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Necessary and sufficient conditions for existence of stationary and periodic solutions of a stochastic difference equation in Hilbert space. (English) Zbl 0699.60054

The paper deals with stochastic difference equations of the form \[ x(n+1)=A(n)x(n)+e(n) \] in a Hilbert space H, with bounded linear operators A(n). Random processes generated by equations of this form are of interest for the theory of random processes, for statistical analysis, and for various applications, such as radio engineering and models in mathematical economics. An important concept for these applications is that of a periodic process, considered in the paper.
The aim of the work is to discuss questions of existence and uniqueness of stationary and periodic (in a wide sense defined in terms of certain mean characteristics) solutions of these equations. First, conditions characterizing the existence of stationary (when A(n) is constant) and periodic (when A(n) is periodic) solutions are given. Then, a sufficient condition for existence and uniqueness of a periodic solution is proved.
Reviewer: F.Flandoli

MSC:

60H99 Stochastic analysis
60G10 Stationary stochastic processes
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
91B84 Economic time series analysis
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References:

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