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.
[1] It therefore provides the solution of the nonlinear filtering problem in estimation theory.
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.
[6][clarification needed] 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.
One can use the Kushner equation to derive the Kalman–Bucy filter for a linear diffusion process.
is given by The conditional probability is then given at every instant by a normal distribution