have been observed because when the bacterium tumbles it can effectuate several revolutions before resuming to a new direction. Therefore, we do not restrict our change of direction to (0,2π) to allow the bacteria to have the possibility to effectuate change of directions of magnitude greater than 2π. To this end, we differentiate between the probability of having directional change of magnitude α and α +2π . Thus we do not use angle change distributions that are defined modulo 2π such as the von Mises distribution or the wrapped normal distribution. The chemotactic ability of the bacterium is modelled by representing the CCW-bias of a single flagellum as a function of the chemoattractant concentration. The model includes the temporal memory of chemoattractant concentration that the bacterium has, which usually spans about 4s. The information about the quality of the current direction of the bacterium is transmitted to the flagellar motor by assuming that this one varies with the chemoattractant concentration level. In addition, the saturation of the bias is incorporated by assuming that the bacterium performs a temporal comparison of the receptor occupancy. The present CCW-bias-Model accounts for the chemotactic ability of the bacterium as well as its adaptation to uniform chemoattractant environment. The models of one-flagellated and multi-flagellated bacterial motion, are used to investigate two main problems. The first one consists of determining the optimal tumbling angle strategy of the bacteria. The second one consists of looking at the effects of the tumble variation on the chemotactic efficiency of the bacteria. In order to address these questions, the chemotactic efficiency measure is defined in such a way that it reflects the ability of the bacteria to converge and to stay in a near neighbourhood of the source so that they gain more nutrients. Since its movement is entirely governed by its single flagellum, the one flagellated bacterium is more able to effectuate a run motion. Tumbling events are modelled to be all equivalent because there is not any fraction of flagella to consider. On the other hand, the tumble variation of the multi-flagellated bacteria is modelled by assuming that the directional change during a tumble is a function of the fraction of CW-rotating motors. By assuming that the number of CW-rotating flagella follows a binomial distribution, we suppose that the multi-flagellated bacteria are less able to effectuate a run motion. This also implies that the change of direction produced by fewer CW-rotating flagella are more likely to happen, and this compensates the lack of run. The models show that the optimal tumbling angle change for the bacteria is less than 2π and that higher flagellated bacteria have higher chemotacitc efficiency. As the number of flagella of the bacteria increases, there can be more tumble variation, in this case the bacteria are more capable of adjusting their direction. There could be some situation were the bacteria are not moving to the right direction, but do not require a large change of direction. This ability to adjust their direction accordingly allows them to converge nearer to the source and to gain more nutrients. In addition, the dependence of the tumbling angle on the fraction of CW-rotating flagella of the mutli-flagellated bacteria, implies that there is a correlation between the tumbling angle deviation and the external environment, because the rotational states CCW-CW of the flagella depends on the external cue. Consequently, it would not be impossible that the average magnitude of tumbling angle change depends on the external environment. To investigate this possibility we analyse the distribution of the tumbling tendency of a single bacterium over time, which is the distribution over time of the average positive tumbling change of the bacterium, within zerogradient environment and within non-zero-gradient environment. We defined the average of these tumbling tendency over time as the directional persistence. We observe that the directional persistence within these different nonzero- gradient environment remains the same. However, the difference between the directional persistence within zero-gradient and non-zeros gradient environment gets larger as the number of flagella of the cell increases. There is more correlation between the external environment and the tumbling tendency of the bacterium. Which is the reason why the higher flagellated bacteria responds the best to the external environment by having the higher chemotactic performance. Finally, the total directional persistence generated by the optimal tumbling angle change of the bacteria is the average directional persistence of the bacteria regardless of their number of flagella. Its value, predicted by the model is 1.54 rad within a non-zero-gradient environment and 1.63 rad within a zero-gradient environment. / AFRIKAANSE OPSOMMING: Bakteriese chemotakse, soos waargeneem word vir Escherichia coli, in ’n veld van chemiese lokmiddel molekules word gekenmerk deur ’n hardloopen- tuimel beweging. Die beweging word bewerkstellig deur die regsom of linksom rotasie van flagella; filamentagtige aanhangsels geheg aan molekulêre motors op die selliggaam. ’n Hardloop aksie kom voor as al die flagella linksom roteer en word gebruik om ’m voordelige koers te handhaaf. Tuimels kom voor sodra een van die flagella regsom draai en word gebruik vir heroriënteering. Van wee die variasie wat waargeneem word tussen individuele bakterieë wat hardloop-en-tuimel bewegiging vertoon, verkies ons ’n probabilistiese raamwerk om in te werk. ’n Belangrike eienskap van die chemotakse vermoë van E. coli is dat die sel meer gereeld hardloop terwyl dit in die regte rigting beweeg