In this article, 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.
|Original language||English (US)|
|Number of pages||40|
|Journal||Stochastic Analysis and Applications|
|State||Published - Jun 19 2022|
Bibliographical noteKAUST Repository Item: Exported on 2022-06-21
Acknowledged KAUST grant number(s): URF/1/2584-01-01
Acknowledgements: This work is supported by the KAUST Office of Sponsored Research (OSR) under Award No. URF/1/2584-01-01 in the KAUST Competitive Research Grants Program-Round 4 (CRG2015) and the Alexander von Humboldt Foundation.
ASJC Scopus subject areas
- Applied Mathematics
- Statistics and Probability
- Statistics, Probability and Uncertainty