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Error Estimation for Quasi-Monte Carlo Methods

  • Giray Ökten
Part of the Lecture Notes in Statistics book series (LNS, volume 127)

Abstract

A hybrid-Monte Carlo method designed to obtain a statistical error analysis using deterministic sequences is introduced. The method, which is called “random sampling from low-discrepancy sequences”, produces estimates that satisfy deterministic error bounds yet confidence interval analysis can be applied to measure the accuracy of them.

A particular implementation of the hybrid method is applied to three problems; one from mathematical finance and others from particle transport theory. The method is compared with the pseudorandom method and two quasirandom methods. Encouraging numerical results in favor of the hybrid method are obtained.

Keywords

Sample Standard Deviation Monte Carlo Sequence European Call Option Mixed Sequence Root Sequence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    H. Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods, CBMS-SIAM, 1992.CrossRefzbMATHGoogle Scholar
  2. [2]
    E. Braaten and G. Weller, “An Improved Low-Discrepancy Sequence for Multidimensional Quasi-Monte Carlo Integration”, J. Comp. Physics, 33, 249–258, 1979.CrossRefzbMATHGoogle Scholar
  3. [3]
    H. Faure, “Using permutations to reduce discrepancy”, J. Comp. Appl. Math., 31, 97–103, 1990.CrossRefzbMATHMathSciNetGoogle Scholar
  4. [4]
    J. Spanier, “Quasi-Monte Carlo Methods for Particle Transport Problems”, Proc. Conf. on Monte Carlo Methods in Scientific Computing, Univ. Las Vegas, 1994.Google Scholar
  5. [5]
    B. S. Moskowitz, “Application of quasi-random sequences to Monte Carlo methods”, Ph. D. Dissertation, UCLA, 1993.Google Scholar
  6. [6]
    G. Ökten, “A Probabilistic Result on the Discrepancy of a Hybrid-Monte Carlo Sequence and Applications”, Monte Carlo Methods and Applications, Vol. 2, No. 4, 255–270, 1996.CrossRefzbMATHMathSciNetGoogle Scholar
  7. [7]
    A. B. Owen, “Randomly Permuted (t, m, s)—Nets and (t,s)—Sequences”, Proc. Conf. on Monte Carlo Methods in Scientific Computing, Univ. Las Vegas, 1994.Google Scholar
  8. [8]
    R. Cranley and T. N. L. Patterson, “Randomization of Number Theoretic Methods for Multiple Integration”, SIAM J. Numer. Anal., Vol. 13, No. 6, 904–914, 1976.CrossRefzbMATHMathSciNetGoogle Scholar
  9. [9]
    S. Joe, “Randomization of lattice rules for numerical multiple integration”, J. Comp. Appl. Math., 31, 299–304, 1990.CrossRefzbMATHMathSciNetGoogle Scholar
  10. [10]
    J. C. Hull, Options,Futures, And Other Derivative Securities, Prentice-Hall, Inc., 1989.Google Scholar
  11. [11]
    Y. R. Rubinstein, Simulation And The Monte Carlo Method, John Wiley & Sons, 1981.CrossRefzbMATHGoogle Scholar
  12. [12]
    G. Pagès and Y. J. Xiao, “Sequences with low discrepancy and pseudorandom numbers: theoretical remarks and numerical tests”, Technical Report, Laboratoire de Mathematiques et Modelisation, Paris, 1991.Google Scholar
  13. [13]
    B. Efron and R. Tibshirani, An Introduction to the Bootstrap, Chapman & Hall, London, 1993.CrossRefzbMATHGoogle Scholar
  14. [14]
    J. Spanier and E. M. Gelbard, Monte Carlo Principles And Neutron Transport Problems, Addison-Wesley, 1969.zbMATHGoogle Scholar
  15. [15]
    J. Spanier, “An Analytic Approach to Variance Reduction”, SIAM J. Appl. Math., 18, 172–190, 1970.CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Giray Ökten
    • 1
  1. 1.Department of MathematicsThe Claremont Graduate SchoolClaremontUSA

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