Recentemente, Guiol, Machado e Schinazi propuseram um modelo estocástico para a evolução de espécies. Nesse modelo, as intensidades de nascimentos de novas espécies e de ocorrências de extinções são invariantes ao longo do tempo. Ademais, no instante de nascimento de uma nova espécie, a mesma é rotulada com um número aleatório gerado de uma distribuição absolutamente contínua. Toda vez que ocorre uma extinção, apenas uma espécie morre - a com o menor número vinculado. Quando a intensidade com que surgem novas espécies é maior que a com que ocorrem extinções, existe um valor crítico f_c tal que todas as espécies rotuladas com números menores que f_c morrerão quase certamente depois de um tempo aleatório finito, e as rotuladas com números maiores que f_c terão probabilidades positivas de se tornarem perpétuas. No entanto, espécies menos aptas continuam a aparecer durante o processo evolutivo e não há a garantia do surgimento de uma espécie imortal. Consideramos um caso particular do modelo de Guiol, Machado e Schinazi e abordamos estes dois últimos pontos. Caracterizamos o processo pontual limite vinculado às espécies na fase subcrítica do modelo e discorremos sobre a existência de espécies imortais. / Recently, Guiol, Machado and Schinazi proposed a stochastic model for species evolution. In this model, births and deaths of species occur with intensities invariant over time. Moreover, at the time of birth of a new species, it is labeled with a random number sampled from an absolutely continuous distribution. Each time there is an extinction event, exactly one existing species disappears: that with the smallest number. When the birth rate is greater than the extinction rate, there is a critical value f_c such that all species that come with number less than f_c will almost certainly die after a finite random time, and those with numbers higher than f_c survive forever with positive probability. However, less suitable species continue to appear during the evolutionary process and there is no guarantee the emergence of an immortal species. We consider a particular case of Guiol, Machado and Schinazi model and approach these last two points. We characterize the limit point process linked to species in the subcritical phase of the model and discuss the existence of immortal species.

Neste trabalho, aplicamos o algoritmo Breadth-First Search para encontrar o tamanho de uma componente conectada no grafo aleatório de Erdos-Rényi. Uma cadeia de Markov é obtida deste procedimento. Apresentamos alguns resultados bem conhecidos sobre o comportamento dessa cadeia de Markov. Combinamos alguns destes resultados para obter uma proposição sobre a probabilidade da componente atingir um determinado tamanho e um resultado de convergência do estado da cadeia neste instante. Posteriormente, aplicamos o teorema de convergência de Darling (2002) a sequência de cadeias de Markov reescaladas e indexadas por N, o número de vértices do grafo, para mostrar que as trajetórias dessas cadeias convergem uniformemente em probabilidade para a solução de uma equação diferencial ordinária. Deste resultado segue a bem conhecida lei fraca dos grandes números para a componente gigante do grafo aleatório de Erdos-Rényi, no caso supercrítico. Além disso, obtemos o limite do fluído para um modelo epidêmico que é uma extensão daquele proposto em Kurtz et al. (2008). / In this work, we apply the Breadth-First Search algorithm to find the size of a connected component of the Erdos-Rényi random graph. A Markov chain is obtained of this procedure. We present some well-known results about the behavior of this Markov chain, and combine some of these results to obtain a proposition about the probability that the component reaches a certain size and a convergence result about the state of the chain at that time. Next, we apply the convergence theorem of Darling (2002) to the sequence of rescaled Markov chains indexed by N, the number of vertices of the graph, to show that the trajectories of these chains converge uniformly in probability to the solution of an ordinary dierential equation. From the latter result follows the well-known weak law of large numbers of the giant component of the Erdos-Renyi random graph, in the supercritical case. Moreover, we obtain the uid limit for an epidemic model which is an extension of that proposed in Kurtz et al. (2008).

