A high order time discretization of the solution of the non-linear filtering problem

Abstract

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 ∈ N. The result paves the way for constructing high order numerical approximation for the solution of the filtering problem.

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

“…Thus, the present paper is a continuation of the previous work [5] by two of the authors which gives a new high-order time discretisation for the filtering functional. We extend this result to produce the robust version, of any order, of the discretisation from [5]. The implementation of the resulting numerical method remains open and is subject of future research.…”

“…Therefore, accurate and fast numerical algorithms for the approximate solution of the filtering problem are essential. In this contribution we analyse a recently developed high order time discretisation of the solution of the filtering problem from the literature [5] and prove that the so discretised solution possesses a property known as robustness. Thus, the present paper is a continuation of the previous work [5] by two of the authors which gives a new high-order time discretisation for the filtering functional.…”

“…In this contribution we analyse a recently developed high order time discretisation of the solution of the filtering problem from the literature [5] and prove that the so discretised solution possesses a property known as robustness. Thus, the present paper is a continuation of the previous work [5] by two of the authors which gives a new high-order time discretisation for the filtering functional. We extend this result to produce the robust version, of any order, of the discretisation from [5].…”

“…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 [5] where a discretisation of the solution of the filtering problem of arbitrary order has been established. We expand the work in [5] by showing that robust approximations can be derived from the discretisations therein.…”

“…This work is a continuation of [5] where a discretisation of the solution of the filtering problem of arbitrary order has been established. We expand the work in [5] by showing that robust approximations can be derived from the discretisations therein.…”