Dependence in risk theory

保险公司通过购买大量的保单来承担重大风险。为了避免未来面临财务困难，保险公司必须能够对其机构的财务健康状况有足够的评估。为此，风险理论可以用来衡量保险合同组合的全球风险。这种评估可以使用固定时间段的短期模型进行，也可以使用长期模型进行。对于长期模型，可以通过考察盈余过程在几个时期内的演变来评估全球风险，无论是离散的还是连续的。在这种情况下，破产理论最常被用来完成这一任务。传统上，对盈余过程特征的描述依赖于索赔数额与到达间隔时间之间独立的假设。在许多应用中，这种假设是不充分的，需要具有相依结构的泛化，以避免对保险公司所面临的全球风险的错误评估。在我的研究计划中，我将提出并包括在风险模型中可以在实践中观察到的不同依赖结构。离散和连续时间风险模型都将被考虑。不同的量，如破产概率、破产前的盈余和破产时的赤字，在评估保险投资组合的风险时是有意义的。在所提出的扩展中，将使用基于Gerber-Shiu惩罚函数的统一方法来研究这些与破产相关的量。向股东支付股息和投资收益的可能性也将在建议的具有相关性的风险模型中进行研究。

An insurance company submits itself to an important risk by subscribing a high number of insurance policies. To avoid facing financial difficulties in the future, insurance companies must be able to have an adequate appraisal of the financial health of their institution. For that purpose, risk theory can be used to measure the global risk of a portfolio of insurance contracts. This evaluation can be made with short term models over a fixed period of time and with long term models. With long term models, the global risk can be assessed by examining the evolution of the surplus process over several periods, either on a discrete or continuous time basis. In such a case, ruin theory is most oftenly used to accomplish this task. The description of the characteristics of the surplus process has relied traditionally on an assumption of independence between the amount of claims and the interarrival times. In many applications, this assumption is inadequate and generalizations with a dependence structure is needed to avoid a misevaluation of the global risk faced by an insurance company. In my research program, I will propose and include different dependence constructions which can be observed in practice in risk models. Both discrete and continuous time risk models will be considered. Different quantities such as the ruin probability, the surplus immediately prior to ruin and the deficit at ruin are of interest in the assessment of the risk of an insurance portfolio. The unified approach based on the Gerber-Shiu penalty function will be used to study these ruin related quantities in the extensions proposed. The possibility of dividend payments to shareholders and investment income will also be studied within the suggested risk models with dependence.