The Natural Banach Space for Version Independent Risk Measures

Abstract

Risk measures, or coherent measures of risk are often considered on the space $L^\infty$, andimportant theorems on risk measures build on that space. Other risk measures, among themthe most important risk measure – the Average Value-at-Risk – are well-defined on the largerspace $L^1$ and this seems to be the natural domain space for this risk measure. Spectral riskmeasures constitute a further class of risk measures of central importance, and they are oftenconsidered on some $L^p$ space. But in many situations this is possibly unnatural, because any$L^p$ with $p > p_0$, say, is suitable to define the spectral risk measure as well. In addition tothat risk measures have also been considered on Orlicz and Zygmund spaces. So it remains fordiscussion and clarification, what the natural domain to consider a risk measure is?This paper introduces a norm, which is built from the risk measure, and a Banach space,which carries the risk measure in a natural way. It is often strictly larger than its originaldomain, and obeys the key property that the risk measure is finite valued and continuous onthat space in an elementary and natural way.