Breuer, Lothar
(2012)
*
Exit problems for reflected Markov-modulated Brownian motion.
*
Journal of Applied Probability,
49
(3).
pp. 697-709.
ISSN 0021-9002.
(doi:10.1239/jap/1346955327)
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Official URL http://dx.doi.org/10.1239/jap/1346955327 |

## Abstract

Let (?, ?) denote a Markov-modulated Brownian motion (MMBM) and denote its supremum process by S. For some a > 0, let ? (a) denote the time when the reflected process ? := S -- ? first surpasses the level a. Furthermore, let ?_(a) denote the last time before ? (a) when ? attains its current supremum. In this paper we shall derive the joint distribution of S?(a), ?_(a), and ?(a), where the latter two will be given in terms of their Laplace transforms. We also provide some remarks on scale matrices for MMBMs with strictly positive variation parameters. This extends recent results for spectrally negative Lévy processes to MMBMs. Due to well-known fluid embedding and state-dependent killing techniques, the analysis applies to Markov additive processes with phase-type jumps as well. The result is of interest to applications such as the dividend problem in insurance mathematics and the buffer overflow problem in queueing theory. Examples will be given for the former.

Item Type: | Article |
---|---|

DOI/Identification number: | 10.1239/jap/1346955327 |

Uncontrolled keywords: | exit problem, Markov additive process, Markov-modulated Brownian motion, reflection |

Subjects: | Q Science > QA Mathematics (inc Computing science) > QA273 Probabilities |

Divisions: | Faculties > Sciences > School of Mathematics Statistics and Actuarial Science > Statistics |

Depositing User: | Lothar Breuer |

Date Deposited: | 04 Oct 2012 07:19 UTC |

Last Modified: | 29 May 2019 09:28 UTC |

Resource URI: | https://kar.kent.ac.uk/id/eprint/31237 (The current URI for this page, for reference purposes) |

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