The solution of the continuous time filtering problem can be represented as a ratio of two expectations of certain functionals of the signal process that are parametrized by the observation path. We introduce a class of discretization schemes of these functionals of arbitrary order. The result generalizes the classical work of Picard, who introduced first order discretizations to the filtering functionals. For a given time interval partition, we construct discretization schemes with convergence rates that are proportional with the m-power of the mesh of the partition for arbitrary \(m\in {\mathbb {N}}\). The result paves the way for constructing high order numerical approximation for the solution of the filtering problem.

连续时间滤波问题的解决方案可以表示为信号处理的某些功能的两个期望值的比率，这些期望值是由观察路径参数化的。我们介绍了这些任意阶函数的离散化方案。结果概括了Picard的经典著作，他将一阶离散化引入了滤波功能。对于给定的时间间隔分区，我们构造离散化方案，其收敛速度与任意\（m \ in {\ mathbb {N}} \）中的分区网格的m幂成比例。结果为构造高阶数值逼近铺平了道路，以解决滤波问题。