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Randomized Dimension Reduction for Monte Carlo Simulations

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  • Nabil Kahalé

    (ESCP Europe, LabEx ReFi, 75011 Paris, France)

Abstract

We present a new unbiased algorithm that estimates the expected value of f ( U ) via Monte Carlo simulation, where U is a vector of d independent random variables, and f is a function of d variables. We assume that f does not equally depend on all its arguments. Under certain conditions, we prove that, for the same computational cost, the variance of our estimator is lower than the variance of the standard Monte Carlo estimator by a factor of order d . Our method can be used to obtain a low-variance unbiased estimator for the expectation of a function of the state of a Markov chain at a given time step. We study applications to volatility forecasting and time-varying queues. Numerical experiments show that our algorithm dramatically improves on the standard Monte Carlo method for large values of d and is highly resilient to discontinuities.

Suggested Citation

  • Nabil Kahalé, 2020. "Randomized Dimension Reduction for Monte Carlo Simulations," Management Science, INFORMS, vol. 66(3), pages 1421-1439, March.
    • Handle: RePEc:inm:ormnsc:v:66:y:2020:i:3:p:1421-1439
    DOI: 10.1287/mnsc.2018.3250
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    References listed on IDEAS

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    1. Zdravko I. Botev & Pierre L'Ecuyer & Gerardo Rubino & Richard Simard & Bruno Tuffin, 2013. "Static Network Reliability Estimation via Generalized Splitting," INFORMS Journal on Computing, INFORMS, vol. 25(1), pages 56-71, February.
    2. Paul Glasserman & Philip Heidelberger & Perwez Shahabuddin, 1999. "Asymptotically Optimal Importance Sampling and Stratification for Pricing Path‐Dependent Options," Mathematical Finance, Wiley Blackwell, vol. 9(2), pages 117-152, April.
    3. Pierre L'Ecuyer & Christiane Lemieux, 2000. "Variance Reduction via Lattice Rules," Management Science, INFORMS, vol. 46(9), pages 1214-1235, September.
    4. Kai Nagel & Peter Wagner & Richard Woesler, 2003. "Still Flowing: Approaches to Traffic Flow and Traffic Jam Modeling," Operations Research, INFORMS, vol. 51(5), pages 681-710, October.
    5. 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.
    6. Peter W. Glynn & Ward Whitt, 1992. "The Asymptotic Efficiency of Simulation Estimators," Operations Research, INFORMS, vol. 40(3), pages 505-520, June.
    7. Imry Rosenbaum & Jeremy Staum, 2017. "Multilevel Monte Carlo Metamodeling," Operations Research, INFORMS, vol. 65(4), pages 1062-1077, August.
    8. 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.
    9. Fredrik Åkesson & John P. Lehoczky, 2000. "Path Generation for Quasi-Monte Carlo Simulation of Mortgage-Backed Securities," Management Science, INFORMS, vol. 46(9), pages 1171-1187, September.
    10. Michael B. Giles, 2008. "Multilevel Monte Carlo Path Simulation," Operations Research, INFORMS, vol. 56(3), pages 607-617, June.
    11. 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.
    12. Liu, Ruixue & Owen, Art B., 2006. "Estimating Mean Dimensionality of Analysis of Variance Decompositions," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 712-721, June.
    13. Xiaoqun Wang, 2006. "On the Effects of Dimension Reduction Techniques on Some High-Dimensional Problems in Finance," Operations Research, INFORMS, vol. 54(6), pages 1063-1078, December.
    14. Zohar Feldman & Avishai Mandelbaum & William A. Massey & Ward Whitt, 2008. "Staffing of Time-Varying Queues to Achieve Time-Stable Performance," Management Science, INFORMS, vol. 54(2), pages 324-338, February.
    15. Chang-Han Rhee & Peter W. Glynn, 2015. "Unbiased Estimation with Square Root Convergence for SDE Models," Operations Research, INFORMS, vol. 63(5), pages 1026-1043, October.
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    Cited by:

    1. Nabil Kahale, 2023. "Simulating Gaussian vectors via randomized dimension reduction and PCA," Papers 2304.07377, arXiv.org.

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