Risk processes analyzed using fluid queues

集体风险理论作为保险数学的一部分，研究与保险业务相关的随机模型。在这样的模型中，索赔的发生被描述为一个点过程，保险公司因索赔而必须支付的金额是一个随机变量。在这个随机的框架内，风险是一个关键概念，而不是确定性的世界观。破产概率、破产前的盈余和破产时的赤字只是衡量保险业务风险的一部分。在大多数实际情况下，能够预测一些未来事件的结果是很有意义的。在保险中，能够预测索赔的频率和索赔成本是非常重要的。需要数学模型来帮助做出这些预测。然而，模型越逼真，分析起来就越困难。特别是，这项研究将集中在一个数学方程的分析，其解是非常复杂的评估。类似的方程也出现在许多其他应用的概率模型中，如排队论。因此，来自其他研究领域的一些结果可能适用于风险理论，反之亦然。我将用来研究这种复杂模型的方法更多的是以概率为导向，而不是以分析为导向。在排队论中被广泛研究的流体流动的首次通过时间也将应用于风险模型。将与现有的分析方法进行比较，差异主要通过计算机密集的计算通过数值分析来说明。

Collective risk theory, as a part of insurance mathematics, deals with stochastic models pertinent to an insurance business. In such a model, the occurrence of claims is described by a point process and the amount of money that an insurer has to pay due to a claim as a random variable. Within this stochastic framework, rather than a deterministic view of the world, risk is a key concept. The probability of ruin, the surplus prior to ruin and the deficit at the time of ruin are just some of the quantities that measure the risk of an insurance business. In most of the real situations it is of interest to be able to predict the outcome of some future events. In insurance, it is of huge importance to be able to predict the frequency of the claims and also the claim costs. Mathematical models are needed to help to make these predictions. However, the more realistic the model, the more difficult it is to analyze. In particular, this research will focus on the analysis of a mathematical equation whose solution is very complicated to evaluate. Similar equations appear in many other applied probability models such as queueing theory. Consequently, some of the results from other areas of research may be applied in Risk Theory and vice versa. The methods that I will apply to study such complicated models are rather probabilistically oriented than analytically oriented. First passage times for fluid flows that are extensively studied in queueing theory will be applied to the risk models as well. Comparison with the existing analytical approaches will be carried out, the differences being illustrated via numerical analysis primarily through computer-intensive calculations.