## Abstract

Let X be a real valued Lévy process and set R_{t}=X_{t}−inf_{s≤t}X_{s}. This paper addresses the asymptotic behavior of the sample paths of the reflected process R on first passage over an arbitrarily high level u. We show that under the convolution equivalent condition of Klüppelberg et al. (2004), the sample paths of R on the first excursion which crosses over a high level u can be decomposed into two processes. The first describes the paths in a neighborhood of the origin. The process then takes a large jump into a neighborhood of u. The second process describes the subsequent paths. This sample path behavior is similar to that of X conditioned to cross level u. Using this connection many results concerning, for example, undershoots and overshoots can be easily obtained.

Original language | English (US) |
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Pages (from-to) | 29-47 |

Number of pages | 19 |

Journal | Stochastic Processes and their Applications |

Volume | 145 |

DOIs | |

State | Published - Mar 2022 |

## Keywords

- Convolution equivalence
- First passage time
- Lévy process
- Overshoot
- Reflected process

## ASJC Scopus subject areas

- Statistics and Probability
- Modeling and Simulation
- Applied Mathematics