Coupling Importance Sampling and Multilevel Monte Carlo using Sample Average Approximation

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Abstract

In this work, we propose a smart idea to couple importance sampling and Multilevel Monte Carlo (MLMC). We advocate a per level approach with as many importance sampling parameters as the number of levels, which enables us to handle the different levels independently. The search for parameters is carried out using sample average approximation, which basically consists in applying deterministic optimisation techniques to a Monte Carlo approximation rather than resorting to stochastic approximation. Our innovative estimator leads to a robust and efficient procedure reducing both the discretization error (the bias) and the variance for a given computational effort. In the setting of discretized diffusions, we prove that our estimator satisfies a strong law of large numbers and a central limit theorem with optimal limiting variance, in the sense that this is the variance achieved by the best importance sampling measure (among the class of changes we consider), which is however non tractable. Finally, we illustrate the efficiency of our method on several numerical challenges coming from quantitative finance and show that it outperforms the standard MLMC estimator.

Keywords

Sample average approximation Multilevel Monte Carlo Variance reduction Uniform strong large law of numbers Central limit theorem Importance sampling 

Mathematics Subject Classification (2010)

60F05 62F12 65C05 60H35 

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Notes

Acknowledgments

We are grateful to the anonymous referees for their valuable comments and suggestions, which helped us greatly improve the paper.

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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Sorbonne Paris Cité, LAGA, CNRS (UMR 7539)Université Paris 13VilletaneuseFrance
  2. 2.Laboratoire Jean KuntzmannUniversity Grenoble AlpesGrenobleFrance

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