Hindawi Publishing Corporation Journal of Probability and Statistics Volume 2010, Article ID 357321, 19 pages doi:10.1155/2010/357321 Research Article On Some Layer-Based Risk Measures with Applications to Exponential Dispersion Models Olga Furman 1 and Edward Furman 2 1 Actuarial Research Center, University of Haifa, Haifa 31905, Israel 2 Department of Mathematics and Statistics, York University, Toronto, ON, Canada M3J 1P3 Correspondence should be addressed to Edward Furman, efurman@mathstat.yorku.ca Received 13 October 2009; Revised 21 March 2010; Accepted 10 April 2010 Academic Editor: Johanna Neˇ slehov´ a Copyright q 2010 O. Furman and E. Furman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Layer-based counterparts of a number of well-known risk measures have been proposed and studied. Namely, some motivations and elementary properties have been discussed, and the analytic tractability has been demonstrated by developing closed-form expressions in the general framework of exponential dispersion models. 1. Introduction Denote by X the set of actuarial risks, and let 0 ≤ X ∈X be a random variable rv with cumulative distribution function cdf Fx, decumulative distribution function ddf Fx 1 - Fx, and probability density function pdf f x. The functional H : X → 0, ∞ is then referred to as a risk measure, and it is interpreted as the measure of risk inherent in X. Naturally, a quite significant number of risk measuring functionals have been proposed and studied, starting with the arguably oldest Value-at-Risk or VaR cf. 1, and up to the distorted cf. 2–5 and weighted cf. 6, 7 classes of risk measures. More specifically, the Value-at-Risk risk measure is formulated, for every 0 <q< 1, as VaR q X inf x : F X x ≥ q , 1.1 which thus refers to the well-studied notion of the qth quantile. Then the family of distorted risk measures is defined with the help of an increasing and concave function g : 0, 1 → 0, 1, such that g 0 0 and g 1 1, as the following Choquet integral: H g X R g Fx dx. 1.2