# On Banach spaces of vector-valued random variables and their duals motivated by risk measures

## TL;DR

This paper introduces specific Banach spaces of vector-valued random variables inspired by financial risk measures, demonstrating their properties, dual space representations, and the Lipschitz continuity of associated risk functionals.

## Contribution

It defines new Banach spaces tailored for risk measures, analyzes their properties, and characterizes their duals using vector measures, highlighting their relevance in mathematical finance.

## Key findings

- Risk functionals are Lipschitz continuous on these spaces.
- Risk functionals cannot be extended to larger spaces of random variables.
- Dual spaces are represented via vector measures with values in the dual of the state space.

## Abstract

We introduce Banach spaces of vector-valued random variables motivated from mathematical finance. So-called risk functionals are defined in a natural way on these Banach spaces and it is shown that these functionals are Lipschitz continuous. The risk functionals cannot be defined on strictly larger spaces of random variables which creates a particular interest for the spaces presented. We elaborate key properties of these Banach spaces and give representations of their dual spaces in terms of vector measures with values in the dual space of the state space.