2014/03/11 by Guy Latouche, Latouche, Guy, Giang T. Nguyen +1
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.PR
paper · pdf · doi:10.48550/arxiv.1403.2522
arxiv created 2014/03/11 · openalex publication_date 2014/03/11 · arxiv updated 2014/03/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We extend to Markov-modulated Brownian motion (MMBM) the renewal approach which has been successfully applied to the analysis of Markov-modulated fluid models. It has recently been shown that MMBM may be expressed as the limit of a parameterized family of Markov-modulated fluid models. We prove that the weak convergence also holds for systems with two reflecting boundaries, one at zero and one at b >0, and that the stationary distributions of the approximating fluid models converge to the stationary distribution of the two-sided reflected MMBM. Thus, we obtain a new representation for the stationary distribution, effectively separating the limiting behaviour of the process at the boundaries from its behaviour in the interior of (0,b).