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. However, the correct equation in terms of Itō calculus was first derived by Kushner although a more heuristic Stratonovich version of it appeared already in Stratonovich's works in late fifties. However, the derivation in terms of Itō calculus is due to Richard Bucy.
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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
The term
Kalman-Bucy filter
One can simply use the Kushner equation to derive the Kalman-Bucy filter for a linear diffusion process. Suppose we have
where
Likewise, the variation of the variance
The conditional probability is then given at every instant by a normal distribution