论文标题
一维McKean-Vlasov方程和与不连续漂移的相互作用粒子系统的适应性和数值方案
Well-posedness and numerical schemes for one-dimensional McKean-Vlasov equations and interacting particle systems with discontinuous drift
论文作者
论文摘要
在本文中，我们首先为一维McKean-Vlasov随机微分方程（SDE）和相关粒子系统建立了良好的结果，其依赖于度量的漂移系数在空间成分中是不连续的，并且仅是国家的LIPSCHITZ功能。我们只需要在扩散系数上有相当温和的条件，即在漂移的不连续点处是非零的，而我们需要对漂移的度量依赖性施加某些结构性假设。其次，我们研究了粒子系统的Euler-Maruyama类型方案，以近似一维McKean-Vlasov SDE的解决方案。在这里，我们将根据时间步骤的数量和粒子数量来证明强大的收敛结果。由于漂移的不连续性，收敛分析是非标准化的，并且通常无法为Lipschitz案例闻名的通常强的收敛订单$ 1/2 $，无法为所有方案恢复。
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.