en minder gereeld in die teenoorgestelde geval. Dit beklemtoon dat tuimels gebruik word vir heroriënteering. Dit is al eksperimenteel vasgestel dat daar betekenisvolle variasie kan wees in die tuimel wyse, wat afhang van die breukdeel regsom roterende motors (Turner et al., 2000). Die hoekverskil afkomstig van minder regsom roterende flagella was vasgestel om kleiner te wees in vergelyking met die menig regsom roterende geval. Verder word die bydrae tot die hoekverskil van ’n klein breukdeel regsom roterende flagella selde beduidend vir bakterieë met baie flagella. As gevolg van hierdie waarnemings, tref ons onderskeid tussen modelle vir een-flagella en multiflagella gevalle. Aangesien die tuimel hoeksverskil vergroot saam met die breukdeel regsom roterende motore, is dit nie onmoontlik om gevalle te hê waar die hoeveelheid draaiaksie gegenereer deur die regsom roterende motore ’n rigtingsverskil groter as 2π kan bewerkstellig nie. Dit was nie moontlik om hierdie eienskap waar te neem nie aangesien die bakterieë ’n paar keer kan tuimel voordat ’n nuwe rigting vasgestel word. Vir hierdie rede beperk ons nie die hoeksverskil tot (0,2π) nie om die bakterieë toe te laat om rigtings veranderinge groter as 2π te ondergaan. Vir hierdie doel, onderskei ons tussen die waarskynlikheid van ’n rigtinsverskil met grootte α en α + 2π. Dus, gebruik ons nie hoekverskil verspreidings wat modulo 2 gedefinieer is nie, soos die von Mises verspreiding of omwinde normaalverdeling. Die chemotakse vermoë van die bakterium word gemodelleer deur die linksom sydigheid van ’n enkele flagellum as ’n funksie van die chemotakse lokmiddel konsentrasie voor te stel. Die model sluit in die tydelike geheue wat die bakterium besit oor chemotakse lokmiddel konsentrasie, wat gewoonlik oor 4s strek. Die informasie oor die kwaliteit van die huidige rigting van die bakterium word deur gegee na die flagella motor toe deur die aanname te maak dat dit wissel met die chemotakse lokmiddel konsentrasie vlak. Die versadiging van die sydigheid word geinkorporeer deur aan te neem dat die bakterium ’n temporale vergelyking maak tussen reseptor okkupasie. Die huidige linksom sydige model neem die bakterium chemotakse vermoë in ag, as ook aanpassing tot ’n uniforme chemotakse lokmiddel omgewing. Die modelle van een-flagella en multi-flagella bakteriële beweging word gebruik om twee hoof probleme te bestudeer. Die eerste, bestaan daaruit om vas te stel wat die optimale tuimel hoek strategie van die bakterieë is. Die tweede kyk na die uitwerking van tuimel variasie op chemotakse effektiwiteit. In orde om hierdie vra te adreseer word die chemotakse effektiwiteit op so mannier gedefinieer dat dit die bakteriese vermoë om die buurt om die oorsprong te nader en daar te bly. Aangesien die beweging heeltemal vasgestel word deur een flagella, in die een-flagella geval, is ’n bakterium meer in staat daartoe om ’n hardloop aksie te bewerkstellig. Tuimel voorvalle word as ekwivalent gemodeleer omdat daar geen breukdeel roterende flagella is om in ag te neem nie. In teenstelling, word die tuimel variasie van multi-flagella bakterieë gemodeleer deur die aanname te maak dat rigtingsverandering gedurende ’n tuimel ’n funksie is van die breukdeel regsom roterende motore. Deur die aanname te maak dat die getal regsom roterende flagella ’n binomiese verspreiding volg, veronderstel ons dat multi-flagella bakterieë minder in staat daartoe is om ’n hardloop aksie te onderneem. Hierdie impliseer ook dat rigtingverandering wat geproduseer word deur minder regsom roterende flagella meer geneig is om voor te kom en dan kompenseer vir ’n tekortkoming aan hardloop gebeure. Die modelle wys dat die optimale tuimelhoek verandering minder as 2 is en dat bakterieë met meer flagella meer chemotaksies effektief is. Soos die getal flagella vermeder, kan daar meer tuimel variasie wees, en in die geval is die bakterieë meer in staat om hul rigting te verander. Daar kan omstandighede wees waar die bakterieë nie in die regtige rigting beweeg nie, maar nie ’n groot rigtingsverskil nodig het nie. Hierdie vermoë om hul rigting byvolglik te verander stel hul in staat om nader aan die oorsprong te konvergeer en dus meer voedingstowwe op te neem. Die afhanklikheid van die tuimel hoek op die breukdeel regsom roterende flagella van multi-flagella bakterieë dui daarop dat daar ’n korrelasie is tussen die tuimel hoek afwyking en die eksterne omgewing, omdat die roterings toestand, regs- of linksom, van die flagella afhanklik is van die eksterne sein. As ’n gevolg, is dit nie onmoontlik dat die gemiddelde grootte van die tuimel hoek verandering van die eksterne omgewing afhang nie. Om hierdie moontlikheid te bestudeer, analiseer ons die verspreiding van die tuimel neiging van ’n enkele bakterium oor tyd, wat die verspreiding oor tyd van die gemiddelde positiewe tuimel verandering is, in ’n nulgradient en nie-nul-gradient omgewing. Ons het hierdie gemiddelde tuimel neigings oor tyd gedefinieer as die rigtings volharding. Ons het waargeneem dat die rigtings volharding binne verskillende nienul- gradient omgewings dieselfde bly. Nogtans is die verskil tussen die rigtings volharding binne nul-gradient en nie-nul-gradient omgewings groter soos die getal flagella vermeder. Daar is meer korrelasie tussen die eksterne omgewing en tuimel neiging van die bakterium. Dit is die rede hoekom bakterieë met meer flagella die beste reageer op die eksterne omgewing deur beter chemotakse effektiwiteit. Ten slotte, die totale rigtings volharding gegenereer deur die optimale tuimel hoek verandering is die gemiddelde rigtings volharding ongeag van die getal flagella. Die waarde wat deur die model voorspel word is 1.54 rad binne ’n nie-nul-gradient omgewing en 1.63 rad binne ’n nul-gradient omgewing.

