  In this paper, we implement a weak Milstein Scheme to simulate
low-dimensional stochastic differential equations (SDEs). We prove that
combining the antithetic multilevel Monte-Carlo (MLMC) estimator introduced by
Giles and Szpruch with the MLMC approach for weak SDE approximation methods by
Belomestny and Nagapetyan, we can achieve a quadratic computational complexity
in the inverse of the Root Mean Square Error (RMSE) when estimating expected
values of smooth functionals of SDE solutions, without simulating Levy areas
and without requiring any strong convergence of the underlying SDE
approximation method. By using appropriate discrete variables this approach
allows us to calculate the expectation on the coarsest level of resolution by
enumeration, which results in a reduced computational effort compared to
standard MLMC sampling. These theoretical results are also confirmed by a
numerical experiment.
