# Stochastic processes, filtering of

filtration of stochastic processes

The problem of estimating the value of a stochastic process $Z ( t)$ at the current moment $t$ given the past of another stochastic process related to it. For example, estimate a stationary process $Z ( t)$ given the values $X ( s)$, $s \leq t$, of a stationary process stationarily related to it (see , for example). Usually one considers the estimator $\widehat{Z} ( t)$ which minimizes the mean-square error, ${\mathsf E} | \widehat{Z} ( t) - Z ( t) | ^ {2}$. The use of the term "filter" goes back to the problem of isolating a signal from a "mixture" of a signal and a random noise. An important case of this is the problem of optimal filtering, when the connection between $Z ( t)$ and $X ( t)$ is described by a stochastic differential equation

$$d X ( t) = Z ( t) d t + d Y ( t) ,\ \ t > t _ {0} ,$$

where the noise is assumed to be independent of $Z ( t)$ and is given by a standard Wiener process $Y ( t)$.

A widely used filtering method is the Kalman–Bucy method, which applies to processes $Z ( t)$ that are described by linear stochastic differential equations. For example, if, in the above scheme,

$$d X ( t) = a ( t) Z ( t) d t + d Y ( t)$$

with zero initial conditions, then

$$\widehat{Z} ( t) = \int\limits _ {t _ {0} } ^ { t } c ( t , s ) d X ( t) ,$$

where the weight function $c ( t , s )$ is obtained from the equations:

$$\frac{d}{dt} c ( t , s ) = \ [ a ( t) - b ( t) ] c ( t , s ) ,\ \ t > s ,$$

$$c( s, s) = b( s),$$

$$\frac{d}{dt} b ( t) = 2 a ( t) b ( t) - [ b ( t) ] ^ {2} + 1 ,\ t > t _ {0} ,\ b ( t _ {0} ) = 0 .$$

The generalization of this method to non-linear equations is called the general stochastic filtering problem or the non-linear filtering problem (see ).

In the case when

$$Z ( t) = \sum _ { k= } 1 ^ { n } c _ {k} Z _ {k} ( t)$$

depends on the unknown parameters $c _ {1} \dots c _ {n}$, one can obtain the interpolation estimator $\widehat{Z} ( t)$ by estimating these parameters given $X ( s)$, $s \leq t$; the method of least squares applies here, along with its generalizations (see , for example).

How to Cite This Entry:
Stochastic processes, filtering of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stochastic_processes,_filtering_of&oldid=48862
This article was adapted from an original article by Yu.A. Rozanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article