Thesis (MSc (Mathematics))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: Extracting data from video streams and using the data to better understand the observed world allows many systems to automatically perform tasks that ordinarily needed to be completed by humans. One such problem with a wide range of applications is that of detecting and tracking people in a video sequence. This thesis looks speci cally at the problem of estimating the positions of players on a sports eld, as observed by a multi-view camera setup. Previous attempts at solving the problem are discussed, after which the problem is broken down into three stages: detection, 2D tracking and 3D position estimation. Possible solutions to each of the problems are discussed and compared to one another. Motion detection is found to be a fast and e ective solution to the problem of detecting players in a single view. Tracking players in 2D image coordinates is performed by implementing a hierarchical approach to the particle lter. The hierarchical approach is chosen as it improves the computational complexity without compromising on accuracy. Finally 3D position estimation is done by multiview, forward projection triangulation. The components are combined to form a full system that is able to nd and locate players on a sports eld. The overall system that is developed is able to detect, track and triangulate player positions. The components are tested individually and found to perform well. By combining the components and introducing feedback between them the results of the individual components as well as those of the overall system are improved. / AFRIKAANSE OPSOMMING: Deur data uit 'n video-stroom te ontrek, en die data te gebruik om die wêreld wat waargeneem word beter te verstaan, kan baie rekenaarstelsels take outomaties voltooi wat voorheen deur 'n mens sou gedoen moes word. Een so 'n probleem wat 'n wye toepassingsveld het, is om mense te vind en te volg in 'n video. Hierdie tesis kyk spesi ek daarna om die posisie van spelers op 'n sportveld te vind, gegee 'n klomp kameras wat na die veld kyk. Daar word na vorige stelsels wat hierdie probleem probeer oplos gekyk, waarna die probleem in drie dele opgedeel word: vind die spelers, volg die spelers in 2D en skat die posisie van die spelers in 3D. Moontlike oplossings vir elk van hierdie dele word bespreek en vergelyk met mekaar. Daar word gevind dat om beweging te identi seer 'n eenvoudige manier is om die spelers te vind. Hulle word dan gevolg in 2D beeldkoördinate deur gebruik te maak van 'n hiërargiese implementasie van die partikel- lter. Die hiërargiese implementering word gekies omdat dit die spoed van die partikel- lter verbeter, sonder om die akkuraatheid te verswak. Laastens word die 3D posisie gevind deur multi-sigpunt, voorwaartse projeksie triangulering. Die verskillende komponente word kombineer om 'n volledige stelsel te vorm wat spelers kan vind en plaas op 'n veld. Die volledige stelsel wat ontwikkel is, is in staat om spelers te vind, volg en hulle posisies te bepaal. Elk van die individuele komponente word getoets, en daar word gevind dat hulle goed op hulle eie werk. Deur die komponente te kombineer en terugvoer tussen verskillende komponente te bewerkstellig word die resultate van die individuele komponente, sowel as dié van die volledige stelsel nog verbeter.

Thesis (MSc (Mathematics))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: See full text for abstract / AFRIKAANSE OPSOMMING: Sien volteks vir opsomming

Thesis (MSc (Mathematics))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: The first approach initiated by Merton [Mer69, Mer71] to solve utility maximization portfolio problems in continuous time is based on stochastic control theory. The idea of Merton was to interpret the maximization portfolio problem as a stochastic control problem where the trading strategies are considered as a control process and the portfolio wealth as the controlled process. Merton derived the Hamilton-Jacobi-Bellman (HJB) equation and for the special case of power, logarithm and exponential utility functions he produced a closedform solution. A principal disadvantage of this approach is the requirement of the Markov property for the stocks prices. The so-called martingale method represents the second approach for solving utility maximization portfolio problems in continuous time. It was introduced by Pliska [Pli86], Cox and Huang [CH89, CH91] and Karatzas et al. [KLS87] in di erent variant. It is constructed upon convex duality arguments and allows one to transform the initial dynamic portfolio optimization problem into a static one and to resolve it without requiring any \Markov" assumption. A de nitive answer (necessary and su cient conditions) to the utility maximization portfolio problem for terminal wealth has been obtained by Kramkov and Schachermayer [KS99]. In this thesis, we study the convex duality approach to the expected utility maximization problem (from terminal wealth) in continuous time stochastic markets, which as already mentioned above can be traced back to the seminal work by Merton [Mer69, Mer71]. Before we detail the structure of our thesis, we would like to emphasize that the starting point of our work is based on Chapter 7 in Pham [P09] a recent textbook. However, as the careful reader will notice, we have deepened and added important notions and results (such as the study of the upper (lower) hedge, the characterization of the essential supremum of all the possible prices, compare Theorem 7.2.2 in Pham [P09] with our stated Theorem 2.4.9, the dynamic programming equation 2.31, the superhedging theorem 2.6.1...) and we have made a considerable e ort in the proofs. Indeed, several proofs of theorems in Pham [P09] have serious gaps (not to mention typos) and even aws (for example see the proof of Proposition 7.3.2 in Pham [P09] and our proof of Proposition 3.4.8). In the rst chapter, we state the expected utility maximization problem and motivate the convex dual approach following an illustrative example by Rogers [KR07, R03]. We also brie y review the von Neumann - Morgenstern Expected Utility Theory. In the second chapter, we begin by formulating the superreplication problem as introduced by El Karoui and Quenez [KQ95]. The fundamental result in the literature on super-hedging is the dual characterization of the set of all initial endowments leading to a super-hedge of a European contingent claim. El Karoui and Quenez [KQ95] rst proved the superhedging theorem 2.6.1 in an It^o di usion setting and Delbaen and Schachermayer [DS95, DS98] generalized it to, respectively, a locally bounded and unbounded semimartingale model, using a Hahn-Banach separation argument. The superreplication problem inspired a very nice result, called the optional decomposition theorem for supermartingales 2.4.1, in stochastic analysis theory. This important theorem introduced by El Karoui and Quenez [KQ95], and extended in full generality by Kramkov [Kra96] is stated in Section 2.4 and proved at the end of Section 2.7. The third chapter forms the theoretical core of this thesis and it contains the statement and detailed proof of the famous Kramkov-Schachermayer Theorem that addresses the duality of utility maximization portfolio problems. Firstly, we show in Lemma 3.2.1 how to transform the dynamic utility maximization problem into a static maximization problem. This is done thanks to the dual representation of the set of European contingent claims, which can be dominated (or super-hedged) almost surely from an initial endowment x and an admissible self- nancing portfolio strategy given in Corollary 2.5 and obtained as a consequence of the optional decomposition of supermartingale. Secondly, under some assumptions on the utility function, the existence and uniqueness of the solution to the static problem is given in Theorem 3.2.3. Because the solution of the static problem is not easy to nd, we will look at it in its dual form. We therefore synthesize the dual problem from the primal problem using convex conjugate functions. Before we state the Kramkov-Schachermayer Theorem 3.4.1, we present the Inada Condition and the Asymptotic Elasticity Condition for Utility functions. For the sake of clarity, we divide the long and technical proof of Kramkov-Schachermayer Theorem 3.4.1 into several lemmas and propositions of independent interest, where the required assumptions are clearly indicate for each step of the proof. The key argument in the proof of Kramkov-Schachermayer Theorem is an in nitedimensional version of the minimax theorem (the classical method of nding a saddlepoint for the Lagrangian is not enough in our situation), which is central in the theory of Lagrange multipliers. For this, we have stated and proved the technical Lemmata 3.4.5 and 3.4.6. The main steps in the proof of the the Kramkov-Schachermayer Theorem 3.4.1 are: We show in Proposition 3.4.9 that the solution to the dual problem exists and we characterize it in Proposition 3.4.12. From the construction of the dual problem, we nd a set of necessary and su cient conditions (3.1.1), (3.1.2), (3.3.1) and (3.3.7) for the primal and dual problems to each have a solution. Using these conditions, we can show the existence of the solution to the given problem and characterize it in terms of the market parameters and the solution to the dual problem. In the last chapter we will present and study concrete examples of the utility maximization portfolio problem in speci c markets. First, we consider the complete markets case, where closed-form solutions are easily obtained. The detailed solution to the classical Merton problem with power utility function is provided. Lastly, we deal with incomplete markets under It^o processes and the Brownian ltration framework. The solution to the logarithmic utility function as well as to the power utility function is presented. / AFRIKAANSE OPSOMMING: Die eerste benadering, begin deur Merton [Mer69, Mer71], om nutsmaksimering portefeulje probleme op te los in kontinue tyd is gebaseer op stogastiese beheerteorie. Merton se idee is om die maksimering portefeulje probleem te interpreteer as 'n stogastiese beheer probleem waar die handelstrategi e as 'n beheer-proses beskou word en die portefeulje waarde as die gereguleerde proses. Merton het die Hamilton-Jacobi-Bellman (HJB) vergelyking afgelei en vir die spesiale geval van die mags, logaritmies en eksponensi ele nutsfunksies het hy 'n oplossing in geslote-vorm gevind. 'n Groot nadeel van hierdie benadering is die vereiste van die Markov eienskap vir die aandele pryse. Die sogenaamde martingale metode verteenwoordig die tweede benadering vir die oplossing van nutsmaksimering portefeulje probleme in kontinue tyd. Dit was voorgestel deur Pliska [Pli86], Cox en Huang [CH89, CH91] en Karatzas et al. [KLS87] in verskillende wisselvorme. Dit word aangevoer deur argumente van konvekse dualiteit, waar dit in staat stel om die aanvanklike dinamiese portefeulje optimalisering probleem te omvorm na 'n statiese een en dit op te los sonder dat' n \Markov" aanname gemaak hoef te word. 