On one-dimensional Riccati diffusions
- Publisher:
- Institute of Mathematical Statistics
- Publication Type:
- Journal Article
- Citation:
- Annals of Applied Probability, 2017, 29, (2), pp. 1127-1187
- Issue Date:
- 2017
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This article is concerned with the fluctuation analysis and the stability
properties of a class of one-dimensional Riccati diffusions. This class of
Riccati diffusion is quite general, and arises, for example, in data
assimilation applications, and more particularly in ensemble (Kalman-type)
filtering theory. 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. In this context, the time-uniform
convergence results developed in this work do not require a stable signal. 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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