# Kalman Filter and its Modern Extensions for the Continuous-time Nonlinear Filtering Problem

## TL;DR

This paper reviews classical and modern filtering algorithms for continuous-time nonlinear problems, highlighting their differences, advantages, and a new approximation method based on optimal transport, with numerical comparisons.

## Contribution

It introduces a novel approximation algorithm for nonlinear filtering using optimal transport and coupling, expanding the existing filtering framework.

## Key findings

- EnKBF and FPF avoid resampling, reducing variance.
- Feedback control structure enhances stability and error correction.
- Numerical example demonstrates algorithm performance improvements.

## Abstract

This paper is concerned with the filtering problem in continuous-time. Three algorithmic solution approaches for this problem are reviewed: (i) the classical Kalman-Bucy filter which provides an exact solution for the linear Gaussian problem, (ii) the ensemble Kalman-Bucy filter (EnKBF) which is an approximate filter and represents an extension of the Kalman-Bucy filter to nonlinear problems, and (iii) the feedback particle filter (FPF) which represents an extension of the EnKBF and furthermore provides for an consistent solution in the general nonlinear, non-Gaussian case. The common feature of the three algorithms is the gain times error formula to implement the update step (to account for conditioning due to the observations) in the filter. In contrast to the commonly used sequential Monte Carlo methods, the EnKBF and FPF avoid the resampling of the particles in the importance sampling update step. Moreover, the feedback control structure provides for error correction potentially leading to smaller simulation variance and improved stability properties. The paper also discusses the issue of non-uniqueness of the filter update formula and formulates a novel approximation algorithm based on ideas from optimal transport and coupling of measures. Performance of this and other algorithms is illustrated for a numerical example.