A perturbation analysis of stochastic matrix Riccati diffusions

TL;DR

This paper develops a perturbation theory for stochastic matrix Riccati diffusions, providing detailed expansions, convergence results, and fluctuation analysis relevant to filtering and control applications.

Contribution

It introduces non-asymptotic Taylor-type expansions for stochastic Riccati flows using stochastic analysis and semigroup techniques, advancing understanding of their fluctuations and convergence.

Findings

Derived non-asymptotic Taylor expansions of Riccati flows.

Proved convergence of sample covariance matrices to deterministic flows.

Established a functional central limit theorem for the stochastic process.

Abstract

Matrix differential Riccati equations are central in filtering and optimal control theory. The purpose of this article is to develop a perturbation theory for a class of stochastic matrix Riccati diffusions. Diffusions of this type arise, for example, in the analysis of ensemble Kalman-Bucy filters since they describe the flow of certain sample covariance estimates. In this context, the random perturbations come from the fluctuations of a mean field particle interpretation of a class of nonlinear diffusions equipped with an interacting sample covariance matrix functional. The main purpose of this article is to derive non-asymptotic Taylor-type expansions of stochastic matrix Riccati flows with respect to some perturbation parameter. These expansions rely on an original combination of stochastic differential analysis and nonlinear semigroup techniques on matrix spaces. The results here quantify the fluctuation of the stochastic flow around the limiting deterministic Riccati equation, at any order. The convergence of the interacting sample covariance matrices to the deterministic Riccati flow is proven as the number of particles tends to infinity. Also presented are refined moment estimates and sharp bias and variance estimates. These expansions are also used to deduce a functional central limit theorem at the level of the diffusion process in matrix spaces.