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

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.