A Wasserstein Coupled Particle Filter for Multilevel Estimation

TL;DR

This paper introduces a novel Wasserstein coupled particle filter for multilevel estimation of partially observed diffusions, demonstrating reduced computational effort and improved accuracy through theoretical analysis and numerical experiments.

Contribution

It proposes a new multilevel particle filter using Wasserstein coupling, with proven CLT and efficiency gains over existing methods.

Findings

Reduced asymptotic variance in some scenarios.

Theoretical proof of CLT for the new method.

Numerical validation showing efficiency improvements.

Abstract

In this paper, we consider the filtering problem for partially observed diffusions, which are regularly observed at discrete times. We are concerned with the case when one must resort to time-discretization of the diffusion process if the transition density is not available in an appropriate form. In such cases, one must resort to advanced numerical algorithms such as particle filters to consistently estimate the filter. It is also well known that the particle filter can be enhanced by considering hierarchies of discretizations and the multilevel Monte Carlo (MLMC) method, in the sense of reducing the computational effort to achieve a given mean square error (MSE). A variety of multilevel particle filters (MLPF) have been suggested in the literature, e.g., in Jasra et al., SIAM J, Numer. Anal., 55, 3068--3096. Here we introduce a new alternative that involves a resampling step based on the optimal Wasserstein coupling. We prove a central limit theorem (CLT) for the new method. On considering the asymptotic variance, we establish that in some scenarios, there is a reduction, relative to the approach in the aforementioned paper by Jasra et al., in computational effort to achieve a given MSE. These findings are confirmed in numerical examples. We also consider filtering diffusions with unstable dynamics; we empirically show that in such cases a change of measure technique seems to be required to maintain our findings.