IDEAS home Printed from https://ideas.repec.org/a/bpj/mcmeap/v26y2020i1p33-47n4.html
   My bibliography  Save this article

Approximation of Euler–Maruyama for one-dimensional stochastic differential equations involving the maximum process

Author

Listed:
  • Hiderah Kamal

    (Department of Mathematics, Faculty of Science, University of Aden, Aden, Yemen)

Abstract

The aim of this paper is to show the approximation of Euler–Maruyama Xtn{X_{t}^{n}} for one-dimensional stochastic differential equations involving the maximum process. In addition to that it proves the strong convergence of the Euler–Maruyama whose both drift and diffusion coefficients are Lipschitz. After that, it generalizes to the non-Lipschitz case.

Suggested Citation

  • Hiderah Kamal, 2020. "Approximation of Euler–Maruyama for one-dimensional stochastic differential equations involving the maximum process," Monte Carlo Methods and Applications, De Gruyter, vol. 26(1), pages 33-47, March.
    • Handle: RePEc:bpj:mcmeap:v:26:y:2020:i:1:p:33-47:n:4
    DOI: 10.1515/mcma-2020-2057
    as

    Download full text from publisher

    File URL: https://doi.org/10.1515/mcma-2020-2057
    Download Restriction: For access to full text, subscription to the journal or payment for the individual article is required.
    File URL: https://libkey.io/10.1515/mcma-2020-2057?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. Chan, K. S. & Stramer, O., 1998. "Weak consistency of the Euler method for numerically solving stochastic differential equations with discontinuous coefficients," Stochastic Processes and their Applications, Elsevier, vol. 76(1), pages 33-44, August.
    2. Chaumont, L. & Doney, R. A., 2000. "Some calculations for doubly perturbed Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 85(1), pages 61-74, January.
    3. Rainer Avikainen, 2009. "On irregular functionals of SDEs and the Euler scheme," Finance and Stochastics, Springer, vol. 13(3), pages 381-401, September.
    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. Mark Podolskij & Bezirgen Veliyev & Nakahiro Yoshida, 2018. "Edgeworth expansion for Euler approximation of continuous diffusion processes," CREATES Research Papers 2018-28, Department of Economics and Business Economics, Aarhus University.
    2. Yao Tung Huang & Qingshuo Song & Harry Zheng, 2015. "Weak Convergence of Path-Dependent SDEs in Basket CDS Pricing with Contagion Risk," Papers 1506.00082, arXiv.org, revised May 2016.
    3. Hideyuki Tanaka & Toshihiro Yamada, 2012. "Strong Convergence for Euler-Maruyama and Milstein Schemes with Asymptotic Method," Papers 1210.0670, arXiv.org, revised Nov 2013.
    4. F Bourgey & S de Marco & Emmanuel Gobet & Alexandre Zhou, 2020. "Multilevel Monte-Carlo methods and lower-upper bounds in Initial Margin computations," Post-Print hal-02430430, HAL.
    5. Yuta Otsuki & Shotaro Yagishita, 2024. "Optimal reinsurance and investment via stochastic projected gradient method based on Malliavin calculus," Papers 2411.05417, arXiv.org.
    6. F Bourgey & S de Marco & Emmanuel Gobet & Alexandre Zhou, 2020. "Multilevel Monte-Carlo methods and lower-upper bounds in Initial Margin computations," Working Papers hal-02430430, HAL.
    7. Giulia Di Nunno & Anton Yurchenko-Tytarenko, 2022. "Sandwiched Volterra Volatility model: Markovian approximations and hedging," Papers 2209.13054, arXiv.org, revised Jul 2024.
    8. Yaskov, Pavel, 2019. "On pathwise Riemann–Stieltjes integrals," Statistics & Probability Letters, Elsevier, vol. 150(C), pages 101-107.
    9. Christian Bayer & Chiheb Ben Hammouda & Raul Tempone, 2020. "Multilevel Monte Carlo with Numerical Smoothing for Robust and Efficient Computation of Probabilities and Densities," Papers 2003.05708, arXiv.org, revised Oct 2023.
    10. Daphné Giorgi & Vincent Lemaire & Gilles Pagès, 2020. "Weak Error for Nested Multilevel Monte Carlo," Methodology and Computing in Applied Probability, Springer, vol. 22(3), pages 1325-1348, September.
    11. Mike Giles & Lukasz Szpruch, 2012. "Multilevel Monte Carlo methods for applications in finance," Papers 1212.1377, arXiv.org.
    12. Dereich, Steffen & Heidenreich, Felix, 2011. "A multilevel Monte Carlo algorithm for Lévy-driven stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 121(7), pages 1565-1587, July.
    13. Michael B. Giles & Kristian Debrabant & Andreas Ro{ss}ler, 2013. "Analysis of multilevel Monte Carlo path simulation using the Milstein discretisation," Papers 1302.4676, arXiv.org, revised Jun 2019.
    14. Bourgey Florian & De Marco Stefano & Gobet Emmanuel & Zhou Alexandre, 2020. "Multilevel Monte Carlo methods and lower–upper bounds in initial margin computations," Monte Carlo Methods and Applications, De Gruyter, vol. 26(2), pages 131-161, June.
    15. Laukkarinen, Eija, 2020. "Malliavin smoothness on the Lévy space with Hölder continuous or BV functionals," Stochastic Processes and their Applications, Elsevier, vol. 130(8), pages 4766-4792.
    16. Ma, Xiaocui & Yue, Haitao & Xi, Fubao, 2022. "The averaging method for doubly perturbed distribution dependent SDEs," Statistics & Probability Letters, Elsevier, vol. 189(C).
    17. Holly Brannelly & Andrea Macrina & Gareth W. Peters, 2019. "Quantile Diffusions for Risk Analysis," Papers 1912.10866, arXiv.org, revised Sep 2021.
    18. Hideyuki Tanaka & Toshihiro Yamada, 2013. "Strong Convergence for Euler-Maruyama and Milstein Schemes with Asymptotic Method (Forthcoming in "International Journal of Theoretical and Applied Finance")," CARF F-Series CARF-F-333, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    19. Antoine Lejay & Paolo Pigato, 2017. "A threshold model for local volatility: evidence of leverage and mean reversion effects on historical data," Working Papers hal-01669082, HAL.
    20. Dirk Becherer & Plamen Turkedjiev, 2014. "Multilevel approximation of backward stochastic differential equations," Papers 1412.3140, 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:bpj:mcmeap:v:26:y:2020:i:1:p:33-47:n:4. 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: Peter Golla (email available below). General contact details of provider: https://www.degruyter.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.