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

Al Gerbi Anis (), Jourdain Benjamin () and Clément Emmanuelle ()
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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

Monte Carlo Methods and Applications, 2016, vol. 22, issue 3, 197-228

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.

Keywords: Discretisation of SDEs; multilevel Monte Carlo methods; strong convergence; Ninomiya–Victoir scheme (search for similar items in EconPapers)
Date: 2016
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Citations: View citations in EconPapers (2)

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DOI: 10.1515/mcma-2016-0109

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