Application of Girsanov Theorem to Particle Filtering of Discretely Observed Continuous-Time Non-Linear Systems

TL;DR

This paper explores how Girsanov's theorem can enhance particle filtering techniques for continuous-time systems with discrete observations, especially in complex models with lower-dimensional noise processes.

Contribution

It introduces a novel application of Girsanov's theorem to evaluate likelihood ratios in particle filtering for non-linear stochastic systems with discrete measurements.

Findings

Girsanov's theorem facilitates likelihood ratio computation in importance sampling.

Method extends particle filtering to models with lower-dimensional noise processes.

Rao-Blackwellization improves filtering in Gaussian and static parameter models.

Abstract

This article considers the application of particle filtering to continuous-discrete optimal filtering problems, where the system model is a stochastic differential equation, and noisy measurements of the system are obtained at discrete instances of time. It is shown how the Girsanov theorem can be used for evaluating the likelihood ratios needed in importance sampling. It is also shown how the methodology can be applied to a class of models, where the driving noise process is lower in the dimensionality than the state and thus the laws of state and noise are not absolutely continuous. Rao-Blackwellization of conditionally Gaussian models and unknown static parameter models is also considered.