### Abstract

Motivated by classical considerations from risk theory, we investigate boundary crossing problems for refracted Levy processes. The latter is a Levy process whose dynamics change by subtracting off a fixed linear drift (of suitable size) whenever the aggregate process is above a pre-specified level. More formally, whenever it exists, a refracted Levy process is described by the unique strong solution to the stochastic differential equation dU(t) = -delta 1({Ut > b})dt + dX(t), where X = {X-t: t >= 0) is a Levy process with law P and b, delta is an element of R such that the resulting process U may visit the half line (b, infinity) with positive probability. We consider in particular the case that X is spectrally negative and establish a suite of identities for the case of one and two sided exit problems. All identities can be written in terms of the q-scale function of the driving Levy process and its perturbed version describing motion above the level b. We remark on a number of applications of the obtained identities to (controlled) insurance risk processes.

Original language | English |
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Pages (from-to) | 24-44 |

Number of pages | 21 |

Journal | Annales de l'Institut Henri Poincaré: Probabilités et Statistiques |

Volume | 46 |

Issue number | 1 |

DOIs | |

Publication status | Published - Feb 2010 |

### Keywords

- Levy processes
- Stochastic control
- fluctuation theory

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## Cite this

Kyprianou, A. E., & Loeffen, R. L. (2010). Refracted Levy processes.

*Annales de l'Institut Henri Poincaré: Probabilités et Statistiques*,*46*(1), 24-44. https://doi.org/10.1214/08-aihp307