ON ONE-DIMENSIONAL RICCATI DIFFUSIONS

• 成果类型: Article
• 署名作者: Bishop, A. N.; Del Moral, P.; Kamatani, K.; Remillard, B.
• 署名单位: University of Technology Sydney; Commonwealth Scientific & Industrial Research Organisation (CSIRO); Inria; Centre National de la Recherche Scientifique (CNRS); University of New South Wales Sydney; University of Osaka; Universite de Montreal; HEC Montreal
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/18-AAP1431
• 发表日期: 2019
• 页码: 1127-1187
• 关键词: ensemble kalman filter; sobolev inequalities; nonlinear stability; Data assimilation; CONVERGENCE; probability; propagation; ergodicity; EQUATIONS; accuracy

摘要

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.