The Natural Banach Space for Version Independent Risk Measures

TL;DR

This paper introduces a new Banach space constructed from a risk measure, providing a natural, larger domain where the risk measure is finite and continuous, clarifying the appropriate functional setting for various risk measures.

Contribution

It defines a natural Banach space from a risk measure, extending its domain and ensuring finiteness and continuity in a unified framework.

Findings

The new space is often strictly larger than traditional domains.

The risk measure is finite and continuous on this space.

Provides a unified framework for different classes of risk measures.

Abstract

Risk measures, or coherent measures of risk are often considered on the space L^\infty, and important theorems on risk measures build on that space. Other risk measures, among them the most important risk measure---the Average Value-at-Risk---are well defined on the larger space L^1 and this seems to be the natural domain space for this risk measure. Spectral risk measures constitute a further class of risk measures of central importance, and they are often considered 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 to that risk measures have also been considered on Orlicz and Zygmund spaces. So it remains for discussion and clarification, what the natural domain to consider a risk measure is? Abstract 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 original domain, and obeys the key property that the risk measure is finite valued and continuous on that space in an elementary and natural way.