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

经典齐次泊松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。我打算在这个研究项目中对齐次泊松过程进行推广。一种推广是非齐次泊松（NHP）过程，其中索赔强度是反映风险季节性的时间函数。本研究项目重点研究马尔可夫环境下具有周期性的Cox过程，该过程是NHP过程的自然延伸，可用于表征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。在某些情况下，这种季节性影响与社会或其他自然现象相结合，产生周期性甚至更普遍的环境。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。总的来说，对拟议模型的风险相关数量、统计推断和与废墟有关的问题所做的工作将为保险公司提供一种有效的定量方法，通过使用时间相关的强度率，而不是分段恒定的强度率，更准确地衡量风险。考虑过程的周期性和马尔可夫成分将使这些模型在季节性和随机环境下的计数过程建模中更加实用和有用。允许索赔强度、索赔严重性和泊松风险过程的保费随时间变化的可能性是朝着更现实的模型迈出的重要一步，并且比许多其他扩展（如续期到达过程）更有动力。

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.