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

TL;DR

This paper introduces new multilevel particle filtering methods for efficiently estimating expectations in partially observed McKean-Vlasov SDEs, reducing computational costs compared to traditional approaches.

Contribution

The paper develops a novel multilevel particle filter for McKean-Vlasov SDEs, providing theoretical analysis and demonstrating improved computational efficiency over standard particle filters.

Findings

MLPF achieves lower computational cost than PF for the same accuracy.

Theoretical cost bounds are established for both PF and MLPF.

Numerical experiments validate the theoretical results.

Abstract

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.