Markov-Modulated Brownian Motion with Temporary Change of Regime at Level Zero
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We determine the stationary distribution of a one-sided Markov-Modulated Brownian Motion (MMBM) of which the behaviour is modified during the intervals between a visit to level zero and the next visit to a fixed positive level b. We use the semi-regenerative structure of the process, and we also use the fluid approximation for MMBMs introduced by Latouche and Nguyen in 2015. Finally, we show how the expressions can be simplified in some interesting special cases and we conclude by providing some numerical illustrations.
KeywordsStochastic processes Markov-modulated Brownian motion Stationary distribution Matrix analytic methods Semi-regenerative processes
Mathematics Subject Classification (2010)60J65 66K37 60J25 60K15
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We thank the referee for a careful analysis and a detailed report. The research of M. Simon was supported by the Belgian F.R.S.-FNRS through an FRIA research grant. We acknowledge the support of the Australian Research Council Center of Excellence for Mathematical and Statistical Frontiers (ACEMS).
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