Stochastic volatility models have long provided a popular alternative to the Black- Scholes-Merton framework. They provide, in a self-consistent way, an explanation for the presence of implied volatility smiles/skews seen in practice. Incorporating jumps into the stochastic volatility framework gives further freedom to nancial mathematicians to t both the short and long end of the implied volatility surface. We present three stochastic volatility models here - the Heston model, the Bates model and the SVJJ model. The latter two models incorporate jumps in the stock price process and, in the case of the SVJJ model, jumps in the volatility process. We analyse the e ects that the di erent model parameters have on the implied volatility surface as well as the returns distribution. We also present pricing techniques for determining vanilla European option prices under the dynamics of the three models. These include the fast Fourier transform (FFT) framework of Carr and Madan as well as two Monte Carlo pricing methods. Making use of the FFT pricing framework, we present calibration techniques for tting the models to option data. Speci cally, we examine the use of the genetic algorithm, adaptive simulated annealing and a MATLAB optimisation routine for tting the models to option data via a leastsquares calibration routine. We favour the genetic algorithm and make use of it in tting the three models to ALSI and S&P 500 option data. The last section of the dissertation provides hedging techniques for the models via the calculation of option price sensitivities. We nd that a delta, vega and gamma hedging scheme provides the best results for the Heston model. The inclusion of jumps in the stock price and volatility processes, however, worsens the performance of this scheme. MATLAB code for some of the routines implemented is provided in the appendix.

A contextualização de eventos próximos da realidade dos alunos aliada a utilização da informática como ferramenta auxiliar no aprendizado da probabilidade, pode ser um dos caminhos para a melhoria do ensino de Matemática. Assim, este trabalho buscou a modelagem matemática de eventos temporais do dia a dia dos alunos do ensino básico. A modelagem se baseou no conceito de Cadeias de Markov e teve o objetivo de auxiliar o professor dos ensinos fundamental e médio a introduzir o conceito de probabilidade. As aplicações das Cadeias de Markov também possibilitam apresentar aos alunos dos ensinos médio e fundamental como a Matemática pode resolver problemas do cotidiano. Para introduzir os conceitos de Cadeias de Markov foi necessário uma revisão teórica dos conceitos da teoria da probabilidade e os conceitos de Cadeias de Markov foram estudados em literatura em língua inglesa. Considerando o interesse e curiosidade demonstrado pelos alunos em experiência prévia com o material, as atividades mostraram-se muito eficientes. Espera-se que esse trabalho possa contribuir para a prática docente de outros professores. / The contextualization of events close to the reality of the students allied to the use of information technology as an auxiliary tool in the learning of probability, can be one of the ways to improve the teaching of Mathematics. Thus, this paper sought the mathematical modeling of temporal events from the daily of students of basic Education. The modeling was based on the concept of Markov Chains and aimed to help the middle and high school teachers to introduce the concept of probability. The applications of the Markov Chains also make it possible to present to the students of the middle and high school teachings how Mathematics can solve daily problems. To introduce the concepts of Markov Chains, a theoretical revision of the concepts of probability theory was necessary and the concepts of Markov Chains were studied in literature in English Language. Considering the interest and curiosity demonstrated by the students in previous experience with the material, the activities were very efficient. It is hoped that this paper may contribute to the teaching practice of other teachers.

Increasing the efficiency and performance of internal combustion engines is always of interest in the automotive industry. One limiting factor to achieve this in gasoline combustion engines is the ignition timing which can not always be set where optimal ignition efficiency and performance is obtained. This is due to the knock phenomenon which is an abnormal combustion process that can damage the engine. Due to knock, a feedback controller which sets the ignition timing at the best possible value without the risk of harming the engine is required. In this thesis, a statistically driven knock intensity simulation environment based on the Burr Type XII distribution model was set up. In the simulation environment, different stochastic knock feedback controllers were implemented along with background noise estimation techniques used in the knock detection system. The feedback controllers were evaluated against the conventional knock controller commonly used in today’s engines in terms of ignition angle and transient response. The results from the simulation environment showed that a more advanced mean ignition angle can be achieved with stochastic based knock control strategies with the same knock-rate and without lessening the fast transient response achieved from the conventional strategy. To evaluate the results, some of the controllers were implemented in a four cylinder two-liter four stroke Volvo engine with similar results.

