A primer on generalized weighted risk functionals

Weighted risk functionals have been extensively studied to date. Indeed, this versatile class of risk functionals has enjoyed a variety of applications in risk management and insurance, and it has been generalized along a number of directions. Some applications of the class of weighted risk functionals in risk management and insurance include pricing, valuation, risk measurement, risk capital allocation and risk sharing, while variations of weighted risk functionals have allowed for such enhancements as multivariate probability weighting and the augmentation of utility functions. We propose a class of generalized weighted risk functionals that incorporates the possibility of arbitrary aggregations. To this end, we introduce the notion of an aggregation function in the context of the aforementioned weighted risk functionals. We then delineate the ways in which distinct orders on the weight functions and on the aggregation functions affect the order of our generalized weighted risk functionals. We conclude with several observations that facilitate applications of the generalized weighted risk functionals.