A set of some predictable fuzzy stochastic processes is a complete metric space for some metric. A set of some square integrable fuzzy martingales is a complete metric space for some metric and the set of all almost surely continuous square integrable martingales is its closed subset.

A variable upper the definition of the It integral of standard Wiener process with respect to fuzzy stochastic process and its properties are given, A variable upper limit It integral of fuzzy stochastic process with respect to the standard Wiener process is proven to be an almost surely continuous square integral fuzzy martingale.

In this paper,set-valued squire integral martingale and real-valued squire integral martingale are given.It will take an important role in the deeper research of set-valued stochastic analysis.

The stochastic integral of simple real-valued predictable processes with respect to bounded convex fuzzy set-valued square integrable martingales is given at beginning.Then we prove the integration is a fuzzy set-valued square integtable martingale.

In this paper, existence theorems of variance of mean integration in respect to set-valued stochastic process are given.It will take a important role in the deeper research of set-valued stochastic analysis.

It integrals of the real predicable processes respect to fuzzy set-valued Wiener stochastic(process) and to bounded integrable compact convex Wiener stochastic process are associated by the level sets and support sets.Corresponding definition and characters are given.At last,there is an(introduction) of the application of such an It integral in stochastic differential coefficient equation.

In this paper,the concept and qualities of set-valued class(D) processes and set-valued and set valued local martingale are introduced,then the concept and qualities of set valued local square integral martingale are discussed.

为研究实值可料过程关于可积有界紧凸集值Wiener过程的伊藤积分,先从简单实值可料过程入手,以支撑函数为工具,给出相关定义及性质.然后借助于简单过程构造出一般过程的轨道,把结论推广到一般实值可料过程,给出其关于可积有界紧凸集值Wiener过程的伊藤积分的定义及性质.

研究了简单实值可料过程关于有界闭凸集值平方可积鞅的随机积分,证明了它是一个集值平方可积鞅;然后将被积函数推广为一般实可料过程,证明了在一定假设条件下其随机积分是存在的,且是一个集值平方可积鞅.

In order to study the derivative and integral theories of the set-valued stochastic processes, an introduction is firstly made of the concepts of the strong (weak) mean square integral and derivative of the bounded closed convex set-valued stochastic processes. Then the relations between the mean square derivative and the mean square integral were discussed by means of support functions and Hausdorff measure. Based on them, the Newton-Leibniz formulas of the mean square integral of the bounded closed convex set-valued stochastic processes were proved. Fimally an example was presented. The conclusions are important to the further studying of the set-valued stochastic derivative equations.

本文给出了集值随机分析中的两个完备性定理,它对集值随机过程的进一步研究将起到很重要的作用.

为研究集值随机过程的微积分理论,首先利用支撑函数定义了二阶矩有界闭凸集值随机过程的均方Riemann积分,其次利用支撑函数以及均方收敛的性质证明了二阶矩有界闭凸集值随机过程的均方Riemann积分的线性性、同数学期望的可交换性等性质.

为了研究集值随机过程的微积分理论,利用有界闭凸集合弱收敛的性质和集合"开"的概念,给出了有界闭凸集值随机过程的均方导数的定义,建立了均方导数的若干性质,并讨论了集值随机过程均方可导与均方连续的关系.

集值马尔可夫过程在集值随机过程理论中是非常重要的,但到目前为止,有关集值马尔可夫过程的研究成果还不多见,就连它的定义,至今还没有统一的处理.从集值马尔可夫过程的最简单定义出发,证明了它的一系列定义的等价性,从而对这类过程的本质有较为完整的认识.

本文证明了弱集值Amart可Riesz分解的充要条件.

集值随机过程的积分理论是随机过程理论的一个新分支.利用随机集的位似及集值随机过程的均方收敛理论,创造性的定义了集值随机过程的均方Riemann-Stieljes积分,随后证明了均方Riemann-Stieljes积分存在性的判定定理,最后给出了分部积分计算公式,为今后进一步研究集值随机过程的微积分奠定了良好的理论基础.

为了得到关于弱集值渐近鞅的收敛性质，首先证明了支撑函数列的极限亦为一支撑函数。利用支撑函数的性质以及集值鞅的Doob停止定理，证明得到了两个结论：①在一定条件下，弱集值渐近鞅存在无限逼近的闭凸集值鞅；②在弱收敛意义下，弱集值渐近鞅收敛的两个等价条件。

如同马尔可夫过程在随机过程理论中的重要性一样，集值马尔可夫过程在集值随机过程理论中也是非常重要的。但由于种种原因，到目前为止，集值马尔可夫过程能见到的研究成果还不多见，就连它的定义，至今还没有统一的处理，使研究工作感到困难。从最直观的角度给出集值马尔可夫过程的定义，在此基础上，证明了它的一系列定义的等价性，从而对这类过程的本质有较为完整的认识。类似于经典情形一样，引进了转移函数和集值齐次马尔可夫过程的概念。运用柯尔莫哥洛夫定理，讨论了集值齐次马尔可夫过程的判定定理。在上述讨论的基础上，证明了齐次集值马尔可夫过程的存在性。

讨论了与分枝过程相关联的一些随机过程,证明了:与多型分与下临界非零状态互通的多型Galton-Watson分枝过程相关联的的随机过程是齐次不可分正常返的Markov链.与临界非零状态互通的多型Galton-Watson分枝过程相关联的随机过程是齐次不可分非常返的Markov链.

给出了宽平稳集值过程和Gauss集值过程的定义,证明了二阶矩平稳闭凸集值过程必是宽平稳集值过程,还证明了宽平稳Gauss有界闭凸集值过程是平稳集值过程.