'n Bepalende antwoord (met die nodige en voldoende voorwaardes) tot die nutsmaksimering portefeulje probleem vir terminale vermo e is verkry deur Kramkov en Schachermayer [KS99]. In hierdie proefskrif bestudeer ons die konveks dualiteit benadering tot die verwagte nuts maksimering probleem (van terminale vermo e) in kontinue tyd stogastiese markte, wat soos reeds vermeld is teruggevoer kan word na die seminale werk van Merton [Mer69, Mer71]. Voordat ons die struktuur van ons tesis uitl^e, wil ons graag beklemtoon dat die beginpunt van ons werk gebaseer is op Hoofstuk 7 van Pham [P09] se onlangse handboek. Die noukeurige leser sal egter opmerk, dat ons belangrike begrippe en resultate verdiep en bygelas het (soos die studie van die boonste (onderste) verskansing, die karakterisering van die noodsaaklike supremum van alle moontlike pryse, vergelyk Stelling 7.2.2 in Pham [P09] met ons verklaarde Stelling 2.4.9, die dinamiese programerings vergelyking 2.31, die superverskansing stelling 2.6.1...) en ons het 'n aansienlike inspanning in die bewyse gemaak. Trouens, verskeie bewyse van stellings in Pham cite (P09) het ernstige gapings (nie te praat van setfoute nie) en selfs foute (kyk byvoorbeeld die bewys van Stelling 7.3.2 in Pham [P09] en ons bewys van Stelling 3.4.8). In die eerste hoofstuk, sit ons die verwagte nutsmaksimering probleem uit een en motiveer ons die konveks duaale benadering gebaseer op 'n voorbeeld van Rogers [KR07, R03]. Ons gee ook 'n kort oorsig van die von Neumann - Morgenstern Verwagte Nutsteorie. In die tweede hoofstuk, begin ons met die formulering van die superreplikasie probleem soos voorgestel deur El Karoui en Quenez [KQ95]. Die fundamentele resultaat in die literatuur oor super-verskansing is die duaale karakterisering van die versameling van alle eerste skenkings wat lei tot 'n super-verskans van' n Europese voorwaardelike eis. El Karoui en Quenez [KQ95] het eers die super-verskansing stelling 2.6.1 bewys in 'n It^o di usie raamwerk en Delbaen en Schachermayer [DS95, DS98] het dit veralgemeen na, onderskeidelik, 'n plaaslik begrensde en onbegrensde semimartingale model, met 'n Hahn-Banach skeidings argument. Die superreplikasie probleem het 'n prag resultaat ge nspireer, genaamd die opsionele ontbinding stelling vir supermartingales 2.4.1 in stogastiese ontledings teorie. Hierdie belangrike stelling wat deur El Karoui en Quenez [KQ95] voorgestel is en tot volle veralgemening uitgebrei is deur Kramkov [Kra96] is uiteengesit in Afdeling 2.4 en bewys aan die einde van Afdeling 2.7. Die derde hoofstuk vorm die teoretiese basis van hierdie proefskrif en bevat die verklaring en gedetailleerde bewys van die beroemde Kramkov-Schachermayer stelling wat die dualiteit van nutsmaksimering portefeulje probleme adresseer. Eerstens, wys ons in Lemma 3.2.1 hoe om die dinamiese nutsmaksimering probleem te omskep in 'n statiese maksimerings probleem. Dit kan gedoen word te danke aan die duaale voorstelling van die versameling Europese voorwaardelike eise, wat oorheers (of super-verskans) kan word byna seker van 'n aanvanklike skenking x en 'n toelaatbare self- nansierings portefeulje strategie wat in Gevolgtrekking 2.5 gegee word en verkry is as gevolg van die opsionele ontbinding van supermartingale. In die tweede plek, met sekere aannames oor die nutsfunksie, is die bestaan en uniekheid van die oplossing van die statiese probleem gegee in Stelling 3.2.3. Omdat die oplossing van die statiese probleem nie maklik verkrygbaar is nie, sal ons kyk na die duaale vorm. Ons sintetiseer dan die duale probleem van die prim^ere probleem met konvekse toegevoegde funksies. Voordat ons die Kramkov-Schachermayer Stelling 3.4.1 beskryf, gee ons die Inada voorwaardes en die Asimptotiese Elastisiteits Voorwaarde vir Nutsfunksies. Ter wille van duidelikheid, verdeel ons die lang en tegniese bewys van die Kramkov-Schachermayer Stelling ref in verskeie lemmas en proposisies op, elk van onafhanklike belang waar die nodige aannames duidelik uiteengesit is vir elke stap van die bewys. Die belangrikste argument in die bewys van die Kramkov-Schachermayer Stelling is 'n oneindig-dimensionele weergawe van die minimax stelling (die klassieke metode om 'n saalpunt vir die Lagrange-funksie te bekom is nie genoeg in die geval nie), wat noodsaaklik is in die teorie van Lagrange-multiplikators. Vir die, meld en bewys ons die tegniese Lemmata 3.4.5 en 3.4.6. Die belangrikste stappe in die bewys van die die Kramkov-Schachermayer Stelling 3.4.1 is: Ons wys in Proposisie 3.4.9 dat die oplossing vir die duale probleem bestaan en ons karaktiriseer dit in Proposisie 3.4.12. Uit die konstruksie van die duale probleem vind ons 'n versameling nodige en voldoende voorwaardes (3.1.1), (3.1.2), (3.3.1) en (3.3.7) wat die prim^ere en duale probleem oplossings elk moet aan voldoen. Deur hierdie voorwaardes te gebruik, kan ons die bestaan van die oplossing vir die gegewe probleem wys en dit karakteriseer in terme van die mark parameters en die oplossing vir die duale probleem. In die laaste hoofstuk sal ons konkrete voorbeelde van die nutsmaksimering portefeulje probleem bestudeer vir spesi eke markte. Ons kyk eers na die volledige markte geval waar geslote-vorm oplossings maklik verkrygbaar is. Die gedetailleerde oplossing vir die klassieke Merton probleem met mags nutsfunksie word voorsien. Ten slotte, hanteer ons onvolledige markte onderhewig aan It^o prosesse en die Brown ltrering raamwerk. Die oplossing vir die logaritmiese nutsfunksie, sowel as die mags nutsfunksie word aangebied.

