Abstract

Periodic non-homogeneous Poisson processes and Poisson models under Markovian environments are studied in this thesis. By accounting for periodic seasonal variations and random fluctuations in the underlying risk, these models generalize the classical homogeneous Poison risk model. Non-homogenous Poisson processes with periodic claim intensity rates are proposed for the claim counting process of risk theory. We introduce a doubly periodic Poisson model with short and long-term trends. Beta-type intensity functions are presented as illustrations. Doubly periodic Poisson models are appropriate when the seasonality does not repeat the exact same short-term pattern every year, but has a peak intensity that varies over a longer period. This reflects periodic environments like those forming hurricanes, in alternating El Niño/La Niña years. The properties of the model and the statistical inference of the model parameters are discussed. An application of the model to the dataset of Atlantic Hurricanes Affecting the United States (1899-2000) is discussed in detail. Further we introduce a periodic regime-switching Cox risk model by considering both, seasonal variations and stochastic fluctuations in the claims intensity. The intensity process, governed by a periodic function with a random peak level, is proposed. The periodic intensity function follows a deterministic pattern in each short-term period, and is illustrated by a beta-type function. A finite-state Markov chain defines the level process, explaining the random effect due to different underlying risk years. The properties of this regime-switching claim counting process are discussed in detail. By properly defining the Lundberg coefficient; Lundberg-type bounds for finite time ruin probabilities in the two-state risk model case are derived. A detailed derivation of the likelihood function and the maximum likelihood estimates of the model parameters is also given. Statistical applications of the model to the Atlantic hurricanes affecting the United States dataset are discussed under two different level classifications schemes. The Markov-modulated risk model is considered to reflect a risk process or insurance business alternating between a finite number of Poisson models. Here we assume that the claim inter-arrivals, claim severities and premiums of the model are influenced by an external Markovian environment. The effect of this external environment may be characterized, at any time, by a state variable, representing for example, certain types of epidemics, a variety of weather conditions or of different states of the economy