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Markov modulation of a two-sided reflected Brownian motion with application to fluid queues

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  • D’Auria, Bernardo
  • Kella, Offer

Abstract

In this paper, we study a reflected Markov-modulated Brownian motion with a two sided reflection in which the drift, diffusion coefficient and the two boundaries are (jointly) modulated by a finite state space irreducible continuous time Markov chain. The goal is to compute the stationary distribution of this Markov process, which in addition to the complication of having a stochastic boundary can also include jumps at state change epochs of the underlying Markov chain because of the boundary changes. We give the general theory and then specialize to the case where the underlying Markov chain has two states.

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  • D’Auria, Bernardo & Kella, Offer, 2012. "Markov modulation of a two-sided reflected Brownian motion with application to fluid queues," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1566-1581.
    • Handle: RePEc:eee:spapps:v:122:y:2012:i:4:p:1566-1581
    DOI: 10.1016/j.spa.2012.02.001
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    References listed on IDEAS

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    1. Z. Jiang & M. R. Pistorius, 2008. "On perpetual American put valuation and first-passage in a regime-switching model with jumps," Papers 0803.2302, arXiv.org.
    2. Sotomayor, Luz R. & Cadenillas, Abel, 2011. "Classical and singular stochastic control for the optimal dividend policy when there is regime switching," Insurance: Mathematics and Economics, Elsevier, vol. 48(3), pages 344-354, May.
    3. Zhengjun Jiang & Martijn Pistorius, 2008. "On perpetual American put valuation and first-passage in a regime-switching model with jumps," Finance and Stochastics, Springer, vol. 12(3), pages 331-355, July.
    4. D'Auria, Bernardo & Kella, Offer & Ivanovs, Jevgenijs & Mandjes, Michel, 2010. "First passage of a Markov additive process and generalized Jordan chains," DES - Working Papers. Statistics and Econometrics. WS ws103923, Universidad Carlos III de Madrid. Departamento de Estadística.
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    Cited by:

    1. Peter W. Glynn & Rob J. Wang, 2019. "On the rate of convergence to equilibrium for two-sided reflected Brownian motion and for the Ornstein–Uhlenbeck process," Queueing Systems: Theory and Applications, Springer, vol. 91(1), pages 1-14, February.
    2. Dianetti, Jodi & Ferrari, Giorgio & Tzouanas, Ioannis, 2023. "Ergodic Mean-Field Games of Singular Control with Regime-Switching (extended version)," Center for Mathematical Economics Working Papers 681, Center for Mathematical Economics, Bielefeld University.
    3. Guy Latouche & Matthieu Simon, 2018. "Markov-Modulated Brownian Motion with Temporary Change of Regime at Level Zero," Methodology and Computing in Applied Probability, Springer, vol. 20(4), pages 1199-1222, December.
    4. Ren'e Aid & Matteo Basei & Giorgio Ferrari, 2023. "A Stationary Mean-Field Equilibrium Model of Irreversible Investment in a Two-Regime Economy," Papers 2305.00541, arXiv.org.

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