Techn. Hochsch., Diss., 2002--Aachen.

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Two different sets of problems are addressed in this thesis. The first one is on partially observed semi-Markov Games (POSMG) and the second one is on semi-Markov modulated financial market model. In this thesis we study a partially observable semi-Markov game in the infinite time horizon. The study of a partially observable game (POG) involves three major steps: (i) construct an equivalent completely observable game (COG), (ii) establish the equivalence between POG and COG by showing that if COG admits an equilibrium, POG does so, (iii) study the equilibrium of COG and find the corresponding equilibrium of original partially observable problem. In case of infinite time horizon game problem there are two different payoff criteria. These are discounted payoff criterion and average payoff criterion. At first a partially observable semi-Markov decision process on general state space with discounted cost criterion is studied. An optimal policy is shown to exist by considering a Shapley’s equation for the corresponding completely observable model. Next the discounted payoff problem is studied for two-person zero-sum case. A saddle point equilibrium is shown to exist for this case. Then the variable sum game is investigated. For this case the Nash equilibrium strategy is obtained in Markov class under suitable assumption. Next the POSMG problem on countable state space is addressed for average payoff criterion. It is well known that under this criterion the game problem do not have a solution in general. To ensure a solution one needs some kind of ergodicity of the transition kernel. We find an appropriate ergodicity of partially observed model which in turn induces a geometric ergodicity to the equivalent model. Using this we establish a solution of the corresponding average payoff optimality equation (APOE). Thus the value and a saddle point equilibrium is obtained for the original partially observable model. A value iteration scheme is also developed to find out the average value of the game. Next we study the financial market model whose key parameters are modulated by semi-Markov processes. Two different problems are addressed under this market assumption. In the first one we show that this market is incomplete. In such an incomplete market we find the locally risk minimizing prices of exotic options in the Follmer Schweizer framework. In this model the stock prices are no more Markov. Generally stock price process is modeled as Markov process because otherwise one may not get a pde representation of price of a contingent claim. To overcome this difficulty we find an appropriate Markov process which includes the stock price as a component and then find its infinitesimal generator. Using Feynman-Kac formula we obtain a system of non-local partial differential equations satisfied by the option price functions in the mildsense. .Next this system is shown to have a classical solution for given initial or boundary conditions. Then this solution is used to have a F¨ollmer Schweizer decomposition of option price. Thus we obtain the locally risk minimizing prices of different options. Furthermore we obtain an integral equation satisfied by the unique solution of this system. This enable us to compute the price of a contingent claim and find the risk minimizing hedging strategy numerically. Further we develop an efficient and stable numerical method to compute the prices. Beside this work on derivative pricing, the portfolio optimization problem in semi-Markov modulated market is also studied in the thesis. We find the optimal portfolio selections by optimizing expected utility of terminal wealth. We also obtain the optimal portfolio selections under risk sensitive criterion for both finite and infinite time horizon.

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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.

Submitted by Irene Nascimento (irene.kessia@ufpe.br) on 2015-03-11T19:10:26Z No. of bitstreams: 2 DISSERTAÇÃO Flávio Leandro Alves dos Santos.pdf: 3045919 bytes, checksum: 6187bd40862e0817909589819cf96483 (MD5) license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) / Made available in DSpace on 2015-03-11T19:10:26Z (GMT). No. of bitstreams: 2 DISSERTAÇÃO Flávio Leandro Alves dos Santos.pdf: 3045919 bytes, checksum: 6187bd40862e0817909589819cf96483 (MD5) license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Previous issue date: 2013-04-10 / O presente trabalho propõe uma metodologia que permite obter métricas de Disponibilidade e Indisponibilidade de um funcionário que trabalha em uma das seis regiões de atuação de uma Companhia de geração de energia elétrica, através da integração entre o processo semi-Markoviano (PSM) e Análise de variabilidade populacional Bayesiana. A Análise de variabilidade populacional Bayesiana é um método para se chegar a uma distribuição a priori para avaliação Bayesiana dos parâmetros de confiabilidade baseado em dados parcialmente relevante. Já o processo semi-Markoviano pode ser visto como um processo cujas sucessivas transições de estados são governadas pelas probabilidades de transição do processo Markoviano (PM), mas sua permanência em qualquer estado é descrita por uma variável aleatória que depende do estado atual ocupado e do estado em que a próxima transição será feita. A integração do PSM e da Análise de variabilidade populacional Bayesiana origina um modelo híbrido o qual é capaz de representar o comportamento do trabalhador que sofre diversos tipos de acidentes do trabalho com diferentes tempos de recuperação e taxas de falhas. Diante deste contexto, será utiliza a Análise de variabilidade populacional Bayesiana para a estimação da distribuição da taxa de acidentes e de recuperação para o processo semi- Markoviano. Após aplicação da metodologia, é exposto métricas de um operário em cada uma das seis regiões de atuação da companhia elétrica como: Tempo operacional médio,Disponibilidade média, Disponibilidade instantânea ao final da missão, Tempo falho médio, Indisponibilidade média, Indisponibilidade instantânea ao final da missão e a Probabilidade do funcionário não acidentado e acidentado por acidente de trabalho.

