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April 2019 On one-dimensional Riccati diffusions
A. N. Bishop, P. Del Moral, K. Kamatani, B. Rémillard
Ann. Appl. Probab. 29(2): 1127-1187 (April 2019). DOI: 10.1214/18-AAP1431


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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A. N. Bishop. P. Del Moral. K. Kamatani. B. Rémillard. "On one-dimensional Riccati diffusions." Ann. Appl. Probab. 29 (2) 1127 - 1187, April 2019.


Received: 1 November 2017; Revised: 1 June 2018; Published: April 2019
First available in Project Euclid: 24 January 2019

zbMATH: 07047446
MathSciNet: MR3910025
Digital Object Identifier: 10.1214/18-AAP1431

Primary: 60G52, 60G99, 60H10, 93E11

Rights: Copyright © 2019 Institute of Mathematical Statistics


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Vol.29 • No. 2 • April 2019
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