On the depletion problem for an insurance risk process: new non-ruin quantities in collective risk theory

The field of risk theory has traditionally focused on ruin-related quantities. In particular, the so-called expected discounted penalty function (Gerber and Shiu. N Am Actuar J, 2(1):48–78, 1998) has been the object of a thorough study over the years. Although interesting in their own right, ruin related quantities do not seem to capture path-dependent properties of the reserve. In this article we aim at presenting the probabilistic properties of drawdowns and the speed at which an insurance reserve depletes as a consequence of the risk exposure of the company. These new quantities are not ruin related yet they capture important features of an insurance position and we believe it can lead to the design of a meaningful risk measures. Studying drawdowns and speed of depletion for Lévy insurance risk processes represent a novel and challenging concept in insurance mathematics. In this paper, all these concepts are formally introduced in an insurance setting. Moreover, we specialize recent results in fluctuation theory for Lévy processes (Mijatovic and Pistorius. Stoch Process Their Appl, 22:3812–3836, 2012) in order to derive expressions for the distribution of several quantities related to the depletion problem. Of particular interest are the distribution of drawdowns and the Laplace transform for the speed of depletion. These expressions are given for several examples of Lévy insurance risk processes for which they can be calculated, in particular for the classical Cramér–Lundberg model.