Risk measures on incomplete markets: a new non-solid paradigm
Vasily Melnikov
TL;DR
Risk measures on incomplete markets are analyzed on a non-solid vector space $E\subset L^{0}$ with a proper convex functional $\varphi:E\to\mathbb{R}\cup\{\infty\}$. The authors connect a Fatou-type property of $\varphi$ to local lower semicontinuity in a topology induced by a dual price space $F$, yielding dual representations $\varphi(f)=\sup_{g\in F}(\int f g\,dP-\varphi^{*}(g))$ and enabling avoidance of non-$\sigma$-additive measures. They introduce lifts to extend $\varphi$ to the solid hull $\mathrm{span}(\mathrm{sol}(K))$, proving a nontrivial extension theorem under reductive lifts and establishing monotone-extension conditions. Applications to semimartingale market models show dual representations for risk measures on bounded stochastic integrals and provide an extension framework for incomplete markets. Overall, the work broadens the risk-measures theory beyond lattice frameworks, addressing solidity and completeness issues central to financial economics.
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-$σ$-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.