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，对与风险组合相关的全球风险进行适当评估。在精算学中，有效的风险管理依赖于适当的风险模型，通过总索赔额的分布总结投资组合的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度量的贡献是另一个关键因素。除了对个体风险进行建模之外，找到描述其相互关系的适当依赖结构的问题也具有挑战性。 我在未来几年的研究计划是提出和研究具有依赖性的风险模型，并在其中定义适当的风险度量，研究其属性并提出计算方法。这些创新将更适合精算师的现代任务。在短期内，我将提出单期相依风险模型，并研究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构建分层copula，并研究其性质和估计。我还将介绍基于泊松分布及其扩展的多变量计数分布。然后，我将定义和研究具有不同依赖结构的集体风险模型，这些依赖结构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。我将使用层次连接函数来定义这样的结构。之后，我将研究理想的性质，如同质性，次可加性，凸性，并就不同的随机过程的随机顺序的一致性，以破产为基础的风险措施。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属性。我将介绍离散时间风险模型，允许时间依赖和业务线之间的依赖。我将研究这两种依赖形式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。 我的研究计划有望对精算学中相关风险模型的构建和研究产生重大影响，从而在精算学及相关领域的顶级期刊上做出贡献。从长远来看，我预计我的研究计划的目标的实现将大大有助于精算科学的重要进步和创新。 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.