Discrete-Parameter Controlled Stochastic Processes



Let two sets X and U with σ-algebras of measurable subsets \( \mathfrak{A}\;and\;\mathfrak{B} \) respectively, i.e. two measurable spaces \( (X,\mathfrak{A})\;and\;(U,\mathfrak{B}) \) be given. The first space is called the phase space of the basic process and the second the phase space of control. Let N be the set of non-negative integers. In this Chapter all the processes are defined on the set N. To define a controlled process it is necessary to define the probability distribution of a random process with values in X provided a sequence of controls at each instant of time is given and also to define a rule according to which these controls are selected. We shall now describe the components of a controlled process in a more precise manner.


Markov Chain Basic Process Control Object Borel Function Lower Semicontinuous Function 
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Copyright information

© Springer-Verlag New York Inc. 1979

Authors and Affiliations

  1. 1.Institute of Applied Mathematics and MechanicsAcademy of Sciences of the Ukranian SSRDonetskUSSR
  2. 2.Institute of MathematicsAcademy of Sciences of the Ukranian SSRKievUSSR

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