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This dissertation includes two topics. For the first topic about the housing price jump risk, we use EM gradient algorithms to estimate parameters of the jump diffusion model and test whether the US monthly housing price have jump risk during 1986 to 2006. Then, in order to obtain a viable pricing framework of mortgage insurance contracts, this paper uses the jump diffusion processes of Merton (1976) to model the dynamic process of housing price. Using this model, we investigate the impact of price jump risk on the valuation of mortgage insurance premium from jump intensity, abnormal volatility of jump size and normal volatility. Empirical results indicate that the abnormal volatility of jump size has the most significant impact on the mortgage insurance premium. For the second topic about the catastrophe risk, we investigate that, for catastrophic events, the assumption that catastrophe claims occur in terms of the Poisson process seems inadequate as it has constant intensity. We propose Markov Modulated Poisson process to model the arrival process for catastrophic events. Under this process, the underlying state is governed by a homogenous Markov chain, and it is the generalization of Cummins and Geman (1993, 1995), Chang, Chang, and Yu (1996), Geman and Yor (1997) and Vaugirard (2003a, 2003b). We apply Markov jump diffusion model to derive pricing formulas for catastrophe insurance products, included catastrophe futures call option, catastrophe PCS call spread and catastrophe bond. We use the data of PCS index and the annual number of hurricane events during 1950 to 2004 to test the quality of the fitting under the Markov Modulated Poisson process and the Poisson process. We reach the conclusion that the Markov Modulated Poisson process is fitter than the Poisson process and Weiner process in modeling the arrival rate of hurricane events when pricing three insurance products. Hence, if different status of climate environment has significant different arrival intensity in real economy, using jump diffusion model to evaluate CAT insurance products could cause significant mispricing.

Traffic modeling in computer networks has been researched for decades. A good model should reflect the features of real-world network traffic. With a good model, synthetic traffic data can be generated for experimental studies; network performance can be analysed mathematically; service provisioning and scheduling can be designed aligning with traffic changes. An important part of traffic modeling is to capture the dependence, either the dependence among different traffic flows or the temporal dependence within the same traffic flow. Nevertheless, the power of dependence models, especially those that capture the functional dependence, has not been fully explored in the domain of computer networks. This thesis studies copula theory, a theory to describe dependence between random variables, and applies it for better performance evaluation and network resource provisioning. We apply copula to model both contemporaneous dependence between traffic flows and temporal dependence within the same flow. The dependence models are powerful and capture the functional dependence beyond the linear scope. With numerical examples, real-world experiments and simulations, we show that copula modeling can benefit many applications in computer networks, including, for example, tightening performance bounds in statistical network calculus, capturing full dependence structure in Markov Modulated Poisson Process (MMPP), MMPP parameter estimation, and predictive resource provisioning for cloud-based composite services. / Graduate / 0984 / fdong@uvic.ca

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本論文中，我們提出基於機場安全檢查的分層排隊理論模型，模型中的旅客基於歷史的安全數據被分成三組。我們運用二維馬可夫過程（two-dimensional Markov process）以及馬可夫調控卜瓦松過程（Markov modulated Poisson process）構建模型的排隊系統並加以分析。我們收集了台灣桃園國際機場和其它兩個機場的旅客數據以驗證我們提出的模型，並運用模擬退火法（simulated annealing）求得近似最佳解（near-optimum solution）。最後我們通過模型的旅客平均等候時間和另外兩種等候模型進行比較，之後得出我們的模型確實可以在不增加成本，甚至提升安全性的同時能夠有效地減少平均等候時間。 / This thesis proposes a tiered inspection system for airport security, wherein passengers are divided into three classes based on historical security records. A two-dimensional Markov process and a Markov modulated Poisson process (MMPP) queue were used in the formulation of the security inspection system. Simulated annealing was then used to obtain near-optimum solution for the model. The efficacy of the proposed model was evaluated using the arrival data of passengers at Taoyuan International Airport and other two international airports. A comparison with two conventional queueing models with regard to the average waiting time demonstrated the effectiveness of the proposed security inspection system in enhancing service efficiency and boosting the level of security.

