Computational Doob's h-transforms for Online Filtering of Discretely Observed Diffusions

TL;DR

This paper introduces a neural network-based computational framework for approximating Doob's h-transforms in online filtering of discretely observed nonlinear diffusions, significantly improving efficiency in challenging scenarios.

Contribution

It proposes a novel method combining neural networks and nonlinear Feynman-Kac formulas to approximate intractable h-transforms for particle filtering.

Findings

Achieves orders of magnitude efficiency gains over existing filters.

Effective in high-dimensional and highly informative observation regimes.

Pre-trains locally optimal filters before data assimilation.

Abstract

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.