• 主题
高级搜索  >
历史搜索 清空历史
首页> 外文学位 >A class of bivariate Erlang distributions and ruin probabilities in multivariate risk models.
【24h】

A class of bivariate Erlang distributions and ruin probabilities in multivariate risk models.

机译：多元风险模型中的一类二元Erlang分布和破产概率。

获取原文
获取原文并翻译 | 示例
页面导航
  • 摘要
  • 著录项
  • 相似文献
  • 相关主题

摘要

This dissertation is devoted to modeling dependence with potential applications in actuarial science and is divided in two parts: the first part considers dependence in the context of bivariate survival data analysis and the second, related to risk theory, deals with dependence between classes of an insurance business.;The first part is presented in the form of a research paper in Chapter 3. In this contribution, we introduce a new class of bivariate distributions of Marshall-Olkin type, called bivariate Erlang distributions. It is shown that the bivariate Erlang distribution has both an absolutely continuous and a singular part. The Laplace transform, product moments and conditional densities are derived and also, the finite mixture of the bivariate Erlang distributions is described. Potential applications of bivariate Erlang distributions in life insurance and finance are considered. The maximum likelihood estimators of the parameters are computed via an Expectation-Maximization algorithm. Simulations are carried out to measure the performance of the estimator.;The second part related to risk theory is presented in Chapters 4 and 5 of this thesis and is devoted to the study of multivariate risk processes, which may be useful in analyzing ruin problems for insurance companies with a portfolio of dependent classes of business. We apply results from the theory of piecewise deterministic Markov processes in order to derive exponential martingales needed to establish computable upper bounds for the ruin probabilities, as their exact expressions are intractable.;As an extension of the multivariate risk model proposed by Asmussen and Albrecher (2010), we first consider an m-dimensional risk process obtained by modeling the dependence through the number of claims using the Poisson model with common shocks. We assume that in addition to the individual shocks, a common shock affects all classes of business and that another common shock has an impact on each pair of classes. Also, dependence between claims sizes across classes is allowed. The asymptotic behavior of the the probability that ruin occurs in all classes simultaneously before a fixed time t, in both cases of dependent heavy-tailed claims and independent heavy-tailed claims, is investigated.;Inspired by the work of Dufresne and Gerber (1991) and of Li, Liu and Tang (2007), we embrace the idea of adding a diffusion process characterized by an m-dimensional correlated Brownian motion.;For each of these two multivariate models an expression for the probability that ruin occurs in at least one class of business and an upper bound for the probability that ruin occurs in all classes simultaneously are derived. Numerical results regarding the upper bounds are reported for these models assuming three classes of insurance business, where the dependence between claims sizes is modeled using the notion of copula. It is established that adding a diffusion process leads to increasing these upper bounds.;Further, in a more realistic setting, our research project is outlined by investigating ruin probabilities associated to an m-dimensional risk process which assumes that in addition to the individual claim arrivals for each class of business, governed by Poisson processes, there are aggregate claims produced by a common renewal counting process that affects all classes of business.;In this multivariate context, the surplus vector process is Markovianized by introducing a supplementary process, and tools from the theory of piecewise deterministic Markov processes are applied in order to obtain exponential martingales. Based on these martingales, we derive an upper bound for the probability that ruin occurs in all classes simultaneously. Also, an upper bound for this type of ruin probability is derived in a special case where the individual shocks are absent and the claims across classes are produced only by the renewal process. The latter upper bound is illustrated by numerical results, where a bivariate version is considered and the dependence in claim sizes is captured using copula techniques.;Keywords: Erlang distribution, Expectation-Maximization algorithm, Piecewise deterministic Markov processes, Multivariate risk model, Ruin probability, Poisson model with common shocks, Renewal processes, Copulas.
机译：本文致力于对保险精算中潜在应用的依赖关系进行建模，分为两部分：第一部分在双变量生存数据分析的背景下考虑依赖关系，第二部分与风险理论相关，处理保险类别之间的依赖关系第一部分以研究论文的形式在第3章中进行介绍。在此贡献中，我们介绍了一类新的Marshall-Olkin类型的双变量分布，称为双变量Erlang分布。结果表明，二元Erlang分布具有绝对连续和奇异部分。推导了Laplace变换，乘积矩和条件密度，并描述了二元Erlang分布的有限混合。考虑了二元Erlang分布在人寿保险和金融中的潜在应用。参数的最大似然估计器是通过Expectation-Maximization算法计算的。进行了仿真计算，以评估估计器的性能。第二部分与风险理论相关，在本论文的第4章和第5章中进行了介绍，致力于研究多元风险过程，这可能有助于分析破产的破产问题。具有相关业务类别投资组合的保险公司。我们应用分段确定性马尔可夫过程理论的结果，以推导建立毁灭概率的可计算上限所需的指数as，因为它们的精确表达是难解的。;作为Asmussen和Albrecher提出的多元风险模型的扩展（ 2010年），我们首先考虑一个 m 风险过程，该过程是通过使用具有常见冲击的Poisson模型通过索赔数量对依赖关系进行建模而获得的。我们假设，除了个别冲击外，共同冲击还会影响所有业务类别，并且另一共同冲击会影响每对业务类别。同样，允许跨类别的索赔大小之间的依赖性。在从属重尾索赔和独立重尾索赔的情况下，都研究了在固定时间 t 之前，所有类别同时发生破坏的概率的渐近行为。在Dufresne和Gerber（1991）以及Li，Liu和Tang（2007）的工作中，我们接受了添加一个以 m 维相关的布朗运动为特征的扩散过程的想法。通过两个多元模型，得出了至少一类业务中发生破产的概率的表达式，并得出了所有类中同时发生破产的概率的上限。在假设保险业务分为三类的情况下，报告了有关这些模型的上限的数值结果，其中索赔大小之间的依存关系使用copula概念建模。建立扩散过程会导致这些上限的增加。此外，在更现实的环境中，我们的研究项目通过研究与 m 维风险过程相关的破产概率来概述，该假设假定在Poisson流程的控制下，除了每种业务类别的单个索偿到达之外，通用续签计数流程还产生了影响所有业务类别的合计索偿。;在这种多变量情况下，剩余向量流程是Markovian化的引入补充过程，并应用分段确定性马尔可夫过程理论中的工具以获得指数mar。基于这些mar，我们得出了所有类别同时发生破坏的概率的上限。同样，在不存在个别冲击并且仅通过续签过程产生跨类索赔的特殊情况下，得出了这种破产概率的上限。数值结果说明了后者的上限，其中考虑了双变量版本，并使用copula技术捕获了索赔金额的依存关系。关键字： Erlang分布，期望最大化算法，分段确定性马尔可夫过程，多元风险模型，破产概率，具有常见冲击的泊松模型，更新过程，Copulas。

著录项

  • 作者

    Groparu-Cojocaru, Ionica.;

  • 作者单位

    Universite de Montreal (Canada).;

  • 授予单位 Universite de Montreal (Canada).;
  • 学科 Education Mathematics.
  • 学位 Ph.D.
  • 年度 2013
  • 页码 216 p.
  • 总页数 216
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类 肿瘤学;
  • 关键词

相似文献

  • 外文文献
  • 中文文献
  • 专利
获取原文

客服邮箱：kefu@zhangqiaokeyan.com

京公网安备：11010802029741号 ICP备案号：京ICP备15016152号-6 六维联合信息科技 (北京) 有限公司©版权所有

  • 客服微信

  • 服务号