Convergence rate of Euler-Maruyama scheme for McKean-Vlasov SDEs with density-dependent drift

TL;DR

This paper establishes the convergence rate of the Euler-Maruyama scheme for McKean-Vlasov SDEs with density-dependent drift, ensuring weak well-posedness and providing quantitative error estimates.

Contribution

It introduces a novel analysis of the Euler-Maruyama scheme's convergence for McKean-Vlasov SDEs with density-dependent drift, combining stability estimates and Fokker-Planck equation techniques.

Findings

Proves weak well-posedness of the SDEs.

Derives convergence rate in weighted L^1 norm.

Establishes stability estimates for the Euler-Maruyama scheme.

Abstract

In this paper, we study weak well-posedness of a McKean-Vlasov stochastic differential equations (SDEs) whose drift is density-dependent and whose diffusion is constant. The existence part is due to H\"older stability estimates of the associated Euler-Maruyama scheme. The uniqueness part is due to that of the associated Fokker-Planck equation. We also obtain convergence rate in weighted $L^1$ norm for the Euler-Maruyama scheme.