  The work of Kalman and Bucy has established a duality between filtering and
optimal estimation in the context of time-continuous linear systems. This
duality has recently been extended to time-continuous nonlinear systems in
terms of an optimization problem constrained by a backward stochastic partial
differential equation. Here we revisit this problem from the perspective of
appropriate forward-backward stochastic differential equations. This approach
sheds new light on the estimation problem and provides a unifying perspective.
It is also demonstrated that certain formulations of the estimation problem
lead to deterministic formulations similar to the linear Gaussian case as
originally investigated by Kalman and Bucy. Finally, optimal control of
partially observed diffusion processes is discussed as an application of the
proposed estimators.
