An Analytical Study of Norms and Banach Spaces Induced by the Entropic Value-at-Risk

This paper addresses the Entropic Value-at-Risk (EVaR), a recently introduced coherent risk measure. It is demonstrated that the norms induced by EVaR induce the same Banach spaces, irrespective of the confidence level. Three spaces, called the primal, dual, and bidual entropic spaces, corresponding with EVaR are fully studied. It is shown that these spaces equipped with the norms induced by EVaR are Banach spaces. The entropic spaces are then related to $L^p$ spaces as well as to specific Orlicz hearts and Orlicz spaces. This analysis indicates that the entropic spaces can be used as very flexible model spaces, larger than $L^\infty$, over which all $L^p$-based risk measures are well-defined. The dual EVaR norm and corresponding Hahn–Banach functionals are presented explicitly, which are not explicitly known for the Orlicz and Luxemburg norms that are equivalent to the EVaR norm. The duality relationships among entropic spaces are investigated. The duality results are also used to develop an extended Donsker–Varadhan variational formula and to explicitly provide the dual and Kusuoka representations of EVaR, as well as the corresponding maximizing densities in both representations. Our results indicate that financial concepts can be successfully used to develop insightful tools for not only modern risk theory but also other fields of stochastic analysis and modeling.