IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v33y2020i1d10.1007_s10959-018-0848-7.html
   My bibliography  Save this article

Uniform Ergodicity for Brownian Motion in a Bounded Convex Set

Author

Listed:
  • Jackson Loper

    (Columbia University)

Abstract

We consider an n-dimensional Brownian motion trapped inside a bounded convex set by normally reflecting boundaries. It is well known that this process is uniformly ergodic. However, the rates of this ergodicity are not well understood, especially in the regime of very high-dimensional sets. Here we present new bounds on these rates for convex sets with a given diameter. Our bounds do not depend upon the smoothness of the boundary nor the value of the ambient dimension, n.

Suggested Citation

  • Jackson Loper, 2020. "Uniform Ergodicity for Brownian Motion in a Bounded Convex Set," Journal of Theoretical Probability, Springer, vol. 33(1), pages 22-35, March.
    • Handle: RePEc:spr:jotpro:v:33:y:2020:i:1:d:10.1007_s10959-018-0848-7
    DOI: 10.1007/s10959-018-0848-7
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-018-0848-7
    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/s10959-018-0848-7?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. Qin Hu & Yongjin Wang & Xuewei Yang, 2012. "The Hitting Time Density for a Reflected Brownian Motion," Computational Economics, Springer;Society for Computational Economics, vol. 40(1), pages 1-18, June.
    2. Vadim Linetsky, 2004. "Lookback options and diffusion hitting times: A spectral expansion approach," Finance and Stochastics, Springer, vol. 8(3), pages 373-398, August.
    3. Ding, Xiaodong & Wu, Rangquan, 1998. "A new proof for comparison theorems for stochastic differential inequalities with respect to semimartingales," Stochastic Processes and their Applications, Elsevier, vol. 78(2), pages 155-171, November.
    4. Fitzsimmons, P. J. & Pitman, Jim, 1999. "Kac's moment formula and the Feynman-Kac formula for additive functionals of a Markov process," Stochastic Processes and their Applications, Elsevier, vol. 79(1), pages 117-134, January.
    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. Herrmann, Samuel & Massin, Nicolas, 2023. "Exact simulation of the first passage time through a given level of jump diffusions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 203(C), pages 553-576.
    2. Dell'Era Mario, M.D., 2008. "Pricing of the European Options by Spectral Theory," MPRA Paper 17429, University Library of Munich, Germany.
    3. Alistair N Boettiger & Peter L Ralph & Steven N Evans, 2011. "Transcriptional Regulation: Effects of Promoter Proximal Pausing on Speed, Synchrony and Reliability," PLOS Computational Biology, Public Library of Science, vol. 7(5), pages 1-14, May.
    4. Masaaki Fukasawa, 2010. "Asymptotic analysis for stochastic volatility: Edgeworth expansion," Papers 1004.2106, arXiv.org.
    5. Vadim Linetsky, 2004. "Spectral Expansions for Asian (Average Price) Options," Operations Research, INFORMS, vol. 52(6), pages 856-867, December.
    6. Decamps, Marc & De Schepper, Ann & Goovaerts, Marc, 2009. "Spectral decomposition of optimal asset-liability management," Journal of Economic Dynamics and Control, Elsevier, vol. 33(3), pages 710-724, March.
    7. Matthew Lorig, 2011. "Pricing Derivatives on Multiscale Diffusions: an Eigenfunction Expansion Approach," Papers 1109.0738, arXiv.org, revised Apr 2012.
    8. Peter Carr & Vadim Linetsky, 2006. "A jump to default extended CEV model: an application of Bessel processes," Finance and Stochastics, Springer, vol. 10(3), pages 303-330, September.
    9. Campi, Luciano & Polbennikov, Simon & Sbuelz, Alessandro, 2009. "Systematic equity-based credit risk: A CEV model with jump to default," Journal of Economic Dynamics and Control, Elsevier, vol. 33(1), pages 93-108, January.
    10. Qin Hu & Yongjin Wang & Xuewei Yang, 2012. "The Hitting Time Density for a Reflected Brownian Motion," Computational Economics, Springer;Society for Computational Economics, vol. 40(1), pages 1-18, June.
    11. Kokou Essiomle & Franck Adékambi, 2023. "Valuation of Equity-Linked Death Benefits on Two Lives with Dependence," Risks, MDPI, vol. 11(1), pages 1-26, January.
    12. Masaaki Fukasawa, 2010. "Central limit theorem for the realized volatility based on tick time sampling," Finance and Stochastics, Springer, vol. 14(2), pages 209-233, April.
    13. Shane Miller, 2007. "Pricing of Contingent Claims Under the Real-World Measure," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 2-2007, January-A.
    14. Damien Ackerer & Damir Filipović, 2020. "Option pricing with orthogonal polynomial expansions," Mathematical Finance, Wiley Blackwell, vol. 30(1), pages 47-84, January.
    15. Kevin Z. Tong & Allen Liu, 2019. "Option pricing in a subdiffusive constant elasticity of variance (CEV) model," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 6(02), pages 1-21, June.
    16. Depperschmidt, Andrej & Pfaffelhuber, Peter, 2010. "Asymptotics of a Brownian ratchet for protein translocation," Stochastic Processes and their Applications, Elsevier, vol. 120(6), pages 901-925, June.
    17. Vladislav Krasin & Ivan Smirnov & Alexander Melnikov, 2018. "Approximate option pricing and hedging in the CEV model via path-wise comparison of stochastic processes," Annals of Finance, Springer, vol. 14(2), pages 195-209, May.
    18. Gongqiu Zhang & Lingfei Li, 2021. "A General Approach for Lookback Option Pricing under Markov Models," Papers 2112.00439, arXiv.org.
    19. Liming Feng & Vadim Linetsky, 2009. "Computing exponential moments of the discrete maximum of a Lévy process and lookback options," Finance and Stochastics, Springer, vol. 13(4), pages 501-529, September.
    20. Shane Miller, 2007. "Pricing of Contingent Claims Under the Real-World Measure," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 25, July-Dece.

    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:jotpro:v:33:y:2020:i:1:d:10.1007_s10959-018-0848-7. 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.