## Abstract

We investigate the behavior of Lévy processes with convolution equivalent Lévy measures, up to the time of first passage over a high level u. Such problems arise naturally in the context of insurance risk where u is the initial reserve. We obtain a precise asymptotic estimate on the probability of first passage occurring by time T . This result is then used to study the process conditioned on first passage by time T . The existence of a limiting process as uis demonstrated, which leads to precise estimates for the probability of other events relating to first passage, such as the overshoot. A discussion of these results, as they relate to insurance risk, is also given.

Original language | English (US) |
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Pages (from-to) | 1506-1543 |

Number of pages | 38 |

Journal | Annals of Applied Probability |

Volume | 23 |

Issue number | 4 |

DOIs | |

State | Published - Aug 2013 |

## Keywords

- Convolution equivalence
- First passage time
- Insurance risk
- Lévy process
- Probability of ruin in finite time

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty