On the convergence of the Euler-Maruyama scheme for McKean-Vlasov SDEs
Noufel Frikha, Xuanye Song
TL;DR
The paper studies the convergence of the Euler–Maruyama time discretization for McKean–Vlasov SDEs via an interacting particle system. It develops a semigroup-based, measure-valued regularity framework on the Wasserstein space and proves explicit strong and weak convergence rates in two regimes: (i) irregular coefficients with non-degenerate diffusion, where both trajectory and semigroup errors are quantified, and (ii) smooth coefficients, where an optimal weak rate $N^{-1}+h$ is established. The approach hinges on backward Kolmogorov PDEs on the Wasserstein space, Gaussian density estimates, and a regularization argument to handle second-order measure derivatives. These results provide dimension-free rates under suitable regularity and offer guidance for choosing the number of particles $N$ and time step $h$ in simulations of mean-field dynamics with applications to propagation of chaos and mean-field games.
Abstract
Building on the well-posedness of the backward Kolmogorov partial differential equation in the Wasserstein space, we analyze the strong and weak convergence rates for approximating the unique solution of a class of McKean-Vlasov stochastic differential equations via the Euler-Maruyama time discretization scheme applied to the associated system of interacting particles. We consider two distinct settings. In the first, the coefficients and test function are irregular, but the diffusion coefficient remains non-degenerate. Leveraging the smoothing properties of the underlying heat kernel, we establish the strong and weak convergence rates of the scheme in terms of the number of particles N and the mesh size h. In the second setting, where both the coefficients and the test function are smooth, we demonstrate that the weak error rate at the level of the semigroup is optimal, achieving an error of order N -1 + h.