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 epsilon greater than 0 ε > 0 $\varepsilon>0$ , to obtain a mean square error of script upper O left parenthesis epsilon squared right parenthesis O ( ε 2 ) $\mathcal{O}(\varepsilon^2)$ the PF has a cost per observation time of script upper O left parenthesis epsilon Superscript negative 5 Baseline right parenthesis O ( ε − 5 ) $\mathcal{O}(\varepsilon^{-5})$ , and the MLPF costs script upper O left parenthesis epsilon Superscript negative 4 Baseline right parenthesis O ( ε − 4 ) $\mathcal{O}(\varepsilon^{-4})$ (best case) or script upper O left parenthesis epsilon Superscript negative 4 Baseline log left parenthesis epsilon right parenthesis squared right parenthesis O ( ε − 4 log ⁡ ( ε ) 2 ) $\mathcal{O}(\varepsilon^{-4}\log(\varepsilon)^2)$ (worst case). Our theoretical results are supported by numerical experiments.