Pathwise approximations for the solution of the non-linear filtering problem

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.