An Approach to Duality in Nonlinear Filtering

TL;DR

This paper extends the classical duality between estimation and control from linear Gaussian systems to nonlinear filtering, providing new theoretical insights and deriving the Wonham filter for finite state-space Markov chains.

Contribution

It establishes a duality result for nonlinear filtering, mirroring the Kalman-Bucy duality, and applies it to derive the Wonham filter for finite state-space Markov chains.

Findings

Duality between nonlinear filtering and control established

Derivation of the classical Wonham filter from the duality

Extension of linear Gaussian duality to nonlinear systems

Abstract

This paper revisits the question of duality between minimum variance estimation and optimal control first described for the linear Gaussian case in the celebrated paper of Kalman and Bucy. A duality result is established for nonlinear filtering, mirroring closely the original Kalman-Bucy duality of control and estimation for linear systems. The result for the finite state-space continuous time Markov chain is presented. It's solution is used to derive the classical Wonham filter.