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A Characterization of the n-th Degree Bounded Stochastic Dominance

Bar Light, Andres Perlroth

TL;DR

This work characterizes the n-th degree bounded stochastic dominance (BSD) order by linking it to decision-makers' risk tolerance and providing a decision-theoretic foundation. It identifies maximal generators for BSD through intersections of curvature-constrained utility sets, showing BSD is equivalent to lower partial moment inequalities for all utilities in $U_{n,[a,b]} \cap LP_{n,[a,b]}$ (and $AP_{n,[a,b]}$). Key contributions include a complete generator characterization, Jensen-type inequalities for n-convex functions, and comparative statics results for global bounds on Arrow-Pratt risk aversion and prudence, with applications to portfolio optimization under BSD constraints and savings under uncertainty. The results illuminate both the flexibility and limitations of BSD, notably its dependence on the interval $[a,b]$ and boundary risk behavior, and they provide practical tools for analyzing risk-sensitive decision making.

Abstract

We provide a novel characterization of the $n$th degree bounded stochastic dominance (BSD) order, linking it to the risk tolerance of decision makers and providing a decision theoretic foundation for these stochastic orders. Our results reveal two contrasting implications, on the positive side, they show that BSD reflects specific risk preferences through the choice of the interval $[a,b]$, by characterizing it in terms of utility functions with globally bounded Arrow Pratt risk aversion or that satisfy an $n$ convexity condition. On the negative side, they highlight limitations of BSD, including the dependence of BSD on the chosen interval and the peculiar risk aversion behavior of decision-makers included in the generator of BSD. We illustrate our results through a portfolio optimization model with stochastic dominance constraints. Additionally, using our characterization, we present comparative statics results for decision making under uncertainty with globally bounded risk aversion measures and savings decisions under globally bounded prudence measures, and derive inequalities for $n$ convex functions.

A Characterization of the n-th Degree Bounded Stochastic Dominance

TL;DR

This work characterizes the n-th degree bounded stochastic dominance (BSD) order by linking it to decision-makers' risk tolerance and providing a decision-theoretic foundation. It identifies maximal generators for BSD through intersections of curvature-constrained utility sets, showing BSD is equivalent to lower partial moment inequalities for all utilities in (and ). Key contributions include a complete generator characterization, Jensen-type inequalities for n-convex functions, and comparative statics results for global bounds on Arrow-Pratt risk aversion and prudence, with applications to portfolio optimization under BSD constraints and savings under uncertainty. The results illuminate both the flexibility and limitations of BSD, notably its dependence on the interval and boundary risk behavior, and they provide practical tools for analyzing risk-sensitive decision making.

Abstract

We provide a novel characterization of the th degree bounded stochastic dominance (BSD) order, linking it to the risk tolerance of decision makers and providing a decision theoretic foundation for these stochastic orders. Our results reveal two contrasting implications, on the positive side, they show that BSD reflects specific risk preferences through the choice of the interval , by characterizing it in terms of utility functions with globally bounded Arrow Pratt risk aversion or that satisfy an convexity condition. On the negative side, they highlight limitations of BSD, including the dependence of BSD on the chosen interval and the peculiar risk aversion behavior of decision-makers included in the generator of BSD. We illustrate our results through a portfolio optimization model with stochastic dominance constraints. Additionally, using our characterization, we present comparative statics results for decision making under uncertainty with globally bounded risk aversion measures and savings decisions under globally bounded prudence measures, and derive inequalities for convex functions.
Paper Structure (11 sections, 13 theorems, 61 equations, 1 figure)

This paper contains 11 sections, 13 theorems, 61 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main Results
  3. The Maximal Generator of Bounded Stochastic Dominance
  4. Examples and Applications
  5. Portfolio Optimization
  6. Decision Making and the Arrow-Pratt Risk Aversion Measure
  7. Prudence and Savings Decisions
  8. Relation to $n$-convex Functions and Inequalities
  9. Proof of Theorem \ref{['Main:Thm']}
  10. Summary
  11. Appendix

Key Result

Proposition 1

Let $F$ and $G$ be two distributions over $[a,b]$ and let $n$ be a positive integer. Then $F \succeq_{n} G$ if and only if for all $u \in U_{n,[a,b]}$.

Figures (1)

  • Figure 1: We observe that the expected return (blue line) increases as $b$ increases. Correspondingly, the optimal portfolio reallocates weight away from the safer asset toward riskier assets, which have higher expected returns.

Theorems & Definitions (19)

  • Definition 1
  • Proposition 1
  • Definition 2
  • Theorem 1
  • Proposition 2
  • Remark 1
  • Lemma 1
  • Remark 2
  • Remark 3
  • Proposition 3
  • ...and 9 more