Dependence in compound distributions - with insurance applications

这个博士项目的目的是研究复合分布中随机相关性的影响。在保险的背景下，让Y = X1 +. + XN为索赔总额，其中Xi为单项索赔额，N为索赔件数。这是对复合分布的解释，复合分布是随机变量的随机数之和。复合分布是一种非常强大的数学工具，在真实的世界中有许多应用，特别是在保险索赔中。例如，在进行索赔时，我们通常假设每个索赔的金额Xi是独立和同分布的，遵循一个指定的分布，如泊松。在最简单的情况下，这些假设使我们能够相对容易地计算总索赔额S的统计量，在更复杂的情况下，我们可以使用中心极限定理和其他类似的渐近结果来近似这些统计量，但是，独立性假设有实际的限制。这是因为在声明之间可能会出现依赖性。这种依赖性的例子包括保险周期和经济因素，最明显的是冠状病毒大流行。因此，重要的是要看看索赔之间的依赖关系，因为它将使保险公司能够满足偿付能力II法规。这个项目将着重于使用方差和协方差矩阵以及Copula来讨论和测量随机相关性。使用极限定理，如中心极限定理和极值定理在确定单变量情况下的尾部依赖和条件copula在多变量情况下，将使随机依赖的准确模拟。这些模拟将允许敏感性分析，使用蒙特卡罗方法，以确定随机依赖性对索赔的影响。因此，索赔之间的相关性的影响，高斯近似的中心极限定理的质量可以调查。

The aim of this PhD project is to investigate the effects of stochastic dependency in compound distribution. In an insurance context, let Y = X1 + ... + XN be the total claim amount, where the Xi are the individual claim amounts and N is the number of claims. This is the interpretation of a compound distribution, which is a sum of a random number of random variables. The compound distribution is a very powerful mathematical tool that has many real world applications, in particular to insurance claims. For example, when making a claim, we often assume the amount of each individual claim Xi are independent and identically distributed, follows a specified distribution such as Poisson. In the simplest cases these assumptions allow us to calculate statistics of the aggregate claims S with relative ease, and in more complicated situations we can use central limit theorems and other such asymptotic results to approximate these statistics.However, there are practical limitations to the independence assumption. This is because dependencies can arise between claims. Examples of such dependencies include insurance cycles, and economic factors most noticeably the coronavirus pandemic. Hence, it is important to look at the dependency between claims, as it will enable insurers to meet the Solvency II regulations. The discussion and measurement of stochastic dependence that this project willfocus on is using variance and co-variance matrices, and copulas. Using limiting theorems, such as the central limit theorem and extreme value theorem in determining tail dependencies in the uni-variate case and conditional copulas in the multivariate case, will enable an accurate simulation of stochastic dependence. These simulations will allow for a sensitivity analysis, using Monte Carlo methods, to determine the impact of stochastic dependency onclaims. As a result, the impact of correlation between claims on the quality of Gaussian approximation in the central limit theorem can be investigated.