The combined survival status of the insured lives is a critical problem when pricing and reserving insurance products with more than one life. Our preliminary experience examination of bivariate annuity data from a large Canadian insurance company shows that the relative risk of mortality for an individual increases after the loss of his/her spouse, and that the increase is especially dramatic shortly after bereavement. This preliminary result is supported by the empirical studies over the past 50 years, which suggest dependence between a husband and wife. The dependence between a married couple may be significant in risk management of joint-life policies. This dissertation progressively explores Markovian models in pricing and risk management of joint-life policies, illuminating their advantages in dependent modeling of joint time-until-death (or other exit time) random variables. This dissertation argues that in the dependent modeling of joint-life dependence, Markovian models are flexible, transparent, and easily extended. Multiple state models have been widely used in historic data analysis, particularly in the modeling of failures that have event-related dependence. This dissertation introduces a ¡°common shock¡± factor into a standard Markov joint-life mortality model, and then extends it to a semi-Markov model to capture the decaying effect of the "broken heart" factor. The proposed models transparently and intuitively measure the extent of three types of dependence: the instantaneous dependence, the short-term impact of bereavement, and the long-term association between lifetimes. Some copula-based dependence measures, such as upper tail dependence, can also be derived from Markovian approaches. Very often, death is not the only mode of decrement. Entry into long-term care and voluntary prepayment, for instance, can affect reverse mortgage terminations. The semi-Markov joint-life model is extended to incorporate more exit modes, to model joint-life reverse mortgage termination speed. The event-triggered dependence between a husband and wife is modeled. For example, one spouse's death increases the survivor's inclination to move close to kin. We apply the proposed model specifically to develop the valuation formulas for roll-up mortgages in the UK and Home Equity Conversion Mortgages in the US. We test the significance of each termination mode and then use the model to investigate the mortgage insurance premiums levied on Home Equity Conversion Mortgage borrowers. Finally, this thesis extends the semi-Markov joint-life mortality model to having stochastic transition intensities, for modeling joint-life longevity risk in last-survivor annuities. We propose a natural extension of Gompertz' law to have correlated stochastic dynamics for its two parameters, and incorporate it into the semi-Markov joint-life mortality model. Based on this preliminary joint-life longevity model, we examine the impact of mortality improvement on the cost of a last survivor annuity, and investigate the market prices of longevity risk in last survivor annuities using risk-neutral pricing theory.

The combined survival status of the insured lives is a critical problem when pricing and reserving insurance products with more than one life. Our preliminary experience examination of bivariate annuity data from a large Canadian insurance company shows that the relative risk of mortality for an individual increases after the loss of his/her spouse, and that the increase is especially dramatic shortly after bereavement. This preliminary result is supported by the empirical studies over the past 50 years, which suggest dependence between a husband and wife. The dependence between a married couple may be significant in risk management of joint-life policies. This dissertation progressively explores Markovian models in pricing and risk management of joint-life policies, illuminating their advantages in dependent modeling of joint time-until-death (or other exit time) random variables. This dissertation argues that in the dependent modeling of joint-life dependence, Markovian models are flexible, transparent, and easily extended. Multiple state models have been widely used in historic data analysis, particularly in the modeling of failures that have event-related dependence. This dissertation introduces a ¡°common shock¡± factor into a standard Markov joint-life mortality model, and then extends it to a semi-Markov model to capture the decaying effect of the "broken heart" factor. The proposed models transparently and intuitively measure the extent of three types of dependence: the instantaneous dependence, the short-term impact of bereavement, and the long-term association between lifetimes. Some copula-based dependence measures, such as upper tail dependence, can also be derived from Markovian approaches. Very often, death is not the only mode of decrement. Entry into long-term care and voluntary prepayment, for instance, can affect reverse mortgage terminations. The semi-Markov joint-life model is extended to incorporate more exit modes, to model joint-life reverse mortgage termination speed. The event-triggered dependence between a husband and wife is modeled. For example, one spouse's death increases the survivor's inclination to move close to kin. We apply the proposed model specifically to develop the valuation formulas for roll-up mortgages in the UK and Home Equity Conversion Mortgages in the US. We test the significance of each termination mode and then use the model to investigate the mortgage insurance premiums levied on Home Equity Conversion Mortgage borrowers. Finally, this thesis extends the semi-Markov joint-life mortality model to having stochastic transition intensities, for modeling joint-life longevity risk in last-survivor annuities. We propose a natural extension of Gompertz' law to have correlated stochastic dynamics for its two parameters, and incorporate it into the semi-Markov joint-life mortality model. Based on this preliminary joint-life longevity model, we examine the impact of mortality improvement on the cost of a last survivor annuity, and investigate the market prices of longevity risk in last survivor annuities using risk-neutral pricing theory.

Zugl.: Münster (Westfalen), University, Diss., 2006.

Thesis (Ph. D.)--University of Texas Health Science Center at Houston, School of Public Health, 2008. / Source: Dissertation Abstracts International, Volume: 69-02, Section: B, page: 1086. Adviser: Wenyaw Chan. Includes bibliographical references.

