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

06/26/2020
by   Gunther Leobacher, et al.
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In this paper, we first establish well-posedness results of McKean-Vlasov stochastic differential equations (SDEs) and related particle systems with a drift coefficient that is discontinuous in the spatial component. The analysis is restricted to the one-dimensional case, and we only require a fairly mild condition on the diffusion coefficient, namely to be non-zero in a point of discontinuity, while we need to impose certain structural assumptions on the measure-dependence of the drift. Second, we study fully implementable Euler-Maruyama-type schemes for the particle system to approximate the solution of the McKean-Vlasov SDEs. Here, we will prove strong convergence results in terms of number of time-steps and number of particles. Due to the discontinuity of the drift, the convergence analysis is non-standard and the usual 1/2 strong convergence order known for the Lipschitz case cannot be recovered for all presented schemes.

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