Kushner 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 + \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

References

1. 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. [[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. 892–901.
3. [[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. 223–225.
4. [[Ruslan L. Stratonovich. Stratonovich, R.L.]] (1960) ''Application of the Markov processes theory to optimal filtering''. Radio Engineering and Electronic Physics, 5:11, pp. 1–19.
5. [[Ruslan L. Stratonovich. Stratonovich, R.L.]] (1960). ''Conditional Markov Processes''. Theory of Probability and Its Applications, 5, pp. 156–178.
6. Bucy, R. S.. (1965). "Nonlinear filtering theory". IEEE Transactions on Automatic Control.