Well-posedness and numerical schemes for one-dimensional McKean-Vlasov equations and interacting particle systems with discontinuous drift

TL;DR

This paper proves well-posedness for one-dimensional McKean-Vlasov SDEs with discontinuous drifts, and analyzes Euler-Maruyama schemes for particle approximation, highlighting challenges due to drift discontinuities.

Contribution

It establishes well-posedness under mild conditions and provides convergence analysis for numerical schemes with discontinuous drifts, which is novel.

Findings

Well-posedness results for McKean-Vlasov SDEs with discontinuous drift.

Strong convergence of Euler-Maruyama schemes under measure-dependent discontinuous drift.

Standard convergence order 1/2 does not hold due to drift discontinuity.

Abstract

In this paper, we first establish well-posedness results for one-dimensional McKean-Vlasov stochastic differential equations (SDEs) and related particle systems with a measure-dependent drift coefficient that is discontinuous in the spatial component, and a diffusion coefficient which is a Lipschitz function of the state only. We only require a fairly mild condition on the diffusion coefficient, namely to be non-zero in a point of discontinuity of the drift, while we need to impose certain structural assumptions on the measure-dependence of the drift. Second, we study Euler-Maruyama type schemes for the particle system to approximate the solution of the one-dimensional McKean-Vlasov SDE. Here, we will prove strong convergence results in terms of the number of time-steps and number of particles. Due to the discontinuity of the drift, the convergence analysis is non-standard and the usual strong convergence order $1/2$ known for the Lipschitz case cannot be recovered for all schemes.