On one-dimensional Riccati diffusions
Résumé
This article is concerned with the fluctuation analysis and the stability properties of a class of one-dimensional Riccati diffusions. This class of Riccati diffusion is quite general, and arises, for example, in data assimilation applications, and more particularly in ensemble (Kalman-type) filtering theory. 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. In this context, the time-uniform convergence results developed in this work do not require a stable signal. 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.