Session A7 - Stochastic Computation
July 12, 15:00 ~ 15:25 - Room B2
Strong convergence properties of the Ninomiya Victoir scheme and applications to multilevel Monte Carlo
University Paris-Est, Ecole des Ponts, CERMICS, France - firstname.lastname@example.org
We prove that the strong convergence rate of the Ninomiya-Victoir scheme is 1/2. The normalized error converges stably to the solution of an affine SDE with a source term involving the commutators between the Brownian vector fields. When the Brownian vector fields commute, this limit vanishes and we show that the strong convergence rate improves to 1. We also show that averaging the order of integration of the Brownian fields leads to a coupling with strong order 1 with the scheme proposed by Giles and Szpruch (2014) in order to achieve the optimal complexity in the multilevel Monte Carlo method. Last, we are 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 so that the convergence properties of our multilevel estimators are preserved 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 for the Brownian ODEs needed by Ninomiya and Ninomiya (2009) to obtain the same order of strong convergence.
Joint work with Anis Al Gerbi (Ecole des Ponts, CERMICS) and Emmanuelle Clément (Ecole Centrale Paris, MICS).