A note on convex stochastic dominance and diversification with applications

Abstract.

In this note, we present an overlooked result derived from Fishburn’s Convex Stochastic Dominance (CSD) (Fishburn (1974a; 1974b). We demonstrate that CSD is a sufficient condition for establishing second-order stochastic dominance in the context of linear combinations of random variables. While this result may appear straightforward, to the best of our knowledge, it has not been explicitly presented as we have done in this work. Furthermore, the implications of this finding are extensive, as it can be applied to various areas of research, as we illustrate in this study.

Keywords:

• Stochastic dominance
• second-order stochastic dominance
• cumulative distribution functions
• convex stochastic dominance
• risk analysis

Acknowledgments

We are grateful for the valuable comments and feedback provided by Michel Denuit and Luis Fuentes García, which greatly contributed to the improvement of this article. We also acknowledge Xu Guo for refining the proofs in this work. We are grateful for the support of our university, as well as to the Agencia Nacional de Investigación e Innovación (ANII).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1. Recalling that X first-order stochastically dominates Y if and only if F(x)≤G(x) for all $x\in [a,b]$.

2. Related to further the derivation of this result, see Sub-Section 3.4.2.5 in Denuit et al. (2006) or Theorem 7.3.10 in Kaas et al. (2008).