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 $ x _ {t} $ is called an innovation process for a stochastic process $ \xi _ {t} $ with $ {\mathsf E} | \xi _ {t} | ^ {2} < \infty $ if $ x _ {t} $ has non-correlated components with non-correlated increments and if

$$ H _ {t} ( \xi ) = H _ {t} ( x) \ \textrm{ for all } t , $$

where $ H _ {t} ( \xi ) $ and $ H _ {t} ( x) $ are the mean-square closed linear hulls of all the values $ \xi _ {s} $( $ s \leq t $) and $ x _ {s} $( $ s \leq t $), respectively (in a suitable space of functions on the probability space $ \Omega $). The number of components $ N $( $ N \leq \infty $) of $ x _ {t} $ is called the multiplicity of the innovation process, and is uniquely determined by $ \xi _ {t} $. In the case of one-dimensional $ \xi _ {t} $ in discrete time, $ N = 1 $, and in the case of continuous time $ N < \infty $ only under certain special assumptions about the correlation function of $ \xi _ {t} $( see [4], [5]). In applications one may take advantage of the fact that $ \xi _ {t} $ can be represented as a linear functional of the values of $ x _ {s} $, $ s \leq t $.

In the non-linear theory (see [5], [6]), the term innovation stochastic process usually refers to a Wiener process $ x _ {t} $ such that

$$ {\mathcal F} _ {t} ^ \xi = {\mathcal F} _ {t} ^ {x} , $$

where $ {\mathcal F} _ {t} ^ \xi $, $ {\mathcal F} _ {t} ^ {x} $ are the $ \sigma $- algebras of events generated by the values of $ \xi _ {s} $, $ x _ {s} $, $ s \leq t $, respectively. In the case when $ \xi _ {t} $( $ 0 \leq t \leq T $) is an Itô process with stochastic differential

$$ d \xi _ {t} = a ( t) d t + d w _ {t} , $$

the Wiener process $ \overline{w}\; _ {t} $ defined by

$$ \overline{w}\; _ {t} = \xi _ {t} - \int\limits _ { 0 } ^ { t } {\mathsf E} ( a ( s) \mid {\mathcal F} _ {s} ^ \xi ) d s $$

is an innovation stochastic process for $ \xi _ {t} $, for example, when

$$ {\mathsf E} \int\limits _ { 0 } ^ { T } a ^ {2} ( s) d s < \infty $$

and the processes $ a $ and $ w $ form a Gaussian system (see [6]).

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