Monte Carlo and Quasi-Monte Carlo Methods 2010

Volume 23 of the series Springer Proceedings in Mathematics & Statistics pp 471-486


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

  • Reiichiro KawaiAffiliated withDepartment of Mathematics, University of Leicester Email author 
  • , Junichi ImaiAffiliated withFaculty of Science and Technology, Keio University


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.