Submitted by HUGO TADEU DELATORRE null (hugomath@uol.com.br) on 2016-02-21T19:31:03Z No. of bitstreams: 1 Aplicação das Cadeias de Markov no Ensino Médio - Hugo Tadeu Delatorre.pdf: 1086069 bytes, checksum: 6440bb8fa43488471cc064085e427f5e (MD5) / Approved for entry into archive by Ana Paula Grisoto (grisotoana@reitoria.unesp.br) on 2016-02-23T13:42:56Z (GMT) No. of bitstreams: 1 delatorre_ht_me_sjrp.pdf: 1086069 bytes, checksum: 6440bb8fa43488471cc064085e427f5e (MD5) / Made available in DSpace on 2016-02-23T13:42:56Z (GMT). No. of bitstreams: 1 delatorre_ht_me_sjrp.pdf: 1086069 bytes, checksum: 6440bb8fa43488471cc064085e427f5e (MD5) Previous issue date: 2016-01-22 / As Cadeias de Markov são um instrumento poderosíssimo para previsão de eventos do futuro baseado apenas em um passado relativamente recente. São muito utilizadas nas situações mais diversas como na bolsa de valores, na fidelização de clientes, previsão do tempo, dentre outras. O objetivo principal deste trabalho é mostrar, em nível de Ensino Médio algumas interessantes aplicações, bem como estimular os alunos à pesquisa, coleta e processamento de dados, mostrando-nos o quanto a Matemática está presente no cotidiano de cada um deles. / The Markov Chains are a powerful tool for predicting future events based only on a relatively recent past. They are widely used in different situations such as on the stock exchange in customer loyalty, weather, among others. The main objective of this work is to show high school level in some interesting applications, as well as encourage students to research, collecting and processing data, showing us how mathematics is present in the daily life of each one of them.

Submitted by Jean Akio Shida Hashioka (jeanhashioka@hotmail.com) on 2018-07-15T18:07:38Z No. of bitstreams: 1 JEAN AKIO SHIDA HASHIOKA - DISSERTAÇÃO.pdf: 1481230 bytes, checksum: 0f68b8e12c0a1f82eeaf421be75f5c17 (MD5) / Approved for entry into archive by Elza Mitiko Sato null (elzasato@ibilce.unesp.br) on 2018-07-17T13:30:41Z (GMT) No. of bitstreams: 1 hashioka_jas_me_sjrp.pdf: 1481230 bytes, checksum: 0f68b8e12c0a1f82eeaf421be75f5c17 (MD5) / Made available in DSpace on 2018-07-17T13:30:41Z (GMT). No. of bitstreams: 1 hashioka_jas_me_sjrp.pdf: 1481230 bytes, checksum: 0f68b8e12c0a1f82eeaf421be75f5c17 (MD5) Previous issue date: 2018-06-15 / Este trabalho consiste em apresentar a abordagem das cadeias de Markov como ferramenta auxiliadora na prática docente da matemática no Ensino Médio, tornando o processo mais tangível à realidade dos alunos. A contextualização dos conteúdos de matrizes, sistemas lineares e probabilidade poderá ser feita com exemplos práticos do cotidiano, considerando o meio social em que vivem os estudantes, resgatando assim o desejo pela aprendizagem e pelas aplicações da matemática. Espera-se desta forma maior receptividade da disciplina por parte dos discentes e, potencialmente, melhor resposta ao aprendizado pretendido. Assim sendo, o estudo aborda, num primeiro momento, a convergência de distribuição de probabilidade de uma cadeia de Markov de dois estados por meio de limites no in nito de uma função de probabilidade. Desta primeira cadeia de Markov de dois estados, é elaborado um roteiro de aula a ser abordado como exemplo a ser trabalhado em sala de aula relacionado às probabilidades de um time de futebol vencer as suas próximas partidas. Prosseguindo, observa-se a aplicabilidade das cadeias de Markov para calcular a distribuição de probabilidades de um jogador estar perdido em diferentes salas de um labirinto para cada tentativa de encontrar a saída. A m de evidenciar outro exemplo de aplicação das cadeias de Markov, há a construção de um modelo de preci cação de ativos com o objetivo de prever os preços de algumas ações de empresas negociadas na BM&FBOVESPA, a bolsa de valores do Brasil. Tal modelo de preci cação de ativos mostrou-se adequado estatisticamente como ferramenta de análise e cálculo dos retornos médios esperados de alguns dos ativos estudados. Por meio do conteúdo apresentado neste estudo, espera-se contribuir com o aprofundamento de alguns recursos e conceitos para a prática docente com aulas sobre cadeias de Markov no Ensino Médio. Esses aspectos direcionam esta pesquisa para um relevante processo de desenvolvimento do raciocínio, senso crítico e tomada de decisões em situações progressivamente mais complexas vividas pelos alunos. / This work presents the Markov chain approach as a useful tool in the teaching practice of mathematics in High School, making the process more tangible to the students' reality. The contextualization of matrix contents, linear systems and probability can be done with practical examples of daily life, considering the social environment in which students live, thus recovering the desire for learning and the applications of mathematics. It is expected in this way more receptivity of the discipline on the part of the students and, potentially, better response to the intended learning. Thus, the study addresses, rst, the convergence of probability distribution of a two-state Markov chain by means of in nite limits of a probability function. From this rst Markov chain of two states, a lesson script is elaborated to be approached as example to be worked in classroom related to the probabilities of a soccer team to win its next matches. Proceeding, we observe the applicability of Markov chains to calculate the probability distribution of a player being lost in di erent rooms of a maze for each attempt to nd the exit. In order to highlight another example of the application of the Markov chains, an asset pricing model is designed to predict the prices of some shares of companies traded on BM&FBOVESPA, the Brazilian stock exchange. Such an asset pricing model proved to be statistically adequate as a tool for analysis and calculation of the expected average returns of some of the assets studied. Through the content presented in this study, it is hoped to contribute with the deepening of some resources and concepts for the teaching practice with classes on Markov chains in High School. These aspects direct this research to a relevant process of development of reasoning, critical sense and decision making in progressively more complex situations experienced by students.

