Path decomposition of ruinous behavior for a general Lévy insurance risk process

Abstract

We analyze the general Lévy insurance risk process for Lévy 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.

Published in at http://dx.doi.org/10.1214/11-AAP797 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• Ruin Probabilities and Overshoots for General Levy Insurance Risk Processes
• Overshoots and undershoots of Lévy processes

Cited by in corpus (5)

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• Asymptotic Distributions of the Overshoot and Undershoots for the Lévy Insurance Risk Process in the Cramér and Convolution Equivalent Cases
• Pricing Two-asset Options under Exponential Lévy Model Using a Finite Element Method