Stochastic filtering is a vibrant area of research in both control theory and statistics, with broad applications in many scientific fields. Despite its extensive historical development, there still lacks an effective method for joint parameter-state estimation in SDEs. The state-of-the-art particle filtering methods suffer from either sample degeneracy or information loss, with both issues stemming from the dynamics of the particles generated to represent system parameters. This paper provides a novel and effective approach for joint parameter-state estimation in SDEs via Rao-Blackwellization and modularization. Our method operates in two layers: the first layer estimates the system states using a bootstrap particle filter, and the second layer marginalizes out system parameters explicitly. This strategy circumvents the need to generate particles representing system parameters, thereby mitigating their associated problems of sample degeneracy and information loss. Moreover, our method employs a modularization approach when integrating out the parameters, which significantly reduces the computational complexity. All these designs ensure the superior performance of our method. Finally, a numerical example is presented to illustrate that our method outperforms existing approaches by a large margin.

翻译：随机滤波是控制理论与统计学中一个充满活力的研究领域，在众多科学领域具有广泛应用。尽管其发展历史久远，针对随机微分方程中联合参数-状态估计问题仍缺乏有效方法。现有最优粒子滤波方法存在样本退化或信息损失问题，这两种问题均由表征系统参数的粒子动态特性引发。本文提出一种基于Rao-Blackwellization与模块化的新型有效方法，用于随机微分方程中的联合参数-状态估计。该方法采用双层结构：第一层通过自举粒子滤波器估计系统状态，第二层显式边缘化系统参数。这种策略规避了生成表征系统参数的粒子需求，从而缓解了与之相关的样本退化与信息损失问题。此外，本方法在参数积分过程中采用模块化方法，显著降低了计算复杂度。这些设计共同保障了本方法的卓越性能。最后通过数值算例表明，本方法在性能上显著优于现有方法。