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 [1], 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 [2]).

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 [3], for example).

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In the filtering of stochastic processes one distinguishes two problems. The linear filtering problem is to estimate a stationary stochastic process given a linear function of the past of a real stationary process such that a least-squares criterion is minimized. The stochastic filtering problem or non-linear filtering problem is to determine the conditional probability distribution of a process given the past of a related process.

The linear filtering problem has first been formulated and solved by N. Wiener [a18] and A.N. Kolmogorov [a20]. R.E. Kalman has reformulated the linear filtering problem for a stochastic system in state space form. The solution to that problem is known as the Kalman filter for discrete-time processes [a7] and as the Kalman–Bucy filter for continuous-time processes [a8]. The new elements in the problem formulation are the emphasis on recursive filters and on the finite-dimensionality of the state space.

In Wiener–Kolmogorov filtering (cf. [a12], [a20]), one is given a pair of jointly stationary zero-mean normally-distributed stochastic processes $ \{ {y ( t) , z ( t) } : {t \in \mathbf R } \} $ and one would like to obtain the optimal least-square estimator of $ z ( t) $ from the observed past of $ y $: $ \{ y ( t ^ \prime ) \textrm{ for } t ^ \prime < t \} $. The optimal estimator, $ \widehat{z} ( t) $, will be given by the convolution

$$ \widehat{z} ( t) = \int\limits _ {- \infty } ^ { t } G ( t - t ^ \prime ) y ( t ^ \prime ) d t ^ \prime . $$

The convolution kernel is determined by the integral equation

$$ \int\limits _ { 0 } ^ \infty G ( t ^ \prime ) R _ {yy} ( t - t ^ \prime ) d t ^ \prime = \ R _ {zy} ( t) ,\ t > 0, $$

where $ R _ {yy} ( t) = {\mathsf E} \{ y ( t) y ^ {T} ( 0) \} $ and $ R _ {zy} ( t) = {\mathsf E} \{ z ( t) y ^ {T} ( 0) \} $.

This integral equation is a so-called Wiener–Hopf equation, and determines $ G $ as a function of $ R _ {yy} $ and $ R _ {zy} $. The most effective way of solving it is by means of the method of spectral decomposition of a random function.

In Kalman–Bucy filtering (cf. [a7], [a8]), the model is given by the linear stochastic differential equation

$$ d x = A ( t) x d t + B ( t) d w ( t) , $$

$$ d y = C ( t) x d t + D ( t) d w ( t) , $$

with $ w $ a Wiener process which generates the observed vector process $ y $ through the state vector process $ x $. The matrices $ A ( t) $, $ B ( t) $, $ C ( t) $, $ D ( t) $ are assumed to be known and of suitable dimension, with $ D( t) $ and $ D ^ {T} ( t) $ strictly positive-definite. The initial state $ x ( t _ {0} ) $ is normally distributed with known mean $ m _ {0} $ and covariance $ \Pi _ {0} $, and is assumed to be independent of $ w $. Further, $ y ( t _ {0} ) $ is taken to be zero.

The problem is to estimate $ x ( t) $ from the observations $ y ( t ^ \prime ) $ for $ t _ {0} \leq t ^ \prime < t $. The Kalman filter, which generates this estimator, is given by

$$ d \widehat{x} = A ( t) \widehat{x} d t + L ( t) ( d y - C ( t) \widehat{x} d t ) $$

on $ t \geq t _ {0} $ with initial condition $ \widehat{x} ( t _ {0} ) = m _ {0} $ and the Kalman gain $ L $ is defined by

$$ L ( t) = ( \Sigma ( t) C ^ {T} ( t) + B ( t) D ^ {T} ( t) ) ( D ( t) D ^ {T} ( t ) ) ^ {-1} , $$

where $ \Sigma $ is the solution of the Riccati equation

$$ \dot \Sigma = A ( t) \Sigma + \Sigma A ^ {T} ( t) - ( \Sigma C ^ {T} ( t) + B ( t) D ^ {T} ( t) ) ( D ( t) D ^ {T} ( t) ) ^ {-1} \times $$

$$ \times ( C ( t) \Sigma + D ( t) B ^ {T} ( t)) + B ( t) B ^ {T} ( t) , $$

with $ \Sigma ( t _ {0} ) = \Pi _ {0} $. This differential equation in the ( $ n \times n $)- symmetric matrix $ \Sigma $ can be shown to have a unique symmetric non-negative definite solution. Its solution is in fact equal to the estimation error:

$$ \Sigma ( t) = {\mathsf E} \{ ( x ( t) - \widehat{x} ( t) ) ( x ( t) - \widehat{x} ( t) ) ^ {T} \} . $$

In the time-invariant Kalman filter one assumes that the initial time since observations were taken goes to minus infinity: $ t _ {0} \rightarrow - \infty $. If one assumes that the matrices $ A $, $ B $, $ C $, $ D $ are independent of $ t $ and satisfy certain observability and controllability conditions, then the infinite Kalman filter becomes

$$ d \widehat{x} = A \widehat{x} d t + L ( d y - \widehat{x} d t ) , $$

where $ L $ is defined by

$$ L = ( \Sigma ^ {+} C ^ {T} + B D ^ {T} ) ( D D ^ {T} ) ^ {-1} $$

when $ \Sigma ^ {+} $ is the (unique) symmetric non-negative definite solution of the algebraic Riccati equation

$$ 0 = A \Sigma + \Sigma A ^ {T} - ( \Sigma C ^ {T} + B D ^ {T} ) ( D D ^ {T} ) ^ {-1} \times $$

$$ \times ( C \Sigma + D B ^ {T} ) + B B ^ {T} . $$

The Kalman filter, in particular its time-invariant version, is one of the most basic results in control theory and signal processing and has found wide application in process control, aerospace engineering, econometrics, etc. Many of these applications involve non-linear systems, and the Kalman filter is applied in a non-rigorous way by a procedure of successive linearization. Such algorithms are known as extended Kalman filters and have proved remarkably effective in practice [a11]. See [a21], [a22] for general surveys of linear filtering theory.

The study of the stochastic filtering problem, or non-linear filtering, has been initiated by R.L. Stratonovich [a16] and H.J. Kushner [a9]. A generalization and a proof using martingale theory is due to M. Fujisaki, G. Kallianpur and H. Kunita [a4]. See also [a26] and [2]. An approach leading to dynamical equations for a non-normalized conditional density has been developed by Kallianpur, C. Striebel [a6], R.E. Mortensen [a12], M. Zakai [a19], and E. Pardoux [a13]. See also [a25]. None of these filtering formulas is directly implementable, since all are "infinite-dimensional" , i.e. describe the time evolution of conditional distribution or density functions in the form of measure-valued or stochastic partial differential equations. In 1980, V.E. Benes [a1] discovered a class of non-linear systems for which the conditional density admits a finite-dimensional parametrization, and this has led to extensive research on characterizing such systems and exploring the connection, uncovered by R.W. Brockett and J.M.C. Clark [a3], between non-linear filtering and certain Lie algebras of differential operators; see [a25], [a18]. Further work, e.g. [a23], [a24], has been concerned with establishing the existence of smooth conditional density functions, using methods based on the Malliavin calculus.

Stochastic filtering problems for counting process observations have first been considered by D.L. Snyder, see [a15]. Generalizations may be found in [a2], [a14], [a17].

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