Error estimate for the approximate solution to multivariate feedback particle filter

In this paper, based on the assumption that the gain function K has been optimally obtained in the multivariate feed-back 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.