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Analyses of Insurance Risk Models with Dividend Payments
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Dividend payment strategy has become an increasingly important issue for financial institutions. Due to its practical importance, the impact of dividend payments on insurance business needs to be studied carefully and thoroughly. This requires in depth analyses of various ruin problems in the presence of a suitably chosen strategy. In the classical risk model, the surplus of an insurance company can grow without bounds. To avoid this unrealistic feature, one can incorporate dividend payments into the model. In fact, the problem of optimal dividend payment strategy has been around for half a century. Although many dividend strategies have been proposed in the actuarial literature, we only considered three of them in the project. Under the so-called barrier strategy, if the surplus grows beyond a certain level called barrier, the difference between the surplus and the barrier is paid out as dividends until a new claim arrives. Despite its popularity on the theoretical side, such a strategy has little practical acceptance as it leads to the certainty of ultimate ruin. A modified version of the barrier strategy called the threshold strategy assumes that dividends are paid at a rate smaller than the premium income rate whenever the surplus is above the threshold level, and that no dividends are paid whenever the surplus is below the threshold. Another interesting dividend strategy is called the randomized dividend strategy in which the insurer pays a fixed amount of dividends with a certain probability when the surplus is larger than the predetermined threshold level.

To make the ruin analysis more interesting but challenging, we not only incorporated one of the three dividend strategies into a basic insurance risk model, but also added a special feature to the basic risk model. The basic risk models under consideration were the compound binomial model, the compound Poisson model, and the renewal model, while the special features included debit interest, delayed claims, dependence structure, diffusion component, and return on investments. In this project, we focused on investigating the expected discounted

penalty function for the abovementioned risk models. It should be pointed out that the expected discounted penalty function is a very useful tool in modern insurance risk theory as it embraces important ruin quantities such as ruin probability, Laplace transform of ruin time, and distributions related to surplus immediately before ruin and deficit at ruin. For each model of study, we were able to derive explicit expressions for the expected discounted penalty function. Moreover, we discussed the issue of optimal dividends in some cases, and examined the impact of dependence on optimal dividends for the models with dependence structure.

To establish the main results of the project, we used various advanced mathematical techniques such as classical risk theory, generating function technique, integrodifferential equation, Laplace transform, Markov theory, renewal theory, and so on. The outcomes of the project were summarized in a series of papers which appeared in major international actuarial and probability journals. So far, quite a number of citations of these papers can be found in the literature. It is expected that the results are not only useful for further research on the topic, but also provide the insurance profession with some useful insights into the dividend-payment problem.

Professor Kam Chuen Yuen
Department of Statistics & Actuarial Science
The University of Hong Kong