实值非负函数关于集值序增函数的集值Riemann-Stieltjes积分

本文首先建立了实值非负函数关于集值序增函数的集值Riemann-Stieltjes积分,并讨论了集值Riemann-Stieltjes积分的性质,给出了集值Riemann-Stieltjes可积的充要条件,最后引入了集值Riemann-Stieltjes随机积分．     

集值Lebesgue—Stieltjes积分

本文首先刻划了B(R_ )上的集值测度,其次建立了(R_ B(R_ ))上的集值Lebesgue-Stieltjes积分.最后,进—步建立了集值随机Lebesgue-Stietjes积分的理论.     

集值局部平方可积鞅

介绍集值类（D）过程和集值局部鞅的概念和有关性质，进而讨论了集值局部平方可积鞅的概念和性质。     

实值可料过程关于模糊集值Wiener过程的伊藤积分

通过利用水平集和承集将实值可料过程关于模糊集值W iener过程的伊藤积分与实值可料过程关于可积有界紧凸集值W iener过程的伊藤积分联系起来,给出相应的定义和性质,并初步探讨了该伊藤积分在随机微分方程方面的应用。     

有界闭凸值测度的集值积分

引入实值函数关于有界闭凸值测度的集值积分,并讨论了集值积分的收敛定理,证明了当集值测度为有界闭凸集值的有界变差集值测度时,关于弱紧凸集值测度的积分性质对有界闭凸集值测度仍然保持.推广了实值函数关于弱紧凸值测度的积分.     

集值随机过程的投影

本文研究了集值可积变差随机过程的可选和可料对偶投影.当Banach空间X具有RNP,其对偶空间X*可分时,证明了Pwkc(X)值的可积变差过程存在唯一的可选和可料对偶投影.最后讨论了集值随机过程对偶投影的性质.     

关于集值Bartle积分新的极限定理

给出集值Bartle积分一个新的定义,并进一步讨论了数值函数关于有界闭凸集值可数可加集值测度的积分的性质,建立了集值Bartle积分的新的极限定理.     

集值随机过程的投影

本文研究了集值可积变差随机过程的可选和可料对偶投影．当Ｂａｎａｃｈ空间Ｘ具有ＲＮＰ,其对偶空间Ｘ＊可分时,证明了Ｐｗｋｃ（Ｘ）值的可积变差过程存在唯一的可选和可料对偶投影．最后讨论了集值随机过程对偶投影的性质．     

集值测度的RadOn-Nikodym定理及集值算子的Pettis-Aumanri积分表示

本文建立了Banach空间集值测度的Radon-Nikodym定理,并给出了两类集值算子的Pettis-Aumann积分表示.     

集值映射Pettis-Aumann积分的一些性质

对集值映射的Pettis-Aumann型积分的性质作了讨论，给出了Pettis-Aumann积分型积分的Fatou引理形式以及其它一些收敛定理，探讨了带参数集值映射的积分性质。     

Properties of set-valued stochastic differential equations

The paper is devoted to properties of set-valued stochastic differential equations. The main result of the paper deals with existence and uniqueness of solutions. Furthermore, a connection between solutions of stochastic differential inclusions and solutions of set-valued stochastic differential equations are given. The result of the paper extends a lot of particular results dealing with such type equations.     

在Pbkc（c[0,1]）与Pbkc（Lp[0,1]）取值的集值随机变量（1）

对拟连续测度空间(G,β,u)的一致有界等度连续函数族,通过包含关系,取凸包和闭包,构造了在Pbkc(c[0,1])与Pbkc(Lp[0,1])取值的集值随机变量及连续的集值映射,深化了集值随机过程理论研究.     

Strong solution of Itô type set-valued stochastic differential equation

In this paper, we shall firstly illustrate why we should introduce an It5 type set-valued stochastic differential equation and why we should notice the almost everywhere problem. Secondly we shall give a clear definition of Aumann type Lebesgue integral and prove the measurability of the Lebesgue integral of set-valued stochastic processes with respect to time t. Then we shall present some new properties, especially prove an important inequality of set-valued Lebesgue integrals. Finally we shall prove the existence and the uniqueness of a strong solution to the It5 type set-valued stochastic differential equation.     

Approximation theorems for set-valued stochastic integrals

The article is devoted to new properties of Aumann, Lebesgue, and Itô set-valued stochastic integrals considered in papers [1 Kisielewicz, M. (2014). Properties of generalized set-valued stochastic integrals. Discuss. Math. (DICO) 34:131–147. [Google Scholar],2 Kisielewicz, M., Michta, M. (2017). Integrably bounded set-valued stochastic integrals. J. Math. Anal. Appl. 449:1893–1910.[Crossref], [Web of Science ®] , [Google Scholar]]. In particular, it contains some approximation theorems for Aumann and Itô set-valued stochastic integrals. Hence, in particular, it follows that Aumann and Lebesgue set-valued stochastic integrals cover a.s., both for measurable and IF-nonanticipative integrably bounded set-valued stochastic processes.     

On a Theorem of Deutsch and Singer

We introduce the notion of a starshaped set-valued function, and relate it to convex set-valued functions. We prove a theorem which, in the convex case, implies that single-valuedness is exceptional if the function is not everywhere single-valued. A second result shows a remarkable costantness of images for starshaped set-valued functions. Both theorems were inspired from a result of Deutsch and Singer, which they strengthen.     

集值映射的Epi-导数及其应用

在本文里，集值映射的Epi-导数被引入，它可以认作是实值Lipschitz函数的Clarke-广义方向导数的推广，同时它的一些性质也被研究．进一步地，利用这个Epi-导数集值映射的次微分被定义并研究它的性质．作为其应用，我们给出了集值优化问题的一些(必要或充分)最优性条件．     

Stochastic recursive inclusions with non-additive iterate-dependent Markov noise

In this paper we study the asymptotic behaviour of stochastic approximation schemes with set-valued drift function and non-additive iterate-dependent Markov noise. We show that a linearly interpolated trajectory of such a recursion is an asymptotic pseudotrajectory for the flow of a limiting differential inclusion obtained by averaging the set-valued drift function of the recursion w.r.t. the stationary distributions of the Markov noise. The limit set theorem by Benaim is then used to characterize the limit sets of the recursion in terms of the dynamics of the limiting differential inclusion. We then state two variants of the Markov noise assumption under which the analysis of the recursion is similar to the one presented in this paper. Scenarios where our recursion naturally appears are presented as applications. These include controlled stochastic approximation, subgradient descent, approximate drift problem and analysis of discontinuous dynamics all in the presence of non-additive iterate-dependent Markov noise.