BISHOP, Adrian; DEL MORAL, Pierre; KAMATANI, Kengo; RÉMILLARD, Bruno

doi:10.1214/18-AAP1431

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BISHOP, Adrian
University of Technology Sydney [UTS]

DEL MORAL, Pierre
Quality control and dynamic reliability [CQFD]
Institut de Mathématiques de Bordeaux [IMB]

KAMATANI, Kengo
Osaka University [Osaka]

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The Annals of Applied Probability. 2019, vol. 29, n° 2, p. 1127-1187

Institute of Mathematical Statistics (IMS)

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This article is concerned with the fluctuation analysis and the stabil-ity properties of a class of one-dimensional Riccati diffusions. These one-dimensional stochastic differential equations exhibit a quadratic drift func-tion and a non-Lipschitz continuous diffusion function. We present a novelapproach, combining tangent process techniques, Feynman–Kac path inte-gration and exponential change of measures, to derive sharp exponential de-cays to equilibrium. We also provide uniform estimates with respect to thetime horizon, quantifying with some precision the fluctuations of these dif-fusions around a limiting deterministic Riccati differential equation. Theseresults provide a stronger and almost sure version of the conventional centrallimit theorem. We illustrate these results in the context of ensemble Kalman–Bucy filtering. To the best of our knowledge, the exponential stability and thefluctuation analysis developed in this work are the first results of this kind forthis class of nonlinear diffusions.< Leer menos

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Ensemble Kalman filters

Quadratic stochastic differential equations

Ri-catti diffusions

Uniform fluctuation estimates

Uniform stability estimates

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On one-dimensional Riccati diffusions

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