Filtering, Stability, and Robustness
Filtering, Stability, and Robustness
复制标题
过滤、稳定性和鲁棒性
作者：
R. Handel
非线性滤波理论研究的是噪声观测中马尔可夫信号的最优估计问题。这种估计必然取决于为信号和观测过程选择的模型。本文在连续时间滤波的框架下，研究了白色噪声类型观测值下滤波器对基础模型选择的长期敏感性。
本论文的第一个主题是滤波器的渐近稳定性，利用条件扩散理论来研究。这导致改进的路径稳定性界限，并在一个完全概率设置现有的稳定性结果的新见解。此外，我详细地发展了有限状态马尔可夫信号的条件扩散理论，并澄清了估计和随机控制之间的对偶性。
本文的第二个主题是非线性滤波器对信号和观测过程的模型参数的灵敏度。本节集中讨论有限状态的情况，其中相应的模型参数是信号的跳跃率、观测函数和初始测量。主要结果是，与真实的和修改后的模型参数的滤波器之间的期望差异是有界的无限时间间隔上的一致，提供的信号处理满足混合属性。证明使用的过滤器上的单纯形，以及Malliavin演算和预期的随机演算产生的随机流的属性。
本论文的第三个也是最后一个主题是量子滤波器的渐近稳定性。我开始发展量子滤波理论使用参考概率方法。所得到的过滤器的稳定性是不容易研究使用上述方法，平滑违反了nondemolition的要求。幸运的是，可以通过随机化滤波器的初始状态来取得进展。使用这种技术，我证明了过滤估计的测量观察是稳定的，无论底层模型，只要初始状态是绝对连续的在适当的意义上。
The theory of nonlinear filtering concerns the optimal estimation of a Markov signal in noisy observations. Such estimates necessarily depend on the model that is chosen for the signal and observations processes. This thesis studies the sensitivity of the filter to the choice of underlying model over long periods of time, within the framework of continuous time filtering with white noise type observations.
The first topic of this thesis is the asymptotic stability of the filter, which is studied using the theory of conditional diffusions. This leads to improvements on pathwise stability bounds, and to new insight into existing stability results in a fully probabilistic setting. Furthermore, I develop in detail the theory of conditional diffusions for finite-state Markov signals and clarify the duality between estimation and stochastic control in this context.
The second topic of this thesis is the sensitivity of the nonlinear filter to the model parameters of the signal and observations processes. This section concentrates on the finite state case, where the corresponding model parameters are the jump rates of the signal, the observation function, and the initial measure. The main result is that the expected difference between the filters with the true and modified model parameters is bounded uniformly on the infinite time interval, provided that the signal process satisfies a mixing property. The proof uses properties of the stochastic flow generated by the filter on the simplex, as well as the Malliavin calculus and anticipative stochastic calculus.
The third and final topic of this thesis is the asymptotic stability of quantum filters. I begin by developing quantum filtering theory using reference probability methods. The stability of the resulting filters is not easily studied using the preceding methods, as smoothing violates the nondemolition requirement. Fortunately, progress can be made by randomizing the initial state of the filter. Using this technique, I prove that the filtered estimate of the measurement observable is stable regardless of the underlying model, provided that the initial states are absolutely continuous in a suitable sense.
影响因子：
4
作者：
Apte, A.;Hairer, M.;Voss, J.
通讯作者：
Voss, J.