Risk models with dependence: construction, properties, and risk measurement

保险公司和金融机构必须开发有效的风险管理工具，以便对与风险组合相关的全球风险进行适当评估。在精算学中，有效的风险管理有赖于适当的风险模型、通过总索赔额分布概括投资组合全球风险的适当风险度量以及计算这种分布的有效方法。根据特定的分配规则，投资组合的组成部分对风险衡量的贡献是另一个关键因素。除了对单个风险进行建模外，找到描述它们相互关系的适当依赖结构的问题也是具有挑战性的。 我未来几年的研究计划是提出和研究具有相关性的风险模型，并在这些模型中定义适当的风险度量，研究它们的性质，并提出计算它们的方法。这些创新将更适合精算师的现代任务。在短期内，我将提出具有相依性的单周期风险模型，并考察这些模型中的风险聚集和风险分配。为此，对于具有大量风险的投资组合，我将提出新的方法来建立分层Copula，并研究它们的性质和估计。我还将介绍基于泊松分布及其扩展的多元计数分布。然后，我将定义和研究具有不同依赖结构的集体风险模型，这些模型将索赔数量与每个索赔金额以及每个索赔金额与其他索赔金额联系起来。我将使用等级联结来定义这样的结构。然后，我将研究基于破产的风险度量的理想性质，如齐性、次可加性、凸性和关于随机过程的不同随机序的一致性。值得注意的是，我将研究定义为伦德伯格系数的乘法逆的风险指数的性质。我将引入离散时间风险模型，既允许时间相关性，也允许业务之间的相关性。我将研究这两种形式的依赖对基于破产的风险度量的影响。我还将研究基于欧拉原理的风险分配。 我的研究计划承诺将对精算学中相关风险模型的构建和研究产生重大影响，从而在精算学和相关领域的顶级期刊上发表论文。从长远来看，我预计我的研究计划目标的实现将对精算科学的重要进步和创新做出重大贡献。我的研究计划也将为HQP提供极好的培训机会。

It is crucial for insurers and financial institutions to develop efficient risk management tools to have an appropriate assessment of the global risk associated to a portfolio of risks. In actuarial science, effective risk management relies on appropriate risk models, adequate risk measures that summarize the global risk of the portfolio through the distribution of the aggregate claim amount, and efficient methods to compute such a distribution. Contributions of the components of the portfolio to the risk measure according to a specific allocation rule is another key element. Besides modeling individual risks, the problem of finding the appropriate dependence structure describing their interrelations is challenging. My research program over the next few years is to propose and study risk models with dependence and, within them, to define appropriate risk measures, study their properties and propose methods to compute them. These innovations will be better suited for modern tasks of actuaries. In short-term, I will propose single-period risk models with dependence, and examine risk aggregation and risk allocation in those models. To this end, for portfolios with a large number of risks, I will propose new approaches to build hierarchical copulas, and investigate their properties and their estimation. I will also introduce multivariate counting distributions, based on the Poisson distribution and its extensions. Then, I will define and investigate collective risk models with different dependence structures that link the number of claims to each claim amount and each claim amount to the others. I will use hierarchical copulas to define such structures. Afterwards, I will examine desirable properties for ruin-based risk measures such as homogeneity, subadditivity, convexity, and consistency regarding different stochastics orders for stochastic processes. Notably, I will investigate the properties of an index of riskiness defined as the multiplicative inverse of the Lundberg coefficient. I will introduce discrete-time risk models, allowing both temporal dependence and dependence between lines of business. I will examine the impact of these two forms of dependence on ruin-based risk measures. I will also investigate risk allocation based on Euler's principle. My research program promises to have a significant impact on the construction and investigation of dependent risk models in actuarial science, leading to contributions in top journals in actuarial science and related fields. In the long term, I anticipate that the accomplishment of the objectives of my research program will significantly contribute to important advances and innovations in actuarial science. My research program will also offer excellent training opportunities for HQP.