Journal of Applied Probability

Exit problems for reflected Markov-modulated Brownian motion

Lothar Breuer

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Abstract

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.

Article information

Source
J. Appl. Probab., Volume 49, Number 3 (2012), 697-709.

Dates
First available in Project Euclid: 6 September 2012

Permanent link to this document
https://projecteuclid.org/euclid.jap/1346955327

Digital Object Identifier
doi:10.1239/jap/1346955327

Mathematical Reviews number (MathSciNet)
MR3012093

Zentralblatt MATH identifier
1256.60027

Subjects
Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60G51: Processes with independent increments; Lévy processes 60J55: Local time and additive functionals

Keywords
Markov-modulated Brownian motion reflection exit problem Markov additive process

Citation

Breuer, Lothar. Exit problems for reflected Markov-modulated Brownian motion. J. Appl. Probab. 49 (2012), no. 3, 697--709. doi:10.1239/jap/1346955327. https://projecteuclid.org/euclid.jap/1346955327


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