IDEAS home Printed from https://ideas.repec.org/a/bpj/mcmeap/v13y2007i1p37-70n3.html
   My bibliography  Save this article

Multi-step Richardson-Romberg Extrapolation: Remarks on Variance Control and Complexity

Author

Listed:
  • Pagès Gilles

    (Email: gpa@ccr.jussieu.fr)

Abstract

We propose a multi-step Richardson-Romberg extrapolation method for the computation of expectations Ef(XT ) of a diffusion (Xt)t∈[0,T] when the weak time discretization error induced by the Euler scheme admits an expansion at an order R ≥ 2. The complexity of the estimator grows as R2 (instead of 2R in the classical method) and its variance is asymptotically controlled by considering some consistent Brownian increments in the underlying Euler schemes. Some Monte Carlo simulations were carried with path-dependent options (lookback, barrier) which support the conjecture that their weak time discretization error also admits an expansion (in a different scale). Then an appropriate Richardson-Romberg extrapolation seems to outperform the Euler scheme with Brownian bridge.

Suggested Citation

  • Pagès Gilles, 2007. "Multi-step Richardson-Romberg Extrapolation: Remarks on Variance Control and Complexity," Monte Carlo Methods and Applications, De Gruyter, vol. 13(1), pages 37-70, April.
    • Handle: RePEc:bpj:mcmeap:v:13:y:2007:i:1:p:37-70:n:3
    DOI: 10.1515/MCMA.2007.003
    as

    Download full text from publisher

    File URL: https://doi.org/10.1515/MCMA.2007.003
    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.2007.003?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. Arturo Kohatsu & Roger Pettersson, 2002. "Variance reduction methods for simulation of densities on Wiener space," Economics Working Papers 597, Department of Economics and Business, Universitat Pompeu Fabra.
    2. Bernard Lapeyre & Emmanuel Temam, 2001. "Competitive Monte Carlo methods for the pricing of Asian options," Post-Print hal-01667057, HAL.
    3. Gobet, Emmanuel, 2000. "Weak approximation of killed diffusion using Euler schemes," Stochastic Processes and their Applications, Elsevier, vol. 87(2), pages 167-197, June.
    4. Gobet, Emmanuel & Menozzi, Stéphane, 2004. "Exact approximation rate of killed hypoelliptic diffusions using the discrete Euler scheme," Stochastic Processes and their Applications, Elsevier, vol. 112(2), pages 201-223, August.
    5. Guyon, Julien, 2006. "Euler scheme and tempered distributions," Stochastic Processes and their Applications, Elsevier, vol. 116(6), pages 877-904, June.
    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. Vincent Lemaire & Thibaut Montes & Gilles Pagès, 2020. "New Weak Error bounds and expansions for Optimal Quantization," Post-Print hal-02361644, HAL.
    2. Vincent Lemaire & Thibaut Montes & Gilles Pagès, 2019. "New Weak Error bounds and expansions for Optimal Quantization," Working Papers hal-02361644, HAL.

    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. Aurélien Alfonsi & Benjamin Jourdain & Arturo Kohatsu-Higa, 2014. "Pathwise optimal transport bounds between a one-dimensional diffusion and its Euler scheme," Post-Print hal-00727430, HAL.
    2. Lejay, Antoine & Maire, Sylvain, 2007. "Computing the principal eigenvalue of the Laplace operator by a stochastic method," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 73(6), pages 351-363.
    3. Benjamin Jourdain & Mohamed Sbai, 2013. "High order discretization schemes for stochastic volatility models," Post-Print hal-00409861, HAL.
    4. Rey, Clément, 2019. "Approximation of Markov semigroups in total variation distance under an irregular setting: An application to the CIR process," Stochastic Processes and their Applications, Elsevier, vol. 129(2), pages 539-571.
    5. Lucia Caramellino & Barbara Pacchiarotti & Simone Salvadei, 2015. "Large Deviation Approaches for the Numerical Computation of the Hitting Probability for Gaussian Processes," Methodology and Computing in Applied Probability, Springer, vol. 17(2), pages 383-401, June.
    6. Gobet, Emmanuel & Menozzi, Stéphane, 2010. "Stopped diffusion processes: Boundary corrections and overshoot," Stochastic Processes and their Applications, Elsevier, vol. 120(2), pages 130-162, February.
    7. Cetin, Umut, 2018. "Diffusion transformations, Black-Scholes equation and optimal stopping," LSE Research Online Documents on Economics 87261, London School of Economics and Political Science, LSE Library.
    8. Rey Clément, 2017. "Convergence in total variation distance of a third order scheme for one-dimensional diffusion processes," Monte Carlo Methods and Applications, De Gruyter, vol. 23(1), pages 1-12, March.
    9. repec:hal:wpaper:hal-00727430 is not listed on IDEAS
    10. Gobet, Emmanuel & Labart, Céline, 2007. "Error expansion for the discretization of backward stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 117(7), pages 803-829, July.
    11. Guyon, Julien, 2006. "Euler scheme and tempered distributions," Stochastic Processes and their Applications, Elsevier, vol. 116(6), pages 877-904, June.
    12. Emmanuel Gobet, 2009. "Advanced Monte Carlo methods for barrier and related exotic options," Post-Print hal-00319947, HAL.
    13. Taguchi, Dai & Tanaka, Akihiro, 2020. "Probability density function of SDEs with unbounded and path-dependent drift coefficient," Stochastic Processes and their Applications, Elsevier, vol. 130(9), pages 5243-5289.
    14. Casella, Bruno & Roberts, Gareth O., 2011. "Exact Simulation of Jump-Diffusion Processes with Monte Carlo Applications," MPRA Paper 95217, University Library of Munich, Germany.
    15. Hoel Håkon & von Schwerin Erik & Szepessy Anders & Tempone Raúl, 2014. "Implementation and analysis of an adaptive multilevel Monte Carlo algorithm," Monte Carlo Methods and Applications, De Gruyter, vol. 20(1), pages 1-41, March.
    16. Diana Dorobantu & Yahia Salhi & Pierre-Emmanuel Thérond, 2018. "Modelling net carrying amount of shares for market consistent valuation of life insurance liabilities," Working Papers hal-01840057, HAL.
    17. Valentin Konakov & Stéphane Menozzi, 2011. "Weak Error for Stable Driven Stochastic Differential Equations: Expansion of the Densities," Journal of Theoretical Probability, Springer, vol. 24(2), pages 454-478, June.
    18. Bruno Casella & Gareth O. Roberts, 2011. "Exact Simulation of Jump-Diffusion Processes with Monte Carlo Applications," Methodology and Computing in Applied Probability, Springer, vol. 13(3), pages 449-473, September.
    19. repec:hal:wpaper:hal-00400666 is not listed on IDEAS
    20. Bouchard, Bruno & Touzi, Nizar, 2004. "Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 111(2), pages 175-206, June.
    21. Gibson, Rajna & Lhabitant, Francois-Serge & Talay, Denis, 2010. "Modeling the Term Structure of Interest Rates: A Review of the Literature," Foundations and Trends(R) in Finance, now publishers, vol. 5(1–2), pages 1-156, December.
    22. Madalina Deaconu & Samuel Herrmann, 2023. "Strong Approximation of Bessel Processes," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-24, March.

    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:13:y:2007:i:1:p:37-70:n:3. 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.