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. The equation is sometimes referred to as the Stratonovich–Kushner^[2][3][4][5] (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.^[6]

Overview

Assume the state of the system evolves according to

${\displaystyle dx=f(x,t)\,dt+\sigma \,dw}$

and a noisy measurement of the system state is available:

${\displaystyle dz=h(x,t)\,dt+\eta \,dv}$

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:

${\displaystyle dp(x,t)=L[p(x,t)]dt+p(x,t){\big (}h(x,t)-E_{t}h(x,t){\big )}^{\top }\eta ^{-\top }\eta ^{-1}{\big (}dz-E_{t}h(x,t)dt{\big )}.}$

where

${\displaystyle L[p]:=-\sum {\frac {\partial (f_{i}p)}{\partial x_{i}}}+{\frac {1}{2}}\sum (\sigma \sigma ^{\top })_{i,j}{\frac {\partial ^{2}p}{\partial x_{i}\partial x_{j}}}}$

is the Kolmogorov forward operator and

${\displaystyle dp(x,t)=p(x,t+dt)-p(x,t)}$

is the variation of the conditional probability.

The term ${\displaystyle dz-E_{t}h(x,t)dt}$ is the innovation, i.e. the difference between the measurement and its expected value.

See also

• Zakai equation

References