An Euler scheme for McKean SDEs with Besov drift: convergence rate and implementation

TL;DR

This paper introduces a novel Euler scheme for one-dimensional McKean-Vlasov SDEs with Besov drift, proving convergence rates and demonstrating its effectiveness through numerical experiments.

Contribution

It presents the first implementable numerical scheme for McKean-Vlasov SDEs with Besov drift, combining mollification, Euler-Maruyama, and PDE methods.

Findings

Proven strong convergence of the scheme with explicit rate.

Numerical experiments validate the scheme's applicability.

Demonstrated the impact of McKean interaction on the law of the solution.

Abstract

We study a one-dimensional McKean-Vlasov stochastic differential equation (SDE) with a drift equal to a product of a distribution depending on the state of the process and a non-linear function depending pointwise on the law density of the solution. Building on recent well-posedness results, we propose the first implementable numerical scheme for this class of SDEs. Our approach combines mollification of the distributional drift with the Euler-Maruyama scheme and a PDE-based approximation of the law via the associated Fokker-Planck equation. We prove strong convergence of the scheme and derive an explicit rate, showing how to balance the smoothing parameter with the time discretisation. Numerical experiments confirm the applicability of our scheme and demonstrate the significant influence of the McKean interaction term on the law of the solution.