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Variance Reduction via Lattice Rules

Author

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  • Pierre L'Ecuyer

    (Département d'Informatique et de Recherche Opérationnelle, Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montréal, H3C 3J7, Canada, www.iro.umontreal.ca/~lecuyer)

  • Christiane Lemieux

    (Département d'Informatique et de Recherche Opérationnelle, Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montréal, H3C 3J7, Canada, www.math.vcalgary.ca/~lemieux)

Abstract

This is a review article on lattice methods for multiple integration over the unit hypercube, with a variance-reduction viewpoint. It also contains some new results and ideas. The aim is to examine the basic principles supporting these methods and how they can be used effectively for the simulation models that are typically encountered in the area of management science. These models can usually be reformulated as integration problems over the unit hypercube with a large (sometimes infinite) number of dimensions. We examine selection criteria for the lattice rules and suggest criteria which take into account the quality of the projections of the lattices over selected low-dimensional subspaces. The criteria are strongly related to those used for selecting linear congruential and multiple recursive random number generators. Numerical examples illustrate the effectiveness of the approach.

Suggested Citation

  • Pierre L'Ecuyer & Christiane Lemieux, 2000. "Variance Reduction via Lattice Rules," Management Science, INFORMS, vol. 46(9), pages 1214-1235, September.
    • Handle: RePEc:inm:ormnsc:v:46:y:2000:i:9:p:1214-1235
    DOI: 10.1287/mnsc.46.9.1214.12231
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    References listed on IDEAS

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    Cited by:

    1. 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.
    2. Raimova, Gulnora, 2011. "Variance reduction methods at the pricing of weather options," Applied Econometrics, Russian Presidential Academy of National Economy and Public Administration (RANEPA), vol. 21(1), pages 3-15.
    3. Jan Baldeaux, 2011. "Exact Simulation of the 3/2 Model," Papers 1105.3297, arXiv.org, revised May 2011.
    4. 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.
    5. Lemieux, Christiane & L’Ecuyer, Pierre, 2001. "On selection criteria for lattice rules and other quasi-Monte Carlo point sets," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 55(1), pages 139-148.
    6. Hatem Ben-Ameur & Michèle Breton & Pierre L'Ecuyer, 2002. "A Dynamic Programming Procedure for Pricing American-Style Asian Options," Management Science, INFORMS, vol. 48(5), pages 625-643, May.
    7. Brown, Paul T. & Joshi, Chaitanya & Joe, Stephen & Rue, Håvard, 2021. "A novel method of marginalisation using low discrepancy sequences for integrated nested Laplace approximations," Computational Statistics & Data Analysis, Elsevier, vol. 157(C).
    8. 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.
    9. Nabil Kahalé, 2020. "Randomized Dimension Reduction for Monte Carlo Simulations," Management Science, INFORMS, vol. 66(3), pages 1421-1439, March.
    10. 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.
    11. L'Ecuyer, Pierre, 2004. "Random number generation," Papers 2004,21, Humboldt University of Berlin, Center for Applied Statistics and Economics (CASE).
    12. Dingeç, Kemal Dinçer & Hörmann, Wolfgang, 2013. "Control variates and conditional Monte Carlo for basket and Asian options," Insurance: Mathematics and Economics, Elsevier, vol. 52(3), pages 421-434.
    13. Nico Achtsis & Ronald Cools & Dirk Nuyens, 2011. "Conditional sampling for barrier option pricing under the LT method," Papers 1111.4808, arXiv.org, revised Dec 2012.
    14. Jan Baldeaux & Dale Roberts, 2012. "Quasi-Monte Carlo methods for the Heston model," Papers 1202.3217, arXiv.org, revised May 2012.
    15. Julien Keutchayan & Michel Gendreau & Antoine Saucier, 2017. "Quality evaluation of scenario-tree generation methods for solving stochastic programming problems," Computational Management Science, Springer, vol. 14(3), pages 333-365, July.
    16. 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.
    17. Xing Jin & Allen X. Zhang, 2006. "Reclaiming Quasi-Monte Carlo Efficiency in Portfolio Value-at-Risk Simulation Through Fourier Transform," Management Science, INFORMS, vol. 52(6), pages 925-938, June.
    18. L’Ecuyer, Pierre & Granger-Piché, Jacinthe, 2003. "Combined generators with components from different families," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 62(3), pages 395-404.
    19. Han, Chuan-Hsiang & Lai, Yongzeng, 2010. "A smooth estimator for MC/QMC methods in finance," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 81(3), pages 536-550.
    20. Munger, D. & L’Ecuyer, P. & Bastin, F. & Cirillo, C. & Tuffin, B., 2012. "Estimation of the mixed logit likelihood function by randomized quasi-Monte Carlo," Transportation Research Part B: Methodological, Elsevier, vol. 46(2), pages 305-320.
    21. Yu, Jie & Goos, Peter & Vandebroek, Martina, 2010. "Comparing different sampling schemes for approximating the integrals involved in the efficient design of stated choice experiments," Transportation Research Part B: Methodological, Elsevier, vol. 44(10), pages 1268-1289, December.

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