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.
"Non-asymptotic error bounds for the multilevel Monte Carlo Euler method applied to SDEs with constant diffusion coefficient." Electron. J. Probab. 24 1 - 34, 2019. https://doi.org/10.1214/19-EJP271