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# Ninomiya-Victoir scheme: strong convergence, antithetic version and application to multilevel estimators

2 MATHRISK - Mathematical Risk Handling
UPEM - Université Paris-Est Marne-la-Vallée, ENPC - École des Ponts ParisTech, Inria de Paris
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 ${\mathcal O}(\epsilon^{-2})$ for the precision $\epsilon$. 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 discretization levels.
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https://hal-enpc.archives-ouvertes.fr/hal-01188675
Contributeur : Benjamin Jourdain <>
Soumis le : lundi 31 août 2015 - 13:42:04
Dernière modification le : mardi 21 avril 2020 - 10:16:04

### Identifiants

• HAL Id : hal-01188675, version 1
• ARXIV : 1508.06492

### Citation

Anis Al Gerbi, Benjamin Jourdain, Emmanuelle Clément. Ninomiya-Victoir scheme: strong convergence, antithetic version and application to multilevel estimators. Monte Carlo Methods and Applications, De Gruyter, 2016, 22 (3), pp.197-228. ⟨hal-01188675⟩

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