Convergence of set valued sub- and supermartingales in the Kuratowski-Mosco sense

Abstract

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.