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.     

一类集值有界变差过程及其对偶投影

本文给出了一类集值有界变差过程的定义,并在有限维情况下证明了集值有界变差过程的可选(可料)对偶投影的存在唯一性。     

自反B空间中集值增过程的对偶投影

假定Ａ是以自反Ｂａｎａｃｈ空间中弱紧凸集为值的集值增过程，本文研究了非负有界可测过程关于Ａ的积分以及Ａ在乘积可测空间上生成的集值测度，证明了每个可积集值增过程存在唯一对偶可选（可料）投影．     

自反B空间中可测集值随机过程的可选（可料）投影

设Ｘ是可分自反Ｂａｎａｃｈ空间，（Ｆ，Ａ，ｔ∈Ｒ＋）是取值于Ｘ中闭凸集的可测集值随机过程，该文证明了，若任给停时Ｔ，ＦＴＩ（Ｔ〈∞）关于ＡＴ（相应地关于Ａ）是σ一可积，则（Ｆ，ｔ∈Ｒ＋）必存在可选（相应地可料）投影过程。     

有界可料过程关于集值平方可积鞅的集值随机积分

本文定义了一类有界可料过程关于集值平方可积鞅的集值随机积分,并研究了集植随机积分的性质。此为建立集值随机分析的理论奠定了基础。     

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。     

局部平方可积鞅的Chung重对数律(英文)

设X＝（Xt，t0）为局部平方可积鞅，且Xo＝0，＜X，X＞t为其二阶可料变差．利用连续半鞅的强逼近结果，我们证明了在较弱的条件下，X的Chung重对数律成立，即     

平方可积鞅

讨论了集值平方可积鞅和实值平方可积鞅的性质.它对集值随机过程的进一步研究将起到很重要的作用.     

过程的投影理论注记

<正> σ-可积性的概念在离散参数的过程论中的作用,我们已经在[1]里进行了讨论.在连续参数过程论中能否起类似的作用呢?至少在鞅论方面要遇到一些困难,例如我们就未能不借助可选时的概念给出局部上鞅和局部鞅的定义.不过,在随机过程一般理论方面,还是可以起一点作用的,它可以给出可测过程可选投影或可料投影存在的充要条件,     

局部平方可积鞅Chung重对数律的下界

设Ｘ＝（Ｘｔ，ｔ≥０）为零初值的局部平方可积鞅〈Ｘ，Ｘ〉＝（〈Ｘ，Ｘ〉ｔ，ｔ≥０）为具可料二阶交差，在类似于Ｋｏｌｍｏｎｏｒｏｖ最初给出的条件下，证明了局部平方可积鞅的Ｃｈｕｎｇ重对数律的下界成立，即     

Projections and regularity of abstract processes

In this note we shall prove the existence of optional and predictable projections of stochastic processes X taking values in a Banach space E. Furthermore, if the range of X is contained in a compact set and if X is cadlag (respectively caglad), then the optional (respectively predictable) projection possesses the same property. Finally, we shall prove that every E-valued martingale has a cadlag modification     

On the Theory of (Dual) Projection for Fuzzy Stochastic Processes

Abstract

In a theory similar to one of real-valued stochastic processes, in this paper, we investigate the projection and dual projection for fuzzy stochastic processes. First, the related concepts of fuzzy stochastic processes are introduced, such as adaption, measurability, optionality, predictability, etc. Subsequently, we study fuzzy stochastic integral and fuzzy measure generated by increasing fuzzy stochastic processes. Moreover, (dual) projection w.r.t. (increasing) fuzzy stochastic processes are discussed. We prove the existence and uniqueness of (dual) optional (predictable) projection for (increasing) fuzzy stochastic processes.     

Integration with Respect to Hilbert Space‐Valued Semimartingales via Jacod‐Grigelionis Characteristics

Abstract

The main aim of this paper is to describe the space of operator‐valued predictable processes, which are integrable with respect to a Hilbert space valued quasi‐left continuous semimartingale. The space of integrable processes is a randomized Musielak‐Orlicz space with a modular explicitely expressed in terms of Jacod‐Grigelionis characteristics.     

Convex integral functionals of regular processes

This article gives dual representations for convex integral functionals on the linear space of regular processes. This space turns out to be a Banach space containing many more familiar classes of stochastic processes and its dual can be identified with the space of optional random measures with essentially bounded variation. Combined with classical Banach space techniques, our results allow for a systematic treatment of stochastic optimization problems over BV processes and, in particular, yields a maximum principle for a general class of singular stochastic control problems.     

Frechet-valued martingales and stochastic integrals

Stochastic processes with values in a separable Frechet space whose a itinuous linear functional are real-valued square integrable martingales are investigated. The coordinate measures on the Fréchet space are obtained from cylinder set measures on a Hilbert space that is dense in the Fréchet space. Real-valued stochastic integrals are defined from the Fréchet-valued martingales using integrands from the topological dual of the aforementioned Hilbert space. An increasing process with values in the self adjoint operators on the Hilbert space plays a fundamental role in the definition of stochastic integrals. For Banach-valued Brownian motion the change of variables formula of K. Itô is generalized. A converse to the construction of the measures on the Fréchet space from cylinder set measures on a Hilbert space is also obtained.     

Fatou's Lemma for Gelfand Integrable Mappings

We provide a version of Fatou's lemma for mappings taking their values in E ^*, the topological dual of a separable Banach space. The mappings are assumed to be Gelfand integrable, a difference with previous papers, which, in infinite dimensional spaces, are mainly considering Bochner integrable mappings. This result is motivated by a general equilibrium model with locations studied by Cornet and Medecin (1999) and directly applies to it, since the space E ^* considered by Cornet and Medecin is the space of (Radon) vector measures defined on a compact metric space.     

A convergence theorem for convex set valued supermartingales ∗

Functions whose values are convex subsets provide a natural setting for the study of goal uncertainty in decision making. In fact under reasonable assumptions the successive estimates of the convex valued conditional expectations of the utility function form a supermartingale. It is of course important to determine if such estimates converge

It is shown that if {F_n, H_n} is a supermartingale where F_n has as values convex subsets of a Banach space X with separable dual and provided that F_n is “uniformly integrable” then F_n converges in some appropriate mode made precise in the work     

Decomposition of a class of functionals and the predictable representation theorem on Banach spaces

LetB be a separable real Banach space andX(t) be a symmetric conservative diffusion process taking values inB. In this paper, we decompose the functionalu(X(t),t) into a sum of a square integrable martingale and a regular 0-quadratic variation process. On this basis, we establish the predictable representation theorem ofX(t).This project is supported by the National Natural Science Foundation of China.