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Discrete-Parameter Controlled Stochastic Processes

In: Controlled Stochastic Processes

Author

Listed:
  • Iosif Il’ich Gihman

    (Academy of Sciences of the Ukranian SSR, Institute of Applied Mathematics and Mechanics)

  • Anatoliĭ Vladimirovich Skorohod

    (Academy of Sciences of the Ukranian SSR, Institute of Mathematics)

Abstract

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.

Suggested Citation

  • Iosif Il’ich Gihman & Anatoliĭ Vladimirovich Skorohod, 1979. "Discrete-Parameter Controlled Stochastic Processes," Springer Books, in: Controlled Stochastic Processes, chapter 1, pages 1-78, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4612-6202-2_1
    DOI: 10.1007/978-1-4612-6202-2_1
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