IDEAS home Printed from https://ideas.repec.org/a/eee/stapro/v70y2004i1p19-24.html
   My bibliography  Save this article

On oscillations of the geometric Brownian motion with time-delayed drift

Author

Listed:
  • Gushchin, Alexander A.
  • Küchler, Uwe

Abstract

The geometric Brownian motion is the solution of a linear stochastic differential equation in the Itô sense. If one adds to the drift term a possible nonlinear time-delayed term and starts with a non-negative initial process then the process generated in this way, may hit zero and may oscillate around zero infinitely many times depending on properties of both the drift terms and the diffusion constant.

Suggested Citation

  • Gushchin, Alexander A. & Küchler, Uwe, 2004. "On oscillations of the geometric Brownian motion with time-delayed drift," Statistics & Probability Letters, Elsevier, vol. 70(1), pages 19-24, October.
    • Handle: RePEc:eee:stapro:v:70:y:2004:i:1:p:19-24
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167-7152(04)00194-4
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

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

    Other versions of this item:

    References listed on IDEAS

    as
    1. David G. Hobson & L. C. G. Rogers, 1998. "Complete Models with Stochastic Volatility," Mathematical Finance, Wiley Blackwell, vol. 8(1), pages 27-48, January.
    2. Appleby, John A. D. & Buckwar, Evelyn, 2003. "Noise Induced Oscillation in Solutions of Stochastic Delay Differential Equations," SFB 373 Discussion Papers 2003,9, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    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. Sekine, Jun, 2008. "Marginal distribution of some path-dependent stochastic volatility model," Statistics & Probability Letters, Elsevier, vol. 78(13), pages 1846-1850, September.
    2. L. Lin & M. Schatz & D. Sornette, 2019. "A simple mechanism for financial bubbles: time-varying momentum horizon," Quantitative Finance, Taylor & Francis Journals, vol. 19(6), pages 937-959, June.
    3. Andrea Pascucci & Marco Di Francesco, 2005. "On the complete model with stochastic volatility by Hobson and Rogers," Finance 0503013, University Library of Munich, Germany.
    4. Zhao, Hui & Rong, Ximin & Zhao, Yonggan, 2013. "Optimal excess-of-loss reinsurance and investment problem for an insurer with jump–diffusion risk process under the Heston model," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 504-514.
    5. F. Fornari & A. Mele, 1998. "ARCH Models and Option Pricing : The Continuous Time Connection," THEMA Working Papers 98-30, THEMA (THéorie Economique, Modélisation et Applications), Université de Cergy-Pontoise.
    6. Samuel Chege Maina, 2011. "Credit Risk Modelling in Markovian HJM Term Structure Class of Models with Stochastic Volatility," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 1-2011, January-A.
    7. Tak Siu, 2006. "Option Pricing Under Autoregressive Random Variance Models," North American Actuarial Journal, Taylor & Francis Journals, vol. 10(2), pages 62-75.
    8. Zongwu Cai & Hongwei Mei & Rui Wang, 2024. "A model specification test for nonlinear stochastic diffusions with delay," Statistical Inference for Stochastic Processes, Springer, vol. 27(3), pages 795-812, October.
    9. Zheng, Xiaoxiao & Zhou, Jieming & Sun, Zhongyang, 2016. "Robust optimal portfolio and proportional reinsurance for an insurer under a CEV model," Insurance: Mathematics and Economics, Elsevier, vol. 67(C), pages 77-87.
    10. Emilio Barucci & Paul Malliavin & Maria Elvira Mancino & Roberto Renò & Anton Thalmaier, 2003. "The Price‐Volatility Feedback Rate: An Implementable Mathematical Indicator of Market Stability," Mathematical Finance, Wiley Blackwell, vol. 13(1), pages 17-35, January.
    11. Andrea Pascucci & Paolo Foschi, 2005. "Calibration of the Hobson&Rogers model: empirical tests," Finance 0509020, University Library of Munich, Germany.
    12. Huang, Wei & Goto, Satoru & Nakamura, Masatoshi, 2004. "Decision-making for stock trading based on trading probability by considering whole market movement," European Journal of Operational Research, Elsevier, vol. 157(1), pages 227-241, August.
    13. Alexander Lipton, 2024. "Hydrodynamics of Markets:Hidden Links Between Physics and Finance," Papers 2403.09761, arXiv.org.
    14. Christian Gourieroux & Razvan Sufana, 2004. "Derivative Pricing with Multivariate Stochastic Volatility : Application to Credit Risk," Working Papers 2004-31, Center for Research in Economics and Statistics.
    15. Marcel Nutz & Andr'es Riveros Valdevenito, 2023. "On the Guyon-Lekeufack Volatility Model," Papers 2307.01319, arXiv.org, revised Jul 2024.
    16. Sotirios Sabanis, 2012. "A class of stochastic volatility models and the q -optimal martingale measure," Quantitative Finance, Taylor & Francis Journals, vol. 12(7), pages 1111-1117, February.
    17. Alexander Lipton, 2023. "Kelvin Waves, Klein-Kramers and Kolmogorov Equations, Path-Dependent Financial Instruments: Survey and New Results," Papers 2309.04547, arXiv.org.
    18. Peter K. Friz & Thomas Wagenhofer, 2023. "Reconstructing volatility: Pricing of index options under rough volatility," Mathematical Finance, Wiley Blackwell, vol. 33(1), pages 19-40, January.
    19. Lakshithe Wagalath, 2016. "Feedback effects and endogenous risk in financial markets," Finance, Presses universitaires de Grenoble, vol. 37(2), pages 39-74.
    20. John A. D. Appleby & John A. Daniels & Katja Krol, 2012. "A Black--Scholes Model with Long Memory," Papers 1202.5574, arXiv.org.

    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:eee:stapro:v:70:y:2004:i:1:p:19-24. 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: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

    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.