Doklady Mathematics

, Volume 98, Issue 2, pp 494–497 | Cite as

Variance Reduction in Monte Carlo Estimators via Empirical Variance Minimization

  • D. V. Belomestny
  • L. S. IosipoiEmail author
  • N. K. Zhivotovskiy


For Monte Carlo estimators, a variance reduction method based on empirical variance minimization in the class of functions with zero mean (control functions) is described. An upper bound for the efficiency of the method is obtained in terms of the properties of the functional class.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    D. Belomestny, S. Häfner, and M. Urusov, J. Math. Anal. Appl. 458, 393–418 (2018).MathSciNetCrossRefGoogle Scholar
  2. 2.
    R. Christian and G. Casella, Monte Carlo Statistical Methods (Springer, New York, 1999).zbMATHGoogle Scholar
  3. 3.
    S. Clemencon, G. Lugosi, and N. Vayatis, Ann. Stat. 36 (2), 844–874 (2008).CrossRefGoogle Scholar
  4. 4.
    I. T. Dimov, Monte Carlo Methods for Applied Scientists (World Scientific, Singapore, 2008).zbMATHGoogle Scholar
  5. 5.
    P. Glasserman, Monte Carlo Methods in Financial Engineering (Springer Science & Business Media, New York, 2013).zbMATHGoogle Scholar
  6. 6.
    W. Hoeffding, J. Am. Stat. Assoc. 58 (301), 13–30 (1963).CrossRefGoogle Scholar
  7. 7.
    R. Nickl and B. M. Pötscher, J. Theor. Probab. 20 (2), 177–199 (2007).CrossRefGoogle Scholar
  8. 8.
    C. J. Oates, J. Cockayne, F.-X. Briol, and M. Girolami, “Convergence rates for a class of estimators based on Stein’s identity” (2016). arXiv:1603.03220.Google Scholar
  9. 9.
    C. J. Oates, M. Girolami, and N. Chopin, J. R. Stat. Soc.: Ser. B (Stat. Method.) 79 (3), 695–718 (2017).CrossRefGoogle Scholar
  10. 10.
    R. Y. Rubinstein and D. P. Kroese, Simulation and the Monte Carlo Method (Wiley, New York, 2016).CrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  • D. V. Belomestny
    • 1
    • 2
  • L. S. Iosipoi
    • 1
    • 3
    Email author
  • N. K. Zhivotovskiy
    • 1
    • 2
    • 4
  1. 1.National Research University Higher School of EconomicsMoscowRussia
  2. 2.University of Duisburg-EssenDuisburg and EssenGermany
  3. 3.Faculty of Mechanics and MathematicsMoscow State UniversityMoscowRussia
  4. 4.Institute for Information Transmission ProblemsRussian Academy of SciencesMoscowRussia

Personalised recommendations