  In this paper, we are interested in deriving non-asymptotic error bounds for
the multilevel Monte Carlo method. As a first step, we deal with the explicit
Euler discretization of stochastic differential equations with a constant
diffusion coefficient. We prove that, as long as the deviation is below an
explicit threshold, a Gaussian-type concentration inequality optimal in terms
of the variance holds for the multilevel estimator. To do so, we use the
Clark-Ocone representation formula and derive bounds for the moment generating
functions of the squared difference between a crude Euler scheme and a finer
one and of the squared difference of their Malliavin derivatives.

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