Computational Doob's h-transforms for Online Filtering of Discretely Observed Diffusions
This work addresses computational bottlenecks in particle filtering for diffusion processes, offering significant efficiency gains in specific challenging regimes, though it is incremental as it builds on existing auxiliary particle filter frameworks.
The paper tackled the problem of online filtering for discretely observed nonlinear diffusion processes by approximating intractable Doob's h-transforms using neural networks and nonlinear Feynman-Kac formulas, resulting in a particle filter that can be orders of magnitude more efficient than state-of-the-art methods in scenarios with highly informative observations, extreme data, or high state dimensions.
This paper is concerned with online filtering of discretely observed nonlinear diffusion processes. Our approach is based on the fully adapted auxiliary particle filter, which involves Doob's $h$-transforms that are typically intractable. We propose a computational framework to approximate these $h$-transforms by solving the underlying backward Kolmogorov equations using nonlinear Feynman-Kac formulas and neural networks. The methodology allows one to train a locally optimal particle filter prior to the data-assimilation procedure. Numerical experiments illustrate that the proposed approach can be orders of magnitude more efficient than state-of-the-art particle filters in the regime of highly informative observations, when the observations are extreme under the model, or if the state dimension is large.