Ninomiya-Victoir scheme : Multilevel Monte-Carlo estimators and discretization of the involved Ordinary Differential Equations

International audienceIn this paper, we summarize the results about the strong convergence rate ofthe Ninomiya-Victoir scheme and the stable convergence in law of its normalizederror that we obtained in previous papers. We then recall the properties of themultilevel Monte Carlo estimators involving this scheme that we introduced andstudied before. Last, we are interested in the error introduced by discretizingthe ordinary differential equations involved in the Ninomiya-Victoir scheme. Weprove that this error converges with strong order 2 when an explicitRunge-Kutta method with order 4 (resp. 2) is used for the ODEs corresponding tothe Brownian (resp. Stratonovich drift) vector fields. We thus relax the order5 for the Brownian ODEs needed by Ninomiya and Ninomiya (2009) to obtain thesame order of strong convergence. Moreover, the properties of our multilevelMonte-Carlo estimators are preserved when these Runge-Kutta methods are used