# Perturbations and projections of Kalman-Bucy semigroups

## TL;DR

This paper investigates how perturbations and projections affect Kalman-Bucy semigroups and Riccati equations, providing error characterizations and introducing new models relevant to filtering, control, and stochastic processes.

## Contribution

It offers new insights into the effects of perturbations and projections on Kalman-Bucy and Riccati equations, including error analysis and novel projection models like Bose-Mesner projections.

## Key findings

- Error bounds between nominal and regularised Riccati flows
- Characterization of perturbation effects on Kalman-Bucy distributions
- Introduction of new projection models such as Bose-Mesner projections

## Abstract

We analyse various perturbations and projections of Kalman-Bucy semigroups and Riccati equations. For example, covariance inflation-type perturbations and localisation methods (projections) are common in the ensemble Kalman filtering literature. In the limit of these ensemble methods, the regularised sample covariance tends toward a solution of a perturbed/projected Riccati equation. With this motivation, results are given characterising the error between the nominal and regularised Riccati flows and Kalman-Bucy filtering distributions. New projection-type models are also discussed; e.g. Bose-Mesner projections. These regularisation models are also of interest on their own, and in, e.g., differential games, control of stochastic/jump processes, and robust control.