On the Depletion Problem for an Insurance Risk Process: New Non-ruin Quantities in Collective Risk Theory

TL;DR

This paper introduces and analyzes new path-dependent risk measures such as drawdowns and depletion speed for Le9vy insurance risk processes, offering insights beyond traditional ruin-based metrics.

Contribution

It presents the probabilistic properties of drawdowns and depletion speed in insurance risk, extending fluctuation theory to these new quantities in a novel way.

Findings

Derived distributions for drawdowns and depletion speed.

Provided explicit formulas for Le9vy processes, including Cramer-Lundberg model.

Highlighted potential for new risk measures in insurance mathematics.

Abstract

The field of risk theory has traditionally focused on ruin-related quantities. In particular, the socalled Expected Discounted Penalty Function 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 L\'evy 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, using recent results in fluctuation theory for L\'evy processes, we 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 some examples of L\'evy insurance risk processes for which they can be calculated, in particular for the classical Cramer-Lundberg model.