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Speeding up Monte Carlo Integration: Control Neighbors for Optimal Convergence

Author

Listed:
  • Leluc, Rémi

    (CMAP)

  • Portier, François

    (CREST)

  • Zhuman, Aigerim

    (Université catholique de Louvain, LIDAM/ISBA, Belgium)

  • Segers, Johan

    (Université catholique de Louvain, LIDAM/ISBA, Belgium)

Abstract

A novel linear integration rule called control neighbors is proposed in which nearest neighbor estimates act as control variates to speed up the convergence rate of the Monte Carlo procedure. The main result is the O(n−1/2n−1/d) convergence rate – where n stands for the number of evaluations of the integrand and d for the dimension of the domain – of this estimate for Lipschitz functions, a rate which, in some sense, is optimal. Several numerical experiments validate the complexity bound and highlight the good performance of the proposed estimator.

Suggested Citation

  • Leluc, Rémi & Portier, François & Zhuman, Aigerim & Segers, Johan, 2023. "Speeding up Monte Carlo Integration: Control Neighbors for Optimal Convergence," LIDAM Discussion Papers ISBA 2023019, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    • Handle: RePEc:aiz:louvad:2023019
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    References listed on IDEAS

    as
    1. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
    2. Kohler, Michael & Krzyżak, Adam, 2013. "Optimal global rates of convergence for interpolation problems with random design," Statistics & Probability Letters, Elsevier, vol. 83(8), pages 1871-1879.
    3. Portier, Francois & Segers, Johan, 2019. "Monte Carlo integration with a growing number of control variates," LIDAM Reprints ISBA 2019035, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    4. Kohler, Michael & Krzyzak, Adam & Walk, Harro, 2006. "Rates of convergence for partitioning and nearest neighbor regression estimates with unbounded data," Journal of Multivariate Analysis, Elsevier, vol. 97(2), pages 311-323, February.
    5. Leluc, Rémi & Portier, François & Segers, Johan, 2021. "Control variate selection for Monte Carlo integration," LIDAM Reprints ISBA 2021024, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    6. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    7. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
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    Cited by:

    1. Leluc, Rémi & Dieuleveut, Aymeric & Portier, François & Segers, Johan & Zhuman, Aigerim, 2024. "Sliced-Wasserstein Estimation with Spherical Harmonics as Control Variates," LIDAM Discussion Papers ISBA 2024003, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).

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