Analysis of a defective renewal equation arising in ruin theory

This paper studies in detail the solution of a defective renewal equation which involves the time of ruin, the surplus immediately before ruin, and the deficit at the time of ruin. The analysis is simplified by introduction and analysis of a related compound geometric distribution, which is studied in detail. Tijms approximations and bounds for these quantities are also discussed. Examples are given for the cases when the claim size distribution is exponential, combinations of exponentials and mixtures of Erlangs. In a subsequent paper, we will extend our analysis to the moments of the time of ruin, the moments of the surplus before the time of ruin, the moments of the deficit at the time of ruin, and correlations between them.