Does a multivariate Riccati diffusion forget its initial covariance under common noise?
Let $d \ge 2$, let $A \in \mathbb{R}^{d \times d}$, let $R, S$ be real symmetric positive definite $d \times d$ matrices, and write $\mathcal{S}_d^+$ for the positive definite symmetric matrices. For an integer $N \ge d + 1$ and $Q \in \mathcal{S}_d^+$, let $P_t^{N,Q}$ be the unique strong positive definite solution of the Itô equation $$dP_t = (AP_t + P_tA^\top + R - P_tSP_t) \, dt + \frac{2}{\sqrt{N}} \bigl[P_t^{1/2} \, dM_t \, R^{1/2}\bigr]_{\mathrm{sym}}, \qquad P_0 = Q,$$ where $[B]_{\mathrm{sym}} = (B + B^\top)/2$, square roots are the principal positive ones, and $M$ is a $d \times d$ matrix of independent standard real Brownian motions, the same for all $Q$. This is the $\kappa = 0$ covariance diffusion of the ensemble Kalman–Bucy filter in the notation of Bishop and Del Moral, who give the existence and uniqueness of the strong solution, and, for $N$ large enough that $\mathbb{E}\|P_t\|$ and $\mathbb{E}\|P_t^{-1}\|$ are uniformly bounded, exponential convergence of its law to the unique invariant law in a weighted norm. For $d = 1$, Bishop, Del Moral, Kamatani and Rémillard proved an exponential $L^n$ contraction estimate for the stochastic flow, in a weighted distance between initial values.
Problem. Is it true, for all such $d, A, R, S$, that for every $p \in [1, \infty)$ and every compact $K \subset \mathcal{S}_d^+$ there is an integer $N_0 \ge d + 1$ such that for every integer $N \ge N_0$ there are constants $C < \infty$ and $\lambda > 0$ with $$\Bigl( \mathbb{E} \bigl\| P_t^{N,Q_1} - P_t^{N,Q_2} \bigr\|_F^p \Bigr)^{1/p} \le C e^{-\lambda t} \|Q_1 - Q_2\|_F \qquad (t \ge 0, \; Q_1, Q_2 \in K)?$$
Source: Stated by AIM — Open Applied Problems, after Bishop and Del Moral (2023), §8.4, on multivariate versions of the scalar contraction estimate (6.27); AIM's is a formulation for their κ = 0 case on compact sets of initial covariances.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- ST
- Keywords
- AIM Open Applied Problems, contraction estimates, ensemble Kalman filters, Riccati diffusions
- Literature checked
- Nov. 5, 2026 by clemens-admin
References
@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},
}
@article{bishop2019scalar,
author = {Bishop, A. N. and Del Moral, P. and Kamatani, K. and R{\'e}millard, B.},
title = {On one-dimensional {R}iccati diffusions},
journal = {Ann. Appl. Probab.},
volume = {29},
number = {2},
year = {2019},
pages = {1127--1187},
doi = {10.1214/18-AAP1431},
eprint = {1711.10065},
archivePrefix = {arXiv},
}
@article{bishop2019matrix,
author = {Bishop, Adrian N. and Del Moral, Pierre},
title = {On the stability of matrix-valued {R}iccati diffusions},
journal = {Electron. J. Probab.},
volume = {24},
year = {2019},
pages = {1--40},
doi = {10.1214/19-EJP342},
eprint = {1808.00235},
archivePrefix = {arXiv},
}
@misc{aim-does-a-multivariate-riccati-diffusion-forget-its-initial-covariance-under-common-noise,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Does a multivariate Riccati diffusion forget its initial covariance under common noise?}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/4f5b1659355cb61d194da94ef14615214e1363a8/problems/887-does-a-multivariate-riccati-diffusion-forget-its-initial-covariance-under-common-noise.md},
note = {Entry 887 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-01113, "Does a multivariate Riccati diffusion forget its initial covariance under common noise?", clopen.solutions.