Nesta tese abordamos o problema do hedging de mínima variância de derivativos em mercados incompletos usando a teoria de controle ótimo estocástico com critério quadrático de otimização. Desenvolvemos um modelo geral de apreçamento e hedging de derivativos em mercados incompletos, a tempo discreto, que é capaz de acomodar qualquer tipo de payoff com característica européia que dependa de n ativos de risco. Nesse modelo, o mercado pode apresentar diferentes modos de operação, o que foi formalizado matematicamente por meio de uma cadeia de Markov. Também desenvolvemos um modelo geral de apreçamento e hedging de derivativos em mercados incompletos, a tempo discreto e espaço de estados contínuo, que é capaz de acomodar qualquer tipo de payoff com característica européia que dependa de um ativo de risco cujos retornos sejam representados por um processo de difusão com saltos. Desenvolvemos, ainda, expressões analíticas fechadas para o apreçamento e hedging de uma opção de compra européia vanilla em duas situações: (1) quando os retornos do ativo de risco são representados por um processo de difusão com saltos, e (2) quando os retornos do ativo de risco são representados por um processo de Wiener. Por fim, realizamos simulações numéricas para o controle (hedging) de uma opção de compra européia vanilla quando os retornos do ativo de risco são representados por um processo de Wiener, e comparamos os resultados obtidos com a estratégia de controle derivada do modelo de Black & Scholes. / In this thesis we approach the mean-variance hedging problem of derivatives in incomplete markets employing the theory of stochastic optimal control with quadratic optimization criteria. We developed a general derivatives pricing and hedging model in incomplete markets, in discrete time, capable of accommodating any type of European payoff contingent on n risky assets. In this model, the market may exhibit different operating modes, which were mathematically formalized by means of a Markov chain. We also developed a general derivatives pricing and hedging model in incomplete markets, in discrete time and continuous state space, capable of accommodating any type of European payoff contingent on one risky asset whose returns are described by a jump diffusion process. Even further, we developed closed-form analytical expressions for the pricing and hedging of a European vanilla call option in two situations: (1) when the risky asset returns are described by a jump diffusion process, and (2) when the risky asset returns are described by a Wiener process. Finally, we simulated the control (hedging) of a European vanilla call option when the risky asset returns are described by a Wiener process, and compared the results to those obtained with the control strategy derived from the Black & Scholes model.

Survey is time-consuming and expensive. Therefore, it needs to be both effective and efficient. Some archaeologists have argued that current survey techniques are not effective (Shott 1985, 1989), but most archaeologists continue to employ these methods and therefore must believe they are effective. If our survey techniques are effective, why do simulations suggest otherwise? If they are ineffective, can we improve them? The answers to these practical questions depend on the topological characteristics of archaeological site distributions. In this study I analyze archaeological site distributions in the Valley of Oaxaca, Mexico, using lacunarity and fractal dimension. Fractal dimension is a parameter of fractal patterns, which are complex, space-filling designs exhibiting self-similarity and power-law scaling. Lacunarity is a statistical measure that describes the texture of a spatial dispersion. It is useful in understanding how archaeological tests should be spaced during surveys. Between these two measures, I accurately describe the regional topology and suggest new considerations for archaeological survey design. / Includes bibliography. / Thesis (M.A.)--Florida Atlantic University, 2014. / FAU Electronic Theses and Dissertations Collection

O presente trabalho é uma formulação teórica que permite estudar num modelo estocástico, a influência dos erros de classificação na mensuração da prevalência da esquistossomose mansônica. Os erros de classificação são desagregados e identificados como: falhas de leitura por parte do examinador ou preparo inadequado da lâmina; contingências biológicas que possibilitam o aparecimento de ovos não viáveis e a eliminação de ovos contínua por parte dos indivíduos. É apresentada uma solução geral para o problema, bem como soluções para os casos em que se conhece a distribuição de probabilidades do número de ovos de S.mansoni. Uma solução aproximada e independente da forma e dependente dos dois primeiros momentos da distribuição do número de ovos é sugerida. A influência dos erros de classificação pode quantitativamente ser apreciada, através de um conjunto de tabelas elaboradas com diversos valores dos parâmetros intervenientes no problema. / The present paper is a theoretical approach which will, allow studying the influence - in a stochastic model - of errors in classifying the measurement of the prevalence of Schistosomiasis mansoni. The misclassification errors considered are due to: (A) failure of the examiner in either (1) reading or (2) poor technique. (B) biological contingences which will allow for the appearence of (1) sterile eggs, or (2) discontinuity in the elimination of eggs by the carriers. An exact general solution of the problem is presented, as well as solutions for the particular cases in which the probability distribution of S.mansoni eggs counts in known. An approximate solution is suggested, which is independent from the way in which the number of eggs is distributed, but depends upon the first two moments of the probability distribution of the eggs counts. The influence of misclassification errors can be judged in a quantitative way, by means of a set of tables mande up for the different parametric values of the problem.

