On Stop-Loss Order and the Distortion Pricing Principle

A number of more or less well-known, but quite complex, characterizations of stop-loss order are reviewed and proved m an elementary way. Two recent proofs of the stop-loss order preserving property for the distortion pricing principle are invalidated through a simple counterexample A new proof is presented. It is based on the important Hardy-Littlewood transform, which is known to characterize the stop-loss order by reduction to the usual stochastic order, and the dangerousness characterization of stop-loss order under a finite crossing condition Finally, we complete and summarize the main properties of the distortion pricing principle. Keywords: Pricing theory, distortion function, quantile function, stop-loss order, stochastic order, Hardy-Littlewood transform