  This article is concerned with the fluctuation analysis and the stability
properties of a class of one-dimensional Riccati diffusions. These
one-dimensional stochastic differential equations exhibit a quadratic drift
function and a non-Lipschitz continuous diffusion function. We present a novel
approach, combining tangent process techniques, Feynman-Kac path integration,
and exponential change of measures, to derive sharp exponential decays to
equilibrium. We also provide uniform estimates with respect to the time
horizon, quantifying with some precision the fluctuations of these diffusions
around a limiting deterministic Riccati differential equation. These results
provide a stronger and almost sure version of the conventional central limit
theorem. We illustrate these results in the context of ensemble Kalman-Bucy
filtering. To the best of our knowledge, the exponential stability and the
fluctuation analysis developed in this work are the first results of this kind
for this class of nonlinear diffusions.
