Exponentially Bounded C-semigroup and the Cauchy Initial Value Problems in Complete Random Normed Modules

In this paper,the notion of C-semigroup of continuous module homomorphisms on a complete random normal(RN)module is introduced and investigated.The existence and uniqueness of solution to the Cauchy problem with respect to exponentially bounded C-semigroups of continuous module homomorphisms in a complete RN module are established.

A Characterization of Continuous Module Homomorphisms on Random Semi—Normed Modules and Its Applications

In this paper, a characterization of continuous module homomorphisms on random semi-normed modules is first given; then the characterization is further used to show that the Hahn-Banach type of extension theorem is still true for continuous module homomorphisms on random semi-normed modules.

L^0-CONVEX COMPACTNESS AND RANDOM NORMAL STRUCTURE IN L^0(F,B)

Let(B,||·||)be a Banach space,(?,F,P)a probability space,and L^0(F,B)the set of equivalence classes of strong random elements(or strongly measurable functions)from(?,F,P)to(B,||·||).It is well known that L^0(F,B)becomes a complete random normed module,which has played an important role in the process of applications of random normed modules to the theory of Lebesgue-Bochner function spaces and random operator theory.Let V be a closed convex subset of B and L^0(F,V)the set of equivalence classes of strong random elements from(?,F,P)to V.The central purpose of this article is to prove the following two results:(1)L^0(F,V)is L^0-convexly compact if and only if V is weakly compact;(2)L^0(F,V)has random normal structure if V is weakly compact and has normal structure.As an application,a general random fixed point theorem for a strong random nonexpansive operator is given,which generalizes and improves several well known results.We hope that our new method,namely skillfully combining measurable selection theorems,the theory of random normed modules,and Banach space techniques,can be applied in the other related aspects.

A Characterization for a Complete Random Normed Module to Be Mean Ergodic

In this paper, we first study the mean ergodicity of random linear operators using some techniques of measure theory and L;-convex analysis. Then, based on this, we give a characterization for a complete random normed module to be mean ergodic.

Representation theorems of the dual of Lebesgue-Bochner function spaces

By representing random conjugate spaces a general representation theorem on classical duals is proved. For application, we unify and improve many known important representation theorems of the dual of Lebesgue-Bochner function spaces.

Recent progress in random metric theory and its applications to conditional risk measures

The purpose of this paper is to give a selective survey on recent progress in random metric theory and its applications to conditional risk measures.This paper includes eight sections.Section 1 is a longer introduction,which gives a brief introduction to random metric theory,risk measures and conditional risk measures.Section 2 gives the central framework in random metric theory,topological structures,important examples,the notions of a random conjugate space and the Hahn-Banach theorems for random linear functionals.Section 3 gives several important representation theorems for random conjugate spaces.Section 4 gives characterizations for a complete random normed module to be random reflexive.Section 5 gives hyperplane separation theorems currently available in random locally convex modules.Section 6 gives the theory of random duality with respect to the locally L0-convex topology and in particular a characterization for a locally L0-convex module to be L0-pre-barreled.Section 7 gives some basic results on L0-convex analysis together with some applications to conditional risk measures.Finally,Section 8 is devoted to extensions of conditional convex risk measures,which shows that every representable L∞-type of conditional convex risk measure and every continuous Lp-type of convex conditional risk measure(1 ≤ p < +∞) can be extended to an L∞F(E)-type of σ,λ(L∞F(E),L1F(E))-lower semicontinuous conditional convex risk measure and an LpF(E)-type of T,λ-continuous conditional convex risk measure(1 ≤ p < +∞),respectively.

The relations among the three kinds of conditional risk measures

Let（Ω , E, P） be a probability space, F a sub-σ-algebra of E, L^p（E）（1 p ＋∞） the classical function space and LF^p（E） the L^0（F）-module generated by L^p（E）, which can be made into a random normed module in a natural way. Up to the present time, there are three kinds of conditional risk measures, whose model spaces are L^∞（E）, L^p（E）（1 p ＋∞） and LF^p（E）（1 p ＋∞） respectively, and a conditional convex dual representation theorem has been established for each kind. The purpose of this paper is to study the relations among the three kinds of conditional risk measures together with their representation theorems. We first establish the relation between L^p（E） and LF^p（E）, namely LF^p（E） = Hcc（L^p（E））, which shows that LF^p（E）is exactly the countable concatenation hull of L^p（E）. Based on the precise relation, we then prove that every L^0（F）-convex L^p（E）-conditional risk measure（1 p ＋∞） can be uniquely extended to an L^0（F）-convex LF^p（E）-conditional risk measure and that the dual representation theorem of the former can also be regarded as a special case of that of the latter, which shows that the study of L^p-conditional risk measures can be incorporated into that of LF^p（E）-conditional risk measures. In particular, in the process we find that combining the countable concatenation hull of a set and the local property of conditional risk measures is a very useful analytic skill that may considerably simplify and improve the study of L^0-convex conditional risk measures.

Several applications of the theory of random conjugate spaces to measurability problems

The central purpose of this paper is to illustrate that combining the recently developed theory of random conjugate spaces and the deep theory of Banach spaces can, indeed, solve some difficult measurability problems which occur in the recent study of the Lebesgue (or more general, Orlicz)-Bochner function spaces as well as in a slightly different way in the study of the random functional analysis but for which the measurable selection theorems currently available are not applicable. It is important that this paper provides a new method of studying a large class of the measurability problems, namely first converting the measurability problems to the abstract existence problems in the random metric theory and then combining the random metric theory and the relative theory of classical spaces so that the measurability problems can be eventually solved. The new method is based on the deep development of the random metric theory as well as on the subtle combination of the random metric theory with classical space theory.

An L^0(F,R)-valued Function's Intermediate Value Theorem and Its Applications to Random Uniform Convexity

Let （Ω, F, P） be a probability space and L0（F,R） the algebra of equivalence classes of real- valued random variables on （Ω, F, P）. When L0（F,R） is endowed with the topology of convergence in probability, we prove an intermediate value theorem for a continuous local function from L0（F, R） to L0（F,R）. As applications of this theorem, we first give several useful expressions for modulus of random convexity, then we prove that a complete random normed module （S, ｜｜·｜｜） is random uniformly convex iff LP（S） is uniformly convex for each fixed positive number p such that 1 〈 p 〈 ＋∞.

Various Expressions for Modulus of Random Convexity

We first prove various kinds of expressions for modulus of random convexity by using an L^0（F, R）-valued function＇s intermediate value theorem and the well known Hahn-Banach theorem for almost surely bounded random linear functionals, then establish some basic properties including continuity for modulus of random convexity. In particular, we express the modulus of random convexity of a special random normed module L^0（F, X） derived from a normed space X by the classical modulus of convexity of X.