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In [[Filtering problem (stochastic processes)|filtering theory]] the '''Kushner equation'''<ref>[[Harold J. Kushner|Kushner H.J.]] (1964) ''On the differential equations satisfied by conditional probability
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densities of Markov processes, with applications.''. J. SIAM Control Ser. A, 2(1), pp. 106-119.</ref> (after Harold Kushner) is an equation for the [[conditional probability]] [[probability density function|density]] of the state of a [[stochastic process|stochastic]] non-linear [[dynamical system]], given noisy measurements of the state.  It therefore provides the solution of the [[nonlinear filter]]ing problem in [[estimation theory]]. The equation is sometimes referred to as the '''Stratonovich–Kushner'''<ref>[[Ruslan L. Stratonovich|Stratonovich, R.L.]] (1959). ''Optimum nonlinear systems which bring about a separation of a signal with constant parameters from noise''. Radiofizika, 2:6, pp.&nbsp;892&ndash;901.</ref><ref>[[Ruslan L. Stratonovich|Stratonovich, R.L.]] (1959). ''On the theory of optimal non-linear filtering of random functions''. Theory of Probability and its Applications, 4, pp.&nbsp;223&ndash;225.</ref><ref>[[Ruslan L. Stratonovich|Stratonovich, R.L.]] (1960) ''Application of the Markov processes theory to optimal filtering''. Radio Engineering and Electronic Physics, 5:11, pp.&nbsp;1&ndash;19.</ref><ref>[[Ruslan L. Stratonovich|Stratonovich, R.L.]] (1960). ''Conditional Markov Processes''. Theory of Probability and its Applications, 5, pp.&nbsp;156&ndash;178.</ref> (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 Bucy.<ref>[[Bucy, R. S.]] (1965) ''Nonlinear filtering theory''. IEEE Transactions on Automatic Control,
10, pp. 198–198. [http://ieeexplore.ieee.org/xploreAssets/images/absImages/01098109.png]</ref>


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Assume the state of the system evolves according to
 
:<math>dx = f(x,t) \, dt + \sigma dw</math>
 
and a noisy measurement of the system state is available:
 
:<math>dz = h(x,t) \, dt + \eta dv</math>
 
where ''w'', ''v'' are independent [[Wiener process]]es. Then the conditional probability density ''p''(''x'',&nbsp;''t'') of the state at time ''t'' is given by the Kushner equation:
 
:<math>dp(x,t) = L[p(x,t)] dt + p(x,t) [h(x,t)-E_t h(x,t) ]^T \eta^{-\top}\eta^{-1} [dz-E_t h(x,t) dt].</math>
 
where <math>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}</math> is the Kolmogorov Forward operator and <math>dp(x,t) = p(x,t + dt) - p(x,t)</math> is the variation of the conditional probability.
 
The term <math>dz-E_t h(x,t) dt</math> is the [[innovation (signal processing)|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 <math> f(x,t) = a x</math> and <math> h(x,t) = c x </math>. The Kushner equation will be given by
:<math>
dp(x,t) = L[p(x,t)] dt + p(x,t) [c x- c \mu(t)]^T \eta^{-\top}\eta^{-1} [dz-c \mu(t) dt],
</math>
where <math> \mu(t) </math> is the mean of the conditional probability at time <math> t</math>. Multiplying by <math> x</math> and integrating over it, we obtain the variation of the mean
:<math>
d\mu(t) = a \mu(t) dt + \Sigma(t) c^\top \eta^{-\top}\eta^{-1} \left(dz - c\mu(t) dt\right).
</math>
Likewise, the variation of the variance <math>\Sigma(t)</math> is given by
:<math>
\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).
</math>
The conditional probability is then given at every instant by a normal distribution <math>\mathcal{N}(\mu(t),\Sigma(t))</math>.
 
== References ==
<references />
 
== See also ==
 
*[[Zakai equation]]
 
[[Category:Signal processing]]
[[Category:Estimation theory]]
[[Category:Stochastic processes]]
[[Category:Nonlinear filters]]

Latest revision as of 21:36, 17 June 2014

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