In this thesis, it is shown that the application of a threshold on the surplus level of a particular discrete-time delayed Sparre Andersen insurance risk model results in a process that can be analyzed as a doubly infinite Markov chain with finite blocks. Two fundamental cases, encompassing all possible values of the surplus level at the time of the first claim, are explored in detail. Matrix analytic methods are employed to establish a computational algorithm for each case. The resulting procedures are then used to calculate the probability distributions associated with fundamental ruin-related quantities of interest, such as the time of ruin, the surplus immediately prior to ruin, and the deficit at ruin. The ordinary Sparre Andersen model, an important special case of the general model, with varying threshold levels is considered in a numerical illustration.

In this thesis, it is shown that the application of a threshold on the surplus level of a particular discrete-time delayed Sparre Andersen insurance risk model results in a process that can be analyzed as a doubly infinite Markov chain with finite blocks. Two fundamental cases, encompassing all possible values of the surplus level at the time of the first claim, are explored in detail. Matrix analytic methods are employed to establish a computational algorithm for each case. The resulting procedures are then used to calculate the probability distributions associated with fundamental ruin-related quantities of interest, such as the time of ruin, the surplus immediately prior to ruin, and the deficit at ruin. The ordinary Sparre Andersen model, an important special case of the general model, with varying threshold levels is considered in a numerical illustration.

Bibliography: pages 117-[124]. / This thesis investigates the use of the Black-Scholes option pricing model for long term options for the purposes of costing long term maturity guarantees. The maturity guarantees concerned are typically given on endowment policies issued by life offices. These endowment policies have terms usually in excess of five years. The thesis investigates whether the assumptions underlying the Black-Scholes model, which was developed for pricing short term traded options, are still acceptable when applied to long term options, and if not, what adjustments need to be made. The paper focuses on the pricing of European put options which are equivalent to the payoff of a maturity guarantee.

When seeking to identify a profitable technical analysis (TA) strategy, a na¨ıve investigation will compare a large number of possible strategies using the same set of historical market data. This process can give rise to a significant data mining bias, which can cause spurious results. There are various methods which account for this bias, with each one providing a different set of advantages and disadvantages. This dissertation compares three of these methods, the step wise Superior Predictive Ability (step-SPA) method of P.-H. Hsu, Y.-C. Hsu and Kuan (2010), the False Discovery Rate (FDR) method of Benjamini and Hochberg (1995) and the Monte Carlo Permutations (MCP) method of Masters (2006). The MCP method is also extended, using a step wise algorithm, to allow it to identify multiple profitable strategies. The results of the comparison show that while both the FDR and extended MCP methods can be useful under certain circumstances, the stepSPA method is ultimately the most robust, making it the best choice in spite of its significant computational requirements and stricter set of assumptions.

Includes bibliographical references. / Sexually transmitted infections (STIs) have been shown to increase the probability of HIV transmission, but there remains much uncertainty regarding the role of STI treatment in HIV prevention. This thesis aims to develop a mathematical model to estimate the prevalence of STIs in South Africa, the contribution of STIs to the spread of HIV, and the effects of changes in sexual behaviour and changes in STI treatment. A deterministic model is developed to simulate the transmission of HIV and six other STIs (syphilis, genital herpes, chancroid, gonorrhoea, chlamydial infection and trichomoniasis), as well as the incidence of bacterial vaginosis and vaginal candidiasis in women. The model is fitted to national HIV prevalence survey data, STI prevalence data from sentinel surveys and data from sexual behaviour surveys, using Bayesian techniques. Model results suggest that South Africa has some of the highest STI prevalence levels in the world, but that certain STIs – notably syphilis, chancroid, gonorrhoea and trichomoniasis – have declined in prevalence since the mid-1990s, following the introduction of syndromic management programmes and increases in condom use. STIs account for more than half of new HIV infections, and genital herpes is the most significant STI promoting the transmission of HIV. Syndromic management programmes reduced HIV incidence in South Africa by 3-10% over the decade following their introduction (1994-2004). Further reductions in HIV incidence could be achieved by promoting patient-initiated treatment of genital herpes, by addressing rising levels of drug resistance in gonococcal isolates, and by encouraging prompt health seeking for STIs. Concurrent partnerships are a major factor driving HIV transmission, accounting for 74-87% of new HIV infections over the 1990-2000 period. Halving unprotected sex in non-spousal relationships would reduce HIV incidence over the 2010 -2020 period by 32-43%. This thesis contributes to the understanding of HIV/AIDS epidemiology in South Africa by quantifying the contribution of various behavioural and biological factors to HIV transmission. This thesis also high lights several opportunities for reducing the future incidence of HIV. In addition, this thesis advances the assessment of uncertainty in STI models by proposing a Bayesian approach to incorporating sexual behaviour data and STI prevalence data into the parameter estimation process

