Filtering problem (stochastic processes)

In the theory of stochastic processes, the filtering problem is a mathematical model for a number of filtering problems in signal processing and the like. The general idea is to form some kind of "best estimate" for the true value of some system, given only some (potentially noisy) observations of that system. The problem of optimal non-linear filtering (even for non-stationary case) was solved by Ruslan L. Stratonovich (1959 [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.] , 1960 [Stratonovich, R.L. (1960). "Application of the Markov processes theory to optimal filtering". Radio Engineering and Electronic Physics, 5:11, pp.1-19.] ). 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.

If the separation principle applies, then filtering also arises as part of the solution of a optimal control problem, i.e. 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"_"t" in "n"-dimensional Euclidean space R^"n" of a system of interest at time "t" is a random variable "Y"_"t" : Ω → R^"n" given by the solution to an Itō stochastic differential equation of the form

:$mathrm\{d\}\; Y\_\{t\}\; =\; b(t,\; Y\_\{t\})\; ,\; mathrm\{d\}\; t\; +\; sigma\; (t,\; Y\_\{t\})\; ,\; mathrm\{d\}\; B\_\{t\},$

where "B" denotes standard "p"-dimensional Brownian motion, "b" : [0, +∞) × R^"n" → R^"n" is the drift field, and "σ" : [0, +∞) × R^"n" → R^{"n"×"p"} is the diffusion field. It is assumed that observations "H"_"t" in R^"m" (note that "m" and "n" may, in general, be unequal) are taken for each time "t" according to

:$H\_\{t\}\; =\; c(t,\; Y\_\{t\})\; +\; gamma\; (t,\; Y\_\{t\})\; cdot\; mbox\{noise\}.$

Adopting the Itō interpretation of the stochastic differential and setting

:$Z\_\{t\}\; =\; int\_\{0\}^\{t\}\; H\_\{s\}\; ,\; mathrm\{d\}\; s,$

this gives the following stochastic integral representation for the observations "Z"_"t":

:$mathrm\{d\}\; Z\_\{t\}\; =\; c(t,\; Y\_\{t\})\; ,\; mathrm\{d\}\; t\; +\; gamma\; (t,\; Y\_\{t\})\; ,\; mathrm\{d\}\; W\_\{t\},$

where "W" denotes standard "r"-dimensional Brownian motion, independent of "B" and the initial condition "X"₀, and "c" : [0, +∞) × R^"n" → R^"n" and "γ" : [0, +∞) × R^"n" → R^{"n"×"r"} satisfy

:

for all "t" and "x" and some constant "C".

The filtering problem is the following: given observations "Z"_"s" for 0 ≤ "s" ≤ "t", what is the best estimate "Ŷ"_"t" of the true state "Y"_"t" of the system based on those observations?

By "based on those observations" it is meant that "Ŷ"_"t" is measurable with respect to the "σ"-algebra "G"_"t" generated by the observations "Z"_"s", 0 ≤ "s" ≤ "t". Denote by "K" = "K"("Z", "t") be collection of all R^"n"-valued random variables "Y" that are square-integrable and "G"_"t"-measurable:

:$K\; =\; K(Z,\; t)\; =\; L^\{2\}\; (Omega,\; G\_\{t\},\; mathbf\{P\};\; mathbf\{R\}^\{n\}).$

By "best estimate", it is meant that "Ŷ"_"t" minimizes the mean-square distance between "Y"_"t" 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"_"t" of the minimization problem (M) is given by

:

where "P"_"K"("Z","t") denotes the orthogonal projection of "L"²(Ω, Σ, P; R^"n") onto the linear subspace "K"("Z", "t") = "L"²(Ω, "G"_"t", P; R^"n"). Furthermore, it is a general fact about conditional expectations that if "F" is any sub-"σ"-algebra of Σ then the orthogonal projection

:$P\_\{F\}\; :\; L^\{2\}\; (Omega,\; Sigma,\; mathbf\{P\};\; mathbf\{R\}^\{n\})\; o\; L^\{2\}\; (Omega,\; F,\; mathbf\{P\};\; mathbf\{R\}^\{n\})$

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.

References

* cite book
last = Øksendal
first = Bernt K.
authorlink = Bernt Øksendal
title = Stochastic Differential Equations: An Introduction with Applications
edition = Sixth edition
publisher=Springer
location = Berlin
year = 2003
id = ISBN 3-540-04758-1 (See Section 6.1)