  Duality between estimation and optimal control is a problem of rich
historical significance. The first duality principle appears in the seminal
paper of Kalman-Bucy, where the problem of minimum variance estimation is shown
to be dual to a linear quadratic (LQ) optimal control problem. Duality offers a
constructive proof technique to derive the Kalman filter equation from the
optimal control solution. This paper generalizes the classical duality result
of Kalman-Bucy to the nonlinear filter: The state evolves as a continuous-time
Markov process and the observation is a nonlinear function of state corrupted
by an additive Gaussian noise. A dual process is introduced as a backward
stochastic differential equation (BSDE). The process is used to transform the
problem of minimum variance estimation into an optimal control problem. Its
solution is obtained from an application of the maximum principle, and
subsequently used to derive the equation of the nonlinear filter. The classical
duality result of Kalman-Bucy is shown to be a special case.
