We introduce a weighted particle representation for the solution of the filtering problem based on a suitably chosen variation of the classical de Finetti theorem. This representation has important theoretical and numerical applications. In this paper, we explore some of its theoretical consequences. The first is to deduce the equations satisfied by the solution of the filtering problem in three different frameworks: the signal independent Brownian measurement noise model, the spatial observations with additive white noise model and the cluster detection model in spatial point processes. Secondly we use the representation to show that a suitably chosen filtering discretisation converges to the filtering solution. Thirdly we study the leading error coefficient for the discretisation. We show that it satisfies a stochastic partial differential equation by exploiting the weighted particle representation for both the approximation and the limiting filtering solution.
翻译:我们根据古典德费内蒂理论的恰当选择变量,为过滤问题的解决办法引入加权粒子代表法。这种代表法具有重要的理论和数字应用。在本文中,我们探讨了其理论后果。首先,我们从三个不同框架中推断出过滤问题解决办法所满足的方程式:信号独立的布朗测量噪音模型、带添加白噪音模型的空间观测和空间点过程中的集束探测模型。第二,我们使用该代表法表明,选择得当的过滤离散方法与过滤解决方案相汇合。第三,我们研究了离散的主要误差系数。我们通过利用加权粒子代表法近似和限制过滤解决方案,表明它满足了随机偏差部分方程式。