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

• Main Authors: Crisan, Dan, Lobbe, Alexander, Ortiz-Latorre, Salvador
• Format: Journal Article
• Language: English
• Published: 11-01-2021
• Subjects: Computer Science - Numerical Analysis; Mathematics - Numerical Analysis; Mathematics - Probability
• DOI: 10.48550/arxiv.2101.03957

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.