We obtain non asymptotic bounds for the Monte Carlo algorithm associated to the Euler discretization of some diffusion processes. The key tool is the Gaussian concentration satisfied by the density of the discretization scheme. This Gaussian concentration is derived from a Gaussian upper bound of the density of the scheme and a modification of the so-called "Herbst argument" used to prove Logarithmic Sobolev inequalities. We eventually establish a Gaussian lower bound for the density of the scheme that emphasizes the concentration is sharp.
"On Some non Asymptotic Bounds for the Euler Scheme." Electron. J. Probab. 15 1645 - 1681, 2010. https://doi.org/10.1214/EJP.v15-814