IDEAS home Printed from https://ideas.repec.org/a/bpj/mcmeap/v12y2006i2p143-170n2.html
   My bibliography  Save this article

Balanced Milstein Methods for Ordinary SDEs

Author

Listed:
  • Kahl Christian

    (1. Department of Mathematics, University of Wuppertal, Gaußstraße 20, Wuppertal, D-42119, Germany)

  • Schurz Henri

    (2. Department of Mathematics, Southern Illinois University, 1245 Lincoln Drive, Carbondale, IL 62901-4408, USA)

Abstract

Convergence, consistency, stability and pathwise positivity of balanced Milstein methods for numerical integration of ordinary stochastic differential equations (SDEs) are discussed. This family of numerical methods represents a class of highly efficient linear-implicit schemes which generate mean square converging numerical approximations with qualitative improvements and global rate 1. 0 of mean square convergence, compared to commonly known numerical methods for SDEs with Lipschitzian coefficients.

Suggested Citation

  • Kahl Christian & Schurz Henri, 2006. "Balanced Milstein Methods for Ordinary SDEs," Monte Carlo Methods and Applications, De Gruyter, vol. 12(2), pages 143-170, April.
    • Handle: RePEc:bpj:mcmeap:v:12:y:2006:i:2:p:143-170:n:2
    DOI: 10.1515/156939606777488842
    as

    Download full text from publisher

    File URL: https://doi.org/10.1515/156939606777488842
    Download Restriction: For access to full text, subscription to the journal or payment for the individual article is required.
    File URL: https://libkey.io/10.1515/156939606777488842?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. G. N. Milstein & Eckhard Platen & H. Schurz, 1998. "Balanced Implicit Methods for Stiff Stochastic Systems," Published Paper Series 1998-1, Finance Discipline Group, UTS Business School, University of Technology, Sydney.
    2. John C. Cox & Jonathan E. Ingersoll Jr. & Stephen A. Ross, 2005. "A Theory Of The Term Structure Of Interest Rates," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 5, pages 129-164, World Scientific Publishing Co. Pte. Ltd..
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Kang, Ting & Li, Qiang & Zhang, Qimin, 2019. "Numerical analysis of the balanced implicit method for stochastic age-dependent capital system with poisson jumps," Applied Mathematics and Computation, Elsevier, vol. 353(C), pages 166-177.
    2. Yin, Zhengwei & Gan, Siqing, 2015. "An error corrected Euler–Maruyama method for stiff stochastic differential equations," Applied Mathematics and Computation, Elsevier, vol. 256(C), pages 630-641.
    3. Li, Yan & Zhang, Qimin, 2020. "The balanced implicit method of preserving positivity for the stochastic SIQS epidemic model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 538(C).
    4. Renata Rendek, 2013. "Modeling Diversified Equity Indices," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 23, July-Dece.
    5. Benjamin Jourdain & Mohamed Sbai, 2013. "High order discretization schemes for stochastic volatility models," Post-Print hal-00409861, HAL.
    6. Nicola Bruti-Liberati, 2007. "Numerical Solution of Stochastic Differential Equations with Jumps in Finance," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 1-2007.
    7. Weng, Lihui & Liu, Wei, 2019. "Invariant measures of the Milstein method for stochastic differential equations with commutative noise," Applied Mathematics and Computation, Elsevier, vol. 358(C), pages 169-176.
    8. Sun, Xianming & Gan, Siqing & Vanmaele, Michèle, 2015. "Analytical approximation for distorted expectations," Statistics & Probability Letters, Elsevier, vol. 107(C), pages 246-252.
    9. Tan, Jianguo & Men, Weiwei & Pei, Yongzhen & Guo, Yongfeng, 2017. "Construction of positivity preserving numerical method for stochastic age-dependent population equations," Applied Mathematics and Computation, Elsevier, vol. 293(C), pages 57-64.
    10. Tan, Jianguo & Chen, Yang & Men, Weiwei & Guo, Yongfeng, 2021. "Positivity and convergence of the balanced implicit method for the nonlinear jump-extended CIR model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 182(C), pages 195-210.
    11. Nikolaos Halidias, 2016. "On construction of boundary preserving numerical schemes," Papers 1601.07864, arXiv.org, revised Feb 2016.
    12. Halidias Nikolaos, 2016. "On the construction of boundary preserving numerical schemes," Monte Carlo Methods and Applications, De Gruyter, vol. 22(4), pages 277-289, December.
    13. Nikolaos Halidias & Ioannis Stamatiou, 2015. "Approximating explicitly the mean reverting CEV process," Papers 1502.03018, arXiv.org, revised May 2015.
    14. Christian Kahl & Peter Jackel, 2006. "Fast strong approximation Monte Carlo schemes for stochastic volatility models," Quantitative Finance, Taylor & Francis Journals, vol. 6(6), pages 513-536.
    15. Nicola Bruti-Liberati, 2007. "Numerical Solution of Stochastic Differential Equations with Jumps in Finance," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 1, July-Dece.
    16. Yao, Jinran & Gan, Siqing, 2018. "Stability of the drift-implicit and double-implicit Milstein schemes for nonlinear SDEs," Applied Mathematics and Computation, Elsevier, vol. 339(C), pages 294-301.
    17. Renata Rendek, 2013. "Modeling Diversified Equity Indices," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 4-2013.
    18. Xianming Sun & Siqing Gan, 2014. "An Efficient Semi-Analytical Simulation for the Heston Model," Computational Economics, Springer;Society for Computational Economics, vol. 43(4), pages 433-445, April.
    19. Ahmadian, D. & Farkhondeh Rouz, O. & Ballestra, L.V., 2019. "Stability analysis of split-step θ-Milstein method for a class of n-dimensional stochastic differential equations," Applied Mathematics and Computation, Elsevier, vol. 348(C), pages 413-424.
    20. Eckhard Platen & Renata Rendek, 2009. "Exact Scenario Simulation for Selected Multi-dimensional Stochastic Processes," Research Paper Series 259, Quantitative Finance Research Centre, University of Technology, Sydney.
    21. H. A. Mardones & C. M. Mora, 2020. "First-Order Weak Balanced Schemes for Stochastic Differential Equations," Methodology and Computing in Applied Probability, Springer, vol. 22(2), pages 833-852, June.
    22. Xiaoling Wang & Xiaofei Guan & Pei Yin, 2020. "A New Explicit Magnus Expansion for Nonlinear Stochastic Differential Equations," Mathematics, MDPI, vol. 8(2), pages 1-17, February.

