A note on conditional risk measures of Orlicz spaces and Orlicz-type modules

Abstract

We consider conditional and dynamic risk measures of Orlicz spaces and study their robust representation. For this purpose, given a probability space $(\Omega,\mathcal{E},\mathbb{P})$, a sub-$\sigma$-algebra $\mathcal{F}$ of $\mathcal{E}$, and a Young function $\varphi$, we study the relation between the classical Orlicz space $L^\varphi(\mathcal{E})$ and the modular Orlicz-type module $L^\varphi_\mathcal{F}(\mathcal{E})$; based on conditional set theory, we describe the conditional order continuous dual of a Orlicz-type module; and by using scalarization and modular extensions of conditional risk measures together with elements of conditional set theory, we finally characterize the robust representation of conditional risk measures of Orlicz spaces