Orientador: José Gilberto Spasiani Rinaldi / Banca: Luiz Carlos Benini / Banca: Teresa Cristina Martins Dias / Resumo: As Cadeias de Markov são um instrumento poderosíssimo para previsão de eventos do futuro baseado apenas em um passado relativamente recente. São muito utilizadas nas situações mais diversas como na bolsa de valores, na fidelização de clientes, previsão do tempo, dentre outras. O objetivo principal deste trabalho é mostrar, em nível de Ensino Médio algumas interessantes aplicações, bem como estimular os alunos à pesquisa, coleta e processamento de dados, mostrando-nos o quanto a Matemática está presente no cotidiano de cada um deles / Abstract: The Markov Chains are a powerful tool for predicting future events based only on a relatively recent past. They are widely used in different situations such as on the stock exchange in customer loyalty, weather, among others. The main objective of this work is to show high school level in some interesting applications, as well as encourage students to research, collecting and processing data, showing us how mathematics is present in the daily life of each one of them / Mestre

A fibrilação atrial (FA) é um dos tipos mais freqüentes de arritmia cardíaca e é caracterizada principalmente pela aleatoriedade na ocorrência dos batimentos do coração. Sob este aspecto, a fibrilação atrial pode ser considerada um processo estocástico e por isso tem sido freqüentemente modelada através de cadeias de Markov. Seguindo trabalhos anteriores sobre este tópico, este trabalho modela seqüências temporais de batimentos cardíacos como um processo markoviano de três estados para detecção automática de FA. O modelo foi treinado e desenvolvido através dos sinais da base de dados MIT-BIH. Outro método mais consolidado na detecção de FA, denominado \"Razão RR\", também foi implementado, com o objetivo de comparar os resultados do Modelo Markoviano. A avaliação de desempenho para ambos os métodos implementados fo i realizada medindo-se a sensibilidade (Se) e o valor preditivo positivo (+P) para a detecção de FA. Estes dois métodos - Modelos Markovianos e \"Razão RR\" - tiveram seus coeficientes e limiares otimizados com o objetivo de maximizar, ao mesmo tempo, os valores de Se e +P. Após a otimização, ambos os métodos foram testados com uma nova base de dados, independente da base de dados de desenvolvimento. Os resultados obtidos com a base de dados de teste foram Se=84,940% e +P=81,579%, consolidando os Modelos Markoviano s para detecção de batimentos aleatórios. / Atrial fibrillation (AF) is one of the most common cardiac arrhythmia and it is mainly characterized by the presence of random RR intervals. In this way, atrial fibrillation has been studied as a stochastic process and it has been often modeled through Markov chains. Following previous studies on this subject, this work models time sequences of heartbeats as a three states Markov process for AF automatic detection. The model was trained and developed using signals from MIT-BIH database. Another consolidated method for AF detection, called \"RR Ratios\", was also applied to compare Markov Model\'s results. The performance evaluation of both methods was measured through sensitivity (Se) and positive predictive (+P) for AF detection. These two methods - Markov Model and \"RR Ratio\" - had their coefficients and thresholds optimized in order to maximize the values of Se and +P at the same time. After optimization, both methods were tested with another database, independent of development database. The obtained results were Se = 84,940% and +P = 81,579%, consolidating Markov Models for detecting random heartbeats.