Multivariate stochastic dominance for risk averters and risk seekers
RAIRO - Operations Research - Recherche Opérationnelle, Volume 50 (2016) no. 3, pp. 575-586.

This paper first extends some well-known univariate stochastic dominance results to multivariate stochastic dominances (MSD) for both risk averters and risk seekers, respectively, to n order for any n1 when the attributes are assumed to be independent and the utility is assumed to be additively and separable. Under these assumptions, we develop some properties for MSD for both risk averters and risk seekers. For example, we prove that MSD are equivalent to the expected-utility maximization for both risk averters and risk seekers, respectively. We show that the hierarchical relationship exists for MSD. We establish some dual relationships between the MSD for risk averters and risk seekers. We develop some properties for non-negative combinations and convex combinations random variables of MSD and develop the theory of MSD for the preferences of both risk averters and risk seekers on diversification. At last, we discuss some MSD relationships when attributes are dependent and discuss the importance and the use of the results developed in this paper.

Received:
Accepted:
DOI: 10.1051/ro/2016026
Classification: D81, G11
Keywords: Multivariate stochastic dominance, risk averters, risk seekers, ascending stochastic dominance, descending stochastic dominance, utility function
Guo, Xu 1, 2; Wong, Wing-Keung 3, 4, 5

1 School of Statistics, Beijing Normal University, Beijing, P.R. China.
2 College of Economics and Management, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, P.R. China.
3 Department of Finance, Asia University, Taiwan.
4 Faculty of Economics, Hong Kong Baptist University, Hong Kong, P.R. China.
5 Department of Economics, Lingnan University, Hong Kong, P.R. China.
@article{RO_2016__50_3_575_0,
     author = {Guo, Xu and Wong, Wing-Keung},
     title = {Multivariate stochastic dominance for risk averters and risk seekers},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     pages = {575--586},
     publisher = {EDP-Sciences},
     volume = {50},
     number = {3},
     year = {2016},
     doi = {10.1051/ro/2016026},
     mrnumber = {3538841},
     zbl = {1390.91104},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ro/2016026/}
}
TY  - JOUR
AU  - Guo, Xu
AU  - Wong, Wing-Keung
TI  - Multivariate stochastic dominance for risk averters and risk seekers
JO  - RAIRO - Operations Research - Recherche Opérationnelle
PY  - 2016
SP  - 575
EP  - 586
VL  - 50
IS  - 3
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/ro/2016026/
DO  - 10.1051/ro/2016026
LA  - en
ID  - RO_2016__50_3_575_0
ER  - 
%0 Journal Article
%A Guo, Xu
%A Wong, Wing-Keung
%T Multivariate stochastic dominance for risk averters and risk seekers
%J RAIRO - Operations Research - Recherche Opérationnelle
%D 2016
%P 575-586
%V 50
%N 3
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/ro/2016026/
%R 10.1051/ro/2016026
%G en
%F RO_2016__50_3_575_0
Guo, Xu; Wong, Wing-Keung. Multivariate stochastic dominance for risk averters and risk seekers. RAIRO - Operations Research - Recherche Opérationnelle, Volume 50 (2016) no. 3, pp. 575-586. doi : 10.1051/ro/2016026. http://archive.numdam.org/articles/10.1051/ro/2016026/

Z.D. Bai, H.X. Liu and W.K. Wong, Enhancement of the Applicability of Markowitz’s Portfolio Optimization by Utilizing Random Matrix Theory. Math. Finance 19 (2009) 639–667. | DOI | MR | Zbl

Z.D. Bai, H. Li, M. Mcaleer and W.K. Wong, Stochastic Dominance Statistics for Risk Averters and Risk Seekers: An Analysis of Stock Preferences for USA and China. Quant. Finance 15 (2015) 889–900. | DOI | Zbl

J.C. Cox, A Theorem on Additively-Separable Quasi-Concave Functions. J. Econ. Theory 6 (1973) 210–212. | DOI | MR

R. Davidson and J.Y. Duclos, Statistical inference for stochastic dominance and for the measurement of poverty and inequality. Econometrica 68 (2000) 1435–1464. | DOI | MR | Zbl

D. Dentcheva and A. Ruszczynski, Optimization with stochastic dominance constraints. SIAM J. Optim. 14 (2003) 548–566. | DOI | MR | Zbl

D. Dentcheva and A. Ruszczynski, Optimality and duality theory for stochastic optimization problems with nonlinear dominance constraints. Math. Program. 99 (2004) 329–350. | DOI | MR | Zbl

D. Dentcheva and A. Ruszczynski, Optimization with multivariate stochastic dominance constraints, Math. Program. 117 (2009) 111–127. | DOI | MR | Zbl

M. Denuit and L. Eeckhoudt, Bivariate stochastic dominance and substitute risk (in)dependent utilities. Decis. Anal. 7 (2010) 302–312. | DOI