Thesis (PhD (Mathematics))--University of Stellenbosch, 2010. / Contains bibliography. / ENGLISH ABSTRACT: See full text for summary. / AFRIKAANSE OPSOMMING: Sien volteks vir opsomming

Thesis (MSc (Mathematics))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: In this thesis, higher order numerical methods for weak approximation of solutions of stochastic differential equations (SDEs) are presented. They are motivated by option pricing problems in finance where the price of a given option can be written as the expectation of a functional of a diffusion process. Numerical methods of order at most one have been the most used so far and higher order methods have been difficult to perform because of the unknown density of iterated integrals of the d-dimensional Brownian motion present in the stochastic Taylor expansion. In 2001, Kusuoka constructed a higher order approximation scheme based on Malliavin calculus. The iterated stochastic integrals are replaced by a family of finitely-valued random variables whose moments up to a certain fixed order are equivalent to moments of iterated Stratonovich integrals of Brownian motion. This method has been shown to outperform the traditional Euler-Maruyama method. In 2004, this method was refined by Lyons and Victoir into Cubature on Wiener space. Lyons and Victoir extended the classical cubature method for approximating integrals in finite dimension to approximating integrals in infinite dimensional Wiener space. Since then, many authors have intensively applied these ideas and the topic is today an active domain of research. Our work is essentially based on the recently developed higher order schemes based on ideas of the Kusuoka approximation and Lyons-Victoir “Cubature on Wiener space” and mostly applied to option pricing. These are the Ninomiya-Victoir (N-V) and Ninomiya- Ninomiya (N-N) approximation schemes. It should be stressed here that many other applications of these schemes have been developed among which is the Alfonsi scheme for the CIR process and the decomposition method presented by Kohatsu and Tanaka for jump driven SDEs. After sketching the main ideas of numerical approximation methods in Chapter 1 , we start Chapter 2 by setting up some essential terminologies and definitions. A discussion on the stochastic Taylor expansion based on iterated Stratonovich integrals is presented, we close this chapter by illustrating this expansion with the Euler-Maruyama approximation scheme. Chapter 3 contains the main ideas of Kusuoka approximation scheme, we concentrate on the implementation of the algorithm. This scheme is applied to the pricing of an Asian call option and numerical results are presented. We start Chapter 4 by taking a look at the classical cubature formulas after which we propose in a simple way the general ideas of “Cubature on Wiener space” also known as the Lyons-Victoir approximation scheme. This is an extension of the classical cubature method. The aim of this scheme is to construct cubature formulas for approximating integrals defined on Wiener space and consequently, to develop higher order numerical schemes. It is based on the stochastic Stratonovich expansion and can be viewed as an extension of the Kusuoka scheme. Applying the ideas of the Kusuoka and Lyons-Victoir approximation schemes, Ninomiya- Victoir and Ninomiya-Ninomiya developed new numerical schemes of order 2, where they transformed the problem of solving SDE into a problem of solving ordinary differential equations (ODEs). In Chapter 5 , we begin by a general presentation of the N-V algorithm. We then apply this algorithm to the pricing of an Asian call option and we also consider the optimal portfolio strategies problem introduced by Fukaya. The implementation and numerical simulation of the algorithm for these problems are performed. We find that the N-V algorithm performs significantly faster than the traditional Euler-Maruyama method. Finally, the N-N approximation method is introduced. The idea behind this scheme is to construct an ODE-valued random variable whose average approximates the solution of a given SDE. The Runge-Kutta method for ODEs is then applied to the ODE drawn from the random variable and a linear operator is constructed. We derive the general expression for the constructed operator and apply the algorithm to the pricing of an Asian call option under the Heston volatility model. / AFRIKAANSE OPSOMMING: In hierdie proefskrif, word ’n hoërorde numeriese metode vir die swak benadering van oplossings tot stogastiese differensiaalvergelykings (SDV) aangebied. Die