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 fully implementable 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.
翻译:在本文中,我们首先为单维McKan-Vlassov Stochatic 差异方程式及相关粒子系统建立良好的监测结果,其空间组成部分中不连续的量性漂移系数,以及只有国家功能Lipschitz的传播系数。我们只需要对扩散系数有一个相当温和的条件,即在漂移不连续的点上为非零,而我们需要对漂移的衡量依赖性施加某些结构性假设。第二,我们研究微粒系统完全可执行的欧勒山型微粒系统方案,以近似单维McKan-Vlassov SDE的解决方案。在这里,我们将证明在时间步数和粒子数量方面有很强的趋同结果。由于漂移的不连续,趋同分析是非标准性的,通常无法在所有方案中恢复Lipschitz案已知的强趋同1/2美元。