Asymptotics for the normalized error of the Ninomiya–Victoir scheme.

Summary n Gerbi et al. (2016) we proved strong convergence with order 1/2 of the Ninomiya–Victoir schemeXN V,ηwith time stepT/Nto the solutionXof the limiting SDE. In this paper we check that thenormalized error defined by√N(X−XN V,η)converges to an affine SDE with source terms involvingthe Lie brackets between the Brownian vector fields. The limit does not depend on the Rademacher randomvariablesη. This result can be seen as a first step to adapt to the Ninomiya–Victoir scheme the centrallimit theorem of Lindeberg Feller type, derived in Ben Alaya and Kebaier (2015) for the multilevel MonteCarlo estimator based on the Euler scheme. When the Brownian vector fields commute, the limit vanishes.This suggests that the rate of convergence is greater than 1/2 in this case and we actually prove strongconvergence with order 1 and study the limit of the normalized errorN(X−XN V,η). The limiting SDEinvolves the Lie brackets between the Brownian vector fields and the Stratonovich drift vector field. Whenall the vector fields commute, the limit vanishes, which is consistent with the fact that the Ninomiya–Victoirscheme coincides with the solution to the SDE on the discretization grid.