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

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


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.



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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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of LeicesterLeicesterUK
  2. 2.Faculty of Science and TechnologyKeio UniversityYokohamaJapan

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