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New Weak Error bounds and expansions for Optimal Quantization

Author

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  • Vincent Lemaire

    (LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation - UPD7 - Université Paris Diderot - Paris 7 - SU - Sorbonne Université - CNRS - Centre National de la Recherche Scientifique)

  • Thibaut Montes

    (LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation - UPD7 - Université Paris Diderot - Paris 7 - SU - Sorbonne Université - CNRS - Centre National de la Recherche Scientifique, ICA - The Independent Calculation Agent)

  • Gilles Pagès

    (LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation - UPD7 - Université Paris Diderot - Paris 7 - SU - Sorbonne Université - CNRS - Centre National de la Recherche Scientifique)

Abstract

We propose new weak error bounds and expansion in dimension one for optimal quantization-based cubature formula for different classes of functions, such that piecewise affine functions, Lipschitz convex functions or differentiable function with piecewise-defined locally Lipschitz or α-Hölder derivatives. This new results rest on the local behaviors of optimal quantizers, the L r-L s distribution mismatch problem and Zador's Theorem. This new expansion supports the definition of a Richardson-Romberg extrapolation yielding a better rate of convergence for the cubature formula. An extension of this expansion is then proposed in higher dimension for the first time. We then propose a novel variance reduction method for Monte Carlo estimators, based on one dimensional optimal quantizers.

Suggested Citation

  • Vincent Lemaire & Thibaut Montes & Gilles Pagès, 2020. "New Weak Error bounds and expansions for Optimal Quantization," Post-Print hal-02361644, HAL.
    • Handle: RePEc:hal:journl:hal-02361644
    DOI: 10.1016/j.cam.2019.112670
    Note: View the original document on HAL open archive server: https://hal.science/hal-02361644v3
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    References listed on IDEAS

    as
    1. Delattre Sylvain & Graf Siegfried & Luschgy Harald & Pagès Gilles, 2004. "Quantization of probability distributions under norm-based distortion measures," Statistics & Risk Modeling, De Gruyter, vol. 22(4), pages 261-282, April.
    2. 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.
    3. T. A. McWalter & R. Rudd & J. Kienitz & E. Platen, 2018. "Recursive marginal quantization of higher-order schemes," Quantitative Finance, Taylor & Francis Journals, vol. 18(4), pages 693-706, April.
    4. BALLY Vlad & PAGÈS Gilles & PRINTEMS Jacques, 2001. "A stochastic quantization method for nonlinear problems," Monte Carlo Methods and Applications, De Gruyter, vol. 7(1-2), pages 21-34, December.
    5. Vlad Bally & Gilles Pagès & Jacques Printems, 2005. "A Quantization Tree Method For Pricing And Hedging Multidimensional American Options," Mathematical Finance, Wiley Blackwell, vol. 15(1), pages 119-168, January.
    6. Pagès, Gilles & Sagna, Abass, 2018. "Improved error bounds for quantization based numerical schemes for BSDE and nonlinear filtering," Stochastic Processes and their Applications, Elsevier, vol. 128(3), pages 847-883.
    7. Pagès Gilles & Printems Jacques, 2003. "Optimal quadratic quantization for numerics: the Gaussian case," Monte Carlo Methods and Applications, De Gruyter, vol. 9(2), pages 135-165, April.
    8. P. Carr & D. Madan, 2001. "Optimal positioning in derivative securities," Quantitative Finance, Taylor & Francis Journals, vol. 1(1), pages 19-37.
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    Cited by:

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    3. Yang, Jun & He, Ping & Fang, Kai-Tai, 2022. "Three kinds of discrete approximations of statistical multivariate distributions and their applications," Journal of Multivariate Analysis, Elsevier, vol. 188(C).

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