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In this paper, the theory of mean-field backward doubly stochastic Volterra integral equations (MF-BDSVIEs) is studied. First, we derive the well-posedness of M-solutions to MFBDSVIEs, and prove the comparison theorem for such a type of equations. Furthermore, the regularity result of the M-solution for MF-BDSVIEs is established by virtue of Malliavin calculus. Finally, as an application of the comparison theorem, we obtain the properties of dynamic risk measures governed by MF-BDSVIEs., Comment: 27 pages

In 2010, the first author of this paper introduced the notion of $\sigma$--stability for a nonempty subset of an $L^0(\mathcal{F},K)$--module in [T.X. Guo, Relations between some basic results derived from two kinds of topologies for a random locally convex module, J. Funct. Anal. 258(2010), 3024--3047], this kind of $\sigma$--stability is purely algebraic and leads to a series of deep developments of random normed modules and random locally convex modules. Motivated by this, A. Jamneshan, M. Kupper and J. M. Zapata recently introduced another kind of $\sigma$--stability for a nonempty subset of a random metric space $(E,d)$, called $d$--$\sigma$--stability since it depends on the random metric $d$. $d$--$\sigma$--stability coincides with the previous $\sigma$--stability in the case of random normed modules, which motivates us in this paper to generalize the precise form of Ekeland's variational principle from a complete random normed module to a complete $d$--$\sigma$--stable random metric space. Besides, this paper also utilize $d$--$\sigma$--stability to generalize Nadler's fixed point theorem for a multivalued contraction mapping from a complete metric space to a complete random metric space. To our surprise, our simple fixed point theorem, however, can derive the known basic fixed point theorems of contraction type for both random operators and $\sigma$--stable mappings on a complete random normed module. A lot of examples shows the study of random metric spaces is more complicated than that of random normed modules., Comment: 19 pages

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., Comment: 28 pages