Multilevel particle filters for partially observed McKean–Vlasov stochastic differential equations

Abstract We consider the filtering problem associated with partially observed McKean–Vlasov stochastic differential equations (SDEs). The model consists of data that are observed at regular and discrete times; the objective is to compute the conditional expectation of (functionals) of the solutions of the SDE at the current time. This problem is challenging even in the ordinary SDE case and requires numerical approximations. Based on the ideas in Ben Rached et al. (2024) and dos Reis et al. (2023), we develop a new particle filter (PF) and multilevel particle filter (MLPF) to approximate the aforementioned expectations. We prove under assumptions that, for $\varepsilon>0$ , to obtain a mean square error of $\mathcal{O}(\varepsilon^2)$ the PF has a cost per observation time of $\mathcal{O}(\varepsilon^{-5})$ , and the MLPF costs $\mathcal{O}(\varepsilon^{-4})$ (best case) or $\mathcal{O}(\varepsilon^{-4}\log(\varepsilon)^2)$ (worst case). Our theoretical results are supported by numerical experiments.