Overview
Assume the state of the system evolves according to
and a noisy measurement of the system state is available:
where are independent Wiener processes. Then the unnormalized conditional probability density of the state at time t is given by the Zakai equation:
where the operator
As previously mentioned p is an unnormalized density, i.e. it does not necessarily integrate to 1. After solving for p we can integrate it and normalize it if desired (an extra step not required in the Kushner approach).
Note that if the last two terms on the right hand side are omitted (by choosing h identically zero), we are left with a nonstochastic PDE: the familiar Kolmogorov Forward Equation, which describes the evolution of the state when no measurement information is available.
Read more about this topic: Zakai Equation