Multilevel Particle Filters for Partially Observed McKean-Vlasov Stochastic Differential Equations

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.