连续参数集值下鞅的收敛定理  

Convergence Theorem of Continuous Parameter Set Valued Submartingale

作　　者：朱永忠[1] 

机构地区：[1]河海大学数学物理系

出　　处：《河海大学学报（自然科学版）》1999年第1期89-92,共4页Journal of Hohai University(Natural Sciences)

基　　金：河海大学青年科学基金

摘　　要：首先给出了连续参数集值下鞅的定义．继而证明了连续参数集值下鞅的三个等价定理：（ａ）Ｌ１ｗｋｃ（Ｘ）值下鞅等价于任给τ１＜τ２，τ１，τ２∈Ｔ，∫ΩＦτ１ｄＰ∫ΩＦτ２ｄＰ；（ｂ）Ｌ１ｆｃ（Ｘ）值下鞅等价于任给ｓ，ｔ∈Ｒ＋，ｓ＜ｔ，Ｓ１Ｆｓ（Ｆｓ）ｃｌ｛Ｅ（ｇ｜Ｆｓ），ｇ∈Ｓ１Ｆｔ（Ｆｔ）｝；（ｃ）Ｘ可分时，闭凸集值下鞅等价于任给ｓ，ｔ∈Ｒ＋，ｓ＜ｔ，Ａ∈Ｆｓ，ｃｌ∫ＡＦｓｄＰｃｌ∫ＡＦｔｄＰ．最后给出了弱紧凸集值随机集族的弱收敛定理和Ｘ有ＲＮＰ，Ｘ可分时闭凸集值右连续下鞅的弱收敛定理．The definition of continuous parameter set valued submartingale is given.Then,three equivalent theorems of continuous parameter set valued submartingale is proved:(1)L 1 wkc (X) valued submartingale is equal to ∫ ΩF τ 1 d p∫ ΩF τ 2 d p for any τ 1,τ 2∈T and τ 1<τ 2;(2)L 1 fc (X) valued submartingale is equal to S 1 F s (F s)cl{E(g/F s);g∈S 1 F t (F t)} for any s,t∈R + and s<t;(3)When X  is separable,closed convex set valued submartingale is equal to  cl ∫ AF s d p cl ∫ AF t d pfor any s,t∈R + and s<t.In the end ,the weak convergence theorem of weak compact convex set valued random sets and weak convergence theorem of closed convex set valued right continuous submartingale are given.

关 键 词：弱紧凸集 闭凸集 连续参数 集值下鞅 收敛定理 

分 类 号：O174.13[理学—数学] O211.6[理学—基础数学]