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

TL;DR

This paper develops high-order pathwise approximation methods for the nonlinear filtering problem, demonstrating their robustness and Lipschitz continuity, which are crucial for practical numerical applications and future machine learning approaches.

Contribution

It extends previous discretization techniques by establishing robust, Lipschitz continuous high-order pathwise filtering functionals suitable for numerical and machine learning applications.

Findings

High-order discretized filtering functionals are Lipschitz continuous.

Pathwise representations are robust and practical for numerical implementation.

The work lays groundwork for machine learning methods in filtering.

Abstract

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.