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### Joint asymptotic distribution of certain path functionals of the reflected process

File | Description | Size | Format | |
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Levy-max-6.pdf | Accepted version | 238.87 kB | Adobe PDF | View/Open |

euclid.ecp.1464033418.pdf | Published version | 349.9 kB | Adobe PDF | View/Open |

Title: | Joint asymptotic distribution of certain path functionals of the reflected process |

Authors: | Mijatovic, A Pistorius, MR |

Item Type: | Journal Article |

Abstract: | Let τ(x) be the first time that the reflected process Y of a L´evy process X crosses x > 0. The main aim of this paper is to investigate the joint asymptotic distribution of Y (t) = X(t) − inf0≤s≤t X(s) and the path functionals Z(x) = Y (τ(x)) − x and m(t) = sup0≤s≤t Y (s) − y ∗(t), for a certain non-linear curve y ∗(t). We restrict to L´evy processes X satisfying Cram´er’s condition, a non-lattice condition and the moment conditions that E[|X(1)|] and E[exp(γX(1))|X(1)|] are finite (where γ denotes the Cram´er coefficient). We prove that Y (t) and Z(x) are asymptotically independent as min{t, x} → ∞ and characterise the law of the limit (Y∞, Z∞). Moreover, if y ∗(t) = γ−1 log(t) and min{t, x} → ∞ in such a way that t exp{−γx} → 0, then we show that Y (t), Z(x) and m(t) are asymptotically independent and derive the explicit form of the joint weak limit (Y∞, Z∞, m∞). The proof is based on excursion theory, Theorem 1 in [7] and our characterisation of the law (Y∞, Z∞). |

Issue Date: | 23-May-2016 |

Date of Acceptance: | 9-May-2016 |

URI: | http://hdl.handle.net/10044/1/32321 |

DOI: | http://dx.doi.org/10.1214/16-ECP4359 |

ISSN: | 1083-589X |

Publisher: | Institute of Mathematical Statistics (IMS) |

Journal / Book Title: | Electronic Communications in Probability |

Volume: | 21 |

Copyright Statement: | This paper is made available under a Creative Commons Attribution 4.0 International License. |

Keywords: | Statistics & Probability 0104 Statistics |

Publication Status: | Published |

Article Number: | 43 |

Appears in Collections: | Financial Mathematics Mathematics Faculty of Natural Sciences |