Convergence of Densities for the Euler--Maruyama Approximation of Brownian-Driven McKean--Vlasov Equations
Abstract
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.
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BibTeX
@misc{yin2026convergence,
title = {{Convergence of Densities for the Euler--Maruyama Approximation of Brownian-Driven McKean--Vlasov Equations}},
author = {Zhenghan Yin and Xu Sun},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19769},
eprint = {2609.19769},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19769},
}
RIS
TY - UNPB
TI - Convergence of Densities for the Euler--Maruyama Approximation of Brownian-Driven McKean--Vlasov Equations
AU - Zhenghan Yin
AU - Xu Sun
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.19769
UR - https://arxiv.org/abs/2609.19769
AB - 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.
ER -