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Monte Carlo and Quasi-Monte Carlo Methods 2010 pp 471–486Cite as

On Monte Carlo and Quasi-Monte Carlo Methods for Series Representation of Infinitely Divisible Laws

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 23))

Abstract

Infinitely divisible random vectors and Lévy processes without Gaussian component admit representations with shot noise series. To enhance efficiency of the series representation in Monte Carlo simulations, we discuss variance reduction methods, such as stratified sampling, control variates and importance sampling, applied to exponential interarrival times forming the shot noise series. We also investigate the applicability of the generalized linear transformation method in the quasi-Monte Carlo framework to random elements of the series representation. Although implementation of the proposed techniques requires a small amount of initial work, the techniques have the potential to yield substantial improvements in estimator efficiency, as the plain use of the series representation in those frameworks is often expensive. Numerical results are provided to illustrate the effectiveness of our approaches.

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Acknowledgements

The authors would like to thank an anonymous referee for various valuable comments and Japan Society for the Promotion of Science for Grant-in-Aid for Scientific Research 21340024 and 21710157.

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Authors and Affiliations

  1. Department of Mathematics, University of Leicester, Leicester, LE1 7RH, UK

    Reiichiro Kawai

  2. Faculty of Science and Technology, Keio University, Yokohama, 223–8522, Japan

    Junichi Imai

Authors
  1. Reiichiro Kawai
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  2. Junichi Imai
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Correspondence to Reiichiro Kawai .

Editor information

Editors and Affiliations

  1. , Institute of Applied Mathematics, University of Warsaw, Banacha 2, Warsaw, 02-097, Poland

    Leszek Plaskota

  2. , Institute of Applied Mathematics, University of Warsaw, Banacha 2, Warsaw, 02-097, Poland

    Henryk Woźniakowski

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Kawai, R., Imai, J. (2012). On Monte Carlo and Quasi-Monte Carlo Methods for Series Representation of Infinitely Divisible Laws. In: Plaskota, L., Woźniakowski, H. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2010. Springer Proceedings in Mathematics & Statistics, vol 23. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27440-4_26

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