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Reflecting Brownian motion in three dimensions: a new proof of sufficient conditions for positive recurrence

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  • J. Dai
  • J. Harrison

Abstract

Let Z = {Z(t), t ≥ 0} be a semimartingale reflecting Brownian motion that lives in the three-dimensional non-negative orthant. A 2002 paper by El Kharroubi, Ben Tahar and Yaacoubi gave sufficient conditions for positive recurrence of Z. Recently Bramson, Dai and Harrison have shown that those conditions are also necessary for positive recurrence. In this paper we provide an alternative proof of sufficiency, the salient feature of which is its use of a linear Lyapunov function. Copyright Springer-Verlag 2012

Suggested Citation

  • J. Dai & J. Harrison, 2012. "Reflecting Brownian motion in three dimensions: a new proof of sufficient conditions for positive recurrence," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 75(2), pages 135-147, April.
    • Handle: RePEc:spr:mathme:v:75:y:2012:i:2:p:135-147
    DOI: 10.1007/s00186-010-0304-7
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    References listed on IDEAS

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    1. Ahmed El Kharroubi & Abdelghani Ben Tahar & Abdelhak Yaacoubi, 2002. "On the stability of the linear Skorohod problem in an orthant," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 56(2), pages 243-258, November.
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    Cited by:

    1. S. Franceschi, 2021. "Green’s Functions with Oblique Neumann Boundary Conditions in the Quadrant," Journal of Theoretical Probability, Springer, vol. 34(4), pages 1775-1810, December.
    2. Andrey Sarantsev, 2017. "Reflected Brownian Motion in a Convex Polyhedral Cone: Tail Estimates for the Stationary Distribution," Journal of Theoretical Probability, Springer, vol. 30(3), pages 1200-1223, September.
    3. Wenpin Tang, 2019. "Exponential ergodicity and convergence for generalized reflected Brownian motion," Queueing Systems: Theory and Applications, Springer, vol. 92(1), pages 83-101, June.

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