随机凸分析(Ⅱ):L^0-准桶的随机局部凸模中的连续性和次可微性定理

Random convex analysis(Ⅱ): Continuity and subdifferentiability theorems in L^0-pre-barreled random locally convex modules

本文继续研究随机凸分析.首先,引入L0-准桶模的概念;接着,在赋予局部L0-凸拓扑的随机局部凸模的框架下,为了建立随机局部凸模为L0-准桶模的特征,本文发展了随机对偶理论,尤其是证明用于条件风险度量的模途径中的模型空间LpF(E)是L0-准桶的,这形成本文最困难的部分.最后,本文证明L0-准桶的随机局部凸模上的真下半连续L0-凸函数的连续性和次可微性定理.因此,本文的主要结果可以很好地适用于L0-凸条件风险度量的连续性和次可微性的研究.

In this paper, we continue to study random convex analysis. First, we introduce the notion of an L0-pre-barreled module. Then, we develop the theory of random duality under the framework of a random locally convex module endowed with the locally L0-convex topology in order to establish a characterization for a random locally convex module to be L0-pre-barreled, in particular we prove that the model space Lp（ζ） employed in the module approach to conditional risk measures is L0-pre-barreled, which forms the most difficult part of this paper. Finally, we prove the continuity and subdifferentiability theorems for a proper lower semicontinuous L0-convex function on an L0-pre-baxreled random locally convex module. So the principal results of this paper may be well suited to the study of continuity and subdifferentiability for L0-convex conditional risk measures.