    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. Renata Rendek, 2013. "Modeling Diversified Equity Indices," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 23, July-Dece.
    2. Nikolaos Halidias & Ioannis Stamatiou, 2015. "Approximating explicitly the mean reverting CEV process," Papers 1502.03018, arXiv.org, revised May 2015.
    3. Nicola Bruti-Liberati, 2007. "Numerical Solution of Stochastic Differential Equations with Jumps in Finance," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 1, July-Dece.
    4. Nicola Bruti-Liberati, 2007. "Numerical Solution of Stochastic Differential Equations with Jumps in Finance," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 1-2007.
    5. Renata Rendek, 2013. "Modeling Diversified Equity Indices," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 4-2013.
    6. Chantal Labb'e & Bruno R'emillard & Jean-Franc{c}ois Renaud, 2010. "A simple discretization scheme for nonnegative diffusion processes, with applications to option pricing," Papers 1011.3247, arXiv.org.
    7. Christian Kahl & Peter Jackel, 2006. "Fast strong approximation Monte Carlo schemes for stochastic volatility models," Quantitative Finance, Taylor & Francis Journals, vol. 6(6), pages 513-536.
    8. Kau, James B. & Keenan, Donald C., 1999. "Patterns of rational default," Regional Science and Urban Economics, Elsevier, vol. 29(6), pages 765-785, November.
    9. Camilla LandÊn, 2000. "Bond pricing in a hidden Markov model of the short rate," Finance and Stochastics, Springer, vol. 4(4), pages 371-389.
    10. Álvarez Echeverría Francisco & López Sarabia Pablo & Venegas Martínez Francisco, 2012. "Valuación financiera de proyectos de inversión en nuevas tecnologías con opciones reales," Contaduría y Administración, Accounting and Management, vol. 57(3), pages 115-145, julio-sep.
    11. Hisashi Nakamura & Wataru Nozawa & Akihiko Takahashi, 2009. "Macroeconomic Implications of Term Structures of Interest Rates Under Stochastic Differential Utility with Non-Unitary EIS," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 16(3), pages 231-263, September.
    12. Darren Shannon & Grigorios Fountas, 2021. "Extending the Heston Model to Forecast Motor Vehicle Collision Rates," Papers 2104.11461, arXiv.org, revised May 2021.
    13. Matsumura, Marco & Moreira, Ajax & Vicente, José, 2011. "Forecasting the yield curve with linear factor models," International Review of Financial Analysis, Elsevier, vol. 20(5), pages 237-243.
    14. Ivanova, Vesela & Puigvert Gutiérrez, Josep Maria, 2014. "Interest rate forecasts, state price densities and risk premium from Euribor options," Journal of Banking & Finance, Elsevier, vol. 48(C), pages 210-223.
    15. Lin, Bing-Huei, 1999. "Fitting the term structure of interest rates for Taiwanese government bonds," Journal of Multinational Financial Management, Elsevier, vol. 9(3-4), pages 331-352, November.
    16. Gollier, Christian, 2002. "Time Horizon and the Discount Rate," Journal of Economic Theory, Elsevier, vol. 107(2), pages 463-473, December.
    17. Robert R. Bliss & Ehud I. Ronn, 1997. "Callable U.S. Treasury bonds: optimal calls, anomalies, and implied volatilities," FRB Atlanta Working Paper 97-1, Federal Reserve Bank of Atlanta.
    18. Henry, Olan T. & Olekalns, Nilss & Suardi, Sandy, 2007. "Testing for rate dependence and asymmetry in inflation uncertainty: Evidence from the G7 economies," Economics Letters, Elsevier, vol. 94(3), pages 383-388, March.
    19. Ammann, Manuel & Kind, Axel & Wilde, Christian, 2003. "Are convertible bonds underpriced? An analysis of the French market," Journal of Banking & Finance, Elsevier, vol. 27(4), pages 635-653, April.
    20. Sergio Zúñiga, 1999. "Modelos de Tasas de Interés en Chile: Una Revisión," Latin American Journal of Economics-formerly Cuadernos de Economía, Instituto de Economía. Pontificia Universidad Católica de Chile., vol. 36(108), pages 875-893.

    More about this item

    Statistics

    Access and download statistics

    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:12:y:2006:i:2:p:143-170:n:2. 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.