An analytical study of norms and Banach spaces induced by the entropic value-at-risk

This paper addresses the Entropic Value-at-Risk (\({{\mathrm{\mathsf {EV@R}}}}\)), a recently introduced coherent risk measure. It is demonstrated that the norms defined by \({{\mathrm{\mathsf {EV@R}}}}\) induce the same Banach spaces, irrespective of the confidence level. Three vector spaces, called the primal, dual, and bidual entropic spaces, corresponding with \({{\mathrm{\mathsf {EV@R}}}}\) are fully studied. It is shown that these spaces equipped with the norms induced by \({{\mathrm{\mathsf {EV@R}}}}\) are Banach spaces. The entropic spaces are then related to the \(L^p\) spaces, as well as specific Orlicz hearts and Orlicz spaces. This analysis indicates that the primal and bidual 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 \({{\mathrm{\mathsf {EV@R}}}}\) norm and corresponding Hahn–Banach functionals are presented in closed form, which are not explicitly known for the Orlicz and Luxemburg norms that are equivalent to the \({{\mathrm{\mathsf {EV@R}}}}\) norm. The duality relationships among the 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 \({{\mathrm{\mathsf {EV@R}}}}\), 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 the theory of modern risk measurement but also other fields of stochastic analysis and modeling.