Asian Journal of Mathematics

Volume 27 (2023)

Number 1

On the stability of linear feedback particle filter

Pages: 95 – 120



Xiuqiong Chen (School of Mathematics, Renmin University of China, Beijing, China)

Stephen S.-T. Yau (Department of Mathematical Sciences, Tsinghua University, Beijing, China; and Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Beijing, China)


In this paper, we study the stability of feedback particle filter (FPF) for linear filtering systems with Gaussian noises. We first provide some local contraction estimates of the exact linear FPF, whose conditional distribution is exactly the posterior distribution of the state as long as their initial values are equal. Then we study the convergence of the linear FPF formed by $N$ particles, and prove that the mean squared errors between the actual moments $(m_t, P_t)$ and their approximations $(m^{(N)}_t , P^{(N)}_t)$ by FPF are of order $\mathcal{O}(1/N)$ and decay exponentially fast as time $t$ goes to infinity.


feedback particle filter, Kalman–Bucy filter, linear system, convergence

2010 Mathematics Subject Classification

34A12, 93C05, 93D20

This work is supported by National Natural Science Foundation of China (NSFC) grant 12201631, and by the Tsinghua University Education Foundation fund 042202008.

Received 9 August 2022

Accepted 21 December 2022

Published 16 June 2023