Kushner equation

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:

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

where

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

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 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

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

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

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

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

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

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