Convergence of Densities for the Euler--Maruyama Approximation of Brownian-Driven McKean--Vlasov Equations

RESUMO

Convergence of the transition density of the Euler-Maruyama scheme for a Brownian-driven McKean-Vlasov stochastic differential equation is established, the coefficients being uniformly elliptic, spatially smooth, and Lipschitz with respect to the law in the 2-Wasserstein distance. The density p_n of the n-step scheme is shown to converge to the density p of the McKean-Vlasov equation in a Gaussian-weighted supremum norm at rate O(n^{-1/2}), that is, of order one half in the step size, and hence also uniformly in space. The proof develops a discrete parametrix (Levi-Hadamard) expansion for the Euler chain and compares it with the continuous parametrix of the limiting equation along a hierarchy of measure-flow, time-discretisation and lattice corrections; the discrepancy between the discrete and the continuous measure flows is of order O(n^{-1/2}) in the 2-Wasserstein distance and does not degrade the final rate.