  The multi-level Monte Carlo method proposed by M. Giles (2008) approximates
the expectation of some functionals applied to a stochastic process with
optimal order of convergence for the mean-square error. In this paper, a
modified multi-level Monte Carlo estimator is proposed with significantly
reduced computational costs. As the main result, it is proved that the modified
estimator reduces the computational costs asymptotically by a factor
$(p/\alpha)^2$ if weak approximation methods of orders $\alpha$ and $p$ are
applied in case of computational costs growing with same order as variances
decay.

Probability