M. Denuit, L. Eeckhoudt and B. Rey, Some consequences of correlation aversion in decision science. Ann. Oper. Res. 176 (2010) 259–269. | DOI | MR | Zbl

M. Egozcue and W.K. Wong, Gains from Diversification: A Majorization and Stochastic Dominance Approach. Eur. J. Oper. Res. 200 (2010) 893–900. | DOI | MR | Zbl

R. Eisner and R. Strotz, Flight insurance and the theory of choice. J. Political Economy 69 (1961) 355–368. | DOI

W.M. Fong, W.K. Wong and H.H. Lean, International Momentum Strategies: A Stochastic Dominance Approach. J. Financ. Mark. 8 (2005) 89–109. | DOI

J. Hadar and W.R. Russell, Stochastic Dominance and Diversification. J. Econ. Theory 3 (1971) 288–305. | DOI | MR

J.S. Hammond, Simplifying the Choice between Uncertain Prospects where Preference is Nonlinear. Manage. Sci. 20 (1974) 1047–1072. | DOI | Zbl

G.H. Hardy, J.E. Littlewood and G. Pólya, Inequalities, Cambridge University Press. Cambridge, MA (1934). | Zbl

G.B. Hazen, Partial information, dominance, and potential optimality in multiattribute utility theory. Oper. Res. 34 (1986) 296–310. | DOI | MR | Zbl

T. Homem-De-Mello and S. Mehrotra, A cutting surface method for uncertain linear programs with linear stochastic dominance constraints. SIAM J. Optim. 20 (2009) 1250–1273. | DOI | MR | Zbl

J. Hu, T. Homem-De-Mello and S. Mehrotra, Sample average approximation of stochastic dominance constrained programs. Math. Program. 133 (2012) 171–201. | DOI | MR | Zbl

W.H. Jean, The geometric mean and stochastic dominance. J. Finance 35 (1980) 151–158. | DOI | MR

N. Jegadeesh and S. Titman, Returns to Buying Winners and Selling Losers: Implications for Stock Market Efficiency. J. Finance 48 1993 65–91. | DOI

R.L. Keeney and H. Raiffa, Decisions with Multiple Objectives: Preferences and Value Tradeoffs. Wiley, New York (1976). | MR | Zbl

H. Levy, Stochastic Dominance, Efficiency Criteria, and Efficient Portfolios: The Multi-Period Case. Am. Econ. Rev. 43 (1973) 986-994.

H. Levy, Stochastic dominance and expected utility: Survey and analysis. Management Science 38 (1992) 555–593. | DOI | Zbl

C.K. Li and W.K. Wong, Extension of Stochastic Dominance Theory to Random Variables. RAIRO: OR 33 (1999) 509–524. | DOI | Numdam | MR | Zbl

H.M. Markowitz, The Utility of Wealth. J. Political Economy 60 (1952a) 151–156. | DOI

H.M. Markowitz, Portfolio Selection. J. Finance 7 (1952b) 77–91.

E. Ok and L. Kranich, The Measurement of Opportunity Inequality: a Cardinality-Based Approach. Social Choice Welfare 15 (1998) 263–287. | DOI | MR | Zbl

Z. Qiao, E. Clark and W.K. Wong, Investors’ Preference towards Risk: Evidence from the Taiwan Stock and Stock Index Futures Markets. Accounting & Finance 54 (2014) 251–274. | DOI

J.P. Quirk and R. Saposnik, Admissibility and Measurable Utility Functions. Rev. Econ. Stud. 29 (1962) 140–146. | DOI

P.A. Samuelson, General Proof that Diversification Pays. J. Finance Quant. Anal. 2 (1967) 1–13. | DOI

S. Sriboonchita, W.K. Wong, D. Dhompongsa and H.T. Nguyen, Stochastic Dominance and Applications to Finance, Risk and Economics. Chapman and Hall/CRC, Boca Raton, Florida (2009). | MR | Zbl

J. von Neumann and O. Morgenstern, Theory of Games and Economic Behavior. Princeton University Press, Princeton N.J. (1944). | MR | Zbl

W.K. Wong, Stochastic Dominance and Mean-Variance Measures of Profit and Loss for Business Planning and Investment. Eur. J. Oper. Res. 182 (2007) 829–843. | DOI | Zbl

W.K. Wong and R. Chan, Markowitz and Prospect Stochastic Dominances. Ann. Finance 4 (2008) 105–129. | DOI | Zbl

W.K. Wong and C.K. Li, A Note on Convex Stochastic Dominance Theory. Econ. Lett. 62 (1999) 293–300. | DOI | MR | Zbl

W.K. Wong and C. Ma, Preferences over Location-Scale Family. Econom. Theory 37 (2008) 119–146. | DOI | MR | Zbl

Cited by Sources: