minimal for the risk measures generated by duality.

The paper is organized as follows. Section 2 contains the description of the minimal penalty functions for a general probability space, providing necessary and sufficient conditions, the last one rectricted to a subset of equivalent probability measures. Section 3 reports the structure of the densities for a probability space that supports a Lévy processes and the convergence properties needed to prove the lower semicontinuity of the set of penalty functions defined in Section 4. In this last section we show that these penalty functions are minimal.

Minimal penalty function of risk measures concentrated in $\mathcal{Q}_{\ll }\left( \mathbb{P}\right) $. \[Sect Minimal Penalty Function of CMR\]
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Any penalty function $\psi $ induce a convex risk measure $\rho $, which in turn has a representation by means of a minimal penalty function $\psi _{\rho }^{\ast }.$ Starting with a penalty function $\psi $ concentrated in a convex and closed subset of the set of absolutely continuous probability measures with respect to some reference measure $\mathbb{P}$, in this section we give necessary and sufficient conditions in order to guarantee that $\psi $ is the minimal penalty within this set. We begin recalling briefly some known 