We study conditional expectiles, defined as a natural generalisation of conditional expectations by means of the minimisation of an asymmetric quadratic loss function. We show that conditional expectiles can be equivalently characterised by a conditional first order condition and we derive their...

We introduce the concepts of φ-complete mixability and φ-joint mixability and we investigate some necessary and sufficient conditions to the φ-mixability of a set of distribution functions for some supermodular functions φ. We give examples and numerical verifications which confirm our findings

We give analytical bounds on the Value-at-Risk and on convex risk measures for a portfolio of random variables with fixed marginal distributions under an additional positive dependence structure. We show that assuming positive dependence information in our model leads to reduced dependence...

We introduce the concepts of ϕ-complete mixability and ϕ-joint mixability and we investigate some necessary and sufficient conditions to the ϕ-mixability of a set of distribution functions for some supermodular functions ϕ. We give examples and numerical verifications which confirm our findings.

We give analytical bounds on the Value-at-Risk and on convex risk measures for a portfolio of random variables with fixed marginal distributions under an additional positive dependence structure. We show that assuming positive dependence information in our model leads to reduced dependence...