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 socalled 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 dividendpayment problem. 
Professor Kam Chuen Yuen
Department of Statistics & Actuarial Science
The University of Hong Kong
kcyuen@hku.hk

