Particle representation for the solution of the filtering problem. Application to the error expansion of filtering discretizations

TL;DR

This paper introduces a weighted particle representation for filtering problems, enabling theoretical analysis and numerical approximation, including convergence and error expansion of discretizations in various models.

Contribution

It develops a novel weighted particle representation based on a variation of de Finetti's theorem, facilitating analysis of filtering equations and discretization errors.

Findings

Derived filtering equations for multiple models.

Proved convergence of filtering discretizations.

Characterized leading error coefficient via SPDE.

Abstract

We introduce a weighted particle representation for the solution of the filtering problem based on a suitably chosen variation of the classical de Finetti theorem. This representation has important theoretical and numerical applications. In this paper, we explore some of its theoretical consequences. The first is to deduce the equations satisfied by the solution of the filtering problem in three different frameworks: the signal independent Brownian measurement noise model, the spatial observations with additive white noise model and the cluster detection model in spatial point processes. Secondly we use the representation to show that a suitably chosen filtering discretisation converges to the filtering solution. Thirdly we study the leading error coefficient for the discretisation. We show that it satisfies a stochastic partial differential equation by exploiting the weighted particle representation for both the approximation and the limiting filtering solution.