In principle either approach allows one to estimate a quantity function (the state of a dynamical system) from noisy measurements, even when the system is non-linear (thus generalizing the earlier results of Wiener and Kalman for linear systems and solving a central problem in estimation theory).
The application of this approach to a specific engineering situation may be problematic however, as these equations are quite complex.
of the state at time t is given by the Zakai equation: where is a Kolmogorov forward operator.
, integration and normalization can be done if desired (an extra step not required in the Kushner approach).
Note that if the last term on the right hand side is omitted (by choosing h identically zero), the result is a nonstochastic PDE: the familiar Fokker–Planck equation, which describes the evolution of the state when no measurement information is available.