MV-SDEs灵活的分步灵活计划 (A flexible split-step scheme for MV-SDEs)

We present an implicit Split-Step explicit Euler type Method (dubbed SSM) for the simulation of McKean-Vlasov Stochastic Differential Equations (MV-SDEs) with drifts of super-linear growth in space, Lipschitz in measure and non-constant Lipschitz diffusion coefficient. The scheme is designed to leverage the structure induced by the interacting particle approximation system, including parallel implementation and the solvability of the implicit equation. The scheme attains the classical one-half root mean square error (rMSE) convergence rate in stepsize and closes the gap left by [18, "Simulation of McKean-Vlasov SDEs with super-linear growth" in IMA Journal of Numerical Analysis, 01 2021. draa099] regarding efficient implicit methods and their convergence rate for this class of McKean-Vlasov SDEs. A sufficient condition for the mean-square contractivity of the scheme is presented. Several numerical examples are presented including a comparative analysis of other known algorithms for this class (taming and adaptive time-stepping) across parallel and non-parallel implementations.

翻译：我们提出了一个隐含的分解分流直线 Euler 类型方法(dubbed SSM),用于模拟McKan-Vlasov Stochatic Equations(MV-SDEs),模拟空间超线性增长的流动,Lipschitz 测量和非contant Lipschitz 扩散系数,Lipschitz 测量和非contant Lipschitz 扩散系数,这个办法旨在利用由交互式粒子近似系统引发的结构,包括平行实施和隐含方程式的可溶性。这个办法在步骤化和缩小[18,“Mckean-Vlasov SDEs模拟超线性增长”后留下的差距方面达到了典型的半根均差正方差率(rMSE),在IMA Nual Numerical 分析杂志,01 2021. 德拉099,关于高效的隐含方法及其这一类的趋同率。提出了该办法中平均值的充足条件。这个办法的折中合同性条件。提出了几个数字例子,包括对这一类的已知的其他算法的比较分析。