In this paper we consider the filtering problem associated to partially observed McKean-Vlasov stochastic differential equations (SDEs). The model consists of data that are observed at regular and discrete times and the objective is to compute the conditional expectation of (functionals) of the solutions of the SDE at the current time. This problem, even the ordinary SDE case is challenging and requires numerical approximations. Based upon the ideas in [3, 12] we develop a new particle filter (PF) and multilevel particle filter (MLPF) to approximate the afore-mentioned expectations. We prove under assumptions that, for $\epsilon>0$, to obtain a mean square error of $\mathcal{O}(\epsilon^2)$ the PF has a cost per-observation time of $\mathcal{O}(\epsilon^{-5})$ and the MLPF costs $\mathcal{O}(\epsilon^{-4})$ (best case) or $\mathcal{O}(\epsilon^{-4}\log(\epsilon)^2)$ (worst case). Our theoretical results are supported by numerical experiments.

翻译：本文研究部分观测McKean-Vlasov随机微分方程（SDE）的滤波问题。模型包含在规则离散时间点观测的数据，目标在于计算当前时刻SDE解（泛函）的条件期望。该问题即便在普通SDE情形下也具有挑战性，需要借助数值近似方法。基于[3,12]中的思想，我们提出了一种新型粒子滤波（PF）与多层粒子滤波（MLPF）来逼近上述期望。我们在一定假设下证明：对于$\epsilon>0$，为达到均方误差$\mathcal{O}(\epsilon^2)$，PF在每个观测时间步的计算成本为$\mathcal{O}(\epsilon^{-5})$，而MLPF的计算成本为$\mathcal{O}(\epsilon^{-4})$（最优情形）或$\mathcal{O}(\epsilon^{-4}\log(\epsilon)^2)$（最差情形）。数值实验支持了我们的理论结果。