On Levy's convergence theorems of two-parameter multivalued random processes (Q2466267)

\textit{Z. Wang} and \textit{X.-H. Xue} [Trans. Am. Math. Soc. 341, No. 2, 807--827 (1994; Zbl 0806.60034)] and \textit{W. Dong} and \textit{Z. Wang} [Proc. Am. Math. Soc. 126, No. 6, 1799--1810 (1998; Zbl 0927.60015)] obtained convergence results for one-parameter multivalued martingales. In the present paper these results are extended to the two-parameter case. Let \((\Omega,\Sigma,P)\) be a complete probability space. Let \({\mathfrak X}\) be a real separable Banach space and let \(L^p_c[\Omega,{\mathfrak X}]\) \((p\geq 1)\) denote the closure of the set of simple functions in \(L^p_c[\Omega,{\mathfrak X}]\) (denoting the family of measurable multifunctions \(F:\Omega\to P_c({\mathfrak X})\) satisfying \(\int_\Omega|F(\omega)|^p P(d\omega)< \infty\) (\(P_c({\mathfrak X})\) being the family of all nonempty bounded closed convex subsets of \({\mathfrak X}\))). For the set \(N\) of nonnegative integers let \(J:= N\times N\). For \(s= (s_1,s_2)\in J\) and \(t= (t_1,t_2)\in J\), \(s\leq t\) means \(s_1\leq t_1\) and \(s_2\leq t_2\); let \(s\wedge t:= (s_1\wedge t_1, s_2\wedge t_2)\). \({\mathcal F}= ({\mathcal F}_t)\) \((t\in J)\) is called a two-parameter filtration if, for each \(t\), \({\mathcal F}_t\) is a sub-\(\sigma\)-algebra of \(\Sigma\), and \(s\leq t\) implies \({\mathcal F}_s\subset{\mathcal F}_t\). \({\mathcal F}\) is called commuting if, for all \(s\), \(t\), \({\mathcal F}_s\) and \({\mathcal F}_t\) are conditionally independent given \({\mathcal F}_{s\wedge t}\). A two-parameter process \((M_t)\) \((t\in J)\) is called a martingale w.r.t. the filtration \({\mathcal F}\) if, for each \(t\), \(M_t\) is \({\mathcal F}_t\)-measurable, \(E[|M_t|]<\infty\) and \(E[M_t|{\mathcal F}_s]= M_s\) a.s. if \(s\leq t\). The upward case of Lévy's convergence theorem derived by the authors is then Theorem. Let \(p> 1\) and \(F\in L^p_c[\Omega, {\mathfrak X}]\). Let \({\mathcal F}\) be a commuting filtration and put \(F_t:= E[F|{\mathcal F}_t]\), \(F_\infty:= E[F|{\mathcal F}_\infty]\) where \({\mathcal F}_\infty\) is the \(\sigma\)-algebra generated by \({\mathcal F}_{(0, 0)}\cup{\mathcal F}_{(1,1)}\cup\cdots\). Then \(F_t\to F\) a.s. w.r.t. the Hausdorff metric. The authors also obtain a variant of Lévy's convergence theorem in the downward case.