Risk processes analyzed using fluid queues

点击翻译按钮获取中文摘要

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.