In filtering theory the Kushner equation (after Harold Kushner) is an equation for the conditional probability density of the state of a stochastic non-linear dynamical system, given noisy measurements of the state. It therefore provides the solution of the nonlinear filtering problem in estimation theory. The equation is sometimes referred to as the Stratonovich–Kushner (or Kushner–Stratonovich) equation.
Overview
Assume the state of the system evolves according to
and a noisy measurement of the system state is available:
where w, v are independent Wiener processes. Then the conditional probability density p(x, t) of the state at time t is given by the Kushner equation:
where is the Kolmogorov Forward operator and is the variation of the conditional probability.
The term is the innovation i.e. the difference between the measurement and its expected value.
Kalman–Bucy filter
One can simply use the Kushner equation to derive the Kalman–Bucy filter for a linear diffusion process. Suppose we have and . The Kushner equation will be given by
where is the mean of the conditional probability at time . Multiplying by and integrating over it, we obtain the variation of the mean
Likewise, the variation of the variance is given by
The conditional probability is then given at every instant by a normal distribution .