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