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Finite-state, discrete-parameter Markov chains are used to provide a model of the population of software use to support statistical testing of software. Once a Markov chain usage model has been constructed, any number of statistically typical tests can be obtained from the model. Markov mathematics can be applied to obtain values, such as the long run probabilities, that provide information for test planning and analysis of test results. Because Markov chain usage models of industrial-sized systems are often very large, the time and memory required to compute the long run probabilities can be prohibitive. This thesis describes a procedure for automatically decomposing a large Markov chain model 'into several smaller models from which the original model's long run probabilities can be calculated. The procedure supports both parallel processing to reduce the elapsed time, and sequential processing to reduce memory requirements.

Call number: LD2668 .T4 1963 W66 / Master of Science

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Master of Science. Johannesburg, 2014. / This dissertation considers the pricing of European and American options. European option prices are determined by the market and can be veri ed by a closed-form solution to the Black-Scholes model. These options can only be exercised at the maturity date. American option prices are not derived from the market and cannot be priced using the same closed-form solution as in the case of the European options because American options can be exercised at any time on or before the maturity date. An initial method was investigated in pricing a European option but could not price American options. Improvements were made producing two robust option pricing models. The results of which were compared to the closed-form solution in the case of European options and a numerical approximation solution in the case of American options. The improved models showed two signi cant bene ts. The rst bene t is the ability to price both European and American options and the second is the ability to calibrate the models to market prices using market data. Changes to the parameters of the models showed the limitations of each improved model. In conclusion, the improved methods are e ective procedures for solving the European and American option pricing problem. Keywords: European options, American options, Markov Decision Processes, Kernel-Based Reinforcement Learning, Calibration.