This thesis is a study on stable distributions and some of their applications in understanding financial markets. Three broad problems are explored: First, we study a parameter and density estimation problem for stable distributions using commodity market data. We investigate and compare the accuracy of the quantile, logarithmic, maximum likelihood (ML) and empirical characteristic function (ECF) methods. It turns out that the ECF is the most recommendable method, challenging literature that instead suggests the ML. Secondly, we develop an affine theory for subordinated random processes and apply the results to pricing commodity futures in markets where the spot price includes jumps. The jumps are introduced by subordinating Brownian motion in the spot model by an α-stable process, α ε (0; 1] which leads to a new pricing approach for models with latent variables. The third problem is the pricing of general derivatives and risk management based on Malliavin calculus. We derive a Bismut-Elworthy-Li (BEL) representation formula for computing financial Greeks under the framework of subordinated Brownian motion by an inverse α-stable process with α ε (0; 1]. This subordination by an inverse α-stable process allows zero returns in the model rendering it fit for illiquid emerging markets. In addition, we demonstrate that the model is best suited for pricing derivatives with irregular payoff functions compared to the traditional Euler methods.

Includes abstract. / Includes bibliographical references (p. 228-245). / This thesis applies multivariate statistical techniques to six data sets to account for past and present-day features underlying ethnic fertility differentials in Zambia.

Includes bibliographical references. / Analysis of census data is important to uncover new insights as well as highlight where improvements in future data collection are required. The study provides an assessment of the fertility estimates derived from census data in comparison to those derived from the Zimbabwe Demographic and Health Surveys. Robust methods are used to estimate fertility levels and to identify the trends in fertility in Zimbabwe. Fertility decline in Zimbabwe is observed to have started in the early 1980s. The greatest level of decline occurred between the 1980s and the mid-1990s. In more recent years fertility in Zimbabwe has stalled at roughly four children per woman. Using projected parity progression ratios fertility decline has been observed to be in part a result of parity limitation, as fewer women progress to higher parities. A comparison of the census and Zimbabwe Demographic and Health Survey fertility measures show that for the same cohort of women, the measures of fertility are strongly congruent. While there are problems with census data, it has been shown that using robust estimation the census fertility estimates are comparable to those from the Demographic and Health Surveys.

The forecast accuracy of a hidden Markov model (HMM) may be low due first, to the measure of forecast accuracy being ignored in the parameterestimation method and, second, to overfitting caused by the large number of parameters that must be estimated. A general approach to forecasting is described which aims to resolve these two problems and so improve the forecast accuracy of the HMM. First, the application of extremum estimators to the HMM is proposed. Extremum estimators aim to improve the forecast accuracy of the HMM by minimising an estimate of the forecast error on the observed data. The forecast accuracy is measured by a score function and the use of some general classes of score functions is proposed. This approach contrasts with the standard use of a minus log-likelihood score function. Second, penalised estimation for the HMM is described. The aim of penalised estimation is to reduce overfitting and so increase the forecast accuracy of the HMM. Penalties on both the state-dependent distribution parameters and transition probability matrix are proposed. In addition, a number of cross-validation approaches for tuning the penalty function are investigated. Empirical assessment of the proposed approach on both simulated and real data demonstrated that, in terms of forecast accuracy, penalised HMMs fitted using extremum estimators generally outperformed unpenalised HMMs fitted using maximum likelihood.

Estimating the level and trend in population mortality rates at advanced ages in South Africa is complicated by problems with both the population and death data. Population and death data, particularly in developing countries, often suffer from age misreporting - age exaggeration and digit preference. Also, censuses may under- or overestimate the population and registration of deaths is usually incomplete in developing countries (Dorrington, Moultrie and Timæus 2004). To avoid these problems, the research in this dissertation relies on the method of extinct generations and its extensions (Thatcher, Kannisto and Andreev 2002) to re-estimate the population using only the death data, which is often recorded more accurately than the population data. Since deaths are not reported completely in South Africa, the death data must be corrected before use. Death Distribution Methods (Moultrie, Dorrington, Hill et al. 2013) are used to correct the death data for incomplete registration of deaths. After correction, Near Extinct Generation methods (NEG) are used to re-estimate the population by projecting future deaths of nearly extinct cohorts. After showing that mortality rates produced using the original NEG methods are biased because of age and year of birth heaping present in the South African death data, the NEG methods are adapted to the South African context. The adapted NEG model smooths the age and year of birth heaping in the death data and produces mortality rates that are less biased than the original NEG methods. This model - referred to as the NEG-GAM model in this research - is used to re-estimate the population at each age from 70 and above and to calculate mortality rates since 1985. The population estimates aged 70+ produced using the NEG-GAM model match those from the 2011 census well. It is found that both the population and death data suffer from the same pattern of heaping, that the population and death data are affected by age exaggeration and that the death data are less affected by age exaggeration than the population data. The level and trend in mortality rates calculated using the NEG-GAM model are discussed and compared to the mortality rates in the Human Mortality Database and other studies of South African mortality. The mortality rates produced for the African and Coloured population groups appear too low at the older ages due to age exaggeration in the death data, while those for the Indian and White population groups appear to be reasonable over the entire age range. Mortality appears to be improving in the age range 70-79 for the Coloured, Indian and White population groups and deteriorating slowly for the African population group.