Exact asymptotics of the ruin probability in the Sparre Andersen model
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866917545064267776 |
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| author | Promyslov, Platon |
| author_facet | Promyslov, Platon |
| contents | For the Sparre Andersen non-life insurance model with investments in an arbitrary Lévy process, we establish the exact power-law asymptotics of the ruin probability $Ψ(u)\sim C^* u^{-β}$ as $u\to\infty$ with a positive finite constant $C^*$; the exponent $β$ is the Cramér root of the Laplace exponent of the Lévy process describing the logarithm of the risky asset price. This strengthens previously known two-sided estimates of the order -- the existence of an exact limit had remained an open question. The proof combines a reduction to discrete time, the one-dimensional Kesten-Goldie theorem for the stationary measure of the associated affine recursion, and Goldie's result on the asymptotics of the supremum of a perpetuity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_30499 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Exact asymptotics of the ruin probability in the Sparre Andersen model Promyslov, Platon Probability 60G50, 60G51, 60K05, 91G05 For the Sparre Andersen non-life insurance model with investments in an arbitrary Lévy process, we establish the exact power-law asymptotics of the ruin probability $Ψ(u)\sim C^* u^{-β}$ as $u\to\infty$ with a positive finite constant $C^*$; the exponent $β$ is the Cramér root of the Laplace exponent of the Lévy process describing the logarithm of the risky asset price. This strengthens previously known two-sided estimates of the order -- the existence of an exact limit had remained an open question. The proof combines a reduction to discrete time, the one-dimensional Kesten-Goldie theorem for the stationary measure of the associated affine recursion, and Goldie's result on the asymptotics of the supremum of a perpetuity. |
| title | Exact asymptotics of the ruin probability in the Sparre Andersen model |
| topic | Probability 60G50, 60G51, 60K05, 91G05 |
| url | https://arxiv.org/abs/2605.30499 |