Asymptotics for the normalized error of the Ninomiya–Victoir scheme

In Gerbi et al. (2016) we proved strong convergence with order 1∕2 of the Ninomiya–Victoir scheme X^NV,η with time step T∕N to the solution X of the limiting SDE. In this paper we check that the normalized error defined by NX−X^NV,η converges to an affine SDE with source terms involving the Lie brackets between the Brownian vector fields. The limit does not depend on the Rademacher random variables η. This result can be seen as a first step to adapt to the Ninomiya–Victoir scheme the central limit theorem of Lindeberg Feller type, derived in Ben Alaya and Kebaier (2015) for the multilevel Monte Carlo 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 strong convergence with order 1 and study the limit of the normalized error NX−X^NV,η. The limiting SDE involves the Lie brackets between the Brownian vector fields and the Stratonovich drift vector field. When all the vector fields commute, the limit vanishes, which is consistent with the fact that the Ninomiya–Victoir scheme coincides with the solution to the SDE on the discretization grid.