This dissertation provides a brief theoretical introduction to the Malliavin calculus leading to a particular application in finance. The Malliavin calculus concepts are used to aid in the simulation of the Greeks for financial contingent claims. Particular focus is placed on creating efficiency in the more exotic type option simulations, where no closed solution pricing formulae exist. Copyright / Dissertation (MSc)--University of Pretoria, 2009. / Mathematics and Applied Mathematics / unrestricted

In the first part of the project the elementary development of an extended number system called Hyperreals is discussed. The second half of this project develops the basics of Nonstandard Analysis, including the theory of ultrafilters, and the formal construction of the Hyperreals.

published_or_final_version / Mathematics / Master / Master of Philosophy

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This project proceeds from the premise that the historical and logical value of Boole's logical calculus and its connection with Frege's logic remain to be recognised. It begins by discussing Gillies' application of Kuhn's concepts to the history oflogic and proposing the use of the concept of research programme as a methodological tool in the historiography oflogic. Then it analyses'the development of mathematical logic from Boole to Frege in terms of overlapping research programmes whilst discussing especially Boole's logical calculus. Two streams of development run through the project: 1. A discussion and appraisal of Boole's research programme in the context of logical debates and the emergence of symbolical algebra in Britain in the nineteenth century, including the improvements which Venn brings to logic as algebra, and the axiomatisation of 'Boolean algebras', which is due to Huntington and Sheffer. 2. An investigation of the particularity of the Fregean research programme, including an analysis ofthe extent to which certain elements of Begriffsschrift are new; and an account of Frege's discussion of Boole which focuses on the domain common to the two formal languages and shows the logical connection between Boole's logical calculus and Frege's. As a result, it is shown that the progress made in mathematical logic stemmed from two continuous and overlapping research programmes: Boole's introduction ofmathematics in logic and Frege's introduction oflogic in mathematics. In particular, Boole is regarded as the grandfather of metamathematics, and Lowenheim's theorem ofl915 is seen as a revival of his research programme.

The primary purpose of this thesis is to carefully develop and prove some of the fundamental, classical theorems of the differential calculus for functions of two real variables.

A Dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in ful lment of the requirements for the degree of Master of Science in Mathematics. 26 August 2016. / In this thesis we analyse the issue of decidability for two modal logics which contain least binders. Towards this goal, we begin the work with a brief survey of modal logic, PDL, the modal -calculus and algebraic filtrations as exposited by Conradie et al. The first such modal logic we analyse is the fragment of the modal -calculus corresponding to PDL; the second logic is the equational theory of the class of -algebras (motivated by the least root calculus of Pratt). We offer a new, algebraic, proof for the decidability of PDL by showing that PDL has the finite model property with respect to the class of dynamic algebras. We then show that the equational theory of the class of -algebras has the finite model property with respect to the class of -algebras; this is based on the proof of Pratt but differs in an important detail. The finite model property results for these two modal logics are achieved by an algebraic filtration method based on that of Conradie et al. / LG2017

by Xiao-Qing Jin. / Thesis (M.Ph.)--Chinese University of Hong Kong, 1986 / Bibliography: leaves 47-48