Joint densities involving the time to ruin in the Sparre Andersen risk model under exponential assumptions

Recent research into the nature of the distribution of the time of ruin in some Sparre Andersen risk models has resulted in series expansions for the associated density function. Examples include Dickson and Willmot (2005) in the classical Poisson model with exponential interclaim times, and Borovkov and Dickson (2008), who used a duality argument in the case with exponential claim amounts. The aim of this paper is not only to unify previous methodology through the use of Lagrange’s expansion theorem, but also to provide insight into the nature of the series expansions by identifying the probabilistic contribution of each term in the expansion through analysis involving the distribution of the number of claims until ruin. The (defective) distribution of the number of claims until ruin is then further examined. Interestingly, a connection to the well-known extended truncated negative binomial (ETNB) distribution is also established. Finally, a closed-form expression for the joint density of the time to ruin, the surplus prior to ruin, and the number of claims until ruin is derived. In the last section, the formula of Dickson and Willmot (2005) for the density of the time to ruin in the classical risk model is re-examined to identify its individual contributions based on the number of claims until ruin.