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Complementing existing results on minimal ruin probabilities, we minimizeexpected discounted penalty functions (or Gerber-Shiu functions) in aCramer-Lundberg model by choosing optimal reinsurance. Reinsurance strategiesare modelled as time dependant control functions, which leads to a setting fromthe theory of optimal stochastic control and ultimately to the problem'sHamilton-Jacobi-Bellman equation. We show existence and uniqueness of thesolution found by this method and provide numerical examples involving lightand heavy tailed claims and also give a remark on the asymptotics.

Optimal Reinsurance for Gerber-Shiu Functions in the Cramer-Lundberg Model