Kushner equation explained

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.

Overview

Assume the state of the system evolves according to

dx = f(x,t) dt + \sigma dw

and a noisy measurement of the system state is available:

dz = h(x,t) dt + η 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:

dp(x,t) = L[p(x,t)] dt + p(x,t) ( h(x,t) - E _{t} h(x,t) ) ^{\top} η ^{-\top} η ^{-1} ( dz - E _{t} h(x,t) dt ) .

where

L[p] : = - \sum (\partial (f _{i} p))/(\partial x _{i}) + (1)/(2) \sum (\sigma \sigma ^{\top)} _{i,j} (\partial ^{2} p)/(\partial x _{i} \partial x _{j}) is the Kolmogorov forward operator and dp(x,t) = p(x,t + dt) - p(x,t) is the variation of the conditional probability.

The term

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

f(x,t) = A x and h(x,t) = C x . The Kushner equation will be given bydp(x,t) = L[p(x,t)] dt + p(x,t) ( C x - C \mu(t) ) ^{\top} η ^{-\top} η ^{-1} ( dz - C \mu(t) dt ) , where \mu(t) is the mean of the conditional probability at time t . Multiplying by x and integrating over it, we obtain the variation of the meand\mu(t) = A \mu(t) dt + \Sigma(t) C ^{\top} η ^{-\top} η ^{-1} ( dz - C\mu(t) dt ) . Likewise, the variation of the variance \Sigma(t) is given by\tfrac{d}{dt}\Sigma(t) = A\Sigma(t) + \Sigma(t) A ^{\top} + \sigma ^{\top} \sigma - \Sigma(t) C ^{\topη} ^{-\top} η ^{-1} C \Sigma(t). The conditional probability is then given at every instant by a normal distribution l{N}(\mu(t),\Sigma(t)) .

See also

• Zakai equation

Notes and References

1. Harold J. Kushner . Kushner . H. J. . 1964 . On the differential equations satisfied by conditional probability densities of Markov processes, with applications . Journal of the Society for Industrial and Applied Mathematics, Series A: Control. 2 . 1 . 106–119 . 10.1137/0302009 .
2. [Ruslan L. Stratonovich|Stratonovich, R.L.]
3. [Ruslan L. Stratonovich|Stratonovich, R.L.]
4. [Ruslan L. Stratonovich|Stratonovich, R.L.]
5. [Ruslan L. Stratonovich|Stratonovich, R.L.]