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

TL;DR

This paper investigates the minimal penalty functions for convex risk measures, providing necessary and sufficient conditions in static and Levy process settings, with implications for risk assessment and measure selection.

Contribution

It characterizes minimal penalties for convex risk measures and extends results to Levy processes, linking penalty minimality to process coefficients.

Findings

Necessary and sufficient conditions for minimal penalties in static frameworks

Guarantees for minimality of penalties in Levy process models

Analysis of convergence of density process quadratic variations

Abstract

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.