本项目致力于发展适合于随机赋范模理论及其应用的紧性理论。首先, 系统发展最近由申请人提出的L^0凸紧性理论，进而解决动态风险中引人瞩目的条件均值—条件凸风险度量的优化问题，将Banach空间几何理论与完备随机赋范模的几何理论结合起来建立定义在随机自反的随机赋范模上的L^0值函数的极大极小理论。在此基础上，试图提出一种适合于随机赋范模理论发展的紧性理论，目标是最终建立完备随机赋范模上著名的J.Schauder不动点定理与KKM映像原理。

This project is devoted to developing a kind of compactness suitable for the theory of random normed modules and its applications. First, we systematically develop the theory of L^0-convex compactness recently presented by us and aim at solving the remarkable conditional mean-conditional convex risk measure optimization problem,by combining the geometric theories of Banach spaces and complete random normed modules we also establish the minimax theorem for an L^0-valued function defined on random reflexive random normed modules.Then, based on the work, we attempt to present and develop a kind of compactness suitable for the theory of random normed modules, our target is to eventually establish the famous J.Schauder fixed point theorem and KKM mapping principle in complete random normed modules