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A dual representation theorem on the conditional Orlicz space generated from a random normed module

Xia Zhang, Ke Qian, Ming Liu

TL;DR

The paper advances the theory of conditional Orlicz spaces on random normed modules by (i) introducing random Orlicz functions and defining the conditional Orlicz spaces $L_{\\mathcal{F}}^{\Phi}(E)$ and $H_{\\mathcal{F}}^{\Phi}(E)$, (ii) proving the denseness of the Orlicz heart in the RN module under the $(\varepsilon,\lambda)$-topology and (iii) establishing a dual representation linking the dual of the conditional Orlicz heart to the conditional Orlicz space built from the random conjugate space, with isometric isomorphism given by $[T f](x)=E[f(x)\mid \mathcal{F}]$ for $f\in L_{\\mathcal{F}}^{\Psi}(E^*)$. These results generalize and refine existing dualities in the literature, including special cases that recover classical $L^p$ and Orlicz dualities, and provide a robust framework for conditional risk measures in random environments.

Abstract

In this paper, we first introduce the notion of a random Orlicz function, and further present the conditional Orlicz space generated from a random normed module. Second, we prove the denseness of the Orlicz heart of a random normed module $E$ in $E$ with respect to the $(\varepsilon, λ)$-topology. Finally, based on the above work, we establish a dual representation theorem on the conditional Orlicz space generated from a random normed module, which extends and improves some known results.

A dual representation theorem on the conditional Orlicz space generated from a random normed module

TL;DR

The paper advances the theory of conditional Orlicz spaces on random normed modules by (i) introducing random Orlicz functions and defining the conditional Orlicz spaces and , (ii) proving the denseness of the Orlicz heart in the RN module under the -topology and (iii) establishing a dual representation linking the dual of the conditional Orlicz heart to the conditional Orlicz space built from the random conjugate space, with isometric isomorphism given by for . These results generalize and refine existing dualities in the literature, including special cases that recover classical and Orlicz dualities, and provide a robust framework for conditional risk measures in random environments.

Abstract

In this paper, we first introduce the notion of a random Orlicz function, and further present the conditional Orlicz space generated from a random normed module. Second, we prove the denseness of the Orlicz heart of a random normed module in with respect to the -topology. Finally, based on the above work, we establish a dual representation theorem on the conditional Orlicz space generated from a random normed module, which extends and improves some known results.
Paper Structure (4 sections, 8 theorems, 49 equations)

This paper contains 4 sections, 8 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Terminology and Notation
  3. The Conditional Orlicz Space Generated from a Random Normed Module
  4. Dual Representation of the Random Conjugate Space of $H_{\mathcal{F}}^{\Phi}(E)$

Key Result

Proposition 3.3

Let $(E,\|\cdot\|)$ be an $RN$ module, $\phi$ a continuous Orlicz function, and $(L^{\phi}\left(E\right),||\cdot||_{\phi L})$ and $(H^{\phi}(E),||\cdot||_{\phi L})$ defined as above. If $(L^{\phi}\left(E\right),||\cdot||_{\phi L})$ and $(H^{\phi}(E),||\cdot||_{\phi L})$ are complete, then $E$ is als

Theorems & Definitions (21)

  • Definition 2.1: 13
  • Definition 2.2: 1
  • Definition 2.3
  • Definition 3.1
  • Example 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • ...and 11 more