Core Concepts
This paper develops new particle filter (PF) and multilevel particle filter (MLPF) methodologies to efficiently estimate the filtering expectations associated with partially observed McKean-Vlasov stochastic differential equations (SDEs). The authors prove that the PF has a cost per-observation time of O(ϵ^-5) and the MLPF has a cost of O(ϵ^-4) (best case) or O(ϵ^-4 log(ϵ)^2) (worst case) to achieve a mean square error of O(ϵ^2).
Abstract
The paper considers the filtering problem associated with partially observed McKean-Vlasov SDEs, where the objective is to compute the conditional expectation of the solution of the SDE (or functionals thereof) given all the observations recursively at every observation time.
Key highlights:
The authors develop new particle filter (PF) and multilevel particle filter (MLPF) methodologies to approximate the filtering expectations.
For the PF, the authors prove that to obtain a mean square error of O(ϵ^2), the cost per-observation time is O(ϵ^-5).
For the MLPF, the authors prove that the cost is O(ϵ^-4) (best case) or O(ϵ^-4 log(ϵ)^2) (worst case) to achieve the same mean square error.
The increased costs compared to ordinary Monte Carlo methods are primarily due to the need to approximate the laws of the McKean-Vlasov SDE.
The theoretical results are supported by numerical experiments on the Kuramoto model and a modified Kuramoto model.
Stats
The paper does not contain any explicit numerical data or statistics. The key results are the theoretical bounds on the computational cost of the PF and MLPF methods.