the Zakai and Kushner-Stratonovich equations
/ zah-KYE; KOOSH-ner strah-toh-NOH-vich /
When the signal or observation is nonlinear, or the noise is non-Gaussian, the conditional distribution of the hidden state is no longer a Gaussian summarised by a mean and covariance — it is a full probability measure that must itself be propagated. The Zakai and Kushner-Stratonovich equations are the exact stochastic PDEs that this conditional distribution obeys. They are the rigorous heart of nonlinear filtering: they say precisely how your belief about the hidden state should evolve as each new noisy observation arrives, generalising the Kalman-Bucy filter to arbitrary (Markovian) signals.
Set up a signal dX_t = b(X_t) dt + sigma(X_t) dW_t with generator L, observed through dY_t = h(X_t) dt + dV_t. The Kushner-Stratonovich equation governs the NORMALISED conditional density (or measure) pi_t, i.e. pi_t(f) = E[ f(X_t) given F^Y_t ]: d pi_t(f) = pi_t(L f) dt + ( pi_t(h f) - pi_t(h) pi_t(f) ) ( dY_t - pi_t(h) dt ). The first term is the prediction (the signal's own dynamics push the belief forward via L), and the second term is the correction by the innovation dY_t - pi_t(h) dt; crucially that correction is NONLINEAR in pi_t (it contains the product pi_t(h)pi_t(f)), which is exactly why no finite set of moments closes and why nonlinear filtering is hard. The Zakai equation is a clever linearisation: it governs an UNNORMALISED density rho_t (related to pi_t by a change of measure that removes the observation drift), d rho_t(f) = rho_t(L f) dt + rho_t(h f) dY_t. This equation is LINEAR in rho_t, far easier to analyse and discretise; you recover pi_t at the end by normalising, pi_t(f) = rho_t(f)/rho_t(1).
These equations are the exact, infinite-dimensional generalisation of the Kalman filter — and they reveal exactly why filtering is hard: the conditional distribution is in general an infinite-dimensional object, so the filter is an SPDE, not a finite system of ODEs. The Kalman-Bucy filter is the special, finite-dimensional collapse that happens only when everything is linear-Gaussian (the Benes filter is one of the very few other exactly-finite-dimensional cases). Honest caveats: these are genuine stochastic PDEs driven by the observation, and except for the linear-Gaussian and a handful of Benes-type models they have no closed-form solution and no finite-dimensional sufficient statistic — which is the whole motivation for numerical methods like the particle filter that approximate rho_t or pi_t by a swarm of weighted samples. Also the derivation uses the innovations approach / a Girsanov change of measure (the Kallianpur-Striebel formula), and the well-posedness of the Zakai SPDE requires regularity conditions on b, sigma, h.
Tracking a target whose bearing only (an angle) is observed, dY = arctan(position) dt + noise — a classic nonlinear observation. The conditional density can become bimodal (the target could be on either side), so no mean-plus-covariance summary suffices. The Zakai equation propagates the full unnormalised density exactly; in practice you discretise it on a grid or approximate it with a particle filter, then normalise to read off the estimate.
Nonlinear observations make the belief multimodal — only the full conditional density (Zakai/KS), not a Gaussian, captures it.
The Kushner-Stratonovich equation is nonlinear (hence no closed moment recursion); the Zakai equation linearises it by dropping normalisation. Except for linear-Gaussian and Benes models there is no finite-dimensional filter — that infinite-dimensionality is the reason particle methods exist.