On the analysis of ruin-related quantities in the delayed renewal risk model

This paper first explores the Laplace transform of the time of ruin in the delayed renewal risk model. We show that Gδd(u), the Laplace transform of the time of ruin in the delayed model, also satisfies a defective renewal equation and use this to study the Cramer-Lundberg asymptotics and bounds of Gδd(u). Next, the paper considers the stochastic decomposition of the residual lifetime of maximal aggregate loss and more generally Lδd in the delayed renewal risk model, using the framework equation introduced in Kim and Willmot (2011) and the defective renewal equation for the Laplace transform of the time of ruin. As a result of the decomposition, we propose a way to calculate the mean and the moments of the proper deficit in the delayed renewal risk model. Lastly, closed form expressions are derived for the Gerber-Shiu function in the delayed renewal risk model with the distributional assumption of time until the first claim and simulation results are included to assess the impact of different distributional assumptions on the time until the first claim.