随机McKean-Vlasov微分方程分裂步θ方法的长时间性态分析

Analysis of Long-Time Behavior of the Split-Step θ Method for Stochastic McKean-Vlasov Differential Equations

摘要: 本文主要研究漂移项系数满足超线性增长及单边Lipschitz条件的随机McKean-Vlasov微分方程的分裂步$\theta$ 方法的长时间性态。基于对分布的交互粒子逼近，引入了分裂步$\theta$ 方法对相应的粒子系统进行离散化处理。研究表明：当参数$\theta$ 满足$\frac{1}{2}\le \theta \le 1$ 时，该分裂步$\theta$ 方法具有均方收缩性；而当$\frac{1}{2}<\theta \le 1$ 时，该方法具有均方稳定性；而对于$0\le \theta \le \frac{1}{2}$ 的情形，则需要附加线性增长条件以保证均方稳定性。最后，我们通过数值实验验证了理论结果。

Abstract: This paper focuses on the long-time behavior of the split-step θ-method for stochastic McKean-Vlasov differential equations with drift coefficients satisfying super-linear growth and a one-sided Lipschitz condition. Based on the interacting-particle approximation of the distribution, a split-step θ-method is introduced to discretize the corresponding particle system. The analysis shows that the method possesses mean-square contraction when the parameter θ lies in the interval $\frac{1}{2}\le \theta \le 1$ , and that it is mean-square stable for $\frac{1}{2}<\theta \le 1$ . For the case $0\le \theta \le \frac{1}{2}$ , an additional linear growth condition is required to ensure mean-square stability. Finally, numerical experiments are provided to validate the theoretical findings.