2016/03/07 by Guy Latouche, Latouche, Guy, Giang T. Nguyen +1
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR
paper · pdf · doi:10.48550/arxiv.1603.01945
34 pages
arxiv created 2016/03/07 · arxiv updated 2016/03/08
We consider a Markov-modulated Brownian motion \Y(t), ρ(t)\ with two boundaries at 0 and b > 0, and allow for the controlling Markov chain \ρ(t)\ to instantaneously undergo a change of phase upon hitting either of the two boundaries at semi-regenerative epochs defined to be the first time the process reaches a boundary since it last hits the other boundary. We call this process a flexible Markov-modulated Brownian motion. Using the recently-established links between stochastic fluid models and Markov-modulated Brownian motions, we determine important characteristics of first exit times of a Markov-modulated Brownian motion from an interval with a regulated boundary. These results allow us to follow a Markov-regenerative approach and obtain the stationary distribution of the flexible process. This highlights the effectiveness of the regenerative approach in analyzing Markov-modulated Brownian motions subject to more general boundary behaviors than the classic regulated boundaries.