Path Decomposition of Ruinous Behaviour for a General Lévy Insurance Risk Process

Mathematics – Probability

Scientific paper

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Scientific paper

We analyse the general L\'evy insurance risk process for L\'evy measures in the convolution equivalence class ${\cal S}^{(\alpha)}$, $\alpha > 0$, 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 ${\cal S}^{(\alpha)}$ assumption are synthesised 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.

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