A finite dimensional filter with exponential conditional density

TL;DR

This paper develops new classes of scalar nonlinear filtering problems with finite-dimensional solutions by constructing specific drifts, expanding the limited set of known systems where optimal filters are finite dimensional.

Contribution

It introduces a method to design nonlinear filtering problems with finite-dimensional exponential family filters by tailoring the state equation's drift.

Findings

Constructed finite-dimensional filters for new nonlinear systems.

Extended the class of systems with known finite-dimensional filters.

Demonstrated the filter's evolution within a specified exponential family.

Abstract

In this paper we consider the continuous--time nonlinear filtering problem, which has an infinite--dimensional solution in general, as proved by Chaleyat--Maurel and Michel. There are few examples of nonlinear systems for which the optimal filter is finite dimensional, in particular Kalman's, Benes', and Daum's filters. In the present paper, we construct new classes of scalar nonlinear filtering problems admitting finite--dimensional filters. We consider a given (nonlinear) diffusion coefficient for the state equation, a given (nonlinear) observation function, and a given finite--dimensional exponential family of probability densities. We construct a drift for the state equation such that the resulting nonlinear filtering problem admits a finite--dimensional filter evolving in the prescribed exponential family augmented by the observaton function and its square.