Affine storage and insurance risk models

The aim of this paper is to analyze a general class of storage processes, in which the rate at which the storage level increases or decreases is assumed to be an affine function of the current storage level, and, in addition, both upward and downward jumps are allowed. To do so, we first focus on a related insurance risk model, for which we determine the ruin probability at an exponentially distributed epoch jointly with the corresponding undershoot and overshoot, given that the capital level at time 0 is exponentially distributed as well. The obtained results for this insurance risk model can be translated in terms of two types of storage models, in one of those two cases by exploiting a duality relation. Many well-studied models are shown to be special cases of our insurance risk and storage models. We conclude by showing how the exponentiality assumptions can be greatly relaxed.