motivering vir hierdie werk word gegee deur ’n probleem in finansies, naamlik om opsiepryse vas te stel, waar die prys van ’n gegewe opsie beskryf kan word as die verwagte waarde van ’n funksionaal van ’n diffusie proses. Numeriese metodes van orde, op die meeste een, is tot dus ver in algemene gebruik. Dit is moelik om hoërorde metodes toe te pas as gevolg van die onbekende digtheid van herhaalde integrale van d-dimensionele Brown-beweging teenwoordig in die stogastiese Taylor ontwikkeling. In 2001 het Kusuoka ’n hoërorde benaderings skema gekonstrueer wat gebaseer is op Malliavin calculus. Die herhaalde stogastiese integrale word vervang deur ’n familie van stogastiese veranderlikes met eindige waardes, wat se momente tot ’n sekere vaste orde bestaan. Dit is al gedemonstreer dat hierdie metode die tradisionele Euler-Maruyama metode oortref. In 2004 is hierdie metode verfyn deur Lyons en Victoir na volumeberekening op Wiener ruimtes. Lyons en Victoir het uitgebrei op die klassieke volumeberekening metode om integrale te benader in eindige dimensie na die benadering van integrale in oneindige dimensionele Wiener ruimte. Sedertdien het menige outeurs dié idees intensief toegepas en is die onderwerp vandag ’n aktiewe navorsings gebied. Ons werk is hoofsaaklik gebaseer op die onlangse ontwikkelling van hoërorde skemas, wat op hul beurt gebaseer is op die idees van Kusuoka benadering en Lyons-Victoir "Volumeberekening op Wiener ruimte". Die werk word veral toegepas op die prysvastelling van opsies, naamlik Ninomiya-Victoir en Ninomiya-Ninomiya benaderings skemas. Dit moet hier beklemtoon word dat baie ander toepassings van hierdie skemas al ontwikkel is, onder meer die Alfonsi skema vir die CIR proses en die ontbinding metode wat voorgestel is deur Kohatsu en Tanaka vir sprong aangedrewe SDVs. Na ’n skets van die hoof idees agter metodes van numeriese benadering in Hoofstuk 1 , begin Hoofstuk 2 met die neersetting van noodsaaklike terminologie en definisies. ’n Diskussie oor die stogastiese Taylor ontwikkeling, gebaseer op herhaalde Stratonovich integrale word uiteengeset, waarna die hoofstuk afsluit met ’n illustrasie van dié ontwikkeling met die Euler-Maruyama benaderings skema. Hoofstuk 3 bevat die hoofgedagtes agter die Kusuoka benaderings skema, waar daar ook op die implementering van die algoritme gekonsentreer word. Hierdie skema is van toepassing op die prysvastelling van ’n Asiatiese call-opsie, numeriese resultate word ook aangebied. Ons begin Hoofstuk 4 deur te kyk na klassieke volumeberekenings formules waarna ons op ’n eenvoudige wyse die algemene idees van "Volumeberekening op Wiener ruimtes", ook bekend as die Lyons-Victoir benaderings skema, as ’n uitbreiding van die klassieke volumeberekening metode gebruik. Die doel van hierdie skema is om volumeberekening formules op te stel vir benaderings integrale wat gedefinieer is op Wiener ruimtes en gevolglik, hoërorde numeriese skemas te ontwikkel. Dit is gebaseer op die stogastiese Stratonovich ontwikkeling en kan beskou word as ’n ontwikkeling van die Kusuoka skema. Deur Kusuoka en Lyon-Victoir se idees oor benaderings skemas toe te pas, het Ninomiya-Victoir en Ninomiya- Ninomiya nuwe numeriese skemas van orde 2 ontwikkel, waar hulle die probleem omgeskakel het van een waar SDVs opgelos moet word, na een waar gewone differensiaalvergelykings (GDV) opgelos moet word. Hierdie twee skemas word in Hoofstuk 5 uiteengeset. Alhoewel die benaderings soortgelyk is, is daar ’n beduidende verskil in die algoritmes self. Hierdie hoofstuk begin met ’n algemene uiteensetting van die Ninomiya-Victoir algoritme waar ’n arbitrêre vaste tyd horison, T, gebruik word. Dié word toegepas op opsieprysvastelling en optimale portefeulje strategie probleme. Verder word numeriese simulasies uitgevoer, die prestasie van die Ninomiya-Victoir algoritme was bestudeer en vergelyk met die Euler-Maruyama metode. Ons maak die opmerking dat die Ninomiya-Victoir algoritme aansienlik vinniger is. Die belangrikste resultaat van die Ninomiya-Ninomiya benaderings skema word ook voorgestel. Deur die idee van ’n Lie algebra te gebruik, het Ninomiya en Ninomiya ’n stogastiese veranderlike met GDV-waardes gekonstrueer wat se gemiddeld die oplossing van ’n gegewe SDV benader. Die Runge-Kutta metode vir GDVs word dan toegepas op die GDV wat getrek is uit die stogastiese veranderlike en ’n lineêre operator gekonstrueer. ’n Veralgemeende uitdrukking vir die gekonstrueerde operator is afgelei en die algoritme is toegepas op die prysvasstelling van ’n Asiatiese opsie onder die Heston onbestendigheids model.

