Risk measures on \(\mathcal{P}(\mathbb R)\) and value at risk with probability/loss function (Q2875724)

The authors define a new class of law invariant risk measures \(\Phi:\mathcal P(\mathbb R)\rightarrow \mathbb R\cup \{+\infty\}\) on the set \(\mathcal P(\mathbb R)\) of probability measures on \(\mathbb R\). These measures are based on an appropriate family of acceptance sets. They are monotone and quasi-convex. The proposition is a generalization of the classical notion of \(\mathrm{VaR}_\lambda\), that takes into account the balance between the probability and the amount of the loss. The dual representation of the risk measures on \(\mathcal P(\mathbb R)\) is also proven.