Short Communication: An Axiomatization of $\Lambda$-Quantiles

We give an axiomatic foundation to $\Lambda$-quantiles, a family of generalized quantiles introduced in [M. Frittelli, M. Maggis, and I. Peri, Math. Finance, 24 (2014), pp. 442--463] under the name Lambda Value at Risk. Under mild assumptions, we show that these functionals are characterized by property call “locality,” which means any change distribution probability mass arises entirely above or below value $\Lambda$-quantile does not modify its value. make comparisons with related axiomatization usual given Chambers [Math. 19 (2009), 335--342], based on stronger “ordinal covariance,” meaning covariant respect increasing transformations. Further, present systematic treatment properties refining some results Peri Burzoni, C. Ruffo, Quant. 17 (2017), 1735--1743] showing case nonincreasing $\Lambda$ $\Lambda$-quantiles closely resemble those quantiles.