Filtering problem (stochastic processes)
In the theory of stochastic processes, the filtering problem is a mathematical model for a number of state estimation problems in signal processing and related fields. The general idea is to establish a "best estimate" for the true value of some system from an incomplete, potentially noisy set of observations on that system. The problem of optimal non-linear filtering (even for the non-stationary case) was solved by Ruslan L. Stratonovich (1959, 1960), see also Harold J. Kushner's work and Moshe Zakai's, who introduced a simplified dynamics for the unnormalized conditional law of  the filter known as Zakai equation. The solution, however, is infinite-dimensional in the general case. Certain approximations and special cases are well understood: for example, the linear filters are optimal for Gaussian random variables, and are known as the Wiener filter and the Kalman-Bucy filter. More generally, as the solution is infinite dimensional, it requires finite dimensional approximations to be implemented in a computer with finite memory. A finite dimensional approximated nonlinear filter may be more based on heuristics, such as the Extended Kalman Filter or the Assumed Density Filters, or more methodologically oriented such as for example  the Projection Filters, some sub-families of which are shown to coincide with the Assumed Density Filters. In general, if the separation principle applies, then filtering also arises as part of the solution of an optimal control problem. For example, the Kalman filter is the estimation part of the optimal control solution to the linear-quadratic-Gaussian control problem. The mathematical formalism. Consider a probability space (Ω, Σ, P) and suppose that the (random) state "Y" in "n"-dimensional Euclidean space R of a system of interest at time "t" is a random variable "Y" : Ω → R given by the  solution to an Itō stochastic differential equation of the form where "B" denotes standard "p"-dimensional Brownian motion, "b" : [0, +∞) × R → R is the drift field, and "σ" : [0, +∞) × R → R is the diffusion field. It is assumed that observations "H" in R (note that "m" and "n" may, in general, be unequal) are taken for each time "t" according to Adopting the Itō interpretation of the stochastic differential and setting this gives the following stochastic integral representation for the observations "Z": where "W" denotes standard "r"-dimensional Brownian motion, independent of "B" and  the initial condition "Y", and "c" : [0, +∞) × R → R and "γ" : [0, +∞) × R → R satisfy for all "t" and "x" and some constant "C". The filtering problem is the following: given observations "Z" for 0 ≤ "s" ≤ "t", what is the best estimate "Ŷ" of the true state "Y" of the system based on those observations? By "based on those observations" it is meant that "Ŷ" is measurable with respect to the "σ"-algebra "G" generated by the observations "Z", 0 ≤ "s" ≤ "t". Denote by "K" = "K"("Z", "t") the  collection of all R-valued random variables "Y" that are square-integrable and "G"-measurable: By "best estimate", it is meant that "Ŷ" minimizes the mean-square distance between "Y" and all candidates in "K": Basic result: orthogonal projection. The space "K"("Z", "t") of candidates is a Hilbert space, and the general theory of Hilbert spaces implies that the solution "Ŷ" of the minimization problem (M) is given by where "P" denotes the orthogonal projection of "L"(Ω, Σ, P; R) onto the linear subspace "K"("Z", "t") = "L"(Ω, "G", P; R). Furthermore, it is a general fact about conditional expectations that  if "F" is any sub-"σ"-algebra of Σ then the orthogonal projection is exactly the conditional expectation operator E[·|"F"], i.e., Hence, This elementary result is the basis for the general Fujisaki-Kallianpur-Kunita equation of filtering theory.