This work develops a particle system addressing the approximation of McKean-Vlasov stochastic differential equations (SDEs). The novelty of the approach lies in involving low discrepancy sequences nontrivially in the construction of a particle system with coupled noise and initial conditions. Weak convergence for SDEs with additive noise is proven. A numerical study demonstrates that the novel approach presented here doubles the respective convergence rates for weak and strong approximation of the mean-field limit, compared with the standard particle system. These rates are proven in the simplified setting of a mean-field ordinary differential equation in terms of appropriate bounds involving the star discrepancy for low discrepancy sequences with a group structure, such as Rank-1 lattice points. This construction nontrivially provides an antithetic multilevel quasi-Monte Carlo estimator. An asymptotic error analysis reveals that the proposed approach outperforms methods based on the classic particle system with independent initial conditions and noise.

翻译：本研究构建了一个用于逼近McKean-Vlasov随机微分方程（SDE）的粒子系统。该方法的创新之处在于，将低差异序列非平凡地应用于构建具有耦合噪声和初始条件的粒子系统。我们证明了加性噪声SDE的弱收敛性。数值研究表明，与标准粒子系统相比，本文提出的新方法将平均场极限的弱逼近和强逼近的收敛速率分别提高了一倍。这些速率在平均场常微分方程的简化设定下得到了证明，其证明依赖于对具有群结构的低差异序列（如Rank-1格点）的星差异的适当界。该构造非平凡地提供了一个对偶多水平拟蒙特卡洛估计量。渐近误差分析表明，所提出的方法优于基于具有独立初始条件和噪声的经典粒子系统的方法。