Zakai equation


In filtering theory the Zakai equation is a linear stochastic partial differential equation for the un-normalized density of a hidden state. In contrast, the Kushner equation gives a non-linear stochastic partial differential equation for the normalized density of the hidden state. In principle either approach allows one to estimate a quantity function from noisy measurements, even when the system is non-linear. The application of this approach to a specific engineering situation may be problematic however, as these equations are quite complex. The Zakai equation is a bilinear stochastic partial differential equation. It was named after Moshe Zakai.

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, is an unnormalized density and thus does not necessarily integrate to 1. After solving for, integration and normalization can be done if desired.
Note that if the last two terms on the right hand side are omitted, the result is a nonstochastic PDE: the familiar Kolmogorov Forward Equation, which describes the evolution of the state when no measurement information is available.