Advanced ruin theory models and applications

该建议考虑了所谓的“双重风险模型”的扩展。与通常的保险破产模型不同，对偶模型假设公司有稳定的费用流和随机的收入，收入在随机时间到达。扩展，我们探讨涉及推广的拉普拉斯变换的破产时间，是standardised研究在这个模型下。更确切地说，我们构建了一个类似的广泛研究的预期贴现罚款函数，引入了更多的数量，除了破产时间的利益。由于对偶模型的性质显着不同的保险破产模型中，破产的结果缺乏收入，而不是作为一个索赔的结果，这些公司的管理层的利益的数量不同，包括在现在经典的期望折现罚款函数。因此，我们需要更有创造性地确定应该研究的对象。我们打算探索的对偶模型的另一个扩展与随机过程的推广有关，该随机过程模拟了落入该框架的企业的盈余演变。 我打算继续的另一条研究路线与保险公司盈余过程中实施的各种依赖结构有关。这种依赖性允许保险公司将某些类型的保单的特定特征纳入模型中。例如，众所周知，地震区的财产损失程度与连续地震之间的时间间隔高度相关。盈余过程中依赖性的另一个例子是，被保险人根据他们的风险因素水平被分成不同的群体。更确切地说，有人指出，一个群体的被保险人往往比另一个群体的被保险人提出更低的索赔额。 另一种研究探索了通过考虑在零处修改的计数分布来推广索赔数量过程的可能性。这类修改允许提前研究引入免赔额或增加现有保险单免赔额的可能性。现行保单合约与经修订合约的盈利能力及风险性之间的比较依赖于过往合约库所累积的数据。 本提案中所述的最后一项研究考虑了保险费率的更详细的变化。一个更现实的方式来实现变化的保险费率是通过纳入一个保险公司的盈余过程中的随机变量，触发保费变化的决定。目前关于这一主题的文献的主要部分假设，保费的变化是确定性的，通过固定一些阈值，公司的盈余应该满足。

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.