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Ninomiya–Victoir scheme: Strong convergence, antithetic version and application to multilevel estimators

Author

Listed:
  • Al Gerbi Anis

    (Université Paris-Est, Cermics (ENPC), INRIA, F-77455, Marne-la-Vallée, France)

  • Jourdain Benjamin

    (Université Paris-Est, Cermics (ENPC), INRIA, F-77455, Marne-la-Vallée, France)

  • Clément Emmanuelle

    (Université Paris-Est, LAMA (UMR 8050), UPEMLV, UPEC, CNRS, F-77454, Marne-la-Vallée, France)

Abstract

In this paper, we are interested in the strong convergence properties of the Ninomiya–Victoir scheme which is known to exhibit weak convergence with order 2. We prove strong convergence with order 1/2${1/2}$. This study is aimed at analysing the use of this scheme either at each level or only at the finest level of a multilevel Monte Carlo estimator: indeed, the variance of a multilevel Monte Carlo estimator is related to the strong error between the two schemes used on the coarse and fine grids at each level. Recently, Giles and Szpruch proposed a scheme permitting to construct a multilevel Monte Carlo estimator achieving the optimal complexity O⁢(ϵ-2)${O(\epsilon^{-2})}$ for the precision ϵ. In the same spirit, we propose a modified Ninomiya–Victoir scheme, which may be strongly coupled with order 1 to the Giles–Szpruch scheme at the finest level of a multilevel Monte Carlo estimator. Numerical experiments show that this choice improves the efficiency, since the order 2 of weak convergence of the Ninomiya–Victoir scheme permits to reduce the number of discretisation levels.

Suggested Citation

  • Al Gerbi Anis & Jourdain Benjamin & Clément Emmanuelle, 2016. "Ninomiya–Victoir scheme: Strong convergence, antithetic version and application to multilevel estimators," Monte Carlo Methods and Applications, De Gruyter, vol. 22(3), pages 197-228, September.
    • Handle: RePEc:bpj:mcmeap:v:22:y:2016:i:3:p:197-228:n:1
    DOI: 10.1515/mcma-2016-0109
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    References listed on IDEAS

    as
    1. Syoiti Ninomiya & Nicolas Victoir, 2008. "Weak Approximation of Stochastic Differential Equations and Application to Derivative Pricing," Applied Mathematical Finance, Taylor & Francis Journals, vol. 15(2), pages 107-121.
    2. Michael B. Giles, 2008. "Multilevel Monte Carlo Path Simulation," Operations Research, INFORMS, vol. 56(3), pages 607-617, June.
    3. Anis Al Gerbi & Benjamin Jourdain & Emmanuelle Cl'ement, 2015. "Ninomiya-Victoir scheme: strong convergence, antithetic version and application to multilevel estimators," Papers 1508.06492, arXiv.org, revised Oct 2015.
    4. Michael B. Giles & Lukasz Szpruch, 2012. "Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without L\'{e}vy area simulation," Papers 1202.6283, arXiv.org, revised May 2014.
    5. Aurélien Alfonsi, 2015. "Affine Diffusions and Related Processes: Simulation, Theory and Applications," Post-Print hal-03127212, HAL.
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    Cited by:

    1. Giorgi Daphné & Lemaire Vincent & Pagès Gilles, 2017. "Limit theorems for weighted and regular Multilevel estimators," Monte Carlo Methods and Applications, De Gruyter, vol. 23(1), pages 43-70, March.
    2. Al Gerbi, A. & Jourdain, B. & Clément, E., 2018. "Asymptotics for the normalized error of the Ninomiya–Victoir scheme," Stochastic Processes and their Applications, Elsevier, vol. 128(6), pages 1889-1928.

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