On discrete-time risk models with dependence based on integer-valued time series processes

Abstract

In the actuarial literature, dependence structures in risk models have been extensively studied. The main theme of this thesis is to investigate some discrete-time risk models with claim numbers modeled by integer-valued time series processes.

The first model is a common shock risk model with temporal dependence between the claim numbers in each individual class of business. Specifically the Poisson MA(1) process and Poisson AR(1) process are considered for the temporal dependence. To study the ruin probability, the equations associated with the adjustment coefficients are derived. Comparisons are also made to assess the impact of the dependence structures on the ruin probability.

Another model involving both the correlated classes of business and the time series approach is then studied. Thinning dependence structure is adopted to model the dependence among classes of business. The Poisson MA(1) and Poisson AR(1) processes are used to describe the claim-number processes. Adjustment coefficients and ruin probabilities are examined.

Finally a discrete-time risk model with the claim number following a Poisson ARCH process is proposed. In this model, the mean of the current claim number depends on the previous observations. Within this framework, the equation for finding the adjustment coefficient is derived. Numerical studies are also carried out to examine the effect of the Poisson ARCH dependence structure on several risk measures including ruin probability, Value at Risk, and conditional tail expectation.