On the Convergence of the Euler-Maruyama Scheme for McKean-Vlasov SDEs

Abstract

On the convergence of the Euler-Maruyama scheme for McKean-Vlasov SDEs: 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 $N^{-1} + h$.