# Kushner equation

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]}^{[clarification needed]}

## Overview

Assume the state of the system evolves according to

- [math]\displaystyle{ dx = f(x,t) \, dt + \sigma dw }[/math]

and a noisy measurement of the system state is available:

- [math]\displaystyle{ dz = h(x,t) \, dt + \eta dv }[/math]

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:

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

where [math]\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} }[/math] is the Kolmogorov Forward operator and [math]\displaystyle{ dp(x,t) = p(x,t + dt) - p(x,t) }[/math] is the variation of the conditional probability.

The term [math]\displaystyle{ dz-E_t h(x,t) dt }[/math] 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 [math]\displaystyle{ f(x,t) = A x }[/math] and [math]\displaystyle{ h(x,t) = C x }[/math]. The Kushner equation will be given by

- [math]\displaystyle{ dp(x,t) = L[p(x,t)] dt + p(x,t) [C x- C \mu(t)]^\top \eta^{-\top}\eta^{-1} [dz-C \mu(t) dt], }[/math]

where [math]\displaystyle{ \mu(t) }[/math] is the mean of the conditional probability at time [math]\displaystyle{ t }[/math]. Multiplying by [math]\displaystyle{ x }[/math] and integrating over it, we obtain the variation of the mean

- [math]\displaystyle{ d\mu(t) = A \mu(t) dt + \Sigma(t) C^\top \eta^{-\top}\eta^{-1} \left(dz - C\mu(t) dt\right). }[/math]

Likewise, the variation of the variance [math]\displaystyle{ \Sigma(t) }[/math] is given by

- [math]\displaystyle{ \frac{d\Sigma(t)}{dt} = A\Sigma(t) + \Sigma(t) A^\top + \sigma^\top \sigma-\Sigma(t) C^\top\eta^{-\top} \eta^{-1} C \Sigma(t). }[/math]

The conditional probability is then given at every instant by a normal distribution [math]\displaystyle{ \mathcal{N}(\mu(t),\Sigma(t)) }[/math].

## See also

## References

- ↑ Kushner, H. J. (1964). "On the differential equations satisfied by conditional probability densities of Markov processes, with applications".
*J. SIAM Control Ser. A***2**(1): 106-119. doi:10.1137/0302009. - ↑ Stratonovich, R.L. (1959).
*Optimum nonlinear systems which bring about a separation of a signal with constant parameters from noise*. Radiofizika, 2:6, pp. 892–901. - ↑ Stratonovich, R.L. (1959).
*On the theory of optimal non-linear filtering of random functions*. Theory of Probability and Its Applications, 4, pp. 223–225. - ↑ Stratonovich, R.L. (1960)
*Application of the Markov processes theory to optimal filtering*. Radio Engineering and Electronic Physics, 5:11, pp. 1–19. - ↑ Stratonovich, R.L. (1960).
*Conditional Markov Processes*. Theory of Probability and Its Applications, 5, pp. 156–178. - ↑ Bucy, R. S. (1965). "Nonlinear filtering theory".
*IEEE Transactions on Automatic Control***10**(2): 198. doi:10.1109/TAC.1965.1098109.

Original source: https://en.wikipedia.org/wiki/Kushner equation.
Read more |