Conditional Value-at-Risk (CVaR) Norm: Stochastic Case

The concept of Conditional Value-at-Risk (CVaR) is used in various applications in uncertain environment. This paper introduces CVaR norm for a random variable, which is by de nition CVaR of absolute value of this random variable. It is proved that CVaR norm is indeed a norm in the space of random variables. CVaR norm is de ned in two variations: scaled and non-scaled. L-1 and L-in nity norms are limiting cases of the CVaR norm. In continuous case, scaled CVaR norm is a conditional expectation of the random variable. A similar representation of CVaR norm is valid for discrete random variables. Several properties for scaled and non-scaled CVaR norm, as a function of con dence level, were proved. Dual norm for CVaR norm is proved to be the maximum of L-1 and scaled L-in nity norms. CVaR norm, as a Measure of Error, generates a Regular Risk Quadrangle. Negative CVaR function, which is a non convex extension for CVaR norm, is introduced analogously to function L-p for p < 1. Linear regression problems were solved by minimizing CVaR norm of regression residuals.