Orderings and Probability Functionals Consistent with Preferences

Abstract

This paper unifies the classical theory of stochastic dominance and investor preferences with the recent literature on risk measures applied to the choice problem faced by investors. First, we summarize the main stochastic dominance rules used in the finance literature. Then we discuss the connection with the theory of integral stochastic orders and we introduce orderings consistent with investors' preferences. Thus, we classify them, distinguishing several categories of orderings associated with different classes of investors. Finally, we show how we can use risk measures and orderings consistent with some preferences to determine the investors' optimal choices.

Keywords:

• Stochastic dominance
• probability functionals
• integral orderings
• coherent and convex measures
• utility theory
• efficient choices

Keywords:

• G11
• C44
• C61

Acknowledgements

The authors gratefully acknowledge the guidance provided by the Associate Editor, William Ziemba, and the referee, Leonard MacLean. Sergio Ortobelli's research was partially supported under Murst 60% 2006, 2007, 2008. Svetlozar Rachev's research was supported by grants from the Division of Mathematical, Life and Physical Sciences, College of Letters and Science, University of California, Santa Barbara, and the Deutsche Forschungsgemeinschaft.

Notes

1. In particular, it is well known that and orders are equivalent to the respective ≥₋₁ and ≥₋₂ orders. However, it is still not clear if, for any α, there is a correspondence between and the respective inverse order ≥_−α. This question should be the subject of future research.

2. We denote by [Z/G] any random variable that has as its distribution the conditional distribution of Z given G.

3. FORS is an abbreviation for the initials of the surnames of the authors of this paper, Fabozzi, Ortobelli, Rachev, and Shalit.