Advanced ruin theory models and applications

该提案考虑了所谓的“双重风险模型”的扩展。与通常的保险破产模型不同，双重模型假设一家公司有稳定的支出流和随机时间到达的随机数量的收入。我们探索的扩展包括推广在这个模型下被标准研究的毁灭时间的拉普拉斯变换。更准确地说，我们构建了一个类似于广泛研究的期望折扣惩罚函数的模型，该函数被引入，以包含除破产时间之外的更多数量的利息。由于双重模型的性质与保险破产模型有很大不同，因为破产是由于缺乏收入而不是索赔的结果，因此这些公司管理层的利益数量与现在经典的预期贴现惩罚函数中包含的利益数量不同。因此，我们在确定应该研究的对象时需要更有创造性。我们打算探索的双重模型的另一个扩展与随机过程的概括有关，该过程模拟了属于该框架的企业剩余的演变。

This proposal considers extensions of the so-called "dual risk model." In contrast to the usual insurance ruin models, the dual model assumes that a company has a steady stream of expenses and random amounts of income arriving at random times. The extensions that we explore involve generalizing the Laplace transform of the time to ruin that is standardly studied under this model. More precisely, we construct an analogue to the widely studied expected discounted penalty function that was introduced to incorporate more quantities of interest apart from the time to ruin. Since the nature of the dual model significantly differs from the insurance ruin models in that ruin results from the lack of income rather than as a consequence of a claim, the quantities of interest to the management of such companies differ from those that are included in the now classical expected discounted penalty function. Consequently, we need to be more creative in identifying the objects that should be studied. Another extension of the dual model that we intend to explore is related to generalizations of the stochastic process that models the evolution of the surplus of businesses that fall in this framework. Another line of research that I intend to pursue is related to various dependence structures that are implemented into the surplus process of insurance companies. Such dependences allow the insurer to incorporate into the model specific features of certain types of policies. For instance, it is well known that the extent of damages to property in earthquake zones is highly correlated to the amount of time that elapses between consecutive earthquakes. Another example of dependence in the surplus process is when insureds are divided into groups depending on their level of some risk factor. More precisely, it is noted that insureds from one group tend to make claims for lower amounts than insureds from another group. Yet another line of research explores the possibility to generalize the claim-number process by considering counting distributions that are modified at zero. This type of modifications allows for studying in advance the possibility of introducing a deductible or increasing the deductible of already existing insurance policies. The comparison between the profitability and riskiness of the current policy contracts and the revised contracts relies on the accumulated data by the pool of former contracts. The last line of research that is described in this proposal considers more elaborate changes of premium rates. A more realistic way of implementing changes to the premium rate is by incorporating into the surplus process of an insurance company a random variable that triggers the decision of premium changing. The major part of the current literature on this topic assumes that the change in premium is made deterministically by fixing some thresholds that the company's surplus should meet.