Along with playing an ever-increasing role in the integration of other communication networks and expanding in application diversities, the current Internet suffers from serious overuse and congestion bottlenecks. Efficient congestion control is fundamental to ensure the Internet reliability, satisfy the specified Quality-of-Service (QoS) constraints and achieve desirable performance in response to varying application scenarios. Active Queue Management (AQM) is a promising scheme to support end-to-end Transmission Control Protocol (TCP) congestion control because it enables the sender to react appropriately to the real network situation. Analytical performance models are powerful tools which can be adopted to investigate optimal setting of AQM parameters. Among the existing research efforts in this field, however, there is a current lack of analytical models that can be viewed as a cost-effective performance evaluation tool for AQM in the presence of heterogeneous traffic, generated by various network applications. This thesis aims to provide a generic and extensible analytical framework for analyzing AQM congestion control for various traffic types, such as non-bursty Poisson and bursty Markov-Modulated Poisson Process (MMPP) traffic. Specifically, the Markov analytical models are developed for AQM congestion control scheme coupled with queue thresholds and then are adopted to derive expressions for important QoS metrics. The main contributions of this thesis are listed as follows: • Study the queueing systems for modeling AQM scheme subject to single-class and multiple-classes Poisson traffic, respectively. Analyze the effects of the varying threshold, mean traffic arrival rate, service rate and buffer capacity on the key performance metrics. • Propose an analytical model for AQM scheme with single class bursty traffic and investigate how burstiness and correlations affect the performance metrics. The analytical results reveal that high burstiness and correlation can result in significant degradation of AQM performance, such as increased queueing delay and packet loss probability, and reduced throughput and utlization. • Develop an analytical model for a single server queueing system with AQM in the presence of heterogeneous traffic and evaluate the aggregate and marginal performance subject to different threshold values, burstiness degree and correlation. • Conduct stochastic analysis of a single-server system with single-queue and multiple-queues, respectively, for AQM scheme in the presence of multiple priority traffic classes scheduled by the Priority Resume (PR) policy. • Carry out the performance comparison of AQM with PR and First-In First-Out (FIFO) scheme and compare the performance of AQM with single PR priority queue and multiple priority queues, respectively.

Along with playing an ever-increasing role in the integration of other communication networks and expanding in application diversities, the current Internet suffers from serious overuse and congestion bottlenecks. Efficient congestion control is fundamental to ensure the Internet reliability, satisfy the specified Quality-of-Service (QoS) constraints and achieve desirable performance in response to varying application scenarios. Active Queue Management (AQM) is a promising scheme to support end-to-end Transmission Control Protocol (TCP) congestion control because it enables the sender to react appropriately to the real network situation. Analytical performance models are powerful tools which can be adopted to investigate optimal setting of AQM parameters. Among the existing research efforts in this field, however, there is a current lack of analytical models that can be viewed as a cost-effective performance evaluation tool for AQM in the presence of heterogeneous traffic, generated by various network applications. This thesis aims to provide a generic and extensible analytical framework for analyzing AQM congestion control for various traffic types, such as non-bursty Poisson and bursty Markov-Modulated Poisson Process (MMPP) traffic. Specifically, the Markov analytical models are developed for AQM congestion control scheme coupled with queue thresholds and then are adopted to derive expressions for important QoS metrics. The main contributions of this thesis are listed as follows: iii ¿ Study the queueing systems for modeling AQM scheme subject to single-class and multiple-classes Poisson traffic, respectively. Analyze the effects of the varying threshold, mean traffic arrival rate, service rate and buffer capacity on the key performance metrics. ¿ Propose an analytical model for AQM scheme with single class bursty traffic and investigate how burstiness and correlations affect the performance metrics. The analytical results reveal that high burstiness and correlation can result in significant degradation of AQM performance, such as increased queueing delay and packet loss probability, and reduced throughput and utlization. ¿ Develop an analytical model for a single server queueing system with AQM in the presence of heterogeneous traffic and evaluate the aggregate and marginal performance subject to different threshold values, burstiness degree and correlation. ¿ Conduct stochastic analysis of a single-server system with single-queue and multiple-queues, respectively, for AQM scheme in the presence of multiple priority traffic classes scheduled by the Priority Resume (PR) policy. ¿ Carry out the performance comparison of AQM with PR and First-In First-Out (FIFO) scheme and compare the performance of AQM with single PR priority queue and multiple priority queues, respectively.

為了改進Black-Scholes模式的實證現象，許多其他的模型被建議有leptokurtic特性以及波動度聚集的現象。然而對於其他的模型分析的處理依然是一個問題。在本論文中，我們建議使用馬可夫跳躍擴散過程，不僅能整合leptokurtic與波動度微笑特性，而且能產生波動度聚集的與長記憶的現象。然後，我們應用Lucas的一般均衡架構計算選擇權價格，提供均衡下當跳躍的大小服從一些特別的分配時則選擇權價格的解析解。特別地，考慮當跳躍的大小服從兩個情況，破產與lognormal分配。當馬可夫跳躍擴散模型的馬可夫鏈有兩個狀態時，稱為轉換跳躍擴散模型，當跳躍的大小服從lognormal分配我們得到選擇權公式。使用轉換跳躍擴散模型選擇權公式，我們給定一些參數下研究公式的數值極限分析以及敏感度分析。 / To improve the empirical performance of the Black-Scholes model, many alternative models have been proposed to address the leptokurtic feature of the asset return distribution, and the effects of volatility clustering phenomenon. However, analytical tractability remains a problem for most of the alternative models. In this dissertation, we propose a Markov jump diffusion model, that can not only incorporate both the leptokurtic feature and volatility smile, but also present the economic features of volatility clustering and long memory. Next, we apply Lucas's general equilibrium framework to evaluate option price, and to provide analytical solutions of the equilibrium price for European call options when the jump size follows some specific distributions. In particular, two cases are considered, the defaultable one and the lognormal distribution. When the underlying Markov chain of the Markov jump diffusion model has two states, the so-called switch jump diffusion model, we write an explicit analytic formula under the jump size has a lognormal distribution. Numerical approximations of the option prices as well as sensitivity analysis are also given.