IDEAS home Printed from https://ideas.repec.org/a/spr/queues/v102y2022i1d10.1007_s11134-022-09853-9.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11134-022-09853-9
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s11134-022-09853-9?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. 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.
    2. 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.
    3. Ivanovs, Jevgenijs & Boxma, Onno, 2015. "A bivariate risk model with mutual deficit coverage," Insurance: Mathematics and Economics, Elsevier, vol. 64(C), pages 126-134.
    4. 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.
    5. 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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Avi Mandelbaum & Kavita Ramanan, 2010. "Directional Derivatives of Oblique Reflection Maps," Mathematics of Operations Research, INFORMS, vol. 35(3), pages 527-558, August.
    2. Avram, F. & Badescu, A.L. & Pistorius, M.R. & Rabehasaina, L., 2016. "On a class of dependent Sparre Andersen risk models and a bailout application," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 27-39.
    3. 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.
    4. Hansjoerg Albrecher & Pablo Azcue & Nora Muler, 2015. "Optimal Dividend Strategies for Two Collaborating Insurance Companies," Papers 1505.03980, arXiv.org.
    5. 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.
    6. Boxma, Onno & Frostig, Esther & Perry, David & Yosef, Rami, 2017. "A state dependent reinsurance model," Insurance: Mathematics and Economics, Elsevier, vol. 74(C), pages 170-181.
    7. Pablo Azcue & Nora Muler & Zbigniew Palmowski, 2016. "Optimal dividend payments for a two-dimensional insurance risk process," Papers 1603.07019, arXiv.org, revised Apr 2018.
    8. Deuschel, Jean-Dominique & Nishikawa, Takao, 2007. "The dynamic of entropic repulsion," Stochastic Processes and their Applications, Elsevier, vol. 117(5), pages 575-595, May.
    9. Gordienko, E. & Vázquez-Ortega, P., 2018. "Continuity inequalities for multidimensional renewal risk models," Insurance: Mathematics and Economics, Elsevier, vol. 82(C), pages 48-54.
    10. 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.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:queues:v:102:y:2022:i:1:d:10.1007_s11134-022-09853-9. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.