We consider high order approximations of the solution of the stochastic filtering problem, derive their pathwise representation in the spirit of the earlier work of Clark and Davis and prove their robustness property. In particular, we show that the high order discretised filtering functionals can be represented by Lipschitz continuous functions defined on the observation path space. This property is important from the practical point of view as it is in fact the pathwise version of the filtering functional that is sought in numerical applications. Moreover, the pathwise viewpoint will be a stepping stone into the rigorous development of machine learning methods for the filtering problem. This work is a continuation of a recent work by two of the authors where a discretisation of the solution of the filtering problem of arbitrary order has been established. We expand the previous work by showing that robust approximations can be derived from the discretisations therein.

翻译：我们考虑了随机过滤问题解决方案的高度顺序近似值,根据克拉克和戴维斯先前的工作精神得出了它们的路径代表,并证明了它们的稳健性。特别是,我们表明,高顺序分解过滤功能可由Lipschitz在观察路径空间上定义的连续功能来代表。从实际的角度来看,这一属性是重要的,因为它实际上是数字应用中寻求的过滤功能的路径版本。此外,路径正确的观点将是严格发展过滤问题的机器学习方法的跳板。这项工作是两位作者最近的一项工作的继续,其中已经确定了对任意秩序过滤问题解决办法的分解。我们扩大了先前的工作,显示从其中的离散中可以产生稳健的近似值。