发表机构
California Institute of Technology（加州理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过研究Zakai方程，利用规范变换分离有限维随机与无限维确定性动力学，在线性可检测条件下建立了最优滤波器在χ^p-散度、全变差范数和2-Wasserstein距离下的定量渐近指数稳定性。
AI 中文摘要
滤波问题要求在给定信号过程的部分含噪观测的情况下估计该信号过程，其初始状态未知。从概率论的角度看，每个时刻的最佳估计是最优滤波器，即给定截至该时刻的所有观测时信号的条件分布。最优滤波器的渐近稳定性问题，即询问以错误初始分布初始化的滤波器是否收敛到正确初始化的滤波器，具有基本的理论和实践重要性。本文研究了具有线性漂移和加性布朗噪声的信号-观测模型的最优滤波器的定量渐近稳定性。当初始分布额外假设为高斯分布时，最优滤波器退化为著名的Kalman-Bucy滤波器，对于该滤波器，信号和观测矩阵上的可检测性条件是确保渐近稳定性的尖锐条件。在可检测性条件成立但初始分布不必为高斯分布的线性设置中，我们通过研究未归一化滤波器满足的Zakai方程做出以下贡献。首先，我们在具有非线性漂移的一般情况下研究Zakai方程的确定性部分的渐近行为，并将可检测性条件置于此背景下。其次，在线性漂移的情况下，我们通过一个规范变换来刻画Zakai方程的解（从而刻画最优滤波器），该变换将有限维随机动力学与无限维确定性动力学分离。第三，我们利用这种表示来建立滤波器在$\chi^p$-散度（相对于参考最优滤波器）、全变差范数和$2$-Wasserstein距离下的渐近指数稳定性。
英文摘要
The filtering problem asks to estimate a signal process, the initial state of which is unknown, given its partial and noisy observations. From a probabilistic perspective, the best estimate at each time is the optimal filter, which is the conditional law of the signal given all observations up to the time. The problem of asymptotic stability of the optimal filter, which asks whether the filter initialized with the incorrect initial law converges to the correctly initialized filter, is of fundamental practical and theoretical importance. This paper studies the quantitative asymptotic stability of the optimal filter for signal-observation models with linear drift and additive Brownian noise. When the initial law is additionally assumed to be Gaussian, the optimal filter reduces to the celebrated Kalman-Bucy filter, for which the detectability condition on the signal and observation matrices is the sharp condition ensuring asymptotic stability. In the linear setting where the detectability condition holds but the initial law need not be Gaussian, we make the following contributions by studying the Zakai equation satisfied by the unnormalized filter. First, we study the asymptotic behavior of the deterministic part of the Zakai equation in the general case with nonlinear drift, and place the detectability condition in this context. Second, in the case of linear drift, we characterize the solution of the Zakai equation (hence the optimal filter) by a gauge transformation that separates the finite-dimensional stochastic dynamics from the infinite-dimensional deterministic dynamics. Third, we use this representation to establish asymptotic exponential filter stability in the $χ^p$-divergence (with respect to a reference optimal filter), the total variation norm, and the $2$-Wasserstein distance.
Comments39 pages
