集值下鞅的收敛性与Riesz分解

假定（X,·）为可分的Banach空间, X^*为其对偶空间, X^*可分. 设(Ω,B,P)为完备的概率空间, {B_n, n≥1}为B_n的上升子σ域族, 且B=∨B_n, 首先研究了支撑函数的几个性质, 利用支撑函数及实值鞅（上鞅、 下鞅）的收敛定理与Riesz分解定理, 证明了集值下鞅在弱收敛意义下的收敛定理, 在此基础上, 给出集值下鞅可Riesz分 解的一个充要条件.

Convergence and Riesz Decomposition of Set-valued Submartingale

Throughout this paper, （X,·） is assumed to be a real separable Banach space with a dual and separable X^*, we let (Ω，B,P) be a complete probability space, further, {B_n, n≥1} is an increase sub σ fields filtration of B, and B=∨ B _n. First of all, we discussed some properties of support function, we used the support function and Riesz decomposition theorem and convergence theorem of real valued martingale (supermartingale, submartingale) to prove its Riesz decomposition theorem and convergence theorem, on the basis of which a sufficien t condition of Riesz decomposition of set-valued was given.