• DOI:10.1007/s10543-022-00920-4
• Corpus ID: 239998054

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

@article{Leobacher2020WellposednessAN,
  title={Well-posedness and numerical schemes for one-dimensional McKean–Vlasov equations and interacting particle systems with discontinuous drift},
  author={Gunther Leobacher and Christoph Reisinger and Wolfgang Stockinger},
  journal={BIT Numerical Mathematics},
  year={2020},
  volume={62},
  pages={1505 - 1549}
}

• G. Leobacher, C. Reisinger, Wolfgang Stockinger
• Published 26 June 2020
• Mathematics
• BIT Numerical Mathematics

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 presented schemes.