Geodesic
Ninomiya-Victoir scheme : Multilevel Monte Carlo estimators and discretization of the involved Ordinary Differential Equations
A. Al Gerbi1 ; B. Jourdain1 ; E. Clément2
1 Université Paris-Est, Cermics (ENPC), INRIA, F-77455, Marne-la-Vallée, France,
2 Université Paris-Est, LAMA (UMR 8050), UPEMLV, UPEC, CNRS, F-77454, Marne-la-Vallée, France,
ESAIM. Proceedings, Tome 59 (2017), pp. 1-14.

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In this paper, we recall the result about the strong convergence rate of the Ninomiya-Victoir scheme and the properties of the multilevel Monte Carlo estimators involving this scheme that we introduced and studied in [2]. We are also interested in the error introduced by discretizing the ordinary differential equations involved in the Ninomiya-Victoir scheme. We prove that this error converges with strong order 2 when an explicit Runge-Kutta method with order 4 (resp. 2) is used for the ODEs corresponding to the Brownian (resp. Stratonovich drift) vector fields. We thus relax the order 5 needed in [11] for the Brownian ODEs to obtain the same order of strong convergence. Moreover, the properties of our multilevel Monte-Carlo estimators are preserved when these Runge-Kutta methods are used.
DOI : 10.1051/proc/201759001

A. Al Gerbi 1 ; B. Jourdain 1 ; E. Clément 2

1 Université Paris-Est, Cermics (ENPC), INRIA, F-77455, Marne-la-Vallée, France,
2 Université Paris-Est, LAMA (UMR 8050), UPEMLV, UPEC, CNRS, F-77454, Marne-la-Vallée, France, 
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     author = {A. Al Gerbi and B. Jourdain and E. Cl\'ement},
     title = {Ninomiya-Victoir scheme : {Multilevel} {Monte} {Carlo} estimators and discretization of the involved {Ordinary} {Differential} {Equations}},
     journal = {ESAIM. Proceedings},
     pages = {1--14},
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     doi = {10.1051/proc/201759001},
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A. Al Gerbi; B. Jourdain; E. Clément. Ninomiya-Victoir scheme : Multilevel Monte Carlo estimators and discretization of the involved Ordinary Differential Equations. ESAIM. Proceedings, Tome 59 (2017), pp. 1-14. doi : 10.1051/proc/201759001. http://geodesic.mathdoc.fr/articles/10.1051/proc/201759001/

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