We consider the discrete-time filtering problem in scenarios where the observation noise is degenerate or low. We focus on the case where the observation equation is a linear function of the state and that additive noise is low or degenerate, however, we place minimal assumptions on the hidden state process. In this scenario we derive new particle filtering (PF) algorithms and, under assumptions, in such a way that as the noise becomes more degenerate a PF which approximates the low noise filtering problem provably inherits the properties of the PF used in the degenerate case. We extend our framework to the case where the hidden states are drawn from a diffusion process. In this scenario we develop new PFs which are robust to both low noise and fine levels of time discretization. We illustrate our algorithms numerically on several examples.

翻译：本文研究观测噪声退化或较低情况下的离散时间滤波问题。我们重点关注观测方程为状态线性函数且加性噪声较低或退化的情形，同时对隐状态过程仅施加最小化假设。在此框架下，我们推导出新的粒子滤波算法，并在特定假设条件下证明：当噪声趋于退化时，用于近似低噪声滤波问题的粒子滤波算法能够继承退化情形下所用粒子滤波的理论性质。我们将该框架拓展至隐状态服从扩散过程的情形，并开发出对低噪声与精细时间离散化均具有鲁棒性的新型粒子滤波算法。最后通过多个数值算例对所提算法进行验证。