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A dissertation submitted to the Faculty of Science, University of the Witwatersrand, in requirement for the degree Master of Science, January 2017. / This dissertation contains evolutionary partial differential equations (PDEs). The PDEs are used to investigate ecological phenomena. The main goal is to determine Lie point symmetries, perform Lie reduction, obtain analytical solutions and visualize the solutions in 3D plots using the help of Mathematica. Drift diffusion, biased diffusion and the Kierstead, Slobodkin and Skellam (KiSS) models arising in population ecology are discussed. The importance of these PDEs in ecology is to analyse the movements of organisms and their long-term existence especially in heterogeneous environments. / XL2018

Digitized by Kansas Correctional Industries

In this thesis, we study geometric intersection representations of graphs. For a fixed class, the well-known recognition problem asks whether a given graph belongs to this class. We study a generalization of this problem called partial representation extension. Its input consists of a graph with a partial representation, so a part of the graph is pre-drawn. The problems asks whether this partial representation can be extended to a representation of the entire graph. We study this problem for classes of interval graphs, proper interval graphs, unit interval graphs and chordal graphs (in the setting of subtrees-in-tree representations). We give linear-time algorithms for the first two classes and an almost quadratic-time algorithm for unit interval graphs. For chordal graphs, we consider different versions of the problem and show that almost all cases are NP-complete. Even though the classes of proper and unit interval graphs are known to be equal, the partial representation extension problem distinguishes them. For unit interval graphs, it poses additional restrictions concerning precise positions of intervals, and we describe a new structure of unit interval representations to deal with this. 1

Cette thèse se compose de cinq parties indépendantes dédiées à la modélisation et à l’étude des problèmes liés au risque du défaut, en information partielle. La première partie constitue l’Introduction. La deuxième partie est dédiée au calcul de la probabilité de survie d’une firme, conditionnellement à l’information à disposition de l’investisseur, dans un modèle structurel en information partielle. On utilise une technique numérique hybride basée sur la méthode Monte Carlo et la quantification optimale. Dans la troisième partie on traite, avec l’approche Programmation Dynamique, un problème en temps discret de maximisation de l’utilité de la richesse terminale, dans un marché où des titres soumis au risque du défaut sont négociés. Le risque de contagion entre les défauts est modélisé, ainsi que la possible incertitude du modèle. Dans la quatrième partie on s’intéresse au problème de l’incertitude liée à l’horizon temporel d’investissement. Dans un marché complet soumis au risque du défaut, on résout, soit avec la méthode martingale, soit avec la Programmation Dynamique, trois problèmes de maximisation de l’utilité de la consommation: quand l’horizon temporel est fixe, fini mais incertain et infini. Enfin, dans la cinquième partie on traite un problème purement théorique. Dans le contexte du grossissement de filtrations, notre but est de redémontrer, dans un cadre spécifique, les résultats déjà connus sur la caractérisation des martingales, la décomposition des martingales par rapport à la filtration de référence comme semimartingales dans les filtrations progressivement et initialement grossies et le Théorème de Représentation Prévisible. / This Ph.D. thesis consists of five independent parts devoted to the modeling and to studying problems related to default risk, under partial information. The first part constitutes the Introduction. The second part is devoted to the computation of survival probabilities of a firm, conditionally to the information available to the investor, in a structural model, under partial information. We exploit a numerical hybrid technique based on the application of the Monte Carlo method and of optimal quantization. In the third part we deal, by means of the Dynamic Programming, with a discrete time maximization of the expected utility from terminal wealth problem, in a market where defaultable assets are traded. Contagion risk between the default times is modeled, as well as model uncertainty, by working under partial information. In the fourth part we are interested in studying the problem linked to the uncertainty of the investment horizon. In a complete market model subject to default risk, we solve, both with the martingale method and with the Dynamic Programming, three different problems of maximization of expected utility from consumption: when the investment horizon is fixed, finite but uncertain, and infinite. Finally, in the fifth part we deal with a purely theoretical problem. In the context of enlargement of filtrations our aim is to retrieve, in a specific setting, the already known results on martingales’ characterization, on the decomposition of martingales with respect to the reference filtration as semi-martingales in the progressively and in the initially enlarged filtrations and the Predictable Representation Theorem.

Thesis (M.D.S.)--University of Hong Kong, 1996. / Includes bibliographical references (leaves 133-140) Also available in print.

Thèse--Universit́e de Paris.