## The Annals of Applied Probability

### 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.

#### Article information

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

Dates
Revised: June 2018
First available in Project Euclid: 24 January 2019

https://projecteuclid.org/euclid.aoap/1548298938

Digital Object Identifier
doi:10.1214/18-AAP1431

Mathematical Reviews number (MathSciNet)
MR3910025

Zentralblatt MATH identifier
07047446

#### Citation

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. https://projecteuclid.org/euclid.aoap/1548298938

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