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：本文考虑与部分观测的McKean-Vlasov随机微分方程（SDEs）相关的滤波问题。模型由规则离散时间观测的数据构成，目标是计算当前时刻SDE解（泛函）的条件期望。该问题即使对于普通SDE情形也颇具挑战性，需要数值近似方法。基于文献[3,12]的思想，我们开发了一种新的粒子滤波（PF）和多层粒子滤波（MLPF）来逼近上述期望。在假设条件下证明了：对于$\epsilon>0$，为达到$\mathcal{O}(\epsilon^2)$的均方误差，PF每次观测时间的计算成本为$\mathcal{O}(\epsilon^{-5})$，而MLPF在最优情形下成本为$\mathcal{O}(\epsilon^{-4})$，最差情形为$\mathcal{O}(\epsilon^{-4}\log(\epsilon)^2)$。我们的理论结果得到了数值实验的支持。