Kushner equation

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.

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

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

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

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

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

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

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

• Zakai equation