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An Antithetic Approach of Multilevel Richardson-Romberg Extrapolation Estimator for Multidimensional SDES

  • Cheikh MbayeEmail author
  • Gilles Pagès
  • Frédéric Vrins
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10187)

Abstract

The Multilevel Richardson-Romberg (ML2R) estimator was introduced by Pagès & Lemaire in [1] in order to remove the bias of the standard Multilevel Monte Carlo (MLMC) estimator in the 1D Euler scheme. Milstein scheme is however preferable to Euler scheme as it allows to reach the optimal complexity \(O(\varepsilon ^{-2})\) for each of these estimators. Unfortunately, Milstein scheme requires the simulation of Lévy areas when the SDE is driven by a multidimensional Brownian motion, and no efficient method is currently available to this purpose so far (except in dimension 2). Giles and Szpruch [2] recently introduced an antithetic multilevel correction estimator avoiding the simulation of these areas without affecting the second order complexity. In this work, we revisit the ML2R and MLMC estimators in the framework of the antithetic approach, thereby allowing us to remove the bias whilst preserving the optimal complexity when using Milstein scheme.

Keywords

Multilevel Monte Carlo Antithetic Multilevel Monte Carlo Richardson-Romberg extrapolation Milstein scheme Option pricing 

Notes

Acknowledgements

The work of Cheikh Mbaye is supported by the Fédération Wallonie-Bruxelles and the National Bank of Belgium via an FSR grant.

The opinions expressed in this paper are those of the authors and do not necessarily reflect the views of the National Bank of Belgium.

References

  1. 1.
    Lemaire, V., Pagès, G.: Multilevel Richardson-Romberg extrapolation. pre-pub. LPMA 1603 arXiv:1401.1177 (2014). (forthcoming in Bernoulli)
  2. 2.
    Giles, M.B., Szpruch, L.: Antithetic multilevel Monte Carlo for multi-dimensional SDEs without Lévy area simulation. Ann. Appl. Prob. 24(4), 1585–1620 (2014)CrossRefzbMATHGoogle Scholar
  3. 3.
    Giles, M.B.: Multilevel Monte Carlo path simulation. Oper. Res. 56(3), 607–617 (2008)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Université Catholique de LouvainLouvain-la-NeuveBelgium
  2. 2.Université Pierre et Marie CurieParisFrance

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