The Annals of Applied Probability

On one-dimensional Riccati diffusions

A. N. Bishop, P. Del Moral, K. Kamatani, and B. Rémillard

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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.

Article information

Ann. Appl. Probab., Volume 29, Number 2 (2019), 1127-1187.

Received: November 2017
Revised: June 2018
First available in Project Euclid: 24 January 2019

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Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60G52: Stable processes 93E11: Filtering [See also 60G35] 60G99: None of the above, but in this section

Ensemble Kalman filters quadratic stochastic differential equations Ricatti diffusions uniform fluctuation estimates uniform stability estimates


Bishop, A. N.; Del Moral, P.; Kamatani, K.; Rémillard, B. On one-dimensional Riccati diffusions. Ann. Appl. Probab. 29 (2019), no. 2, 1127--1187. doi:10.1214/18-AAP1431.

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