This paper is devoted to filtering, smoothing, and prediction of polynomial processes that are partially observed. These problems are known to allow for an explicit solution in the simpler case of linear Gaussian state space models. The key insight underlying the present piece of research is that in filtering applications polynomial processes and their discrete counterpart are indistinguishable from Gaussian processes sharing their first two moments. We describe the construction of these Gaussian equivalents of polynomial processes and explicitly compute optimal linear filters, predictors and smoothers for polynomial processes in discrete and continuous time. The consideration of Gaussian equivalents also opens the door to parameter estimation and linear-quadratic optimal control in the context of polynomial processes.

翻译：本文致力于研究部分观测多项式过程的滤波、平滑与预测问题。在较为简单的线性高斯状态空间模型情形下，此类问题已知存在显式解。本研究的关键洞见在于：在滤波应用中，多项式过程及其离散对应物与具有相同一阶矩和二阶矩的高斯过程是不可区分的。我们描述了这些多项式过程的高斯等价形式的构造方法，并显式计算了离散与连续时间下多项式过程的最优线性滤波器、预测器和平滑器。高斯等价形式的引入也为多项式过程背景下的参数估计与线性二次最优控制问题开辟了新的途径。