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September 2012 Exit problems for reflected Markov-modulated Brownian motion
Lothar Breuer
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J. Appl. Probab. 49(3): 697-709 (September 2012). DOI: 10.1239/jap/1346955327


Let (X, J) denote a Markov-modulated Brownian motion (MMBM) and denote its supremum process by S. For some a > 0, let σ(a) denote the time when the reflected process Y := S - X first surpasses the level a. Furthermore, let σ-(a) denote the last time before σ(a) when X attains its current supremum. In this paper we shall derive the joint distribution of Sσ(a), σ-(a), and σ(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.


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Lothar Breuer. "Exit problems for reflected Markov-modulated Brownian motion." J. Appl. Probab. 49 (3) 697 - 709, September 2012.


Published: September 2012
First available in Project Euclid: 6 September 2012

zbMATH: 1256.60027
MathSciNet: MR3012093
Digital Object Identifier: 10.1239/jap/1346955327

Primary: 60J25
Secondary: 60G51, 60J55

Rights: Copyright © 2012 Applied Probability Trust


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Vol.49 • No. 3 • September 2012
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