Some actuarial problems on risk models with thinning dependence

Abstract

The optimal reinsurance problem and the dividend problem are concerned in this thesis for some risk models with dependence. Specifically, the models of our study consist of multiple classes of insurance business which are correlated due to the so-called thinning-dependence structure.

In the first part, we study the optimal reinsurance problem based on a continuous-time risk model with thinning dependence for two typical reinsurance premium principles, namely the expected value premium principle and the variance premium principle. The optimal reinsurance strategies are derived under the criterion of maximizing the adjustment coefficient or minimizing the well-known Lundberg upper bound for the ruin probability. Numerical examples are also provided to illustrate the impact of the model parameters on the optimal reinsurance strategies.

In the second part, we consider the expected discounted dividends until ruin for a discrete-time risk model with thinning dependence, where dividends are paid according to a barrier strategy. Under a barrier strategy, when the surplus of each class is above a certain barrier level, the excess part is paid immediately as dividends. To compute the expected discounted dividends for each class, we need to consider the vector of the claim-size random variables from all classes and derive their joint probability mass function. With the help of the multivariate Panjer recursion for the joint probability mass function of the claim sizes, several equations related to the expected discounted dividends are obtained according to the levels of initial capital. The common shock structure is discussed as a special case and the corresponding equations are derived. We also carry out some numerical examples to show the impact of model parameters on the dividend payments.