Duality for Nonlinear Filtering I: Observability

TL;DR

This paper develops a duality framework for nonlinear filtering in hidden Markov models using backward stochastic differential equations, linking observability to controllability in a novel way.

Contribution

It introduces a dual control system for nonlinear filtering via BSDEs, extending classical linear duality to nonlinear systems and providing explicit controllability criteria.

Findings

Duality links stochastic observability to controllability.

Explicit formulas for controllability gramian are derived.

The framework generalizes classical linear duality to nonlinear systems.

Abstract

This paper is concerned with the development and use of duality theory for a hidden Markov model (HMM) with white noise observations. The main contribution of this work is to introduce a backward stochastic differential equation (BSDE) as a dual control system. A key outcome is that stochastic observability (resp. detectability) of the HMM is expressed in dual terms: as controllability (resp. stabilizability) of the dual control system. All aspects of controllability, namely, definition of controllable space and controllability gramian, along with their properties and explicit formulae, are discussed. The proposed duality is shown to be an exact extension of the classical duality in linear systems theory. One can then relate and compare the linear and the nonlinear systems. A side-by-side summary of this relationship is given in a tabular form (Table~II).