A characterization of ruin-inducing probability measures in a renewal risk model

TL;DR

This paper characterizes all ruin-inducing probability measures in a renewal risk model, enabling explicit ruin probability calculations without requiring moment generating functions, thus applicable to heavy-tailed distributions.

Contribution

It provides a complete characterization of ruin-inducing measures using pairs of functions, generalizing the Esscher transform and applicable to heavy-tailed claim sizes.

Findings

Explicit representation of ruin probability as an expectation under ruin-inducing measures

Framework includes classical Esscher transform as a special case

Applicable to heavy-tailed claim size distributions

Abstract

In this work, we derive a complete characterization of all ruin-inducing probability measures that preserve the structure of a given compound renewal process in terms of suitable pairs of functions $(\gamma,\delta)$. This result allows us to obtain an explicit representation of the infinite-time ruin probability as an expectation under any ruin-inducing probability measure. A key feature of our approach is that the construction of these measures does not rely on the existence of moment generating functions, and is therefore applicable to heavy-tailed claim size distributions. The proposed framework includes the classical Esscher transform as a special case.