  This paper is concerned with the development and use of duality theory for a
nonlinear filtering model with white noise observations. The main contribution
of this paper 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 (BSDE) 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.
