On the Discounted Kth Moment of the Deficit at Ruin in the Delayed Renewal Risk Model

In this paper, we derive a very general expression for the discounted kth moment of the deficit at ruin in the delayed renewal risk model. The formula would yield some of the earlier relevant results as special cases, and reduce to a mathematically tractable form if the distributions for claim sizes and interclaim times are of exponential variants including a mixture of Erlang distributions. We provide numerical examples with an emphasis on the impact of the time until the first claim on the kth moment of the deficit at ruin.