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Higher order measures of risk and stochastic dominance

Alois Pichler

Abstract

Higher order risk measures are stochastic optimization problems by design, and for this reason they enjoy valuable properties in optimization under uncertainties. They nicely integrate with stochastic optimization problems, as has been observed by the intriguing concept of the risk quadrangles, for example. Stochastic dominance is a binary relation for random variables to compare random outcomes. It is demonstrated that the concepts of higher order risk measures and stochastic dominance are equivalent, they can be employed to characterize the other. The paper explores these relations and connects stochastic orders, higher order risk measures and the risk quadrangle. Expectiles are employed to exemplify the relations obtained.

Higher order measures of risk and stochastic dominance

Abstract

Higher order risk measures are stochastic optimization problems by design, and for this reason they enjoy valuable properties in optimization under uncertainties. They nicely integrate with stochastic optimization problems, as has been observed by the intriguing concept of the risk quadrangles, for example. Stochastic dominance is a binary relation for random variables to compare random outcomes. It is demonstrated that the concepts of higher order risk measures and stochastic dominance are equivalent, they can be employed to characterize the other. The paper explores these relations and connects stochastic orders, higher order risk measures and the risk quadrangle. Expectiles are employed to exemplify the relations obtained.
Paper Structure (14 sections, 21 theorems, 131 equations)

This paper contains 14 sections, 21 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Outline of the paper.
  3. Mathematical Framework
  4. Higher order spectral risk
  5. General stochastic dominance relations
  6. Characterization of stochastic dominance relations
  7. Stochastic dominance in numerical computations
  8. Characterization of stochastic dominance for spectral risk measures
  9. Higher order stochastic dominance
  10. Comparison of stochastic order relations
  11. Example: the expectile
  12. The dual norm of expectiles
  13. Higher order expectiles
  14. Summary

Key Result

Proposition 2.3

Let $(\mathcal{Y},\|\cdot\|)$ be a normed space of random variables. For the functional $\mathcal{R}_\beta$ defined in eq:3 it holds that so that $\mathcal{R}_\beta(\cdot)$ is indeed well-defined on $(\mathcal{Y},\,\|\cdot\|)$ for every $\beta\in[0,1)$.

Theorems & Definitions (58)

  • Definition 2.1: Risk functional
  • Definition 2.2: Higher order risk measure
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • Proposition 2.5: Duality
  • Remark 2.6
  • proof
  • Example 2.7: Hölder spaces
  • ...and 48 more