On One-Dimensional Riccati Diffusions

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.

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Cited by in corpus (8)

• A perturbation analysis of stochastic matrix Riccati diffusions
• On the stability of matrix-valued Riccati diffusions
• On the continuous time limit of the Ensemble Kalman Filter
• On the Mathematical Theory of Ensemble (Linear-Gaussian) Kalman-Bucy Filtering
• Mean field limit of Ensemble Square Root Filters -- discrete and continuous time
• Stability Properties of Systems of Linear Stochastic Differential Equations with Random Coefficients
• Asymptotic behavior of the forecast-assimilation process with unstable dynamics
• Emergence of phantom cold dark matter from spacetime diffusion