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]

  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 (1): 106–119. doi:10.1137/0302009.
  2. ^ 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. ^ 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. ^ Stratonovich, R.L. (1960) Application of the Markov processes theory to optimal filtering. Radio Engineering and Electronic Physics, 5:11, pp. 1–19.
  5. ^ 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. 10 (2): 198. doi:10.1109/TAC.1965.1098109.