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Teoriya Veroyatnostei i ee Primeneniya, 1992, Volume 37, Issue 1, Pages 57–63 (Mi tvp4124)  

Set-Indexed Stochastic Processes and Predictability

B. G. Ivanova, E. Merzach

University of Ottawa
Abstract: The objective of this work is to propose a beginning of a general theory for stochastic processes indexed by a family of subsets of a topological space. The following concepts will be defined and intensively studied: Random sets, stopping sets, announceable stopping sets, and $\sigma $-fields associated with stopping sets. These notions lead to the study of the predictable $\sigma $-field and its different characterizations. Different kinds of martingales will be defined, as well as some extensions (quasi-martingales, local martingales) and the optional sampling theorems will be discussed. A Doob–Meyer decomposition will be given. The notions of stochastic integral and local time will be introduced. The Markov properties for set-indexed processes will be discussed in our context such as the germ-field property and the strong Markov property. We will study convergence theorems for sequences of set-indexed processes. Finally, the results obtained will be applied to Gaussian fields and to jump processes.
English version:
Theory of Probability and its Applications, 1993, Volume 37, Issue 1, Pages 61–66
DOI: https://doi.org/10.1137/1137011
Bibliographic databases:
Language: Russian
Citation: B. G. Ivanova, E. Merzach, “Set-Indexed Stochastic Processes and Predictability”, Teor. Veroyatnost. i Primenen., 37:1 (1992), 57–63; Theory Probab. Appl., 37:1 (1993), 61–66
Citation in format AMSBIB
\Bibitem{IvaMer92}
\by B.~G.~Ivanova, E.~Merzach
\paper Set-Indexed Stochastic Processes and Predictability
\jour Teor. Veroyatnost. i Primenen.
\yr 1992
\vol 37
\issue 1
\pages 57--63
\mathnet{http://mi.mathnet.ru/tvp4124}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1211201}
\zmath{https://zbmath.org/?q=an:0794.60044}
\transl
\jour Theory Probab. Appl.
\yr 1993
\vol 37
\issue 1
\pages 61--66
\crossref{https://doi.org/10.1137/1137011}
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