On One-Dimensional Riccati Diffusions

TL;DR

This paper investigates the stability and fluctuation behaviors of one-dimensional Riccati diffusions, introducing new analytical techniques to establish exponential decay to equilibrium and precise fluctuation estimates, with applications to Kalman filtering.

Contribution

It presents the first exponential stability and fluctuation analysis for this class of nonlinear Riccati diffusions, using novel combined analytical methods.

Findings

Established sharp exponential decay to equilibrium.

Provided uniform fluctuation estimates over time.

Applied results to ensemble Kalman-Bucy filtering.

Abstract

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.