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# Pure and Applied Mathematics Quarterly

## Volume 18 (2022)

### Number 3

### Error estimate for the approximate solution to multivariate feedback particle filter

Pages: 1113 – 1146

DOI: https://dx.doi.org/10.4310/PAMQ.2022.v18.n3.a9

#### Authors

#### Abstract

In this paper, based on the assumption that the gain function $K$ has been optimally obtained in the multivariate feedback particle filter (FPF), we focus on the error estimate for the approximate solutions to the particle’s density evolution equation, which is actually the forward Kolmogorov equation (FKE) satisfied by the “particle population”. The approximation is essentially the unnormalized density of the states conditioning on the discrete observations with the given time discretization. Mainly owing to the representation of Brownian bridges for the Brownian motion, and the assumption on the coercivity condition, we prove that the mean square error of the approximate solution is of order equal to the square root of the time interval.

#### Keywords

feedback particle filter, forward Kolmogorov equation, error estimate, Brownian bridges

The author is supported by the Fundamental Research Funds for the Central Universities under grant no. GK202103002 and the start-up fund from Shaanxi Normal University.

Received 3 January 2022

Received revised 24 February 2022

Accepted 25 April 2022

Published 24 July 2022