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Exit Problems for Reflected Markov-Modulated Brownian Motion

Part of:Markov processesStochastic processes

Published online by Cambridge University Press:  04 February 2016

Lothar Breuer
Lothar Breuer*
Affiliation:
University of Kent
*
∗ Postal address: Institute of Mathematics and Statistics, University of Kent, Canterbury, CT2 7NF, UK. Email address: l.breuer@kent.ac.uk
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Abstract

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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 S σ(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.

Keywords

Markov-modulated Brownian motionreflectionexit problemMarkov additive process

MSC classification

Primary:60J25: Continuous-time Markov processes on general state spaces
Secondary:60G51: Processes with independent increments; L'evy processes60J55: Local time and additive functionals

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 3 , September 2012, pp. 697 - 709
Copyright
© Applied Probability Trust 

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