Non-differentiable transformations preserving stochastic dominance

Abstract

In this paper, we solve the following problem: when does a stochastic improvement in one risk maintain itself under a non everywhere continuously differentiable transformation of this risk? Using the notion of divided differences, we show that stochastic dominance at the third (and higher) order, and sometimes at the second one, is not preserved after simple piecewise linear transformation of the initial risk. Our analysis complements the one that exists for everywhere continuously differentiable transformations.

Keywords:

• s-concavity
• s-increasing concavity
• divided differences
• Ekern increase in risk

Acknowledgements

The authors would like to express their gratitude to two anonymous referees whose comments have been extremely useful to revise a previous version of the present work. The financial support of PARC ‘Stochastic Modelling of Dependence’ 2012–2017 awarded by the Communauté française de Belgique is gratefully acknowledged by Michel Denuit.

Notes

1 By improvement, we basically mean non-deterioration, ie either statu quo or improvement.

2 As pointed out by one referee, the decision-maker might instead decide to change his demand for newspapers.