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.{{cite journal |authorlink=Harold J. Kushner |last=Kushner |first=H. J. |year=1964 |title=On the differential equations satisfied by conditional probability densities of Markov processes, with applications |journal= Journal of the Society for Industrial and Applied Mathematics, Series A: Control|volume=2 |issue=1 |pages=106–119 |doi=10.1137/0302009 }} It therefore provides the solution of the nonlinear filtering problem in estimation theory. The equation is sometimes referred to as the Stratonovich–KushnerStratonovich, R.L. (1959). Optimum nonlinear systems which bring about a separation of a signal with constant parameters from noise. Radiofizika, 2:6, pp. 892–901.Stratonovich, R.L. (1959). On the theory of optimal non-linear filtering of random functions. Theory of Probability and Its Applications, 4, pp. 223–225.Stratonovich, R.L. (1960) Application of the Markov processes theory to optimal filtering. Radio Engineering and Electronic Physics, 5:11, pp. 1–19.Stratonovich, R.L. (1960). Conditional Markov Processes. Theory of Probability and Its Applications, 5, pp. 156–178. (or Kushner–Stratonovich) equation. {{clarify|text=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.{{cite journal |last=Bucy |first=R. S. |year=1965 |title=Nonlinear filtering theory |journal=IEEE Transactions on Automatic Control |volume=10 |issue=2 |page=198 |doi=10.1109/TAC.1965.1098109 }}|date=September 2020|reason=It's unclear who derived what and how.}}

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:

: $dz = h(x,t) \, dt + \eta\, dv$

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:

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

where

: $L[p] := -\sum \frac{\partial (f_i p)}{\partial x_i} + \frac{1}{2} \sum (\sigma \sigma^\top)_{i,j} \frac{\partial^2 p}{\partial x_i \partial x_j}$

is the Kolmogorov forward operator and

: $dp(x,t) = p(x,t + dt) - p(x,t)$

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 by

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

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 mean

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

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

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

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

References

Category:Signal estimation

Category:Nonlinear filters

Kushner equation

Overview

= Kalman–Bucy filter =

See also

References