Stochastic process, renewable
innovation stochastic process
A stochastic process with a fairly "simple" structure, constructed from an input process and containing all necessary information about this process. Innovation stochastic processes have been used in the problem of linear prediction of stationary time series, in non-linear problems of statistics of stochastic processes, and elsewhere (see [1]–[3]).
The concept of an innovation stochastic process can be introduced into the theory of linear and non-linear stochastic processes in various ways. In the linear theory (see [4]), a vector stochastic process is called an innovation process for a stochastic process
with
if
has non-correlated components with non-correlated increments and if
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where and
are the mean-square closed linear hulls of all the values
(
) and
(
), respectively (in a suitable space of functions on the probability space
). The number of components
(
) of
is called the multiplicity of the innovation process, and is uniquely determined by
. In the case of one-dimensional
in discrete time,
, and in the case of continuous time
only under certain special assumptions about the correlation function of
(see [4], [5]). In applications one may take advantage of the fact that
can be represented as a linear functional of the values of
,
.
In the non-linear theory (see [5], [6]), the term innovation stochastic process usually refers to a Wiener process such that
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where ,
are the
-algebras of events generated by the values of
,
,
, respectively. In the case when
(
) is an Itô process with stochastic differential
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the Wiener process defined by
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is an innovation stochastic process for , for example, when
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and the processes and
form a Gaussian system (see [6]).
References
[1] | A.N. Kolmogorov, "Interpolation and extrapolation of stationary random sequences" Rand Coorp. Memorandum , RM-3090-PR (April 1962) Izv. Akad. Nauk. SSSR Ser. Mat. , 5 (1941) pp. 3–14 |
[2] | A.N. Shiryaev, "Stochastic equations of nonlinear filtering of Markovian jumps" Probl. of Inform. Transmission , 2 : 3 (1966) pp. 1–18 Probl. Pered. Inform. , 2 : 3 (1966) pp. 3–22 |
[3] | T. Kailath, "A view of three decades of linear filtering theory" IEEE Trans. Inform. Theory , 20 : 2 (1974) pp. 146–181 |
[4] | Yu.A. Rozanov, "Innovation processes" , Wiley (1977) (Translated from Russian) |
[5] | A.N. Shiryaev, "Reduction of data with saving of information and innovation processes" , Trans. School-Seminar Theory of Stochastic Processes (Druskininka, 1974) , 2 , Vilnius (1975) pp. 235–267 (In Russian) |
[6] | R.S. Liptser, A.N. Shiryaev, "Statistics of stochastic processes" , 1 , Springer (1977) (Translated from Russian) |
Comments
References
[a1] | Yu.A. Rozanov, "Innovation processes" , Winston (1977) (Translated from Russian) |
Stochastic process, renewable. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stochastic_process,_renewable&oldid=48859