Zakai equation

In filtering theory the Zakai equation is a linear recursive filtering equation for the un-normalized density of a hidden state. In contrast, the Kushner equation gives a non-linear recursive equation for the normalized density of the hidden state. In principle either approach allows one to estimate a quantity (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. The Zakai equation is a bilinear stochastic partial differential equation.

Overview

Assume the state of the system evolves according to

<math>dx = f(x,t) dt + dw</math>

and a noisy measurement of the system state is available:

<math>dz = h(x,t) dt + dv</math>

where <math>dw, dv</math> are independent Wiener processes. Then the unnormalized conditional probability density <math>p(x,t)</math> of the state at time t is given by the Zakai equation:

<math>dp = L(p) dt + p h^T dz</math>

where the operator <math>L = -sum frac + sum frac</math>

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...
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