The main purpose for an aggressive investor is to maximize the return in theinvestments. But in order to do so the risk should be taken into consideration.In this thesis, we utilize Markowitz portfolio theory, one of the standard modelsfor maximizing return while considering risk. The model allows the investorto balance risk tolerance and expected return on the stock market based onhistorical data. Simply put, the goal is to allocate capital to stocks in a mannerthat maximizes expected return while considering risk. The tested cases involvedmodels utilizing historical data from five and ten years ago, respectively. Theresulting allocation distribution of these portfolios depended on the varianceconstraint and, to a large extent, on how many years of historical data were used.These results emphasize the significance of data in portfolio optimization.

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This thesis consists of five scientific papers dealing with equations related to the optimal switching problem, mainly backward stochastic differential equations and variational inequalities. Besides the scientific papers, the thesis contains an introduction to the optimal switching problem and a brief outline of possible topics for future research. Paper I concerns systems of variational inequalities with operators of Kolmogorov type. We prove a comparison principle for sub- and supersolutions and prove the existence of a solution as the limit of solutions to iteratively defined interconnected obstacle problems. Furthermore, we use regularity results for a related obstacle problem to prove Hölder continuity of this solution. Paper II deals with systems of variational inequalities in which the operator is of non-local type. By using a maximum principle adapted to this non-local setting we prove a comparison principle for sub- and supersolutions. Existence of a solution is proved using this comparison principle and Perron's method. In Paper III we study backward stochastic differential equations in which the solutions are reflected to stay inside a time-dependent domain. The driving process is of Wiener-Poisson type, allowing for jumps. By a penalization technique we prove existence of a solution when the bounding domain has convex and non-increasing time slices. Uniqueness is proved by an argument based on Ito's formula. Paper IV and Paper V concern optimal switching problems under incomplete information. In Paper IV, we construct an entirely simulation based numerical scheme to calculate the value function of such problems. We prove the convergence of this scheme when the underlying processes fit into the framework of Kalman-Bucy filtering. Paper V contains a deterministic approach to incomplete information optimal switching problems. We study a simplistic setting and show that the problem can be reduced to a full information optimal switching problem. Furthermore, we prove that the value of information is positive and that the value function under incomplete information converges to that under full information when the noise in the observation vanishes.

We introduce random fluctuations on contact and recovery rates in three basic deterministic models in mathematical epidemiology and obtain stochastic counterparts. This paper addresses qualitative and quantitative analysis of stochastic SIS model with disease deaths and demographic effects, and stochastic SIR models with/without disease deaths and demographic effects. We prove the global existence of a unique strong solution and discuss stochastic asymptotic stability of disease free and endemic equilibria. We also investigate numerical properties of these models and prove the convergence of the Balanced Implicit Method approximation to the analytic solution. We simulate the models with fairly realistic parameters to visualize our conclusions.

Orientador: Pedro José Catuogno / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-16T07:50:09Z (GMT). No. of bitstreams: 1 Albuquerque_RobertaRodrigues_M.pdf: 577918 bytes, checksum: 382f6bfc15bbbcfa2efbd32b5ec398e7 (MD5) Previous issue date: 2010 / Resumo: Neste trabalho estamos interessados no transporte paralelo da geometria diferencial no contexto do cálculo estocástico. Inicialmente resumimos os pontos fundamentais da geometria riemmaniana como as idéias de conexão, curvatura, transporte paralelo, a identidade de Bochner-Weitenböck e o mapa de desenvolvimento de Cartan, em seguida desenvolvemos alguns resultados da geometria estocástica como a fórmula geométrica de Itô, mas para isto inserimos brevemente a chamada geometria de segunda ordem. Ao final, examinaremos o transporte paralelo estocástico em algumas circunstâncias como no mapa de desenvolvimento estocástico, mapa de rolamento estocástico, construção do movimento Browniano em variedades e ainda com fluxos estocásticos na solução da equação de Stratonovich / Abstract: This dissertation is about the stochastic version of the parallel translation in the differential geometry. In the beginning it provides some basic background to Riemannian geometry, for example, the definiton of conexion, curvature, parallel translation, the Bochner-Weitenböck identity and the Cartan's rolling map theorem. After that, it is to dedicate to development of some results on stochastic geometry as the geometric Itô formula, but to do that it is important to study the second order geometry. In the end, it is essential to give attention to stochastic parallel transport in some environment as the Cartan's rolling map in the stochastic context, stochastic rolling constuctions, Brownian motion on manifolds and the stochastic flow as the solution of the Stratonovich equation / Mestrado / Geometria Estocastica / Mestre em Matemática

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