Discrete-time insurance risk models with dependence structures

Abstract

Regarding the relationships among different insurance claims, especially in non-life insurance, the dependence behaviour in various models has been studied extensively. In this thesis, some discrete-time risk models with dependence structures would be investigated. One traditional discrete-time risk model is the time series risk model, in which the dependence would be on two aspects: time correlated claims and dependent business classes. A general vector (multivariate) autoregressive moving average (VARMA) model would be adopted to analyze the ruin probability of a surplus process. An upper bound for the ruin probability is derived for the general order of multivariate time series models in claims. Simulation studies are carried out for model comparison for finite time ruin probabilities. Another class of risk model is the compound binomial risk model, where the dependence structure would be based on the existence of a so-called by-claim in the claim process. The by-claim could be incurred in the same period as the main insurance claim, or it would be incurred in the next period, depending on a certain probability. A randomized dividend payment scheme with some fixed threshold value in surplus level would also be considered in this thesis. A methodology is discovered to obtain the Gerber-Shiu expected penalty function for the extended model. The final model investigated in this thesis is the periodic time series risk model. The periodic structure of the model gives a practical interpretation of the business cycle, in which there are high season and low season for the business. Some lower order periodic time series models are considered for the claim structures.