Robust representation of convex risk measures by probability measures

Artzner et al. [1] initiated a new direction to assess risks of financial positions by an axiomatic approach. It relies fundamentally on the concept of risk measures, which are functionals defined on sets of financial positions and satisfying some basic properties. The convex risk measures are exactly those which guarantee that diversification does not increase risk. From the standpoint of individual investors risk measures may be interpreted as loss functions expressing the preferences on the respective sets of financial positions. Starting from this point of view, Föllmer and Schied succeeded in finding a kind of robust Savage representation for convex risk measures by probability contents [3]. They also gave a sufficient condition to achieve a representation by probability measures. One aim of the paper is to show the reverse direction of their result. Another subject of the paper is to weaken the criterion within the setting when the sets of scenarios are endowed with a metrizable topology, restoring an earlier attempt by Föllmer and Schied.