Filtering Problem (stochastic Processes)

In the theory of stochastic processes, filtering describes the problem of determining the state of a system from an incomplete and potentially noisy set of observations. While originally motivated by problems in engineering, filtering found applications in many fields from signal processing to finance.

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. Damiano Brigo, Bernard Hanzon and François Le Gland, Approximate Nonlinear Filtering by Projection on Exponential Manifolds of Densities, Bernoulli, Vol. 5, N. 3 (1999), pp. 495--534

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''_''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

:$\mathrm Y_ = b(t, Y_) \, \mathrm t + \sigma (t, Y_) \, \mathrm B_,$

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

:$H_ = c(t, Y_) + \gamma (t, Y_) \cdot \mbox.$

Adopting the Itō interpretation of the stochastic differential and setting

:$Z_ = \int_^ H_ \, \mathrm s,$

this gives the following stochastic integral representation for the observations ''Z''_''t'':

:$\mathrm Z_ = c(t, Y_) \, \mathrm t + \gamma (t, Y_) \, \mathrm W_,$

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

:$\big, c (t, x) \big, + \big, \gamma (t, x) \big, \leq C \big( 1 + , x , \big)$

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 function, measurable with respect to the sigma algebra, ''σ''-algebra ''G''_''t'' generated by the observations ''Z''_''s'', 0 ≤ ''s'' ≤ ''t''. Denote by ''K'' = ''K''(''Z'', ''t'') the collection of all R^''n''-valued random variables ''Y'' that are square-integrable and ''G''_''t''-measurable:

:$K = K(Z, t) = L^ (\Omega, G_, \mathbf; \mathbf^).$

By "best estimate", it is meant that ''Ŷ''_''t'' minimizes the mean-square distance between ''Y''_''t'' and all candidates in ''K'':

:\mathbf \left Y_ - \hat_ \big, ^ \right= \inf_ \mathbf \left Y_ - Y \big, ^ \right \qquad \mbox

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

:$\hat_ = P_ \big( Y_ \big),$

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

:$P_ : L^ (\Omega, \Sigma, \mathbf; \mathbf^) \to L^ (\Omega, F, \mathbf; \mathbf^)$

is exactly the conditional expectation operator E ''F'' i.e.,

:P_ (X) = \mathbf \big F \big

Hence,

:\hat_ = P_ \big( Y_ \big) = \mathbf \big G_ \big

This elementary result is the basis for the general Fujisaki-Kallianpur-Kunita equation of filtering theory.

See also

* The smoothing problem, closely related to the filtering problem
* Filter (signal processing)
* Kalman filter, a well-known filtering algorithm related both to the filtering problem and the smoothing problem
* Smoothing

References

Further reading

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

Control theory
Signal estimation
Stochastic differential equations