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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

##### MSC:
 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)