Thesis (MSc (Mathematical Sciences))--University of Stellenbosch, 2011. / Includes bibliography. / ENGLISH ABSTRACT: This thesis takes into account the different levels of infectiousness of the human immunodeficiency virus (HIV) infected individuals throughout their period of infection. Infectiousness depends on the time since infection. It is high shortly after the infection occurs and then much lower for several years, and thereafter a higher plateau is reached before the acquired immunodeficiency syndrome (AIDS) phase sets in. In line with this, we formulated a mathematical model which is structured according to the age of infection. To understand the dynamics of the disease, we first discuss and analyse a simple model in which the age of infection is not considered, but progression of the HIV-AIDS transmission is taken into consideration by introducing three stages of infection. Analysis of these models tells us that the disease can be eradicated from the population only if on average one infected individual infects less than one person in his or her infectious period, otherwise the disease persists. To investigate the reduction of the number of infections caused by a single infectious individual to less than one, we introduce different treatment strategies for a model which depends on the age of infection, and we analyse it numerically. Current strategies amount to introducing treatment only at a late stage of infection when the infected individual has already lived through most of the infectious period. From our numerical results, this strategy does not result in eradication of the disease, even though it does reduce the burden for the individual. To eradicate the disease from the population, everyone would need to be HIV tested regularly and undergo immediate treatment if found positive. / AFRIKAANSE OPSOMMING: Hierdie tesis hou rekening met die verskillende aansteeklikheidsvlakke van die menslike immuniteitsgebreksvirus (MIV) deur besmette individue gedurende hulle aansteeklikheidstydperk. Die graad van aansteeklikheid hang af van die tydperk sedert infeksie. Dit is hoog kort nadat die infeksie plaasvind en daarna heelwat laer vir etlike jare, en dan volg n hoer plato voordat uiteindelik die Verworwe-Immuniteitsgebreksindroom (VIGS) fase intree. In ooreenstemming hiermee, formuleer ons n wiskundige model van MIV-VIGSoordrag met n struktureer waarin die tydperk sedert infeksie bevat is. Om die dinamika van die siekte te verstaan, bespreek en analiseer ons eers n eenvoudige model sonder inagneming van die tydperk sedert infeksie, terwyl die progressie van MIV-VIGS-oordrag egter wel in ag geneem word deur die beskouing van drie stadiums van infeksie. Analise van die modelle wys dat die siekte in die bevolking slegs uitgeroei kan word as elke besmette mens gemiddeld minder as een ander individu aansteek gedurende die tydperk waarin hy of sy self besmet is, anders sal die siekte voortduur. Vir die ondersoek oor hoe om die aantal infeksies per besmette individu tot onder die waarde van een te verlaag, beskou ons verskeie behandelingsstrategiee binne die model, wat afhang van die tydperk sedert infeksie, en ondersoek hulle numeries. Die huidige behandelingstrategiee kom neer op behandeling slegs gedurende die laat sta- dium van infeksie, wanneer die besmette individu reeds die grootste deel van die aansteeklikheidsperiode deurleef het. Ons numeriese resultate toon dat hierdie strategie nie lei tot uitroeiing van die siekte nie, alhoewel dit wel die las van die siekte vir die individu verminder. Om die siekte binne die bevolking uit te roei, sou elkeen gereeld vir MIV getoets moes word en indien positief gevind, dadelik met behandeling moes begin.

Thesis (MSc (Mathematical Sciences)) University of Stellenbosch, 2011. / ENGLISH ABSTRACT: Cellular Automata (CA) have become a popular vehicle to study complex dynamical behaviour of systems. CA can be used to model a wide variety of physical, biological, chemical and other systems. Such systems typically consist of subparts that change their state independently, based on the state of their immediate surroundings and some generally shared laws of change. When the CA approach was used to solve the LEGO construction problem, the best solution was found when using a variant of CA allowing for the clustering of cells. The LEGO construction problem concerns the optimal layout of a set of LEGO bricks. The advantages found for using the CA method with clustering in this case are the ease of implementation, the significantly smaller memory usage to previously implemented methods, and its trivial extension to construct multicoloured LEGO sculptures which were previously too complex to construct. In our research we propose to explore the definitions of clustering in CA and investigate the implementation and application of this method. We look at the ant sorting method described by Lumer and Faieta, and compare the implementation of this algorithm using regular CA as well as the clustering variation. The ant sorting model is a simple model, in which ants move randomly in space and pick up and deposit objects on the basis of local information. / AFRIKAANSE OPSOMMING: Sellulêre Outomate (SO) het ’n populêre metode geword om die komplekse dinamiese gedrag van sisteme bestudeer. SO kan gebruik word om ’n groot verskeidenheid fisiese, biologiese, chemiese en ander tipe sisteme te modelleer. Sulke sisteme bestaan tipies uit subafdelings wat, gebaseer op die status van hulle omgewing en ’n paar algemene gedeelde reëls van verandering, hulle status onafhanklik verander. Met die gebruik van die SO benadering om the LEGO konstruksieprobleem op te los, is die beste oplossing bereik deur gebruik te maak van ’n variant van SO, waar selle saamgroepeer kan word. Die LEGO konstruksieprobleem behels die optimale uitleg van ’n stel LEGO blokkies. In hierdie geval is die voordele van die SO met sel groepering die maklike implementasie, ’n beduidende kleiner geheuegebruik teenoor voorheen geïmplementeerde metodes, en die triviale uitbreiding daarvan om gekleurde LEGO beelde wat voorheen te kompleks was, te kan bou. In ons ondersoek verken ons die definisies van selgroepering in SO en ondersoek die implementasie en toepassing van die metode. Ons kyk na die miersorteringsmetode beskryf deur Lumer en Faieta, en vergelyk die implementasie van hierdie algoritme deur gewone SO asook die groeperingsvariasie te gebruik. Die miersorteringsmodel is ’n eenvoudige model waarin miere lukraak in ’n omgewing beweeg en voorwerpe optel of neersit volgens plaaslike inligting.