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Recent Advances in Randomized Quasi-Monte Carlo Methods

In: Modeling Uncertainty

Author

Listed:
  • Pierre L’Ecuyer

    (Université de Montréal)

  • Christiane Lemieux

    (University of Calgary)

Abstract

We survey some of the recent developments on quasi-Monte Carlo (QMC) methods, which, in their basic form, are a deterministic counterpart to the Monte Carlo (MC) method. Our main focus is the applicability of these methods to practical problems that involve the estimation of a high-dimensional integral. We review several QMC constructions and different randomizations that have been proposed to provide unbiased estimators and for error estimation. Randomizing QMC methods allows us to view them as variance reduction techniques. New and old results on this topic are used to explain how these methods can improve over the MC method in practice. We also discuss how this methodology can be coupled with clever transformations of the integrand in order to reduce the variance further. Additional topics included in this survey are the description of figures of merit used to measure the quality of the constructions underlying these methods, and other related techniques for multidimensional integration.

Suggested Citation

  • Pierre L’Ecuyer & Christiane Lemieux, 2002. "Recent Advances in Randomized Quasi-Monte Carlo Methods," International Series in Operations Research & Management Science, in: Moshe Dror & Pierre L’Ecuyer & Ferenc Szidarovszky (ed.), Modeling Uncertainty, chapter 0, pages 419-474, Springer.
    • Handle: RePEc:spr:isochp:978-0-306-48102-4_20
    DOI: 10.1007/0-306-48102-2_20
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    Citations

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    Cited by:

    1. Vladimir K. Kaishev & Dimitrina S. Dimitrova, 2009. "Dirichlet Bridge Sampling for the Variance Gamma Process: Pricing Path-Dependent Options," Management Science, INFORMS, vol. 55(3), pages 483-496, March.
    2. L’Ecuyer, P. & Sanvido, C., 2010. "Coupling from the past with randomized quasi-Monte Carlo," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 81(3), pages 476-489.
    3. Vandewoestyne, Bart & Chi, Hongmei & Cools, Ronald, 2010. "Computational investigations of scrambled Faure sequences," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 81(3), pages 522-535.
    4. Yumiao Tian & Maorong Ge & Frank Neitzel, 2020. "Variance Reduction of Sequential Monte Carlo Approach for GNSS Phase Bias Estimation," Mathematics, MDPI, vol. 8(4), pages 1-15, April.
    5. Pierre L'Ecuyer & Christian Lécot & Bruno Tuffin, 2008. "A Randomized Quasi-Monte Carlo Simulation Method for Markov Chains," Operations Research, INFORMS, vol. 56(4), pages 958-975, August.
    6. Philipp N. Baecker, 2007. "Real Options and Intellectual Property," Lecture Notes in Economics and Mathematical Systems, Springer, number 978-3-540-48264-2, December.
    7. Athanassios N. Avramidis & Pierre L'Ecuyer, 2006. "Efficient Monte Carlo and Quasi-Monte Carlo Option Pricing Under the Variance Gamma Model," Management Science, INFORMS, vol. 52(12), pages 1930-1944, December.
    8. Baldeaux Jan, 2008. "Quasi-Monte Carlo methods for the Kou model," Monte Carlo Methods and Applications, De Gruyter, vol. 14(4), pages 281-302, January.
    9. Bastin, Fabian & Cirillo, Cinzia & Toint, Philippe L., 2006. "Application of an adaptive Monte Carlo algorithm to mixed logit estimation," Transportation Research Part B: Methodological, Elsevier, vol. 40(7), pages 577-593, August.
    10. Pierre L’Ecuyer, 2009. "Quasi-Monte Carlo methods with applications in finance," Finance and Stochastics, Springer, vol. 13(3), pages 307-349, September.
    11. Xiaoqun Wang & Ian H. Sloan, 2011. "Quasi-Monte Carlo Methods in Financial Engineering: An Equivalence Principle and Dimension Reduction," Operations Research, INFORMS, vol. 59(1), pages 80-95, February.
    12. Pierre L’Ecuyer & Florian Puchhammer & Amal Ben Abdellah, 2022. "Monte Carlo and Quasi–Monte Carlo Density Estimation via Conditioning," INFORMS Journal on Computing, INFORMS, vol. 34(3), pages 1729-1748, May.
    13. Hatem Ben-Ameur & Pierre L'Ecuyer & Christiane Lemieux, 2004. "Combination of General Antithetic Transformations and Control Variables," Mathematics of Operations Research, INFORMS, vol. 29(4), pages 946-960, November.
    14. Okten, Giray & Eastman, Warren, 2004. "Randomized quasi-Monte Carlo methods in pricing securities," Journal of Economic Dynamics and Control, Elsevier, vol. 28(12), pages 2399-2426, December.
    15. H. Heitsch & H. Leövey & W. Römisch, 2016. "Are Quasi-Monte Carlo algorithms efficient for two-stage stochastic programs?," Computational Optimization and Applications, Springer, vol. 65(3), pages 567-603, December.
    16. Yang Huang & Yongdao Zhou, 2022. "Convergence of Uniformity Criteria and the Application in Numerical Integration," Mathematics, MDPI, vol. 10(19), pages 1-20, October.
    17. Xiaoqun Wang & Ken Seng Tan, 2013. "Pricing and Hedging with Discontinuous Functions: Quasi-Monte Carlo Methods and Dimension Reduction," Management Science, INFORMS, vol. 59(2), pages 376-389, July.
    18. Jank, Wolfgang, 2005. "Quasi-Monte Carlo sampling to improve the efficiency of Monte Carlo EM," Computational Statistics & Data Analysis, Elsevier, vol. 48(4), pages 685-701, April.
    19. Xing Jin & Michael C. Fu & Xiaoping Xiong, 2003. "Probabilistic Error Bounds for Simulation Quantile Estimators," Management Science, INFORMS, vol. 49(2), pages 230-246, February.
    20. Lucio Barabesi, 2003. "A Monte Carlo integration approach to Horvitz-Thompson estimation in replicated environmental designs," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(3), pages 355-374.
    21. F. Y. Kuo & W. T. M. Dunsmuir & I. H. Sloan & M. P. Wand & R. S. Womersley, 2008. "Quasi-Monte Carlo for Highly Structured Generalised Response Models," Methodology and Computing in Applied Probability, Springer, vol. 10(2), pages 239-275, June.
    22. Zhijian He & Xiaoqun Wang, 2021. "An Integrated Quasi-Monte Carlo Method for Handling High Dimensional Problems with Discontinuities in Financial Engineering," Computational Economics, Springer;Society for Computational Economics, vol. 57(2), pages 693-718, February.

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