英文标题：
《On random convex analysis》
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作者：
Tiexin Guo, Erxin Zhang, Mingzhi Wu, Bixuan Yang, George Yuan and
Xiaolin Zeng
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最新提交年份：
2017
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英文摘要：
Recently, based on the idea of randomizing space theory, random convex analysis has been being developed in order to deal with the corresponding problems in random environments such as analysis of conditional convex risk measures and the related variational problems and optimization problems. Random convex analysis is convex analysis over random locally convex modules. Since random locally convex modules have the more complicated topological and algebraic structures than ordinary locally convex spaces, establishing random convex analysis will encounter harder mathematical challenges than classical convex analysis so that there are still a lot of fundamentally important unsolved problems in random convex analysis. This paper is devoted to solving some important theoretic problems. First, we establish the inferior limit behavior of a proper lower semicontinuous $L^0$--convex function on a random locally convex module endowed with the locally $L^0$--convex topology, which makes perfect the Fenchel--Moreau duality theorem for such functions. Then, we investigate the relations among continuity, locally $L^0$--Lipschitzian continuity and almost surely sequent continuity of a proper $L^0$--convex function. And then, we establish the elegant relationships among subdifferentiability, G\\^ateaux--differentiability and Fr\\\'ech\\\'et--differentiability for a proper $L^0$--convex function defined on random normed modules. At last, based on the Ekeland\'s variational principle for a proper lower semicontinuous $\\bar{L}^0$--valued function, we show that $\\varepsilon$--subdifferentials can be approximated by subdifferentials. We would like to emphasize that the success of this paper lies in simultaneously considering the $(\\varepsilon, \\lambda)$--topology and the locally $L^0$--convex topology for a random locally convex module.
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中文摘要：
近年来，基于随机空间理论的思想，人们发展了随机凸分析，以处理随机环境中的相应问题，如条件凸风险测度分析以及相关的变分问题和优化问题。随机凸分析是随机局部凸模上的凸分析。由于随机局部凸模具有比普通局部凸空间更复杂的拓扑和代数结构，因此建立随机凸分析将遇到比经典凸分析更困难的数学挑战，因此在随机凸分析中仍然存在许多重要的未解决问题。本文致力于解决一些重要的理论问题。首先，我们建立了一个适当的下半连续的$L^0$--凸函数在一个随机局部凸模上的下极限行为，该随机局部凸模具有局部$L^0$--凸拓扑，这完善了这类函数的Fenchel--Moreau对偶定理。然后，我们研究了适当的凸函数的连续性、局部的$L^0$--李普希兹连续性和几乎确定的序列连续性之间的关系。然后，我们建立了定义在随机赋范模上的适当的$L^0$凸函数的次可微性、G \\^ateaux--可微性和Fr \\\'ech \\\'et--可微性之间的优雅关系。最后，基于Ekeland关于适当的下半连续$\\bar{L}^0$--值函数的变分原理，我们证明了$\\varepsilon$--次微分可以近似为次微分。我们想强调的是，本文的成功在于同时考虑了随机局部凸模的$（\\varepsilon，\\lambda）$--拓扑和局部$L^0$--凸拓扑。
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分类信息：

一级分类：Mathematics 数学
二级分类：Functional Analysis 功能分析
分类描述：Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory
Banach空间，函数空间，实函数，积分变换，分布理论，测度理论
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一级分类：Quantitative Finance 数量金融学
二级分类：Mathematical Finance 数学金融学
分类描述：Mathematical and analytical methods of finance, including stochastic, probabilistic and functional analysis, algebraic, geometric and other methods
金融的数学和分析方法，包括随机、概率和泛函分析、代数、几何和其他方法
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