Dependence in compound distributions - with insurance applications

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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.