P. Artzner [Math. Finance 9, No. 3, 203-228 (1999; Zbl 0980.91042)] has introduced the notion of a coherent measure of risk. It is defined as a mapping \(\rho\colon{\mathcal X}\to\mathbb R\) from a certain space \(\mathcal X\) of functions \(X\) (describing the resulting discounted net worth at the end of a given period) on some set \(\Omega\) of possible scenarios that has the subadditivity, positive homogeneity, monotonicity and translation invariance properties. The authors introduce the notion of a convex measure of risk, an extension of the concept of a coherent risk measure. A mapping \(\rho\colon{\mathcal X}\to\mathbb R\) from a convex space \(\mathcal X\) of functions is called convex measure of risk if it satisfies the condition of convexity, monotonicity and translation invariance. In the case where \(\mathcal X\) is the space of all real-valued functions on a finite set \(\Omega\) any convex measure of risk is of the form \[ \rho(X)=\sup_{Q\in\mathcal P}\left(E_Q[-X]-\alpha(Q)\right), \] where \(\mathcal P\) is the set of all probability measures on \(\Omega\) and \(\alpha(Q)\) is a penalty function on \(\mathcal P\).
An appropriate extension of the representation theorem is proved where the finite set \(\Omega\) is replaced by a probability space \((\Omega,{\mathcal F},P)\) and \({\mathcal X}\) is the space of bounded random variables \({\mathcal X}=L^{\infty}(\Omega,{\mathcal F},P)\). (For relevant references see W. Schachermayer [Optimal investment in incomplete financial markets. Helyette (ed.) et al., Mathematical finance - Bachelier congress 2000,. Berlin: Springer, 427–462 (2002; Zbl 1002.91033)].)
A convex measure of risk can be characterized in terms of properties of the associated acceptance set \({\mathcal A}=\left\{x\in{\mathcal X}|\rho(X)\leq0\right\}\). Two main case studies of convex risk measures proposed in the article are defined in terms of such acceptance sets. The situation is considered where the acceptance set is defined in terms of a robust notion of bounded shortfall risk. In this case the associated penalty function can be described by a functional of the type of a dual Orlicz space. In the context of a financial market model it turns out that the representation theorem is closely related to the superhedging duality under convex constraints.