Risk Measures on $\mathcal{P}(\mathbb{R})$ and Value At Risk with Probability/Loss function

Abstract

We propose a generalization of the classical notion of the $V@R_{\lambda }$ that takes into account not only the probability of the losses, but the balance between such probability and the amount of the loss. This is obtained by defining a new class of law invariant risk measures based on an appropriate family of acceptance sets. The $V@R_{\lambda }$ and other known law invariant risk measures turn out to be special cases of our proposal. We further prove the dual representation of Risk Measures on $\mathcal{P}($

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• Complete duality for quasiconvex dynamic risk measures on modules of the $L^p$-type