AI 中文总结
针对McKean-Vlasov随机微分方程，提出一种带额外粗粒子的单系综多层蒙特卡洛方法，在全局Lipschitz条件下证明L^p误差O(ε)且成本O(ε^{-2-δ})，实验验证速率。
AI 中文摘要
数值求解McKean-Vlasov随机微分方程在计算上具有挑战性，因为时间离散化和解分布离散化的成本会叠加。多层思想已被提出以提供加速。本文研究了Ricketson（2015）提出的多层蒙特卡洛方法，该方法适用于漂移项和扩散项通过期望E[R(X_t)]依赖于解X_t的分布的方程。该方案遵循单系综范式，其中粒子在每个时间步跨层相互作用。虽然跨层反馈使该方案在实践中具有吸引力，但它引入的相关性迄今已将其成本-误差分析限制在一个具有线性漂移、确定性扩散和R为恒等映射的模型问题上。我们使用额外的粗粒子来强制耦合误差向更粗层几何衰减。这使我们能够证明我们的主要贡献，即对于任意p≥2和δ>0，在成本O(ε^{-2-δ})下达到O(ε)的L^p误差（常数随δ→0而增长），仅假设漂移、扩散和R具有全局Lipschitz界。一项探索性实验与推导的速率一致，并表明在实践中，对于小δ，常数不会显著增长。我们的方法和证明策略也可能适用于其他单系综多层方案，例如多层系综卡尔曼滤波器。
英文摘要
Numerically solving McKean-Vlasov stochastic differential equations is computationally challenging due to the compounding costs of discretizing in time and in the distribution of the solution. Multilevel ideas have been proposed to provide speed-ups. In this work, we study the multilevel Monte Carlo method proposed by Ricketson (2015) for equations whose drift and diffusion terms depend on the law of the solution $X_t$ through the expectation $E[R(X_t)]$. The scheme follows the single-ensemble paradigm, where particles interact across levels at each timestep. While cross-level feedback makes this scheme attractive in practice, the correlations it introduces have so far confined its cost-error analysis to a model problem with linear drift, deterministic diffusion, and $R$ the identity. We use additional coarse particles to enforce geometrically decaying coupling errors towards the coarser levels. This allows us to prove our main contribution, an $L^p$-error of $O(ε)$ at cost $O(ε^{-2-δ})$ for any $p\ge2$ and $δ>0$ (with a constant that grows as $δ\to0$), assuming only global Lipschitz bounds on the drift, the diffusion, and $R$. An exploratory experiment is consistent with the derived rates and suggests that in practice the constant does not grow significantly for small $δ$. Our methodology and proof strategy may also be useful for other single-ensemble multilevel schemes, such as multilevel ensemble Kalman filters.
