On Poisson models under Markovian environments and their applications in risk theory

经典的齐次Poisson计数过程及其相应的风险模型在精算文献中得到了广泛的研究。该风险模型假设索赔发生的强度率恒定，这在某些实际情况下是不现实的。 在这个研究项目中，我打算致力于齐次泊松过程的推广。一种推广是非齐次泊松（NHP）过程，其中索赔强度是时间的函数，反映了风险的季节性。本文研究马尔可夫环境下的周期性考克斯过程，它是NHP过程的自然推广，可以用来刻画索赔强度的潜在风险波动。 受季节性条件影响的保险风险显然是在周期性随机环境中演变的。在有些情况下，这种季节性效应联合收割机与社会或其他自然现象结合起来，产生周期性的或甚至更普遍的环境。例如，众所周知，天气因素会影响汽车或火灾保险索赔，而北方的季节性暴风雪和南方的飓风或洪水会影响财产保险。一般而言，预计将完成的工作，对风险相关的数量，统计推断和破产相关的问题，建议的模型将提供一个有效的和定量的方法，保险公司更准确地衡量风险，使用时间依赖，而不是分段恒定的强度率。周期性和马尔可夫分量考虑的过程将使这些模型更实用和有用的建模计数过程下的季节性和随机环境。允许索赔强度，索赔严重程度和保费的泊松风险过程随时间变化的可能性是一个重要的一步，更现实的模型和更好的动机比许多其他扩展，如更新到达过程。

The classical homogeneous Poisson counting process and its corresponding risk model have been investigated extensively in the actuarial literature. This risk model assumes a constant intensity rate for the occurrence of claims which is not realistic in some practical situations.  I intend to work on the generalization of the homogeneous Poisson process in this research project. One generalization is the non-homogeneous Poisson (NHP) process, in which the claim intensity is a function of time reflecting the seasonality of the risk. This research project focuses on the Cox process with periodicity and under Markovian environment, which is a natural extension of the NHP process and can be used to characterize the underlying risk fluctuations in the claims intensity.          Insurance risks that are subject to seasonal conditions clearly evolve in periodic random environments. There are instances where such seasonal effects combine with social or other natural phenomena to produce periodic or even more general environments. For example, weather factors are known to affect automobile or fire insurance claims, while seasonal snow storms in the north and hurricanes or floods in the south affect property insurance. In general, the work expected to be done on the risk-related quantities, statistical inferences and ruin-related problems for the proposed models would provide an effective and quantitative method for insurance companies to measure risk more accurately by using time dependent, rather than piecewise constant, intensity rates. The periodicity and Markovian components considered for the processes would make these models more practical and useful in modeling counting processes under seasonality and random environments. The possibility of allowing the claim intensity, the claim severity and the premium of the Poisson risk process to vary in time is a major step towards more realistic models and better motivated than many other extensions like renewal arrival processes.