  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.
