On the time value of Parisian ruin in (dual) renewal risk processes with exponential jumps

This paper studies the Parisian ruin problem first proposed by Dassios and Wu (2008a,b), where the Parisian ruin time is defined to be the first time when the surplus process has stayed below zero continuously for a pre-specified time length d. Both the insurance risk process and the dual model will be considered under exponential distributional assumption on the jump sizes while keeping the inter-arrival times arbitrary. In these two models, the Laplace transform of the Parisian ruin time is derived by extending the excursion techniques in Dassios and Wu (2008a) and taking advantage of the memoryless property of exponential distributions. Our results are represented in integral forms, which are expressed in terms of the (joint) densities of various ruin-related quantities that are available in the literature or obtainable using the Lagrange's expansion theorem. As a by-product, we also provide the joint distribution of the numbers of periods of negative surplus that are of duration more than d and less than d, which can be obtained using some of our intermediate results. The case where the Parisian delay period d is replaced by a random time is also discussed, and it is applied to find the Laplace transform of the occupation time when the surplus is negative. Numerical illustrations concerning an Erlang(2) insurance risk model are given at the end.