离散时间正规鞅算子值函数的Bochner积分  

Bochner Integration of Operator-Valued Functions in Terms of Discrete-Time Normal Martingales

作　　者：陈金淑[1] 唐玉玲 CHEN Jinshu;TANG Yuling(School of Science,Lanzhou University of Technology,Lanzhou,730050,China;School of Mathematics and Statistics,Hexi University,Zhangye,734000,China)

机构地区：[1]兰州理工大学理学院,兰州730050 [2]河西学院数学与统计学院,张掖734000

出　　处：《应用概率统计》2023年第3期436-448,共13页Chinese Journal of Applied Probability and Statistics

基　　金：国家自然科学基金项目(批准号:12161050)资助.

摘　　要：设M是一个具有混沌表示性质的离散时间正规鞅,φ(M)■L2(M)■φ*(M)是基于M泛函的Gel’fand三元组.从φ(M)到φ*(M)的连续线性算子可称为M泛函上的广义算子,以φ表示此类算子的全体.本文的主要目的在于建立φ值函数关于φ值测度的积分运算.为此,本文首先讨论φ值测度的基本性质,在此基础上定义了φ值函数关于φ值测度在卷积意义下的Bochner积分,并建立了相应的控制收敛定理和卷积意义下的Fubini定理.Let M be a discrete-time normal martingale satisfying some mild conditions,■(M)■L2(M)■(M)be the Gel'fand triple constructed from the functionals of M.L denote the space of continuous linear operators from the testing functional space ■(M)to the generalized functional space ■(M).As is known,the usual product in L may not make sense.However,by using the 2D-Fock transform,one can introduce convolution in L,then one can try to introduce a Bochner-style integral for L-valued functions with respect to L-valued measures in the sense of convolution.This paper just studies such a type of integral.First,a class of L-valued measures are introduced and their basic properties are examined.Then,an integral of an L-valued function with respect to an L-valued measure is de ned and a dominated convergence theorem is established for this integral.Finally,a convolution measure of two L-valued measures is also discussed and a Fubini type theorem is proved for this integral.

关 键 词：离散时间正规鞅 BOCHNER积分 算子值测度 FUBINI定理 

分 类 号：O211.6[理学—概率论与数理统计]