Apresentamos um modelo estocástico para a evolução de uma espécie pelo processo de seleção natural. Compreender bem o processo evolutivo é de fundamental importância para a biologia, pois é através dele que as espécies e a vida se transformaram ao longo do tempo até chegarmos no mundo como conhecemos hoje. Detalhamos um resultado encontrado na literatura, e também introduzimos algumas variações e sugestões para aprimorar a modelagem original. O modelo proposto é interessante por conta de sua simplicidade e capacidade de capturar aspectos qualitativos esperados segundo as teorias biológicas. / We present a stochastic model for the evolution of a species by natural selection. A good understanding of the evolutionary process is fundamental for the biological sciences, since it describes how life and all species developed until we reached the world as we know today. We show in details a result available on the literature, and also introduce some variations and suggestions in order to improve the original modeling. The model presented here is interesting due to its simplicity and ability to reproduce some qualitative aspects expected from the biological theories.

Wai-tung Ho. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1992. / Includes bibliographical references (leaves 81-83). / Chapter CHAPTER 1 --- OVERVIEW OF CONSTRAINTED ESTIMATION OF STRUCTURAL EQUATION MODEL --- p.1 / Chapter CHAPTER 2 --- MULTISAMPLE ANALYSIS OF STRUCTURAL EQUATION MODELS WITH STOCHASTIC CONSTRAINTS --- p.4 / Chapter 2.1 --- The Basic Model --- p.4 / Chapter 2.2 --- Bayesian Approach to Nuisance Parameters --- p.5 / Chapter 2.3 --- Estimation and Algorithm --- p.8 / Chapter 2.4 --- Asymptotic Properties of the Bayesian Estimate --- p.11 / Chapter CHAPTER 3 --- MULTISAMPLE ANALYSIS OF STRUCTURAL EQUATION MODELS WITH EXACT AND STOCHASTIC CONSTRAINTS --- p.17 / Chapter 3.1 --- The Basic Model --- p.17 / Chapter 3.2 --- Bayesian Approach to Nuisance Parameters and Estimation Procedures --- p.18 / Chapter 3.3 --- Asymptotic Properties of the Bayesian Estimate --- p.20 / Chapter CHAPTER 4 --- SIMULATION STUDIES AND NUMERICAL EXAMPLE --- p.24 / Chapter 4.1 --- Simulation Study for Identified Models with Stochastic Constraints --- p.24 / Chapter 4.2 --- Simulation Study for Non-identified Models with Stochastic Constraints --- p.29 / Chapter 4.3 --- Numerical Example with Exact and Stochastic Constraints --- p.32 / Chapter CHAPTER 5 --- DISCUSSION AND CONCLUSION --- p.34 / APPENDICES --- p.36 / TABLES --- p.66 / REFERENCES --- p.81

Thesis (M.Phil.)--Chinese University of Hong Kong. / Bibliography: leaf 34.

Stochastic analysis on fractals is, as one might expect, a subfield of analysis on fractals. An intuitive starting point is to observe that on many fractals, one can define diffusion processes whose law is in some sense invariant with respect to the symmetries and self-similarities of the fractal. These can be interpreted as fractal-valued counterparts of standard Brownian motion on Rd. One can study these diffusions directly, for example by computing heat kernel and hitting time estimates. On the other hand, by associating the infinitesimal generator of the fractal-valued diffusion with the Laplacian on Rd, it is possible to pose stochastic partial differential equations on the fractal such as the stochastic heat equation and stochastic wave equation. In this thesis we investigate a variety of questions concerning the properties of diffusions on fractals and the parabolic and hyperbolic SPDEs associated with them. Key results include an extension of Kolmogorov's continuity theorem to stochastic processes indexed by fractals, and existence and uniqueness of solutions to parabolic SPDEs on fractals with Lipschitz data.