Dependence in compound distributions - with insurance applications

这个博士项目的目的是调查随机相依性在复合分布中的影响。在保险上下文中，设Y=X1+...+XN是总索赔金额，其中Xi是个人索赔金额，N是索赔数量。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。复合分布是一个非常强大的数学工具，它在现实世界中有许多应用，特别是在保险索赔方面。例如，当提出索赔时，我们经常假设每个索赔的金额xi是独立的、同分布的，服从指定的分布，如泊松分布。在最简单的情况下，这些假设使我们能够相对容易地计算出S索赔总额的统计量，而在更复杂的情况下，我们可以使用中心极限定理和其他类似的渐近结果来逼近这些统计量。然而，独立性假设存在实际限制。这是因为声明之间可能会出现依赖关系。这种依赖的例子包括保险周期和经济因素，最引人注目的是冠状病毒大流行。因此，重要的是要研究索赔之间的相关性，因为这将使保险公司能够满足偿付能力II的规定。本项目将关注的随机相关性的讨论和测量是使用方差和协方差矩阵以及联系数。使用极限定理，如中心极限定理和极值定理，在单变量情况下确定尾部相关性，在多变量情况下使用条件Copula，将使随机相关性的准确模拟成为可能。这些模拟将允许使用蒙特卡罗方法进行敏感性分析，以确定随机相依性对索赔的影响。因此，可以研究索赔之间的相关性对中心极限定理中的高斯近似的质量的影响。

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.