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The minimality of the penalization function associated with a convex riskmeasure is analyzed in this paper. First, in a general static framework, weprovide necessary and sufficient conditions for a penalty function defined in aconvex and closed subset of the absolutely continuous measures with respect tosome reference measure $\mathbb{P}$ to be minimal. When the probability spacesupports a L\'{e}vy process, we establish results that guarantee the minimalityproperty of a penalty function described in terms of the coefficientsassociated with the density processes. The set of densities processes isdescribed and the convergence of its quadratic variation is analyzed.

Characterization of the minimal penalty of a convex risk measure with applications to Levy processes