A New Coherent Risk Measure of Entropic Value at Risk for Uncertain Systems

Abstract

Uncertain risk analysis was initially explored by Liu, who introduced the concepts of loss function and risk index within uncertainty theory. Various risk measures, such as value at risk (VaR), tail VaR (TVaR), expected loss and entropic VaR (EVaR), have been proposed within probability theory. To date, Peng extended the concepts of VaR and TVaR, while Liu and Ralescu extended the notion of expected loss for uncertain systems. This paper mainly extends the concept of EVaR as a novel uncertain coherent risk measure. It will be demonstrated as the tightest upper bound one can find using Chernoff inequality for the VaR. In addition, we verify the properties of EVaR under uncertain measure, including positive homogeneity, translation invariance, monotonicity and subadditivity under independence. Furthermore, this paper provides a comparison of VaR, TVaR and EVaR under uncertain measure. Two examples will be given to illustrate this comparison.