---
abstract: 'The minimality of the penalty 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 on this set. When the probability space supports a Lévy 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.'
author:
- '[Daniel Hernández–Hernández[^1]   Leonel Pérez-Hernández[^2]]{}'
title: ' Characterization of the minimal penalty of a convex risk measure with applications to Lévy processes.'
---

**Key words:** Convex risk measures, Fenchel-Legendre transformation, minimal penalization, Lévy process.\
**Mathematical Subject Classification:** 91B30, 46E30.

Introduction
============

The definition of coherent risk measure was introduced by Artzner *et al.* in their fundamental works [@ADEH; @1997], [@ADEH; @1999] for finite probability spaces, giving an axiomatic characterization that was extended later by Delbaen [@Delbaen; @2002] to general probability spaces. In the papers mentioned above one of the fundamental axioms was 