Risk measures on incomplete markets: a new non-solid paradigm

TL;DR

This paper develops a new framework for risk measures on incomplete markets by exploring their dual representations and extension properties in vector spaces lacking lattice structures, broadening the scope of risk theory.

Contribution

It introduces a novel approach to risk measures on non-lattice spaces, establishing dual representation conditions and extension theorems relevant for incomplete markets.

Findings

Dual representation linked to Fatou-like property

Extension theorems under regular lift conditions

Applicable to incomplete financial markets

Abstract

We study risk measures $\varphi:E\longrightarrow\mathbb{R}\cup\{\infty\}$, where $E$ is a vector space of random variables which a priori has no lattice structure$\unicode{x2014}$a blind spot of the existing risk measures literature. In particular, we address when $\varphi$ admits a tractable dual representation (one which does not contain non-$\sigma$-additive signed measures), and whether one can extend $\varphi$ to a solid superspace of $E$. The existence of a tractable dual representation is shown to be equivalent, modulo certain technicalities, to a Fatou-like property, while extension theorems are established under the existence of a sufficiently regular lift, a potentially non-linear mechanism of assigning random variable extensions to certain linear functionals on $E$. Our motivation is broadening the theory of risk measures to spaces without a lattice structure, which are ubiquitous in financial economics, especially when markets are incomplete.