On the behavior of the high order stop-loss transform for convolutions with some applications

Abstract

High order stop-loss transforms provide a risk measure that enables some flexibility on the weight given to high or low values of the risk. We interpret stop-loss transforms as iterated distributions and prove a recursive representation for risks expressed as convolutions. We apply this to the case of gamma distributions with integer shape parameter, the Erlang distributions, proving that high order stop-loss transforms are equivalent to the tails of the exponential distribution. The latter result is also extended to general gamma distributions. Furthermore, we prove that this equivalence to exponential tails does not hold in general, by proving that the stop-loss transform for a Weilbull distribution degenerates, unless, of course, in the exponential case.

Key Words:

• Stop-loss transform
• iterated distributions
• convolution

Acknowledgments

The authors wish to express their thanks to the anonymous Referee and to the Associate Editor for their careful reading and comments that helped to improve on an earlier version of the paper.

Additional information

Funding

This work was partially supported by the Centre for Mathematics of the University of Coimbra - UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES.