Duality for Nonlinear Filtering II: Optimal Control

This article is concerned with the development and use of duality theory for a nonlinear filtering model with white noise observations. The main contribution of this article is to introduce a stochastic optimal control problem as a dual to the nonlinear filtering problem. The mathematical statement of the dual relationship between the two problems is given in the form of a duality principle. The constraint for the optimal control problem is the backward stochastic differential equation introduced in the companion paper. The optimal control solution is obtained from an application of the maximum principle, and subsequently used to derive the equation of the nonlinear filter. The proposed duality is shown to be an exact extension of the classical Kalman–Bucy duality, and different from other types of optimal control and variational formulations given in literature.