# 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

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

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(xt) of the state at time t is given by the Kushner equation:

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

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.

### Kalman–Bucy filter

One can simply use the Kushner equation to derive the Kalman–Bucy filter for a linear diffusion process. Suppose we have $\displaystyle{ f(x,t) = A x }$ and $\displaystyle{ h(x,t) = C x }$. The Kushner equation will be given by

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

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

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

Likewise, the variation of the variance $\displaystyle{ \Sigma(t) }$ is given by

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

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