In this paper we extend some results in Cramér (1955) by considering the expected discounted penalty function as a generalization of the infinite time ruin probability. We consider his ruin theory model that allows the claim sizes to take positive as well as negative values. Depending on the sign of these events, they are called claims or premiums, respectively. We then demonstrate that when the events' arrival process is a renewal process, the Gerber-Shiu function satisfies a defective renewal equation. Subsequently, we consider some special cases such as when claims have exponential distribution or the arrival process is a compound Poisson process and premiums have Erlang(n,) distribution. We are then able to specify the parameter and the functions involved in the above-mentioned defective renewal equation.
Published September 2009 , 32 pages