arXiv subjects
Publications and source records attributed to Mingzhi Wu.
Based on both the fundamental theorem of affine geometry in regular $L^0$-modules and the recent progress in random convex analysis, this paper characterizes the stable fully order preserving and order reversing operators acting on the class of proper lower semicontinuous $L^0$-convex functions in complete random normed modules.
For the study of some typical problems in finance and economics, Žitković %[G. Žitković, Convex compactness and its applications, Math. Finan. Eco., 3(1)(2010) 1--12] introduced convex compactness and gave many remarkable applications. Recently, motivated by random convex optimization and random variational inequalities, Guo, et al introduced $L^0$-convex compactness, developed the related theory of $L^0$-convex compactness in random normed modules and further applied it to backward stochastic equations. %[T.X. Guo, et al, Two fixed point theorems in complete random normed modules and their applications to backward stochastic equations, J. Math. Anal. Appl., 483(2020) 123644]. In this paper, we extensively study $L^0$-convexly compact sets in random locally convex modules so that a series of fundamental results are obtained. First, we show that every $L^0$-convexly compact set is complete (hence is also closed and has the countable concatenation property). Then, we prove that any $L^0$-convexly compact set is linearly homeomorphic to a weakly compact subset of some locally convex space, and simultaneously establish the equivalence between $L^0$-convex compactness and convex compactness for a closed $L^0$-convex set. Finally, we establish Tychonoff type, James type and Banach-Alaoglu type theorems for $L^0$-convex compactness, respectively.
Let $(Ω,{\mathcal F},P)$ be a probability space and $L^0({\mathcal F})$ the algebra of equivalence classes of real-valued random variables defined on $(Ω,{\mathcal F},P)$. A left module $M$ over the algebra $L^0({\mathcal F})$(briefly, an $L^0({\mathcal F})$-module) is said to be regular if $x=y$ for any given two elements $x$ and $y$ in $M$ such that there exists a countable partition $\{A_n,n\in \mathbb N\}$ of $Ω$ to $\mathcal F$ such that ${\tilde I}_{A_n}\cdot x={\tilde I}_{A_n}\cdot y$ for each $n\in \mathbb N$, where $I_{A_n}$ is the characteristic function of $A_n$ and ${\tilde I}_{A_n}$ its equivalence class. The purpose of this paper is to establish the fundamental theorem of affine geometry in regular $L^0({\mathcal F})$-modules: let $V$ and $V^\prime$ be two regular $L^0({\mathcal F})$-modules such that $V$ contains a free $L^0({\mathcal F})$-submodule of rank $2$, if $T:V\to V^\prime$ is stable and invertible and maps each $L^0$-line segment to an $L^0$-line segment, then $T$ must be $L^0$-affine.
Let $L^0$ be the algebra of equivalence classes of real valued random variables on a given probability space, and $(L^0)^n$ the $n$-ary Cartesian power of $L^0$ for each integer $n\geq 2$. We consider $(L^0)^n$ as a free module over $L^0$ and study affine geometry in $(L^0)^n$. One of our main results states that: an injective mapping $T: (L^0)^n\to (L^0)^n$ which is local and maps each $L^0$-line onto an $L^0$-line must be an $L^0$-affine linear mapping. The other main result states that: a bijective mapping $T: (L^0)^n\to (L^0)^n$ which is local and maps each $L^0$-line segment onto an $L^0$-line segment must be an $L^0$-affine linear mapping. These results extend the fundamental theorem of affine geometry from $\mathbb R^n$ to $(L^0)^n$.
In 2010, Gordan Žitković introduced the notion of convex compactness for a convex subset of a linear topological space and gave some important applications to both nonlinear analysis and mathematical economics in [ Gordan Žitković, Convex compactness and its applications, Math. Finance Econom. 3(1) (2010) 1--12 ]. Motivated by Gordan Žitković's idea, in this paper we introduce the notion of $L^0$--convex compactness for an $L^0$--convex subset of a topological module over the topological algebra $L^0(\mathcal{F},K)$, where $L^0(\mathcal{F},K)$ is the algebra of equivalence classes of random variables from a probability space $(Ω,\mathcal{F},P)$ to the scalar field $K$ of real numbers or complex numbers, endowed with the topology of convergence in probability. This paper continues to develop the theory of $L^0$--convex compactness by establishing various kinds of characterization theorems for $L^0$--convex subsets of a class of important topological modules--complete random normed modules, in particular, we make use of the theory of random conjugate spaces to give a characterization theorem of James type for a closed $L^0$--convex subset of a complete random normed module. As applications, we successfully generalize some basic theorems of classical convex optimization and variational inequalities from a convex function on a reflexive Banach space to an $L^0$--convex function on a random reflexive random normed module. Since the usual weak compactness method fails in the random setting of this paper and in particular, since the difficulties caused by the partial order structure of the range of an $L^0$--valued function also frequently occurs in the study of problems involved in this paper, we are forced to discover a series of new skills to meet the needs of this paper.
Noting the important role the abstract $L^p$ space has played in the development of random normed modules, in this paper we introduce and study the Orlicz space generated from a random normed module. First, we give a basic dual space representation theorem which identify the dual of the Orlicz heart of a random normed module with the Orlicz space generated from the random conjugate space. Then, we establish the respective equivalence relations of the strict convexity and uniform convexity of this abstract Orlicz space to the random strict convexity and random uniform convexity of the underlying random normed module. These results demonstrate that it is possible to use the Orlicz space theory in the further development of random nomed modules.
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, Gâteaux--differentiability and Fréchét--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 $(\varepsilon, λ)$--topology and the locally $L^0$--convex topology for a random locally convex module.
In this paper, we embed each $L^\infty$-normed module $E$ into an appropriate and unique complete random normed module $E_0$ so that the properties of $E$ are closely related to the properties of $E_0$.
This paper constructs a counterexample showing that not every locally $L^0$--convex topology is necessarily induced by a family of $L^0$--seminorms. Random convex analysis is the analytic foundation for $L^0$--convex conditional risk measures, this counterexample, however, shows that a locally $L^0$--convex module is not a proper framework for random convex analysis. Further, this paper also gives a necessary and sufficient condition for a locally $L^0$--convex topology to be induced by a family of $L^0$--seminorms. Finally, we give some comments showing that based on random locally convex modules, we can establish a perfect random convex analysis to meet the needs of the study of $L^0$--convex conditional risk measures.