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Regions of stability for random decision systems with complete connections. (English) Zbl 0573.93077
Let (X,\({\mathcal X})\), (Y,\({\mathcal Y})\), (W,\({\mathcal W})\) be measurable spaces, \(\Theta \subset R^ k\), \(\{\) P(\(\theta\) ;.,.,.);\(\theta \in \Theta \}^ a \)set of stochastic kernels defined on \(W\times Y\times {\mathcal X}\), \(\{\) z(\(\theta\) ;.,.,.): \(W\times Y\times {\mathcal X}\to W;\theta \in \theta \}\), z being measurable, \(Y: \Theta\) \(\to {\mathcal Y}\), q and h known real valued functions defined on \(\Theta\) \(\times W\times Y\). The paper gives two results concerning the regions of stability of the functional equation: \[ f(\theta;w)=\sup_{y\in Y(\theta)}[q(\theta;w,y)+C+h(\theta;w,y)\int_{X}P(\theta;w,y,dx)\quad f(\theta;z(\theta;w,y,x))] \] The system \(\{\) X,\({\mathcal X})\), (W,\({\mathcal W})\), (Y,\({\mathcal Y})\), \(\Theta\), Y(\(\cdot)\), z, \(P\}\) is named parametric random decision system with complete connections and is regarded as the formalization of a decision problem.
Reviewer: H.N.Teodorescu

93E15 Stochastic stability in control theory
62C99 Statistical decision theory
60B05 Probability measures on topological spaces
39B52 Functional equations for functions with more general domains and/or ranges
93E03 Stochastic systems in control theory (general)