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A dual skew symmetry for transient reflected Brownian motion in an orthant

Author

Listed:
  • Sandro Franceschi

    (Institut Polytechnique de Paris)

  • Kilian Raschel

    (Université d’Angers)

Abstract

We introduce a transient reflected Brownian motion in a multidimensional orthant, which is either absorbed at the apex of the cone or escapes to infinity. We address the question of computing the absorption probability, as a function of the starting point of the process. We provide a necessary and sufficient condition for the absorption probability to admit an exponential product form, namely that the determinant of the reflection matrix is zero. We call this condition a dual skew symmetry. It recalls the famous skew symmetry introduced by Harrison (Adv Appl Probab 10:886–905, 1978), which characterizes the exponential stationary distributions in the recurrent case. The duality comes from that the partial differential equation satisfied by the absorption probability is dual to the one associated with the stationary distribution in the recurrent case.

Suggested Citation

  • Sandro Franceschi & Kilian Raschel, 2022. "A dual skew symmetry for transient reflected Brownian motion in an orthant," Queueing Systems: Theory and Applications, Springer, vol. 102(1), pages 123-141, October.
    • Handle: RePEc:spr:queues:v:102:y:2022:i:1:d:10.1007_s11134-022-09853-9
    DOI: 10.1007/s11134-022-09853-9
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    References listed on IDEAS

    as
    1. Ernst, Philip A. & Franceschi, Sandro & Huang, Dongzhou, 2021. "Escape and absorption probabilities for obliquely reflected Brownian motion in a quadrant," Stochastic Processes and their Applications, Elsevier, vol. 142(C), pages 634-670.
    2. Harrison, J. Michael, 2022. "Reflected Brownian motion in the quarter plane: An equivalence based on time reversal," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 1189-1203.
    3. Deuschel, Jean-Dominique & Zambotti, Lorenzo, 2005. "Bismut-Elworthy's formula and random walk representation for SDEs with reflection," Stochastic Processes and their Applications, Elsevier, vol. 115(6), pages 907-925, June.
    4. Ivanovs, Jevgenijs & Boxma, Onno, 2015. "A bivariate risk model with mutual deficit coverage," Insurance: Mathematics and Economics, Elsevier, vol. 64(C), pages 126-134.
    5. 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.
    Full references (including those not matched with items on IDEAS)

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