We consider the discrete-time filtering problem in scenarios where the observation noise is low or degenerate. We focus on the case where the observation equation is a linear function of the state and the data involve additive noise. However, we place minimal assumptions on the hidden state process. For such a class of models we derive new particle filters (PFs) with the key property that their performance is robust to the size of the observation noise. As a consequence, the developed PFs are well-defined in the limiting case of degenerate observation noise. Indicatively, we prove (under assumptions) that the PF applied in this low noise setting 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：我们考虑观测噪声较低或退化场景下的离散时间滤波问题。重点研究观测方程为状态线性函数且数据包含加性噪声的情形，但对隐状态过程仅作极简假设。针对这类模型，我们推导出新型粒子滤波器（PFs），其关键特性在于性能对观测噪声大小具有鲁棒性。因此，所提出的粒子滤波器在退化观测噪声的极限情形下仍具有良好定义。我们证明（在假设条件下）应用于低噪声场景的粒子滤波器继承了退化情形粒子滤波器的性质。进一步将框架扩展至隐状态由扩散过程生成的情形，在此场景中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器。最后通过多个数值算例验证了所提算法的有效性。