Uniform finite-ensemble accuracy under detectability
Let $d, q \ge 1$, let $A \in \mathbb{R}^{d \times d}$ and $H \in \mathbb{R}^{q \times d}$ with $(A, H)$ detectable, let $R, P_0 \in \mathbb{R}^{d \times d}$ be symmetric positive definite, and let $m_0 \in \mathbb{R}^d$. The signal and observation satisfy $$dX_t = AX_t \, dt + R^{1/2} \, dB_t, \qquad dZ_t = HX_t \, dt + dW_t, \qquad X_0 \sim \mathcal{N}(m_0, P_0), \quad Z_0 = 0,$$ where $B, W$ are independent standard Brownian motions of dimensions $d, q$, independent of $X_0$, and $(m_t, P_t)$ denotes the Kalman–Bucy conditional mean and covariance of $X_t$ given $(Z_s)_{s \le t}$. For an integer $N \ge 2$, the square-root ensemble Kalman–Bucy filter is $$d\xi_t^i = A\xi_t^i \, dt + R^{1/2} \, dB_t^i + P_t^N H^\top \Bigl[dZ_t - \tfrac12 H(\xi_t^i + m_t^N) \, dt\Bigr], \qquad 1 \le i \le N,$$ with sample mean $m_t^N$ and sample covariance $P_t^N = \frac{1}{N-1} \sum_{i=1}^N (\xi_t^i - m_t^N)(\xi_t^i - m_t^N)^\top$. The $\xi_0^i$ are independent samples of $\mathcal{N}(m_0, P_0)$ and the $B^i$ are mutually independent standard $d$-dimensional Brownian motions; these are independent of each other, of the signal and of $B, W$.
Taghvaei and Mehta expect a uniform bound $\mathbb{E}\|m_t^N - m_t\|^2 + \mathbb{E}\|P_t^N - P_t\|^2 \le C/\sqrt{N}$ to hold under detectability. When $H^\top H$ is positive definite, Bishop and Del Moral state that, from fixed initial sample mean and covariance, the covariance has error of order $1/\sqrt{N}$ in every $L^n$, uniformly in time; that the mean has it uniformly in time if every eigenvalue of $A$ has negative real part; and that the mean has it otherwise for times up to a horizon tending to infinity with $N$, and for all times when $d = 1$.
Problem. Do there always exist constants $C < \infty$ and an integer $N_0 \ge d + 2$, depending on the fixed model and initial law but not on $t$ or $N$, such that $$\sup_{t \ge 0} \Bigl( \mathbb{E}|m_t^N - m_t| + \mathbb{E}\|P_t^N - P_t\|_{\mathrm{op}} \Bigr) \le \frac{C}{\sqrt{N}}$$ for every integer $N \ge N_0$? The expectations average over both the observations and the ensemble.
Source: Taghvaei and Mehta (2023), Remark 3.7, who expect their uniform error bound (16) to hold under detectability. Stated by AIM — Open Applied Problems for the square-root filter with positive definite R and P₀, and with first moments of the errors.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- ST
- Keywords
- AIM Open Applied Problems, detectability, ensemble Kalman filters, uniform-in-time estimates
- Literature checked
- Nov. 5, 2026 by clemens-admin
References
@article{taghvaei2023,
author = {Taghvaei, Amirhossein and Mehta, Prashant G.},
title = {A survey of feedback particle filter and related controlled interacting particle systems ({CIPS})},
journal = {Annu. Rev. Control},
volume = {55},
year = {2023},
pages = {356--378},
doi = {10.1016/j.arcontrol.2023.03.006},
}
@article{bishop2023,
author = {Bishop, Adrian N. and Del Moral, Pierre},
title = {On the mathematical theory of ensemble (linear-{G}aussian) {K}alman--{B}ucy filtering},
journal = {Math. Control Signals Systems},
volume = {35},
number = {4},
year = {2023},
pages = {835--903},
doi = {10.1007/s00498-023-00357-2},
}
@misc{aim-uniform-finite-ensemble-accuracy-under-detectability,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Uniform finite-ensemble accuracy under detectability}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/4f5b1659355cb61d194da94ef14615214e1363a8/problems/886-uniform-finite-ensemble-accuracy-under-detectability.md},
note = {Entry 886 at commit 4f5b165, 2026-11-04},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01127, "Uniform finite-ensemble accuracy under detectability", clopen.solutions.