What question did this study set out to answer?

This research aims to address the filtering problem in partially observed McKean-Vlasov stochastic differential equations.

February 2, 2026

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

Key Points

This research aims to address the filtering problem in partially observed McKean-Vlasov stochastic differential equations.
Developed a new particle filter (PF) and multilevel particle filter (MLPF) for approximating expectations.
Analyzed numerical costs associated with the PF and MLPF.
Proved performance guarantees under specific assumptions.
The PF achieves a mean square error of O(ε²) with a computational cost of O(ε⁻⁵) per observation time.
The MLPF can have costs of O(ε⁻⁴) in the best case and O(ε⁻⁴ log(ε)²) in the worst case.
The theoretical findings are validated through numerical experiments.

Abstract

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 >0, to obtain a mean square error of O (²) the PF has a cost per observation time of O (^-5), and the MLPF costs O (^-4) (best case) or O (^-4 () ²) (worst case). Our theoretical results are supported by numerical experiments.

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

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