The filtering equations revisited

TL;DR

This paper revisits the change-of-measure method for deriving nonlinear filtering equations, relaxing previous technical conditions and applying it to general Markov processes with two specific examples.

Contribution

It demonstrates that the change-of-measure approach can be extended to broader classes of Markov processes by relaxing earlier technical constraints.

Findings

The filtering equations are valid for general Markov signal processes.

The change-of-measure method can be applied under less restrictive conditions.

Two specific applications illustrate the extended approach.

Abstract

The problem of nonlinear filtering has engendered a surprising number of mathematical techniques for its treatment. A notable example is the change-of--probability-measure method originally introduced by Kallianpur and Striebel to derive the filtering equations and the Bayes-like formula that bears their names. More recent work, however, has generally preferred other methods. In this paper, we reconsider the change-of-measure approach to the derivation of the filtering equations and show that many of the technical conditions present in previous work can be relaxed. The filtering equations are established for general Markov signal processes that can be described by a martingale-problem formulation. Two specific applications are treated.