Path decomposition of ruinous behavior for a general L\'{e}vy insurance risk process

TL;DR

This paper introduces a new path decomposition approach for general Lévy insurance risk processes, providing comprehensive limit theorems and explicit results for key risk functionals as initial reserves grow large.

Contribution

It develops a novel path decomposition method for Lévy risk processes, extending existing asymptotic results and simplifying proofs for functionals like time to ruin and penalty functions.

Findings

Limit distribution for time to ruin conditioned on ruin

Extended asymptotic results under convolution equivalence

Simplified proofs for functional limit theorems

Abstract

We analyze the general L\'{e}vy insurance risk process for L\'{e}vy measures in the convolution equivalence class $\mathcal{S}^{(\alpha)}$, $\alpha>0$, via a new kind of path decomposition. This yields a very general functional limit theorem as the initial reserve level $u\to \infty$, and a host of new results for functionals of interest in insurance risk. Particular emphasis is placed on the time to ruin, which is shown to have a proper limiting distribution, as $u\to \infty$, conditional on ruin occurring under our assumptions. Existing asymptotic results under the $\mathcal{S}^{(\alpha)}$ assumption are synthesized and extended, and proofs are much simplified, by comparison with previous methods specific to the convolution equivalence analyses. Additionally, limiting expressions for penalty functions of the type introduced into actuarial mathematics by Gerber and Shiu are derived as straightforward applications of our main results.