On the joint analysis of discounted aggregate claim costs until ruin and other ruin-related quantities

Abstract

In the context of classical ruin theory, ruin quantities (e.g. ruin probability and the time of ruin) are studied separately. It was not until the publication of the seminal paper by Gerber and Shiu (1998) that researchers began to focus on the joint analysis of several ruin quantities. Gerber and Shiu (1998) proposed the well-known Gerber-Shiu expected discounted penalty function or Gerber-Shiu function which unifies the study of three key ruin quantities, namely the time of ruin, the surplus just before ruin and the deficit at ruin. In this thesis, we further incorporate the discounted aggregate claim costs until ruin as well as the discounted aggregate dividends until ruin if any dividend strategy is implemented into the Gerber-Shiu function. The generalized Gerber-Shiu function is examined under the compound Poisson model with a barrier dividend strategy or a threshold dividend strategy and a risk model with 2-sided jumps.

In a risk model with a barrier dividend strategy or a threshold dividend strategy, dividends are paid to shareholder whenever the surplus reaches a predetermined level. The dividends and the claims can be deemed as the payments to the shareholders and policyholders, respectively. As it will be seen in this thesis, if the cost function is specified, the generalized Gerber-Shiu function allows us study the time of ruin, the discounted total dividends until ruin and the discounted aggregate claims until ruin at the same time. Numerical examples are also provided to illustrate the relationship between these quantities.

In traditional risk models, insurance companies are assumed to receive premiums at a constant rate. However, some premiums are paid in lump sums, such as insurances of traveling art collections. The assumptions in a risk model with 2-sided jumps are more realistic in this regard since the upward jumps represent the random premiums. To separate the contributions of random gains and losses, both the discounted aggregate downward jump `costs' until ruin and the discounted aggregate upward jump `costs' until ruin are incorporated into the Gerber-Shiu function. Aside from the general analysis of the generalized Gerber-Shiu function, explicit solutions of two special cases under specific downward jump distribution assumptions are derived along with numerical illustrations related to the first case.