In this paper, we prove that the weak error between a stochastic differential equation with nonlinearity in the sense of McKean given by moments and its approximation by the Euler discretization with time-step h of a system of N interacting particles is O(N −1 + h). We provide numerical experiments confirming this behaviour and showing that it extends to more general mean-field interaction and study the efficiency of the antithetic sampling technique on the same examples.

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We are interested in the Euler-Maruyama discretization of a stochastic differential equation in dimension d with constant diffusion coefficient and bounded measurable drift coefficient. In the scheme, a randomization of the time variable is used to get rid of any regularity assumption of the drift in this variable. We prove weak convergence with order 1/2 in total variation distance. When the drift has a spatial divergence in the sense of distributions with ρ-th power integrable with respect to the Lebesgue measure in space uniformly in time for some ρ ≥ d, the order of convergence at the terminal time improves to 1 up to some logarithmic factor. In dimension d = 1, this result is preserved when the spatial derivative of the drift is a measure in space with total mass bounded uniformly in time. We confirm our theoretical analysis by numerical experiments.