  Complementing existing results on minimal ruin probabilities, we minimize
expected discounted penalty functions (or Gerber-Shiu functions) in a
Cramer-Lundberg model by choosing optimal reinsurance. Reinsurance strategies
are modelled as time dependant control functions, which leads to a setting from
the theory of optimal stochastic control and ultimately to the problem's
Hamilton-Jacobi-Bellman equation. We show existence and uniqueness of the
solution found by this method and provide numerical examples involving light
and heavy tailed claims and also give a remark on the asymptotics.
