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

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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.