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Convergence of set valued sub- and supermartingales in the Kuratowski-Mosco sense. (English) Zbl 0938.60031
Summary: The purpose of this paper is to prove some convergence theorems of closed and convex set valued sub- and supermartingales in the Kuratowski-Mosco sense. To get submartingale convergence theorems, we give sufficient conditions for the Kudo-Aumann integral and Hiai-Umegaki conditional expectation to be closed both for compact convex set valued random variables and for closed convex set valued random variables. We also give an example of a bounded closed convex set valued random variable whose Kudo-Aumann integral is not closed.

MSC:
60G42 Martingales with discrete parameter
28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections
60G48 Generalizations of martingales
60D05 Geometric probability and stochastic geometry
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