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

  • Anis Al Gerbi EMAIL logo , Benjamin Jourdain and Emmanuelle Clément

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. 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) 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.

MSC 2010: 65C30; 60H35; 65C05

Funding statement: This research benefited from the support of the “Chaire Risques Financiers”, Fondation du Risque.

References

[1] Alfonsi A., Affine Diffusions and Related Processes: Simulation, Theory and Applications, Bocconi Springer Ser. 6, Springer, Cham, 2015. 10.1007/978-3-319-05221-2Search in Google Scholar

[2] Al Gerbi A., Jourdain B. and Clément E., Ninomiya–Victoir scheme: Strong convergence, antithetic version and application to multilevel estimators, preprint 2015, http://arxiv.org/abs/1508.06492. 10.1515/mcma-2016-0109Search in Google Scholar

[3] Debrabant K. and Rößler A., On the acceleration of the multi-level Monte Carlo method, J. Appl. Probab. 52 (2015), 307–322. 10.1239/jap/1437658600Search in Google Scholar

[4] Duffie D. and Glynn P., Efficient Monte Carlo simulation of security prices, Ann. Appl. Probab. 5 (1995), 897–905. 10.1214/aoap/1177004598Search in Google Scholar

[5] Fujiwara T., Sixth order methods of Kusuoka approximation, preprint 2006, http://kyokan.ms.u-tokyo.ac.jp/users/preprint/pdf/2006-7.pdf. Search in Google Scholar

[6] Giles M. B., Multilevel Monte Carlo path simulation, Oper. Res. 56 (2008), 607–617. 10.1287/opre.1070.0496Search in Google Scholar

[7] Giles M. B. and Szpruch L., Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without Lévy area simulation, Ann. Appl. Probab. 24 (2014), 1585–1620. 10.1214/13-AAP957Search in Google Scholar

[8] Lemaire V. and Pagès G., Multilevel Richardson–Romberg extrapolation, preprint 2014, http://arxiv.org/abs/1401.1177; to appear in Bernoulli. 10.2139/ssrn.2539114Search in Google Scholar

[9] Ninomiya S. and Victoir N., Weak approximation of stochastic differential equations and application to derivative pricing, Appl. Math. Finance 15 (2008), 107–121. 10.1080/13504860701413958Search in Google Scholar

[10] Oshima K., Teichmann J. and Velušček D., A new extrapolation method for weak approximation schemes with applications, Ann. Appl. Probab. 22 (2012), 1008–1045. 10.1214/11-AAP774Search in Google Scholar

[11] Pagès G., Multi-step Richardson–Romberg extrapolation: Remarks on variance control and complexity, Monte Carlo Methods Appl. 13 (2007), 37–70. 10.1515/MCMA.2007.003Search in Google Scholar

[12] Revuz D. and Yor M., Continuous Martingales and Brownian Motion, 3rd ed., Grundlehren Math. Wiss. 293, Springer, Berlin, 1999. 10.1007/978-3-662-06400-9Search in Google Scholar

Received: 2015-10-14
Accepted: 2016-6-9
Published Online: 2016-6-30
Published in Print: 2016-9-1

© 2016 by De Gruyter

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