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

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

where ${\displaystyle Lp=-\sum {\frac {\partial (f_{i}p)}{\partial x_{i}}}+{\frac {1}{2}}\sum (\sigma \sigma ^{\top })_{i,j}{\frac {\partial ^{2}p}{\partial x_{i}\partial x_{j}}}}$ 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 ${\displaystyle f(x,t)=ax}$ and ${\displaystyle h(x,t)=cx}$. The Kushner equation will be given by

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

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 ${\displaystyle {\mathcal {N}}(\mu (t),\Sigma (t))}$.

See also[]

• Zakai equation

References[]