The growing requirements of a diversity of applications necessitate the deployment of large and powerful computing systems and failures in these systems may cause severe damage in every aspect from loss of human lives to world economy. However, current fault tolerance techniques cannot meet the increasing requirements for reliability. Thus new solutions are urgently needed and research on proactive schemes is one of the directions that may offer better efficiency. This thesis proposes a novel proactive failure management framework. Its goal is to reduce the failure penalties and improve fault tolerance efficiency in supercomputers when running complex applications. The proposed proactive scheme builds on two core components: failure prediction and proactive failure recovery. More specifically, the failure prediction component is based on the assessment of system events and employs semi-Markov models to capture the dependencies between failures and other events for the forecasting of forthcoming failures. Furthermore, a two-level failure prediction strategy is described that not only estimates the future failure occurrence but also identifies the specific failure categories. Based on the accurate failure forecasting, a prediction-based coordinated checkpoint mechanism is designed to construct extra checkpoints just before each predicted failure occurrence so that the wasted computational time can be significantly reduced. Moreover, a theoretical model has been developed to assess the proactive scheme that enables calculation of the overall wasted computational time.The prediction component has been applied to industrial data from the IBM BlueGene/L system. Results of the failure prediction component show a great improvement of the prediction accuracy in comparison with three other well-known prediction approaches, and also demonstrate that the semi-Markov based predictor, which has achieved the precision of 87.41% and the recall of 77.95%, performs better than other predictors.

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In recent times, credit risk analysis has grown to become one of the most important problems dealt with in the mathematical finance literature. Fundamentally, the problem deals with estimating the probability that an obligor defaults on their debt in a certain time. To obtain such a probability, several methods have been developed which are regulated by the Basel Accord. This establishes a legal framework for dealing with credit and market risks, and empowers banks to perform their own methodologies according to their interests under certain criteria. Credit risk analysis is founded on the rating system, which is an assessment of the capability of an obligor to make its payments in full and on time, in order to estimate risks and make the investor decisions easier.Credit risk models can be classified into several different categories. In structural form models (SFM), that are founded on the Black & Scholes theory for option pricing and the Merton model, it is assumed that default occurs if a firm's market value is lower than a threshold, most often its liabilities. The problem is that this is clearly is an unrealistic assumption. The factors models (FM) attempt to predict the random default time by assuming a hazard rate based on latent exogenous and endogenous variables. Reduced form models (RFM) mainly focus on the accuracy of the probability of default (PD), to such an extent that it is given more importance than an intuitive economical interpretation. Portfolio reduced form models (PRFM) belong to the RFM family, and were developed to overcome the SFM's difficulties.Most of these models are based on the assumption of having an underlying Markovian process, either in discrete or continuous time. For a discrete process, the main information is containted in a transition matrix, from which we obtain migration probabilities. However, according to previous analysis, it has been found that this approach contains embedding problems. The continuous time Markov process (CTMP) has its main information contained in a matrix Q of constant instantaneous transition rates between states. Both approaches assume that the future depends only on the present, though previous empirical analysis has proved that the probability of changing rating depends on the time a firm maintains the same rating. In order to face this difficulty we approach the PD with the continuous time semi-Markov process (CTSMP), which relaxes the exponential waiting time distribution assumption of the Markovian analogue.In this work we have relaxed the constant transition rate assumption and assumed that it depends on the residence time, thus we have derived CTSMP forward integral and differential equations respectively and the corresponding equations for the particular cases of exponential, gamma and power law waiting time distributions, we have also obtained a numerical solution of the migration probability by the Monte Carlo Method and compared the results with the Markovian models in discrete and continuous time respectively, and the discrete time semi-Markov process. We have focused on firms from U.S.A. and Canada classified as financial sector according to Global Industry Classification Standard and we have concluded that the gamma and Weibull distribution are the best adjustment models.